A SYSTEMATIC APPROACH FOR LINEAR PARAMETER VARYING MODELING
AND CONTROL OF INTERNAL COMBUSTION ENGINES
By
Andrew Philip White

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Mechanical Engineering
2012

ABSTRACT
A SYSTEMATIC APPROACH FOR LINEAR PARAMETER VARYING
MODELING AND CONTROL OF INTERNAL COMBUSTION ENGINES
By
Andrew Philip White
In this dissertation, the modeling and control of time-varying automotive systems is
addressed with the use of linear-parameter varying (LPV) control. In practice, many automotive systems with time-varying physical processes are operated with gain-scheduled
controllers whose gains are manually calibrated through engine dynamometer and vehicle
field tests for the best performance as functions of system operational conditions. However,
this approach for gain scheduling is not only expensive and time consuming, but also does
not guarantee the stability and performance of the closed-loop system for all time-varying
parameters. LPV control techniques, on the other hand, can be used to design gain-scheduled
controllers with guaranteed closed-loop stability and performance. The goal of this dissertation is to create a framework to design gain-scheduled controllers using LPV techniques,
including, most importantly, the ability to select physically meaningful performance design
constraints. As part of this effort, LPV control methods have been applied to the air-to-fuel
ratio control of port-fuel-injection systems and the control of hydraulic and electric variable
valve timing systems.
Gain scheduling controllers designed using the LPV method have traditionally included
H∞ performance constraints. This is largely due to the fact that H∞ controllers can provide
robust stability margins that H2 controllers cannot provide. However, since the H∞ norm
is defined as the root-mean-square gain, or ℓ2 to ℓ2 gain, from the exogenous input to the
regulated output, controllers designed with only H∞ performance constraints are not suitable

for use when hard constraints on responses or actuator signals must be met.
When hard constraints on responses or actuator signals must be met, a controller with
a guaranteed ℓ2 to ℓ∞ gain is needed, which is a special type of H2 controller. When
H2 performance is used to design a controller, normally a quadratic cost function that
balances the output performance with the control input needed to achieve that performance is
considered. However, unlike the conventional H2 performance criterion, the system ℓ2 to ℓ∞
gain provides a hard constraint (ℓ∞ ) on the system output for a class of inputs with bounded
ℓ2 norm. Many practical control problems in automotive and aerospace systems impose hard
constraints (or ℓ∞ norm) on the system output. The existing mixed H2 /H∞ LPV control
method cannot be used to solve this class of the control synthesis problems. To remedy this
gap, this dissertation provides a control synthesis method which provides a guaranteed ℓ2 to
ℓ∞ gain on the system output for LPV systems. The result is a gain-scheduled controller that
can provide hard constraints on multiple system outputs. To demonstrate the effectiveness
of this approach, both a numerical example and a simulation study with an electrical VVT
system are presented.

This dissertation is dedicated to
my parents and my wife, Rebecca.

iv

ACKNOWLEDGMENTS

I would like express my sincere gratitude to my co-advisors Professor Guoming Zhu and
Professor Jonguen Choi for their support, advise, and guidance during my Ph.D. study
at Michigan State University. During this time period, they have both shared their vast
and diverse knowledge of control theory with me. I would especially like to thank Professor
Guoming Zhu for providing me with the opportunity to apply my LPV controllers to practical
control problems in the Engine Research Lab. I am also especially grateful for the guidance
and support provided by Professor Jonguen Choi during the first two years of my graduate
studies as a Masters student.
I would like to thank Professor Clark Radcliffe, Professor Ranjan Mukherjee, and Professor Hassan Khalil for serving on my Ph.D. committee. I have gained a wealth of knowledge
from the lectures taught by these professors.
I would also like to thank Dr. Ryozo Nagamune, from the University of British Columbia,
for his valuable valuable contributions on our collaborative research. His suggestions and
comments on our paper “Gain-scheduling control of port-fuel-injection processes” during the
revision process were immensely helpful.
The work that is performed in the Engine Research Lab is by nature a collaborative effort.
Some of the research performed in this dissertation would not have been possible without
the hard work put in by the people who set up the experiments in the lab. In particular I
would like to thank Zhen Ren, Xiaojian Yang, Xuefei Chen, and Kevin Moran.
Last, but not least, I would like to thank my family and friends for their encouragement
and support over the years.

v

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

x

Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Research Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Providing hard constraints for gain-scheduling controllers . . . . .
1.2.2 Application of gain-scheduling control . . . . . . . . . . . . . . . .
1.2.2.1 Air-to-fuel ratio control for port-fuel-injection processes .
1.2.2.2 Control of variable valve timing actuators . . . . . . . .
1.3 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1
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3
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4
5
5
6

Linear Parameter-Varying Modeling and Control Synthesis
Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Modeling of LPV systems . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 First-order Taylor series approximation of LPV systems . . . . . . . .
2.1.2 Polytopic linear time-varying system with Barycentric coordinates . .
Performance of discrete-time polytopic LPV systems . . . . . . . . . . . . .
Control Synthesis Methods for LPV systems . . . . . . . . . . . . . . . . . .
2.3.1 Static Output Feedback Control Synthesis . . . . . . . . . . . . . . .
2.3.2 H∞ Control Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 Mixed H2 /H∞ Control Synthesis . . . . . . . . . . . . . . . . . . . .

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Chapter 2
2.1

2.2
2.3

Chapter 3 Guaranteed ℓ2 − ℓ∞ gain control of LPV systems
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 ℓ2 − ℓ∞ Performance of Discrete-time Systems . . . . . . . . .
3.3 Controller Synthesis for mixed ℓ2 − ℓ∞ /H∞ Control Problem
3.3.1 Gain-Scheduled State Feedback Control Synthesis . . .
3.3.2 Output Feedback LMIs . . . . . . . . . . . . . . . . . .
3.4 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 State Feedback Control . . . . . . . . . . . . . . . . . .
3.4.2 Dynamic Output-Feedback Control . . . . . . . . . . .
3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 4 Gain-Scheduling Control of Port-Fuel-Injection Processes . . .
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69
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vi

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4.2

4.3

4.4
4.5

4.6

4.7

4.8

Event-based discrete-time system modeling . . . . . . . . . . . . . .
4.2.1 Sampling period of the event-based discrete-time system . .
4.2.2 Dynamics of the port-fuel-injection process . . . . . . . . . .
4.2.3 Dynamics of the oxygen sensor . . . . . . . . . . . . . . . .
4.2.4 An LPV system . . . . . . . . . . . . . . . . . . . . . . . . .
LPV gain-scheduling controller design . . . . . . . . . . . . . . . . .
4.3.1 Control Strategy . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Feed-forward compensated generalized plant . . . . . . . . .
4.3.3 First-order Taylor series expansion of the LPV system . . . .
4.3.4 An augmented LPV system for synthesis . . . . . . . . . . .
4.3.4.1 PI Control . . . . . . . . . . . . . . . . . . . . . .
4.3.4.2 PID Control . . . . . . . . . . . . . . . . . . . . . .
4.3.5 A gain-scheduling control synthesis problem . . . . . . . . .
4.3.6 Controller synthesis for polytopic linear time-varying system
Design of LTI Feedback Controller . . . . . . . . . . . . . . . . . .
Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.1 Case 1: Engine Cold Start . . . . . . . . . . . . . . . . . . .
4.5.2 Case 2: Load Change . . . . . . . . . . . . . . . . . . . . . .
4.5.3 Case 3: Engine Speed Change . . . . . . . . . . . . . . . . .
HIL Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6.1 Mean Value Engine Models . . . . . . . . . . . . . . . . . .
4.6.1.1 Valve Model . . . . . . . . . . . . . . . . . . . . . .
4.6.1.2 Manifold Filling Dynamic Model . . . . . . . . . .
4.6.1.3 Engine Respiration Model . . . . . . . . . . . . . .
4.6.1.4 Crankshaft Dynamic Model . . . . . . . . . . . . .
4.6.2 Event Based Engine Models . . . . . . . . . . . . . . . . . .
4.6.2.1 Event Based Wall-Wetting Dynamics . . . . . . . .
4.6.2.2 Event Based Engine Air-to-fuel Ratio . . . . . . . .
4.6.2.3 Event Based Engine Brake Torque . . . . . . . . .
4.6.3 Set-up and Implementation . . . . . . . . . . . . . . . . . .
HIL Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . .
4.7.1 Case 1: Engine Cold Start . . . . . . . . . . . . . . . . . . .
4.7.2 Case 2: Load Change . . . . . . . . . . . . . . . . . . . . . .
4.7.3 Case 3: Engine Speed Change . . . . . . . . . . . . . . . . .
4.7.4 Case 4: Combined Load and Engine Speed Change . . . . .
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 5
5.1
5.2
5.3

Mixed H2 /H∞ Observer-Based LPV Control of a
Engine Cam Phasing Actuator . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
LPV System Modeling . . . . . . . . . . . . . . . . . . . . . . .
LPV Gain Scheduling Controller Design . . . . . . . . . . . . .
5.3.1 Control Strategy . . . . . . . . . . . . . . . . . . . . . .
5.3.2 Generalized Plant . . . . . . . . . . . . . . . . . . . . . .
5.3.3 A gain-scheduling control synthesis problem . . . . . . .
vii

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Hydraulic
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5.4

5.5

5.3.4 Polytopic linear time-varying system
VVT System Test Bench . . . . . . . . . . .
5.4.1 Bench Test Set-up . . . . . . . . . .
5.4.2 Bench Test Results . . . . . . . . . .
Conclusion . . . . . . . . . . . . . . . . . . .

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139
143
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147

Dynamic, Output-Feedback, Gain-Scheduling Control of an
Electric Variable Valve Timing System . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Plant dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 Local Motor Controller . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.2 Motor and Planetary Gear System Dynamics . . . . . . . . . . . . . .
6.2.3 Planetary Gear System Kinematics . . . . . . . . . . . . . . . . . . .
6.2.4 An LPV System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.4.1 Linear least squares estimation of discrete-time LPV system
LPV gain-scheduling controller design . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Guaranteed ℓ2 to ℓ∞ gain controller . . . . . . . . . . . . . . . . . . .
6.3.2 H∞ (ℓ2 to ℓ2 gain) Controller . . . . . . . . . . . . . . . . . . . . . .
Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 6
6.1
6.2

6.3

6.4
6.5

Chapter 7 Conclusions and Future Research
7.1 Summary . . . . . . . . . . . . . . . . . . .
7.2 Specific Contributions . . . . . . . . . . . .
7.3 Future Research . . . . . . . . . . . . . . . .

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APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
Appendix A
Linear Fractional Transformation . . . . . . . . . . . . . . . . . . 180
Appendix B
Port Fuel Injection System Matrices . . . . . . . . . . . . . . . . . 182
BIBLIOGRAPHY

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viii

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LIST OF TABLES

Table 4.1

Modeling parameters.

. . . . . . . . . . . . . . . . . . . . . . . . .

Table 4.2

Measurable time varying parameters (scheduling parameters).

Table 5.1

Identified Gain Ψ(Ne , p)

Table 5.2

Time-varying parameters (scheduling parameters).

Table 6.1

Electric VVT System Parameters . . . . . . . . . . . . . . . . . . . 169

. . .

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85

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ix

. . . . . . . . . 130

LIST OF FIGURES

Figure 2.1

Examples of possible parameter space polytopes. . . . . . . . . . . .

Figure 2.2

Fastest possible parameter transition between the extreme conditions
with N = 2 and b = 0.5 when the rate limit given by (2.25) is in effect. 18

Figure 3.1

Design 1: The covariance bound Zp achieved compared to the constraint (3.60). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 3.2

Figure 3.3

Figure 3.4

Figure 3.5

Figure 3.6

Figure 3.7

Figure 3.8

Figure 4.1

Design 1: The output response of zp,1 , zp,2 , and zp,3 plotted against
each other for design 1 simulated with a positive (I1 ) and negative
(I2 ) impulse function and compared with the ℓ∞ norm bound. . . .
Design 2: A. The covariance bound Zp,2 achieved compared to the
second constraint in (3.61). B. The output response of zp,2 plotted
against zp,3 for design 2 simulated with a positive and negative impulse function and compared with the ℓ∞ norm bound. . . . . . . .

12

57

58

59

The ℓ2 disturbance (A) and the parameter variation (B) used to simulate each controller design. . . . . . . . . . . . . . . . . . . . . . .

60

Design 1: The covariance bound Zp achieved compared to the constraint (3.60). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

Design 1: The output response of zp,1 , zp,2 , and zp,3 plotted against
each other for design 1 simulated with a positive (I1 ) and negative
(I2 ) impulse function and compared with the ℓ∞ norm bound. . . .
Design 2: A. The covariance bound Zp,2 achieved compared to the
second constraint in (3.61). B. The output response of zp,2 plotted
against zp,3 for design 2 simulated with a positive and negative impulse function and compared with the ℓ∞ norm bound. . . . . . . .

65

66

The ℓ2 disturbance (A) and the parameter variation (B) used to simulate each controller design. . . . . . . . . . . . . . . . . . . . . . .

67

Flowchart of the design and validation process of an LPV controller.

72

x

Figure 4.2

The block diagram of the port-fuel-injection process and sensor dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

Block diagram of the combined dynamics of the exhaust gas and
sensor delays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

Comparison of the response to a unit step function for the 4th order
Taylor series approximation model in Eq. 4.18 (dashed line) and a
model Eq. 4.6 with the engine speed fixed at 3500 rpm (dash-dot
line) to the exact discretized oxygen sensor delay model in Eq. 4.6
(solid line) at 1000 rpm and 6000 rpm. . . . . . . . . . . . . . . . .

83

The proposed control strategy for the fuel injection process (without the weighting functions W1 (q) and W2 (q)). Weighting functions
W1 (q) and W2 (q) are only used for controller synthesis. A first-order
Taylor series expansion is applied to the to the systems inside the
dashed box and the LPV control strategy is applied to the all of the
systems inside of the bold box. . . . . . . . . . . . . . . . . . . . .

86

Figure 4.6

Magnitude plot of the weighting functions W1 and W2 . . . . . . . .

89

Figure 4.7

The feed-forward control compensated generalized plant with the
time-varying parameters included. . . . . . . . . . . . . . . . . . .

91

Figure 4.8

Parameter space polytope. . . . . . . . . . . . . . . . . . . . . . . .

99

Figure 4.9

Case 1 Engine Cold Start: In plots A, B, C, and D the equivalence
ratio y(k), proportional control up (k), integral control ui (k), and the
feed-forward control are compared for the gain-scheduling feedback
controller (solid line) and the fixed H∞ controller (dashed line). The
time varying parameters α (dotted line, left axis) and β (dash-dot
line, right axis) are displayed in plot E. . . . . . . . . . . . . . . . . 104

Figure 4.10

Case 2 Load Change: In plots A, B, C, and D the equivalence ratio y(k), proportional control up (k), integral control ui (k), and the
feed-forward control are compared for the gain-scheduling feedback
controller (solid line) and the fixed H∞ controller (dashed line). The
time varying parameters α (dotted line, left axis) and β (dash-dot
line, right axis) are displayed in plot E. . . . . . . . . . . . . . . . . 106

Figure 4.3

Figure 4.4

Figure 4.5

xi

Figure 4.11

Case 3 Engine Speed Change: In plots A, B, C, and D the equivalence
ratio y(k), proportional control up (k), integral control ui (k), and the
feed-forward control are compared for the gain-scheduling feedback
controller (solid line) and the fixed H∞ controller (dashed line). The
wall-wetting parameters α (dotted line, left axis) and β (dash-dot line,
right axis) are displayed in plot E. The engine speed v is displayed in
plot F. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Figure 4.12

Mean Value Engine Model.

Figure 4.13

HIL engine model and controller setup. . . . . . . . . . . . . . . . . 114

Figure 4.14

HIL timing scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Figure 4.15

Case 1: Engine Cold Start using simple model. . . . . . . . . . . . . 117

Figure 4.16

Case 1: Engine Cold Start using HIL. . . . . . . . . . . . . . . . . . 118

Figure 4.17

Case 2: Engine Load Change using HIL. . . . . . . . . . . . . . . . . 120

Figure 4.18

Case 3: Engine Speed Change using HIL. . . . . . . . . . . . . . . . 121

Figure 4.19

Case 4: Combined Load and Engine Speed Change using HIL. . . . 123

Figure 5.1

Flowchart of the design and validation process of an LPV controller. 127

Figure 5.2

Proposed control architecture for the VVT system . . . . . . . . . . 131

Figure 5.3

Frequency response comparison of the mixed H2 /H∞ dynamic LPV
controller with an OCC (H2 ) controller [52] at the corner points of
the parameter space polytope. . . . . . . . . . . . . . . . . . . . . . 135

Figure 5.4

VVT phase actuator test bench . . . . . . . . . . . . . . . . . . . . 143

Figure 5.5

VVT phase actuator test bench diagram

Figure 5.6

Cam advance response at 900rpm with 310kPa oil pressure. . . . . . 148

Figure 5.7

Mean overshoot for each controller operated at an oil pressure of 45
and 60 psi and engine speeds of 900, 1200, 1500, and 1800 rpm. Plot A
displays the mean overshoot for cam advance and plot B displays the
mean overshoot for cam retard. . . . . . . . . . . . . . . . . . . . . 149

. . . . . . . . . . . . . . . . . . . . . . 109

xii

. . . . . . . . . . . . . . . 144

Figure 5.8

Mean 5% settling time for each controller operated at an oil pressure
of 45 and 60 psi and engine speeds of 900, 1200, 1500, and 1800 rpm.
Plot A displays the mean settling time for cam advance and plot B
displays the mean settling time for cam retard. . . . . . . . . . . . 150

Figure 5.9

Mean 10 to 90% rising time for each controller operated at an oil
pressure of 45 and 60 psi and engine speeds of 900, 1200, 1500, and
1800 rpm. Plot A displays the mean rising time for cam advance and
plot B displays the mean rising time for cam retard. . . . . . . . . . 151

Figure 6.1

The block diagram of the electric motor with the planetary gear system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

Figure 6.2

Proposed control scheme for the electrical VVT system. . . . . . . . 165

Figure 6.3

Cam load disturbance. . . . . . . . . . . . . . . . . . . . . . . . . . 170

Figure 6.4

Electrical VVT simulation with β = 2. . . . . . . . . . . . . . . . . . 172

Figure 6.5

Electrical VVT simulation with β = 1. . . . . . . . . . . . . . . . . . 173

Figure 6.6

Electrical VVT simulation with β = 0.5. . . . . . . . . . . . . . . . . 174

Figure A.1

(a) Diagram of a lower LFT. (b) Diagram of an upper LFT. . . . . . 181

xiii

Chapter 1
Introduction

1.1

Motivation

The goal of this research has been to establish a systematic procedure for the design of gainscheduling controllers. In industry, gain-scheduled controllers are normally developed with
long hours of ad-hoc tuning and calibration through engine dynamometer and vehicle field
tests. While these controllers are often used successfully in many practical applications, the
design process through which they are obtained is less than ideal. Not only is the process
expensive and time consuming, but more importantly it does not guarantee the stability and
performance of the closed-loop system for all possible time-varying parameters. Also, the
performance of the closed-loop system with gain-scheduling controllers designed in this way
is also dependent on the experience of person doing the calibration. In order to meet the
challenges posed by the strict requirements facing many industries these days, a systematic
process for designing gain-scheduled controllers with guaranteed performance and stability
for all time-varying parameters is needed.
One promising promising solution is the advanced control theory known as linear-parameter
varying (LPV) control [57, 44, 3, 2, 77, 81, 1, 23, 76, 10, 11, 12, 13]. LPV systems are
time-varying systems whose time-varying components consist of measurable parameters that
can vary over time. Over the years, many developments have been made in the area of
LPV control theory. In the beginning, LPV control theory mainly consisted of heuris1

tic approaches that were carried over from classical gain-scheduling control, and as such
these controllers provided no guaranteed stability, robustness, or performance. The authors of [57] provided analysis conditions for these heuristic approaches that could provided
guaranteed stability with slowly varying parameters. Thankfully, more advanced methods
based on the convex optimization of linear matrix inequalities (LMI) have been developed
[44, 3, 2, 77, 81, 1, 23, 76, 10, 11, 12, 13].
Initially, the small gain theorem was applied to LPV plants with linear fractional transformational (LFT) dependence on the time-varying parameters [44, 3]. This approach allowed
the parameter variations to be complex (i.e. have both real and imaginary parts). However, since the time-varying parameters in LPV systems rarely have imaginary parts, this
was considered a major source of conservatism in gain-scheduling controller design using
this method. Due to this, another method was developed that used a single or parameterdependent quadratic Lyapunov function in the analysis and control design for LPV plants
[2]. However, since this method allowed for arbitrarily fast parameter variation, it can produce conservative results with slowly varying parameters. To handle this problem, known
bounds on the rate of parameter variation were incorporated into the analysis conditions by
[77, 81, 1]. However, the method used by [77, 81, 1] formulates the control synthesis problem as semi-infinite convex optimization with parameter-dependent LMIs, which requires
that the gridding of the parameter space to provide a finite convex optimization. A unified
scheme was developed in [76] that joins both the small gain theorem and Lyapunov function approaches in an effort to provide a flexible approach for control engineers to trade-off
between performance improvement, controller complexity, and design effort. However, to
incorporate known bounds on the rate of parameter variation, the method developed in [76]
still requires gridding of the parameter space, which increases the complexity of implement2

ing the controller in practice. An alternative method which does not require gridding of
the parameter space was provided by [23, 10, 11, 12, 13] for affine-parameter dependent
Lyapunov functions.
Although there has been a considerable amount of research on the design of gainscheduling controllers via LPV control theory, there is still room for improvement. All of the
LPV methods previously mentioned specify the performance of the LPV system as either
H∞ or H2 performance. The difficulty that arises with these methods is that real system
performance is not easily related to the H∞ and H2 performance criteria. Furthermore, the
H∞ and H2 performance criteria, as will be discussed in more detail in Chapter 3, cannot
provide hard constraints on system outputs. As a result, engineers in industry could have a
hard time utilizing the LPV methods in practical applications.
The open question is, how do we bridge the gap between practices used in industry and
the advanced practices used in academia? The answer is to develop LPV controller synthesis
methods that allow the use of physically meaningful design constraints. By considering the ℓ2
to ℓ∞ gain performance of the closed-loop system, physically meaningful performance design
constraints, that can satisfy hard bounds, can be defined. This addition to the current LPV
control theory is expected to be very useful engineers working on practical applications in
industry.

1.2
1.2.1

Research Overview
Providing hard constraints for gain-scheduling controllers

This research considers the optimal control of polytopic, discrete-time linear parameter varying (LPV) systems with a guaranteed ℓ2 to ℓ∞ gain. Additionally, to guarantee robust
3

stability of the closed-loop system under parameter variations, H∞ performance criterion
is also considered as well. Controllers with a guaranteed ℓ2 to ℓ∞ gain and a guaranteed
H∞ performance (ℓ2 to ℓ2 gain) are mixed H2 /H∞ controllers. Normally, H2 controllers are
obtained by considering a quadratic cost function that balances the output performance with
the control input needed to achieve that performance. However, to obtain a controller with
a guaranteed ℓ2 to ℓ∞ gain (closely related to the physical performance constraint), the cost
function used in the H2 control synthesis minimizes the control input subject to maximal
singular-value performance constraints on the output. This problem can be efficiently solved
by a convex optimization with linear matrix inequality (LMI) constraints. The contribution
of this research is the characterization of the control synthesis LMIs for both gain-scheduled
state-feedback and dynamic output-feedback control used to obtain an LPV controller with
a guaranteed ℓ2 to ℓ∞ gain and H∞ performance. Numerical examples are presented to
demonstrate the effectiveness of the proposed LPV methods.

1.2.2

Application of gain-scheduling control

The methods reviewed in Chapter 2 and developed in Chapter 3 have been applied to real
control problems encountered in the control of internal combustion engines. Specifically, the
methods from Chapter 2 are used to design the gain-scheduling controller for the air-to-fuel
ratio control of port-fuel-injection processes presented in Chapter 4. The methods from
Chapter 2 where also used to develop of the observer-based mixed H2 /H∞ output-feedback
controller for the hydraulic variable valve timing actuator in Chapter 5. The guaranteed
ℓ2 − ℓ∞ gain controller synthesis techniques provided in Chapter 3 are applied to the control
of the electric variable valve timing actuator in Chapter 6.

4

1.2.2.1

Air-to-fuel ratio control for port-fuel-injection processes

An event-based sampled discrete-time linear system representing a port-fuel-injection process
based on wall-wetting dynamics is obtained and formulated as a linear parameter varying
(LPV) system. The system parameters used in the engine fuel system model are engine speed,
temperature, and load. These system parameters can be measured in real-time through
physical or virtual sensors. A gain-scheduling controller for the obtained LPV system is then
designed based on the numerically efficient convex optimization or linear matrix inequality
(LMI) technique. To demonstrate the effectiveness of the proposed scheme, both simulation
and hardware-in-the-loop (HIL) simulation results are presented. The HIL simulations not
only show that the designed gain-scheduling controller is effective on a complex mixed meanvalue and crank-based engine model [79], but it also demonstrates feasibility of implementing
the designed gain-scheduling controller on actual hardware that could be used to control an
engine.

1.2.2.2

Control of variable valve timing actuators

Two different variable valve timing actuators are considered in this dissertation. The first
is the hydraulic cam phaser, which uses pressurized engine oil to phase the cam shaft. The
second is an electric variable valve timing system that uses an electric motor connected to a
planetary gear set to phase the cam shaft.
For the hydraulic cam phasing actuator, a family of linear models previously obtained
from a series of closed-loop system identification tests [48, 52] is used to design a dynamic
gain-scheduling controller. Using engine speed and oil pressure as the system parameters,
the family of linear models was translated into a linear parameter varying (LPV) system.
An observer-based gain-scheduling controller for the LPV system is then designed based on
5

the linear matrix inequality (LMI) technique. A discussion on weighting function selection
for mixed H2 /H∞ controller synthesis is presented, with an emphasis placed on examining
various frequency responses of the system. Test bench results show the effectiveness of the
proposed scheme.
For the electric variable valve timing system, a discrete-time, linear parameter-varying
(LPV) system representing the electric variable valve timing (VVT) system is developed
with engine oil viscosity as the time-varying parameter. A gain-scheduled, dynamic, outputfeedback controller is then designed such that the closed-loop system will have a guaranteed
ℓ2 to ℓ∞ gain. This is done by first constructing a set of linear matrix inequality constraints
and then performing a convex optimization to obtain the controller matrices which satisfy the
constraints. Results from a simulation study demonstrate the effectiveness of the proposed
scheme.

1.3

Organization

This dissertation is organized as follows: a review of existing LPV control synthesis techniques and the modeling required to utilize them is presented in Chapter 2. These techniques
are extended in Chapter 3 such that hard constraints on system outputs can be obtained
with the guaranteed ℓ2 to ℓ∞ gain control problem. In Chapter 4 a gain-scheduled air-to-fuel
ratio controller for port-fuel-injection engines is developed using the wall-wetting parameters
and engine speed as the time-varying parameters for the LPV control synthesis. Results for
both a simulation study and a hardware-in-the loop simulation are presented. In Chapter 5,
a family of LTI systems, representing a variable valve timing actuator, obtained through
closed-loop system identification [48, 52] are converted into an LPV model. LPV control

6

synthesis is then applied to the LPV model to obtain a gain-scheduling controller for the
variable valve timing actuator. The obtained controller is then validated on the variable
valve timing actuator used for the system identification. In Chapter 6, a gain-scheduling
controller with a guaranteed ℓ2 to ℓ∞ gain is designed to provide hard constraints on the
system output of an electrical variable valve timing actuator. Chapter 7 concludes this
dissertation by summarizing the results and providing directions for future research.

7

Chapter 2
Linear Parameter-Varying Modeling
and Control Synthesis Methods
This chapter is split into the following two parts: modeling of LPV systems and control
synthesis methods for LPV systems.

2.1

Modeling of LPV systems

Throughout this dissertation, the control synthesis methods used rely on the existence of
an LPV model with a polytopic parameter dependence. Unfortunately, this is not the most
intuitive form that an LPV model can take. Many physical systems have parameter variations
that can be easily represented with LFTs. For this reason, we will demonstrate how to
convert an LPV model with LFT parameter dependence into an LPV model with a polytopic
parameter dependence.
Consider the following open-loop, discrete-time LPV system with LFT parameter depen-

8

dency:


 x(k + 1)


 l(k)



 z(k)


y(k)





  A Bp Bw Bu
 
 
  Cl Dlp Dlw Dlu
 
=
 
  Cz Dzp Dzw Dzu
 
 
Cy Dyp Dyw Dyu





  x(k) 




  p(k) 






  w(k) 




u(k)

(2.1)

p(k) = Θ(k)l(k)
where x(k) is the state at time k, w(k) is the exogenous input, and u(k) is the control
input. The vectors z(k) and y(k) are the performance output and the measurement for
control. Also, p(k) and l(k) are the pseudo-input and pseudo-output connected by Θ(k).
The time-varying parameter Θ(k) follows the structure

Θ(k) ∈ Θ = diag(θ1 In1 , θ2 In2 , · · · , θN InN ) .

(2.2)

To emphasize the fact that there exists an LFT with respect to the time-varying parameter matrix Θ(k), the state-space matrices can be re-arranged into the following upper
LFT:


 l(k)


 x(k + 1)



 z(k)


y(k)









  Dlp Cl Dlw Dlu   p(k) 


 


 
  Bp A Bw Bu   x(k) 


 


=


 
  Dzp Cz Dzw Dzu   w(k) 


 


 
u(k)
Dyp Cy Dyw Dyu
=:M

p(k) = Θ(k)l(k).

9

(2.3)

The time-varying matrix can be Θ(k) can be absorbed back into the state space matrices
such that the state space matrices would be given by


 x(k + 1)


 z(k)


y(k)





  A(Θ(k)) Bw (Θ(k)) Bu (Θ(k))
 
 
 =  Cz (Θ(k)) Dzw (Θ(k)) Dzu (Θ(k))
 
 
Cy (Θ(k)) Dyw (Θ(k)) Dyu (Θ(k))
=:H(Θ)





  x(k) 




  w(k) 




u(k)

(2.4)

where

H(Θ) := Fu (M, Θ)

 A Bw Bu


=  Cz Dzw Dzu


Cy Dyw Dyu





  Bp
 
 
 +  Dzp
 
 
Dyp






−1
 Θ(k) I − Dlp Θ(k)



(2.5)
Cl Dlw Dlu

.

It is clear from (2.5) that when the matrix Dlp is non-zero, then the system matrices are not
affine functions, i.e., a linear combination of the time-varying parameters plus a constant
translation. Since, as previously mentioned, all control synthesis methods covered in this
book rely on an LPV model with a polytopic parameter dependence, the system matrices
must be affine functions of the time-varying parameters. If the matrix Dlp is non-zero, then
some approximation must be made. If the parameter variation is “small”, then a first-order
Taylor series approximation can be performed.

10

2.1.1

First-order Taylor series approximation of LPV systems

¯
Using the first-order Taylor series expansion at Θ = Θ, the LPV system can be approximated
as
N

ˆ
¯
H(Θ(k)) = H(Θ) +
i=1

¯
¯
▽H(Θ) i (θi (k) − θi )

(2.6)

¯
where θi (k), for i = 1, . . . , N are the individual parameters in Θ(k), and ▽H(Θ) i is the
¯
partial derivative of the LFT system H(Θ) with respect to θi solved at Θ. The ith partial
derivative of the the upper LFT system H(Θ) is computed by [39]

[▽H(Θ)]i = M21 [I − ΘM11 ]−1 Ei [I − M11 Θ]−1 M12 ,
where

M11 = Dlp ,

M12 =

Cl Dlw Dlu

and the matrices Ei are defined such that

,



 Bp


M21 =  Dzp


Dyp

(2.7)






,



(2.8)

N

Θ(k) =

θi (k)Ei .

(2.9)

i=1

After performing this first-order Taylor series approximation, then the approximated
ˆ
system H(Θ(k)) will have affine parameter dependence with respect to Θ(k). As shown in
the next section, a polytopic LPV model can be obtained from an LPV system with affine
parameter dependence.

11

B

A

V4

V2

θ1
V1

V2

θ1
θ3

V1

V3

V1

θ2
V3

V5
V7

V2
V6
C

V4

θ1

θ2

V8
Figure 2.1: Examples of possible parameter space polytopes.

2.1.2

Polytopic linear time-varying system with Barycentric coordinates

The LPV system with affine parameter dependence can be represented by the following
polytopic linear time-varying system


 x(k + 1)


 z(k)


y(k)





  A(λ(k)) Bw (λ(k)) Bu (λ(k))
 
 
 =  Cz (λ(k)) Dzw (λ(k)) Dzu (λ(k))
 
 
Cy (λ(k)) Dyw (λ(k)) Dyu (λ(k))
=:H(λ)

12





  x(k) 




  w(k) 




u(k)

(2.10)

where the system matrices A(λ(k)), Bw (λ(k)), Bu (λ(k)), Cz (λ(k)), Cy (λ(k)), Dzw (λ(k)),
Dzu (λ(k)), Dyw (λ(k)), and Dyu (λ(k)) belong to the polytope

D=

(A, Bw , Bu , Cz , Cy , Dzw , Dzu , Dyw , Dyu )(λk ) :
(2.11)

(A, Bw , Bu , Cz , Cy , Dzw , Dzu , Dyw , Dyu )(λk )
N

=
i=1

λi (k)(A, Bw , Bu , Cz , Cy , Dzw , Dzu, Dyw , Dyu )i , λk ∈ ΛN ,

with (A, Bw , Bu , Cz , Cy , Dzw , Dzu, Dyw , Dyu )i the vertices of the polytope and λk = λ(k) ∈
RN the vector of time-varying barycentric coordinates lying in the unit simplex

ΛN =





ζ ∈ RN :

N
i=1

ζi = 1, ζi ≥ 0, i = 1, · · · , N





.

(2.12)

ˆ
The vertices of the polytope D are obtained by solving the system matrices of H(Θ) at
each of the vertices Vi for i = 1, . . . , N. Then each of the state space matrices in H(λk ) are
computed as the convex combination of the vertice systems of this polytope, such that, for
example, the state matrix would be computed by
N

A(λk ) =

λi (k)Ai .

(2.13)

i=1

Each of the other matrices in H(λk ) are computed in the same way. The convex combination
coefficients {λi (Θ)} for a given Θ and set of vertices {Vi } are also known as the barycentric
coordinates. The barycentric coordinate function is defined in [64] as

λi (Θ) =

Υi (Θ)
,
i Υi (Θ)
13

(2.14)

where Υi (Θ) is the weight function of vertex i for a point Θ inside of the convex polytope.
The weight function is
Υi (Θ) =

vol(Vi )
,
Πj∈ind(V ) (nj · (Vi − Θ))

(2.15)

i

where vol(Vi ) is the volume of the parallelepided span by the normals to the facets incident
on vertex i, i.e., Vi . {nj } is the collection of normal vectors to the facets incident on vertex i.
The volume of a parallelepided can be found as

vol(Vi ) = |det(nind )| .

(2.16)

where nind is a matrix whose rows are the vectors nj where j ∈ ind(Vi ).
Since the polytopic LTV system have been defined, we will now focus our attention in
the next section on the performance specifications for the polytopic LTV system.

2.2

Performance of discrete-time polytopic LPV systems

Consider the H2 or H∞ weighted closed-loop discrete-time LPV system

H :=



 x(k + 1) = A(λ )x(k) + B (λ )w(k),

w k
k




x(0) = 0
(2.17)

z(k) = Cz (λk )x(k) + Dw (λk )w(k)

where x(k) ∈ Rn is the state, w(k) ∈ Rr is the exogenous input, and z(k) ∈ Rp is the
performance output. The system matrices A(λk ), Bw (λk ), Cz (λk ), Dw (λk ), belong to the
polytope similar to D in (2.11).

14

The H∞ performance of the system (2.17) from w(k) to z(k) is defined by the quantity
H ∞=

z(k) 2
w(k) 2 =0 w(k) 2
sup

(2.18)

p

with w(k) ∈ ℓr and z(k) ∈ ℓ2 . By using the bounded real lemma, an upper bound for the
2
H∞ performance is characterized by the following lemma [12].
Lemma 1. Consider the system H given by (2.17). If there exist bounded matrices G(λk )
and P (λk ) = P (λk )T > 0 for all λk ∈ ΛN such that


P (λk+1 )
A(λk )G(λk )
Bw (λk )
0



 G(λk )T A(λk )T G(λk ) + G(λk )T − P (λk )
0
G(λk )T Cz (λk )T




Bw (λk )T
0
ηI
Dw (λk )T


0
Cz (λk )G(λk )
Dw (λk )
ηI








>0





(2.19)

then the system H is exponentially stable and

H ∞≤

inf

P (λk ),G(λk ),η

η.

This lemma is an extension of a standard result provided by [18, 17].
The infinite horizon H2 performance of the system (2.17) from w(k) to z(k) is defined as
H 2 = lim sup E
2
T→∞


1 T
T

k=0

z(k)T z(k)





(2.20)

where w(k) is a zero-mean white noise Gaussian process with identity covariance, E denotes
the expectation operator, and the positive integer T denotes the time horizon. An upper

15

bound for the H2 performance is characterized by the following lemma [12].
Lemma 2. Consider the system H given by (2.17). If there exists bounded matrices G(λk ),
P (λk ) = P (λk )T > 0, and W (λk ) = W (λk )T for all λk ∈ ΛN such that


P (λk+1)
A(λk )G(λk )
Bw (λk )



 G(λk )T A(λk )T G(λk ) + G(λk )T − P (λk )
0


Bw (λk )T
0
I
and



 W (λk ) − Dw (λk )Dw (λk

G(λk )T Cz (λk )T

)T

Cz (λk )G(λk )
G(λk ) + G(λk

)T

− P (λk )






>0





>0

(2.21)

(2.22)

then the system H is exponentially stable and its H2 performance is bounded by ν given by
ν2 =

inf

sup trace {W (λk )}

P (λk ),G(λk ),W (λk ) λ ∈Λ
k N

such that H 2 ≤ ν.
The proof for this lemma can be found in [12].
Note that the parameter dependent LMI conditions in Lemmas 1 and 2 must be evaluated
for all λk in the unit simplex ΛN . This leads to an infinite dimensional problem. By imposing
an affine parameter-dependent structure on the Lyapunov matrix P (λk ), such that
N

P (λk ) =

λi (k)Pi ,

i = 1, . . . , N,

i=1

a finite set of LMIs in terms of the vertices of the polytope D can be obtained.

16

(2.23)

To reduce conservatism, the parameter variation rate

∆λi (k) = λi (k + 1) − λi (k),

i = 1, . . . , N

(2.24)

is assumed to be limited. Two limits have been considered in the literature. The first rate
limit considered in the literature [40, 10, 11] is given by

−bλi (k) ≤ ∆λi (k) ≤ b (1 − λi (k)) ,

i = 1, . . . , N,

(2.25)

with b ∈ [0, 1]. With this parameter variation rate bound and the affine parameter-dependent
structure in (2.23), the H∞ performance criteria in Lemma 1 can be transformed into a finite
number of LMIs, as shown in the next Lemma [10].
Lemma 3. The system H (2.17) has an H∞ performance bounded by η if there exist matrices
Gi ∈ Rn×n and symmetric matrices Pi ∈ Rn×n such that


⋆
⋆
⋆
 (1 − b)Pi + bPℓ



GT A T
Gi + GT − P i
⋆
⋆
i i
i



T

0
ηI
⋆
Bw,i


0
Cz,i Gi
Dw,i ηI

17








>0





(2.26)

A 1

B 0.5
λ1 (k)
λ2 (k)

0.9
0.8

∆λ1 (k)
∆λ2 (k)

0.4
0.3
0.2

0.6

0.1

λi (k)

∆λi (k)

0.7
0.5
0.4

0

-0.1

0.3

-0.2

0.2

-0.3

0.1

-0.4

0
1

2

3

4

5

k

6

7

8

9

10

-0.5
1

2

3

4

5

k

6

7

8

9

10

Figure 2.2: Fastest possible parameter transition between the extreme conditions with N = 2
and b = 0.5 when the rate limit given by (2.25) is in effect.
holds for i = 1, . . . , N and ℓ = 1, . . . , N and


⋆
⋆
⋆
 (1 − b)Pi + (1 − b)Pj + 2bPℓ



⋆
Gi + GT + Gj + GT − P i − P j
GT A T + GT A T
j
i
i j
j i



T
T

Bw,i + Bw,j
0
2ηI
⋆


0
Cz,i Gj + Cz,j Gi
Dw,i + Dw,j 2ηI








>0





(2.27)

holds for ℓ = 1, . . . , N, i = 1, . . . , N − 1, and j = i + 1, . . . , N.
A proof for this lemma can be found in [10].
While the rate limit (2.25) can be useful, it is, however, not very realistic. To see this,
one only needs to consider the example parameter variation with N = 2 and b = 0.5 as
displayed in Fig. 2.2. In this example, the time-varying parameter starts at one extreme and
moves the other extreme as quickly as the parameter variation rate limit (2.25) allows. It is
clear that the maximum parameter variation rate is dependent on the current value of the
18

parameters with the rate limit given by (2.25).
A more realistic parameter variation limit that is not dependent on the current value of
the time-varying parameter is considered in [41, 12]. This limit is given by

−b ≤ ∆λi (k) ≤ b,

i = 1, . . . , N,

(2.28)

with b ∈ [0, 1]. When using this parameter variation rate, the uncertainty domain, where
the vector (λ(k), ∆λ(k))T ∈ R2N takes values, is modeled by the compact set
Γb =
N
i=1

fj
hj

δ ∈ R2N : δ ∈ co{g 1 , . . . , g M }, g j =
j

j

f j ∈ RN ,

hj ∈ RN ,
(2.29)

N

fi = 1 with fi ≥ 0, i = 1, . . . , N,

,

j

hi = 0,

j = 1, . . . , M

i=1

defined as the convex combination of the vectors g j , for j = 1, . . . , M, given a priori. This
definition of Γb ensures that λ(k) ∈ ΛN and that
N

∆λi (k) = 0

(2.30)

i=1

holds for all k ≥ 0. For a given bound b, the columns of Γb can be generated as the columns
of a matrix V as follows [12] (in MATLAB code)
V = zeros(2*N,N^2+(N-1)^2+(N-1));
for i = 1:1:N
V(i,(i-1)*N+1) = 1;
ind = 1;
for j = 1:1:N
19

if j ISNOT i
V(i,(i-1)*N+ind+1) = 1;
V(N+i,(i-1)*N+ind+1) = -b;
V(N+j,(i-1)*N+ind+1) = b;

V(i,N^2+(i-1)*(N-1)+ind) = b;
V(j,N^2+(i-1)*(N-1)+ind) = 1-b;
V(N+i,N^2+(i-1)*(N-1)+ind) = -b;
V(N+j,N^2+(i-1)*(N-1)+ind) = b;

ind = ind + 1;
end
end
end
f = V(1:N,:);
h = V(N+1:2*N,:);
With the uncertainty set Γb , each λi (k) and ∆λi (k) for i = 1, 2, . . . , N are given by
M

λi (k) =

j
fi γj (k)

M
j=1

j=1

20

j

hi γj (k)

and ∆λi (k) =

(2.31)

such that the affine representation of P (λk ) is given by
N

N

P (λk ) =

λi (k)Pi =
i=1



M

=
j=1

˜
with Pj =

j
N
i=1 fi Pi

γj (k) 

i=1
N



M

j



j=1



j

fi γj (k) Pi

(2.32)

M

fi Pi  =

i=1



˜
˜
γj (k)Pj = P (γ(k))
j=1

as shown in [12]. Using the same structure for λk , the system matrices

in H (2.17) are also converted to the new representation in terms of γ(k) ∈ ΛM , such that
M

˜
A(λk ) = A(γ(k)) =

˜
with Aj =

j
N
i=1 fi Ai .

j=1

˜
γj (k)Aj

(2.33)

All other matrices in H are converted the same way. It is also shown

in [12], that by combining (2.31) with the fact that λk+1 = λk + ∆λk ,
N

N

(λi (k) + ∆λi (k))Pi =

P (λk+1 ) =
i=1
M

=
j=1

ˆ
with Pj =

N
i=1

j



γj (k) 

i=1
N
i=1

j

j






M

j

j=1

fi + hi Pi  =

M

j



fi + hi γj (k) Pi

(2.34)

ˆ
ˆ
γj (k)Pj = P (γ(k))
j=1

j

fi + hi Pi . The authors of [12] note that due to these representations of

P (λk ) and P (λk+1 ), the LMIs of Lemma 1 and Lemma 2 can be rewritten with a dependency
on γ(k). They also note that a convenient parameterization of the slack variable G(λk ) is
given by
M

G(λk ) = G(γ(k)) =

γj (k)Gj ,
j=1

γ(k) ∈ ΛM .

(2.35)

Using these parameterizations, the next two lemmas present a finite-dimensional set of LMIs

21

that guarantee the LMI conditions of Lemmas 1 and 2 [12].
Lemma 4. Consider the system H, given by (2.17). Assume that the vectors f j and hj of
Γb are given. If there exist, for j = 1, . . . , M, matrices Gj ∈ Rn×n and, for i = 1, . . . , N,
symmetric positive-definite matrices Pi ∈ Rn×n such that


j
N
i=1 fi












j
+ hi

Pi

˜
GT A T
j j
˜T
Bw,j
0

⋆

⋆



⋆ 


j
N
T −
⋆
⋆ 
fi Pi
Gj + Gj
i=1

>0

0
ηI
⋆ 


˜
˜
Cz,j Gj
Dw,j ηI

(2.36)

for j = 1, . . . , M and












j
N
i=1 fi

j
+ fiℓ + hi

+ hℓ Pi
i

˜
˜
GT A T + GT A T
j ℓ
ℓ j
˜T
˜T
Bw,j + Bw,ℓ
0

⋆

⋆



⋆ 


Θ22,jℓ
⋆
⋆ 

>0

0
2ηI
⋆ 


˜
˜
˜
˜
Cz,j Gℓ + Cz,ℓ Gj Dw,j + Dw,ℓ 2ηI

(2.37)

with
N

Θ22,jℓ = Gj + GT + Gℓ + GT −
j
ℓ

j

fi + fiℓ Pi
i=1

for j = 1, . . . , M − 1 and ℓ = j + 1, . . . , M, then the system H is exponentially stable and
H ∞ ≤ min η.
Pi ,Gj ,η

The proof for this lemma can be found in [12].
Lemma 5. Consider the system H, given by (2.17). Assume that the vectors f j and hj of
22

Γb are given. If there exist, for j = 1, . . . , M, matrices Gj ∈ Rn×n and, for i = 1, . . . , N,
symmetric positive-definite matrices Pi ∈ Rn×n and Wi ∈ Rp×p such that


j
fi

N
i=1








j
+ hi

Pi

˜
GT A T
j j



⋆

⋆ 


j
N
>0
i=1 fi Pi ⋆ 

I

Gj + GT −
j

˜T
Bw,j

0

(2.38)

for j = 1, . . . , M,








N
i=1

j
fi

+ fiℓ

j
+ hi

+ hℓ
i

Pi



⋆

⋆ 


GP jℓ ⋆  > 0


0
2I

˜
˜
GT A T + GT A T
j ℓ
ℓ j
˜T
˜T
Bw,j + Bw,ℓ

(2.39)

where
N

GP jℓ = Gj + GT + Gℓ + GT −
j
ℓ

j

fi + fiℓ Pi
i=1

for j = 1, . . . , M − 1, and ℓ = j + 1, . . . , M,




j
N
i=1 fi Wi

˜
˜T
− Dw,j Dw,j

˜T
GT Cz,j
j

⋆
Gj + GT
j

−

j
N
i=1 fi Pi




>0

(2.40)

for j = 1, . . . , M,




N
i=1

j
fi

+ fiℓ

˜
˜T
˜
˜T
Wi − Dw,j Dw,ℓ + Dw,ℓ Dw,j

˜T
GT Cz,ℓ
j

˜T
+ GT Cz,j
ℓ

⋆
GP jℓ




>0

(2.41)

for j = 1, . . . , M − 1, and ℓ = j + 1, . . . , M, then the system H is exponentially stable and

23

its H2 performance is bound by ν given by
ν2 =

min

Pi ,Gj ,Wi

max trace {Wi } .
i

The proof for this lemma can be found in [12].

2.3
2.3.1

Control Synthesis Methods for LPV systems
Static Output Feedback Control Synthesis

In this section, the gain-scheduled static output feedback controller synthesis results from
[10] and [12] are reviewed.
Consider the following H∞ and H2 weighted, discrete-time polytopic time-varying systems H ∞ and H 2 :


 x(k + 1) = A(λ )x(k) + B∞w (λ )w∞ (k) + Bu (λ )u(k)

k
k
k




H ∞ :=
 z∞ (k) = C∞z (λk )x(k) + D∞w (λk )w∞ (k) + D∞u (λk )u(k)






y(k) = Cy x(k), Cy = Iq , 0



 x(k + 1) = A(λ )x(k) + B (λ )w (k) + Bu (λ )u(k)

2w k 2
k
k




H 2 :=
z2 (k) = C2z (λk )x(k) + D2w (λk )w2 (k) + D2u (λk )u(k)







y(k) = Cy x(k), Cy = Iq , 0

(2.42)

(2.43)

where x(k) ∈ Rn is the state, w∞ (k) ∈ Rr∞ and w2 (k) ∈ Rr2 are the H∞ and H2 exogenous
inputs, z∞ (k) ∈ Rp∞ and z2 (k) ∈ Rp2 are the H∞ and H2 performance outputs, and y ∈ Rq
is the measurement for control. The system matrices of H ∞ and H 2 belong to a polytope
24

similar to the one given in (2.11).

2.3.2

H∞ Control Synthesis

In [10], gain-scheduled static output feedback synthesis LMIs that stabilize the system H ∞
with a guaranteed H∞ performance bound are presented. The rate of variation of the
parameters (2.24) is assumed to be limited by (2.25).
Extending the analysis result presented in Lemma 3, the authors of [10] characterize a
finite set of LMI conditions for the synthesis of a gain scheduled H∞ static output feedback
controller for the system (2.42).
Lemma 6. Consider the system H ∞, given by (2.42). If there exist matrices Gi,1 ∈ Rq×q ,
Gi,2 ∈ Rn−q,q , Gi,3 ∈ Rn−q×n−q , Zi ∈ Rm×q , and symmetric matrices Pi ∈ Rn×n such
that



⋆
⋆
⋆
 (1 − b)Pi + bPℓ

 T T
 G A + Z T BT
Gi + GT − P i
⋆
⋆
i u,i
i
 i i


T

0
ηI
⋆
B∞w,i


0
C∞z,i Gi + D∞u,i Zi D∞w,i ηI

hold for i = 1, . . . , N and ℓ = 1, . . . , N and









>0





⋆
⋆
⋆
 (1 − b)Pi + (1 − b)Pj + 2bPℓ



Θ21,ij
GP ij
⋆
⋆



T
T

B∞w,i + B∞w,j
0
2ηI
⋆


0
Θ42,ij D∞w,i + D∞w,j 2ηI

25

(2.44)








>0





(2.45)

where
GP ij = Gi + GT + Gj + GT − Pi − Pj
i
j
with
T T
T T
Θ21,ij = GT AT + GT AT + Zj Bu,i + Zi Bu,j
j i
i j

Θ42,ij = C∞z,i Gj + C∞z,j Gi + D∞u,i Zj + D∞u,j Zi
hold for ℓ = 1, . . . , N, i = 1, . . . , N − 1, and j = i + 1, . . . , N, with




0 
 Gi,1
Gi = 

Gi,2 Gi,3

and Zi =

Zi,1 0

,

then the parameter-dependent static output feedback gain

ˆ
ˆ
K(λk ) = Z(λk )G(λk )−1 ,

(2.46)

with
N

ˆ
Z(λ(k)) =

N

λi (k)Zi,1

ˆ
and G(λ(k)) =

i=1

λi (k)Gi,1
i=1

stabilizes the system (2.42) with a guaranteed H∞ performance bounded by η for all λ ∈ ΛN
and ∆λ that satisfies (2.25).
A proof for this lemma can be found in [10].

2.3.3

Mixed H2/H∞ Control Synthesis

In [12], gain-scheduled static output feedback synthesis LMIs that stabilize the systems H ∞
and H 2 with guaranteed H∞ and H2 performance bounds are presented. As in [10], the rate
26

of variation of the parameters (2.24) is assumed to be limited by an a priori known bound
b, given by (2.28).
The authors of [12] extend the analysis results of Lemmas 4 and 5 to characterize a finite
set of LMI conditions for the synthesis of a gain scheduled mixed H2 /H∞ static output
feedback controller for the systems H 2 (2.43) and H ∞ (2.42).
Lemma 7. Consider the systems H ∞ and H 2 , given by (2.42) and (2.43). Assume that the
vectors f j and hj of Γb are given. Additionally, assume that a prescribed H∞ performance
bound η is given. If there exist, for i = 1, . . . , N, matrices Gi,1 ∈ Rn×n , Zi,1 ∈ Rm×q and
symmetric positive-definite matrices P∞,i ∈ Rn×n , P2,i ∈ Rn×n , and Wi ∈ Rp×p , and, for
j = 1, . . . , M, matrices G∞j,2 ∈ R(n−q)×q , G2j,2 ∈ R(n−q)×q , G∞j,3 ∈ R(n−q)×(n−q) , and
G2j,3 ∈ R(n−q)×(n−q) such that


N
i=1

j
fi

j
+ hi

P∞,i
⋆
⋆
⋆



j
˜
˜
 GT AT + Z T B T
⋆
⋆
G∞,j + G∞,j − N fi P∞,i
i=1
j u,j
∞,j j



˜T

B∞w,j
0
ηI
⋆


˜
˜
˜
0
C∞z,j G∞,j + D∞u,j Zj
D∞w,j ηI








 = Θj > 0 (2.47)





for j = 1, . . . , M and












N
i=1

j
fi

+ fiℓ

j
+ hi

+ hℓ
i

Θ21,jℓ
˜T
˜T
B∞w,j + B∞w,ℓ
0

P∞,i

⋆

⋆



⋆ 


Θ22,jℓ
⋆
⋆ 

 = Θjl > 0

0
2ηI
⋆ 


˜ ∞w,j + D∞w,ℓ 2ηI
˜
Θ42,jℓ D

27

(2.48)

with
T ˜T
T ˜T
˜
˜
Θ21,jℓ = GT AT + GT AT + Zj Bu,ℓ + Zℓ Bu,j
∞,j ℓ
∞,ℓ j

Θ22,jℓ =

G∞,j + GT
∞,j

+ G∞,ℓ + GT
∞,ℓ

N

−

i=1

j

j

fi + hi P∞,i

˜
˜
˜
˜
Θ42,jℓ = C∞z,j G∞,ℓ + C∞z,ℓ G∞,j + Du,j Zℓ + Du,ℓ Zj
for j = 1, . . . , M − 1 and ℓ = j + 1, . . . , M, and


j
fi

N
i=1



j
+ hi

P2,i
⋆



˜
˜
 GT AT + Z T B T
G2,j + GT −
2,j j
j u,j
2,j


˜T
Bw2,j
0

⋆ 


j
N
 = Φj > 0
i=1 fi P2,i ⋆ 

I

(2.49)

for j = 1, . . . , M, and








N
i=1

j
fi

+ fiℓ

j
+ hi

+ hℓ
i

P2,i

Φ21,jℓ
˜T
˜T
Bw2,j + Bw2,ℓ



⋆

⋆ 


Φ22,jℓ ⋆  = Φjℓ > 0


0
2I

with
T ˜T
T ˜T
˜
˜
Φ21,jℓ = GT AT + GT AT + Zj Bu,ℓ + Zℓ Bu,j
2,j ℓ
2,ℓ j
N

Φ22,jℓ =

G2,j + GT
2,j

+ G2,ℓ + GT
2,ℓ

28

−

j

fi + fiℓ P2,i
i=1

(2.50)

for j = 1, . . . , M − 1 and ℓ = j + 1, . . . , M, and


j
N
i=1 fi Wi

˜T
GT C2z,j
2,j




˜
˜T
− D2w,j D2w,j
T ˜T
+ Zj D2u,j

⋆
G2,j + GT
2,j

j
N
i=1 fi P2,i

−



(2.51)



(2.52)


 = Ψj > 0

for j = 1, . . . , M, and




N
i=1

j
fi

+ fiℓ

˜
˜T
˜
˜T
Wi − D2w,j D2w,ℓ + D2w,ℓ D2w,j

˜T
˜T
GT C2z,ℓ + GT C2z,j
2,j
2,ℓ

T ˜T
+ Zj D2u,ℓ

T ˜T
+ Zℓ D2u,j

⋆
Ψ22,jℓ


 = Ψjℓ > 0

with
N

Ψ22,jℓ =

G2,j + GT
2,j

+ G2,ℓ + GT
2,ℓ

−

j

fi + fiℓ P2,i
i=1

for j = 1, . . . , M − 1 and ℓ = j + 1, . . . , M where



G∞j = 
Zj =

j
N
i=1 fi Gi,1

G∞j,2
j
N
i=1 fi Zi,1

0
G∞j,3
0




,




G2j = 

j
N
i=1 fi Gi,1

G2j,2

0
G2j,3




,

and
(2.53)

,

then the parameter-dependent static output feedback gain

ˆ
ˆ
K(λk ) = Z(λk )G(λk )−1

(2.54)

with
N

ˆ
Z(λk ) =

N

λi (k)Zi,1

ˆ
and G(λk ) =

i=1

λi (k)Gi,1

(2.55)

i=1

stabilizes the system H ∞ with a guaranteed H∞ performance bounded by η and the system

29

H 2 with a guaranteed H2 performance bounded by ν given by
ν2 =

min

P∞,i ,P2,i ,Gi,1 ,G∞j,2 ,G2j,2 ,G∞j,3 ,G2j,3 ,Zi,1 ,Wi

The proof for Lemma 7 is provided by [12].

30

max trace {Wi } .
i

(2.56)

Chapter 3
Guaranteed ℓ2 − ℓ∞ gain control of
LPV systems

3.1

Introduction

The design of multi-objective, mixed H2 /H∞ controllers has been a topic of interest for
sometime [9, 54, 32, 30, 15, 56, 35, 17, 12]. The goal of using both H2 and H∞ performance
criteria is to design a controller which can meet multiple performance objectives. In [9] and
[54] mixed H2 /H∞ control was introduced by minimizing the H2 norm of a closed-loop
transfer function subject to an H∞ norm constraint of another closed-loop transfer function. In [32], mixed H2 /H∞ state-feedback and output-feedback controllers were designed
for continuous-time systems by using a convex optimization approach to solve the coupled
nonlinear matrix Riccati equations and in [30] a similar approach is used for discrete-time
systems. The state-feedback H2 /H∞ design with regional pole placement was addressed
by [15] using the linear matrix inequality (LMI) approach. In [56] and [35], the LMI approach for multi-objective control synthesis for output-feedback controllers is presented. In
[18], an extra instrumental variable was added to the LMI stability conditions to build a
parameter dependent Lyapunov function capable of proving the stability of uncertain lineartime-invariant (LTI) systems. The new extended LMI conditions in [18] were used in [17] to
develop H2 and H∞ LMI conditions for linear state-feedback and output-feedback controllers
31

for uncertain LTI systems. The extended LMI conditions provided by [17] were utilized in
[11] and [10] to develop linear parameter varying (LPV) static output feedback controllers
that meet H2 [11] and H∞ [10] performance bounds for linear time-varying (LTV) systems
with polytopic uncertainty. The results presented in [11] and [10] were extended in [12] to
cover the synthesis of multi-objective H2 /H∞ gain-scheduled output feedback controllers.
Gain scheduling controllers designed using the LPV method have traditionally included
H∞ performance constraints. This is largely due to the fact that H∞ controllers can provide
robust stability margins that H2 controllers cannot provide [83]. However, since the H∞
norm is defined as the root-mean-square gain, or ℓ2 to ℓ2 gain, from the exogenous input
to the regulated output, controllers designed with only H∞ performance constraints are not
suitable for use when hard constraints on responses or actuator signals must be met.
When hard constraints on responses or actuator signals must be met, a controller with
a guaranteed ℓ2 to ℓ∞ gain is needed, which is a special type of H2 controller [87]. A
controller with a guaranteed ℓ2 to ℓ∞ gain provides strict bounds on the regulated output
while minimizing the control input as much as possible. This problem was solved for LTI
systems in [87], where it is referred to as the output covariance constraint (OCC) problem.
The OCC problem defined in [87] is to find a controller for a given system to minimize the
weighted control input cost subject to a set of output constraints. The OCC problem has
two interesting interpretations: stochastic and deterministic. The stochastic interpretation
is obtained by first assuming that the H2 exogenous inputs are uncorrelated zero-mean
white noises with a given intensity. Then the OCC problem minimizes the weighted control
input covariance subject to the output covariance constraints, such that the constraints are
interpreted as constraints on the variance of the performance variables. The deterministic
interpretation is obtained by assuming that the H2 exogenous inputs are unknown but belong
32

to a bounded ℓ2 energy set. Then the OCC problem minimizes the weighted control input
while ensuring that the maximum singular values, or ℓ∞ response, of the regulated outputs
are less than the corresponding output constraints. In other words, the OCC problem is
the problem of minimizing the weighted sum of worst-case peak values on the control signal
subject to the constraints on the worst-case peak values of the performance variables. This
interpretation is important in applications where hard constraints on responses or actuator
signals cannot be ignored, such as space telescope pointing [86], machine tool control, and
system identification of biological systems. For both interpretations, a solution to the OCC
control problem results in a controller with a guaranteed ℓ2 to ℓ∞ gain.
The main contributions of this chapter are the guaranteed ℓ2 −ℓ∞ gain controller synthesis
LMIs for gain-scheduled state-feedback and dynamic output-feedback control for discretetime polytopic LPV systems in Section 3.3. When these LMIs are satisfied, the optimal
state-feedback or dynamic output-feedback LPV controller obtained guarantees that for a
finite disturbance energy, hard constraints on the regulated output are met. The guaranteed
ℓ2 to ℓ∞ gain is achieved by modifying H2 control synthesis LMIs covered in Chapter 2 to
minimize the weighted control input cost while ensuring the output covariances meet the
performance constraints.
The chapter is organized as follows. Section 3.2 formally introduces the ℓ2 − ℓ∞ gain
performance criteria. The mixed ℓ2 − ℓ∞ /H∞ control problem is formulated in Section 3.3
to obtain gain-scheduled state-feedback and dynamic output-feedback controllers that have
a guaranteed ℓ2 to ℓ∞ gain. In Section 3.4, a numerical example is presented to illustrate
the performance of the algorithm. Some concluding remarks for this work are provided in
Section 3.5.

33

3.2

ℓ2 − ℓ∞ Performance of Discrete-time Systems

Consider again the discrete-time, closed-loop, LPV system H in (2.17), which for convenience
is reproduced here

H :=



 x(k + 1) = A(λ )x(k) + B (λ )w(k),

w k
k




x(0) = 0

z(k) = Cz (λk )x(k) + Dw (λk )w(k).

As in Chapter 2, the system matrices A(λk ), Bw (λk ), Cz (λk ), Dw (λk ), also belong to a
polytope similar to D in (2.11).
To define the ℓ2 − ℓ∞ performance criteria for the system H, first the following assumptions are made:
1. The system output z(k) is partitioned into z(k) = [zp (k)T , zu (k)T ]T ,
2. and the feed-through matrix D(λk ) = 0,
such that the system output z(k) is given by








 zp (k)   Cp (λk ) 
z(k) := 
 x(k),
=
Cu (λk )
zu (k)

(3.1)

where the vector zp (k) contains all the variables whose dynamic responses are of interest
and the vector zu (k) contains the weighted control variables to be minimized. The impulse
response of the variables in zp (k) is given by


k−1

h(k) = Cp (λk ) 

ℓ=0



A(λℓ ) Bw (λk ), with
34

k−1
ℓ=k

A(λℓ ) = I,

for k ≥ 0. If we define the ℓ∞ and ℓ2 norms
zp 2
∞

=

T
sup zp (k)zp (k),
k≥0

w

2
2

∞

=

w T (ℓ)w(ℓ),

(3.2)

ℓ=0

then from [74, 85] it can be shown that

zp 2
∞

≤σ

∞

h(k)hT (k)

w 2,
2

(3.3)

k=0

where σ(·) denotes the maximum singular value. In [12] it is demonstrated that
∞
k=0

T
¯
h(k)hT (k) = Cp (λk )P(λk )Cp (λk )

(3.4)

¯
where P(λk ) is the closed-loop controllability Gramian from the disturbance input w(k)
satisfying
¯
¯
P(λk+1 ) = A(λk )P(λk )A(λk )T + B(λk )B(λk )T ,

(3.5)

such that
T
¯
zp 2 ≤ σ Cp (λk )P(λk )Cp (λk )
∞

w 2.
2

(3.6)

Suppose that some a priori information about the constraints on the performance of zp
is known such that an output covariance bound Z p can be constructed. It is the purpose of
this chapter to design an LPV state-feedback or dynamic, output feedback controller with

zp 2 ≤ σ Z p
∞

35

w 2,
2

(3.7)

such that the guaranteed ℓ2 − ℓ∞ gain is given by
zp
w∈ℓ2 , w 2 =0 w
2
sup

2
∞
2
2

≤ σ Zp ,

(3.8)

where
¯
Zp (λk ) = Cp (λk )P(λk )Cp (λk )T ≤ Z p .

(3.9)

This problem, which we call the guaranteed ℓ2 − ℓ∞ gain problem, is defined as follows:
find a static state feedback or full-order dynamic output feedback controller to minimize the
control energy
¯
Zu (λk ) = trace Cu (λk )P(λk )Cu (λk )T ,

(3.10)

of the closed-loop system H, subject to the hard constraint Z p .
Theorem 8. Consider the system H, given by (2.17) with the performance output given by
(3.1). Given the output covariance Z p , if there exist parameter-dependent matrices G(λk ),
P(λk ) = P(λk )T > 0, and W(λk ) = W(λk )T > 0, for all λk ∈ ΛN , such that




P(λk+1 )
A(λk )G(λk )
B(λk ) 





 G(λk )T A(λk )T G(λk ) + G(λk )T − P(λk )
0  > 0,




T
B(λk )
0
I




W(λk )
G(λk )T Cu (λk )T

Cu (λk )G(λk )
G(λk ) + G(λk

)T

− P(λk )

Z p − Cp (λk )P(λk )Cp (λk )T ≥ 0,




 > 0,

(3.11)

(3.12)

(3.13)

then the closed-loop system (2.17) is exponentially stable with a guaranteed ℓ2 to ℓ∞ perfor36

mance given by
zp
w∈ℓ2 , w 2 =0 w
2
sup

2
∞
2
2

≤ σ Zp ,

(3.14)

and a control energy bounded by

Zu =

inf

sup trace {W(λk )} ,

P(λk ),G(λk ),W(λk ) λ ∈Λ
k N

≥ trace Cu (λk )P(λk )Cu (λk )T

(3.15)

≥ Zu (λk ).

Proof. The closed-loop system H is stabilized with the control energy bounded by (3.15) as
a result of applying Lemma 2. However, the guaranteed ℓ2 − ℓ∞ gain performance (3.14) is
a result of the LMI (3.13). Since (3.11) implies that

P(λk+1 ) > A(λk )P(λk )A(λk )T + B(λk )B(λk )T ,

(3.16)

there exist matrices M(λk ) = M(λk )T > 0 such that

P(λk+1 ) = A(λk )P(λk )A(λk )T + B(λk )B(λk )T + M(λk ).

(3.17)

¯
Consequently, P(λk ) > P(λk ) for all k ≥ 0, which shows that
¯
Z p ≥ Cp (λk )P(λk )Cp (λk )T ≥ Cp (λk )P(λk )Cp (λk )T = Zp (k).

(3.18)

Thus, it follows that the guaranteed ℓ2 − ℓ∞ gain (3.14) is satisfied.

Since LMI conditions have been characterized for the ℓ2 − ℓ∞ gain performance of a

37

polytopic LPV system, attention will now be focused on using this LMI conditions to develop
LMIs for the controller synthesis of gain-scheduled state feedback and dynamic, output
feedback controllers.

3.3

Controller Synthesis for mixed ℓ2 − ℓ∞ /H∞ Control
Problem

This section considers the design of minimum energy gain-scheduled controllers that provide
a guaranteed ℓ2 − ℓ∞ gain for the closed-loop system, while also satisfying some other H∞
performance criteria for robustness. Thus, in this section we consider the following discretetime polytopic time-varying systems: the system H h with the ℓ2 − ℓ∞ gain, which provides
hard constraints that must be satisfied, given by

H h :=



 xp (k + 1) = A(λk )xp (k) + Bh (λk )wh (k) + Bu (λk )u(k)









z (k) = C (λ )x (k)
p













p

k

p

(3.19)

zu (k) = Dhu (λk )u(k)

y(k) = Cy (λk )xp (k) + Dyh (λk )wh (k)

and the H∞ weighted system H ∞ given by


 xp (k + 1) = A(λ )xp (k) + B∞ (λ )w∞(k) + Bu (λ )u(k)

k
k
k




H ∞ :=
z∞ (k) = C∞ (λk )xp (k) + D∞ (λk )w∞ (k) + D∞u (λk )u(k)







y(k) = Cy (λk )xp (k) + Dy∞ (λk )w∞ (k)

38

(3.20)

where xp (k) ∈ Rn is the state, wh (k) ∈ Rrh and w∞ ∈ Rr∞ are the exogenous inputs,
and u(k) ∈ Rm is the control input. The outputs zu (k) ∈ Rph and z∞ (k) ∈ Rp∞ are
the weighted system performance outputs for the mixed control synthesis, while the output
zp (k) ∈ Rc contains all variables whose dynamic responses have hard constraints that must
be met. The output vector y(k) ∈ Rq is the measurement to be used for control. The goal
is to provide a finite-dimensional set of LMIs for the synthesis of both gain-scheduled state
feedback controllers of the form
u(k) = K(λk )x(k)

(3.21)

and gain-scheduled, strictly proper, output feedback controllers of the form

xc (k + 1) = Ac (λk )xc (k) + Bc (λk )y(k)

(3.22)

u(k) = Cc (λk )xc (k)
such that the closed-loop systems given by

and



 x(k + 1) = A(λ )x(k) + B (λ )w (k),

k
h k h




h
Hcl :=
zp (k) = Cp (λk )x(k),






 zu (k) = Cu (λ )x(k),
k
∞
Hcl :=



 x(k + 1) = A(λ )x(k) + B (λ )w (k),

∞ k ∞
k




(3.23)

(3.24)

z∞ = C∞ (λk )x(k) + D∞ (λk )w∞ (k),

are exponentially stable and satisfy hard constraints on desired performance outputs for all
possible trajectories of the parameter λk ∈ ΛN , while minimizing the control energy and
satisfying a robustness criteria defined as an H∞ performance bound.
39

3.3.1

Gain-Scheduled State Feedback Control Synthesis

In this section, it is assumed that the state vector, xp (k), is available for feedback without
corruption from the exogenous inputs wh (k) or w∞ (k). This is a standard assumption, and
as in [17] can be enforced on the measurement equation in (3.19) and (3.20) by assigning
to the matrices Cy (λk ), Dyh (λk ), and Dy∞(λk ) the values Cy (λk ) = I, Dyh (λk ) = 0,
and Dy∞(λk ) = 0. The feedback structure provided by the gain-scheduled state-feedback
h
∞
controller (3.21), produces the closed-loop systems Hcl and Hcl in (3.23) and (3.24), where

x(k) = xp (k) and the closed-loop system matrices given by

A(λk ) = A(λk ) + Bu (λk )K(λk ),
Cp (λk ) = Cp (λk ),

Bh (λk ) = Bh (λk ),

B∞ (λk ) = B∞ (λk ),
(3.25)

Cu (λk ) = Dhu (λk )K(λk ),

C∞ (λk ) = C∞ (λk ) + D∞u (λk )K(λk ),

D∞ (λk ) = D∞ (λk ).

When substituting these closed-loop matrices into the first LMI of Theorem 8, the following bilinear matrix inequality results:


 Ph (λk ) (A(λk ) + Bu (λk )K(λk )) G(λk ) Bh (λk )



⋆
G(λk ) + G(λk )T − Ph (λk )
0


⋆
⋆
I






 > 0.



(3.26)

Multiplying the instrumental variable into the parenthesis gives


 Ph (λk ) A(λk )G(λk ) + Bu (λk )K(λk )G(λk ) Bh (λk )



⋆
G(λk ) + G(λk )T − Ph (λk )
0


⋆
⋆
I
40






 > 0,



(3.27)

which clearly shows a multiplication between the state-feedback controller K(λk ) and the
instrumental variable G(λk ). However, by letting L(λk ) = K(λk )G(λk ), the following LMI
is recovered:



 Ph (λk ) A(λk )G(λk ) + Bu (λk )L(λk ) Bh (λk )



⋆
G(λk ) + G(λk )T − Ph (λk )
0


⋆
⋆
I






 > 0.



(3.28)

The same substitution can be made in the second LMI of Theorem 8 to produce the
following LMI:





Dhu (λk )L(λk )

 W (λk )
 > 0.

T − P (λ )
⋆
G(λk ) + G(λk )
h k

(3.29)

The parameter-dependent, full-state feedback controller is solved for by performing a
convex optimization over a set of linear matrix inequalities. The LMIs in this section are
an extension of the work presented in [12], which is covered in Chapter 2. Using the parameterizations (2.32), (2.33), and (2.34) from Chapter 2, the finite-dimensional LMIs in the
following theorem can be solved to obtain a full-state feedback controller (3.21) such that the
closed-loop systems for H h and H ∞ have a guaranteed ℓ2 − ℓ∞ and H∞ gain, respectively.
In the following theorem, the LMI conditions from Lemma 7 are adapted to solve the
mixed controller synthesis problem posed in this section for the gain-scheduled state-feedback
control problem.
Theorem 9. Consider the system H h , given by (3.19). Assume that the vectors f j and hj
of Γb are given. Given Z p , if there exists, for i = 1, 2, . . . , N, matrices, Gi ∈ Rn×n and
Zi ∈ Rm×n , and symmetric positive-definite matrices Ph,i ∈ Rn×n and Wi ∈ Rph ×ph such

41

that


ˆ
Ph,j

⋆
⋆


 T T
˜ ˜
˜ ˜
˜
˜
˜
 G A + Z T B T Gj + GT − Ph,j ⋆
j u,j
j
 j j

˜T
Bh,j
0
I






 = Φj > 0



(3.30)

for j = 1, 2, . . . , M, and


ˆ
ˆ
⋆
⋆
 Ph,j + Ph,ℓ



Φ21,jℓ
Φ22,jℓ ⋆


˜T
˜T
Bh,j + Bh,ℓ
0
2I






 = Φjℓ > 0



(3.31)

with
˜ ˜
˜ ˜
˜T ˜ T
˜T ˜ T
Φ21,jℓ = GT AT + GT AT + Zj Bu,ℓ + Zℓ Bu,j
j ℓ
ℓ j
˜
˜
˜
˜
˜
˜
Φ22,jℓ = Gj + GT + Gℓ + GT − Ph,j − Ph,ℓ
j
ℓ
for j = 1, . . . , M − 1 and ℓ = j + 1, . . . , M, and




˜
Wj

⋆

˜T ˜ T
˜
˜
˜
Zj Dhu,j Gj + GT − Ph,j
j




 = Ψj > 0

(3.32)

for j = 1, 2, . . . , M and




˜
˜
Wj + Wℓ

⋆

˜T ˜ T
˜T ˜ T
Zj Dhu,ℓ + Zℓ Dhu,j Ψ22,jℓ

42




 = Ψjℓ > 0

(3.33)

with
˜
˜
˜
˜
˜
˜
Ψ22,jℓ = Gj + GT + Gℓ + GT − Ph,j − Ph,ℓ
j
ℓ
for j = 1, . . . , M − 1 and ℓ = j + 1, . . . , M and
T
Z p − Cp,i Ph,i Cp,i ≥ 0,

i = 1, 2, . . . , N,

(3.34)

with
N

j
fi

ˆ
Ph,j =
i=1
N

˜
Gj =

j
+ hi

N

˜
Ph,j =

Ph,i ,

j

fi Ph,i ,
i=1

N

j
fi Gi ,

j
fi Zi ,

˜
Zj =

N

˜
and Wj =
i=1

i=1

i=1

j

fi Wi .

then the parameter-dependent full state feedback gain

ˆ
ˆ
K(λ) = Z(λ)G(λ)−1

(3.35)

with
N

ˆ
Z(λ) =

N

λi Z i

ˆ
and G(λ) =

i=1

λi Gi

(3.36)

i=1

stabilizes the the system H h with a guaranteed (weighted) control energy bounded by Z u given
by

Zu =

min

P∞,i ,Pσ,i ,Gi ,Zi ,Wi

max trace{Wi }
i

(3.37)

≥ trace{RK(λ)Ph (λ)K(λ)T } = Zu (λk )
while also ensuring that the hard constraint Z p is satisfied. Additionally, consider the system
43

H ∞ , given by (3.20). If there exist, for i = 1, 2, . . . , N, symmetric positive-definite matrices
P∞,i ∈ Rn×n such that


ˆ
P∞,j

⋆
⋆
⋆


 T T
˜ ˜
˜ ˜
˜
˜
˜
 G A + Z T BT
Gj + GT − P∞,j
⋆
⋆
j u,j
j
 j j


˜T

B∞,j
0
ηI
⋆


˜
˜
˜
˜ ˜
0
C∞,j Gj + D∞u,j Zj D∞,j ηI













(3.38)

= Θj > 0
for j = 1, 2, . . . , M and


ˆ
ˆ
⋆
⋆
⋆
 P∞,j + P∞,ℓ



Θ21,jℓ
Θ22,jℓ
⋆
⋆


 ˜T
˜T
 B∞,j + B∞,ℓ
0
2ηI
⋆


˜
˜
0
Θ42,jℓ D∞,j + D∞,ℓ 2ηI













= Θjℓ > 0
with
˜ ˜
˜ ˜
˜T ˜ T
˜T ˜ T
Θ21,jℓ = GT AT + GT AT + Zj Bu,ℓ + Zℓ Bu,j
j ℓ
ℓ j
˜
˜
˜
˜
˜
˜
Θ22,jℓ = Gj + GT + Gℓ + GT − P∞,j − P∞,ℓ
j
ℓ
˜
˜
˜
˜
˜
˜
˜
˜
Θ42,jℓ = C∞,j Gℓ + C∞,ℓ Gj + D∞u,j Zℓ + D∞u,ℓ Zj
for j = 1, 2, . . . , M − 1 and ℓ = j + 1, . . . , M, where
N

ˆ
P∞,j =

i=1

j
fi

j
+ hi

N

P∞,i ,
44

˜
P∞,j =

i=1

j

fi P∞,i ,

(3.39)

then the parameter-dependent full-state feedback gain K(λ) given by (3.35) also stabilizes the
system H ∞ with a guaranteed H∞ performance bounded by η.
Proof. The following properties are a consequence of applying Lemma 7:
• The system H ∞ is stabilized with a guaranteed H∞ performance bounded by η when
the LMIs (3.38) and (3.39) are satisfied.
• The system H h is stabilized with a guaranteed (weighted) control energy bounded by
Z u (3.37) when the LMIs (3.30),(3.31),(3.32), and (3.33) are satisfied.
However, the fact the output constraint (3.7) is satisfied for i = 1, 2, . . . , N follows from the
LMI constraint (3.34)

T
Z p − Cp,i Ph,i Cp,i ≥ 0,

i = 1, 2, . . . , N.

Since the LMIs (3.30),(3.31),(3.32), and (3.33) are all required to be positive-definite, from
[12] it can be shown that
N

Ph (λk ) =
i=1

¯
λi (k)Ph,i > P(λk ),

∀k ≥ 0,

¯
where P(λk ) is the controllability Gramian satisfying (3.5). Thus, it is also true that
¯
Z p − Cp (λk )P (λk )Cp (λk )T ≥ 0
such that
¯
Zp (λk ) = Cp (λk )P(λk )Cp (λk )T ≤ Z p .

45

3.3.2

Output Feedback LMIs

The feedback structure provided by the gain-scheduled dynamic output-feedback controller
h
∞
(3.22), produces the closed-loop systems Hcl and Hcl in (3.23) and (3.24) where x(k) =

[xT (k) xT (k)]T with the closed-loop system matrices given by
p
c




Bu (λk )Cc (λk ) 
,
Bc (λk )Cy (λk )
Ac (λk )




B∞ (λk )
Bh (λk )




Bh (λk ) = 
,
 , B∞ (λk ) = 
Bc (λk )Dy∞(λk )
Bc (λk )Dyh (λk )

A(λk ) = 

Cp (λk ) =
C∞ (λk ) =

A(λk )

Cp (λk ) 0

,

Cu (λk ) =

C∞ (λk ) D∞u (λk )Cc (λk )

0 Dhu (λk )Cc (λk )
,

(3.40)

,

D∞ (λk ) = D∞ (λk ).

For the full-state feedback problem, the parameter-dependent, strictly proper output
feedback controller is also solved for by performing a convex optimization over a set of linear
matrix inequalities. The LMIs in this section are an extension of the work presented in [13].
It is clear that when substituting the closed-loop matrices (3.40) into the matrix inequalities
of the performance conditions in Lemma 1 and Theorem 8 nonlinear matrix inequalities
result, due to the multiplication between the unknown controller matrices and the slack
variable G(λk ). To obtain set of LMI conditions, the authors in [13] first chose the slack
variable G(λk ) to be independent of the scheduling parameter λk , such that G(λk ) = G.
Then the slack variable G, its inverse G −1 , the Lyapunov matrix P(λk ), and the controller

46

matrices K(λk ) are partitioned as








YT

Z3 

 X Z1 
−1
G := 
,
 , G := 
T Z
V
U Z2
4




 P (λk ) P2 (λk ) 
 Ac (λk ) Bc (λk ) 
P(λk ) := 
 , K(λk ) := 
.
T P (λ )
P2 (λk )
Cc (λk )
0
3 k
From the definition of G and G −1 , it is clear that the following relationship must hold:






YT



XY T

VT

Z3  
+ Z1
 X Z1  
GG −1 = 

=
U Z2
V T Z4
UY T + Z2 V T







XZ3 + Z1 Z4   I 0 
=
,
UZ3 + Z2 Z4
0 I

such that XY T + Z1 V T = I and UY T + Z2 V T = 0. Now, as done in [56, 35, 17, 13], the
parameter-independent transformation matrix


YT




 I
T := 

T
0 V

(3.41)

is introduced and the following nonlinear parameter-dependent change of variables is defined:






 V
 Q(λk ) F (λk ) 
 := 

0
L(λk )
0

 Y
+
0







U
0 
Y Bu (λk ) 


 K(λk ) 
Cy (λk )X I
I


 A(λk )

47

X 0

,

(3.42)



(3.43)

S := Y X + V U.



(3.44)

 P (λk ) J(λk ) 
T
 := T PT ,

J(λk )T H(λk )

The nonlinear matrix inequalities that result from substituting the closed-loop matrices
into the matrix inequalities in Theorem 8 can be transformed into the following LMIs:


A(λk )
Bw (λk )
 P (λk ) J(λk ) A(λk )X + Bu (λk )L(λk )


 ⋆
H(λk )
Q(λk )
Y A(λk ) + F (λk )Cy (λk ) Y Bw (λk )



 ⋆
⋆
X + X T − P (λk )
I + S T − J(λk )
0



 ⋆
⋆
⋆
Y + Y T − H(λk )
0


⋆
⋆
⋆
⋆
I


Dzu (λk )L(λk )
0
 W (λk )



⋆
X + X T − P (λk ) I + S T − J(λk )


⋆
⋆
Y + Y T − H(λk )
Z p − Cp (λk )P (λk )Cp (λk )T > 0






>0












>0







(3.45)

(3.46)

(3.47)

by using the congruence transformations T1 = diag(T , T , I) and T2 = diag(I, T ) on the
first and second nonlinear matrix inequalities, respectively. While the matrix inequalities in
(3.45), (3.46), and (3.47) are now LMIs, they are still infinite dimensional and must be evaluated for all values of time-varying barycentric coordinates λk . To obtain a finite-dimensional
set of LMIs, the parameter dependent structure imposed on the Lyapunov matrix in (2.23) is
imposed on the Lyapunov matrix here as well. In the following, the same parameterizations
used for the state-feedback case are utilized for the gain-scheduled dynamic output-feedback
48

control problem.
Theorem 10. Consider the system H h , given by (3.19). Assume that the vectors f j and
hj of Γb are given. Given Z p , if there exists matrices X ∈ Rn×n , Y ∈ Rn×n , and for
i = 1, 2, . . . , N, matrices, Jh,i ∈ Rn×n , Li ∈ Rm×n , and Fi ∈ Rn×q , and symmetric
positive-definite matrices Ph,i ∈ Rn×n , Hh,i ∈ Rn×n , and Wi ∈ Rph ×ph such that


ˆ
ˆ
˜
˜ ˜
˜
˜
Aj
Bh,j
 Ph,j Jh,j Aj X + Bu,j Lj


 ⋆ H
ˆ h,j
˜
˜
˜ ˜
˜
˜ ˜
Qj
Y Aj + Fj Cy,j Y Bh,j + Fj Dyh,j



˜
˜
 ⋆
⋆
X + X T − Ph,j I + S T − Jh,j
0



˜
 ⋆
⋆
⋆
Y + Y T − Hh,j
0


⋆
⋆
⋆
⋆
I










 = Ψj > 0







(3.48)

for j = 1, . . . , M, and


ˆ
ˆ
ˆ
ˆ
˜
˜
˜
˜
Ψ13,jℓ
Aj + Aℓ
Bh,j + Bh,ℓ
 P j + P ℓ Jj + Jℓ



ˆ
ˆ
˜
˜
⋆
Hj + Hℓ Qj + Qℓ
Ψ24,jℓ
Ψ25,jℓ



˜
˜

⋆
⋆
Ψ33,jℓ
2I + 2S T − Jj − Jℓ
0



˜
˜

⋆
⋆
⋆
2Y + 2Y T − Hj − Hℓ
0


⋆
⋆
⋆
⋆
2I

49










 = Ψjℓ > 0 (3.49)







with
˜
˜
˜ ˜
˜ ˜
Ψ13,jℓ = Aj X + Aℓ X + Bu,j Lℓ + Bu,ℓ Lj
˜
˜
˜ ˜
˜ ˜
Ψ24,jℓ = Y Aj + Y Aℓ + Fj Cy,ℓ + Fℓ Cy,j
˜
˜
˜ ˜
˜ ˜
Ψ25,jℓ = Y Bh,j + Y Bh,ℓ + Fj Dyh,ℓ + Fℓ Dyh,j
˜
˜
Ψ33,jℓ = 2X + 2X T − Pj − Pℓ
for j = 1, . . . , M − 1 and ℓ = j + 1, . . . , M, and




˜
˜
˜
Dhu,j Lj
0
 Wj




T −P
T − J  = Φj > 0
˜j I + S
˜j 
 ⋆ X +X




˜
⋆
⋆
Y + Y T − Hj

(3.50)

for j = 1, . . . , M, and


˜
˜
˜
˜
˜
˜
0
 Wj + Wℓ Dhu,j Lℓ + Dhu,ℓ Lj


˜
˜
˜
˜

⋆
2X + 2X T − Pj − Pℓ 2I + 2S T − Jj − Jℓ


˜
˜
⋆
⋆
2Y + 2Y T − Hj − Hℓ






 = Φjℓ > 0



(3.51)

for j = 1, . . . , M − 1 and ℓ = j + 1, . . . , M, and
T
Z p − Cp,i Pσ,i Cp,i ≥ 0,

50

i = 1, 2, . . . , N,

(3.52)

where
N

ˆ
Ph,j =
i=1
N

˜
Ph,j =

j
(fi

˜
Qj =

j

˜
Aj =

j

j

fi Ai ,
i=1
N

˜
Cz,j =
˜
Cy,j =

N

ˆ
Hh,j =

j

˜
Jh,j =

fi Jh,i ,

N

˜
Hh,j =

j

j

fi Cy,i ,

˜
Fj =

fi Li ,

j

fi Hh,i ,

j

fi Fi ,
i=1

i=1
N

(3.53)

N

j

˜
Bu,j =

fi Bu,i ,

˜
Bw,j =

j

fi Bw,i ,
i=1

N

˜
Dzu,j =

N

j

fi Dzu,i,
i=1
N

˜
Dyw,j =

i=1

j

i=1
N

j

˜
Lj =

j

(fi + hi )Hh,i ,
i=1

N

i=1

fi Cz,i ,
i=1
N

N

j
+ hi )Jh,i ,

i=1

fi Qi ,
i=1
N

j
(fi

ˆ
Jh,j =
i=1

fi Ph,i ,
i=1
N

N

j
+ hi )Ph,i ,

˜
Dzw,j =

j

fi Dzw,i ,
i=1

j

fi Dyw,i ,
i=1

then the parameter-dependent, strictly proper output feedback controller

xc (k + 1) = Ac (λk )xc (k) + Bc (λk )y(k)
u(k) = Cc (λk )xc (k)
with
Ac (λk ) = V −1 Q(λk ) − Y A(λk )X − Y Bu (λk )L(λk ) − F (λk )Cy (λk )X U −1
Bc (λk ) = V −1 F (λk )

(3.54)

Cc (λk ) = L(λk )U −1
stabilizes the system H h with a guaranteed (weighted) control energy bounded by Z u while
also ensuring that the hard constraint Z p is satisfied. Additionally, consider the system H ∞ ,
given by (3.20). If there exist, for i = 1, 2, . . . , N, matrices J∞,i ∈ Rn×n and symmetric
51

positive-definite matrices P∞,i ∈ Rn×n and H∞,i ∈ Rn×n such that




ˆ
ˆ
˜
˜ ˜
˜
˜
Aj
B∞,j
0 
 P∞,j J∞,j Aj X + Bu,j Lj




 ⋆
ˆ ∞,j
˜j
˜j + Fj Cy,j Θ25,j
˜ ˜
H
Q
YA
0 






T −P
˜∞,j I + S T − J∞,j
˜
 ⋆
⋆
X +X
0
Θ36,j 


 = Θj > 0


T −H
˜ ∞,j
˜T 
 ⋆
⋆
⋆
Y +Y
0
C∞,j 





˜T 
⋆
⋆
⋆
ηI
D∞,j 
 ⋆




⋆
⋆
⋆
⋆
⋆
ηI

(3.55)

with
˜
˜ ˜
Θ25,j = Y B∞,j + Fj Dy∞,j
˜T
˜ ˜T
Θ36,j = X T C∞,j + LT D∞u,j
j
for j = 1, . . . , M



ˆ
 P∞,j


















ˆ
+ P∞,ℓ

ˆ
ˆ
J∞,j + J∞,ℓ

⋆

ˆ
ˆ
˜
˜
H∞,j + H∞,ℓ Qj + Qℓ

Θ13,jℓ

˜
˜
Aj + Aℓ Θ15,jℓ
Θ24,jℓ

⋆

⋆

Θ33,jℓ

Θ34,jℓ

⋆

⋆

⋆

Θ44,jℓ

⋆

⋆

⋆

⋆

⋆

⋆

⋆

⋆

52

0






Θ25,jℓ
0 



0
Θ36,jℓ 

 = Θjℓ > 0

0
Θ46,jℓ 



2ηI Θ56,jℓ 


⋆
2ηI

(3.56)

with
˜
˜
˜ ˜
˜ ˜
Θ13,jℓ = Aj X + Aℓ X + Bu,j Lℓ + Bu,ℓ Lj
˜
˜
Θ15,jℓ = B∞,j + B∞,ℓ
˜
˜
˜ ˜
˜ ˜
Θ24,jℓ = Y Aj + Y Aℓ + Fj Cy,ℓ + Fℓ Cy,j
˜
˜
˜ ˜
˜ ˜
Θ25,jℓ = Y B∞,j + Y B∞,ℓ + Fj Dy∞,ℓ + Fℓ Dy∞,j
˜
˜
Θ33,jℓ = 2X + 2X T − P∞,j − P∞,ℓ
˜
˜
Θ34,jℓ = 2I + 2S T − J∞,j − J∞,ℓ
˜T
˜T
˜ ˜T
˜ ˜T
Θ36,jℓ = X T C∞,j + X T C∞,ℓ + LT D∞u,ℓ + LT D∞u,j
j
ℓ
˜
˜
Θ44,jℓ = 2Y + 2Y T − H∞,j − H∞,ℓ
˜T
˜T
Θ46,jℓ = C∞,j + C∞,ℓ
˜T
˜T
Θ56,jℓ = D∞,j + D∞,ℓ
for j = 1, . . . , M − 1 and ℓ = j + 1, . . . , M, and
N

˜
P∞,j =
˜
J∞,j =
˜
H∞,j =

i=1
N
i=1
N
i=1

j
fi P∞,i ,
j

fi J∞,i ,
j

fi H∞,i ,

N

ˆ
P∞,j =
ˆ
J∞,j =

j

i=1
N

j

j

(fi + hi )J∞,i ,

i=1
N

ˆ
H∞,j =

j

(fi + hi )P∞,i ,

i=1

j

(3.57)

j

(fi + hi )H∞,i ,

then the parameter-dependent, strictly proper output feedback controller also stabilizes the
system H ∞ with a guaranteed H∞ performance bounded by η.
The proof of Theorem 10 is similar to the proof for Theorem 9 and is omitted.

53

3.4

Numerical Example

Consider the discrete-time LPV system (originally used in [19], and later used in [4] and
[17])


1
 2 + δ1 0


xp (k + 1) =  1
0.5
0


0
1 −0.5
A(δ1 (k))








 1 + δ2




 xp (k) +  0




0



 0 

 

 

 u(k) +  1  wp (k)
 

 

0

Bu (δ2 (k))







Bh

(3.58)

 1 0 0 




zp (k) =  0 1 0  xp (k)




0 0 1
Cp

zu (k) = u(k)
where δi , i = 1, 2 are the time-varying parameters. In this section, two design examples with
different performance constraints will be considered for both state-feedback and dynamic
output-feedback control.

3.4.1

State Feedback Control

For the state-feedback controller design, the time-varying parameters were assumed to have
the following parameter variation bounds:

δ1 ∈ [−1, 1],

and δ2 ∈ [−0.5, 0.5].

54

(3.59)

The discrete-time LPV system (3.58) is converted to the discrete-time polytopic LPV system
(3.19) by solving A(δ1 ) and Bu (δ2 ) at the vertices of the parameter space polytope of δ1 and
δ2 . The exogenous ℓ2 disturbance wp is a scalar and the performance variable zp has three
components.
In the following, we consider two different ℓ2 to ℓ∞ gain designs. The designs differ in
the grouping of the performance variables inside of zp used to define the constraints (3.9).
The constraints for each design are given as follows:

Design 1: Zp ≤ 1.85 × I3 ,

(3.60)

Design 2: Zp,1 ≤ 1.85,

(3.61)

Zp,2 ≤ 1.85 × I2 ,

where for design 1, Zp denotes the (3 × 3) output covariance matrix corresponding to the
all performance outputs in zp grouped together. In design 2, Zp,1 denotes the (1 × 1)
output variance corresponding to the first performance output of zp and Zp,2 denotes the
(2 × 2) output covariance matrix corresponding to the second and third performance outputs
grouped together.
For each design, to enhance the robustness of the closed-loop system with the controller
K(λk ) with respect to uncertainty in the measurements of the time-varying parameters δ1
and δ2 , the closed-loop H∞ norms of the transfer functions of some appropriately defined
extra inputs and outputs that ‘pull out’ [19, 17] the uncertain parameters are bounded.

55

Specifically, the following H ∞ system is defined:
 
 




 1 
 1 


 
 



 x(k + 1) = A(δ1 (k))x(k) + Bu (δ2 (k))u(k) +  0  w∞,1 (k) +  0  w∞,2 (k)
 
 


 
 


 
 


∞
0
0
H =





 z∞,1 (k) = 1 0 0 x(k)








 z
(k) = u(k)
∞,2

(3.62)

so that the robustness requirement is given by

Hz∞,i w∞,i (λ) ∞ < η = 100,

i = 1, 2,

(3.63)

where η defines the robustness level. Note that the notation used here, specifically w∞,1 (k)
and w∞,2 (k) with the same input matrix, was selected to match what is found in the literature [19, 17].
For each of the ℓ2 -ℓ∞ designs (3.60)-(3.61), the LMIs in Theorem 9 are programmed into
MATLAB using the LMI parser YALMIP [34] and solved with SeDuMi [59] to minimize the
cost function (3.37). As shown in Fig. 3.1 and Fig. 3.3A, each design is feasible and the
achieved covariance bound is tight with the design bound in at least one dimension. The
constraint in design 1 ensures that the covariance bound ellipsoid of Zp remains inside of
the sphere displayed in Fig. 3.1A. Side views of the covariance bound Zp are displayed in
Fig. 3.1B, Fig. 3.1C, and Fig. 3.1D. As displayed in Fig. 3.1C, the output covariance Zp is
tight with the bound in the zp,2 -zp,3 plane.
For design 2, the constraints ensure that the variance of the first output of zp will be

56

2

A

2

B

1

0

zp,2

zp,3

1

-1

-1

-2
2
2
0

-2
-2

0
-2 -2

zp,2

1

2

-1

0
zp,1

1

2

1
zp,3

zp,3

0
zp,1

2

D

1
0
-1
-2
-2

-1

zp,1

2

C

0

0
-1

-1

0

zp,2

1

-2
-2

2

Figure 3.1: Design 1: The covariance bound Zp achieved compared to the constraint (3.60).

57

1.5

A

B

1.5

1

1

zp,3

0.5

0.5
zp,2

0
-0.5

0

-1

-0.5

-1.5

-1
1

zp,2
C

-1.5

1

0

0

-1

-1

0
zp,1

1

-1

0
zp,1

1

zp,1
D

1.5

1.5
1

0.5

0.5
zp,3

1
zp,3

-1

0

0

-0.5

-0.5

-1

-1

-1.5

-1

0

zp,2

-1.5

1

Figure 3.2: Design 1: The output response of zp,1 , zp,2 , and zp,3 plotted against each other
for design 1 simulated with a positive (I1 ) and negative (I2 ) impulse function and compared
with the ℓ∞ norm bound.

58

ts

A 2

B 1.5

1.5

1

1
0.5
zp,3

zp,3

0.5
0
-0.5

I2

0

I1

-0.5

-1
-1

-1.5
-2
-2

-1

0

zp,2

1

-1.5
-1.5

2

-1

-0.5

0
zp,2

0.5

1

1.5

Figure 3.3: Design 2: A. The covariance bound Zp,2 achieved compared to the second
constraint in (3.61). B. The output response of zp,2 plotted against zp,3 for design 2 simulated
with a positive and negative impulse function and compared with the ℓ∞ norm bound.
below 1.85 and the covariance bound of second and third outputs of zp will remain inside of
the circle in Fig. 3.3A. The dashed ellipses in Fig. 3.3A are the obtained output covariances
at each of the vertices for i = 1, . . . , 4, and as shown they are tight with the bound.
To test the performance of each design, we simulate each of the controllers with a positive
impulse (I1 ) followed by a negative impulse (I2 ) as displayed in Fig. 3.4A. To see the effect of
the time-varying parameters, the parameters δ1 and δ2 are varied as displayed in Fig. 3.4B.
The values used to compute the controller at each time step k are the noisy measurements
displayed with a gray dashed line. The response to the ℓ2 disturbance wp (k) for design 1 is
displayed in Fig. 3.2. The response in Fig. 3.2 is plotted inside of the ℓ∞ norm constraint
(the square root of the covariance bound) sphere and the achieved ℓ∞ norm bound ellipsoid.
In Fig. 3.3B, the response of design 2 is plotted inside of the ℓ∞ norm constraint circle and
the achieved ℓ∞ norm bound ellipse. The path of the response, with respect to each of the
impulses (I1 ) and (I2 ), is also displayed in Fig. 3.3B. As shown in Fig. 3.2 and Fig. 3.3B, the
59

A

1

I1

wp (k)

0.5
0
-0.5
I2
-1

δ2 (k)

B

0

50

100

150
k

200

250

300

True
Measured

0.5

0

-0.5
-1

-0.8

-0.6

-0.4

-0.2

0
δ1 (k)

0.2

0.4

0.6

0.8

1

Figure 3.4: The ℓ2 disturbance (A) and the parameter variation (B) used to simulate each
controller design.

60

response for each design stays inside of the ℓ∞ bound.

3.4.2

Dynamic Output-Feedback Control

For the dynamic output-feedback controller design, the measurement equation in [19, 4, 17]
is given as
y(k) =

0 1 + δ3 (k) 0

x(k),

(3.64)

where δ3 is an additional time-varying parameter. As one might guess, this is much more
restrictive than the state feedback case. The good news is that the system is still observable
(assuming δ3 = −1). However, the bad news is that in order to obtain a feasible controller
with the LMIs provided by Theorem 10, some modifications need to be made to the problem.
The modified discrete-time LPV is given by


1
 2 + δ(k) 0


xp (k + 1) = 
1
0.5
0


0
1 −0.5
A(δ(k))







 1 + δ(k)





 xp (k) + 
0




0
Bu (δ(k))



 1 0 0 




zp (k) =  0 1 0  xp (k)




0 0 1
zu (k) = u(k)
0 1 + δ(k) 0





 0 







 u(k) +  0.1  wp (k)






0
Bh

(3.65)

Cp

y(k) =



x(k) + 0.01v(k)

Cy (δ(k))

61

Notice that in the modified system, each of the time-varying parameters δi , i = 1, 2, 3 have
been set equal to each other such that δ = δ1 = δ2 = δ3 , which was originally done by the
authors in [4]. Also, as in [4], the time-varying parameter δ is assumed to have the following
parameter variation bound:
δ ∈ [−0.2,

0.2].

(3.66)

As for the state-feedback design case, the discrete-time LPV system (3.65) is converted to
the discrete-time polytopic LPV system (3.19) by solving A(δ), Bu (δ), and Cy (δ) at the
vertices of the parameter space polytope of δ. For the dynamic output-feedback design, the
exogenous ℓ2 disturbance is given by the process disturbance wp (k) and the measurement
disturbance v(k), such that w(k) = [wp (k), v(k)]T . The performance variable zp , again has
three components.
As we did for the state-feedback control example, we also consider two different ℓ2 to ℓ∞
gain designs for the dynamic output-feedback control example. As before, the designs differ
in the grouping of the performance variables inside of zp used to define the constraints (3.9).
The constraints for each design are given as follows:

Design 1: Zp ≤ 5 × I3 ,

(3.67)

Design 2: Zp,1 ≤ 5,

(3.68)

Zp,2 ≤ 5 × I2 ,

where for design 1, Zp denotes the (3 × 3) output covariance matrix corresponding to the
all performance outputs in zp grouped together. In design 2, Zp,1 denotes the (1 × 1)
output variance corresponding to the first performance output of zp and Zp,2 denotes the
(2 × 2) output covariance matrix corresponding to the second and third performance outputs

62

grouped together.
The robustness of the closed-loop system with the dynamic output-feedback controller
with respect to uncertainty in the measurements of the time-varying parameter δ, the closedloop H∞ norms of the transfer functions of some appropriately defined extra inputs and
outputs that ‘pull out’ [19, 17] the uncertain parameters are bounded. The system H ∞ used
for the dynamic output-feedback design is the same as given in (3.62), with the following
additions:

y(k) = Cy (δ(k))x(k) + w∞,3 (k)
z∞,3 (k) =

0 1 0

(3.69)

x(k)

such that the robustness requirement is now given by

Hz∞,i w∞,i (λ) ∞ < η = 100,

i = 1, 2, 3,

(3.70)

where η defines the robustness level.
For each of the ℓ2 -ℓ∞ designs (3.67)-(3.68), the LMIs in Theorem 10 are programmed
into MATLAB and solved with LMI Lab [22] to minimize the control energy Zu . As shown
in Fig. 3.5 and Fig. 3.7A, each design is feasible and the achieved covariance bound is tight
with the design bound in at least one dimension. The constraint in design 1 ensures that
the covariance bound ellipsoid of Zp remains inside of the sphere displayed in Fig. 3.1A.
Side views of the covariance bound Zp are displayed in Fig. 3.5B, Fig. 3.5C, and Fig. 3.5D.
As displayed in Fig. 3.5C, the output covariance Zp is tight with the bound in the zp,2 -zp,3
plane.
For design 2, the constraints ensure that the variance of the first output of zp will be
63

5

B

0

5

zp,2

zp,3

A

0

-5
5
5
0
zp,2

0
-5 -5

D

0

zp,2

5

0
zp,1

5

5

zp,3

zp,3

0

-5
-5

0
zp,1

zp,1

5

C

-5
-5

0

-5
-5

5

Figure 3.5: Design 1: The covariance bound Zp achieved compared to the constraint (3.60).

64

A

B

2

2
1

0

zp,2

zp,3

1

-1

0
-1

-2

-2

2

2

0

-2

-2

zp,2

-2

-1

0
zp,1

1

2

-2

0

-1

0
zp,1

1

2

zp,1
D

C

1
zp,3

2

1
zp,3

2

0

0

-1

-1

-2

-2
-2

-1

0
zp,2

1

2

Figure 3.6: Design 1: The output response of zp,1 , zp,2 , and zp,3 plotted against each other
for design 1 simulated with a positive (I1 ) and negative (I2 ) impulse function and compared
with the ℓ∞ norm bound.

65

A

B

5

2

zp,3

zp,3

1
0

I2

0

I1

-1
-2

-5
-5

0
zp,2

-2

5

-1

0
zp,2

1

2

Figure 3.7: Design 2: A. The covariance bound Zp,2 achieved compared to the second
constraint in (3.61). B. The output response of zp,2 plotted against zp,3 for design 2 simulated
with a positive and negative impulse function and compared with the ℓ∞ norm bound.
below 5 and the covariance bound of second and third outputs of zp will remain inside of
the circle in Fig. 3.7A. The dashed ellipses in Fig. 3.7A are the obtained output covariances
at each of the vertices for i = 1, 2, and as shown they are tight with the bound.
To test the performance of each design, we again simulate each of the controllers with
a positive impulse (I1 ) followed by a negative impulse (I2 ) as displayed in Fig. 3.8A. To
see the effect of the time-varying parameter, the parameter δ was varied as displayed in
Fig. 3.8B. As before, the values used to compute the controller at each time step k are the
noisy measurements displayed with a gray dashed line. The response to the ℓ2 disturbance
wp (k) for design 1 is displayed in Fig. 3.6. The response in Fig. 3.6 is plotted inside of the
ℓ∞ norm constraint (the square root of the covariance bound) sphere and the achieved ℓ∞
norm bound ellipsoid. In Fig. 3.7B, the response of design 2 is plotted inside of the ℓ∞
norm constraint circle and the achieved ℓ∞ norm bound ellipse. The path of the response,
with respect to each of the impulses (I1 ) and (I2 ), is also displayed in Fig. 3.7B. As shown
66

A

1

I1

wp (k)

0.5
0
-0.5
I2
-1
B

0

50

100

150
k

200

250

300

True
Measured

0.2

δ(k)

0.1
0
-0.1
-0.2
0

50

100

150
k

200

250

300

Figure 3.8: The ℓ2 disturbance (A) and the parameter variation (B) used to simulate each
controller design.

67

in Fig. 3.6 and Fig. 3.7B, the response for each design stays inside of the ℓ∞ bound. It is
interesting to note that the response to the positive impulse (I1 ) is larger than the response
to the negative impulse (I2 ). This is caused by the different time-varying parameter δ(k)
values at the time of each impulse, as is displayed in Fig. 3.8.

3.5

Conclusions

In this chapter, discussion motivating the necessity of the ℓ2 − ℓ∞ gain performance criteria
was provided. Then, the ℓ2 − ℓ∞ gain performance criteria was introduced to allow for the
specification of hard constraints when designing gain-scheduling controllers. Controller synthesis LMIs are provided for the synthesis of state-feedback and dynamic output-feedback
controllers with guaranteed ℓ2 −ℓ∞ gain and H∞ performance. To demonstrate the effectiveness of the controller synthesis LMIs provided in this chapter, gain-scheduled state-feedback
and dynamic output-feedback controllers are designed for a numerical example.

68

Chapter 4
Gain-Scheduling Control of
Port-Fuel-Injection Processes

4.1

Introduction

Increasing concerns about global climate change and ever-increasing demands on fossil fuel
capacity call for reduced emissions and improved fuel economy. Port-fuel-injection (PFI)
fuel systems are widely used in vehicles today; however, direct-injection (DI) fuel systems
have also been introduced to markets globally. To improve the full load performance of DI
engines at high speed, Toyota introduced an engine with a stoichiometric DI system with
a DI injector and an intake port injector for each cylinder (see [29]). The use of gasoline
PFI and ethanol DI dual-fuel system to substantially increase gasoline engine efficiency is
described by [28]. This shows that with the introduction of DI fuel systems for the internal
combustion engine, PFI fuel systems will remain part of the engine fuel system for improved
engine performance, which is the main motivation for revisiting the air-to-fuel (A/F) ratio
control problem for a PFI fuel system.
There have been several fuel control strategies developed for internal combustion engines
to improve the efficiency and exhaust emissions. A key development in the evolution was
the introduction of a closed-loop fuel injection control algorithm [53], followed by the linear quadratic control method [14], and an optimal control and Kalman filtering design [47].
69

Specific applications of A/F ratio control based on observer measurements in the intake
manifold were developed by [8]. Another approach was based on measurements of exhaust
gas A/F ratio measured by the oxygen sensor and on the throttle position [42]. [16] also
developed a nonlinear sliding mode control of A/F ratio based upon the oxygen sensor feedback. Continuing research efforts of A/F ratio control include adaptive approaches [62, 80],
observer-based controllers [46], H∞ controllers [36], model predictive controllers [38], sliding
mode controllers [43], and linear parameter-varying controllers [25, 82, 88]. Conventional
A/F ratio control for automobiles uses both closed-loop feedback and feedforward control to
have good steady state and fast transient responses.
For a spark-ignited engine equipped with a port-fuel-injection system, the wall-wetting
dynamics are commonly used to model the fuel injection process; and the wall-wetting effects
are compensated on the basis of simple time-invariant linear models that are tuned and
calibrated through engine dynamometer and vehicle tests. These models are quite effective
for an engine operated at steady state or slow transition conditions but they are difficult
to use at fast transient and other special operational conditions, for instance, during engine
cold start. One of the approaches to model the wall-wetting dynamics during engine cold
start is to describe it using a family of linear models to approximate the system dynamics
at a given engine coolant temperature, speed and load conditions, that is, to translate the
fuel system model into a linear parameter varying (LPV) system.
As stated earlier, the use of LPV modeling to control the A/F ratio of a port-fuelinjection system has been reported by [25, 82, 88]. In [88], a continuous-time, LPV model is
developed considering only engine speed as a time-varying parameter. Due to the simplicity
of the model used, the issue of engine cold start is not addressed. Furthermore, the control
synthesis method used in [88] relies on gridding the parameter space at a finite number of
70

grid points. In [82], a large variable time delay is present in the A/F ratio control loop for
a lean burn spark ignition engine. LPV control methods are used to compensate for the
variable time delay. In [25], a discrete-time, LPV model is developed with manifold absolute
pressure, exhaust value closing, and inlet value opening as the time-varying parameters.
However, only manifold absolute pressure is used as a scheduling parameter in the gainscheduling control that is synthesized. Also, [25] does not address the issue of engine cold
start. Additionally, all LPV control synthesis methods used by [25] are based in continuous
time, relying on Tustin’s (bilinear) transformation to convert the discrete-time system to a
continuous-time system, thus fixing the engine speed and sampling rate of the discrete-time
system. In contrast to all of these efforts, in this chapter an event-based, gain-scheduling
controller for an event-based, discrete-time LPV system with wall-wetting parameters and
engine speed as time-varying parameters is designed. To cope with practical situations,
the discrete-time LPV control synthesis method in Lemma 6 is used to develop the eventbased, gain-scheduling controller. An affine LPV model including the feedforward control
dynamics is obtained. Gain-Scheduling controllers have been synthesized to guarantee the
robust stability and performance of the affine LPV model.
The contribution of this chapter is as follows. First, an event-based, discrete-time LPV
model for the wall-wetting and oxygen sensor dynamics with wall-wetting parameters and
engine speed as scheduling variables is developed. Then an event-based, gain-scheduling
controller for the derived LPV model is designed. To cope with practical situations, the
discrete-time LPV control synthesis method given by [10] is used to develop the event-based,
gain-scheduling controller.
The control structures used in this study are proportional-integral (PI) and proportionalintegral-derivative (PID). PI controllers are widely used in industry since they are well un71

Unmodeled
engine system

LPV model

Physics-based
modeling

Not satisfied
Finalized LPV
controller
Satisfied

Experimental
validation

LPV controller
design

LPV controller

Analytical and simulation behaviors
Performance requirements
Figure 4.1: Flowchart of the design and validation process of an LPV controller.
derstood by field control engineers. The PI gains are often calibrated in field tests for the
best performance as functions of system operational conditions. However, the system stability and performance are not guaranteed for all time-varying parameters. Therefore, LPV
techniques proposed in this chapter are applied to design gain-scheduling PI controllers for
guaranteed stability and performance for all time-varying parameters. Furthermore, the addition of derivative control to a PI controller adds an extra layer of complexity. The design
of a PID controller at a single operating point can be a difficult iterative procedure, which
would make calibrating PID gains as functions of system operational conditions very time
consuming. However, designing a gain-scheduling PID controller using LPV techniques provided in this chapter is as simple as adding a derivative channel to the control input. The
ability to design either a gain-scheduling PI or PID controller with guaranteed stability and
performance in one shot without requiring hours of calibration is expected to be well received
by industrial control engineers.
The process of designing an LPV controller for any automotive application is depicted

72

in Fig. 4.1. Due to the complexity of internal combustion engines, designing controllers for
specific engine systems using an entire engine model is extremely difficult if even possible.
Therefore, to design a controller for a specific engine subsystem, first a physics-based simplified model is developed to represent the engine subsystem. After the varying parameters are
identified, the physics-based model can be transformed into an LPV model. LPV controller
design can then be carried out on the LPV model to develop an LPV controller. Once the
LPV controller is obtained it must be tested on the original engine to ensure that it meets
all stability and performance requirements. A cost effective way of validating developed
LPV controllers is to implement them in a rapid prototyping real-time control systems and
validate them through hardware-in-the-loop (HIL) simulations.
In this work, we first develop a physics-based model for the port-fuel-injection process
based on the wall-wetting dynamics and formulate it as an LPV system. The system parameters used in the engine fuel system model are engine speed, temperature, and load. These
system parameters can be obtained in real-time through physical or virtual sensors. A gainscheduling controller is then obtained for the derived LPV system based on the numerically
efficient convex optimization (or LMI) techniques. To validate the gain-scheduling PI and
PID controllers, HIL simulations were performed using a mixed mean-value and crank-based
engine model [79].
This chapter is organized as follows. The models and the modeling techniques used are
given in Section 6.2. The design of the gain-scheduling controller in Section 4.3 is covered by
first introducing the control strategy in Section 4.3.1. Then the feed-forward compensated
generalized plant is developed in Section 4.3.2 and its first-order Taylor series expansion is
computed in Section 4.3.3. Next the measurement for control is elaborated in Section 4.3.4.
The gain-scheduling synthesis problem is stated in Section 4.3.5. In Section 4.3.6, the aug73

mented LPV plant obtained in Section 4.3.4 is converted into a polytopic time-varying system, which is an LPV system with a polytopic dependency on a scheduling parameter that
takes values in the unit-simplex, so that the gain-scheduling controller synthesis technique
reviewed in Section 2.3.2 given by [10] can be performed. For comparison, a linear timeinvariant feedback H∞ controller is designed in Section 4.4 using the nominal parameters.
Simulation results from three separate engine operating conditions are presented in Section 4.5. Next, an HIL simulation set up is introduced in Section 4.6 and components of
the mixed mean-value and crank-based engine model [79] are reviewed. In Section 4.7 the
results from the HIL simulations are presented. The concluding remarks are given in the
final section.

4.2

Event-based discrete-time system modeling

In this section, the dynamics of the plant (Fig. 4.2) will be carefully explained and modeled
to develop a control oriented linear parameter varying (LPV) model. The plant given in
Fig. 4.2 shows the port-fuel-injection process for a single cylinder engine. However, the
methods used in this chapter can be extended to a multiple cylinder engine by using the
individual cylinder fuel-gas ratio estimation method developed by [60].

4.2.1

Sampling period of the event-based discrete-time system

The discrete-time linear system is obtained by event-based sampling of the port-fuel-injection
process; hence the sampling time of this discrete-time system is the period of an engine cycle,

ts =

1 min.
Ne rev.

60 sec.
1 min.

2 rev.
1 cycle
74

=

120 sec.
,
Ne cycle

(4.1)

w1 (disturbance fuel-to-air ratio)

Combustion
delay

Wall-wetting dynamics
mi

G(q)

mc

y

cq −1

1/m0
A

-

ys

z

O2 Sensor

Oxygen sensor
w2 (desired equivalence ratio)
Figure 4.2: The block diagram of the port-fuel-injection process and sensor dynamics.
where Ne represents the engine speed in revolutions per minute (rpm) (see general engine
modeling techniques in [7]).

4.2.2

Dynamics of the port-fuel-injection process

The wall-wetting dynamics can be described as follows:

mw (k) = (1 − αk )mw (k − 1) + (1 − βk )mi (k),

(4.2)

mc (k) = αk mw (k − 1) + βk mi (k),
where k ∈ Z≥0 , and mw , mc , and mi denote the amount of fuel, on the wall, in the cylinder,
and injected, respectively. The coefficients α ∈ [0, 1], and β ∈ [0, 1], are the ratios of the fuel
delivered from the wall to the cylinder, and of the fuel entering the cylinder from injection,

75

respectively. For notational simplicity, αk and βk will be used to denote the wall-wetting
parameters at time k, such that αk = α(k) and βk = β(k). These values can be estimated
online through an available set of engine sensors, which allows application of gain-scheduling
control to the plant. Using the discrete-time dynamics in (4.2), the transfer function G(q)
from mi to mc is

G(q) :=

β + (αk − βk )q −1
mc (k)
= k
,
mi (k)
1 − (1 − αk )q −1

(4.3)

where q is the forward shift operator that satisfies qu(k) = u(k + 1). The dotted box in the
block diagram in Fig. 4.2 illustrates the fuel-injection process. The output of G(q) is the
input to the gain block of 1/m0 , which is the nominal value of the inverse of the mass of air
A
trapped in the cylinder mA . The signal w1 represents the deviation
mc
mc
− 0
mA mA

,

which will be treated as a disturbance. Another constant gain factor c = 14.6 in Fig. 4.2
is the value for the air-to-fuel-ratio at stoichiometric. After the combustion delay block the
equivalence ratio y is generated. The diagram of the transfer function from the amount of
fuel injected mi and the disturbance w1 to the equivalence ratio y (inverse of normalized
air-to-fuel ratio) is shown in the dotted box in Fig. 4.2.

4.2.3

Dynamics of the oxygen sensor

To measure y, a full range oxygen sensor is placed in the exhaust manifold at some distance downstream from the exhaust valve. Notice that the continuous-time dynamics and

76

delays will change in the event-based, discrete-time system according to the speed of the
engine (or the sampling time). Therefore, the objective of this section is to obtain oxygen
sensor dynamics in the form of the finite dimensional, event-based, discrete-time LPV system. Finite dimensionality is required for the applicability of most LPV controller design
techniques and the controller design method which will be presented in Section 4.3. To this
end, in general, one can approximate the continuous-time system with a delay by a finite
dimensional event-based, discrete-time LPV system in any standard method. To illustrate
this procedure, we demonstrate how we approximate the oxygen sensor dynamics by Taylor
series approximation in which the approximation error can be minimized by increasing the
order of the Taylor series approximation.
The dynamics of the oxygen sensor are modeled as a first-order sensor delay coupled
d
, of the
with the transport delay of the exhaust gas mixture. The transport delay, TD =
Ne
exhaust gas mixture is both a function of the oxygen sensor placement, which determines
the constant d, and the engine speed, Ne . The combined transfer function in the continuous
time domain is
ys (s) =

exp (−TD s)
y(s),
TO2 s + 1

(4.4)

where ys is the equivalence ratio measured by the sensor and TO2 is the time constant of
the oxygen sensor. Since the delay

TD ∈

d d
,
Ne Ne

77

is small, Equation (4.4) can be approximated by the second-order system

ys (s) =

1
1
y(s),
TD s + 1 TO2 s + 1

which has the state-space representation


1
 −
 TD
xO2 = 
˙

0



1
TD
1
−
TO2




 xO2 + 



=:AO
2

ys =

1 0



0
1
TO2
=:BO
2




 y,


(4.5)

xO2 .

=:CO
2

Using ts as the sampling rate, the corresponding discrete system of Eq. (4.5) is

xO2 (k + 1) = AO2 d xO2 (k) + BO2 d y(k),

(4.6)

ys (k) = CO2 d xO2 (k),
where, due to the invertibility of the matrix AO2 in (4.5),

AO2 d = exp(AO2 ts ),
BO2 d =

ts
0

exp(AO2 τ )dτ

BO2 = A−1 (AO2 d − I)BO2 ,
O
2

CO2 d = CO2 .
Since both TD and ts are functions of engine speed, Ne , naturally AO2 d and BO2 d are as
well. To capture this parameter variation, the matrices AO2 d and BO2 d are now computed

78

80
. To solve for AO2 d , first AO2 is multiplied by ts
for a given transport delay of TD =
Ne



Ne
Ne
 −
80  120

AO2 ts =  80

1  Ne
0
−
TO2


3
3
 −

2

= 2

120  .
0 −
TO2 Ne

(4.7)

Next, the matrix exponent of AO2 ts is computed, which gives


3
 exp − 2
AO2 d = 

0



p1 (Ne ) 
,

p2 (Ne )

(4.8)

where

p1 (Ne ) =

−

3
2

exp −

p2 (Ne ) = exp −

3
2

− exp −

120
3
−
+
TO2 Ne 2
120
TO2 Ne

.

120
TO2 Ne

,

(4.9a)

(4.9b)

To represent the parameter variation in AO2 d , a fourth-order Taylor series approximation of
p1 (Ne ) and p2 (Ne ) is used. To ensure that the coefficients of the Taylor series approximations
of p1 (Ne ) and p2 (Ne ) are numerically stable with respect to the condition number [61],

79

1
1 1
∈[ ,
] is normalized to γ in the following way:
Ne
Ne Ne
1
1
−
Ne Ne,0
γ=
1
1
+
Ne Ne,0

Solving equation (4.10) for

where

1
1
+
1
Ne Ne
.
=
Ne,0
2

(4.10)

1
, and substituting into equations (4.9a) and (4.9b), p1 (γ) and
Ne

p2 (γ) are found to be

p1 (γ) =

−

3
2

exp −

3
2

− exp −

120
−
TO2 Ne,0

p2 (γ) = exp −

120
TO2 Ne,0

120
TO2 Ne,0

1+γ
1−γ

1+γ
1−γ

3
+
2

1+γ
1−γ

.

,

(4.11a)

(4.11b)

Finally, the forth-order Taylor series approximation of AO2 d is represented with the following
lower LFT:
AO2 d = Fℓ (MA
, γI4 )
O d

(4.12)

2

where



3
 exp − 2



0

MA
=
0
O2 d


0


0
0

80



a0 a1 a2 a3 a4 


b0 b1 b2 b3 b4 

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0
0 0 0 1 0

(4.13)

and
1 d1 p1 (0)
,
1! dγ 1
1 d1 p2 (0)
,
b1 =
1! dγ 1

a1 =

1 d2 p1 (0)
,
2! dγ 2
1 d2 p2 (0)
b2 =
,
2! dγ 2

1 d3 p1 (0)
1 d4 p1 (0)
, a4 =
,
3! dγ 3
4! dγ 4
1 d3 p2 (0)
1 d4 p2 (0)
b3 =
, b4 =
.
3! dγ 3
4! dγ 4

a2 =

a3 =

Now focusing attention on Bd , recall that BO2 d = A−1 (AO2 d − I)BO2 (see (4.6)). Since
O2
AO2 d is already found, A−1 is now computed. The inverse of AO2 is given by
O2

A−1
O2



1

 −
 TO2
= TD TO2 

0

1
−
TD
1
−
TD






80

  −TD −TO2   − Ne −TO2 
.
=
=



0
−TO2
0
−TO2




(4.14)

Thus, A−1 can be represented with the following lower LFT:
O
2

1
A−1 = Fℓ M −1 ,
O2
A
N

,

(4.15)


0 −TO2 −80


M −1 =  0 −TO
0 .
A
2

(4.16)

O2

where

O2

To normalize

e



1

0

0

1
to γ, the following upper LFT is used:
Ne

1
= Fu (Mγ , γ),
Ne




1
1
1 .
where Mγ =  2
Ne,0 Ne,0

(4.17)

ˆ
ˆ
The approximated state-space matrices AO2 d and BO2 d are represented in Fig. 4.3 by
ˆ
their respective dotted boxes. The approximated state matrix AO2 d block is formed by the

81

−I2
y

BO2

q −1 I2

M −1
A

MA

O2 d

O2

γk

γ k I4

CO2 d

ys

MA

O2 d

Mγ

ˆ
BO2 d

ˆ
AO2 d

γ k I4

Figure 4.3: Block diagram of the combined dynamics of the exhaust gas and sensor delays.
lower LFT MA

O2 d

connected to the time-varying parameter matrix γk I4 . The approximated

ˆ
input matrix BO2 d block is formed by the matrix multiplications of BO2 d in Eq. (4.6). The
ˆ
ˆ
AO2 d , BO2 d , and CO2 d blocks are then connected in the standard state-space interconnection [58]. After performing the interconnection displayed in Fig. 4.3, the fourth-order
approximated system used for controller design is given by

ˆ
ˆ
xO2 (k + 1) = AO2 d (γk )ˆO2 (k) + BO2 d (γk )y(k),
ˆ
x

(4.18)

ys (k) = CO2 d xO2 (k),
ˆ
ˆ
where

120
) a(γk )
exp(−
ˆ
,
d
AO2 d (γk ) = 
0
b(γk )


d (γk + 1) a(γk )

− b(γk ) + 1 
.
ˆO d (γk ) =  v0 (γk − 1) TO
B 2


2


1 − b(γk )

ˆ
The approximated state matrix AO2 d (γk ) follows directly from (4.8). The approximated
82

Amplitude

Ne = 1000 rpm
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

0.857
0.856
0.855
0.854
0.25

0.35

Ne = 6000 rpm
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

0.67
0.66
0.65
0.64

0.08

0.1

1
2
0
0.1
0.2
0.3
0.4
1.5
2.5
Time (sec)
Time (sec)
Figure 4.4: Comparison of the response to a unit step function for the 4th order Taylor
series approximation model in Eq. 4.18 (dashed line) and a model Eq. 4.6 with the engine
speed fixed at 3500 rpm (dash-dot line) to the exact discretized oxygen sensor delay model
in Eq. 4.6 (solid line) at 1000 rpm and 6000 rpm.
0

0.5

ˆ
input matrix BO2 d (γk ) follows from the matrix operations performed to compute BO2 d in
(4.6). The following polynomial functions a(γk ) and b(γk ):
2
3
4
a(γk ) = 0.3972 − 0.4891γk − 0.0984γk + 0.0608γk + 0.0975γk ,
2
3
4
b(γk ) = 0.3114 − 0.7266γk + 0.1211γk + 0.3095γk + 0.2231γk ,

were found when selecting an oxygen sensor time constant of TO2 = 0.06 seconds and a
80
transport delay of TD =
, by setting d = 80, indicating that the transport delay is about
Ne
54 ms at an engine speed of 1500 rpm. This was determined empirically through engine
calibration tests.
To demonstrate the effectiveness of the proposed model for the event-based sampling of
the oxygen sensor delay, a comparison is made between the proposed 4th order Taylor series
approximation model and a fixed model computed at the nominal engine speed (3500 rpm).

83

In Fig. 4.4, the step response of the 4th order Taylor series approximation model (dashed
line) is compared to the exact discretized model (solid line) at engine speeds of 1000 rpm
and 6000 rpm. The fixed model computed at the nominal engine speed (dash-dot line) is
also compared to the exact model in Fig. 4.4. It is clear that the fixed model computed
at the nominal engine speed either responds too slowly when the engine speed is less than
the nominal speed or too quickly when the engine speed is greater than the nominal speed.
However, the approximated model’s response very closely follows the exact model’s response
as shown in Fig. 4.4.

4.2.4

An LPV system

In summary, by combining the wall-wetting dynamics in (4.2) and the oxygen sensor delay
and dynamics in (4.18) as shown in Fig. 4.2, we obtain the following LPV system for the
event-based discrete-time port-fuel-injection and oxygen sensor dynamics:

1 − αk
0
0


xww (k + 1)
 xww (k)
 cα
k

x
 
0
0
  xcomb (k) 

 comb (k + 1)  =  m0


A
 x (k)

ˆO2
xO2 (k + 1)
ˆ
ˆ
ˆ
0
BO2 d (γk ) AO2 d (γk )


1 − βk
 
 cβ 
0

k 
+
mi (k) +  c  w1 (k),

 m0 
A
0






0

(4.19)




xww (k)


z(k) = 0 0 −CO2 d  xcomb (k)  + w2 (k),
xO2 (k)
ˆ

where xww (k) = mw (k − 1) and xcomb (k) are the wall-wetting state and the combustion
state for the system in the dotted box in Fig. 4.2.
84

Table 4.1: Modeling parameters.
Parameter
TO2 is a constant
TD is a function of engine speed, Ne

Value used in Study
0.06
80
Ne

Table 4.2: Measurable time varying parameters (scheduling parameters).
α(cylinder head temperature(t), manifold absolute pressure(t)) ∈ [0.081, 0.1]
β(cylinder head temperature(t), manifold absolute pressure(t)) ∈ [0.28, 0.89]
1
1
−
Ne (t) Ne,0
∈ [−0.55556, 0.26316]
γ(Ne (t)) =
1
1
+
Ne (t) Ne,0
As can be seen from (Eqs. 4.5, 4.6, 4.18, and 4.19), to apply the model of the LPV system,
one needs to identify TD and TO2 (which are shown in Table 4.1) and measurable time
varying parameters such as α, β, and γ (which are shown in Table 6.1), which will be used
for scheduling the gain of the controller. In particular, the identified bounds of scheduling
variables ([α, α], [β, β] and [γ, γ]) as shown in Table 6.1 will be used in synthesizing the
gain-scheduling controller. From now on, a compact notation Θ will denote an appropriate
gain-scheduling matrix that contains the scheduling variables. The specific structure of Θ
will be presented in Eq. (4.24) of Section 4.3.2. In addition, the LPV system in Eq. (4.19)
is denoted by P (Θ). In the following section, we illustrate how to design the LPV gainscheduling controller as a function of Θ for the LPV model developed in this section.

85

w1
˜

w1

W1 (q)

w2
˜

W2 (q)

w2

z
P (Θ)

mi
w3
˜

w3

Kf (Θ)

eP

L(q)

uf

I(q)
ˆ
H(Θ) ≈ H(Θ)

u

D(q)

eI

eD

K(Θ)

Figure 4.5: The proposed control strategy for the fuel injection process (without the weighting functions W1 (q) and W2 (q)). Weighting functions W1 (q) and W2 (q) are only used for
controller synthesis. A first-order Taylor series expansion is applied to the to the systems
inside the dashed box and the LPV control strategy is applied to the all of the systems inside
of the bold box.

4.3
4.3.1

LPV gain-scheduling controller design
Control Strategy

The objective of the control system is to regulate the equivalence ratio y to a reference input
w2 using feed-forward and feedback control against the disturbance signal w1 (See Fig. 4.2)
and the time-varying wall-wetting dynamics. In particular, we want to guarantee the stability
of the closed-loop system and also minimize the effect of the disturbances for any conceivable
wall-wetting dynamics variations. The proposed control architecture is illustrated in Fig. 6.2.
This scheme has five possible components, that is a feedback controller K(Θ), a feed-forward
controller Kf (Θ), a filter L(q), an integrator I(q), and possibly a differentiator D(q) (if a
PID controller is desired).

86

The feedback controller K(Θ) will be designed for the generalized plant (solid box of
Fig. 6.2), after selecting Kf (Θ), L(q), I(q), possibly D(q), and weighting functions W1 (q)
and W2 (q). Next, we will explain how to select these functions. After the selection, we will
derive the generalized plant in Section 4.3.4 and we will synthesize K(Θ) for the derived
generalized plant in Section 4.3.5.
The feed-forward controller Kf (Θ) is designed using the inverse of cG(q)

Kf (Θ) =

1 − (1 − αk )q −1
βk + (αk − βk )q −1

1
G−1 (q)
=
c
c

.

The selection of the inverse of the plant as a feed-forward controller is a standard technique
[58]. The input to the feed-forward controller is the mass of the air mA , which can be
measured online, multiplied by the equivalence ratio set point w2 . This is denoted by w3 ,
such that w3 = w2 mA . L(q) is designed as a low-pass filter such that the error output z(k)
is filtered with it
L(q) =

0.9999
.
q − 0.0001405

The reason to filter the error output is that the control synthesis technique given by [10]
requires that the output matrix be independent of the time-varying parameters and the
measurement for control must not be corrupted by the unweighted exogenous input, w(k)
˜
of the generalized plant. The low-pass filtering for this purpose is a standard procedure
[2]. The low-pass filter L(q) was obtained from the discretization of the following first-order
continuous transfer function:
Lc (s) =
with a sample period of

2πfc
s + 2πfc

120
. The cut-off frequency fc of Lc (s) was selected to be 20 Hz,
Ne,0
87

which is high enough to obtain low error between the intended output of the continuous-time
filter Lc (q) and the observed output of the discrete-time filter L(q) at different engine speeds,
since the sampling rate is engine speed dependent. The filtered output is also integrated using
the integrator
I(q) =

1
q−1

to obtain zero steady-state error. To enhance the response of the closed-loop system when
large changes in w1 are present, then derivative action [5]
e (k)
F (q − 1)
D(q) := D
=
eP (k)
(F + 1)q − 1
is introduced, where F is chosen to set the location of the pole of the derivative filter. Notice
that I(q) and D(q) are not functions of the sampling rate, ts . This is due to the requirement
ˆ
that, as previously stated, that the output matrix C2 be independent of the time-varying
parameters. For this reason, I(q) is really just a numerical summation and D(q) is a filtered,
numerical differencer.
To use ℓ2 gain or H∞ norm [84] for the performance criterion for shaping the frequency
response of the closed-loop system, weighing functions (which can be considered design
parameters) are also introduced in Fig. 6.2. The weighting functions are selected in the
continuous-time domain as
100
,
50s + 1
2
20
c (s) =
.
W2
50s + 1

c
W1 (s) =

The bandwidth (or cut-off frequency) of each weighting function is very small and the DC

88

60
W1
W2

50

Magnitude (dB)

40
30
20
10
0
-10
10−4

10−3

Frequency (Hz)

10−2

10−1

Figure 4.6: Magnitude plot of the weighting functions W1 and W2 .
gain is large, as shown in Fig. 4.6. The weighting functions are selected to model the
frequency content of their respective input. For the fuel-to-air ratio disturbance w1 , the
c
weighting function W1 (s) is selected as a simple first-order low-pass filter to place an em-

phasis on low frequency disturbances, such as a step throttle change. The weighting function
c
W2 (s) is chosen to be a second-order low-pass filter with a high DC gain (4 times larger than
c
that of W1 (s)) to provide more weight on the low frequency signals since w2 is the step input
c
c
of the desired equivalence ratio. To incorporate the weighting functions W1 (s) and W2 (s)

into the discrete time system, they were discretized at a sample period of

89

120
to obtain the
Ne,0

following discrete-time weighting functions:
0.1411
,
q − 0.9986
0.0003982q + 0.0003979
.
W2 (q) = 2
q − 1.997178q + 0.997180
W1 (q) =

The input to each of the weighting functions are the unweighted exogenous inputs which are
denoted by w1 , w2 , and w3 . Since there is no weighting function for w3 , in this case w3 = w3 ;
˜ ˜
˜
˜
which means that it is weighted equally at all frequencies. Notice that the weighting functions
are chosen by the expected system inputs and their relative (frequency) importance, and they
are only used for controller synthesis [84, 58].

4.3.2

Feed-forward compensated generalized plant

The feed-forward compensated generalized plant is denoted by H(Θ). As depicted in the
dashed box of Fig. 6.2, the feed-forward compensated generalized plant consists of the feedforward controller Kf (Θ), the plant P (Θ), and the weighting functions W1 (q) and W2 (q).
The components of the feed-forward controller Kf (Θ) and the plant P (Θ) are illustrated in
Fig. 4.7. The feed-forward controller Kf (Θ) components are encased inside of the dashed
box in Fig. 4.7 and the plant P (Θ) components are outside of the dashed box.
In the feed-forward control compensated generalized plant H(Θ), the time-varying parameters αk and βk are equivalently transformed to a constant nominal value plus a timeα+α
varying fluctuation. For instance, the parameter variation of αk ∈ [α, α] with α0 =
2
is represented by
αδ (k) = αk − α0 ∈ [α − α0 , α − α0 ],
so that the parameter range of αδ (k) is centered around zero. Hence, αk is replaced by
90

βδ
β0

w1

αδ

u
−

−

α0

1
m0
a

q −1

cq −1

y

βδ
M1/β
∆
q −1

−

Mα/β
−

1
c

w3

γI9
z

− ys

O2 Sensor

w2
Figure 4.7: The feed-forward control compensated generalized plant with the time-varying
parameters included.
α0 + αδ (k). The same is done for βk ∈ [β, β] as well. The parameter variation of v is
represented by γ as shown in Eq. (4.10). The upper LFTs (see Appendix A) inside the
dotted box in Fig. 4.7, M1/β and Mα/β are used to isolate the time-varying parameters
1
by
βδ (k) and αδ (k) [84]. βδ is isolated from
βk
1
1
= Fu M1/β , βδ (k) ,
=
βk
β0 + βδ (k)

91

(4.20)

where


1
1
−β −β 
0
0

.
M1/β = 
1 
1


β0 β0


Both βδ (k) and αδ (k) are isolated from

(4.21)

αk
by
βk

αk
α + αδ (k)
= Fu Mα/β , ∆(k) ,
= 0
βk
β0 + βδ (k)
where




1
1
α0
−β −β − β 
0
0
0 



Mα/β =  0
0
1 ,
 1
1
α0 


β0 β0
β0

(4.22)





0 
 βδ (k)
and ∆(k) = 
.
0
αδ (k)

With the parameter variation represented in this way, the system is written as a discretetime LPV system with LFT parameter dependency,






 x(k + 1)   A B0 B1 B2

 

 
 l(k)  =  C0 D00 D01 D02

 

 
z(k)
C1 D10 D11 D12







 x(k) 




  p(k) 



,


  w(k) 
˜




u(k)

(4.23)

p(k) = Θ(k)l(k),
where x(k) ∈ Rn is the state at time k, w(k) ∈ Rr is the unweighted exogenous input,
˜
z(k) ∈ Rp is the error output, p(k), l(k) ∈ Rnp are the pseudo-input and pseudo-output
connected by Θ(k), and u(k) ∈ Rm is the control input. The state-space matrices for the
LPV system in (4.23) are provided in Appendix B.

92

The time-varying parameter Θ in Eq. (4.23) follows the structure

Θ ∈ Θ = {diag(βδ I3 , αδ I2 , γI9 ) : |αδ | ≤ δ1 , |βδ | ≤ δ2 , |γ| ≤ 1},
where δ1 =

4.3.3

(4.24)

β−β
α−α
and δ2 =
.
2
2

First-order Taylor series expansion of the LPV system

By inspection of the LPV system in Eq. (4.23), D00 was found to be a non-zero matrix.
Hence, the system matrices are not affine functions, i.e., a linear combination of the timevarying parameters plus a constant translation. It is noted at this juncture that LPV control
techniques exist which do handle rational parameter variation, namely the method developed
by [75]. However, for discrete-time systems, no controller formula covering all parameter
variation is given by [75]. Instead, for each set of parameters a controller must be solved
for using the method given by [21]. Since a different controller is needed for each set of
parameters, gridding over the parameter space [1] is necessary, which increases the complexity
of implementing the controller in practice. In contrast, the method developed by [10] does
not require any gridding over the parameter space. Also, as shown in Eq. (4.24) and Table 6.1
each of the parameters are less than 1 at all times. Therefore, neglecting the higher-order
parameter variation is a justifiable approximation. Hence, to utilize the control synthesis
technique given by [10], we calculate the first-order Taylor series approximation of the system
matrices to obtain affine functions in Θ. To find the Taylor series expansion, first the LFT

93

(4.23) is re-arranged to the following representation:






l(k)
D00 C0
 

 x(k + 1)  =  B0 A
D10 C1
z(k)




p(k)
D01 D02 

  x(k) 
B1 B2  
,
 w(k) 
˜
D11 D12
u(k)


(4.25)

=:M

p(k) = Θ(k)l(k).
Notice that Eq. (4.25) is an upper LFT, i.e.,

H(Θ) := Fu (M, Θ)
=

B0
A B1 B2
Θ (I − D00 Θ)−1 C0 D01 D02 .
+
D10
C1 D11 D12

(4.26)

Using the Taylor series expansion at Θ = 0, the system can be approximated as

ˆ
H(Θ) = H(0) + αδ ▽αδ H(0) + βδ ▽β H(0) + γ ▽γ H(0) ,
δ


ˆ
ˆ
ˆ
A(αδ , βδ , γ) B1 (αδ , βδ , γ) B2 (αδ , βδ , γ)
,
=: 
ˆ1 (αδ , βδ , γ) D11 (αδ , βδ , γ) D12 (αδ , βδ , γ)
ˆ
ˆ
C

(4.27)

where the relationship between αδ , βδ , and γ, and Θ can be found in Eq. (4.24) and [▽a H(0)]
is the partial derivative of the LFT system H(Θ) in Eq. (4.26) with respect to a, which can
be calculated as shown by [39]. The state-space representation after performing the Taylor
series expansion is given by

x(k + 1)
z(k)



x(k)
ˆ
ˆ
ˆ
A(αδ , βδ , γ) B1 (αδ , βδ , γ) B2 (αδ , βδ , γ) 
  w(k)  .
˜ 
=
ˆ
ˆ
ˆ
C1 (αδ , βδ , γ) D11 (αδ , βδ , γ) D12 (αδ , βδ , γ)
u(k)


94

(4.28)

4.3.4

An augmented LPV system for synthesis

ˆ
To create an appropriate measurement for gain-scheduling control, the LPV system H(Θ)
must be augmented with the low-pass filter L(q), the integrator I(q), and the numerical
differencer D(q) (when designing a gain-scheduled PID controller).

4.3.4.1

PI Control

After augmenting the affine LPV system with the low pass filter and the integrator the
augmented state space representation is given by

  ˜

˜
˜
A(αδ , βδ , γ) B1 (αδ , βδ , γ) B2 (αδ , βδ , γ)
xAU G (k + 1)
xAU G (k)

  ˜


˜
˜
z(k)
˜
 =  C1 (αδ , βδ , γ) D11 (αδ , βδ , γ) D12 (αδ , βδ , γ)   w(k) 



e(k)
u(k)
˜
C2
0
0


where the augmented states are given by xAU G (k) = x(k)T xL (k) xI (k)

T

(4.29)

∈ RnAU G with

nAU G = n + 2, and the measurement for control is given by e(k) = [eP (k) eI (k)]T ∈ Rq with

95

q = 2. The state-space matrices are given by

ˆ δ , βδ , γ)
A(α
0 0


˜ δ , βδ , γ) =  B C (α , β , γ) A 0  ,
ˆ1 δ δ
A(α
 L
L 
0
CL 1


ˆ
B1 (αδ , βδ , γ)


˜
B1 (αδ , βδ , γ) =  B D (α , β , γ)  ,
ˆ


L

11

δ

δ

0




ˆ
B2 (αδ , βδ , γ)


˜
B2 (αδ , βδ , γ) =  B D (α , β , γ)  ,
ˆ
L 12 δ δ
0

˜
ˆ
C1 (αδ , βδ , γ) = C1 (αδ , βδ , γ) 0 0 ,
0 CL 0
˜
C2 =
,
0 0 1

˜
ˆ
˜
ˆ
and D11 (αδ , βδ , γ) = D11 (αδ , βδ , γ), D12 (αδ , βδ , γ) = D12 (αδ , βδ , γ). The matrices (AL ,
BL , CL ) represent the state-space matrices of the low-pass filter L(q).

4.3.4.2

PID Control

When designing a gain-scheduling PID controller, the augmented the affine LPV system with
the low pass filter, the integrator, and the numerical differencer, the augmented state space
representation is given by


  ˜
˜
˜
A(αδ , βδ , γ) B1 (αδ , βδ , γ) B2 (αδ , βδ , γ)
xAU G (k)
xAU G (k + 1)


  ˜

˜
˜
˜
z(k)
 =  C1 (αδ , βδ , γ) D11 (αδ , βδ , γ) D12 (αδ , βδ , γ)   w(k) 



u(k)
e(k)
˜
C2
0
0


where the augmented states are given by xAU G (k) = x(k)T xL (k) xI (k) xD (k)

T

(4.30)

∈ RnAU G

with nAU G = n+3, and the measurement for control is given by e(k) = [eP (k) eI (k) eD (k)]T ∈
96

Rq with q = 3. The state-space matrices are given by


ˆ
A(αδ , βδ , γ)
0


ˆ
˜
A(αδ , βδ , γ) =  BL C1 (αδ , βδ , γ) AL

0
CL

0
BD CL


ˆ
B1 (αδ , βδ , γ)




ˆ
˜
B1 (αδ , βδ , γ) =  BL D11 (αδ , βδ , γ)  ,


0
0


ˆ
B2 (αδ , βδ , γ)




ˆ
˜
B2 (αδ , βδ , γ) =  BL D12 (αδ , βδ , γ)  ,


0
0

0 0




0 0 ,


1 0 
0 AD

˜
ˆ
C1 (αδ , βδ , γ) = C1 (αδ , βδ , γ) 0 0 0 ,


0 CL 0 0


˜
C2 =  0
0
1 0 ,
0 DD C L 0 C D

˜
ˆ
˜
ˆ
and D11 (αδ , βδ , γ) = D11 (αδ , βδ , γ), D12 (αδ , βδ , γ) = D12 (αδ , βδ , γ). The matrices (AL ,
BL , CL ) represent the state-space matrices of the low-pass filter L(q) and the the matrices
(AD , BD , CD , DD ) represent the state-space matrices of the filtered, numerical differencer
D(q).

4.3.5

A gain-scheduling control synthesis problem

Having augmented all components for the controller synthesis, we need to synthesize the
H∞ gain-scheduling controller K(Θ). The ℓ2 gain of the LPV system in Eq. (4.29) with a

97

gain-scheduling feedback controller is defined as
z ℓ2
.
˜
Θ∈Θ, w ℓ =0 w ℓ2
˜
2
max

(4.31)

Now we formally state the gain-scheduling control design problem.
Problem : The goal is to design a static gain-scheduling control u(k) = K(Θ)e(k) that
stabilizes the closed-loop system and minimizes the worst-case ℓ2 gain (H∞ norm) of the
closed-loop LPV system in Eq. (4.31) for any trajectories of Θ(k) ∈ Θ.
The gain-scheduling method provided by [10] guarantees an H∞ cost such that for an
exogenous input w, the performance output z satisfies
˜

z ℓ2 < η w ℓ2 ,
˜

for any trajectories of Θ(k) ∈ Θ. This method was derived for discrete-time polytopic timevarying systems. Therefore, in the next section, we will transform the augmented, affine
system into a polytopic time-varying system to synthesize the controller.

4.3.6

Controller synthesis for polytopic linear time-varying system

˜
˜
The augmented state-space representation (A(αδ , βδ , γ), B1 (αδ , βδ , γ), ...) in either (4.29) or
¯
¯
(4.30) can be converted into a discrete-time polytopic time-varying system (A[λ(k)], B1 [λ(k)],
...) by using the state-space matrices at vertices {Vi } of the parameter space polytope
displayed in Fig. 4.8. Any system inside of the convex parameter set is represented by a
convex combination of the vertex systems as weighted by the vector λ(k) of barycentric

98

γ
V1
V3

V5
V7

V2
V6

V4

αδ

βδ

V8
Figure 4.8: Parameter space polytope.
coordinates. Barycentric coordinates are used to specify the location of a point as the center
of mass, or barycenter, of masses placed at the vertices of a simplex. [64] provides a formula,
which is covered in Appendix 2.1.2, for computing the barycentric coordinates for any convex
polytope. The discrete-time polytopic linear time-varying system is given by
 
¯
A[λ(k)]
x(k + 1)
 ¯

 z(k)  =  C1 [λ(k)]
¯
C2
e(k)


¯
B1 [λ(k)]
¯
D11 [λ(k)]
0
T

e(k) = eP (k) eI (k)



¯
x(k)
B2 [λ(k)]


¯
D12 [λ(k)]   w(k)  ,
u(k)
0

(4.32)

,

where, for all k ∈ Z≥0 , λ(k) is the vector of time-varying barycentric coordinates that belong
to the unit simplex (2.12). A way to compute the barycentric coordinate vector λ(k) for a
given αδ (k), βδ (k), and γ(k) is provided in Appendix 2.1.2. For all k ∈ Z≥0 , the rate of
variation of the weights

∆λi (k) = λi (k + 1) − λi (k),
99

i = 1, · · · , N

is limited by the calculated bound b such that

−bλi (k) ≤ ∆λi (k) ≤ b(1 − λi (k)),

i = 1, · · · , N

(4.33)

where b ∈ [0, 1].
¯
¯
¯
¯
The system matrices A[λ(k)] ∈ Rn×n , B1 [λ(k)] ∈ Rn×r , B2 [λ(k)] ∈ Rn×m , C1 [λ(k)] ∈
¯
¯
Rp×n , D11 [λ(k)] ∈ Rp×r , D12 [λ(k)] ∈ Rp×m belong to the polytope
¯ ¯ ¯ ¯ ¯
¯
D = {(A, B1 , B2 , C1 , D11 , D12 )(λ(k)) :
¯ ¯ ¯ ¯ ¯
¯
(A, B1 , B2 , C1 , D11 , D12 )(λ(k))
N

=
i=1

¯ ¯ ¯ ¯ ¯
¯
λi (k)(A, B1 , B2 , C1 , D11 , D12 )i , λ(k) ∈ ΛN }.

The system matrices at any time k are the weighted summation of vertex system matrices
{Vi } weighted by their barycentric coordinates λi (k), i.e.
N

¯
A(k) =

¯
λi (k)A(Vi ),
i=1

i = 1, · · · , N.

¯ ¯ ¯ ¯
¯
The same computation holds for B1 , B2 , C1 , D11 , and D12 as well.
Lemma 6 provides a finite set of LMIs that can be used to design the gain-scheduling
controller. Due to Lemma 6, if there exists matrices Gi,1 ∈ Rq×q , Gi,2 ∈ R(nAU G −q)×q ,
Gi,3 ∈ R(nAU G −q)×(nAU G −q) , Zi,1 ∈ Rm×q and symmetric matrices Pi ∈ RnAU G ×nAU G
such that the LMI conditions in Eq. (2.44) and Eq. (2.45) are satisfied, the gain-scheduling
static feedback control is then obtained as shown in (2.46). The LMI’s in Eqs. (2.44) and
(2.45) are solved by programming them into MATLAB using the LMI lab solver [22], which
is included in the Robust Control toolbox. The matrices Gi,1 , Gi,2 , Gi,3 , Zi,1 , Pi , and the
100

H∞ cost η are programmed in MATLAB as free matrix variables for the LMI optimization
to choose. During the solution process, the the H∞ cost η is minimized until the optimal
solution is obtained.

4.4

Design of LTI Feedback Controller

The open-loop state-space plant used for designing this controller is the same as the one in
Fig. 4.7, but has the low-pass filter L(q) and the integrator I(q) added without performing
any Taylor series expansion. Using the nominal parameters, the closed-loop state-space
representation is

x(k + 1) = ACL (K)x(k) + B1 w(k),

(4.34)

z(k) = CCL (K)x(k) + D11 w(k),
where
ACL (K) = A + B2 KC2

and CCL (K) = C1 + D12 KC2 .

Denoting the transfer function from w to z by Hwz , the inequality Hwz 2 < µ holds if,
∞
and only if, there exists a symmetric matrix P such that


P
ACL (K)P B1
0



T
 P AT (K)
P
0 P CCL (K)
CL



T
T

B1
0
I
D11


0
CCL (K)P D11
µI

101








≻0





(4.35)

is feasible [17]. The optimal feedback controller K for the closed-loop system (4.34) is
formulated as the optimization of the bilinear matrix inequality (BMI)

min µ subject to (4.35)

µ,P,K

(4.36)

where P = P T ∈ Rn×n and K ∈ R1×2 for a PI controller or K ∈ R1×3 for a PID controller.
The BMI (4.36) was solved using the PENBMI software [33] as a MATLAB function in
conjunction with the YALMIP [34] programming interface to find the fixed H∞ PI controller
KP I = [1.8260 0.3205] and the PID controller KP ID = [1.4871 0.5009 0.8942].

4.5

Simulation Results

To validate the effectiveness of the proposed gain-scheduling controller, simulations are shown
using the original plant in Eq. (4.23) for the following cases: engine cold start, load change,
and engine speed change.
The necessity of a gain-scheduled controller is demonstrated by comparing its performance with that of a fixed gain H∞ controller for the nominal parameters. The fixed gain
H∞ control synthesis procedure is reviewed in Appendix 4.4.
In each simulation, the time varying parameters α and β are corrupted by low-pass
filtered white noise of up to 10% their nominal values to represent the slowly drifting offset
that might occur in practical situations. To see transient responses, the initial conditions
of the plant for Case 1 were chosen such that a little extra fuel is injected at first, giving
a slightly higher equivalence ratio than 1. The initial conditions in Cases 2 and 3 were set
such that the plant would start with an equivalence ratio of 1. For the following simulation

102

cases, the extracted profiles of time varying parameters from engine dynamometer tests were
used.

4.5.1

Case 1: Engine Cold Start

We simulate an engine operation when it was started at freezing temperatures (0◦ C) and
heated to its normal operation temperature of approximately 100◦ C within about 2 minutes
at an engine speed of 1500 rpm. The purpose of this simulation is to emulate the cold
start of an internal combustion engine when the engine is operated at high idle speed during
the warm-up. Note that during the engine warm-up process the fuel vapor is much less
at low temperature than that at high temperature. Therefore, this leads to quite different
wall wetting dynamics. The wall wetting dynamics coefficients α and β defined in Eq. (4.3)
were obtained from actual engine test data and they are functions of engine cylinder head
temperature, speed and load. Since speed and load are fixed in this simulation, both α and
β are functions of engine temperature and their values are shown in Fig. 4.9E. Notice that
the transient response at 25 seconds in Fig. 4.9 is due to the change in the wall-wetting
parameters as shown in Fig. 4.9E. When the engine has been warming up for about 90
seconds, the closed-loop system with the fixed H∞ controller becomes unstable, while the
LPV controller remains stable. Thus, in Fig. 4.9A, one can readily see the LPV controller’s
advantage of guaranteed stability as the parameters vary with time.

4.5.2

Case 2: Load Change

In this case we simulate an engine dynamometer experiment for an engine operated at a
temperature of 80◦ C with an engine speed of 1500 rpm. After the engine is stably operated

103

A

1.1
.98

y

1.05

25

1

40

0.95
0.9
B

up

0.1
0

.02

-0.1
C

0

-0.2
0.05
0
ui

0

-0.05

-0.08

-0.1
1.04
1.02
1
0.98
0.96
0.94

uf

D

3

0

E

0.12

1

α
β

α 0.1

0.08

0.5 β

0

10

20

30

40 50 60 70
Time (Seconds)

80

90 100 110

0

Figure 4.9: Case 1 Engine Cold Start: In plots A, B, C, and D the equivalence ratio y(k),
proportional control up (k), integral control ui (k), and the feed-forward control are compared
for the gain-scheduling feedback controller (solid line) and the fixed H∞ controller (dashed
line). The time varying parameters α (dotted line, left axis) and β (dash-dot line, right axis)
are displayed in plot E.

104

at this condition with a 32% throttle, the load is increased by a step throttle position from
32% to 46%. Note that in the dynamometer test, the engine speed was maintained by
dynamometer by increasing the load torque. This is similar to the driving condition that a
step throttle is applied to maintain the vehicle speed when the vehicle is driven up a hill.
Note that the step increment of throttle position produces a slight change in the wall-wetting
parameter β as shown in Fig. 4.10E. But in Fig. 4.10, one can find the benefit of guaranteed
performance of the gain-scheduling controller over the time-varying parameters. Note that
the step throttle occurred at the 30th second results in a momentary spike in the equivalence
ratio due to the step air mass flow; but it is quickly pulled back into its target level by the
gain-scheduled controller, while the fixed H∞ controller takes much longer time with a lot
of oscillations and uses more control effort.

4.5.3

Case 3: Engine Speed Change

In this simulation, an engine was operated in a dynamometer with its cylinder head temperature at 80◦ C. To demonstrate the capability for the gain-scheduling controller to handle fast
engine speed variations, smoothed step commands were applied to the engine dynamometer to manipulate the engine speeds shown in Fig. 4.11F. The resulting engine wall-wetting
dynamic parameters, shown in Fig. 4.11E, were used in the simulation. In Fig. 4.11A, one
can see that both controllers, gain-scheduling and fixed H∞ , regulate the engine equivalence
ratio to its target value of one within 5% error except at 25th second when the engine speed
was increased abruptly from 1000 rpm to 4500 rpm. In this case, the engine equivalence ratio
response converges to its target value smoothly for the gain-scheduling controller but with a
lot of oscillations for the fixed H∞ controller. This situation is similar to a transmission gear
shifting where a rapid engine speed change may occur. Again, one can see the advantage of
105

y

A

1.2
1.1
1
0.9
0.8
0.7

B

up

0.2
0

-0.2
C
0
ui

-0.05
-0.1
-0.15

D

uf

1.8
1.6
1.4
1.2
1
0.8
0.12

E

α

α
β

0.5 β

0.1

0.08
20

1

25

30
35
40
Time (Seconds)

45

50

0

Figure 4.10: Case 2 Load Change: In plots A, B, C, and D the equivalence ratio y(k),
proportional control up (k), integral control ui (k), and the feed-forward control are compared
for the gain-scheduling feedback controller (solid line) and the fixed H∞ controller (dashed
line). The time varying parameters α (dotted line, left axis) and β (dash-dot line, right axis)
are displayed in plot E.

106

guaranteed performance over the time-varying parameters as the gain-scheduled controller
regulates the equivalence ratio back into safe limits quicker and with less overshoot than the
fixed H∞ controller.

4.6

HIL Simulation

The engine model used for the HIL simulation is a control oriented four cylinder dual fuel
mean-value engine model developed at Michigan State University [79], which satisfies the
requirements of validating an engine controller. The term “mean-value” indicates that the
developed engine model neglects the reciprocating behavior of the engine, assuming all processes and effects are spread out over the engine cycle. For the HIL simulation, this model
describes the input-output behavior of the physical engine systems with reasonable simulation accuracy using relatively low computational throughput. Ref. [26] provides a good
overview of engine modeling, and most of dynamic equations used in our modeling work are
from this reference book. This engine model also includes all engine transient dynamics.
Figure 4.12 shows the overall mean-valve engine model architecture, along with main subsystem models, such as air-to-fuel ratio, manifold air pressure (MAP), brake mean effective
pressure (BMEP), engine torque, exhaust temperature, etc.

4.6.1

Mean Value Engine Models

The subsystems that are described mathematically by their averaged dynamic behaviors are
given below.

107

y

A

up

B

1.2
1.1
1
0.9
0.8

1.2
1

0.2
0.1
0
-0.1
-0.2

0.8
25

28

0.2
0

C

-0.2
25

ui

28

0.05
0
25

0.1

28

0.05
0

D
uf

1.4
1.2
1
E

α
β

0.12
α

Ne

F

1
0.5 β

0.1

0

0.08
5000
4000
3000
2000
1000
0

5

10

15

20

25

30

35

40

45

50

Time (Seconds)

Figure 4.11: Case 3 Engine Speed Change: In plots A, B, C, and D the equivalence ratio y(k),
proportional control up (k), integral control ui (k), and the feed-forward control are compared
for the gain-scheduling feedback controller (solid line) and the fixed H∞ controller (dashed
line). The wall-wetting parameters α (dotted line, left axis) and β (dash-dot line, right axis)
are displayed in plot E. The engine speed v is displayed in plot F.

108

Ne
Tcoolant
mi (k)

Tcoolant
mi (k) mc (k)

Ne

Tb

Tb

mc (k)

λe

λe

Pin
ts
ts
Ne

BMEP

Fuel
ts
mA (k)

mA (k)
Texh

ma
˙

ma
˙

Air

Pin

BMEP

mexh

Pout
Throttle and Manifold

Combustion

Texh
mexh
Pout
Exhaust Manifold

Pout

Figure 4.12: Mean Value Engine Model.
4.6.1.1

Valve Model

The valve model is used to compute the mass flow rate of air across the valve. The model
used for the intake throttle and the EGR valve follow the governing equations

Pu
mv = Cd (θ)A(θ) √
Ψ
RTu
and
Ψ

Pd
Pu

=











P
2 d
Pu
1
√
2

P
1− d
Pu

Pd
Pu

,

(4.37)

1
P
< d < 1,
2
Pu
1
P
if d < ,
Pu
2

(4.38)

if

where Cd is the valve discharge coefficient; θ is the valve opening angle; R is the gas constant;
A is the valve open area; Pu and Tu are the pressure and temperature upstream from the
valve; and mv is the mass flow rate across the valve. The governing equations (4.37)-(4.38)
follow the assumption that the spacial effects of the connecting pipes before and after the
valve are neglected and that the thermodynamic characteristics of the connecting pipes are
isentropic expansion.
109

4.6.1.2

Manifold Filling Dynamic Model

The manifold pressure of the intake and the exhaust is computed as a function of time by
the governing equation
t

Pm (t) = Pm (0) +

RTm
(min − mout )dt
0 Vm

(4.39)

where Pm is the manifold pressure; Tm is the manifold temperature; Vm is the manifold
volume; min and mout are the inlet and outlet air mass flow rates; and R is the universal
gas constant. The assumptions made by the governing equation (4.39) are that the receiving
behavior is an adiabatic process; the thermodynamic states are uniform over the manifold
volume; and the manifold temperature is averaged over one engine cycle.

4.6.1.3

Engine Respiration Model

The mass flow rate of the air across of the engine cylinders, me , is computed by the engine
respiration model
me =
˙

Pin Vd Ne
κ
RTin 30

Pout
,v
Pin

(4.40)

where κ is a two degree of freedom look-up table; Pin and Tin are the mean pressure and
temperature at the intake manifold; Pout is the mean pressure at the exhaust manifold; Vd
is the engine displacement.

110

4.6.1.4

Crankshaft Dynamic Model

The crankshaft dynamic model, based on Newton’s theory assuming a rigid crankshaft, is
derived as
60 Tb − Tl
˙
Ne =
2π Je

(4.41)

where Je is the rotational inertia of the engine crankshaft; and Tb and Tl are the engine
brake and load torques. The desired engine speed is maintained by an engine dynamometer
model that generates the engine load torque, Tl , using a feedback PID controller.

4.6.2

Event Based Engine Models

The mathematical models used to simulate the cycle-to-cycle varying variables of engine
subsystems are given below. Each variable in this sections is updated based on the engine
cycle (k) and is independent of time, t.

4.6.2.1

Event Based Wall-Wetting Dynamics

When port-fuel-injection is used to deliver fuel to the engine cylinders, some of the fuel
injected after each injector pulse enters the cylinders. However, the remaining fuel sticks
to the walls of the intake port and on the back of the intake valve. The total fuel entering
the engine cylinders then consists of fuel injected from the current injection pulse and fuel
vapor from the fuel mass stored on the walls from previous injection pulses. Knowledge of
this process is necessary to control the metering of fuel for precise air-to-fuel ratio control.
The event based wall-wetting dynamics used in the engine for HIL simulation are the same
as those in Eq. (4.2).

111

4.6.2.2

Event Based Engine Air-to-fuel Ratio

The gas exchange behavior of the engine introduces dynamics into the air-fuel ratio calculation. Since the engine uses exhaust gas recirculation, a substantial amount of the burned gas
remains in the cylinder. The gas fraction carries the air-to-fuel ratio of the previous engine
cycle into the current cycle. Due to this behavior, the air-fuel ratio is modeled cycle-to-cycle
as
˜
λe (k)Mf resh (k) + λe (k − 1)Mburnt (k)
,
λe (k) =
Mf resh (k) + Mburnt (k)

(4.42)

˜
where λe is the normalized air-to-fuel ratio defined as
m (k) 1
˜
.
λe (k) = A
mc (k) c

(4.43)

λe is the normalized air-to-fuel ratio of the gas mixture inside the engine cylinder after the
intake valve is closed. Mf resh is the mass of the fresh gas mixture charge in the cylinder,
which is the summation of the fresh air mass mA and the fresh fuel mass mc , and Mburnt is
the burned gas remaining in the engine cylinder after the exhaust valve closes, which includes
burned gas due to both internal and external EGR (exhaust gas recirculation). Note that
these dynamics are quite different from the LPV design model described in Fig. 4.2.

4.6.2.3

Event Based Engine Brake Torque

Every combustion event, the engine brake torque calculation is triggered using the following
equation:
Tb (k) =

mc (k)Hl n
ηe (Ne , χ, θst , xEGR )
4π

112

(4.44)

where n is the number of engine cylinders; Hl is the lower heat value of the fuel; ηe is
the engine efficiency, which is a function of engine speed, normalized air-to-fuel ratio, spark
timing θst , and the exhaust-gas-recirculation rate xEGR .

4.6.3

Set-up and Implementation

The mean value engine model was implemented into an Opal-RT HIL system using MATLAB/Simulink. The engine model was updated at a sample period of 1 millisecond. Similarly,
the LPV controller, along with feedforward controller, was implemented as an event-based
discrete controller in Simulink into a Mototron Engine Control Module (ECU) sampled every 5 milliseconds as a function call, see HIL simulation scheme shown in Figure 4.13. The
Opal-RT HIL simulator communicates with the Mototron ECU controller through the high
speed controller-area network (CAN), where signals were sent and received with minimal
delay.
The Opal-RT simulation step size of 1 millisecond was chosen in order to emulate a
real-world continuous time engine. Similarly, the Mototron sample rate of 5 milliseconds
for the controller updating is used in many production engine control systems. The CAN
communication between Opal-RT and Mototron has a time delay between the time when
signals are sent from Mototron and the time when they are received by Opal-RT, and vice
versa. This delay was less than 1 millisecond for our setup since only a few variables were
communicated between the HIL simulator and Mototron controller, see the timing scheme
in Figure 4.14. The event based function call was implemented as follows. At each sample
time, the controller checks if the event based sample condition is met; and if so, the function
call will be made to execute the event based control strategy (see Figure 4.14). Since the
sample period of the event-based LPV controller is a function of engine speed and it can
113

Laptop 1
Interface

Laptop 2
Interface
CAN
Opal-RT
HIL Engine Model
Simulator

Mototron
Target Engine
Control Module

Figure 4.13: HIL engine model and controller setup.
executed with a 5ms sample period, the LPV controller can not be updated exactly at each
fuel injection event. This leads to some sample time error between ideal event based sampling
and actual function call implementation.

4.7

HIL Simulation Results

In Fig. 4.15-4.18, the responses for the gain-scheduling PI and PID controllers are given
by, respectively, the solid gray and black lines. The gray dashed line shows the response
of the fixed gain H∞ PID controller. In each of the HIL simulations, white Gaussian noise
was added to each of the measured signals to represent measurement noise. The standard
deviation of the noise added to each signal was set such that the value of the noise would
not be larger than the following percentages of the measured signals: air flow mA ∼ 3%,
equivalence ratio ys ∼ 2%, coolant temperature ∼ 5%, intake pressure ∼ 5%, and engine
speed Ne ∼ 1%. Even though the cycle-to-cycle combustion variations typically present
in internal combustion engines are correlated to engine speed, load and temperatures, the
114

sim en
ul gine
at m
io o
n d
st el
ep

Opal-RT
HIL Engine Model

computational/discretization delays

CAN

1 ms
Mototron
Target
Controller

CAN

controller
sample period

5 ms
Mototron
Function Call
Event

controller
sample period

Time (ms)

5 ms

Cylinder Event
Trigger A

Cylinder Event
Trigger B

Figure 4.14: HIL timing scheme.
sensor measurement noise, due to cycle-to-cycle combustion variations and sensor noise, was
simplified as a Gaussian white noise due to its simplicity and broad bandwidth. Also, in
each of the HIL simulations, the fuel injected is saturated, as a function of the mass air flow,
to ±25% of the fueling that keeps equivalence ratio at one.

4.7.1

Case 1: Engine Cold Start

We simulate an engine cold start process from freezing temperatures (0◦ C) to its normal
operation temperature of approximately 100◦ C within about 2 minutes at an engine speed
of 1500 rpm. The purpose of this simulation is to emulate the cold start of an internal
combustion engine when the engine is operated at high idle speed during the warm-up. Note
that during the engine warm-up process the fuel vapor is much less at low temperature than
that at high temperature. Therefore, this leads to quite different wall-wetting dynamics.
115

The wall-wetting dynamics coefficients α and β defined in Eq. (4.3) were obtained from
actual engine test data and they are functions of engine coolant temperature, speed and
load. Since speed and load were fixed in this simulation, both α and β were functions of
engine temperature and their values are shown in Fig. 4.16C. The responses of the gain
scheduling PI and PID controllers during this simulation, given in Fig. 4.16, are nearly
identical. However, at between 100 and 110 seconds, the fixed gain H∞ PID controller
becomes saturated causing the measured equivalence ratio to oscillate between 0.8 and 1.2,
while both LPV controllers continue to regulate the equivalence ratio to the desired value
of 1. Also, in Fig. 4.16B, the mass of the fuel injected when using the fixed gain H∞ PID
controller has noticeable perturbations due to the noise added to the measured equivalence
ratio. However, the gain scheduling PI and PID controllers have no noticeable perturbations
which demonstrates that not only do they remain stable over the entire operating range of
the engine, but they are also robust to the added measurement noise.
For comparison purposes, a simulation was carried out using the control model described
in Section 6.2 for the engine cold start problem with the response displayed in Fig. 4.15. In
this simulation, no measurement noise is added to the measured signals. Also, a saturation
level is not imposed on the feedback control input.

4.7.2

Case 2: Load Change

In this case we simulate an engine dynamometer experiment for an engine operated at a
temperature of 80◦ C with an engine speed of 1500 rpm. After the engine is stably operated
at this condition with a 32% throttle, the load is increased by a step throttle position from
32% to 46%. Note that in the dynamometer test, the engine speed was maintained by
dynamometer through torque regulation. This is similar to the driving condition that a step
116

A

F H∞ Controller
GS PI Controller
GS PID Controller

Equivalence Ratio

1.01
1
1

0.99

0.99
0.98

0.98

B

20

22

24

26

28

20

22

24

26

28

Injected Fuel

0.02

0.0195

0.019

0.0185
C

0.12

1

α
β

α 0.1

0.08

0.5 β

10 20 30 40 50 60 70 80 90 100 110 120
Time (Seconds)

0

Figure 4.15: Case 1: Engine Cold Start using simple model.

117

Equivalence Ratio

A

1.2
1.15
1.1
1.05
1
0.95
0.9
0.85
0.8

19 20 21 22 23 24

F H∞ Controller
GS PI Controller
GS PID Controller

B 0.028

0.0218

Injected Fuel

0.026

0.0216

0.024

19 20 21 22 23 24

0.022
0.02
0.018
0.016

C

0.12

1

α
β

α 0.1

0.08

0.5 β

0

10

20

30

40 50 60 70
Time (Seconds)

80

90 100 110

Figure 4.16: Case 1: Engine Cold Start using HIL.

118

0

throttle is applied to maintain the vehicle speed when the vehicle is driven up a hill. Note
that the step increment of throttle position produces a slight change in the wall-wetting
parameter β as shown in Fig. 4.17C. The responses of each controller is given in Fig. 4.17A.
Notice that the throttle step occurring at the 30th second results in a drop in the equivalence
ratio due to the step air mass flow. In the detail of Fig. 4.17A, we see that with the gainscheduling PID controller the equivalence ratio only drops to approximately 0.85, while the
gain-scheduling PI and fixed gain H∞ controller both drop to nearly 0.8. Also, notice that
the equivalence ratio with fixed gain H∞ PID controller overshot to over 1.1 with over fueling
as seen in the detail of Fig. 4.17B.

4.7.3

Case 3: Engine Speed Change

In this simulation, an engine was operated on a dynamometer with its coolant temperature
at 80◦ C. To demonstrate the capability for the gain scheduling controller to handle engine
speed variations, a smoothed step command from 1500 rpm to 2500 rpm was applied to the
engine dynamometer to manipulate the engine speed as shown in Fig. 4.18D. The resulting
engine wall-wetting dynamics parameters, shown in Fig. 4.18C, were used in the simulation.
Notice in Fig. 4.18A that the gain-scheduling PID controller regulates the equivalence ratio
of the engine to the target value of 1 within 5% error, while the measured equivalence ratio
of the engine with the gain-scheduling PI controller and the fixed gain H∞ PID controller go
above 1.05. Also, the equivalence ratio with the fixed gain H∞ PID controller drops to below
0.95, while both gain-scheduling controllers only lower the equivalence ratio to about 0.96.
The equivalence ratio with the fixed gain H∞ PID controller also has many oscillations and
uses more control effort as shown in the detail of Fig. 4.18B, which hurts engine transient
fuel economy.
119

A
Equivalence Ratio

1.15
1.1

F H∞ Controller
GS PI Controller
GS PID Controller

1.05
1
0.95
1
0.9
0.8

0.9
0.85
0.8

30

B

31

30

0.75

31

0.05

Injected Fuel

0.045
0.04
0.035
0.03
0.025
0.02
0.015
0.01
C

0.12

α
β

α 0.1

1

0.5 β

0.08

20

25

30
40
35
Time (Seconds)

45

50

Figure 4.17: Case 2: Engine Load Change using HIL.

120

0

Equivalence Ratio

A

B

1.05

1.08
1.06
1.04
1.02
1
0.98
0.96
0.94

1
10

11
F H∞ Controller
GS PI Controller
GS PID Controller

Injected Fuel

0.04
0.02

0.03
0.02

10

0.01
0
0.12

C

11

α
β

α 0.1

0.5 β

0

0.08

Engine Speed

D

1

2600
2400
2200
2000
1800
1600
1400
0

5

Engine Speed
10
20
25
30
15
Time (Seconds)

Figure 4.18: Case 3: Engine Speed Change using HIL.

121

4.7.4

Case 4: Combined Load and Engine Speed Change

In this simulation, an engine was operated on a dynamometer with its coolant temperature
at 80◦ C. To demonstrate the capability for the gain scheduling controller to handle load
changes combined with engine speed variations, the load is increased by a step throttle
position from 32% to 46% and then combined with an engine speed variation generated by a
smoothed step command from 1500 rpm to 2000 rpm as shown in Fig. 4.19D. The resulting
engine wall-wetting dynamics parameters are shown in Fig. 4.19C. Notice in Fig. 4.19A both
of the gain-scheduling controllers drop the measured equivalence ratio to approximately 0.85,
while the fixed gain H∞ PID controller drops the measured equivalence ratio below 0.85.
Also, the fixed gain H∞ PID controller overshoots to nearly 1.15 with over fueling as seen
in the detail of Fig. 4.17B.

4.8

Conclusion

In this chapter, a systematic process for developing gain-scheduling PI and PID controllers
for discrete-time LPV systems is presented. First, a control oriented LPV model is developed by using the dynamics of a port-fuel-injection process. Then the LPV model obtained
is investigated and found to contain parameter variation that is not affine. Due to limitations in current LPV control schemes for discrete-time systems discussed in Section 4.3.3, a
first-order Taylor series approximation is performed on the LPV system H(Θ) in (4.26) to
ˆ
obtain an approximated LPV system H(Θ) in (4.27) with only affine parameter variation.
The measurement for control is generated by augmenting the approximated LPV system
with a low-pass filter and an integrator. The augmented, approximated LPV system is then
converted to a polytopic LPV system so that the synthesis method given by [10] can be uti122

Equivalence Ratio

A

Injected Fuel

B

F H∞ Controller
GS PI Controller
GS PID Controller

1.1
1.05
1
0.95
0.9
0.85
0.8
0.04

5

6

5

6

0.035
0.03
0.025
0.02
0.12

C

α
β

α 0.1

Engine Speed

D

0.08
2100
2000
1900
1800
1700
1600
1500
1400

1

0.5 β

0

Engine Speed
0

5

10
Time (Seconds)

15

Figure 4.19: Case 4: Combined Load and Engine Speed Change using HIL.

123

lized. To validate the gain-scheduling controller found with the finally obtained LPV system
ˆ
H(Θ), simulations are performed using the original LPV system H(Θ). From the simulaˆ
tion results, it is clear that although the approximated LPV system H(Θ) is used to design
the gain-scheduling controller it still performs very well when applied to the original LPV
system H(Θ). Furthermore, not only do the HIL simulation results reaffirm the success of
the simulation results, they also demonstrate the feasibility of implementing of the proposed
LPV scheme on a hardware controller that could be used as an engine control module.
To the authors’ knowledge, this is the first work to develop an air-to-fuel ratio gainscheduling controller for the port-fuel-injection process using the wall-wetting parameters α
and β, and the engine speed Ne as scheduling variables. Future investigation that would
further this research effort includes developing a systematic framework for the off-line estimation of the wall-wetting parameters α and β over all operating conditions for a given
intake manifold.

124

Chapter 5
Mixed H2/H∞ Observer-Based LPV
Control of a Hydraulic Engine Cam
Phasing Actuator

5.1

Introduction

The intake and exhaust valve timing of an internal combustion (IC) engine greatly influence
the fuel economy, emissions, and performance of an IC engine. Conventional valvetrain
systems can only optimize the intake and exhaust valve timing for one given operational
condition. That is, the optimized valve timing can either improve fuel economy and reduce
emissions at low engine speeds or maximize engine power and torque outputs at high engine
speeds. However, with the development of continuously variable valve timing (VVT) systems
[37], the intake and exhaust valve timing can be modified as a function of engine speed and
load to obtain both improved fuel economy and reduced emissions at low engine speeds and
increased power and torque at high engine speeds.
To adjust the intake and exhaust timing, the most common cam phasing system is the
hydraulic van type cam phaser [20]. The control of hydraulic cam phasing systems has been
discussed in [24] and [52]. In [24], a significant nonlinearity in the hydraulic cam phasing

125

system is noted and a nonlinear controller is designed to compensate for it. In [52], an H2
controller is designed using the output covariance constraint (OCC) control design approach
[87]. In this chapter, a gain-scheduling controller is developed using linear parameter varying
(LPV) control design.
In recent years, the use of LPV modeling and control in automotive applications has
received a great deal of attention. LPV modeling and control techniques have been applied
to both diesel engines [65, 55] and gasoline spark-ignition engines [88, 82, 66, 69]. In [65], LPV
control techniques are applied to the air path of turbocharged diesel engines to control the
transient exhaust gas fraction pumped into the cylinders to reduce nitrous oxide emissions.
In [55], an LPV identification technique is applied to a nonlinear turbocharged diesel engine
to obtain an LPV model suitable for control synthesis. In [88], a continuous-time LPV
model is developed considering only engine speed as a time-varying parameter. In [82], a
large variable time delay is present in the air-fuel ratio control loop for a lean burn spark
ignition engine. LPV control methods are used to compensate for the variable time delay.
In [66] and [69], event-based gain-scheduling proportional-integral (PI) and proportionalintegral-derivative (PID) controllers are developed using the wall-wetting parameters and
engine speed as time-varying parameters. In this chapter, the techniques used in [66] and
[69] to obtain the static PI and PID gain-scheduling controllers are augmented to develop
an observer-based dynamic LPV controller using the dynamics of the plant.
To obtain the model of the VVT system, closed-loop system identification was used in
[48] and [52]. A main reason for selecting closed-loop system identification in [48] and [52]
was due to high open-loop gains that makes it difficult to maintain the cam phaser operated
at a fixed location for system identification. During the system identification process, it
was found that the system gain of the VVT actuator is a function of engine speed, load, oil
126

900 rpm, 310 kPa
LTI Systems

Organize Family of LTI LPV System
systems into LPV System

1800 rpm, 414 kPa
LPV Controller Design
System ID Experiments
Revisit Weighting Functions

Select Weighting
Functions

Append Control
Structure and
Weighting Functions

Perform Control
Synthesis

Convert LPV system
to Polytopic System

Finalized
LPV Controller
Not Satisfied
Satisfied

Experimental Validation
on Test Bench

Performance and Stability Requirements
Figure 5.1: Flowchart of the design and validation process of an LPV controller.
pressure, and temperature. Therefore, it seems only natural to exploit the knowledge of how
the system gain of the VVT actuator varies with the time varying parameters. To do this,
the VVT system can be described as a family of linear models to approximate the system
dynamics for a given engine speed, load, oil pressure, and temperature. Thus formulating
an LPV model for the VVT system.
The purpose of this chapter is to develop a dynamic gain-scheduling controller with
guaranteed stability and performance over all time-varying parameters. To do this, the
process depicted in Fig. 5.1 was followed. First, a family of linear-time-invariant (LTI)
models was obtained. Using engine speed and the oil pressure as system parameters, a
family of linear models of the VVT system were obtained by performing multiple system
identifications while maintaining engine speed and oil pressure at specified levels. With the
family of linear models, the LPV model of the VVT system was formulated. To design

127

the dynamic gain-scheduling controller, a standard control structure of observer-based state
feedback with integral control was employed. This control structure, along with H2 and
H∞ performance weighting functions, were then appended onto the LPV model of the VVT
system to obtain the LPV system of the generalized plant. Then the LPV system of the
generalized plant was converted to a polytopic system, which is an LPV system with a
polytopic dependency on a scheduling parameter that takes values in the unit-simplex, so
that the mixed H2 /H∞ discrete-time LPV control synthesis method given by [12] could be
applied to obtain the gain-scheduled state feedback and observer gains. Once a potential
controller was obtained, its performance was experimentally validated on the test bench
used to obtain the family of LTI systems. If the performance and stability requirements
of the VVT system are not satisfied when testing the LPV controller, the selected H2 and
H∞ performance weighting functions are modified and the control synthesis procedure is
performed again. This loop is performed until stability and satisfactory performance are
obtained on the test bench.
As stated previously, a multi-objective, mixed H2 /H∞ control design is performed in this
chapter. The goal of using both H2 and H∞ performance criteria is to design a controller
which can meet multiple performance objectives. In this chapter, a loose H∞ performance
bound is used to guarantee stability of the closed-loop system under parameter variations.
Meanwhile, a tight H2 performance bound is used to make the LPV controller robust to
input disturbances. The selection of H2 and H∞ performance weighting functions is an
important design problem. The selection of H∞ performance weighting functions can be
done as described in [83] and [58]. However, the selection of H2 performance weighting
functions is not covered in such detail. In [87], a systematic way is provided for iteratively
tuning the output H2 weighting functions for robust control of LTI systems. Unfortunately,
128

no such iterative procedure exist yet for LPV systems.
The chapter is organized as follows. The family of linear models obtained from the series
of bench identification tests are introduced in Section 5.2 and the LPV system is formulated.
In Section 6.3, the LPV gain-scheduling controller design method is provided. The bench
test set-up is discussed in Section 5.4.1. In Section 5.4.2, the obtained LPV gain-scheduling
controller is operated on the test bench and compared to the baseline PI and OCC controllers
used in [48]. Concluding remarks are given in the final section.

5.2

LPV System Modeling

To obtain a family of linear models, the closed-loop system identification outlined in [48] was
performed at a series of fixed engine speeds Ne and oil pressures p. The open-loop transfer
functions of the identified family of linear VVT systems sampled at 5 ms are given by
Ψ(Ne , p) (0.0859q − 0.0609)
,
q 2 − 1.9547q + 0.9553
Ψ(Ne , p) (0.0615q − 0.0364)
G(Ne , p = 414 kPa (60 psi)) =
q 2 − 1.9547q + 0.9553

G(Ne , p = 310 kPa (45 psi)) =

(5.1)

where Ψ(Ne , p) is the gain at a specific engine speed Ne and oil pressure p as given in
Table 5.1 and q is the forward shift operator that satisfies qu(k) = u(k + 1).
By inspection of the identified transfer functions in (5.1), the LPV model for the VVT
system is given by
αk q + βk
G(αk , βk ) = 2
q − 1.9547q + 0.9553

(5.2)

where αk and βk are used as the time-varying parameters. For notational simplicity, αk and
βk will be used to denote the parameters at time k, such that αk = α(k) and βk = β(k).

129

Table 5.1: Identified Gain Ψ(Ne , p)
Pressure, p

Engine Speed, Ne (rpm)

Gain Ψ(Ne , p)

310 kPa (45 psi)

900

0.70

1500

0.72

1800

0.68

900

0.95

1500

0.98

1800

0.93

414 kPa (60 psi)

Table 5.2: Time-varying parameters (scheduling parameters).
α(Ne (t), p(t)) ∈ [0.0571, 0.0618]

β(Ne (t), p(t)) ∈ [−0.0438, −0.0339]

The values of αk and βk are found for a specific value of engine speed Ne and oil pressure
p by multiplying the appropriate Ψ value found in Table 5.1 with the appropriate transfer
function in 5.1. The range of values that αk and βk can take are given in Table 6.1.
Using the transfer function in (5.2), a state-space representation of the VVT system is
found to be








 0 −0.9553 
 βk 
xG (k + 1) = 
 xG (k) + 
 uG (k),
1 1.9547
αk
AG

y(k) =

0 1

BG

(5.3)

xG (k).

CG

For convenience, the compact notation Θ = [αk , βk ] will be used to denote the scheduling
variables for the remainder of the chapter.

130

z4

z5

z2

−Wu∞

Wu2

I(q)

w

Wd

uG

z1

Wz2

r
d

y

G(Θ)

Wz∞

z3

−

P (Θ)
y
ˆ
ˆ
P (Θ)
uL

H(Θ)

xP
ˆ

L(Θ)

eL

uP
KS (Θ)

uI

Figure 5.2: Proposed control architecture for the VVT system

5.3
5.3.1

LPV Gain Scheduling Controller Design
Control Strategy

The objective of the control system is to regulate the cam phase y to a reference phase r
using feedback control against the disturbance signal d and the time-varying parameters αk
and βk . In particular, we want to guarantee the stability of the closed-loop system and also
minimize the effect of the disturbances for any conceivable engine speed and oil pressure
variations. The proposed control architecture is illustrated in Fig. 6.2. This scheme has four
ˆ
components, that is a state observer P (Θ), observer gains L(Θ), a state feedback controller
KS (Θ), and an integrator I(q).
The multi-input, single-output LPV plant P (Θ), depicted inside of the dotted box in
Fig. 6.2, is obtained by augmenting the VVT system G(Θ) with the forward Euler method,

131

discrete-time integrator I(q) = ts /(q − 1), where ts is the sample period of the discrete-time
system in seconds. The integrator I(q) introduces integral action into the system to ensure
that the steady-state error between the measured cam phase y and the reference phase r can
be eliminated. By allowing the input to the VVT plant G(Θ) to be equal to

uG (k) = uP (k) +

ts
u (k),
q−1 I

as displayed in the dotted box of Fig. 6.2, one possible state-space representation of P (Θ) is
found to be


√

ts βk
 0 −0.9553


√
xP (k + 1) =  1 1.9547
ts αk


0
0
1
AP (Θ)

y(k) =

0 1 0





0
 βk





 xP (k) +  αk 0




√
ts
0
BP (Θ)






 uP (k)



,

 uI (k)
uS (k)

(5.4)

xP (k).

CP

In (5.4), it is clear that the state matrix AP (Θ) and the input matrix BP (Θ) are both
affected by the time-varying parameters αk and βk .
ˆ
The state observer P (Θ) is used to obtain the estimated states xP of the plant. The
ˆ
ˆ
observer P (Θ) has the standard state-space representation

xP (k + 1) = AP (Θ)ˆP (k) + BP (Θ)uS (k) + L(Θ)eL (k),
ˆ
x
y (k) = CP xP (k),
ˆ
ˆ
where the error input to the plant observer is given by eL (k) = r(k) − (y(k) + d(k)) +
132

(ˆ(k) + d(k)), which simplifies to eL (k) = r(k) − y(k) + y (k). Since we are solving the S/KS
y
ˆ
mixed-sensitivity H∞ optimization using the regulation form, during control synthesis we
let the set point r equal zero as shown in [58], thus further simplifying the observer input
error to eL (k) = −y(k) + y (k). This satisfies the condition in [12] that the measurement for
ˆ
control is not corrupted by the exogenous input w(k). Notice in Fig. 6.2 that the output
disturbance d(k) is connected to the estimated plant output y (k) by dash-dot lines. This
ˆ
is to signify that the exogenous input d(k) is only available to the observer during control
synthesis. However, during implementation since the output disturbance d(k) cannot be
measured it is not available to the observer.
To use mixed H2 /H∞ norms as the performance criteria for shaping the frequency response of the closed-loop system, weighting matrices (which can be considered control design
parameters) are introduced in Fig. 6.2. Oftentimes, the weighting matrices are chosen as
frequency dependent functions; however, for this problem static weighting matrices sufficed.
The weighting matrix Wd was selected to model the signal d using the signal based approach
discussed in [58]. The H∞ performance weighting functions Wz∞ and Wu∞ were selected
to limit the maximum magnitude of the sensitivity function |S(jω)| and the controller multiplied by the sensitivity function |KS(jω)| as discussed in [83]. In this study, the H∞
performance weighting functions were selected primarily for LPV stability. However, the H2
performance weighting functions were selected for LPV performance. The weighting matrices
Wz2 and Wu2 were selected using an iterative trial-and-error process. In the iterative process,
Wz2 and Wu2 started out with values of unity. The control synthesis procedure outlined in
Algorithm 1 was then carried out and the sensitivity function was computed and examined.
The values used in the weighting function Wu2 were then increased and the control synthesis was carried out again and the sensitivity function was examined again. This procedure
133

was executed until desirable characteristics were displayed in the frequency response of the
controller, the sensitivity function, and the controller multiplied by the sensitivity function.
The resulting weighting matrices are as follows:

Wd = 0.01, Wr = 1, Wu∞ = 10,


 15 0 
Wu2 = 
 , Wz∞ = 1, and Wz2 = 1.
0 15
These weighting matrices where tuned to obtain the frequency responses plotted with the
bold lines in Fig. 5.3. For comparison, a full-order dynamic output covariance constraint
(OCC) controller (dashed lines) [52] was used. This controller is known to work well on the
VVT cam phaser test bench at the fixed operational condition of 1500 rpm and 414 kPa
(60 psi) oil pressure, so it was deemed an appropriate starting point.
In Fig. 5.3, the frequency responses of the LPV controller and the OCC controller are
displayed at the corner points of the parameter space polytope (i.e. [α, β], [α, β], [α, β],
[α, β], where α = αmin and α = αmax ). In Fig. 5.3A, the frequency response of each
controller is displayed. At low frequencies, each controller has high gain due to the integral
action built into each controller. In Fig. 5.3B, the sensitivity function of each controller is
displayed. In a typical feedback system, the sensitivity function is linked to the tracking
error performance [83]. At low frequencies, each controller’s sensitivity function is small,
which minimizes tracking error and maximizes disturbance rejection. Fig. 5.3C displays the
frequency response of the controller multiplied by the sensitivity function for each controller.
This plot shows that over the frequency range of 1Hz-20Hz the mixed H2 /H∞ dynamic
LPV controller has lower control effort than the full-order dynamic OCC controller. Since

134

A
20

B 10
LPV Controller
H2 Controller

-10

|S(jω)| (dB)

10

|K(jω)| (dB)

0

-20

0

-30

-10

-40
-50

-20

-60
D
0
-6
-10

C 0
-5

|T (jω)| (dB)

|KS(jω)| (dB)

-10

-20

-20
-25

-6
-10

-40

-30
-35
10−3

0

-30

-15

-15

-50
10−2

10−1
100
Freq. (Hz)

101

102

10−3

100
10−2

101
10−1
100
Freq. (Hz)

101

102

Figure 5.3: Frequency response comparison of the mixed H2 /H∞ dynamic LPV controller
with an OCC (H2 ) controller [52] at the corner points of the parameter space polytope.
this is the frequency range over which the output disturbance d(k) is generally active, it
means that the mixed H2 /H∞ dynamic LPV controller should be robust to the disturbance
d(k). The frequency response of the closed-loop transfer functions with the mixed H2 /H∞
dynamic LPV controller and the OCC controller are displayed in Fig. 5.3D. The benefit of
the mixed H2 /H∞ dynamic LPV controller can be seen in the close-up view in Fig. 5.3D. At
−6 dB, the closed-loop bandwidth with the OCC controller varies between approximately 2
to 4.8 Hz. However, the closed-loop bandwidth with the LPV controller only varies between

135

approximately 1.8 to 2.9 Hz, which is a reduction in span of about 60%.
As displayed in Fig. 6.2, the state feedback gains KS (Θ) and the observer gains L(Θ)
are placed outside of the solid, bold box. This designates that the control synthesis in
Algorithm 1 is performed on only the items inside of the box. By isolating the static gains
KS (Θ) and L(Θ), the design of the observer-based dynamic controller is transformed into
the design of a single static controller K(Θ) by using the following structure:










0   xP (k) 
ˆ
 uS (k)   KS (Θ)



=
0
L(Θ)
eL (k)
uL (k)
u(k)

K(Θ)

(5.5)

e(k)

where xp ∈ R3 , eL ∈ R, uS ∈ R2 , and uL ∈ R3 .
ˆ

5.3.2

Generalized Plant

As shown in Fig. 6.2, the state feedback controller KS (Θ) and observer gains L(Θ) are
designed for the generalized LPV plant H(Θ). The generalized LPV plant H(Θ) is composed
ˆ
by the multi-input, single-output LPV plant P (Θ) and its corresponding state observer P (Θ),
along with the static weighting matrices Wd , Wu∞ , Wu2 , Wz∞ and Wz2 . The state-space
realization of the generalized plant H(Θ) is found by combining the state-space realizations

136

ˆ
of P (Θ) and P (Θ) and performing the connections in Fig. 6.2 to obtain
















0
 xP (k + 1)   AP (Θ)
  xP (k)   BP (Θ) 0   uS (k) 
=



+

xP (k + 1)
ˆ
0
AP (Θ)
BP (Θ) I
xP (k)
ˆ
uL (k)
ˆ
A(Θ)

x(k+1)

ˆ
B(Θ)

x(k)

u(k)

(5.6)

z(k) = Cz x(k) + Dw w(k) + Du u(k)
e(k) = Ce x(k)
where x(k) ∈ Rn is the state at time k, w(k) ∈ Rr is the unweighted exogenous input,
u(k) ∈ Rm is the control input, z(k) ∈ Rp is the performance output, and e(k) ∈ Rq is the
measurement for control. The state matrix AP (Θ) and the input matrix BP (Θ) are both
given in (5.4) and the other state-space matrices are given by









Cz = 








0 1
0 0
0 1
0 0
0 0



0 0 0 


√
−10 ts 0 0 0 



0
0 0 0 ,



0
0 0 0 


0
0 0 0


0
 0


 −10 0



Du =  0
0



 15 0


0 15

0

0 0 0 


0 0 0 



0 0 0 ,



0 0 0 


0 0 0










Dw = 








 0 0


 0 0

Ce = 

 0 0


0 −1

137



1 


0 



1 ,



0 


0



0 1 0 0 


0 0 1 0 

.

0 0 0 1 


0 0 1 0

5.3.3

A gain-scheduling control synthesis problem

Now that the state-space representation of the generalized plant H(Θ) has been obtained,
the mixed H2 /H∞ gain-scheduling controller K(Θ) must be synthesized. The H∞ -norm
from w(k) to Z∞ = [z1 , z2 ]T of the LPV system H(Θ) in (5.6) with the gain-scheduling
controller is defined as
H(Θ) ∞ =

Z∞ ℓ2
.
Θ∈Θ, w ℓ =0 w ℓ2
2
sup

(5.7)

The H2 -norm from w(k) to Z2 = [z3 (k), z4 (k), z5 (k)]T of the LPV system H(Θ) with the
gain-scheduling controller is defined as

H(Θ)

2
2

1
= lim sup E
T
T →∞

T
k=0

T
Z2 Z2 ,

(5.8)

where E denotes the expectation operator and the positive integer T denotes the time horizon.
Now we formally state the gain-scheduling control design problem.
Problem : The goal is to design a static gain-scheduling control u(k) = K(Θ)e(k) that
stabilizes the closed-loop system and minimizes the worst-case H∞ and H2 norms of the
closed-loop LPV system in (5.7) and (5.8) for any trajectories of Θ(k) ∈ Θ.
The gain-scheduling method provided by [12] was derived for discrete-time polytopic
time-varying systems. Therefore, in the next section, the state-space representation of H(Θ)
in (5.6) will be transformed into a polytopic time-varying system so that the controller K(Θ)
can be synthesized.

138

5.3.4

Polytopic linear time-varying system

The state-space representation of the generalized plant H(Θ) in (5.6) can be converted into
ˆ
a discrete-time polytopic time-varying system by solving the state matrix A(Θ) and the
ˆ
input matrix B(Θ) at the vertices of the parameter space polytope, e.g. the state matrix
ˆ
at vertice V2 is given by A2 = A(Θ = [α, β]). Any Θ inside of the convex parameter set is
represented by a convex combination of the vertex systems as weighted by the vector λ(k) of
barycentric coordinates. Barycentric coordinates are used to specify the location of a point
as the center of mass, or barycenter, of masses placed at the vertices of a simplex. A formula
for computing the barycentric coordinates for any convex polytope is provided by [64]. The
discrete-time polytopic time-varying system is given by

x(k + 1) = A(λ(k))x(k) + Bu (λ(k))u(k)
z(k) = Cz x(k) + Dw w(k) + Du u(k)

(5.9)

e(k) = Ce x(k)
where the state matrix A(λ(k)) ∈ Rn×n and the input matrix B(λ(k)) ∈ Rn×m belong to
the polytope
4

D=

(A, Bu )(λ(k)) : (A, Bu )(λ(k)) =
i=1

λi (k)(A, Bu )i , λ(k) ∈ Λ ,

(5.10)

and the other state-space matrices are the same as in (5.6). The state matrix A(λ(k)) and
the input matrix Bu (λ(k)) are the weighted summation of the vertex matrices as weighted

139

by the vector λ(k) of barycentric coordinates, i.e.
4

4

A(λ(k)) =

λi (k)Ai

and Bu (λ(k)) =

i=1

λi (k)Bi
i=1

where Ai and Bi are the vertices of the polytope and λ(k) ∈ R4 is the barycentric coordinate
vector which exists in the unit simplex

Λ=

ζ∈

R4

4

:
i=1

ζi = 1, ζi ≥ 0, i = 1, · · · , 4 .

(5.11)

For all k ∈ Z≥0 , the rate of variation of the barycentric coordinates ∆λi (k) = λi (k + 1) −
λi (k), is limited such that −b ≤ ∆λi (k) ≤ b, with b ∈ [0, 1], which should be selected with
the application in mind. If a worst-case set of parameter variation is known, then this bound
can be calculated.
A finite set of LMIs in [12] can be used to design the H2 /H∞ gain-scheduling controller
K(Θ) in (5.5). Due to Theorems 8 and 9 of [12], if there exists for i = 1, . . . , 4, matrices
Gi,Ks ∈ R3×3 , Gi,L ∈ R(q−3)×(q−3) , Zi,Ks ∈ R(m−3)×3 , and Zi,L ∈ R3×1 assembled as




0 
 Gi,Ks
Gi,1 = 

0
Gi,L





0 
 Zi,Ks
and Zi,1 = 
,
0
Zi,L

(5.12)

along with the other matrix variables defined in Theorems 8 and 9 of [12] satisfying the H∞
LMIs and the H2 LMIs in [12], then the H2 /H∞ controller K(λ(k)) is given by
ˆ
ˆ
K(λ(k)) = Z(λ(k))G(λ(k))−1 ,

140

(5.13)

with
4

ˆ
Z(λ(k)) =

4

λi (k)Zi,1

ˆ
and G(λ(k)) =

i=1

λi (k)Gi,1 .
i=1

This control is proved to stabilize affine parameter-dependent systems such as (5.9) with a
guaranteed H2 and H∞ performance for all λ ∈ Λ and |∆λ| ≤ b. In this work, to ensure that
all possible parameter variations would be covered, we selected b = 0.4. The LMI conditions
of Theorems 8 and 9 of [12] are solved by programming them into MATLAB using the LMI
parser YALMIP [34] and solved using SeDuMi [59]. During the solution process, the goal is
to calculate the gain-scheduled feedback controller K(λ(k)) that minimizes the bound ν on
the H2 performance from w(k) to [z3 , z4 , z5 ]T under a prescribed bound η on the H∞ norm
from w(k) to [z1 , z2 ]T . The procedure for performing the mixed H2 /H∞ control synthesis
is outlined in Algorithm 1.
Note that the minimum feasible H∞ bound ηL can be solved for by using an iterative
algorithm [58], such as the bisection algorithm.
The resulting LPV controller solved at an engine speed and oil pressure of Ne = 1500rpm
and p = 414kPa (60 psi) (for comparison with the H2 output covariance controller) is found
to be
0.109255247q 3 − 0.302866405q 2 + 0.278279285q − 0.0846677044
.
KLP V (q) = 4
q − 3.132121334q 3 + 3.625898107q 2 − 1.853079890q + 0.359303117

(5.14)

As stated previously, the robust H2 controller designed in [52] using the OCC control
design algorithm presented in [87] is used for comparison with the LPV controller. The
robust H2 OCC controller designed in [52] is given by
0.3158302q 3 − 0.9301618q 2 + 0.9129406q − 0.2986088
.
KOCC (q) = 4
q − 3.4051293q 3 + 4.3533113q 2 − 2.4909563q + 0.5427743
141

(5.15)

Algorithm 1 Mixed H2 /H∞ Gain-Scheduling Synthesis

Require: Polytopic LPV system in (5.9), rate of variation bound b ∈ [0, 1], H2 and H∞
input and output channels of (5.9), and a range of prescribed H∞ bounds η ∈ [ηL , ηU ],
where it is assumed that ηL is the minimum feasible H∞ bound.
Ensure: The gain-scheduling controller matrices Gi,1 and Zi,1 needed to compute K(λ(k))
in (5.13).
1: Determine selection matrices Lj and Mj for each performance specification j as in Section 5.3 of [12].
2: Compute Hj using selection matrices Lj and Mj for each performance specification j,
for j = 1, 2.
3: Compute the vectors f j and hj using rate of variation bound b as shown in Appendix 11.3
of [12].
4: Using equation (28) of [12], convert the polytopic LPV system in (5.9) to the form used
in the LMIs of Theorems 8 and 9 of [12].
5: for η = ηL :ηU do
6: Initialize the matrix variables introduced in Theorems 8 and 9 of [12] as free matrix
variables into MATLAB using the YALMIP interface [34].
7: Using Gi,Ks , Gi,L , Zi,Ks , and Zi,L , generate Gi,1 and Zi,1 as shown in (5.12).
8: Using the YALMIP interface [34], program the H∞ LMIs in Theorem 8 of [12] using
prescribed bound η and the H2 LMIs in Theorem 9 of [12] into MATLAB.
9: Using an LMI solver, like SeDuMi [59], solve the system of LMIs with the objective
of minimizing 4 Tr{Wi }, where Wi is a positive-definite H2 free matrix variable
i=1
introduced in Theorem 9 of [12], thus minimizing the H2 norm.
10: end for
11: Select the solution that minimizes the H2 norm the most, yet still has an acceptable
bound η on the H∞ norm.

142

Cam
Position
Sensor

VVT Solenoid

Figure 5.4: VVT phase actuator test bench

5.4
5.4.1

VVT System Test Bench
Bench Test Set-up

The closed-loop system identification outlined in Ref. [52] and the control design testing were
conducted on the VVT test bench displayed in Fig. 5.4. A Ford 5.4L V8 engine head was
modified and mounted on the test bench. The cylinder head has a single cam shaft with a
VVT actuator for one exhaust and two intake valves. These valves introduce a cyclic torque
disturbance to the cam shaft. The cam shaft is driven by an electrical motor (simulating the
crankshaft) through a timing belt, see Fig. 5.5.
An encoder is installed on the motor shaft, which generates the crank angle signal with
one degree resolution, along with a so-called gate signal (one pulse per revolution). A
plate with five magnets adhered is mounted at the other side of the extended cam shaft. As
displayed in Fig. 5.5, one magnet is placed on the edge of the plate and is used to synchronize
143

Hydraulic Oil

Actuator
VVT Pulley
Cam Shaft
Torque Disturbance
Timing Belt
Crank Pulley
Crank Shaft

Magnets
Sensors

Figure 5.5: VVT phase actuator test bench diagram
the top dead center position of the combustion phase. The other four magnets on the face
of the plate are used to determine the cam phase four times per engine cycle. The two
squares in Fig. 5.5 represent hall-effect cam position sensors. As the cam shaft rotates, the
magnets on the plate face pass the hall-effect cam position sensor used to to determine cam
phase and the magnet on the edge of the plate passes the hall-effect cam position sensor
used to determine top dead center position. Within an engine cycle, the cam position sensor
generates four cam position pulses, which are sampled by an Opal-RT real-time controller.
By comparing these pulse locations with respect to the encoder gate signal, the Opal-RT
controller calculates the cam phase with one crank degree resolution.
The cam phase actuator system consists of a solenoid driver circuit, a solenoid actuator,
and a hydraulic cam actuator. The solenoid actuator is controlled by a pulse-width modulation (PWM) signal, whose duty cycle is linearly proportional to the DC voltage command.
An electrical oil pump was used to supply pressurized engine oil to be used for lubrication
and as hydraulic actuating fluid for the cam phase actuator. The cam actuator command
144

voltage signal is generated by the Opal-RT prototype controller and sent to the solenoid
driver. The PWM duty cycle is linearly proportional to input voltage with a maximum duty
cycle 99% corresponding to 5 V and a minimal duty cycle of 1% corresponding to 0 V. The
solenoid actuator controls the hydraulic fluid (engine oil) flow and changes the cam phase.
The cam position sensor signal is sampled by the Open-RT prototype controller and the
corresponding cam phase is calculated within the Opal-RT real-time controller.
A PI controller was tuned for the VVT system on the test bench for comparison purpose
with the LPV and OCC controllers. The PI gains tuning process was completed at different
engine speeds and oil pressures. The following tuned PI controller achieves good balance
between response time and over-shoot oscillations at different conditions:

KP I (q) =

5.4.2

0.2q − 0.1995
.
q−1

(5.16)

Bench Test Results

The mixed H2 /H∞ observer-based dynamic LPV controller was tested on the VVT cam
phaser bench at engine speeds of 900, 1200, 1500, and 1800 rpm for both engine oil pressures
of 310 (45) and 414 kPa (60 psi). The step response of each controller is displayed in Fig. 5.6
for the cam advance (-20◦ to 0◦ ) and the cam retard (0◦ to -20◦ ) at an engine speed of
900 rpm and an oil pressure of 310 kPa (45 psi). In Fig. 5.6B, the control effort of both the
LPV and H2 controllers is visibly lower than the PI controller. Also noticeable in Fig. 5.6B
is that the control effort corrections produced by the LPV controller are smaller than those
produced by the H2 controller. This was anticipated from frequency response plot of each
controller in Fig. 5.3A. Since the LPV controller has lower gain than the H2 controller, it

145

is less sensitive to the change in error signal (which has the resolution of one crank degree
in the experiment), which makes the LPV controller more robust to disturbances in the
cam phase when compared to the H2 controller. This is even more noticeable during cam
retard in Figs. 5.6C and 5.6D. The performance of the LPV controller in comparison with
the H2 and PI controllers can also be shown by computing the control variance once the cam
phase has reached steady state. During cam advance with an engine speed of 900 rpm and
oil pressure of 310 kPa (45 psi), the control variances of the LPV, H2 , and PI controllers
were found to be 0.0048 V2 , 0.0265 V2 and 0.0079 V2 , respectively. During cam retard at
the same engine conditions, the control variances of the LPV, H2 , and PI controllers were
found to be 0.0063 V2 , 0.0281 V2 and 0.0068 V2 , respectively. Similar values for the control
variance for each controller were found at all other engine conditions tested as well. The
control variances of the LPV controller under all engine conditions tested were found to be
approximately anywhere from 6% to 33% of the control variance of the H2 controller.
In Fig. 5.7, the mean of the measured overshoot from ten test runs at each engine condition is plotted for each controller. It is easy to see from Fig. 5.7A and Fig. 5.7B, that
in all cases both the H2 controller and LPV controller obtain lower overshoot than the PI
controller, with the H2 controller displaying the lowest overshoot in most cases. However,
during the cam retard situation displayed in Fig. 5.7B, the overshoot of the LPV controller
is much closer to that of the H2 controller and is even smaller than the H2 controller at
an engine speed of 1800 rpm. The difference in performance between cam advance and cam
retard is attributed to the fact that the dynamics are slightly different. During cam advance,
the actuating torque generated by the oil pressure overcomes the cam load torque causing
the cam phase to advance. However, during cam retard, the oil trapped inside the actuator
bleeds back to the oil reserve when the cam phase is pushed back by the cam load shaft.
146

This difference in dynamics between the cam advance and cam retard, as shown in Fig. 5.7A
and Fig. 5.7B, generally results in lower overshoot and faster settling and rising times for
the cam retard performance compared to the cam advance performance. Also, while the
overshoot performance of all of the controllers in Fig. 5.7A and Fig. 5.7B is above 15%, none
of the controllers include feedforward control. With feedforward control the overshoot would
be significantly reduced.
In Figs. 5.8A and 5.8B the mean of the measured 5% settling time from ten test runs
is displayed. It is observed that for nearly all cases the LPV controller settles quicker than
the H2 controller, with one exception of when the engine is operated with an oil pressure of
310 kPa (45 psi) and at an engine speed of 1800 rpm. For the cam advance, the PI controller
almost uniformly has the quickest settling time. However, as observed in Fig. 5.8B, during
cam retard the settling time of the LPV controller is quicker than the PI controller in most
cases, especially when the engine oil pressure is 414 kPa (60 psi).
The mean 10 to 90% rising time from the ten test runs is displayed in Figs. 5.9A and 5.9B.
The rising time performance during cam advance is very similar for each of the controllers
as displayed in Fig. 5.9A. However, as shown in Fig. 5.9B, during cam retard it is quite
clear that the LPV and PI controllers are faster than the H2 controller by an unmistakable
amount.

5.5

Conclusion

A dynamic gain-scheduling controller was designed by employing an observer-based state
feedback design and static multi-objective H2 /H∞ controller synthesis. By examining the
frequency response of the LPV controller and comparing it to a previously obtained ro-

147

A

C
5

PI Controller
H2 Controller
LPV Controller

y (deg.)

0
-5
-10
-15
-20
-25
B 5

D

uG (V)

4
3
2
1
0

0

1

2
Time (s)

3

4

0

1

2
Time (s)

3

Figure 5.6: Cam advance response at 900rpm with 310kPa oil pressure.

148

4

p = 310 kPa (45 psi)

A

6
(deg.)

5.5
5

4
3.5
6
5.5
5
4.5
4
3.5
3

p = 414 kPa (60 psi)

(deg.)

Cam Advance

4.5

B

5

p = 310 kPa (45 psi)
PI Controller
H2 Controller
LPV Controller

Cam Retard

(deg.)

4.5
4

3.5
3
p = 414 kPa (60 psi)
5.5
5
(deg.)

4.5
4

3.5
3
900

1200
1500
Engine Speed (rpm)

1800

Figure 5.7: Mean overshoot for each controller operated at an oil pressure of 45 and 60 psi
and engine speeds of 900, 1200, 1500, and 1800 rpm. Plot A displays the mean overshoot
for cam advance and plot B displays the mean overshoot for cam retard.

149

2.5

Cam Advance

(s)

A

p = 310 kPa (45 psi)

2

1.5
2.5

p = 414 kPa (60 psi)

PI Controller
H2 Controller
LPV Controller

(s)

2
1.5
1
B

2.5

p = 310 kPa (45 psi)

(s)

2

Cam Retard

1.5
1
2.5

p = 414 kPa (60 psi)

(s)

2
1.5
1
900

1200
1500
Engine Speed (rpm)

1800

Figure 5.8: Mean 5% settling time for each controller operated at an oil pressure of 45 and
60 psi and engine speeds of 900, 1200, 1500, and 1800 rpm. Plot A displays the mean settling
time for cam advance and plot B displays the mean settling time for cam retard.

150

A

0.32

p = 310 kPa (45 psi)
PI Controller
H2 Controller
LPV Controller

(s)

0.31

Cam Advance

0.3

0.29
0.28
0.28

p = 414 kPa (60 psi)

0.26
(s)

0.24

0.22
0.2
0.18

B

0.35

p = 310 kPa (45 psi)

(s)

0.3

Cam Retard

0.25
0.2

0.25

p = 414 kPa (60 psi)

(s)

0.2
0.15
0.1
900

1200
1500
Engine Speed (rpm)

1800

Figure 5.9: Mean 10 to 90% rising time for each controller operated at an oil pressure of 45
and 60 psi and engine speeds of 900, 1200, 1500, and 1800 rpm. Plot A displays the mean
rising time for cam advance and plot B displays the mean rising time for cam retard.

151

bust H2 OCC controller, the LPV controller was found to reduce the operating bandwidth
variation of the closed-loop system by approximately 60%. The frequency response of each
system also demonstrated that the LPV controller had lower control effort over the crucial
frequency range of 1-20 Hz. This was validated by the bench tests run with each controller,
which showed that the LPV controller had much lower control variance than the robust H2
OCC controller. Also, the LPV controller has lower overshoot than the PI controller at all
operating conditions with similar settling and response time characteristics. Additionally,
the LPV controller was designed with a systematic approach while the PI controller was
obtained through ad hoc testing.

152

Chapter 6
Dynamic, Output-Feedback,
Gain-Scheduling Control of an
Electric Variable Valve Timing
System

6.1

Introduction

The fuel economy, emissions, and performance of an internal combustion (IC) engine are
heavily influenced by the intake and exhaust valve timing. With a conventional valvetrain
system, the intake and exhaust valve timing can only be optimized for a single operating
condition. That is, the optimized valve timing can either improve fuel economy and reduce
emissions at low engine speeds or maximize engine power and torque outputs at high engine
speeds. Due to the growing fuel economy demands and emissions regulations, continuously
variable valve timing (VVT) systems [37] were developed. The challenging problem of improving fuel economy and reducing emissions at low engine speed while maintaining engine
performance at high engine speed can be addressed with the use of VVT systems.
To adjust the intake and exhaust timing, two of the most common cam phasing systems

153

are a hydraulic vane type cam phaser [37, 20] and an electric cam phaser [45, 63, 27, 50, 51].
The advantage of hydraulic VVT systems is that only minor changes are required to implement them on conventional valvetrain systems, which reduces the design and engineering
effort. However, the performance of hydraulic VVT systems is significantly degraded at some
engine operating conditions. For instance, during engine cold start before the engine oil is
warmed up the hydraulic VVT system cannot be activated and must remain in the default
position [27]. Also the response time of the electric VVT is significantly shorter than the
hydraulic VVT, which makes it possible to be used for the mode transition control between
spark-ignited and homogeneously charge compression ignition combustion [78]. The electric
cam phaser studied in [50, 51] uses a planetary gear train driven by an electric motor operated with respect to the current engine speed to phase the camshaft. In [50, 51], an H2
controller is designed using the output covariance constraint (OCC) control design approach
[87, 51]. Promising results were obtained, however, in [51] it is noted that the engine oil
viscosity has a significant impact on the performance of the electric cam phaser. Thus in
this chapter, the physics based model developed in [50, 51] will be reconsidered with a timevarying friction coefficient to account for the variation of the engine oil viscosity to obtain a
linear parameter-varying (LPV) system. The obtained LPV system is then used to develop
a gain-scheduled, dynamic, output-feedback controller.
The use of LPV modeling and control in automotive applications has received a great
deal of attention. LPV modeling and control techniques have been applied to both diesel
engines [65, 55] and gasoline spark-ignition engines [88, 82, 66, 69, 71]. In [65], LPV control
techniques are applied to the air path of turbocharged diesel engines to control the transient
exhaust gas fraction pumped into the cylinders to reduce nitrous oxide emissions. In [55],
an LPV identification technique is applied to a nonlinear turbocharged diesel engine to
154

obtain an LPV model suitable for control synthesis. In [88], a continuous-time LPV model is
developed considering only engine speed as a time-varying parameter. In [82], a large variable
time delay is present in the air-fuel ratio control loop for a lean burn spark ignition engine.
LPV control methods are used to compensate for the variable time delay. In [66] and [69],
event-based gain-scheduling proportional-integral (PI) and proportional-integral-derivative
(PID) controllers are developed using the wall-wetting parameters and engine speed as timevarying parameters. In [71], the techniques used in [66] and [69] were augmented to develop
an observer-based dynamic LPV controller using the dynamics of the plant for a hydraulic
VVT system. In this chapter, a dynamic, output-feedback gain-scheduling controller with a
guaranteed ℓ2 to ℓ∞ gain is designed for the electric VVT system considered in [50, 51] by
applying a controller synthesis technique that builds on the techniques provided in [13].
The conventional application of gain-scheduling control involves the calibration of controller gains in a field test for the best performance as functions of system operational
conditions. However, in addition to the fact that this approach can be very expensive and
time consuming the system stability and performance are not guaranteed for all time-varying
parameters. The purpose of this chapter is to efficiently develop a dynamic, output-feedback
gain-scheduling controller with guaranteed stability and performance over all time-varying
parameters. The first step to accomplish this is to obtain a discrete-time LPV model of the
electrical VVT system since the LPV control technique to be used is based in discrete time.
As previously mentioned, the physics based model derived model developed in [50, 51] is
considered with a time-varying friction coefficient to account for the variation of the engine
oil viscosity. Since the model of the electric VVT system developed in [50, 51] is a continuous
time model, the LPV model of the electrical VVT system must then be discretized. Then the
discrete-time LPV system is converted to a polytopic system, which is an LPV system with
155

a polytopic dependency on a scheduling parameter that takes values in the unit-simplex,
so that the guaranteed ℓ2 to ℓ∞ discrete-time LPV control synthesis can be performed to
obtain the gain-scheduled controller matrices of the dynamic, output-feedback controller.
As stated previously, an ℓ2 to ℓ∞ gain control design is performed in this chapter. A
controller with a guaranteed ℓ2 to ℓ∞ gain provides strict bounds on the regulated output
while minimizing the control input. Compared to the H∞ performance specification, also
known as an ℓ2 to ℓ2 gain, used in many LPV gain-scheduled controllers which only guarantee
that the root-mean-square gain from the exogenous input to the regulated output is bounded,
a controller designed with a guaranteed ℓ2 to ℓ∞ gain provides hard constraints on the
performance outputs for any exogenous input with bounded ℓ2 energy. Thus controllers
with an ℓ2 to ℓ∞ gain control design are very useful in applications where hard constraints
on responses or actuator signals cannot be ignored, such as space telescope pointing [86] and
machine tool control.

6.2

Plant dynamics

The electric VVT system studied in this chapter operates with two main components: an
electric motor and a planetary gear set. The planetary gear set consists of an outer ring
gear, a planet gear carrier with planet gears attached, and a sun gear. The ring gear acts
as the VVT pulley, which is driven directly by the crankshaft through a timing belt at half
crankshaft speed. The planet gear carrier is connected to electric motor output shaft. The
planet gears engage the ring and sun gears. The camshaft is connected to the sun gear. The
two inputs to the planetary gear system are the ring gear acting as the VVT pulley and
the planet gear carrier which is driven by the electric motor. The output of the planetary

156

kTcam

u

−

Kmc (s)

Ωr

Motor/Planetary
Dynamics

Local Motor
Controller
Ea

−
Em

Ge (s)

−

T

Gm (s)

Ωc

−

φ

GK (s)
Planetary
Kinematics

Km

Figure 6.1: The block diagram of the electric motor with the planetary gear system.
gear system is the sun gear which connects directly to the camshaft. The modeling of the
electric VVT system was performed in [50, 51]. In the following, an overview of the each of
the transfer functions displayed in Fig. 6.1 is provided. The block diagram will be used to
develop a state space model to be used for controller synthesis. As shown in Fig. 6.1, the
electric VVT system has three main parts that contribute to the system dynamics: the local
motor controller, the motor and planetary gear train dynamics, and the planetary gear train
kinematics.

6.2.1

Local Motor Controller

The electric motor for the electric VVT system has a local PI controller such that it is
operated by providing a desired rotational velocity. The transfer function of the local motor
controller is
Kmc (s) =

kp s + ki
s

157

(6.1)

where kp and ki are the proportional and integral gains. With the inputs and outputs
displayed in Fig. 6.1, the following state-space equation can be found:

xKmc (t) = ki (u(t) − Ωc (t))
˙

(6.2)

Ea (t) = xKmc (t) + kp (u(t) − Ωc (t))
where the state xKmc is the integrated error and the output Ea is the motor armature
voltage.

6.2.2

Motor and Planetary Gear System Dynamics

The motor and and planetary gear train dynamics contained inside the dashed-line box in
Fig. 6.1 are formed by the feedback connection of the transfer functions Ge (s) and Gm (s).
The transfer function Ge (s), which represents the input-output relationship from the voltage
drop across armature circuit to the torque produced by the motor, is given by

Ge (s) =

Kτ
Lm s + Rm

(6.3)

where the armature resistance and inductance are Rm and Lm , respectively, and Kτ is the
motor torque constant. Using the transfer function (6.3), the following state-space equation
can be found:

xe (t) = −
˙

1
Rm
xe (t) +
(Ea (t) − Em (t))
Lm
Lm

T (t) = Kτ xe (t)

158

(6.4)

where the state xe is the current running through the armature circuit, Em is the back-EMF
voltage, and the output T is the torque produced by the motor.
The transfer function Gm (s), which represents the input-output relationship from the
torque acting on the load to the motor speed, is given by

Gm (s) =

1
Js + β(t)

(6.5)

where J is the lumped inertia of the planetary gear system and the motor shaft and B(t) is
the time-varying friction coefficient, to which the oil viscosity is a major contributor. Using
the transfer function (6.5), the following state-space equation can be found:

xm (t) = −
˙

1
β(t)
xm (t) + (T (t) − kTcam (t))
J
J

(6.6)

Ωc (t) = xm (t)
where k, given by k = 1 + 2nr /ns , is the gear ratio from the planetary gear system that
disturbance torque form the camshaft Tcam acts through and the state xm is equal to the
carrier angular velocity Ωc .

6.2.3

Planetary Gear System Kinematics

The planetary gear train kinematics is the final part of the electric VVT model. The sun
gear of the planetary gear system essentially integrates the difference between the planet
carrier rotational velocity, Ωc , and the ring gear rotational velocity, Ωr , to obtain the phase
for the camshaft. The transfer function GK is given by

GK (s) =

2(nr + ns )
ns s

159

(6.7)

where nr and ns are the number of teeth on the ring and run gears, respectively. With the
transfer function (6.7), the following state-space can be found:

xK (t) =
˙

2(ns + nr )
(Ωc − Ωr )
ns

(6.8)

φ(t) = xK (t)
where the state xK is equal to the phase φ of the camshaft.

6.2.4

An LPV System

The state space equations can be gathered together to form the following state space model:

x(t) = A(β(t))x(t) + Bu u(t) + Bd Tcam (t) + Br Ωr
˙

(6.9)

φ(t) = Cx(t)
where x = [xKmc , xe , xm , xK ]T and


0
 0

 1
Rm

−
 L
Lm
A(β(t)) =  m

Kτ
 0

J


0
0

0




0

Br = 


0

 2(n + n )
s
r
−
ns



−ki









0 
 0 
 ki 






k p + Km

 kp 


 0 


−
0 

Lm
 , Bu =  Lm  , Bd = 



,

 k 


β(t)

 − 
 0 
−
0 
 J 


J






2(ns + nr )
0
0
0
ns






, C =





0 0 0 1

160

.

In order to apply the control synthesis in the next section, a discrete-time LPV system is
required. In general, the discretization of the time-varying system (6.9) can be very difficult.
However, the oil viscosity rate of variation is very slow compared to the sample rate ts , such
that they occur on different time scales. With this information, we can assume the friction
coefficient β is constant during a given time step. As such, the following discrete-time system
can be found:

xd (k + 1) = Ad (β)xd (k) + Bu,d (β)u(k) + Bd,d (β)Tcam (k) + Br,d Ωr (k)

(6.10)

φ(k) = Cd xd (k)
where

Ad (β) = exp(A(β)ts ),
Bu,d (β) =
Bd,d (β) =
Br,d =

ts

exp(A(β)τ )dτ

Bu ,

exp(A(β)τ )dτ

Bd ,

exp(A(β)τ )dτ

Br ,

0
ts
0
ts
0

(6.11)

Cd = C.
This computation could be performed with the symbolic toolbox in MATLAB by letting β
be a symbolic variable. Then, for each element in the system matrices that still contains the
friction coefficient β, the 1st -order Taylor series expansion can be performed as done in [66]
to obtain the discrete-time LPV system. However, it was found that with the continuoustime system (6.9), this approach was found to be excessively time consuming. Instead, a
much more efficient numerical approach based on the linear least squares technique [31] was
used to obtain (6.10).
161

6.2.4.1

Linear least squares estimation of discrete-time LPV system

For any given fixed friction coefficient β, the discrete-time system can be easily computed
to obtain the discrete-time state matrix


 a11 (β) a12 (β)


 a21 (β) a22 (β)

Ad (β) = 

 a31 (β) a32 (β)


a41 (β) a42 (β)



a13 (β) 0 


a23 (β) 0 

,

a33 (β) 0 


a43 (β) 1

(6.12)

and input matrices


 bu,1 (β)


 bu,2 (β)

Bu,d (β) = 

 bu,3 (β)


bu,4 (β)















 , Bd,d (β) = 











bd,1 (β) 


bd,2 (β) 

.

bd,3 (β) 


bd,4 (β)

(6.13)

After computing each of these discrete-time matrices over a range of β values, it was noted
that the parameter variation with respect to β appeared to be affine. With this information,
the coefficients of Ad (β), Bu,d (β), and Bd,d (β) were assumed to have the following form

aij (β) = pij (β) + ǫ,
bu,i (β) = pu,i (β) + ǫ,
bd,i (β) = pd,i (β) + ǫ,

162

(6.14)

where ǫ is noise due to numerical error and model mismatch and pij , pu,i , and pd,i are affine
functions with respect to β such that

pij (β) = θij,0 + θij,1 β,
(6.15)

pu,i (β) = θui ,0 + θui ,1 β,
pd,i (β) = θdi ,0 + θdi ,1 β.

The unknown parameters θij,0 , θij,1 , θui ,0 , θui ,1 , θd ,0 , and θd ,1 can be collected into the
i
i
following unknown parameter vectors:




 θij,0 
Θij = 
,
θij,1





 θu ,0 
Θu,i =  i  ,
θui ,1





 θd ,0 
Θd,i =  i 
θdi ,1

(6.16)

such that (6.15) can be re-written with the following linear regression model:

pij (β) = h(β)Θij
pu,i (β) = h(β)Θu,i

(6.17)

pd,i (β) = h(β)Θd,i
where h(β) is given by
h(β) =

1 β

.

(6.18)

Thus, given M randomly selected values for the friction coefficient β, we would like to
ˆ
ˆ
ˆ
determine the linear least square estimates Θij , Θu,i , and Θd,i that minimize the following

163

residual sums of squares:
M

S(Θij ) =
l=1
M

S(Θu,i ) =
l=1
M

S(Θd,i ) =
l=1

|aij (βl ) − h(βl )Θij |2 ,

(6.19)

|bu,i (βl ) − h(βl )Θu,i |2 ,

(6.20)

|bd,i (βl ) − h(βl )Θd,i |2 ,

(6.21)

where | · | denotes the magnitude operator on a complex number. If we let Yij , Yu,i , and
Yd,i be the collection of “measurements” and H be the collection of h(β) for all M friction
coefficients β such that







 aij (β1 ) 






.
.
Yij = 
 , Yu,i = 
.






aij (βM )


 bd,i (β1 ) 




.
.
Yd,i = 
 , and H
.




bd,i (βM )



bu,i (β1 ) 


.
.
,
.


bu,i (βM )


 h(β1 ) 




.
.
=
,
.




h(βM )

(6.22)

then the residual sum of squares equations (6.19)-(6.20) can be re-written as

S(Θij ) = Yij − HΘij

T

Yij − HΘij ,

(6.23)

S(Θu,i ) = Yu,i − HΘu,i

T

Yu,i − HΘu,i ,

(6.24)

S(Θd,i ) = Yd,i − HΘd,i

T

Yd,i − HΘd,i .

(6.25)

In [31], it is shown that the estimates that minimize each residual sum of squares are given
164

v
wp
z

G(q, β)

y

r

u

−

K(q, β)

Figure 6.2: Proposed control scheme for the electrical VVT system.
by
ˆ
Θij = H T H

−1

H T Yij ,

ˆ
Θu,i = H T H

−1

H T Yu,i ,

ˆ
Θd,i = H T H

−1

(6.26)

H T Yd,i .

By using this method, an approximate of the discrete-time LPV system (6.10) can be
efficiently obtained by computing the discrete-time state-space matrices at a finite number
of randomly selected friction coefficients β, and then by solving for the linear least square
estimates as given by (6.26).

6.3

LPV gain-scheduling controller design

The objective of the control system is to regulate the cam phase to a reference phase using
output feedback control against the disturbance signal wp and the time-varying parameters.
In particular, we want to guarantee the stability and performance of the closed-loop system
for any finite energy disturbance and engine oil viscosity variations. The proposed control
165

architecture is illustrated in Fig. 6.2. This scheme has a single component, that being the
dynamic, output-feedback, gain-scheduled controller, K(q, β).

6.3.1

Guaranteed ℓ2 to ℓ∞ gain controller

The plant G(q, β) in Fig. 6.2 can be represented as the following open-loop discrete-time
polytopic LPV system

xp (k + 1) = Ap (λk )xp (k) + Bp (λk )u(k) + Dp (λk )wp (k)
(6.27)

zp (k) = Cp (λk )xp (k)
y(k) = Cy (λk )xp (k) + v(k)

where xp (k) is the state, u(k) is the control input, wp (k) is an exogenous ℓ2 disturbance
weighted by the positive definite symmetric matrix Wp , and zp (k) is the vector of all dynamic
variables of interest, and y(k) is the vector of measurements corrupted by the noise v(k)
weighted by the positive definite symmetric matrix V . The system matrices belong to a
polytope similar to the one introduced in (2.11).
Suppose that we apply to the plant (6.27) a strictly proper, dynamic, output-feedback,
stabilizing control law of the form

xc (k + 1) = Ac (λk )xc (k) + Bc (λk )y(k),
u(k) = Cc (λk )xc (k).

166

(6.28)

Then the resulting closed-loop system is given by

x(k + 1) = A(λk )x(k) + B(λk )w(k),
(6.29)

zp (k) = Cp (λk )x(k),
zu (k) = Cu (λk )x(k),
T
where x = [xT , xT ]T , w = [wp , v T ]T , W = block diag[Wp ,
p
c



V ], and


Bp (λk )Cc (λk ) 
,
Bc (λk )Cy (λk )
Ac (λk )


1/2
0

 Dp (λk )Wp
B(λk ) = 
,
1/2
0
Bc (λk )V


A(λk ) = 

Ap (λk )

Cp (λk ) =

Cp (λk ) 0

Cu (λk ) =

0 Du (λk )Cc (λk )

(6.30)

,
.

Then, as shown in Chapter 3, a controller that minimizes the (weighted) control energy

¯
Zu (λk ) = trace RCu (λk )P(λk )Cu (λk )T , R > 0

(6.31)

of the closed loop system, subject to an ℓ2 to ℓ∞ gain Z p given by
zp 2 ≤ σ Z p
∞

w 2.
2

(6.32)

¯
where Zp (λk ) = Cp (λk )P(λk )Cp (λk )T ≤ Z p , can be efficiently solved for by a convex optimization with the LMI constraints given in Theorem 10.

167

6.3.2

H∞ (ℓ2 to ℓ2 gain) Controller

For comparison with the guaranteed ℓ2 to ℓ∞ gain controller, an H∞ controller is designed as
well. The H∞ controller differs from the ℓ2 to ℓ∞ gain controller in that it has the following
property:
z 2≤η w 2

(6.33)

T
T
where z = [zp , zu ]T . This is also known as the ℓ2 to ℓ2 gain, or root-mean-square gain.

The main difference is the lack of the maximum singular value constraint. This is the reason
why H∞ controllers are not suitable when hard constraints cannot be ignored.
The H∞ controller designed is also a strictly proper, dynamic, output-feedback controller
and has the form given in (6.28). The H∞ controller synthesis was carried out to minimize
the H∞ performance η with LMIs developed based on the work presented in [12] and [13].

6.4

Simulation Results

A simulation study was performed on the electrical VVT system to effectiveness of the
guaranteed ℓ2 to ℓ∞ gain controller compared to the H∞ controller. The parameter values
used in this study for the electrical VVT system are displayed in Table 6.1. During the
simulation, the engine speed was set to 1500 and the cam phase, φ, was stepped from 0◦ to
10◦ for 10 seconds and then back to 0◦ . The ℓ2 disturbance in Fig. 6.3 was used as the cam
load disturbance Tcam .
The guaranteed ℓ2 to ℓ∞ controller was designed with an ℓ2 to ℓ∞ gain given by
Zp = 9.5434 × 10−4 rad2

168

Table 6.1: Electric VVT System Parameters
ts

5 µs

Kτ

ns

30 teeth

Km

Nm
A
1/90 Vs

nr

60 teeth

J

0.2 kgm2

Lm

0.01 H

kp

1 Vs

Rm

1 Ω

ki

0.1 V

45

which is equivalent to a maximum displacement of 1.77◦ from the desired cam phase. The
other control design parameters were chosen to be

Wp = 10,

V = 0.0012,

and R = 1.

The parameters Wp , V , and R were also used to synthesize the H∞ controller as well.
During the gain-scheduled controller design, it was assumed that the friction coefficient
β varied between 0.5 and 2 , with the smaller friction coefficient corresponding to low oil
viscosity (or oil that has been heated up) and the larger friction coefficient corresponding
to high oil viscosity (oil that is cold or at room temperature). Since the oil viscosity varies
slowly, simulations were performed at the following three friction coefficients: 2, 1, and 0.5,
corresponding to the behavior that might occur when engine oil is heated up. The results of
each simulation are displayed in Figs. 6.4-6.6.
In Figs. 6.4-6.6, the cam phase obtained when using the guaranteed ℓ2 to ℓ∞ gain controller is displayed by the solid black line, whereas the cam phase obtained when using the
H∞ controller is displayed by the solid grey line. The reference cam phase is displayed by
a dashed line and can be seen in the close-up details of Figs. 6.4A-6.6A. The bound representing the maximum allowable displacement is displayed by a dash-dot line. Notice in

169

A

B

2

2

1.8

1.6

1.6

1.4

1.4

1.2

1.2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

Tcam (Nm)

1.8

0
0

5

10
Time (s)

15

20 4.5

4.6

4.7
4.8
Time (s)

4.9

5

Figure 6.3: Cam load disturbance.
Figs. 6.4A-6.6A that the cam phase stays within the bounds when using the ℓ2 to ℓ∞ gain
controller. However, for each friction coefficient value, the cam phase violates the bound
when using the H∞ controller.
The activation voltage of the electrical VVT motor is displayed in Figs. 6.4B-6.6B. A saturation of ±14 was placed on the motor activation voltage during the simulation. As clearly
displayed in Figs. 6.4B-6.6B, the ℓ2 to ℓ∞ controller uses large amounts of the activation
voltage potential during the cam phase changes compared to the H∞ controller, which is
why it reaches the reference cam phase about 10 times faster than the H∞ controller. Also
during the cam load disturbance, as displayed in the details in Figs. 6.4B-6.6B, the ℓ2 to ℓ∞
gain controller uses ever so slightly more potential to ensure that the maximum displacement
bound is not violated. Notice also, that as the friction coefficient gets smaller, the activation

170

voltage needed to hold the cam phase at the desired position decreases since the motor does
not have to work against the oil as much with lower oil viscosities.

6.5

Conclusion

In this chapter, a discrete-time, LPV system representing the electric VVT system is developed with engine oil viscosity as the time-varying parameter. A gain-scheduled, dynamic,
output-feedback controller is then designed such that the closed-loop system will have a
guaranteed ℓ2 to ℓ∞ gain, which is very closely related to the physical performance constraints. For comparison, an H∞ controller is also designed to demonstrate the benefit of
guaranteed ℓ2 to ℓ∞ gain controllers. Simulation study results demonstrate the effectiveness
of the proposed scheme.

171

A
10
10
9

φ(deg.)

8

8

4.7

4.8

4.9

14.7

14.8

14.9

6
4
0
2

-1

0

-2

14.7

14.8

14.9

B
12
10

Ea (V )

8
6
4
5.5
5
4.5
4

2
0

5.5
5
4.5
4

-2
-4

4.8

4.7
0

2

4

6

4.9
8

10

12

14

16

Time (s)
Figure 6.4: Electrical VVT simulation with β = 2.

172

18

20

A
10
10
9

φ(deg.)

8

8

4.7

4.8

4.9

14.7

14.8

14.9

6
4
0
2

-1

0

B

-2

14.7

14.8

14.9

12
10
8

Ea (V )

6
4
2
0
3

-4

2.5

2

-6
-8

3

2.5

-2

0

4.8

4.7
2

4

6

2

4.9
8

10

12

14

16

Time (s)
Figure 6.5: Electrical VVT simulation with β = 1.

173

18

20

A
10
10
9

φ(deg.)

8

8

4.7

4.8

4.9

14.7

14.8

14.9

6
4
0
2

-1

0

-2

14.7

14.8

14.9

B
10

Ea (V )

5

0

2.5
2

2

1.5

-5

2.5

1.5
4.8

4.7
-10

0

2

4

6

4.9
8

10

12

14

16

Time (s)
Figure 6.6: Electrical VVT simulation with β = 0.5.

174

18

20

Chapter 7
Conclusions and Future Research

7.1

Summary

In this dissertation, first the current LPV control theory methods were reviewed. Second,
since the current LPV control theory was missing the ability to design LPV controllers with
physically meaningful performance design constraints that could guarantee hard bounds,
the current theory was extended to allow the use of ℓ2 − ℓ∞ gain performance constraints.
Controller synthesis LMIs are provided for the synthesis of state-feedback and dynamic
output-feedback controllers with guaranteed ℓ2 − ℓ∞ gain and H∞ performance. To demonstrate the effectiveness of the controller synthesis LMIs provided a numerical design example
is considered for both the state-feedback and dynamic output-feedback control designs.
Also included in this dissertation is the application of the current and newly developed
LPV methods to engine control problems. In Chapter 4, gain-scheduling PI and PID controllers were developed for the air-to-fuel ratio control of port-fuel-injection processes with
the wall-wetting parameter and engine speed as the time-varying parameters. Simulation and
HIL simulation results demonstrate the successful application of current LPV control theory to obtain a gain-scheduling controller. The HIL simulation results also demonstrate the
feasibility of implementing the obtained gain-scheduling controller on a hardware controller
that could be used as an engine control module.
In Chapter 5, a dynamic output-feedback gain-scheduling controller was designed for a
175

hydraulic variable valve timing actuator. This was done by employing an observer-based
state feedback design and static multi-objective H2 /H∞ controller synthesis. The LPV
controller designed was compared with the robust H2 OCC controller and an ad-hoc PI
controller designed in [52]. The LPV controller was found to have much lower control variance
than the robust H2 OCC controller. Also, while the LPV controller is more complex than
the PI controller in both concept and implementation, it has lower overshoot than the PI
controller at all operating conditions with similar settling and response time characteristics.
Additionally, the LPV controller was designed with a systematic approach while the PI
controller was obtained through ad hoc testing.
The guaranteed ℓ2 − ℓ∞ gain controller synthesis techniques provided in Chapter 3 are
applied to the control of the electric variable valve timing actuator with engine oil viscosity
as the time-varying parameter in Chapter 6. A gain-scheduled, dynamic, output-feedback
controller is designed such that the closed-loop system will have a guaranteed ℓ2 to ℓ∞ gain.
To demonstrate the benefit of guaranteed ℓ2 to ℓ∞ gain controllers, an H∞ controller is also
designed for comparison. The simulation study results demonstrate the effectiveness of the
proposed scheme.

7.2

Specific Contributions

The contributions of this research to systematic gain-scheduling control are as follows:
1. The characterization of the ℓ2 to ℓ∞ gain performance constraint as an LMI constraint
for discrete-time polytopic LPV systems [73].
2. Guaranteed ℓ2 to ℓ∞ controller synthesis LMIs for the state-feedback and dynamic
output-feedback control of discrete-time polytopic LPV systems. These controllers
176

can provide hard constraints on system outputs when the exogenous input is modeled
as a bounded ℓ2 disturbance [73].
3. Chapter 4 is the first work to develop an air-to-fuel ratio gain-scheduling controller for
the port-fuel-injection process using the wall-wetting parameters and the engine speed
as scheduling variables [66, 69, 67, 70, 68].
4. In Chapter 4, a novel approach is developed to express the parameter variation of the
continuous-time oxygen sensor dynamics in the event-based discrete-time model with
minimal error [66].
5. In Chapter 5, a systematic approach is provided for the design of gain-scheduling
controllers when system identification is used to obtain the system model [71].
6. In Chapter 6, an LPV model is developed for the electrical VVT system with engine
oil viscosity as the time-varying parameter [72], which large impact on the response
time according to [49].
7. In Chapter 6, a method for obtaining an approximate discrete-time LPV system from
a continuous-time LPV system based on the linear-least squares estimation method is
provided [72].

7.3

Future Research

The following topics deserve future investigation:
1. The guaranteed ℓ2 − ℓ∞ gain control design considered in this research minimized the
control effort subject to desired system output constraints. It would also be interesting
177

to investigate a guaranteed ℓ2 − ℓ∞ gain control design that finds the best achievable
performance for a fixed amount of control energy, meaning that the control energy
would be constrained and the performance of the system outputs would be optimized
as much as possible.
2. A systematic framework for the off-line estimation of the wall-wetting parameters over
all operating conditions for the a given intake manifold would be desirable for the
practical application of wall-wetting parameters as a scheduling parameter for gainscheduling control.
(a) It may also be necessary to develop on-line estimation procedure to validate the
wall-wetting parameters stored in the engine control unit in case the wall-wetting
parameters change over the life of the engine due to build up of fuel residuals.
3. The hydraulic VVT system studied in Chapter 5 was only capable of being operated
at low engine speeds. To demonstrate the benefits of the LPV approach, a wider range
of engine operating conditions needs to be considered.
4. A mixed ℓ2 −ℓ∞ /H∞ LPV controller design should be performed for the electrical VVT
system and that controller should be tested on an engine equipped with an electrical
VVT actuator.

178

APPENDICES

179

Appendix A
Linear Fractional Transformation
For completeness, we will now give the definition of a linear fractional transformation (LFT).
Linear fractional transformations are used to efficiently formulate the interconnection of
multi-input multi-output sub-systems with multiple sources, such as uncertainties, noises,
disturbances, and varying parameters. As given by [84], the possibly complex coefficient
matrix M is partitioned as

M=

M11 M12
M21 M22

∈ C(p1 +p2 )×(q1 +q2 )

(A.1)

with ∆ℓ ∈ Cq2 ×p2 and ∆u ∈ Cq1 ×p1 . A lower LFT is given with respect to ∆ℓ as
Fℓ (M, ∆ℓ ) = M11 + M12 ∆ℓ (I − M22 ∆ℓ )−1 M21 .

(A.2)

An upper LFT is given with respect to ∆u by

Fu (M, ∆u ) = M22 + M21 ∆u (I − M11 ∆u )−1 M12 .

(A.3)

From the diagrams in Fig. A.1, the reason behind the terminology of lower and upper LFTs
should be clear. The set of equations representing the lower LFT diagram in Fig. A.1a are

180

A

B
w1

z1

∆u

M
u1

y1

w2

y2
M

∆ℓ

z2

u2

Figure A.1: (a) Diagram of a lower LFT. (b) Diagram of an upper LFT.

given by
z1
y1 =

M11 M12
M21 M22

w1
u1 ,

(A.4)

u1 = ∆ℓ y1 ,
and the equations representing Fig. A.1b are given by
y2
z2 =

M11 M12
M21 M22

u2
w2 ,

(A.5)

u2 = ∆u y2 .
The partitioning of M depends on the interconnections with the isolated parameter ∆ℓ
or ∆u and can be determined using the MATLAB function “sysic” [6].

181

Appendix B
Port Fuel Injection System Matrices
The state-space matrices for the LPV system in (4.23) have been found to be

0.91
0
0.0369
0
0
0
0
0

 0.2617
0
0.1544
0
0
1.4352
0
0



 0
0
0.8475
0
0
0
0
0



 0
1.4506
0
0.2231 0.3972
0
0
0
 ∈ R8×8
A=

 0
2.6311
0
0
0.3114
0
0
0



 0
0
0
0
0
0.9986
0
0


 0
0
0
0
0
0
1.9972 −0.9985 
0
0
0
0
0
0
0.9987
0


(B.1)



−0.09 0.0625
0
0
1
0
0
 0.2617 0.2617
0
0
0
0
0

 0
−0.2585 1.6949 1.6949 0
0
0


0
0
0
0 0.0664 −0.0027
B0 =  0
 0
0
0
0
0 0.0436 −0.0073

 0
0
0
0
0
0
0

 0
0
0
0
0
0
0
0
0
0
0
0
0
0


0
0
0
0
0
0
0
0
0
0
0
0
0
0 

0
0
0
0
0
0
0 

−0.0214 −0.0179 −0.0933 −0.4891 −0.0984 0.0608 0.0975  ∈ R8×14

−0.0186 −0.0134
0
−0.7266 0.1211 0.3095 0.2231 

0
0
0
0
0
0
0 

0
0
0
0
0
0
0 
0
0
0
0
0
0
0

182

(B.2)




0
0
0.0043
 0
0
0.0179 


 0
0
−0.0073 


 0

0
0
 ∈ R8×3,
B1 = 
 0

0
0


 0.3756

0
0


 0

0.0266
0
0
0
0


0.0369
 0.1544 


 0 


 0 
8×1

B2 = 
 0 ∈R


 0 


 0 
0


0
0
1
0 0 0 0 0
 0
0
0
0 0 0 0 0


 0
0
0.1525 0 0 0 0 0 


 0
0
−1 0 0 0 0 0 


 −1
0
0.41 0 0 0 0 0 


 0 63.6832
0
0 0 0 0 0


 0
0
0
0 0 0 0 0  ∈ R14×8,

C0 = 
 0
0
0
0 0 0 0 0


 0
0
0
0 0 0 0 0


 0 25.2968
0
0 0 0 0 0


 0
0
0
0 1 0 0 0


 0
0
0
0 0 0 0 0


 0
0
0
0 0 0 0 0
0
0
0
0 0 0 0 0
C1 = 0 0 0 −1 0 0 0.015 0.015 ∈ R1×8

183

(B.3)

(B.4)

(B.5)

(B.6)



0
 0

 0

 0

 −1

 0


D00 =  0
 0

 0

 0

 0

 0

 0

1.6949
0
0
0 0
−1.6949
0
0
0 0
0.2585 −1.6949 −1.6949 0 0
−1.6949
0
0
0 0
0.6949
0
0
0 0
0
0
0
0 0
0
0
0
0 0
0
0
0
0 0
0
0
0
0 0
0
0
0
0 0
0
0
0
0 0
0
0
0
0 0
0
0
0
0 0
0
0
0
0
0 0


0
0
0
0
0 0 0 0
0
0
0
0
0 0 0 0

0
0
0
0
0 0 0 0

0
0
0
0
0 0 0 0

0
0
0
0
0 0 0 0

0
0
0
0
0 0 0 0

1
0
0
0
0 0 0 0  ∈ R14×14,

0
1
0
0
0 0 0 0

0
0
1
0
0 0 0 0

−0.4891 −0.0984 0.0608 0.0975 1 0 0 0 

0
0
0
0
0 0 0 0

0
0
0
0
0 1 0 0

0
0
0
0
0 0 1 0
0
0
0
0
0 0 0 1


0 0 0.1161
1
 0 0 −0.1161 0 


 0 0 0.0073
0 


 0 0 −0.0476 0 


 0 0 0.0476 0.41 


0 0
0
0 


0 0
0
0  ∈ R14×3,


D01 = 
0 0
0
0 


0 0
0
0 


0 0
0
0 


0 0
0
0 


0 0
0
0 


0 0
0
0 
0 0
0
0

184

(B.7)

(B.8)




1
 0 


 0 


 0 


 0.41 


 0 


 0 
14×1

D02 = 
 0 ∈R


 0 


 0 


 0 


 0 


 0 
0

(B.9)

D10 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ∈ R1×14 ,

(B.10)

D10 = 0 0 0 ∈ R1×14 ,

(B.11)

D10 = 0 ∈ R.

(B.12)

185

BIBLIOGRAPHY

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