VIRTUAL CONTROL WITH APPLICATION TO HARD CONSTRAINED CONTROL

PROBLEMS

By

Ali Mohammed Hussein Al-Hajjar

A DISSERTATION

Michigan State University

in partial fulfillment of the requirements

Submitted to

for the degree of

Mechanical Engineering – Doctor of Philosophy

2019

ABSTRACT

VIRTUAL CONTROL WITH APPLICATION TO HARD CONSTRAINED CONTROL

PROBLEMS

By

Ali Mohammed Hussein Al-Hajjar

The saturation problem was and still a challenging research topic in the control community. For
many decades it has attracted and still attracts many researchers and scientists to investigate and
trials to give a resolution to the problem. There are many recent methods and techniques introduced
in the literature as a solutions for the problem, but most of them are introducing an ad hoc methods,
nonlinear or anti-windup schemes were in most, these methods are either incompetent in the sense
of applicability to the broad control problems and/or they are computationally expensive. This
study presents novel solutions modeling and control as a remedy for the saturation problem with
applications. These novel proposed methods are systematic, numerically sounds and are applicable
for a wide range of Linear Time Invariant (LTI), Linear Parameter Varying (LPV), discrete and
continuous systems.

The first novel idea introduced in this dissertation is for LPV systems. The idea of scaling
scheduling parameters is introduced and synthesized. Two novel Linear Parameter Varying (LPV)
modeling and control techniques are proposed for active flutter suppression of a smart airfoil model.
The smart airfoil model is instrumented with a moving mass that can be used to actively control the
airfoil pitching and plunging motions. The first LPV modeling approach makes use of the moving
mass position as a scheduling parameter, and the hard constraint at the boundaries is imposed
by proper selection of the parameter varying function. The second modeling technique utilizes
nonlinear springs and dampers, which are added to both ends of the airfoil groove to confine the
motion of the moving mass. A state feedback based LPV gain-scheduling controller with the
guaranteed H∞ performance is proposed by utilizing the dynamics of the moving mass. In this
novel idea, both the position of the moving mass and the free stream airspeed are considered as the
scheduling parameters.

The second novel idea is the virtual hard constraints control approach. The proposed idea is
introduced to deal with saturation control problems for linear time invariant (LTI) systems using
virtual nonlinear springs and dampers for the optimal control design and then incorporates the
virtual dynamics into the control input for real-time control. The basic idea is to increase the
control cost when these state(s) move close to the hard constraint(s) and saturation limit(s), and
the designed optimal control strategy will avoid moving these state(s) close to the hard constraint
area. The proposed method is applied to a ball and beam system to demonstrate its feasibility
and effectiveness, where virtual spring and damper are introduced during the control design. The
performance of conventional LQR state feedback control and the associated virtual LQR-based
MPC control are compared. The concept is especially relevant when implementing a physical
hardware system, where one of the primary control design objectives is to safe-guard the hardware
constraints during its operations.

The third novel idea is the LPV virtual control (LPVVC) scheme with hard constraints for LPV
system is proposed. The main idea of virtual control and virtual mechanism is introduced here to the
LPV systems to form LPVVC. The proposed control scheme is applied to the flutter suppression of
a smart airfoil to demonstrate its effectiveness and ability of performance enhancement. The state-
feedback LPV (gain-scheduling) control with guaranteed H∞ performance is used for designing
controller based on the model with virtual mechanisms. The virtual mechanisms used in this work
are in terms of springs and dampers located at both ends of the airfoil groove to prevent the control
mass from moving outside of the groove. Noticing that the virtual mechanisms are not limited to
springs and dampers and can be choosing to be in any other form.

ACKNOWLEDGMENTS

This work is a culmination and dedication of many people’s efforts in pursuing my PhD road and
the words are unable to represents my feelings towards them. In addition, this section will fail to
mention them all.

Human can meet and know many people in his life, but the chance of meeting good, humble,
dedicated, understanding and lovely persons are very little.
I would like to present my sincere
thanks and gratitude to my advisor and mentor Professor Guoming Zhu who presents his efforts
and experience and providing supports, excellence guidance, understanding and patience during
these years of my work. His deep experience and perspective of the field has been great source of
knowledge to me. Also, I would like to present my sincere thanks and appreciation to my second
advisor Professor Sean Shan-Min Swei who offered his time and efforts generously and provided
understanding and valuable feedback and patience during these years. His notes, advices and deep
experience was great valuable source to me.

I would like to thank my committee members Professor Hassan Khalil, Professor Ranjan
Mukherjee and Professor Zhaojian Li for they always offer their knowledge and time when I come
to them without any appointment. I have learned a lot from the courses they taught. Further, a
special thank for Professor Weihua Su for his valuable notes and support.

In addition, special thank to my friends Dr. Assaad Alsahlani , Dr. Ali Aljiboory, Tiany He,
Dr. Yifan Men and Yingxu Wang for their help and support. Moreover, I would like to thank my
friends in the ERC Lab in Michigan State University and in the Mechanical engineering department
of Kufa university specially Professor Luay Alansari, Professor Ali S. Alathari, Professor Muhaned
Alwaeli and Dr. Hayder Zainy, Professor Ahmed Alrajihy, Professor Alaa Aljasani, Dr. Moayed
S. Radi (RH), Dr. Noor Muhsen, Dr. Maher Albaghdady, Dr. Yassar Sh., Hayder Hazim, , Sayed
Dawood, Hayder Alsaraf, Karar Khafaji, Dr. Moayed Alharis, Ali Mamory, Ali M. Asadi, Dr.
Moayed Zahid, Dr. Hyder Abdulhussein for their help and support.

iv

Finally, I would like to acknowledge the financial support of the HCED (The Higher Committee
For Education Development in Iraq) and University of Kufa for providing the financial support
during my study at Michigan State University.

TABLE OF CONTENTS

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CHAPTER 1

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LIST OF TABLES .
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LIST OF FIGURES .
KEY TO SYMBOLS AND ABBREVIATIONS . . . . . . . . . . . . . . . . . . . . . . . .
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CHAPTER 2 PRELIMINARIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Linear Parameter Varying (LPV) preliminaries . . . . . . . . . . . . . . . . . . . .
2.2 Linear Quadratic Regulator (LQR) preliminaries . . . . . . . . . . . . . . . . . . .

5
5
7
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1.1 Motivation and Vision .
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1.2 Contribution .
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1.3 Dissertation Outline .
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INTRODUCTION .
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Summary .

2.2.1

1
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3.3.1

CHAPTER 3 NOVEL LINEAR PARAMETER VARYING MODELING AND FLUT-

Introduction .

3.3 LPV Controller Design .

3.1
3.2 LPV Modeling of a Smart Airfoil . . . . . . . . . . . . . . . . . . . . . . . . . .

TER SUPPRESSION CONTROL OF A SMART AIRFOIL . . . . . . . . . 13
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
. 16
3.2.1 LPV plant model: LPV-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2.2 LPV control design model: LPV-1 . . . . . . . . . . . . . . . . . . . . . . 19
3.2.3 LPV control design model with nonlinear springs and dampers: LPV-2 . . 20
3.2.4 LPV control design model with two scheduling parameters: LPV-2A . . . . 21
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3.1.1
Transferring from affine to multi-simplex . . . . . . . . . . . . . 25
3.3.1.2 H∞ control problem . . . . . . . . . . . . . . . . . . . . . . . . 26
PLMI relaxation .
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CHAPTER 4 VIRTUAL CONTROL WITH HARD CONSTRAINTS . . . . . . . . . . . 45
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.1
4.2 Virtual control: a motivational example
. 47
4.3 Virtual modeling and control framework . . . . . . . . . . . . . . . . . . . . . . . 49
4.3.1 Receding horizon LQR control . . . . . . . . . . . . . . . . . . . . . . . . 50
4.3.2 Virtual constraint profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.4 Concept validation through simulation study . . . . . . . . . . . . . . . . . . . . . 53
4.4.1 Virtual control with constant profile . . . . . . . . . . . . . . . . . . . . . 54
4.4.2 Virtual control with piece-wise linear profile
. . . . . . . . . . . . . . . . 55
4.4.3 Virtual control with tangent profile function . . . . . . . . . . . . . . . . . 57

3.4 Numerical Studies .
3.5 Summary .
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Introduction .

3.3.2

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vi

4.5 Summary .

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CHAPTER 5 VIRTUAL LPV FLUTTER CONTROL OF A SMART AIRFOIL WITH

Introduction .

HARD CONSTRAINTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.2 LPV Virtual Control Problem Formulation . . . . . . . . . . . . . . . . . . . . . . 66
5.2.1 A multi-simplex representation . . . . . . . . . . . . . . . . . . . . . . . . 66
5.2.2 H∞ control performance . . . . . . . . . . . . . . . . . . . . . . . . . .
. 69
5.2.3 Virtual control system modeling . . . . . . . . . . . . . . . . . . . . . . . 71
5.3 LPV Modeling and Control of Smart Airfoil . . . . . . . . . . . . . . . . . . . . . 72
5.3.1 Virtual control modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.4 Simulations investigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.4.1 Control design results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.4.2
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
CHAPTER 6 CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORKS . 89
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.1 Conclusions .

Simulation results .

5.5 Summary .

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6.1.1

Scaling scheduling parameter LPV control with application to flutter
suppression of smart airfoil . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.1.2 Virtual control LTI with application to beam and ball system . . . . . . . . 90
6.1.3 Virtual control LPV with application to smart airfoil flutter suppression . . 90
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
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SOME LMI PRELIMINARIES . . . . . . . . . . . . . . . . . . . . 93
SCHEDULING PARAMETER MATRICES . . . . . . . . . . . . . 96
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APPENDICES .
APPENDIX A
APPENDIX B
BIBLIOGRAPHY .

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6.2 Future work .
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vii

LIST OF TABLES

[1, 2] Parameters used for the smart airfoil example.

Table 3.1:
Table 3.2: (cid:107).(cid:107) ∞ comparison; LPV-1 and LPV-2 controllers vs. nonlinear controller [2].

. . . . . . . . . . . . . . . 34

. . 38

Table 3.3: LPV-1 controller vs. LPV-0 controller: αL . . . . . . . . . . . . . . . . . . . . 39
Table 3.4: (cid:107).(cid:107) 2 comparison; LPV-1 and LPV-2 controllers vs. nonlinear controller [2] . . . 40
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Table 4.1: Ball balancing parameters.

Table 4.2: Virtual spring and damping constants for piece-wise linear function . . . . . . . 56

Table 4.3: Virtual nonlinear spring and damping constants for tangent function . . . . . . . 58

[1, 2] Parameters used for the smart airfoil example.

Table 5.1:
Table 5.2: (cid:107).(cid:107) ∞ comparisons between LPVVC, LPVN [3], NLC [2] and LPVR.
. . . . . . 79
Table 5.3: (cid:107).(cid:107) 2 comparisons between LPVVC, LPVN [3] and NLC [2]. . . . . . . . . . . . 79

. . . . . . . . . . . . . . . 77

viii

LIST OF FIGURES

Figure 3.1: The smart airfoil model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Figure 3.2: A saturation function.

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Figure 3.3:

(a) Airfoil groove with nonlinear springs and dampers at the boundaries; (b)
θ1 as function of moving mass position.

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. 21

Figure 3.4: Proposed method to reduce conservativeness. . . . . . . . . . . . . . . . . . . . 22

Figure 3.5: Controller steps.

. .

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Figure 3.6: Comparison between proposed LPV-1 control and [2]: αS.

. . . . . . . . . . . 37

Figure 3.7: Comparison between proposed LPV-1 control and [2]: αL.

. . . . . . . . . . . 38

Figure 3.8: Comparison between LPV-1 control and LPV-0 control: αL.

. . . . . . . . . . 39

Figure 3.9: Comparison between LPV-1 control and LPV-2 control: αL.

. . . . . . . . . . 40

Figure 3.10: Comparison between proposed LPV-2 control and LPV-2A control: αL.

. . . . 41

. 42

. . . 43

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

Figure 3.11: LPV-1 control at V=3.08 and V=3.4.
Figure 3.12: LPV-2A control performance at a defined velocity profile, ¯V ∈ [2.92, 3.4].
Figure 3.13: First entry of K(θ). .
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Figure 3.14: Second entry of K(θ).
Figure 3.15: Fifth entry of K(θ).

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Figure 4.1: A ball on a balancing beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Figure 4.2: A ball on a balancing beam with virtual springs and dampers.

. . . . . . . . . . 48

Figure 4.3: Virtual spring profile functions. . . . . . . . . . . . . . . . . . . . . . . . . .

. 53

Figure 4.4: Virtual damper profile functions.

. . . . . . . . . . . . . . . . . . . . . . . . . 53

Figure 4.5: Comparison of with virtual control and without virtual control (x0= 0.2m). . . . 55

ix

Figure 4.6: Comparison of with virtual control and without virtual control (x0= 0.25m).

. . 56

Figure 4.7: Comparison of constant spring/damper vs. piece-wise profile LVC1 (x0= 0.345m). 57

Figure 4.8: Effect of changing virtual spring constant for piece-wise profile (x0= 0.345m).

. 58

Figure 4.9: Effect of changing virtual damping constant for piece-wise profile (x0= 0.345m). 59

Figure 4.10: Comparison of piece-wise profile LVC1 and tangent profile TANVC1 (x0 =

0.4m).

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. 60

Figure 4.11: Effect of changing γ for tangent profile (x0= 0.4m).

. . . . . . . . . . . . . . . 61

Figure 4.12: Effect of changing β for tangent profile (x0= 0.4m).

. . . . . . . . . . . . . . . 62

Figure 5.1: Smart airfoil. .

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Figure 5.2: Virtual spring and damper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

Figure 5.3: Controller steps.

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Figure 5.4: LPVVC vs LPVN, I.C(0.2)

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. 81

Figure 5.5: LPVVC vs LPVN, I.C(0.6)

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. 83

Figure 5.6: LPVVC vs NLC .

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Figure 5.7: LPVVC vs LPVR .

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x

KEY TO SYMBOLS AND ABBREVIATIONS

Part (A): Symbols and abbreviations used for smart airfoil example.
b
c
CL
CLα
e

Typical section semi-chord (the length of the groove).
Typical section chord.
Lift coefficient.
dCL
dα
Elastic axis (e.a.) from elastic aerodynamic
center, aft positive.
Nondimensional e, e/b.

¯e
Fa(t) Aerodynamic load on the airfoil.

, (CLα

= 2π).

.

Mb2 .

(cid:113) Iα

Gravity constant.
g
Nondimensional g,
ω2
αb
Plunging displacement.
Nondimensional plunging displacement, h
b.
Mass moment of inertia about e.a. per unit span.
Spring constant for plunging mode.
Spring constant for pitching mode.
Mass per unit span of the typical section.
Mass of the control device.
Dynamic pressure, ρaV2
.
2
Radius of gyration about e.a.,
Time.
Control input.
Nondimensional control input,
Free stream airspeed.
Nondimensional free stream airspeed, V
Static unbalance, distance from e.a. to
inertia axis, aft positive.
Nondimensional static unbalance, xα
b .
Displacement, traveling distance of control mass m.
Nondimensional displacement, y
b.
Angel of incidence, positive nose up.
Small initial α.
Large initial α.
Mass ratio, m
M .
First scheduling parameter, a function of ¯y.
Second scheduling parameter, ¯V.
Third scheduling parameter, ¯V2.
Mass ratio of the typical section to the apparent mass, M

u
Mω2
αb

ωαb.

.

xi

g
¯g
h
¯h
Iα
Kh
Kα
M
m
qp
rα
t
u
¯u
V
¯V
xα

¯xα
y
¯y
α
αS
αL
β
θ1
θ2
θ3
µ

π ρab2 .

ωh Natural frequency of uncoupled plunging.
ωα Natural frequency of uncoupled pitching.
ρa Air density.
τ

Nondimensional time, ωαt.

Ball position w.r.t. beam center.
Earth gravity.
Moment of inertia of the beam.
Length of the beam.
Ball mass.

Part (B): Symbols and abbreviations used for beam and ball example.
d
g
I
L
m
k(α) Virtual spring constant.
c(α) Virtual damper constant.
α
τ

Beam rotation angle.
Control torque.

xii

CHAPTER 1

INTRODUCTION

1.1 Motivation and Vision

Saturation problem is inherent problem of the physical systems since actuator motion is always
limited due to physical constraints. Consequently, handling this problem in the linear control
theory and designing a linear controller is still a challenge. The problem in the literature was
handled by using the antiwindup schemes or controller switching techniques [4, 5, 6, 7], to mention
some but not to limit, which could cause substantial performance degradation. Nonlinear Model
Predictive Controllers (MPC) [8, 9, 10] were also used to resolve the hard constrained control
problems with significantly increased control complexity mainly due to the high computational
power required by nonlinear MPC control. Many other methods and techniques were proposed and
used in literature that either propose ad hoc techniques or posses the problems mentioned above.
Therefore, for practical control applications, there is a need to develop practical control strategies,
that can be implemented for real-time control, for systems with hard constraints to keep the system
states staying within hard constraints. To the best of the author knowledge, there is no linear
control design methods can handle the saturation problems efficiently, numerically sound, simple
implementation and in the same time can be applied to various problems and systems which are
LTI, LPV, continuous and discrete systems. This study presents such control design methods.

1.2 Contribution

The following are brief contribution of the dissertation:

1- A novel Linear Parameter Varying LPV modeling and control (LPV-1) is proposed, where
the idea of scaling scheduling parameter introduced and synthesized with the application to smart
airfoil for active flutter suppression control. The smart airfoil model is instrumented with a moving

1

mass that can be used to actively control the airfoil pitching and plunging motions. The LPV
modeling approach makes use of the moving mass position as a scheduling parameter, and the
hard constraint at the boundaries is imposed by proper selection of the parameter varying function.
A state-feedback based LPV gain-scheduling controller with the guaranteed H∞ performance is
proposed by utilizing the dynamics of the moving mass. In this study, both the position of the
moving mass and the free stream airspeed are considered as the scheduling parameters. The nu-
merical simulations demonstrate the effectiveness of the proposed LPV control architectures by
significantly improving the performance, while increasing the flutter speed and reducing the control
effort.

2- A modeling technique utilizing nonlinear springs and dampers (LPV-2) is proposed and
synthesized with the application to smart airfoil for active flutter suppression control, where the
nonlinear springs and dampers are added to both ends of the airfoil groove to confine the motion of
the moving mass. Also a state-feedback based LPV gain-scheduling controller with the guaranteed
H∞ performance is proposed by utilizing the dynamics of the moving mass. A simulation study
is presented shows the efficiency and effectiveness of the proposed method in reducing the control
effort while suppressing the airfoil flutter.

3- A modeling technique (LPV-2A) to reduce the number of scheduling parameters and hence
reduce the conservativeness of the controller is proposed and applied to the smart airfoil for flutter
suppression example.

4- A novel virtual hard constraints control is introduced to deal with saturation control problems
for linear time invariant (LTI) systems using virtual springs and dampers for the optimal control
design (such as linear quadratic regulation (LQR) based model-predictive-control (MPC)) and then
incorporating the virtual dynamics into the control input for real-time control. The idea is to
increase the control cost when these state(s) move close to the hard constraint(s) and saturation

2

limit(s), and the designed optimal control strategy will avoid moving these state(s) close to the
hard constraint area. The proposed method is applied to a ball and beam system to demonstrate
its feasibility and effectiveness, where virtual spring and damper are introduced during the control
design. The performance of conventional LQR state feedback control and the associated vir-
tual LQR-based MPC control are compared. Simulation results show effectiveness of the proposed
method with additional design parameters (virtual spring and damper) introduced for control design.

5- A shape functions represents the profile of the virtual spring and damper is proposed and used
in virtual LQR-based MPC control. The shape functions are used with the beam ball example to
give extra tuning parameters to tune the virtual spring and damper and hence as a tuning parameters
of the virtual controller. Introducing these shape functions opens the door for many other shape
functions to be proposed and used along with the virtual controller as an extra tuning parameters
and to show the flexibility and powerfulness of the proposed virtual control method.

6- A novel Linear Parameter Varying Virtual control (LPVVC) with the application to active
flutter suppression control of smart airfoil. The novel proposed virtual hard constraints control
is proposed and synthesized for the LPV systems. A state-feedback based LPV gain-scheduling
controller with the guaranteed H∞ performance is used. The virtual spring and damper is intro-
duced during the modeling and controller design. Simulation results and numerical investigation
is introduced to show the effectiveness and efficiency of the method.

1.3 Dissertation Outline

The dissertation is organized as following: Chapter two introduces some LPV preliminaries
that will be used for the subsequent chapters. Chapter three presents the novel scaling scheduling
parameter Linear Parameter Varying (LPV) control of active flutter suppression of smart airfoil
along with the scaling scheduling parameter with weighing function (LPV-1), scaling scheduling

3

parameter with nonlinear spring and damper (LPV-2) and reduced number scheduling parameter
modeling (LPV-2A). Chapter four introduces the novel proposed virtual control with application
to the beam and ball system. In addition, the proposed shape functions for the virtual spring and
damper are presented. Then in chapter five the novel proposed Linear Parameter Varying Virtual
control (LPVVC) with application to flutter suppression control of smart airfoil is investigated.
Chapter six summarizes the conclusions and recommendations for future works.

4

CHAPTER 2

PRELIMINARIES

This chapter will briefly presents some LPV modeling preliminaries which will be used for the
chapters three and five. Generally the polytopic representation of linear parameter varying system
is solved by convex optimization, where the system is parametrized and transfered from affine
representation to the multi-simplex representation to be solved using optimization packages [11]
and then utilizing some general equations [3, 12] the system matrices to be reconstructed in the
affine representations for the controller implementation. In doing so, some definitions and lemmas
are needed for this transformation and will be presented in this chapter.

2.1 Linear Parameter Varying (LPV) preliminaries

Generally the set of the natural numbers is referred by N and the set of the real numbers is

referred by R.
Definition 1: [13, 14] A unit simplex Θr is a polytope of r vertices defined as
ρi = 1, ρi ≥ 0, i = 1, 2, .., r

ρ = [ρ1, · · · , ρr] :

r


Θr =

(cid:41)

(cid:40)

i=1
since variable ρi changes inside a unit simplex Θr.
Definition 2: [15] The Cartesian product of a limited number of simplexes is a multi-simplex Θ.
For instance, if there are q simplexes, then

Θ = ΘN1 × ΘN2 × ΘN3 × · · · ΘNq =

ΘNi ,

q


i=1

where the dimension of multi-simplex Θ is the index N = (N1, . . . , Nq), such that ρ in Θ represents
(ρ1, ρ2, ..., ρq) and each ρi in ΘNi represents (ρi(1), ρi(2), ..., ρi(r)).
Definition 3: [13] (Homogeneous Polynomial): Given a unit-simplex ΓN of dimension N ∈ N, a
polynomial P(ρ) defined on RN of degrees g ∈ N is called homogeneous if all of its monomials
have the same total degree g.

5

Definition 4: [13, 14] (Γ-Homogeneous Polynomial): Given a Multi-simplex Γ of dimension
N ∈ N L, a polynomial P(ρ) defined on RN of degrees g ∈ N L is called Γ-homogeneous if for
any given integer i0, with 1 ≤ i0 ≤ L, and for any given ρi ∈ RNi, for 1 ≤ i (cid:44) i0 ≤ L, the partial
application ρi0 ∈ RNi0 (cid:55)→ P(ρ) is a homogeneous polynomial in ρi0.

For N ∈ N and g ∈ N, let KN(g) be the set of N-tuples obtained from all possible combinations
of N nonnegative integers ki, i = 1, ..., N with sum k1 + k2 + .... + kN = g. The number of elements
in KN(g) is given by

JN(g) =

(N + g − 1)!
g!(N − 1)!

Now, let N ∈ N L and g ∈ N L. In this case, the set KN(g) is defined as the Cartesian product
KN(g) = KN1(g1) × ... × KNL(gL). A Γ-homogeneous polynomial matrix A(ρ) of degree g ∈ N L
is defined as follows

A(ρ) = 


ρk Ak

k∈KN(g)

where Ak ∈ Rn×r is a matrix- valued coefficient and ρk the corresponding monomial which is
homogeneous of degree gi in each variable ρi, that is, ρk is given by

ρk = ρ

ki
kL
L , where ρ
i

, ki = (ki,1, ki,2, ..., ki,Ni) such that ki,1 +
k2
2 ...ρ
= gi. Note that the indices k = (k1, k2, ..., kL) are obtained by combining all of the

k1
1 ρ
ki,2 + ... + ki,Ni
N-tuples of the sets KNi(gi), for i = 1, ..., L, yielding a total of JN(g) monomials equal to

ki,1
i,1 ρ

ki,2
i,2 ...ρ

= ρ

ki,Ni
i,Ni

JN(g) =

JNi(gi)

L


i=1

6

Example 2.1.1 If ρ ∈ Γ4, then the following polynomial P(ρ) = 5ρ5
ρ2
1 ρ2 ρ3 ρ4 is a polynomial if degree g = 5

1 + 2ρ3

3 − ρ2
1 ρ1
2 ρ2

2 ρ2

4 + 3ρ3

1 ρ2
2 +

Example 2.1.2 Let ρ take values in the multi-simplex Γ of dimension N=(2,3), so the following
polynomial P(ρ) = ρ3
1,1 ρ1,2 ρ2,2 is a Γ-homogeneous poly-
nomial of degree g=(3,1).

1,1 ρ2,2 − 3ρ1,1 ρ2

1,2 ρ2,3 − ρ3

1,2 ρ2,1 + 6ρ2

Example 2.1.3 For example, a Γ-homogeneous polynomial of degree g=(2,3) defined over the
multi-simplex Γ of dimension N=(2,2) yields KN(g) = K2(2) × K2(3) = (2, 0),(1, 1),(0, 2) ×
(3, 0),(2, 1),(1, 2),(0, 3), with JN(g) = J2(2).J2(3) = 3.4 = 12. Over this multi-simplex, a Γ-
homogeneous matrix valued polynomial A(ρ) takes the form

1,1(ρ3
A(ρ) = ρ2
+ρ1,1 ρ1,2(ρ3
1,2(ρ3

+ρ2

2,1A((2,0),(3,0) + ρ2
2,1A((1,1),(3,0) + ρ2
2,1A((0,2),(3,0) + ρ2

2,1 ρ2,2A((2,0),(2,1) + ρ2,1 ρ2
2,1 ρ2,2A((1,1),(2,1) + ρ2,1 ρ2
2,1 ρ2,2A((0,2),(2,1) + ρ2,1 ρ2

2,2A((2,0),(1,2) + ρ3
2,2A((1,1),(1,2) + ρ3
2,2A((0,2),(1,2) + ρ3

2,2A((2,0),(0,3))
2,2A((1,1),(0,3))
2,2A((0,2),(0,3))

More details will be presented in chapter three and chapter five. Also, for more LPV preliminaries
one can consult [11, 12, 13, 14, 15] and references therein.

2.2 Linear Quadratic Regulator (LQR) preliminaries

In this work, a LQR-based MPC control is utilized in chapter two for the virtual control with
beam and ball system example. In this section, a brief preliminaries of Linear Quadratic Regulation
will be introduced. Most of the formulation are from [16].

7

The linear quadratic optimal control problem approached using Lagrangian and Hamiltonian

functions. Considering a discrete system described as following:

x(k + 1) = A(k)x(k) + B(k)u(k)

(2.1)
where, k = k0, ..., k f −1, x(k) is an (nth) order state vector, u(k) is an (rth) order control vector, and
A(k) and B(k) are matrices of appropriate dimensions. With initial condition x(k = k0) = x(k0)
and a general performance index (PI) with terminal cost as follows:

k f −1


[x(cid:48)(k)Q(k)x(k) + u(cid:48)(k)R(k)u(k)]

(2.2)

J = J(x(k0), u(k0), k0 =

1

2 x(cid:48)(k f )F(k f )x(k f ) +

1
2

k=k0

where, F(k) and Q(k) are symmetric positive semi-definite matrices with appropriate dimensions,
and R(k) is symmetric positive definite matrix. Generally the linear quadratic optimal control
problem is performed with six steps as following: starting with augmented performance index.
Then performing the Lagrangian function. after that, constructing the Euler-Lagrange Equation. In
the fourth step, introducing the Hamiltonian function. Then, finding the open loop optimal control.
Then finally, constructing the state and costate system. These steps will be explained more in details
next.
First step: Augmented Performance Index
Formulating the augmented cost function will be done by combining the cost function 2.2 with the
system 2.1 utilizing Lagrange multiplier function (costate function) λ(k + 1) as following:

Ja =

1

2 x(cid:48)(k f )F(k f )x(k f )
k f −1


+

1
2

k=k0

8

[x(cid:48)(k)Q(k)x(k) + u(cid:48)(k)R(k)u(k)] + λ(k + 1)[A(k)x(k) + B(k)u(k) − x(k + 1)]

(2.3)

Second step: Lagrangian Function
Define the Lagrangian function as

L(x(k), u(k), x(k + 1), λ(k + 1)) =

1

2 x(cid:48)(k)Q(k)x(k)

(cid:48)(k + 1)[A(k)x(k) + B(k)u(k) − x(k + 1)]

(2.4)

+

1

2u(cid:48)(k)R(k)u(k) + λ

Third step: Euler-Lagrange Equations
Then, applying Euler-Lagrange equation to the Lagrangian function (L) with respect to the vari-
ables x(k), u(k) and λ(k + 1). Then, one can get:

∂L(x∗(k), x∗(k + 1), u∗(k), λ∗(k + 1)

∂x∗(k)

∂L(x∗(k), x∗(k + 1), u∗(k), λ∗(k + 1)

∂u∗(k)

∂L(x∗(k), x∗(k + 1), u∗(k), λ∗(k + 1)

∂λ∗(k)

and the final (boundary ) condition will be

∂L(x∗(k − 1), x∗(k), u∗(k − 1), λ∗(k)

∂x∗(k)

= 0

(2.5)

∂L(x∗(k − 1), x∗(k), u∗(k − 1), λ∗(k)

∂u∗(k)

= 0

(2.6)

∂L(x∗(k − 1), x∗(k), u∗(k − 1), λ∗(k)

∂λ∗(k)

= 0

(2.7)

+

+

+

∂L(x∗(k − 1), x∗(k), u∗(k − 1), λ∗(k)

∂x(k)

∂S(x(k),(k)

∂x(k)

+

where from 2.2

δx(k)(cid:12)(cid:12)(cid:12)k=k f

k=k0

= 0

(2.8)

S(x(k f ), k f ) =

1

2 x(cid:48)(k f )F(k f )x(k f )

(2.9)

9

Fourth step: Hamiltonian
Proceeding to get the solution in terms of Hamiltonian which is defined as following

H(x∗(k), u∗(k), λ

∗(k + 1)) =

1

2 X∗(cid:48)(k)Q(k)x∗(k)

1

2u∗(cid:48)(k)R(k)u∗(k) + λ

+

∗(cid:48)(k + 1)[A(k)x∗(k) + B(k)u∗(k)]

(2.10)

Thus the 2.4 and 2.10 are related as

L(x∗(k), x∗(k + 1), u∗(k), λ

∗(k + 1) = H(x∗(k), u∗(k), λ

∗(k + 1)) − λ

∗(k + 1)x∗(k + 1)

(2.11)

then using 2.5 to 2.7 with 2.11 to get the required conditions for the extremum in terms of Hamil-
tonian as follows:

∗(k) =

λ

∂H(x∗(k), u∗(k), λ∗(k + 1)

∂x∗(k)

∂H(x∗(k), u∗(k), λ∗(k + 1)

∂u∗(k)

0 =

x∗(k) =

∂H(x∗(k − 1), u∗(k − 1), λ∗(k)

∂λ∗(k)

note 2.14 also can be written as

x∗(k + 1) =

∂H(x∗(k), u∗(k), λ∗(k + 1)

∂λ∗(k + 1)

10

(2.12)

(2.13)

(2.14)

(2.15)

For the system 2.1 and performance index 2.2 the relations 2.12 2.13 and 2.15 are gotten for the
state, costate and control, and can be written as

x∗(k + 1) = A(k)x∗(k) + B(k)u∗(k)

∗(k) = Q(k)x∗(k) + A(cid:48)(k)λ

∗(k + 1)

λ

0 = R(k)u∗(k) + B(cid:48)(k)λ

∗(k + 1)

(2.16)

(2.17)

(2.18)

Fifth step: Open-Loop Optimal Control

Then utilizing 2.18, the optimal control can be given as

u∗(k) = −R(k)−1B(cid:48)(k)λ

∗(k + 1)

(2.19)

Using the optimal control 2.19 and 2.16 one can get

x∗(k + 1) = A(k)x∗(k) − B(k)R−1(k)B(cid:48)(k)λ

∗(k + 1) = A(k)x∗(k) − E(k)λ

∗(k + 1)

(2.20)

where E(k) = B(k)R−1(k)B(cid:48)(k)
Six step: State and Costate System
The state and costate of 2.17 and 2.20 will be as

11

x∗(k + 1)
 =

λ∗(k)

A(k) −E(k)
 +

Q(k) A(cid:48)(k)

 x∗(k)

λ∗(k + 1)



(2.21)

2.2.1 Summary

This chapter introduced some preliminaries that will be useful for the subsequent chapters. Firstly,
some preliminaries on LPV modeling are introduced. Then, preliminaries on optimal control
problem formulation are introduced. For more general information on LPV one can consult
[11, 12, 13, 14, 15] and more specific details will be presented in chapters three and five. More
general details on LQR control problem can be found in [16] and more specific details on LQR-based
MPC will be presented in chapter four.

12

CHAPTER 3

NOVEL LINEAR PARAMETER VARYING MODELING AND FLUTTER

SUPPRESSION CONTROL OF A SMART AIRFOIL

3.1 Introduction

Active flutter suppression has been a critical research topic in aerospace applications for many
decades. Reducing the aircraft weight, improving the aerodynamic efficiency, and increasing the
critical flight speed continue to be the main thrusts for future aeronautical research, especially,
as the emerging air vehicle structures become highly flexible, active flutter suppression becomes
a key technical design requirement. There is a good body of design methods that are available
in literature concerning suppression of flutter phenomena. Passive methods have been used to
solve this problem for many years, however, these methods lead to increased aircraft mass, which
is undesirable; see references [17, 18, 19, 20]. On the other hand, active control techniques can
provide crucial and liable solutions that would increase the aircraft critical speed and suppress the
oscillations, while enhancing flight efficiency and performance.

There are many active control techniques in literature for flutter suppression. Using piezoelectric
actuation to control flutter was given by Han et al. [21], where experimental and numerical study
were conducted for active flutter suppression of a sweptback cantilevered lifting surface. The finite
element analysis, panel aerodynamics, and the minimum state space realization were used to develop
the equations of motion, which were then used for system analysis and control design by utilizing the
H2 and µ-synthesis control techniques, and subsequently the flutter suppression performance was
evaluated through wind tunnel tests. De Marqui et al. [22] introduced a flexible hanged arrangement
for flutter investigation with hard wings in a wind tunnel. The wing model was a rectangular shape
with NACA 0012 model and a rear edge surface mechanism control. They introduced aeroelastic
model to emulate the aeroelastic behavior of the corresponding system. A full state feedback control
was designed for this model to cancel flutter and retain the stability of the closed-loop system. Zhang

13

and Behal [23] introduced a continuous-time controller to suppress the aeroelastic oscillations of
the wing shape in an unsteady aerodynamic incompressible flow environment. The flap connection
force of the wing rear edge was used to represent the input and the pitching angle to represent the
output. The numerical simulation results demonstrated the effectiveness in canceling aeroelastic
vibrations in before and after flutter travel speed regions subjected to multiple external disturbances.
Additional active flutter suppression techniques included electro-hydraulic mechanical actuation of
control surface [24], reaction jets [25], and micro-flaps [26], to just name a few.

A special attention was given to Linear Parameter-Varying (LPV) modeling and control of an
[27] developed a system identification algorithm for an
airfoil. For example, Wingerden et al.
LPV aeroelastic system outfitted with rear edge control surfaces, where a factorization was used
to form a sensor-like depend on on the old inputs, outputs, and given aeroelastic data, and these
sensor-like were then used to calculate the state progression to form LPV aeroelastic matrices of
the system. As the algorithm can be utilized in a closed-loop arrangement, it can be used to flutter
suppression problems. Barker and Balas [28] designed gain-scheduled depends on two-parameter
LPV controllers for active flutter suppression of the Benchmark Active Control Technology (BACT)
wing section at NASA Langley Research Center. The BACT wing dynamics shifts notably relative
to the Mach number and dynamic pressure. The LPV gain-scheduled controllers depending on
two parameters incorporated these alternations as well as limits on the rate of change of Mach
number and dynamic pressure. The incorporation of the rate limits in the design procedure resulted
in better performance over a vast range of operational conditions when compared to a previously
designed gain-scheduled controller based on linear fractional transformation. Lau and Kerner
[29] utilized a regular linear system for the control of a slim airfoil under subsonic flow. The
two-dimensional airfoil was designed with three degrees of freedoms; flap, plunge, and pitch
angle, resulting a linear system of six dimension with a control input at the flap hinge. The
objective was to utilize feedback for stabilization the airfoil at its flutter speed or above it with many
control procedures. Chen et al. [30] developed LPV aeroservoelastic system utilizing adaptive
method with nonlinear aerodynamics. The LPV controller was able to suppress flutter with perfect

14

precision and validity, moreover, it provided a feasible tool for practical flutter flight tests. Balas et
al. [31] utilizing LPV and H∞ control design procedures to a body-freedom-flutter (BFF) airplane
designed aeroservoelastic controllers and compared the performances of the controllers in both time
domain and frequency simulations. Though the performance from the designed LPV controller
was satisfactory, it did not attain the same degree of performance as from the H∞ controllers. It
should be noted that, from the literature review, one can see that the conventional LPV modeling
considers the scheduling parameter as a physical parameter of the system matrices that changes
with time [12, 32, 33, 34], whereas the quasi-LPV model [35, 36, 37] considers systems with one
or more states in the system matrices.

In this chapter, we consider the smart airfoil model proposed by Swei and Jiang [2] (see Figure
3.1), to investigate the problem of active flutter suppression by utilizing the novel LPV modeling
and control techniques of scaling the scheduling parameter. The smart airfoil is a two-dimensional
airfoil with a groove along its chord that contains a moving light mass. The mass is allowed to move
along the groove, and through its coupling with airfoil aerodynamics, it can control and suppress
the pitching and plunging motion of the airfoil. The airspeed and position of the moving mass
are considered as scheduling parameters in the LPV model and the mass position is utilized as
a scheduling parameter in the control design. In this study, we propose to utilize the scheduling
parameter as part of a scaling factor for the smart airfoil model. In particular, position of the moving
mass is scaled and parametrized such that it is confined within the length of groove. Furthermore,
we propose another novel LPV modeling technique by implementing a pair of nonlinear springs
and dampers at both ends of the groove to gracefully prevent the moving mass from reaching the
hard boundaries. To the best of authors’ knowledge, the integration of scheduling parameter with
the scaled control effector and the parametrization of boundary conditions are novel approach that
have never been reported in the LPV control literature in the past. In this chapter, a number of
LPV control models is developed to best describe the effect of boundary constraints and also to
reduce the level of conservativeness. A full state feedback LPV gain-scheduling controller with
guaranteed H∞ output performance is proposed, in which the controller gains are obtained by

15

solving the numerically tractable Parameterized Linear Matrix Inequality (PLMI).

This chapter is organized as follows. Section 3.2 "LPV Modeling of a Smart Airfoil" presents
the nonlinear model of the smart airfoil, the baseline LPV model (LPV-0), the LPV model with
parameter scaling (LPV-1), the LPV model with nonlinear springs and dampers (LPV-2) at the
boundaries, and the LPV model that is derived from LPV-2 model with less number of scheduling
parameters (LPV-2A). Section 3.3 "LPV Controller Design" contains LPV problem formulation
and controller design. Comparisons and simulation results are presented in Section 3.4 "Numerical
Studies". Conclusions and future work are given in Section 3.5."Conclusion"

Figure 3.1: The smart airfoil model.

3.2 LPV Modeling of a Smart Airfoil

In this section, the mathematical model of the smart airfoil is presented. The linearized equations

of motion of the airfoil aeroservoelastic model can be written as [2]

m + M M xα

M xα

Iα



(cid:220)h(t)

(cid:220)α(t)

 +

Kh

0
0 Kα



h(t)

α(t)

 =

 0

mg

 y(t) + Fa(t)

m(cid:220)y(t) = mgα(t) + u(t)

(3.1)

16

where Fa(t) denotes the aerodynamic loading, m and M the moving mass and airfoil mass, re-
spectively, additional variables and parameters used in equations (3.1) and (3.1) can be found in
Figure 3.1 [2]. It is important to note that the position of moving mass y(t) in equation (3.1) can
be considered as the control input to the airfoil, whereas u(t) in equation (3.1) can be considered
as the control input to the moving mass m. The airfoil with such a control device is called "Smart
Airfoil" [2]. The following quasi-steady aerodynamic load model [38] for Fa(t) is adapted in this
study,

Fa(t) = P(cid:169)(cid:173)(cid:173)(cid:171)

−1

V
e
V



0
0

(cid:219)h(t)

(cid:219)α(t)

 +

0 −1

0

e



h(t)

α(t)



(cid:170)(cid:174)(cid:174)(cid:172)

P = qpcCLα .

(3.2)

Now, substituting equation (3.2) into equation (3.1) and performing nondimensionalization for all
the physical parameters, we obtain the nondimensionalized equations of motion for the smart airfoil
model as follows,

1 + β

¯xα





¯xα
¯r2
α

 +

 2 ¯V

µ
−2 ¯V ¯e
µ

(cid:220)¯h(τ)
(cid:220)α(τ)



0
0



(cid:219)¯h(τ)
(cid:219)α(τ)

 +



ω2
h
ω2
α
0

2 ¯V2
µ
−2 ¯V2 ¯e

µ

+ ¯r2
α



 ¯h(τ)

α(τ)

 =

 0

β ¯g

 ¯y(τ)

(cid:220)¯y(τ) = ¯gα(τ) + ¯u(τ)

(3.3)

where τ = ωαt is the nondimensional time. Note that to simplify the notation, the overhead "dot" in
equations (3.3) and (3.3) represents the time derivative with respect to τ. When the flutter occurs,
the plunging displacement h and pitching angle α are fed back in order to properly position the
moving mass m, which generates a damping effect to the airfoil, hence reducing the flutter vibration
and increasing the critical flutter speed.

17

3.2.1 LPV plant model: LPV-0

Rearranging equations (3.3) and (3.3) yields the following



 +

(cid:220)¯h(τ)
(cid:220)α(τ)

 −2¯r2 ¯V

qµ − 2 ¯V ¯e ¯xα
+ 2 ¯V ¯e(1+β)
2 ¯xα ¯V
qµ

qµ

qµ



0
0



 +

(cid:219)¯h(τ)
(cid:219)α(τ)



−¯r2
αω2
h
qω2
α
¯xαω2
h
qω2
α

−2¯rα ¯V2
qµ − 2 ¯V2 ¯e ¯xα
qµ
+ 2 ¯V2 ¯e(1+β)

+ ¯r2 ¯xα
q
+ ¯r2(1+β)

2 ¯xα ¯V2
qµ

qµ



α(τ)

 ¯h(τ)

 ¯y(τ)



q
− ¯xα βg
(1+β)β ¯g

q

q

=

(cid:220)¯y(τ) = ¯gα(τ) + ¯u(τ)

where q = −[¯r2

α(1 + β) − ¯x2

α]. Now, we define the augmented state x as

where

(3.4)

(3.5)

(3.6)

(3.7)

(3.8)

(cid:105)T

,

(cid:104)
(cid:105)T

x =

¯x, xu

.

¯x =

¯y, (cid:219)¯y

and xu =

(cid:104)
(cid:105)T
(cid:104) ¯h, α, (cid:219)¯h, (cid:219)α
 (cid:219)x(t) = A(θ(t))x(t) + Bu(θ(t)) ¯u(t)

y(t) = C(θ(t))x(t) + Du(θ(t)) ¯u(t)

Then, equations (3.4) and (3.5) can be described in the state space representation as follows,

where y(t) is the controlled output, and the system matrices (A(θ), Bu(θ), C(θ), Du(θ)) are affine in
θ and given by

18



A(θ) =

0
0
¯r2
αω2
h
qω2
α
− ¯xαω2
h
qω2
α
0
0

(cid:104)



,

0
0

0

0

1

a66

0
1

0
0
− ¯xα β
0
q
0 β(1+β)
0
0
0

a65

q

,

Du(θ) = 0 ,

0
0
qµ − ¯r2
α ¯xα
q
α(1+β)
− ¯r2

q

2¯r2
+ 2θ3 ¯e ¯xα
αθ3
qµ
qµ − 2θ3 ¯e(1+β)
−2 ¯xαθ3

qµ

1
0
2¯r2
+ 2θ2 ¯e ¯xα
αθ2
qµ
−2 ¯xθ2
qµ − 2θ2 ¯e(1+β)

qµ

qµ

0
¯g

(cid:105)(cid:48)

0
0

0 0 0 0 1 0

(cid:105)

(cid:104)

,

C(θ) =

Bu(θ) =

0 0 0 0 0 b6

(3.9)
where θ := [θ1, θ2, θ3]T, and θ1 is the scheduling parameter utilized to properly scale and constrain
the position of the moving mass at the boundaries, θ2 and θ3 are the scheduling parameters
representing, respectively, the normalized airspeed ¯V and its square ¯V2. In this study, we consider
θ2 ∈ [0.5, 2.92], hence, θ3 ∈ [0.25, 8.52]. Furthermore, the parameters a65 and a66 are used to
represent nonlinear spring and damping effects at the boundaries. It should be emphasized that θ2
and θ3 are not independent parameters, therefore inevitably it would introduce conservativeness in
the modeling process. However, in Section 3.2.4 an approximate approach is proposed.

To formulate the baseline LPV-0 plant model, we set the parameters b6 = 1 and a65 = 0, a66 = 0.

This LPV-0 model will be used to assess the baseline closed-loop system performance.

3.2.2 LPV control design model: LPV-1

The proposed LPV-1 model is based on the LPV-0 model described in equations ( 3.8) and ( 3.9)
with a65 = 0, a66 = 0, but b6 = θ1, where θ1 is a function of ¯y, i.e. θ1 = f( ¯y), which is used to
constrain the moving mass. In particular, θ1 is devised such that

19



θ1 = 1 ,
θ1 ¯y = 0.35 ,
θ1 ¯y = −0.35 ,

if ¯y ∈ [−0.35 , 0.35]
if ¯y > 0.35
if ¯y < −0.35

It is clear from the first condition above that when ¯y ∈ [−0.35 , 0.35] and θ1 = 1, the LPV-1 model
will be equivalent to the LPV-0 model. The purpose of the second and third conditions is to impose
a constraint on the moving mass when it travels beyond ±0.35, by modulating the control gains
through θ1. The variation of θ1 as function of the position of the moving mass is illustrated in Figure
3.2. To constrain the mass at ±0.5 the damping function must be active starting at ±0.35. This
approach may introduce a slight conservativeness to the control design, it is intuitively appealing
and proved to be effective.

Figure 3.2: A saturation function.

3.2.3 LPV control design model with nonlinear springs and dampers: LPV-2

The second proposed LPV control design model contains nonlinear springs and dampers at both
ends of the groove. They are to gradually stop and reverse the motion of the control mass. Figure
3.3 shows an illustration of the proposed boundary constraints with nonlinear springs and dampers,
and as indicated earlier, the scheduling parameter θ1, which is a function of ¯y, is implemented to
scale the control effector.

20



θ1 = 0 ,
θ1 = ¯y − 0.35 ,
θ1 = −0.35 − ¯y ,

if ¯y ∈ [−0.35 , 0.35]
if ¯y > 0.35
if ¯y < −0.35

It should be emphasized that the choice of Kdamp and Cdamp depends on the location from which
the spring and damping effect are expected to engage, i.e. the closer to ±0.5, the higher the Kdamp
and Cdamp value.

(a)

(b)

Figure 3.3: (a) Airfoil groove with nonlinear springs and dampers at the boundaries; (b) θ1 as
function of moving mass position.

3.2.4 LPV control design model with two scheduling parameters: LPV-2A

A third LPV control design model is modified from LPV-2, in which the parameter θ3 is approx-
imated as a function of θ2, hence the number of scheduling parameters is reduced to two. Since
θ2 and θ3 represent the velocity and its squared value, they are not independent and will introduce
conservativeness due to over bounding the parameters. Kwiatkowski et al. [39] suggested a method
to reduce conservativeness by taking the intersection between the two polytopes. In this section,
we propose to approximate θ3 as a linear function of θ2 with slope β1, as shown in Figure 3.4.

21

Therefore, the resulting LPV model, denoted as LPV-2A, contains two scheduling parameters: θ1
and θ2.

Figure 3.4: Proposed method to reduce conservativeness.

Recall the affine matrix A(θ) is of the form

A(θ) = A0 + θ1A1 + θ2A2 + θ3A3

From Figure 3.4, θ3 can be approximated as

θ3 = θ30 + θ2 tan(β1)

Substituting equation ( 3.11) into equation ( 3.10) yields

(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)

A(θ) = A0 + θ30A3

A0new

+θ1A1 + θ2 (A2 + tan(β1)A3)

(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)

A2new

Therefore, the affine matrix A(θ) in the LPV-2A model can be described by

A(θ) = A0new + θ1A1 + θ2A2new

22

(3.10)

(3.11)

(3.12)

(3.13)

3.3 LPV Controller Design

In this section, we develop state feedback based gain-scheduling LPV controllers for LPV-0,
LPV-1, LPV-2, and LPV-2A models. The goal of LPV control design is to achieve active flutter
suppression with guaranteed H∞ output performance at varying airspeed, through modulating the
dynamics of moving mass in the smart airfoil model, subject to actuation constraints and external
disturbances. In particular, we consider the following LPV systems,

 (cid:219)x(t) = A(θ(t))x(t) + Bu(θ(t)) ¯u(t) + Bw(θ(t))w(t)

y(t) = C(θ(t))x(t) + Du(θ(t)) ¯u(t) + Dw(θ(t))w(t)

(3.14)

where w(t) denotes the disturbance input.

3.3.1 Problem formulation

The LPV model described in equation (3.14) is assumed to have affine parameters. For instance,
matrix A(θ) can be represented by

A(θ(t)) = A0 +

θi(t)Ai

(3.15)

q


i=1

where A0 and Ai are constant matrices, and q denotes the number of scheduling parameters.
Scheduling parameter vector θ(t) is defined as

and each θi is bounded by

θ(t) = [θ1(t), θ2(t), θ3(t), ..., θq(t)]T

η1,i ≤ θi ≤ η2,i

(3.16)

(3.17)

where η1,i and η2,i denote the upper and lower bound. In addition, these scheduling parameters
also have the rate bound given by

(3.18)

µ1,i ≤ (cid:219)θi ≤ µ2,i

23

The proposed LPV control law is a state feedback controller as

¯u(t) = K(θ)x(t)

(3.19)

where K(θ) asymptotically stabilizes the closed-loop system subject to the H∞-norm constraint
from the exogenous input w to the measured output y over the entire parameter variation range. Sub-
stituting controller equation (3.19) into equation (3.14) yields the closed-loop system representation
given by

where

 (cid:219)x(t) = Ac(θ)x(t) + Bw(θ)w(t)

y(t) = Cc(θ)x(t) + Dw(θ)w(t)

Ac(θ) = A(θ) + Bu(θ)K(θ)
Cc(θ) = C(θ) + Du(θ)K(θ)

(3.20)

(3.21)

It should be noted that though (A(θ), Bu(θ), C(θ), Du(θ)) matrices are all affine in θ, the gain-
scheduling feedback K(θ) is not. Before proceeding further, we introduce the following definitions.

(cid:40)

r


(cid:41)

Definition 1: [13, 14] A unit simplex Θr is a polytope of r vertices defined as

Θr =

ρ = [ρ1, · · · , ρr] :

ρi = 1, ρi ≥ 0, i = 1, 2, .., r

i=1
since variable ρi changes inside a unit simplex Θr.
Definition 2: [15] The Cartesian product of a limited number of simplexes is a multi-simplex Θ.
For instance, if there are q simplexes, then

Θ = ΘN1 × ΘN2 × ΘN3 × · · · ΘNq =

ΘNi ,

q


i=1

where the dimension of multi-simplex Θ is the index N = (N1, . . . , Nq), such that ρ in Θ represents
(ρ1, ρ2, ..., ρq) and each ρi in ΘNi represents (ρi(1), ρi(2), ..., ρi(r)).

24

3.3.1.1 Transferring from affine to multi-simplex

To formulate a convex control design problem, we first need to perform a transformation on the
system matrices, from the affine parameter space θ to the multi-simplex convex space Θ. Each
affine scheduling parameter θi is transferred over a unit simplex ρi of two vertices as follows [40],

θi + η2,i
2η2,i

−→ θi = 2η2,i ρi(1) − η2,i
ρi(1) =
η2,i − θi
ρi(2) = 1 − ρi(1) = 1 − θi + η2,i
2η2,i
2η2,i
ρi = (ai(1), ρi(2)) ∈ Θ2, ∀ i = 1, 2, · · · , q .

=

(3.22)

With this transformation, the affine scheduling parameters in equation (3.14) can be converted into
a system with multi-simplex parameters, where the multi-simplex variables are defined as

ρ ∈ Θ = Θ2 × Θ2 × · · · Θ2

(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)

.

q

Moreover, for three scheduling parameters, i.e. q = 3 as the case of our current consideration, the
homogeneous term in the multi-simplex variables can be written as

ρ1 = (ρ1(1), ρ1(2)), ρ2 = (ρ2(1), ρ2(2)), ρ3 = (ρ3(1), ρ3(2)).

Hence, by utilizing the above transformation, the LPV system equation (3.14), which was de-
scribed in affine parameter space θ, can be equivalently described in multi-simplex convex space
Θ as follows,

 (cid:219)x(t) = A(ρ)x(t) + Bu(ρ) ¯u(t) + Bw(ρ)w(t)

y(t) = C(ρ)x(t) + Du(ρ) ¯u(t) + Dw(ρ)w(t)

(3.23)

25

Furthermore, the proposed state feedback gain scheduling controller has the form

¯u(t) = K(ρ)x(t)

(3.24)

where K(ρ) is to be determined. Therefore, the closed-loop system representation in multi-simplex
space is given by

where

 (cid:219)x(t) = Ac(ρ)x(t) + Bw(ρ)w(t)

y(t) = Cc(ρ)x(t) + Dw(ρ)w(t)

Ac(ρ) = A(ρ) + Bu(ρ)K(ρ)
Cc(ρ) = C(ρ) + Du(ρ)K(ρ)

(3.25)

(3.26)

In addition, the rate changes of the scheduling parameters in the unit simplex can also be described
by

(cid:219)ρi(1) + (cid:219)ρi(2) = 0,

i = 1, 2, 3 .

(3.27)

From equations (3.18) and (3.22) we can derive the rate bounds between affine scheduling

parameters and multi-simplex variables, and they are given by

µ1,i
2η2,i

≤ (cid:219)ρi(1) ≤ µ2,i
2η2,i

(3.28)

where µ2,i and µ1,i are rate bounds given in equation (3.18), and η2,i is the upper bound of θi given
in equation (3.17). Furthermore, from equation (3.27), we obtain that (cid:219)ρi(2) = − (cid:219)ρi(1).

3.3.1.2 H∞ control problem

The proposed LPV control problem can be stated as follows: Design a gain-scheduling state
feedback controller of the form equation (3.24), such that for given γ∞ > 0 and any given pair
(ρi, (cid:219)ρi); i = 1, 2, 3, the closed-loop system equation (3.25) is stabilized and satisfies the following

26

H∞ performance constraint,

sup
(ρi, (cid:219)ρi)

sup
w∈l2
w(cid:44)0

||z||2
||w||2

< γ∞

(3.29)

To synthesize the H∞ control problem, we utilize the following theorem.

Theorem 1 [12]: For any given pair (ρi, (cid:219)ρi), i = 1, 2, 3, the closed-loop system equation (3.25)
will be stabilized by the gain scheduling controller K(ρ) = Z(ρ)G(ρ)−1 with guaranteed H∞
performance level γ∞, if there exist a scalar 
 > 0, a continuously differentiable symmetric and
positive-definite matrix P(ρ), and matrices G(ρ) and Z(ρ) satisfying the following parameterized
linear matrix inequality (PLMI),





∗
∗
∗
∗
−I
∗
Dw(ρ)(cid:48) −γ2∞I

< 0

Q(ρ)

∗

P(ρ) − G(ρ) + 
(A(ρ)G(ρ) + Bu(ρ)Z(ρ))(cid:48)

−
(G(ρ)G(ρ)(cid:48))

C(ρ)G(ρ) + Du(ρ)Z(ρ)


C(ρ)G(ρ) + 
 Du(ρ)Z(ρ)

Bw(ρ)(cid:48)

0r×n

(3.30)
where ∗ denotes symmetric entry and Q(ρ) = A(ρ)G(ρ) + Bu(ρ)Z(ρ) + G(ρ)(cid:48)A(ρ)(cid:48) + Z(ρ)(cid:48)Bu(ρ)(cid:48) +
∂P(ρ)
∂ ρ

(cid:219)ρ.
It should be noted that though the H∞ control problem is presented in multi-simplex convex
space Θ, the actual gain-scheduling controllers are implemented in affine θ domain. Roughly
speaking, in order to compute the affine feedback gain matrix described in equation (3.19), we first
need to utilize Theorem 1 to synthesize the solutions Z(ρ) and G(ρ) at the vertices of multi-simplex
space Θ, followed by the inverse transformation process [12] to convert the multi-simplex solutions
into affine representations Z(θ) and G(θ). The detail procedure is given below.

For three scheduling parameters, the parameters Z(θ) and G(θ) in affine θ domain can be

27

described by

Z(θ) = Z0 + θ1Z1 + θ2Z2 + θ3Z3,
G(θ) = G0 + θ1G1 + θ2G2 + θ3G3

(3.31)

where (Z0, G0) and (Zi, Gi), i = 1, 2, 3, are constant matrices to be determined in the sequel.
Moreover, given Z(θ) and G(θ), the affine gain matrix K(θ) in equation (3.19) will then be given by

K(θ) = Z(θ)G−1(θ) .

(3.32)

Therefore, it is clear that the control gain K(θ) is not an affine function of θ. For three scheduling
parameters, there are 8 polytopic solutions for Z(ρ) and G(ρ) in equation (3.30). Let cZi jk and
cGi jk, i, j, k = 1, 2, to denote these solutions, then Z(ρ) and G(ρ) can be represented as

X(ρ) = ρ1(1)ρ2(1)ρ3(1)cX111 + ρ1(1)ρ2(1)ρ3(2)cX112
+ρ1(1)ρ2(2)ρ3(1)cX121 + ρ1(1)ρ2(2)ρ3(2)cX122
+ρ1(2)ρ2(1)ρ3(1)cX211 + ρ1(2)ρ2(1)ρ3(2)cX212
+ρ1(2)ρ2(2)ρ3(1)cX221 + ρ1(2)ρ2(2)ρ3(2)cX222
= X(θ)

(3.33)

where (X, cX) denote (Z, cZ) or (G, cG). Hence, it follows from the inverse transformation process
given in [12] that

2


1
64

j1=1
1

64η2,i

X0 =

Xi =

2

2


j2=1

2

2


j3=1

2


j1=1

j2=1

j3=1

cX j1 j2 j3,

(−1)ji+i cX j1 j2 j3

(3.34)

where the constant matrices (X0, Xi) denote (Z0, Zi) or (G0, Gi), and cX j1 j2 j3 denotes the
polytopic solutions cZ j1 j2 j3 or cG j1 j2 j3 from Theorem 1. Now, substituting (Z0, Zi) and (G0, Gi)
into equation (3.31) yields Z(θ) and G(θ), hence the feedback gain matrix K(θ) can be obtained from
equation (3.32). The inverse transformation formulas for any number of scheduling parameters

28

q > 3 can be found in [12], [13], and [40]. For the purpose of demonstration, a list of (Z0, Zi) and
(G0, Gi) matrices for various LPV models are presented in the Appendix.

It should be noted that PLMI is an infinite dimensional linear matrix inequality (LMI), which
is in general difficult to solve. Many efficient solvers are available in dealing with such problem;
for instance, see references [41, 42, 43, 44]. This work adapts the relaxation method for PLMIs
relaxation [13], where it treats PLMIs with multi-simplex parameters. Numerically, the ROLMIP
package [45] along with YALMIP [11] using SeDuMi [46] solver are used to solve the convex
optimization problem.

Generally, the controller solution process for the LPV control synthesis in this chapter considers
seven steps. Firstly, scaling scheduling parameter introduced. Secondly, change of variables from
affine representation domain to the multi-simplex (convex) domain is performed. Thirdly, modeling
the rates of variation of scheduling parameters in convex domain. In the fourth stage, performing
the PLMI synthesis conditions. The fifth stage is to perform PLMI relaxation to transfer the infinite
dimension PLMI into finite dimension LMI to be solved using optimization package. Then the six
stage involve to utilize inverse transformation to transfer from the multi-simplex representation into
the affine representation. Then finally the seven stage is the controller implementation. The steps
are shown in the following diagram Figure 3.5.

3.3.2 PLMI relaxation

As mentioned above, that the synthesis condition 3.30 is a parameter dependent LMI (PLMI)
which is an infinite dimension LMI and needs to relaxed to finite dimension to be solved in
convex optimization. There are many relaxation methods in the literature to relax a PLMI such
as multi-convexity technique [47], SOS [48, 49], semidefinite programming relaxations for the
generalized problem of moments [50], griding method [51] vertex method [52, 53] dilated LMI
[54] multi-convexity [55] slack variable [44] and many other methods. Most of them either provided
conservative solution, large number of divisions to reach a good results and/or depend on constant

29

Figure 3.5: Controller steps.

Lyapunov functions that prevents using parameter dependent Lyapunov function which will lead for
more conservativeness solution. For that, the method of [13] is used in this work which overcomes
the shortcomings of the mentioned methods. This section will present the solution paradigm of the
method of handling the PLMI and gives general examples for more explanation and most of the
formulation are from [13]. Consider the PLMI can be written as

m


Ni


∂P(ρ(t))
∂ ρi j(t) (cid:219)ρi j(t) < 0

A(ρ(t))T P(ρ(t)) + P(ρ(t))A(ρ(t)) +

(3.35)
for all ρ(t) ∈ Λ and (cid:219)ρ ∈ Ω = Ω1 × ...m. where ρi(t), i = 1, ..., m are independent from each other
this applied also to their derivatives.

i=1

j=1

(cid:219)ρi1(t) + (cid:219)ρi2(t) + ... + (cid:219)ρiNi(t) = 0

(3.36)

Let G(i) denotes the i-th column of matrix G. The sets Ωi, i = 1, ..., m are defined as

30

Mi


l=1

Ωi = δ ∈ RNi : δ =

For example [13], if

Ni


l=1

Hi(l, j) = 0, j = 1, ..., Ni,

ηi ∈ ΛMi

(3.37)

ηi,lH(l)

,

i

− 1 ≤ (cid:219)ρ11(t) ≤ 1,−1 ≤ (cid:219)ρ12(t) ≤ 1,−2 ≤ (cid:219)ρ13(t) ≤ 2

(3.38)

then the solution of 3.36 by 3.38 will give results as

(1, 1,−2),(−1,−1, 2),(1,−1, 0),(−1, 1, 0)

the convex combination of these results will give



in other form

1
1
−2



−1
−1
2

 η11 +


1 −1
1 −1 −1
−2
0

2



−1
1
0

 η14,

ηi ∈ Λ4

(3.39)

H1 ∈ RN1×M1

(3.40)

1
−1
0

 η12 +


1 −1
1
0




 η13 +


,

η11

η12

η13

η14

where N1=3 is the number of parameters in Λ1 and M1=4 is the number of solutions of 3.36 by
3.38. Notice that the number M1 is not known a priori, where it is determined by the number of
extremal solutions. But it could be different from M1=4, if distinct bounds were considered in 3.38.
for instance, if in 3.38 one considers −1 ≤ (cid:219)ρ13(t) ≤ 1, the number of extremal solutions will be
M1=6.

31

Since ρ(t) ∈ Λ implies (cid:219)ρ(t) ∈ Ωfor all t0 and using the definition of a generic ρi j(t) belonging to
Ωi, inequality 3.35 can be rewritten as

m


Ni


i=1

j=1

Mi


l=1

∂P(ρ(t))
∂ ρi j(t)

A(ρ(t))T P(ρ(t)) + P(ρ(t))A(ρ(t)) +

ηilHi(j, l) < 0

(3.41)

For Λ-homogeneous functions P(ρ) and A(ρ) of partial degrees g = (g1, ..., gm) and r = (r1, ..., rm)
respectively, the total degree of the first two terms A(ρ)T P(ρ) + P(ρ)A(ρ) will be ¯g = (g1 +r1, g2 +
r2, ..., gm+rm). Then, the next step is to homogenize the third term in ρ. The general expression for
the derivative of the Lyapunov matrix P(ρ) with respect to the i−th component of the multi-simplex,
i = 1, ..., m and then with respect to its j − th component, j = 1, ..., Ni is given by

= 


k∈KN(g)

∂P(ρ)
∂ ρi j

ki j ρ

k1
1 ...ρ

ki j−1
ki1
i1 ....ρ
i j

....ρ

km
m Pk

kiNi
iNi

.....ρ

= 


k∈KN(g−ei|m)

ρk((k + ei|m ⊕ ej|Ni

)i j Pk+ei|m⊕ej|Ni

)

(3.42)

where ei|m is the vector of dimension m with zero components, except 1 in the i − th position. To
fit (on ρ) with the partial degrees ¯g, the following homogenization is necessary:

m


i=1

(ρi1 + .... + ρiNi)ri+1

Ni


j=1

m


Ni


i=1

j=1

∂P(ρ)
∂ ρi j

=

(cid:32) 




(cid:18)

ρk

k∈KN(g+r)

ˆk∈KN(r+ei|m)

ˆk≤k
(k − ˆk + ei|m ⊕ ej|Ni

×(ri + 1)!
π( ˆki)

)i j Pk− ˆk+ei|m⊕ej|Ni

(cid:19)(cid:33)

(3.43)
where π(ki) = (ki1!)(ki2!)(kiNi!). Now, the third term of 3.41 must be homogenized to become
multi-affine on η. This is done as follows

32

(ηp1 + ... + ηpMp)

Mi


l=1

ηilHi(i, l) =

M1


Mi


...

....

Mm


p1=1

pi=1

pm=1

m


p=1
p(cid:44)i

η1p1 ...ηipi ....ηmpm Hi(j, pi)

(3.44)

Regarding 3.43 and 3.44, the third term in the left-hand side of 3.41 will be as following

m


Ni


i=1

j=1

Mi


l=1

∂P(ρ)
∂ ρi j

ηilHi(j, l) =

M1


P1=1

...

Mm


Pm

(ri + 1)!
π( ˆki) ×

Notice that

(cid:32) 


k∈KN(g+r)

m


Ni


i=1

j=1

ρk

η1p1 ...ηmpm

(cid:18)

(k − ˆk + ei|m ⊕ ej|Ni




(cid:19)

ˆk∈KN(r+ei|m)

ˆk≤k
)i j Pk− ˆk+ei|m⊕ej|Ni

Hi(j, pi)

(cid:33)

(3.45)

(3.46)

m


p=1

(ηp1 + ... + ηpMp)A(ρ(t))T P(ρ(t)) + P(ρ(t))A(ρ(t))

M1


=

Mi


Mm


...

....

p1=1

pi=1

pm=1

η1p1 ...ηipi ....ηmpm × A(ρ(t))T P(ρ(t)) + P(ρ(t))A(ρ(t))

then finally, 3.41 can be tested since all terms have the same partial degrees on both ρ and η.
The following theorem presents LMI relaxations of increasing precision for the problem of robust
stability analysis of matrix A(ρ) with parameters ρ ∈ Λ, (cid:219)ρ ∈ Ω.

Theorem 2 [13] If Λ a multi-simplex of dimension N = (N1, ..., Nm). The Λ-homogeneous poly-
nomial matrix A(ρ) of partial degrees r = (r1, ..., rm) is robustly stable ∀ρ ∈ Λ, (cid:219)ρ ∈ Ω if there
exist g = (g1, ..., gm), k ∈ KN(g) and matrices Pk ∈ Sn such that Pk > 0n,∀k ∈ KN(g) and for all
(i1, ..., im) ∈ 1, ...., M1 × ... × 1, ..., Mm the following LMIs are verified

33

Tk = 




˜k∈KN(r)
˜k≤k

ˆk∈KN(r+ei|m)

ˆk≤k

AT
˜k

Pk− ˜k

+ Pk− ˜k A˜k

+ Ξk < 0n,∀k ∈ KN(g + r)

(3.47)

(cid:18)

(k − ˆk + ei|m ⊕ ej|Ni

)i j Pk− ˆk+ei|m⊕ej|Ni

(cid:19)

Hi(j, pi)

(3.48)

(ri + 1)!
π( ˆki) ×

where

Ξk =

m


Ni


i=1

j=1

for the proof one can consult [13].

3.4 Numerical Studies

The smart airfoil parameters used in this study are given in Table 3.1.

Table 3.1: [1, 2] Parameters used for the smart airfoil example.

(m)
Nondimensionalized

Parameter Value Units
b (m)
0.127
¯e
0.35
¯r2
0.388 Nondimensionalized αS
α
¯xα
0.25
Nondimensionalized αL
airspeed ¯V 2.92
Nondimensionalized
(rad/sec)
64.1
ωα
55.9
(rad/sec)
ωh

initial (cid:219)α
β
µ

Parameter Value Units
initial ¯h
initial (cid:219)¯h

0
0
0.01
0.6
0
0.01
152

Nondimensionalized
Nondimensionalized
(rad)
(rad)
(rad/sec)
Nondimensional
Nondimensional

In addition, in equation (3.14) we choose Bw = [0 0 0 0 0 0.1]T and Dw = 0. The baseline
LPV-0 controller and the proposed LPV-1 and LPV-2 controllers are designed based upon their
corresponding models, with γ∞ = 0.0055 in Theorem 1. For the purpose of comparison, the
scheduling parameter θ2 is varied from 4 m/s to 23.8 m/s, and θ3 from 16 m2/s2 to 566.44 m2/s2.
Note that the LPV-0 controller is designed using the (unsaturated) LPV-0 model. Hence, the
resulting LPV-0 controller is in the form of equation (3.32) without the θ1 terms in equation (3.31).
In developing two-scheduling parameters LPV-2A control model, we choose β1 = 73.3◦ in
Figure 3.4 and θ30 = −1.2. The scheduling gain matrix K(θ) = Z(θ)G(θ)−1 and the affine

34

parameters Z(θ) and G(θ) are given by

Z(θ) = Z0 + θ1Z1 + θ2Z2,
G(θ) = G0 + θ1G1 + θ2G2

(3.49)

where (Z0, G0) and (Zi, Gi), i = 1, 2, are computed by following the similar procedure presented
in Section 3.3.1.2. The design of LPV-2A control model is to reduce the number of scheduling
parameters from three to two, by representing θ3 in terms of θ2, hence reducing the model conser-
vativeness. Note that γ∞ is a measure of robustness performance. The minimum achievable γ∞
for the LPV-2 control is 0.00029, whereas for the LPV-2A control it is 0.0002, a 45% performance
improvement.

Simulation results are compared with those obtained from the nonlinear controller presented
in [2]. Figure 3.6 shows a comparison between the proposed LPV-1 controller and the nonlinear
controller [2], with small initial pitch angle, i.e. αS = 0.01 rad.
It is clearly can be seen that
the proposed LPV-1 control can significantly improve the overall closed-loop performance. As
mentioned, the control mass m is confined to move within the groove between -0.5 and 0.5. In
the proposed LPV-1 control design, the scheduling parameter θ1 is used to constrain the mass
movement. Figure 3.7 shows the same comparisons but with larger initial pitch angle at αL = 0.6
rad. It is apparent that the nonlinear controller [2] cannot handle the large angle of attack, whereas
the proposed LPV-1 control handles this condition effectively, with fast convergence and small
control effort.

Figure 3.8 shows a comparison between the LPV-1 control design and the baseline LPV-0
design. Recall that in LPV-0 control design, no position limitation is imposed on the moving mass
m. Therefore, as can be seen in Figure 3.8, for LPV-0 controller to be effective in suppressing
airfoil vibration, the control mass needs to travel much larger displacement. Furthermore, Figure
3.9 shows a comparison between LPV-1 and LPV-2 controllers when the initial pitch angle is large.
Both controllers, as shown, are able to effectively suppress the airfoil vibration, though the LPV-2
controller, with nonlinear springs and dampers imposed at the boundaries, produces less aggressive
mass motion. Moreover, as shown in Figure 3.10, the mass motion of LPV-2A control is less

35

aggressive compared to that of LPV-2 control. Note that Figure 3.6 to Figure 3.10 are generated
with a fixed airspeed at ¯V = 2.92 (or 23.8 m/s).

For flutter analysis, Figure 3.11 shows the closed-loop system responses with LPV-1 control at
varying airspeed. As shown, the system is unstable when ¯V = 3.4. However, Figure 3.12 shows
the responses of LPV-2A control with starting airspeed of ¯V = 2.92, then increasing to ¯V = 3.08
(or 25.1 m/s) at τ = 250 and to ¯V = 3.4 (or 27.7 m/s) at τ = 450. As shown, the LPV-2A control
can effectively suppress the airfoil vibration beyond the previously identified flutter airspeed of
¯V = 2.96 (or 24.1 m/s) [2].

A quantitative study of the results, in terms of (cid:107).(cid:107)∞ and (cid:107).(cid:107)2 norms of the signals, are presented
in Tables 3.2, 3.3, and 3.4. Table 3.2 shows the comparison between the (cid:107).(cid:107)∞ norms of the proposed
design techniques and [2] with small and large initial conditions. It is clear that the proposed design
techniques render more than 50% improvement for most of the system performance related norms,
with nearly 50 time less control effort. A comparison of (cid:107).(cid:107)∞ norms between LPV-1 control
and baseline LPV-0 control is presented in Table 3.3, which reveals the benefit of modeling the
boundary conditions. Table 3.4 shows a comparison of (cid:107).(cid:107)2 norms between the proposed design
techniques and the one in [2] with small and large initial conditions. It is clear that the proposed
LPV design techniques provide much improved closed-loop responses, with faster convergent rate
as indicated in the percentage improvement in the last column.

Finally, to demonstrate the nonlinear nature of the gain-scheduling feedback matrix K(θ), we
plot the first, second and fifth entries of the LPV-2A control gain, and they are shown in Figure
3.13, Figure 3.14 and Figure 3.15. It is clear that K(θ) is not an affine function of θ. Furthermore,
these figures show that the control gain is clearly more affected by the boundary condition θ1 than
the airspeed θ2.
3.5 Summary

This chapter presented two novel LPV modeling and control design techniques for a smart
airfoil model that utilizes a moving mass for flutter suppression. The LPV gain-scheduling state

36

Figure 3.6: Comparison between proposed LPV-1 control and [2]: αS.

In the first proposed
feedback controllers based on the corresponding models were proposed.
technique, the moving mass position was used as the scaled scheduling parameter, whereas in the
second technique, nonlinear springs and dampers were added to the ends of the groove to constrain
the moving mass. The performance of the proposed LPV controllers was compared with an earlier
nonlinear controller from the literature, and the results clearly demonstrated the advantages and the
effectiveness of the proposed LPV modeling and control techniques in active flutter suppression.
In addition, the proposed LPV control design was also able to significantly increase the flutter
airspeed.

37

Figure 3.7: Comparison between proposed LPV-1 control and [2]: αL.

Table 3.2: (cid:107).(cid:107) ∞ comparison; LPV-1 and LPV-2 controllers vs. nonlinear controller [2].

Case
LPV-1 at αS

LPV-1 at αL

LPV-2 at αL

(cid:107).(cid:107)∞
¯h
α
¯y
¯u
¯h
α
¯y
¯u
¯h
α
¯y
¯u

[2]

0.0040099

0.01

0.24058
0.23176
0.24444

0.6

0.86839
619.82
0.24444

0.6

0.86839
619.82

38

Proposed LPV

0.0029746

0.01

0.083381
0.18602
0.29736

0.6

0.4705
11.347
0.23804

0.6

0.48297
5.5392

Figure 3.8: Comparison between LPV-1 control and LPV-0 control: αL.

Table 3.3: LPV-1 controller vs. LPV-0 controller: αL

LPV-1
0.29736

0.6

0.4705
11.347

(cid:107).(cid:107)∞ LPV-0
¯h
0.14294
0.6
α
¯y
¯u

7.8091
47.979

39

Figure 3.9: Comparison between LPV-1 control and LPV-2 control: αL.

Table 3.4: (cid:107).(cid:107) 2 comparison; LPV-1 and LPV-2 controllers vs. nonlinear controller [2]

Proposed LPV Convergence rate.%

57
60
85
57
8
25
54
92
27
31
74
92

Case
LPV-1 at αS

LPV-1 at αL

LPV-2 at αL

(cid:107).(cid:107)2
¯h
α
¯y
¯u
¯h
α
¯y
¯u
¯h
α
¯y
¯u

[2]

0.4946
0.98788
28.36
26.744
30.216
60.868
254.07
6802.3
30.216
60.868
254.07
6802.3

0.20804
0.38593
4.0848
11.359
27.771
45.592
115.57
519.59
22.143
41.641
65.663
502.97

40

Figure 3.10: Comparison between proposed LPV-2 control and LPV-2A control: αL.

41

Figure 3.11: LPV-1 control at V=3.08 and V=3.4.

42

Figure 3.12: LPV-2A control performance at a defined velocity profile, ¯V ∈ [2.92, 3.4].

Figure 3.13: First entry of K(θ).

43

Figure 3.14: Second entry of K(θ).

Figure 3.15: Fifth entry of K(θ).

44

CHAPTER 4

VIRTUAL CONTROL WITH HARD CONSTRAINTS

4.1 Introduction

Hard constraints in practical control systems are ubiquitous. These constraints constitute
physical limitations on some of the state variables, actuation capacity, performance requirements,
or any of these combinations. In [56] the Hubble telescope pointing error budget is considered as
the constrained performance requirement, [57] and the references therein investigated the missile
control problem subject to hard constraints on rudder/aileron deflections, and [58] considered the
cutting head control for CNC machines, where the cutting head motion needs to be constrained
within the error bound. The input/actuator saturation control problem has been extensively studied
and the research results published in this area are abundant; especially, [59] and [60] offer a
comprehensive coverage on the subject. The saturable servo systems were considered in [61],
modification of the conventional anti-windup was presented in [62], event-triggered scheme was
introduced in [63], [64] proposed the theory of maximal output admissible set of control and state
saturation, and [65] proposed the concept of pseudo-control hedging (PCH) based on nonlinear
dynamic inversion methodology, to just name a few examples. Each of these approaches provides a
specific technique with varying complexity in handling the presence of hard constraints in control
systems. In particular, application of antiwindup scheme or controller switching technique could
potentially cause substantial performance degradation ([4, 5, 6, 7, 66, 67, 68]). In addition, [8, 9, 10]
utilized nonlinear model predictive control (MPC) approach to resolve the hard constrained control
problems, but with significantly increased control complexity due mainly to the high computational
cost. Therefore, for practical control applications, there is still a need to develop an efficient control
strategy that is best suited for real-time control implementation for systems with hard constraints.
This work proposes a novel virtual control concept by utilizing the linear MPC strategy in
dealing with the control systems with hard constraints. The notion of virtual control known in the

45

literature is quite different from what is proposed in this work. For example, [69] introduced the
virtual constraint that was used to limit the one degree-of-freedom joint motion in order to meet the
required switching control rule. In this work, the main idea behind the proposed virtual control is to
introduce the virtual mechanism and incorporate it as part of control model development process.
This idea is motivated by the fact that one can always devise a physical stoppage mechanism to safe-
guard specific states/inputs from ever reaching their saturation limits. In addition, implementation
of this physical mechanism will certainly change how system behaves, especially when nearing
the saturation. This system behavior is precisely what the virtual control is aimed to achieve, but
without actually implementing the stoppage mechanism. Our ultimate control design goal is to
better profiling the closed-loop system responses and keep them away from the imposed constraints,
especially when they get closer to the constrained boundary. The virtual mechanisms considered
in this study are in the forms of virtual nonlinear springs and dampers; of course they can be
any other form of constrained mechanisms. The pairs of proposed virtual springs and dampers
are devised to insert exponentially growing resistant forces in order to prevent the system from
operating near the constrained boundary. The proposed stabilizing controllers are designed for
systems with augmented virtual mechanisms, but in simulation analysis and actual implementation,
these virtual effects will be taken away. In order to ensure that the virtual control is effective only
in the region close to the constrained boundary, the Linear Quadratic Regulator (LQR) based MPC
strategy is adopted for discrete real-time application. The virtual control will become inactive when
the system is operated far away from the constrained boundary, and will exponentially increase its
effect by stiffening both spring and damping effects when nearing the constrained boundary. Based
on the MPC scheme, an optimal finite receding horizon LQR controller is designed at each control
step and operational point. In addition to the LQR weighting matrices that are tunable to achieve
desired performance, the virtual spring and damping parameters can also be used as additional
control design variables, and therefore part of the proposed receding horizon LQR strategy can
be calculated off-line, making the real-time control viable. Moreover, the proposed approach can
easily be incorporated with other advanced control strategies.

46

The MPC approach is a matured methodology that has progressed rapidly in the past decades.
For instance, MPC found its applications in numerous linear systems, [10, 70, 71, 72, 73, 74], and
nonlinear and robust systems, [75, 76, 77, 78, 79], to just name a few. In particular, a nonlinear
MPC requires very high computational load, especially for problems with hard constraints, which
makes the real-time control almost impossible. It should be noted that after the nonlinear springs
and dampers are incorporated in the control design model, the conventional linear control approach
can no longer be used for control design, hence as a remedy the linear quadratic regulator based
MPC method is proposed for solving the virtual control problem. The simplicity of the proposed
approach enables the real-time implementation.

This chapter is organized as follows. Section 4.2 presents a ball balancing example with hard
constraints. The general problem formulation is presented in Section 4.3 along with the control
implementation with the proposed virtual spring and damper profile functions. Section 4.4 presents
the simulation results and discussions. Summaries and conclusions are given in Section 4.5.

4.2 Virtual control: a motivational example

Most of the control systems considered in practical applications are usually subject to a given
level of constraints on some of their states and/or inputs. In this work, these constraints are explicitly
addressed by introducing spring and damper profile functions that are used to not only safe-guard
the constraints from being violated, but also prescribe the system dynamics when nearing the
constraints. The objective is to develop the capability to better tailor the overall system behaviors
as desired, while meeting the practical limitations on the states and/or inputs. In addition, from the
control design perspective, these added spring and damper profile functions can be considered as
"virtual" control and incorporated as part of the state feedback design.

As a motivational example, we consider a balancer problem, [80, 81], as illustrated in Figure 4.1.
We assume that the ball is rolling without slipping and its motion is controlled by an input torque
subject to a constraint on rotational angle. The linearized equations of motion can be described by

47

Figure 4.1: A ball on a balancing beam.

m (cid:220)d = mgαb

I (cid:220)αb = −mgd + τ

(4.1)

where m denotes the mass of moving ball, I the moment of inertia of the beam (assuming that
the ball mass is relatively small), and τ the control torque. In addition, the rotational angle αb is
constrained by

|αb| ≤ αh ,

(4.2)

where αh is a prescribed bound. The essence of this work is to highlight the fact that the constraint
(4.2) can be effectively enforced by introducing a pair of "virtual" stoppage mechanisms of varying
stiffness and damping properties, as shown in Figure 4.2.

Figure 4.2: A ball on a balancing beam with virtual springs and dampers.

As mentioned earlier, these augmented virtual mechanisms are aimed to tailor the system

48

performance when operating near saturation limit, therefore safe-guarding it from reaching the
saturation. And the resultant system behavior is precisely what virtual control system is aspired to
achieve, but with absence of augmented mechanisms.

4.3 Virtual modeling and control framework

Consider a linear time invariant system subjected to an augmented function as follows,

(cid:219)x = Ax + Bu + D f(x) ,

(4.3)

where (A, B) is controllable, B has full column rank, D is a known matrix, and f(x) contains a
vector of continuous profile functions that are to be augmented. It should be noted that the entries
in D are identified apriorily to reflect the constraints on certain states.

Assumption 1: The image of matrix D belongs to the image of matrix B, that is, there exists a
matrix M such that

D = BM .

The above matching condition reveals that the constraints are only imposed upon the system in the
same way as its inputs. Therefore, without loss of generality, the system described in (4.3) can be
rewritten as

(cid:219)x = Ax + Bu + BM f(x)

(4.4)
In the case of balancing ball example, if we let x = [d (cid:219)d αb (cid:219)αb]T and u = τ, then the state space
representation of (4.1) can be written in the form of (4.4), where

A =

, B =

.

(4.5)





0
0
0

1 0 0
0 g 0
0 0 1
−mg/I 0 0 0

49



0
0
0
1/I



In addition, M = 1 and the augmented profile function f(x) can be chosen to be

(cid:104)

0 0 −k(αb) −c(αb)(cid:105)

f(x) =

x = Gx,

(4.6)

where k(αb) and c(αb) are adjustable stiffness and damping parameters as function of αb.
In
particular, these parameters are to be increased, hence spring/damper stiffened, as |αb| gets closer
to its hard limit αh, so as to prevent it from reaching the bound. The mathematical description of
various proposed profile functions for k(αb) and c(αb) will be presented later.

Since the profile function f(x) is known a priori as in (4.6), let Av = BMG then the system

(4.4) can be rewritten as follows,

Σ : (cid:219)x = (A + Av)x + Bu = Ax + B ˆu ,

(4.7)

where ˆu = u + M f(x) is denoted as a "new" control input and M f(x) the virtual control.
In
what follows, the system Σ is used for designing the receding horizon LQR-based model predictive
control law (u), where M f(x) is part of the LQR-MPC design model, and the presence of augmented
profile functions are to ensure the constraints on states are met.

4.3.1 Receding horizon LQR control

Ideally, a continuous LQR-MPC controller should be designed, but due to high computational load
of solving the differential Riccati equation, a finite receding horizon LQR-based discrete-time MPC
approach is proposed for Σ, where the nonlinear model (due to inclusion of M f(x)) is discretized
with a given sample period T for a given f(x(k0T)). The discretized state, control, and the system
matrices are denoted by x(k0), u(k0), and Ad(k0) and Bd(k0), respectively, at sample time k0T.
The performance index at sample instant k0 with k f -step control policy is described by

J =

1
2 x(k f + k0)T F x(k f + k0) +

1
2

x(k)TQx(k) + u(k)T Ru(k) ,

(4.8)

k f +k0−1


k=k0

50

where R is a positive definite symmetric matrix, and F and Q are positive semi-definite symmetric
matrices. The goal is to design a receding horizon controller u of the form

u(k) = L(k)x(k) , k = k0, k0 + 1, ..., k0 + k f − 1 ,

(4.9)

that minimizes J subject to discretized Σ. Following the standard MPC derivation; see for instance
[16] and [82], the control gain matrix L(k) is given by

L(k) = −R−1BT

d(k0)Ad(k0)−T(P(k) − Q),

k = k0, k0 + 1, ..., k0 + k f − 1 ,

where P(k) solves the following discrete-time Riccati equation



P(k) = Ad(k0)T P(k + 1)[I + Bd(k0)R−1BT
P(k0 + k f ) = F ,
k = k0, k0 + 1, ..., k0 + k f − 1

d(k0)P(k + 1)]−1Ad(k0) + Q ,

Recall the defined control ˆu for Σ is given by

ˆu(t) = u(k0) + M f(x) , k0T ≤ t < (k0 + 1)T .

where u(k0) is given in (4.9).

4.3.2 Virtual constraint profiles

(4.10)

(4.11)

As discussed earlier, the augmented virtual mechanisms are aimed to constrain the system responses
within the imposed limitations. In the case of ball balancing problem, these virtual mechanisms can
be devised in terms of parameter dependent springs and dampers. In this study, we propose three
types of profile functions. The first type is a pair of constant spring and damper, i.e. k(αb) = k
and c(αb) = c. The second type is a piece-wise linear symmetric function for k(αb) and c(αb);
a sample of such function is shown in blue solid-line in Figure 4.3, and mathematically it can be
described by

51




k(αb) =

c(αb) =

(|αb|−αm)
αh−αm k2 + k1 ,
(|αb|−αs)
αm−αs k1 ,

0 ,

(|αb|−αm)
αh−αm c2 + c1 ,
(|αb|−αs)
αm−αs c1 ,

0 ,

if αm ≤ |αb| ≤ αh
if αs < |αb| < αm
|αb| ≤ αs

if

if αm ≤ |αb| ≤ αh
if αs < |αb| < αm
|αb| ≤ αs

if

(4.12)

(4.13)

where the coefficients (k1, k2, c1, c2) are tunable parameters according to the desired performance,
αs denotes where the spring/damper begins to affect, and αm denotes the mid-point. It is important
to note that both αs and αm can be chosen according to the desired performance. The third type of
profile function is a scaled continuous tangent function; shown in black dashed-line in Figure 4.3,
and a sample description for a pair of nonlinear spring and damper can be described by

(cid:19)
(cid:19)

2(αh−αs)

(cid:18) π(|αb|−αs)
(cid:18) π(|αb|−αs)

2(αh−αs)

0 ,

 γ tan
 β tan

0 ,

k(αb) =

c(αb) =

,

,

if αs < |αb| ≤ αh
|αb| ≤ αs

if

if αs < |αb| ≤ αh
|αb| ≤ αs

if

(4.14)

(4.15)

where γ and β are the scaling factors used to shape the spring and damping coefficients as desired.
As can be seen, the effect of virtual mechanisms become active when αb is outside of [−αs + αs],
and progressively become stiffened as αb gets closer to the hard bound ±αh. When αb ∈ [−αs +αs]
the virtual springs and dampers are inactive, and the system Σ behaves as a baseline linear time
invariant system. The choice of profile function will depend on the system description and its
constraints, as well as the available control authority. This will be further explored next.

52

Figure 4.3: Virtual spring profile functions.

Figure 4.4: Virtual damper profile functions.

4.4 Concept validation through simulation study

In this section, we first demonstrate the effectiveness of the proposed virtual modeling and
control approach by utilizing the ball balancing problem in Section 4.2. The receding horizon LQR
based MPC controllers are designed with and without the augmented virtual springs and dampers.
Table 4.1 shows the parameters used in the simulations.
Furthermore, the horizon step k f and sample period T are set at 40 and 0.1s, respectively, and the
weighting matrices in the cost function are chosen to be

F = diag [512, 5, 4, 0.02] , Q = diag [120, 1, 80, 1] , R = 1 .

53

Table 4.1: Ball balancing parameters.

Item Parameter Name
Moment of inertia
I
Length of beam
L
Ball mass
m
Sampling rate
T
|αb| lower limit
αs
|αb| mid-point for piece-wise function
αm
|αb| upper limit
αh

Values Units
kg.m2
0.0533

0.8
0.05
0.1
0.1
0.15
0.175

m
kg
s
rad
rad
rad

4.4.1 Virtual control with constant profile

For preliminary analysis, a pair of simple constant spring k and damper c are considered as virtual
constrained mechanism, and they are chosen to be 400 N/rad and 20 N.s/rad, respectively. A hard
constraint on the beam rotational angle is set at αh = 0.175 rad (≈ 10◦). Four initial ball locations
are considered, i.e. x0 = [0.2 0.25 0.345 0.4] m. Figure 4.5 shows the closed-loop responses
when x0 = 0.2 m for the two cases, where the dotted-lines are responses from the LQR controller
designed without virtual spring and dampers (or Av) and the solid-lines are responses from LQR-
based MPC with implementation of virtual constant springs and dampers. As shown, even without
the augmented springs and dampers, the rotational angle αb is within the bound ±0.175 rad, though
with augmented virtual mechanisms the responses are more appreciative, especially the control
effort is much less with the proposed virtual control approach. When the ball starts at a little higher
location, x0 = 0.25 m, it can be seen in Figure 4.6 that, for no augmented virtual spring/damper
case (red dashed-line), the rotational angle needs to travel beyond the imposed constraint αh and it
requires much larger control effort in order to stabilize the motion of the ball, whereas with virtual
mechanisms (blue solid-line), the constraint is met with less control effort. However, when the
ball starts at even higher location, x0= 0.345 m, the proposed constant virtual spring/damper fails
to constrain αb within ±0.175 rad, as shown in Figure 4.7 (red dashed-line). Therefore, in what
follows additional virtual control mechanisms with tunable profile parameters are considered and
their performance assessed.

54

Figure 4.5: Comparison of with virtual control and without virtual control (x0= 0.2m).

4.4.2 Virtual control with piece-wise linear profile

The piece-wise linear profile functions of the form presented in (4.12) and (4.13) are considered.
Five sets of coefficients for (k1, k2, c1, c2) are chosen to study the effect of various linear shape
functions on closed-loop responses. Table 4.2 provides the chosen coefficients, where LVC1,
LVC2, LVC3, LVC4, and LVC5 denote the corresponding virtual receding horizon LQR controllers,
respectively. It should be noted that in developing the virtual controllers, LVC1 to LVC5, the same
weighting matrices and horizon step as defined in previous are used. The initial ball location is set
at x0= 0.345 m.

Figure 4.7 (blue solid-line) shows that the proposed piece-wise linear profile function LVC1 is
able to successfully constrain the closed-loop response on αb. In particular, the mid-point of LVC1
corresponds to the constant spring/damper pair used to generate Figure 4.7 (red dashed-line) that
fails to satisfy the constraint. It should be noted that both LVC1 and constant spring/damper profile

55

0510152025303540-0.100.10.20510152025303540-0.175-0.100.10.1750510152025303540-0.200.20.4Figure 4.6: Comparison of with virtual control and without virtual control (x0= 0.25m).

Table 4.2: Virtual spring and damping constants for piece-wise linear function

Parameter
k1 (N/rad)
k2 (N/rad)
c1 (N.s/rad)
c2 (N.s/rad)

LVC1 LVC2 LVC3 LVC4 LVC5
400
400
500
500
40
20
30
50

1200
1300
20
30

800
900
20
30

400
500
30
40

yield very agreeable control effort, this is critical for making an objective assessment about the
two cases. Figures 4.8 and 4.9 show the closed-loop responses with augmented virtual piece-wise
springs and dampers, and the constraint on αb is met. Figure 4.8 shows that increasing spring
constant reduces the maximum rotational angle αb, but with increased control effort, whereas
Figure 4.9 indicates that increasing damping constant is able to reduce maximum rotational angle
with less control effort. However, when the ball starts at much higher location than x0= 0.345 m,
Figure 4.10 indicates that the proposed virtual piece-wise linear controllers are unable to constrain

56

0510152025303540-0.200.20.40510152025303540-0.175-0.100.10.1750510152025303540-0.500.5Figure 4.7: Comparison of constant spring/damper vs. piece-wise profile LVC1 (x0= 0.345m).
αb within ±0.175 rad (red dashed-line). Therefore, a much aggressive profile function is needed,
especially when |αb| gets closer to αh.

4.4.3 Virtual control with tangent profile function

Here, the continuous tangent profile functions presented in (4.14) and (4.15) are utilized to better
confine the balancing beam. In this simulation, five sets of scaling factors for (β, γ) are chosen to
study their effect on closed-loop responses. These five sets of parameters are listed in Table 4.3,
where TANVC1, TANVC2, TANVC3, TANVC4, and TANVC5 denote the associated virtual
receding horizon LQR controllers, respectively. As shown in Figure 4.3 (black dashed-line),
the effects of spring/damper become very aggressive as |αb| approaches αh, this would prevent
the rotation of the beam from ever reaching ±0.175 rad. Therefore, if there are absolute hard
constraints imposed on the system description, incorporating the proposed virtual tangent profile

57

0510152025303540-0.200.20.40510152025303540-0.100.10.1750510152025303540-0.500.51Figure 4.8: Effect of changing virtual spring constant for piece-wise profile (x0= 0.345m).

function would be a viable approach. Again, the same weighting matrices and horizon step are
used for constructing the virtual controllers. The initial ball location is set at x0 = 0.4 m.
Table 4.3: Virtual nonlinear spring and damping constants for tangent function

Parameter
γ (N/rad)
β (N.s/rad)

TANVC1 TANVC2 TANVC3 TANVC4 TANVC5

300
30

600
30

900
30

300
35

300
40

Figure 4.10 (blue solid-line) shows that the tangent profile function TANVC1 is able to suc-
cessfully constrain the closed-loop response on αb, while the piece-wise profile function LVC1
failed. In addition, as shown in Figure 4.10 that TANVC1 and LVC1 yield similar control effort.
The closed-loop responses with various tangent parameters presented in Table 4.3 are shown in
Figures 4.11 and 4.12, and as shown, all responses meet the αb constraint. The similar trends
as shown in Section 4.4.2 are observed in Figures 4.11 and 4.12, in addition it can be clearly

58

0510152025303540-0.200.20.40510152025303540-0.100.10.1750510152025303540-1012Figure 4.9: Effect of changing virtual damping constant for piece-wise profile (x0= 0.345m).

demonstrated that, as in Figure 4.3 (black dashed-line), as |αb| approaches αh the spring/damper
constant increases exponentially, and this aggressive encounter restricts the motion of the beam.

4.5 Summary

In this chapter, a novel virtual control method in dealing with hard constraints is proposed and
a ball balancing problem is used to demonstrate the efficacy of the proposed concept. A pair of
virtual spring and damper of various profile functions are introduced to accommodate apriorily
identified state constraints and incorporate them into a virtual control model. Subsequently, the
optimal receding horizon LQR-based MPC approach is utilized to design the stabilizing controllers
for both with and without virtual mechanisms. The simulation results clearly demonstrate the
benefits of the proposed virtual control concept, and depending on the nature of constraints, the
virtual mechanisms can be tuned to achieve desired closed-loop system performance, while meeting

59

0510152025303540-0.200.20.40510152025303540-0.100.10.1750510152025303540-1012Figure 4.10: Comparison of piece-wise profile LVC1 and tangent profile TANVC1 (x0 = 0.4m).

the constraint requirements. In particular, as shown in the simulations that as αb is coming closer
to its hard limits αh the virtual spring/damper can be made progressively effective so as to prevent
αb from ever reaching the limit. This concept is especially relevant when implementing a physical
hardware system, where one of the primary control design objectives is to safe-guard the hardware
constraints during its operations.

60

0510152025303540-0.200.20.40510152025303540-0.100.10.1750510152025303540-0.500.51Figure 4.11: Effect of changing γ for tangent profile (x0= 0.4m).

61

0510152025303540-0.200.20.40510152025303540-0.100.10.1750510152025303540-1012Figure 4.12: Effect of changing β for tangent profile (x0= 0.4m).

62

0510152025303540-0.200.20.40510152025303540-0.100.10.1750510152025303540-1012CHAPTER 5

VIRTUAL LPV FLUTTER CONTROL OF A SMART AIRFOIL WITH HARD

CONSTRAINTS

5.1 Introduction

As mentioned in previous chapters that the problems of hard constraints are very important topic
in the industrial applications and control theory since it related to the actuator saturation problem
and especially many real life applications are modeled and behave like an LPV systems. Examples
are varied from Rocket angel of attack to be kept in specific range Rodriguez and Cloutier [57] and
many other applications are considered under this topic.

The control problems with hard constraints exist in almost every physical systems. For instance,
the motion of a linear actuator is always constrained by its mechanical limits. The control problems
with hard constraints were a hot research topic for the last three decades in the control community.
Many control approaches were used for the control problem ranging from antiwindup [83, 84, 85],
conventional LPV [86, 87, 88, 89], switching LPV [90, 91, 92] to mention a few but not to limit.
Most of investigations focus on a specific hard constrained control problem and some are compu-
tationally expensive. Since control design with hard constraints is complicated, there is a need for
a simple and computationally sound control design method for a wide range of systems with hard
constraints. This chapter proposes a new control synthesis approach by introducing the concept of
virtual control for system with hard constraints. The idea is to introduce virtual mechanisms near
hard constraints and to prevent the system from operating close to the hard constrained regions
during the control design and incorporate the virtual mechanisms as part of the control dynamics
during the control implementation.

From existing literature, it can be found that the virtual control notion used is quite different
from what is used in this chapter. They can be divided in two main categories. The first one is

63

centered on Fault Tolerant Control (FTC) (see, for example, [93, 94, 95, 96]), where reconfigurable
control is introduced for faulty systems and named as the virtual actuator. In [93], a fault toler-
ance control scheme, using model reference control as virtual actuator, is proposed for a linear
parameter-varying (LPV) system subject to faulty actuators. More similar work can be found in
[97, 98]. The second category is related to virtual simulation or controlling a virtual object; see [99]
for virtual controlling a numerical machine, [100] for virtual testbed of industrial security systems,
[101] for virtual controlling two layer hydraulic flow, [102] for virtual backstepping neural network
control, and [103] virtual decomposition control (VDC) of large robotics by virtually decomposing
and analyzing coupling force. As a summary, both virtual control categories are different from
what is proposed in this chapter.

In this chapter, a novel LPV hard constrained virtual control (LPVVC) scheme is proposed for
the Linear Parameter Varying (LPV) systems, where the virtual control are casted by introducing
virtual mechanisms (such as springs and dampers) near the hard constraints to keep the system states
from moving close to the posted hard constraints. The usage of term "virtual" is due to the fact that
the introduced virtual mechanisms are not physical and they become part of controller dynamics
when the designed LPV control is implemented physically. A state-feedback LPV gain scheduling
controller is designed with guaranteed H∞ performance and combined with the virtual mechanism
dynamics. This control scheme is applied to suppressing the flutter of a smart airfoil. The target
smart airfoil is an NACA0012 airfoil with a groove in the middle of its chored and a moving mass
along the groove used to suppress the pitching motion caused by flutter phenomena; see [1, 2, 104]
for details. The motion of the actuating moving mass is limited to the groove length. Virtual
mechanisms in term of nonlinear springs and dampers are placed to both ends of the groove to
prevent the mass from moving to both ends of the groove (hard constraints). Note that this approach
is flexible and can be applied to a wide range of LPV systems with various hard constraints. In addi-
tion, the virtual mechanisms can be added to any number of actuation channels with any nonlinear
dynamics; see details in the next two sections. To the best of authors’ knowledge, the LPV virtual

64

control with hard constraints has never been reported before in the literature. Since most of physical
systems are nonlinear with hard constraints and can be modeled as LPV systems, the proposed LPV
virtual control with hard constraints bridges LPV control and practical hard constrained control
problems.
In addition, due to the flexibility of selecting the virtual mechanisms, the proposed
method can be used for many kinds of control problems with hard constraints, and furthermore, it
is fairly inexpensive in terms of computational load, comparing with nonlinear MPC approach [105].

The main contribution of this chapter are four-fold. Firstly, the LPV virtual control (LPVVC) is
formulated for LPV systems with hard constraints. During the LPV control design process, virtual
mechanisms are used to prevent certain states from moving outside of their hard constraints and
during the control implementation phase, the designed LPV control strategy is combined with the
virtual mechanical dynamics to form the LPVVC control law. Note that the proposed control scheme
is radically different from other methods used to handle the hard constrained control problems such
as anti-windup, nonlinear, MPC, etc. Secondly, the proposed method is applied to a smart airfoil
flutter control problem with hard constraints, where the smart airfoil is modeled as an LPV system
to accommodate the nonlinearities in air speed [1] and introduced virtual mechanisms. Thirdly,
introducing virtual variable gains profile function. Where it allows the control designer to choose
different virtual variable gain profile according to the application demands and introduce them in
the controller to prevent actuator saturations. Finally, the performance of the designed LPVVC
control is compared with that of nonlinear control (NLC) [2], conventional LPV control (named
LPVN) with physical springs and dampers [1], LPV control (named LPVR) without considering
hard constraints.

The reminder of the chapter is organised as follows. Section 5.2 presents the virtual control
general problem formulation. The application of the proposed method to the smart airfoil flutter
suppression example is introduced in Section 5.3 along with the control implementation. Section
5.4 presents the simulation results and discussion of comparisons with literatures and other control

65

methods. Finally, the chapter closes in Section 5.5 with conclusions and future work.

5.2 LPV Virtual Control Problem Formulation

In this section, we consider the affine LPV system described by

 (cid:219)x(t) = A(θ(t))x(t) + Bu(θ(t))u(t) + Bw(θ(t))w(t)

y(t) = C(θ(t))x(t) + Du(θ(t))u(t) + Dw(θ(t))w(t)

where θ(t) = (θ1(t), θ2(t), . . . , θq(t)) denotes a vector of scheduling parameters, x(t) the state, u(t)
the control input, w(t) the disturbance input, and y(t) the controlled output. All system matrices
are assumed to be compatible in dimensions and they are in affine parameter-dependent form. For
example, A(θ) is represented by

q


(5.1)

(5.3)

A(θ(t)) = A0 +

θ(t)i Ai ,

(5.2)
where A0 and Ai, i = 1, 2, . . . , q, are constant matrices, and the scheduling parameters θi(t)
are assumed to be measurable in real-time and their magnitude and variational rate are bounded as
follows,

i=1

νi ≤ θi(t) ≤ ¯νi, i = 1, 2, . . . , q,
− σi ≤ (cid:219)θi(t) ≤ σi, i = 1, 2, . . . , q,

(5.4)
Furthermore, we assume that the pair (A(θ), B(θ)) is controllable and B(θ)) has full column rank
for all θ. In this work, the proposed gain-scheduling full state feedback control law is given by

u(t) = K(θ(t))x(t),

(5.5)

where K(θ) is the parameter-dependent gain matrix to be determined.

5.2.1 A multi-simplex representation

Before presenting the affine to multi-simplex transformation, we introduce the following definitions.

66

Definition 1 [13, 14]. A unit simplex Ξs is a polytope of s vertices defined by

(cid:40)

s


Ξs =

ρ = (ρ1, . . . , ρs) :

ρi = 1, ρi ≥ 0, i = 1, 2, .., s

,

(cid:41)

i=1
where the variable ρi varies inside the simplex Ξs.

Definition 2 [15]. A multi-simplex Ξ is the Cartesian product of q number of unit simplex, that is,

Ξ = ΞN1 × ΞN2 × · · · ΞNq =

ΞNi ,

q


i=1

where the index (N1, N2, . . . , Nq) denotes the dimension of the multi-simplex Ξ, hence any
ρ ∈ Ξ can be decomposed into (ρ1, ρ2, . . . , ρq), and each ρi belonging to ΞNi is represented
by (ρi,1, ρi,2, . . . , ρi,Ni) for i = 1, 2, . . . , q.

In order to formulate the affine parameter dependent control problem in multi-simplex rep-
resentation described in ρ-space, we need to first map every scheduling parameter θi into the
corresponding unit simplex form with two vertices (ρi,1, ρi,2).

This can be achieved by utilizing the following general formula [40],

For systems with two scheduling parameters, results in four corresponding vertices can be written
as:

θi − νi
¯νi − νi

¯νi − θi
¯νi − νi

ρi,1 =
ρi,2 = 1 − ρi,1 =
ρi = (ρi,1, ρi,2) ∈ ΞNi , ∀ i = 1, 2, · · · , q,
(cid:18) ¯ν2 − θ2
(cid:18) θ2 − ν2
(cid:18) ¯ν2 − θ2
(cid:18) θ2 − ν2

(cid:18) ¯ν1 − θ1
(cid:18) ¯ν1 − θ1
(cid:18) θ1 − ν1
(cid:18) θ1 − ν1

(ν1, ν2),
(ν1, ¯ν2),
( ¯ν1, ν2),
( ¯ν1, ¯ν2),

¯ν1 − ν1
¯ν1 − ν1
¯ν1 − ν1
¯ν1 − ν1

¯ν2 − ν2
¯ν2 − ν2
¯ν2 − ν2
¯ν2 − ν2

×
×
×
×

ρ1 =

ρ2 =

ρ3 =

ρ4 =

(cid:19)
(cid:19)
(cid:19)
(cid:19)

(cid:19)
(cid:19)
(cid:19)
(cid:19)

For

For

For

For

67

(5.6)

(5.7)

where ΞNi denotes a two-dimensional unit simplex defined by

(cid:40)

ρi = (ρi,1, ρi,2) ∈ R2 :
2

=

ΞNi

j=1 ρi, j = 1, ρi, j ≥ 0

(cid:41)

, i = 1, 2, .., q,

(5.8)

hence, the two-dimensional unit simplex variable ρi is created. Similarly, the rate change of the
scheduling parameter (cid:219)θi(t) given in (5.4) can also be represented using a unit simplex variable. It
follows from the condition that

where

(cid:219)ρi,1(t) + (cid:219)ρi,2(t) = 0 , i = 1, 2, .., q,

(cid:219)ρi,1(t) =

(cid:219)θi − νi
¯νi − νi

, (cid:219)ρi,2(t) = − (cid:219)ρi,1(t) .

Substituting the rate bounds (5.4) into above, we obtain

− σi
¯νi − νi

≤ (cid:219)ρi, j ≤ σi
¯νi − νi

, i = 1, 2, .., q; j = 1, 2.

(5.9)

To establish a two-dimensional unit simplex variable (cid:219)ρi, we first introduce a 2×2 matrix Hi defined
by

Hi =

 − σi
φi ∈ R2 : φi =
2

¯νi−νi
σi
¯νi−νi

(cid:40)

σi
¯νi−νi
− σi
¯νi−νi

 , i = 1, 2, . . . , q ,
(cid:41)

=

ΩNi

j=1 ηi, j H j

i , ηi ∈ ΞNi

, i = 1, 2, . . . , q ,

hence the unit-simplex of dimension 2 can be defined by

(5.10)

(5.11)

where ηi = (ηi,1, ηi,2) and H j
i denotes the jth column of matrix Hi. Therefore, the unit-simplex
variable (cid:219)ρi ∈ ΩNi is created. Finally, the scheduling parameters (θ, (cid:219)θ) with prescribed bounds can
now be transformed into multi-simplex domain through Cartesian product of defined unit-simplexes
as follows,

(ρ, (cid:219)ρ) ∈ Ξ × Ω =

ΞNi ×

ΩNi .

Note that ρ = (ρ1, ρ2, . . . , ρq) ∈ Ξ is defined inside the multi-simplex of dimension (2, 2, . . . , 2).
Therefore, the affine parameter dependent system matrices in (5.1) can be represented in terms of

q


i=1

q


i=1

68

ρ. For instance, from the first equation in (5.6), we obtain that

θi = νi + ρi,1( ¯νi − νi) .

(5.12)

Substituting the above into (5.2) yields

A(θ) = A0 +

q


i=1

q


i=1

νi Ai +

( ¯νi − νi)Ai ρi,1 = A(ρ) .

By following the above parameter transformation, we assume that the LPV control systems (5.1)
described in θ can be transformed into an equivalent system representation in terms of ρ as

 (cid:219)x(t) = A(ρ(t))x(t) + Bu(ρ(t))u(t) + Bw(ρ(t))w(t)

y(t) = C(ρ(t))x(t) + Du(ρ(t))u(t) + Dw(ρ(t))w(t)

(5.13)

The LPV control gain matrix defined in (5.5) will now be depending on the convex scheduling
parameter ρ. Therefore, the closed-loop system in multi-simplex representation can be written as

(cid:219)x(t) = Acl(ρ)x(t) + Bw(ρ)w(t)
y(t) = Ccl(ρ)x(t) + Dw(ρ)w(t)

where

Acl(ρ) = A(ρ) + Bu(ρ)K(ρ) , Ccl(ρ) = C(ρ) + Du(ρ)K(ρ) .

(5.14)

(5.15)

It should be noted that the proposed LPV control system design and analysis process will be
performed based on (5.13), therefore the actual control implementation will require converting
controller gain matrix from ρ-domain to θ-domain [3, 12, 13]. In particular, [3] paper presents
explicitly the conversion of an LPV model with three scheduling parameters.

5.2.2 H∞ control performance

The proposed LPV control problem can be now stated as follows: For system (5.13), a given γ∞ > 0
and any given scheduling parameter pair (ρ, (cid:219)ρ), design a gain-scheduling state-feedback controller
of the form

u(t) = K(ρ)x(t) ,

69



∗

Γ1(ρ)
Γ2(ρ) −ε(V(ρ)V(ρ)(cid:48))
Γ3(ρ)
Bw(ρ)(cid:48)

Γ4(ρ)
0g×n

∗
∗
∗
∗
−I
∗
Dw(ρ)(cid:48) −γ2∞I



< 0

(5.17)

that exponentially stabilizes the closed-loop system (5.14) with guaranteed H∞ performance γ∞,
defined by

< γ∞ .

(5.16)

||z||2
||w||2

sup
(ρi, (cid:219)ρi)

sup
w∈L2
w(cid:44)0

Note that γ∞ is a measure of system robustness. The next theorem provides the synthesis solution
to the H∞ control problem.
Theorem: [12, 106, 107]. The proposed H∞ control problem is solvable for (5.13), if there exist
a scalar ε > 0, symmetric positive-definite continuously differentiable matrix P(ρ), and matrices
V(ρ) and W(ρ), satisfying the following parameterized linear matrix inequality (PLMI)

where ∗ denotes the symmetric entry, and Γ1(ρ), Γ2(ρ), Γ3(ρ) and Γ4(ρ) are given by

Γ1(ρ) =A(ρ)V(ρ) + Bu(ρ)W(ρ) + V(ρ)(cid:48)A(ρ)(cid:48) + W(ρ)(cid:48)Bu(ρ)(cid:48) +
Γ2(ρ) =P(ρ) − V(ρ) + ε(A(ρ)V(ρ) + Bu(ρ)W(ρ))(cid:48)
Γ3(ρ) =C(ρ)V(ρ) + Du(ρ)W(ρ) ,
Γ4(ρ) =εC(ρ)V(ρ) + εDu(ρ)W(ρ) .

,

∂P(ρ)
∂ ρ

(cid:219)ρ ,

Furthermore, if a solution exists to the above PLMI, then the gain-scheduling controller gain is
given by K(ρ) = W(ρ)V(ρ)−1.
Note that the PLMI (5.17) is an infinite dimensional LMI, which is generally difficult to solve. Few
approaches are available in solving a PLMI; see for instance [41, 42, 43, 44], however this work
adapts the method presented in [13].

70

5.2.3 Virtual control system modeling

In this section, we propose a novel modeling approach for a control system with state constraints.
The goal of virtual control design is to develop an effective way to prevent violating the prescribed
state constraints, so as to improve system performance while avoiding hardware to be saturated. It
will be demonstrated that this can be accomplished by introducing augmented profile functions to
the control systems.

Recall the LPV control system described in (5.1). Assuming that there are constrained states
in the system, during the design phase, virtual mechanisms are proposed in the form of augmented
profile function and introduced in the system dynamics where the constrained states are needed.
The resulting system with virtual mechanism are as follows:

(cid:219)x(t) = (A(θ) + Av(θ))x(t) + Bu(θ)u(t) + Bw(θ)w(t)

(5.18)

where A(θ) contains the original system dynamics, and Av(θ) the added virtual mechanism dynam-
ics. In order to make it possible to implement the virtual dynamics into control law, the following
condition needs to be satisfied for and x and uv

Av(θ)x(t) = Bu(θ)uv(t) .

(5.19)

Or equivalently for any θ

Span(Av(θ)) ⊆ Span(Bu(θ)) .

(5.20)
In most practical control applications, the augmented profile function matrix Av(θ) can be chosen in
advance based on the nature of state constraints. Specifically, in this chapter, these profile functions
are in the form of nonlinear springs and dampers so that the associated stiffness and damping
coefficients increase dramatically as the constrained states move closer to their bounds. During the
control implementation phase, the proposed virtual dynamics are augmented in to the control input
ˆu as follows

ˆu = u + (BT

u (θ)Bu(θ))−1BT

u (θ)Av(θ)x(t) ,

(5.21)

71

u Av x(t) denotes the virtual control. Note that Bu(BT

u Bu)−1BT

where uv(t) = (BT
Av x(t) due to (5.20) and u is the LPV control law to be designed.
5.3 LPV Modeling and Control of Smart Airfoil

u Bu)−1BT

u Av x(t) =

This section presents the application of the proposed LPV virtual control approach to the flutter
suppression problem for a smart airfoil; see Figure 5.1. Additional information about the smart
airfoil can be found in [1, 2, 104]. As shown in Figure 5.1, the smart airfoil model is from the
NACA0012 airfoil with a groove along its chord, in which a small moving mass is placed so that
the vibrational motion of the airfoil is suppressed through dynamic coupling to the plunging and
pitching motion of the airfoil.

Figure 5.1: Smart airfoil.

The smart airfoil model in the state-space representation can be described in form of (5.1) (see
[3]), where x(t) = [¯h, α, (cid:219)¯h, (cid:219)α, ¯y, (cid:219)¯y]T representing the normalized plunge, pitch attitude, normalized
plunge rate, pitch rate, normalized control mass displacement and its rate, respectively. The system
matrices are given by (22), where θ1 (will be introduced in the virtual control phase) and θ2 are
two scheduling parameters. The detailed derivation of (22) can be found in [3].

72



0
0
¯r2
αω2
h
qω2
α
− ¯xαω2
h
qω2
α
0
0

(cid:104)

A(θ) =

0
0

α(6.68θ2−2.4)
¯r2

qµ

¯xα(2.4−6.68θ2)

qµ

+

(6.68θ2−2.4) ¯e ¯xα
− (6.68θ2−2.4) ¯e(1+β)

qµ

qµ

− ¯r2
α ¯xα
q
α(1+β)
− ¯r2

q

qµ

1
0
2¯r2
+ 2θ2 ¯e ¯xα
αθ2
qµ
qµ − 2θ2 ¯e(1+β)
−2 ¯xθ2
(cid:104)
(cid:105)(cid:48)

0
0
C(θ) =

qµ

α](cid:105)

(cid:104)−[¯r2

q =

α(1 + β) − ¯x2



(cid:105)

,

,

0
1

0
0
− ¯xα β
0
q
0 β(1+β)
0
0
0
0

q

0
0

0

0

1
0

(5.22)

0 0 0 0 0 0.01

,

0 0 0 0 1 0

(cid:105)

(cid:104)

0
¯g
Bw(θ) =
Dw(θ) = [0],

Bu(θ) =

0 0 0 0 0 1

,
Du(θ) = [0],

5.3.1 Virtual control modeling

The idea of virtual control is to introduce virtual mechanisms into the model during the control
design phase to prevent hard constrained states from moving close to the hard limits, and then in
the implementation phase to transfer the added virtual dynamics (mechanisms) into the controller
law. For the smart airfoil example, the virtual part is introduced in terms of springs and dampers
at both ends of groove to prevents the control mass from hitting the groove boundaries. Note that
the virtual mechanisms can be added freely as long as (5.20) is satisfied. Recalling (5.18), in the
design phase the system model after introducing the virtual mechanisms is
x(t) + Bu(θ)u(t) + Bw(θ)w(t)

(5.23)
Note that Bu has only one nonzero entry (the 6th). To satisfy (5.20), all entries of Av(θ) should

(cid:219)x(t) =

(cid:104)

be zero except the last row. For the airfoil flutter control, Av is selected as

(5.24)

A(θ) + Av(θ)(cid:105)
05×4

Av(θ) =

01×4



05×2

[−k(θ1) − c(θ1)]

73

Note that for the selected Av, it is equivalent to adding virtual springs and dampers at the both
ends of the groove if the nonlinear spring and damper coefficients are modeled as a functions of

the scheduling parameter θ1 as shown in Figure 5.2. The spring stiffness k(θ1) and the damping
coefficient c(θ1) are carefully designed to generate enough force to stop and reverse the motion of
the control mass at both ends of the airfoil groove. According to Figure 5.2, the values of the k(θ1)
and c(θ1) in the vicinity of the airfoil groove between -0.4 and 0.4 are equal to zero and hence the
mass can move freely in this area. But near both ends, i.e., between 0.4 and 0.5 (between -0.5 and
-0.4) their values increase by following a tangent profile to generate forces opposite to the moving
direction and to stop the motion and/or reverse the moving direction.

Figure 5.2: Virtual spring and damper

The virtual spring and damper coefficients were chosen as follows (also see Figure 5.2),



k = 0 , c = 0,
k(θ1) = 80 tan( π(θ1)
c(θ1) = 6 tan( π(θ1)

when | ¯y| ≤ 0.4
0.2 ) , when | ¯y| > 0.4
0.2 ) , when | ¯y| > 0.4

where θ1 = | ¯y| − 0.4. Noted that the functions k(θ1) and c(θ1) can be chosen in different ways,
depending on applications.

Follow up of (5.6) the smart airfoil example has two scheduling parameters, i.e. q = 2, then

variables in the multi-simplex domain can be depicted as

ρ1 = (ρ1,1, ρ1,2), ρ2 = (ρ2,1, ρ2,2)

The LPV H∞ design, based on Theorem 5.17, is performed over multi-simplex convex domain Ξ

74

and the resulting gain-scheduling controller is in terms of

W(ρ) = 
2
V(ρ) = 
2

j1=1

j1=1


2

2
j2=1 Wj1, j2 ρj1, j2,
j2=1 Vj1, j2 ρj1, j2,

(5.25)

In order to calculate the affine feedback gain matrix in (5.5), the inverse transformation described
in [1, 12, 13] are used. For example,

2

2

Wj1, j2,
2

2


j2=1

j1=1

j1=1

j2=1

W0 =

1
16

Wi =

1
16 ¯νi

(−1)ji+i Wj1, j2,

i = 1, 2

(5.26)

(5.27)

Hence, for two scheduling parameters, W(θ) and V(θ) can be written in the affine domain as

W(θ) = W0 + θ1W1 + θ2W2,
V(θ) = V0 + θ1V1 + θ2V2

where Wi (i = 0, 1, 2) are constant matrices determined by (5.26) and Vi (i = 0, 1, 2) can be obtained
in a similar way. Finally, control gain matrix K(θ) defined in (5.5) can be calculated by

K(θ) = W(θ)V−1(θ)

(5.28)

For the case with more than two scheduling parameters, one may consult references [12, 13, 40]. For
the solution of the convex optimization of (3.30), SeDuMi [46] solver with optimization packages
ROLMIP [45] and YALMIP [11] are used. Then, the virtual control law in (5.21) can be formed as

ˆu = W(θ)V−1(θ)x + (BT

u (θ)Bu(θ))−1BT

u (θ)Av(θ)x ,

(5.29)

The virtual LPV controller synthesis solution steps generally can be considered as eight steps.
Firstly, scaling scheduling parameter introduced. Secondly, introducing the virtual dynamics.
Thirdly, change of variables from affine representation domain to the multi-simplex (convex) do-
main is performed. The fourth step is modeling the rates of variation of scheduling parameters in
convex domain. In the fifth stage, performing the PLMI synthesis conditions. The six stage is to

75

perform PLMI relaxation to transfer the infinite dimension PLMI into finite dimension LMI to be
solved using optimization package. Then the seven stage involve to utilize inverse transformation
to transfer from the multi-simplex representation into the affine representation. Then finally the
eight stage is the controller implementation with virtual dynamics compensation. The steps are
depicted in the following diagram Figure 5.3.

Figure 5.3: Controller steps.

76

5.4 Simulations investigation

5.4.1 Control design results

The parameters of the smart airfoil example used in this chapter are shown in Table 5.1 and the
air speed θ2 varies from 0.5 to 2.92 nondimensional speed which is equivalent to from 4m/s to
23.7m/s. The virtual control idea has its unique advantage for designing controller with hard
constraints on state variables.
In order to assess the performance of the proposed LPV virtual
controller (LPVVC) with hard constraints, numerical studies are conducted and compared with the
existing results in the next subsection.

Table 5.1: [1, 2] Parameters used for the smart airfoil example.

(m)
Nondimensionalized

Parameter Value Units
0.127
b (m)
0.35
¯e
¯r2
0.388 Nondimensionalized αS
α
Nondimensionalized αL
0.25
¯xα
airspeed ¯V 2.92
Nondimensionalized
(rad/sec)
64.1
ωα
55.9
(rad/sec)
ωh

initial (cid:219)α
β
µ

Parameter Value Units
initial ¯h
initial (cid:219)¯h

0
0
0.01
0.6
0
0.01
152

Nondimensionalized
Nondimensionalized
(rad)
(rad)
(rad/sec)
Nondimensional
Nondimensional

5.4.2 Simulation results

In order to highlight the significance of the proposed virtual control concept, three comparison
studies are conducted in this section. The first comparison is with the method used in [3], where
a conventional LPV control (LPVN) strategy is utilized. Two pairs of springs and dampers are
physically installed at both ends of the airfoil groove from (|0.35| to |0.5|) [3].

Starting with some mild initial condition (α = 0.2 rad), one can see, from Figure 5.4, that both
LPVN and proposed LPVVC controllers are able to stay within the groove hard constraint, but
for the case of harsh initial conditions (for example, 0.6 rad) it can be seen, from Figure 5.5, that
the LPVN control is not able to keep the moving mass within the groove hard constraints while
the proposed LPVVC does. The system responses shown in Figures 5.4 and 5.5 confirm that the

77

proposed LPVVC virtual control is capable of keeping the actuation mass within its hard constraint
and preventing actuator saturating under both small and large initial conditions.

Furthermore, recall Figure 5.2, where the black curve represents the proposed tangent profile
function used in LPVVC virtual control. Therefore, the actuation mass moves freely between -0.4
and 0.4, but for the LPVN control [3], it reduces to be between -0.35 and 0.35. This indicates
that utilizing the virtual tangent profile function increases free moving range for the control mass,
making the hard constrained control more close to the case without hard constraints. This is
benefited from the selected tangent profile function, which increases the linear region to be between
-0.4 and 0.4 from between -0.35 and 0.35.

The second study is presented in Figure 5.6, where the LPVVC control is compared with a
nonlinear controller (NLC) in [2]. The effectiveness of the proposed controller is clearly demon-
strated, where the nonlinear controller is unable to satisfy the hard constraints for the given initial
condition while the LPVVC does. In addition, the overall performance is also enhanced as well as
the control effort is dramatically reduced from that of the nonlinear controller in [2].

The last comparison is with a regular LPV controller (LPVR) without considering the hard
constraint requirement (see Figure 5.7). It can be clearly observed that since the LPVR control
does not consider the hard constraints, it suppresses the airfoil flutter by regulating the control mass
outside the hard constraints unrealistically. Again, the proposed LPVVC controller fulfills the hard
constraint requirement effectively.

2 = (Þ ∞

A parametric study is also conducted and the results are presented in Tables 5.2 and 5.3 below
in terms of the signal L∞ and L2 norms ((cid:107) ¯y(cid:107)2∞ = supt≥0( ¯yT ¯y), (cid:107) ¯y(cid:107)2
¯yT ¯ydt)) to show the
control and response performance enhancement of the proposed method over LPVN, NLC, and
LPVR. To make a fair comparison, initial conditions for all simulations are set to α = 0.6. One
can see a significant improvement of L2 norm (see Table 5.3). Note that the last two columns of
Table 5.3 show percentages of improvement of the LPVVC control over the NLC [2] and LPVN
[3], respectively. Also, the red boxes in Table 5.2 shows that the other methods violates the hard
constraints limit.

0

78

It is very clear according to the above results that the proposed LPVVC control is a promising
method for dealing with hard constrained control problems with the help of flexibly choosing
various virtual mechanisms.

Table 5.2: (cid:107).(cid:107) ∞ comparisons between LPVVC, LPVN [3], NLC [2] and LPVR.

(cid:107).(cid:107)∞ LPVR NLC [2] LPVN[3] LPVVC
¯h
0.2715
α
¯y
¯u

7.8091
47.979

0.8683
619.82

0.9205
4.917

0.28736

0.6

0.4319
4.494

0.1429

0.6

0.2444

0.6

0.6

Table 5.3: (cid:107).(cid:107) 2 comparisons between LPVVC, LPVN [3] and NLC [2].

(cid:107).(cid:107)2 NLC[2] LPVN[3] LPVVC [2] * % [3]**%
¯h
4.2
-8.7
α
32
¯y
¯u
10.5

27.771
45.592
115.57
519.59

26.59
49.56
54.57
464.99

30.216
60.868
254.07
6802.3

12
18.5
78.5
93.1

* Convergence rate with respect to NLC[2], ** Convergence rate with respect to LPVN[3]

5.5 Summary

This chapter introduced and synthesized a novel LPV virtual control (LPVVC) scheme with
hard constraints. The main idea of virtual control is to introduce virtual mechanisms near the
state hard constraints to confine state movement into hard constrained neighborhood and hence
prevent the states from violating the hard constraints, and during the control implementation,
the virtual mechanisms are augmented with the designed LPV controller to form LPVVC. The
proposed control scheme is applied to the flutter suppression of a smart airfoil to demonstrate its
effectiveness and ability of performance enhancement. The state-feedback LPV (gain-scheduling)
control with guaranteed H∞ performance is used for designing controller based on the model with
virtual mechanisms. The virtual mechanisms used in this chapter are in terms of springs and
dampers located at both ends of the airfoil groove to prevent the control mass from moving outside

79

of the groove. Note that the virtual mechanisms are not limited to springs and dampers. The
performance of the designed LPVVC controller is compared with the conventional LPV control
with hard constraints (LPVN), nonlinear control (NLC), and regular LPV control (LPVR) and
showed significant improvement. For instance, the control mass L2 norm is reduced by 77.5% over
the NLC and 35% over the LPVN.

80

(a)

(b)

Figure 5.4: LPVVC vs LPVN, I.C(α = 0.2).

81

(c)

(d)

Figure 5.4: LPVVC vs LPVN, I.C(α = 0.2).

82

(a)

(b)

Figure 5.5: LPVVC vs LPVN, I.C(α = 0.6).

83

(c)

(d)

Figure 5.5: LPVVC vs LPVN, I.C(α = 0.6).

84

(a)

(b)

Figure 5.6: LPVVC vs NLC.

85

(c)

(d)

Figure 5.6: LPVVC vs NLC.

86

(a)

(b)

Figure 5.7: LPVVC vs LPVR.

87

(c)

(d)

Figure 5.7: LPVVC vs LPVR.

88

CHAPTER 6

CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORKS

The dissertation introduced methods of handling actuator saturation for LTI and LPV systems.
Numerical simulations, comparisons with other methods and parametric studies were introduced
to show the effectiveness and the power of the proposed methods. This chapter provides some
conclusions and proposes some future perspectives for the methods.

6.1 Conclusions

6.1.1 Scaling scheduling parameter LPV control with application to flutter suppression of

smart airfoil

Chapter three presented two novel LPV modeling and control design techniques for a smart airfoil
model that utilizes a moving mass for flutter suppression. The LPV gain-scheduling state feedback
controllers based on the corresponding models were proposed. In the first proposed modeling and
control technique (LPV-1), the moving mass position was used as the scaled scheduling parameter,
whereas in the second modeling and control technique (LPV-2), nonlinear springs and dampers
were added to the ends of the groove to constrain the moving mass. Moreover, a modeling
and control technique to reduce the number of scheduling parameters (LPV-2A) hence to reduce
conservativeness is introduced. The performance of the proposed LPV controllers was compared
with an earlier nonlinear controller from the literature, and the results clearly demonstrated the
advantages and the effectiveness of the proposed LPV modeling and control techniques in active
flutter suppression. In addition, the proposed LPV control design was also able to significantly
increase the flutter airspeed and keeps actuator movements inside its bounds.

89

6.1.2 Virtual control LTI with application to beam and ball system

In chapter four, novel hard constraints virtual controller is proposed and synthesized for LTI
continuous systems as a solution for the saturation problem. The method is applied for Ball
and Beam System with LQR-based MPC state feedback control to show the effectiveness of the
proposed method. The simulations showed the effectiveness and the performance of the proposed
method in handling the hard constraints without saturation of the actuator. The main advantages
of the proposed method is utilizing the linear time invariant design to handle nonlinear hard
constraints, presenting extra tuning parameters to the controller (spring and damper constants), and
the simplicity in controller implementation. Also the method can be extended to the continuous
LTI systems with minor effort.
In addition, the method is aimed to be applied for many other
applications with hard constraints and the other controller design theories.

6.1.3 Virtual control LPV with application to smart airfoil flutter suppression

In chapter five, novel hard constraints virtual controller is proposed and synthesized. The main
idea of virtual control is to introduce virtual mechanisms near the state hard constraints to confine
state movement into hard constrained neighborhood and hence prevent the states from violating the
hard constraints, and during the control implementation, the virtual mechanisms are augmented
with the designed LPV controller to form LPVVC. H∞ state feedback guaranteed performance
control is used. The method is applied for smart airfoil flutter suppression example to show the
effectiveness of the proposed method. Parametric study and numerical simulation are presented.
Comparisons studies with conventional LPV hard constrained control, nonlinear control, and regu-
lar LPV control without considering hard constraints are conducted to assess the performance of the
proposed method and showed advantage over the existing methods. For instance, the control mass
L2 norm is reduced by 77.5% over the nonlinear control and 35% over the conventional LPV control.

90

6.2 Future work

This section proposes some viable future research directions depending on the conclusions of the
work.

It is tempting to extend the idea of scaling scheduling parameter introduced in chapter three which
is applied for affine LPV system with hard constraints to cover other types of LPV system such
as Quasi-LPV. Moreover, it can be applied to other system examples with more hard constraints.
Further, it also can be utilized with other damping functions than the proposed in chapter three.

The promising virtual control idea in chapter four (which is applied to LTI beam and ball system)
and chapter five (which is applied to LPV smart airfoil flutter suppression system) can be applied
to control examples other than beam and ball system and smart airfoil system. Furthermore, it can
be applied with other control methods such as adaptive control and nonlinear control with many
different virtual functions.

Building experimental rig is interesting idea to show the proposed methods in beam and ball system
and smart airfoil flutter suppression system or any other examples.

91

APPENDICES

92

APPENDIX A

SOME LMI PRELIMINARIES

A.1 Hurwitz Stability

In this section some LMI definitions and conditions will be presented. Most of formulation can be
found in [108].

Definition 1 [108]: A polytope P ⊂ Rnis a set which is the convex hull of a nonempty finite set
z1, z2, ...zm ⊂ Rn, that is P = convz1, z2, ...zm.
Proposition 1 [108]: A continuous system

(cid:219)x(t) = Ax(t)

is Hurwitz stable if and only if there exists a matrix P ∈ Sn, such that



P > 0

AT P + PA < 0

A.1.1 Family of systems

Consider the following system

(cid:219)x(t) = A(θ(t))x(t)

where the system coefficient matrix takes the form of

93

(A.1)

(A.2)

(A.3)

For q = 3

q


i=1

A(θ(t)) = A0 +

θ(t)(i,...q)A(i,...q)

A(θ(t)) = A0 + θ(t)A1 + θ(t)A2 + θ(t)A3

(cid:219)x = Ax + Bu

y = Cx + Du

(A.4)

(A.5)

(A.6)

where x ∈ Rn is the state vector,y ∈ Rm is the output vector and x ∈ Rr is the input vector. A,B,C,
and D represents the system matrices with appropriate dimensions.
Definition 2 [108]: The system A.6 is said to be Hurwitz stabilizable if there exists a real matrix
K such that A + BK is Hurwitz stable.
Theorem 2 PBH [108] : The system A.6 is Hurwitz stabilizable if and only if

rank[sI − A
rank[sI − A

B] = n,
B] = n,

∀s ∈ C,
∀s ∈ λ(A),

Re(s) ≥ 0;
Re(s) ≥ 0

thats mean the system is Hurwitz stable if all its uncontrollable modes are Hurwitz stable.
Theorem 3 [108] : The system A.6, is Hurwitz stabilizable if and only if there exist a symmetric
positive definite matrix P and a matrix W satisfying

AP + PAT + BW + WT BT < 0

Definition 3 : The system matrix is called bounded real if it satisfies

GH(s)G(s) ≤ I,

∀s ∈ C,

Re(s) ≥

Preposition 3 : The bounded real of definition 3 is equivalent to

(A.7)

(A.8)

94

(cid:107)G(s)(cid:107)∞ ≤ I

(A.9)

Theorem 3 Bounded-Real Lemma: The system A.6 be controllable then it is non-expansive if and
only if its transfer function is bounded real.
Theorem 4 H∞ LMI conditions: For a system A.6, then (cid:107)G(s)(cid:107)∞ < γ if and only if there exists a
matrix P > 0 such that the following inequality holds



AT P + PA PB CT
−γI DT
D −γI

BT P

C

 < 0

(A.10)

95

APPENDIX B

SCHEDULING PARAMETER MATRICES

B.1 Scheduling Parameter Matrices

In this section, the constant matrices (Z0, G0) and (Zi, Gi) for constructing LPV-0, LPV-1,

LPV-2 and LPV-2A controllers of chapter three are presented.
For LPV-0 controller: (Z0

0, G0

0) and (Z0

i , G0
i )

0 =[−0.051788000, 0.2853700,−0.261390, 0.476160, 0.045888,−8.81260],
Z0
2 =[−0.000028579, 0.0095015,−0.036397, 0.083472,−0.018931,−0.63590],
Z0
3 =[ 0.008960500,−0.1393600, 0.006209,−0.042522,−0.117880, 0.31722] .
Z0

(B.1)

3.6702e − 06
3.6004e − 05 −2.4069e − 06 −2.6686e − 06
1.0487e − 05 −1.3243e − 05

7.1505e − 09 −9.4204e − 05
1.0292e − 05 −7.0926e − 06 −2.0329e − 07
3.9664e − 04
−4.575e − 06
−7.032e − 06
9.1849e − 05
4.0045e − 05
−4.1334e − 07 −2.1419e − 06
4.2533e − 05 −3.7746e − 04 −1.4701e − 04
3.9355e − 06 −3.5161e − 06 −1.3181e − 05
4.9923e − 02 −3.2631e − 01
9.6502e − 05
−1.6558e − 06
−3.961e − 04
−3.288e − 01
1.1691e − 05 −3.0011e − 05
−9.4264e − 05
104.92e − 01

2.3264e − 06
3.9815e − 04

1.0241e − 05 −6.8766e − 06 −1.9139e − 07
−6.8168e − 06
3.54e − 05 −2.4457e − 06
1.063e − 05 −1.3519e − 05
−2.17e − 06
−4.0415e − 07
4.2983e − 05
3.9103e − 06 −3.5054e − 06 −1.3456e − 05
−3.982e − 04
−2.0406e − 06
9.6801e − 05
4.2126e − 06 −5.6076e − 06
−9.4179e − 05

3.642e − 06 −7.7403e − 08 −9.4036e − 05
−2.648e − 06 −3.9416e − 07
3.8971e − 04
9.2068e − 05
2.5043e − 05
−3.792e − 04 −9.1409e − 05
5.1528e − 02 −2.3771e − 01
29.561e − 01
−2.52e − 01

7.7154e − 06
3.9065e − 04




(B.2)

(B.3)




G0
0 =

G0
2 =

96

1.0241e − 05 −6.8766e − 06 −1.9139e − 07
−6.8168e − 06
3.54e − 05 −2.4457e − 06
−4.0415e − 07
−2.17e − 06
3.9103e − 06 −3.5054e − 06 −1.3456e − 05
−2.0393e − 06
−9.4195e − 05

7.7183e − 06
3.9049e − 04

3.6419e − 06
−2.648e − 06 −8.4001e − 07
9.2005e − 05

3.0065e − 08 −9.4368e − 05
3.9108e − 04
2.5621e − 05
4.2983e − 05 −3.7894e − 04 −9.4228e − 05
5.2281e − 02 −2.4613e − 01
30.102e − 01

9.681e − 05 −3.9823e − 04
3.9802e − 06 −4.9266e − 06 −2.6559e − 01

1.063e − 05 −1.3519e − 05

For LPV-1 controller: (Z1

0, G1

0) and (Z1

i , G1
i )



G0
3 =







(B.4)

(B.5)

(B.6)

0 =[ 0.548420,−0.00019372,−0.15522000, 0.71419000, 1.08060000,−109.9000],
Z1
1 =[ 0.553410,−0.00019562,−0.13229000, 0.70671000, 1.08980000,−110.8800],
Z1
2 =[−0.014035,−0.03191200, 0.08397600,−0.05061400,−0.00059914,−0.048978],
Z1
3 =[−0.287940,−0.00022911, 0.00022877,−0.00082886, 0.02613200,−0.121390] .
Z1

G1
0 =

−4.88e − 06 −2.1003e − 07
3.0824e − 06
2.7367e − 05 −1.6313e − 06 −3.1912e − 06
7.9214e − 06 −9.6416e − 06

7.8896e − 06
−4.8329e − 06
−3.6471e − 07 −1.4518e − 06
3.2593e − 06
−4.2384e − 05
−2.8195e − 05

−3.784e − 06 −9.5948e − 06
1.3887e − 04
1.7853e − 04 −5.6427e − 05

8.1301e − 05 −3.6761e − 04

−4.144e − 05 −3.9649e − 05
1.3371e − 04
2.3919e − 04
8.1299e − 05 −5.8287e − 05
7.8805e − 05
1.8855e − 02 −5.0645e − 01
507.95e − 01

3.2516e − 05 −3.6529e − 04

1.6936e − 04 −5.1254e − 01

97





G1
1 =

G1
2 =

G1
3 =

2.7344e − 05 −1.6295e − 06 −3.1935e − 06
7.9138e − 06 −9.6322e − 06

7.8884e − 06 −4.8753e − 06 −2.0747e − 07
−4.8288e − 06
−3.6188e − 07 −1.4515e − 06
3.2555e − 06 −3.7846e − 06 −9.5862e − 06
−4.2359e − 05
−2.8291e − 05

1.388e − 04
1.7829e − 04

3.0788e − 06 −4.1397e − 05 −3.9695e − 05
2.3879e − 04
1.3359e − 04
8.1284e − 05 −5.8183e − 05
7.8771e − 05
1.8855e − 02 −5.0643e − 01
507.95e − 01

1.6924e − 04 −5.1253e − 01

3.2506e − 05 −3.6529e − 04

8.1281e − 05 −3.6761e − 04
−5.63e − 05

2.6788e − 05 −1.6547e − 06 −3.1241e − 06
8.0632e − 06 −9.8762e − 06

7.8555e − 06 −4.7094e − 06 −2.0663e − 07
−4.6633e − 06
−3.6405e − 07 −1.4649e − 06
3.2368e − 06
−4.1803e − 05
−2.1645e − 06

−3.726e − 06 −9.8281e − 06
1.3851e − 04
8.5489e − 05 −6.2007e − 05

3.0568e − 06 −4.0963e − 05 −2.6925e − 06
1.3399e − 04
8.7094e − 05
8.1091e − 05 −6.0543e − 05
1.9392e − 04
−1.305e − 02
2.3088e − 02

3.2861e − 05 −3.6601e − 04
1.398e − 02
1.9973e − 04 −1.3281e − 02

8.111e − 05 −3.6749e − 04

7.8543e − 06 −4.7047e − 06 −2.0406e − 07
−4.6591e − 06
−3.6122e − 07 −1.4646e − 06
3.2329e − 06 −3.7267e − 06 −9.8195e − 06
−4.1778e − 05
−2.2601e − 06

3.0532e − 06
2.6765e − 05 −1.6529e − 06 −3.1264e − 06
8.0557e − 06 −9.8668e − 06
3.2851e − 05
8.109e − 05 −3.6748e − 04
−6.188e − 05

1.3845e − 04
8.525e − 05

−4.092e − 05 −2.7393e − 06
1.3387e − 04
8.6695e − 05
8.1075e − 05 −6.0439e − 05
−3.66e − 04
1.9388e − 04
1.398e − 02 −1.3033e − 02
2.305e − 02

1.9961e − 04 −1.3264e − 02





(B.7)

(B.8)

(B.9)

For LPV-2 controller: (Z2

0, G2

0) and (Z2

i , G2
i )

98

0 =[−2.4213e − 06, 4.2664e − 06, 6.1342e − 06,−3.2752e − 05,−1.9581e − 02,−5.5480e − 01],
Z2
1 =[−5.9946e − 06, 2.3481e − 05, 4.3778e − 05,−1.8995e − 04,−5.1250e − 02,−4.0719e − 01],
Z2
2 =[ 2.2156e − 06,−2.9162e − 05, 1.1126e − 04,−3.9439e − 04,−4.1207e − 02,−5.2469e − 01],
Z2
3 =[−2.7218e − 06,−8.0204e − 06, 2.2373e − 04,−8.4153e − 04,−8.6814e − 02,−6.8411e − 01] .
Z2





G2
0 =

G2
1 =

G2
2 =

2.9948e − 05 −1.9898e − 06 −2.3014e − 06 −1.1892e − 05
8.0774e − 05

8.7657e − 06 −1.0947e − 05

8.6375e − 06 −5.8541e − 06 −1.7052e − 07
−5.8037e − 06
−3.4681e − 07 −1.7695e − 06
3.2827e − 06 −3.0092e − 06 −1.0893e − 05
3.2859e − 07 −4.7744e − 06
3.9208e − 04
−8.6032e − 05

3.061e − 06

1.9876e − 06 −8.5609e − 05
3.8407e − 04
1.3285e − 04
3.5244e − 05 −3.3881e − 04 −4.2294e − 04
5.3732e − 02 −4.4081e − 01
71.034e − 01

8.6082e − 05 −3.6101e − 04
2.2815e − 04 −8.0485e − 04 −3.9729e − 01

8.6356e − 06 −5.8486e − 06 −1.6711e − 07
−5.7989e − 06
2.9927e − 05
−3.4266e − 07 −1.7703e − 06
3.2751e − 06 −3.0071e − 06 −1.0891e − 05
2.0057e − 06 −9.9109e − 06
−9.6388e − 05
4.2388e − 04

−1.989e − 06 −2.3008e − 06 −1.6895e − 05
8.7587e − 06 −1.0942e − 05
8.0278e − 05

3.0549e − 06

3.458e − 06 −9.3032e − 05
4.168e − 04
2.285e − 04
3.5264e − 05 −3.3025e − 04 −9.0546e − 04
5.539e − 02 −3.4871e − 01
67.63e − 01

8.5962e − 05 −3.5333e − 04
1.476e − 04 −5.6966e − 04 −3.7782e − 01

8.6022e − 06 −5.6961e − 06 −1.4859e − 07
−5.6459e − 06
−3.2917e − 07 −1.8423e − 06
3.2152e − 06 −2.7843e − 06
7.7799e − 06
−1.4043e − 06
−7.7589e − 05
3.3782e − 04

2.9851e − 06
2.951e − 05 −2.0785e − 06 −2.0486e − 06
8.9527e − 06 −1.1358e − 05
−1.13e − 05
7.1544e − 05 −3.0434e − 04
1.6989e − 04 −6.2933e − 04 −1.3177e − 01

3.8124e − 07 −7.9024e − 05
3.3505e − 04
8.2653e − 07
6.6592e − 05
8.7906e − 05
3.6103e − 05 −2.8264e − 04 −3.1086e − 04
2.7715e − 02 −1.6798e − 01
30.6e − 01

(B.10)

(B.11)





(B.12)

(B.13)

99

8.6002e − 06 −5.6906e − 06 −1.4519e − 07
−5.6411e − 06
2.9489e − 05 −2.0777e − 06
−3.2503e − 07 −1.8431e − 06
3.2076e − 06 −2.7822e − 06 −1.1298e − 05
2.7281e − 07
−8.7945e − 05

2.6434e − 06
3.6962e − 04

2.9789e − 06
−2.048e − 06 −4.1759e − 06
6.6096e − 05

1.8516e − 06 −8.6448e − 05
3.6778e − 04
1.8355e − 04
3.6122e − 05 −2.7409e − 04 −7.9338e − 04
2.9374e − 02 −7.5872e − 02
−1.123e − 01
27.196e − 01

7.1424e − 05 −2.9666e − 04
8.9333e − 05 −3.9413e − 04

8.9457e − 06 −1.1354e − 05

(B.14)

For LPV-2A controller: (Z2A

0 , G2A

0 ) and (Z2A
i

, G2A

i

)

0 =[9.5650e − 05, 7.7015e − 05,−1.1704e − 04, 1.4560e − 04,−1.3835e − 03,−28.959e − 01],
Z2A
1 =[5.2380e − 05, 9.1385e − 06, 1.0587e − 04,−5.9202e − 04,−5.5273e − 02,−3.0144e − 01],
Z2A
2 =[1.6166e − 05, 4.9277e − 05, 1.2832e − 04,−7.2469e − 04,−1.9366e − 01, 1.6593e − 01] .
Z2A

3.6872e − 05 −1.7063e − 06 −2.5482e − 06 −3.6149e − 06
7.9318e − 05

1.2038e − 05 −1.3235e − 05

1.2067e − 05 −5.7913e − 06 −1.4604e − 07
−5.7468e − 06
−3.8476e − 07 −1.4536e − 06
3.0024e − 06 −3.4095e − 06 −1.3189e − 05
−1.1886e − 06
−8.9113e − 05

3.4651e − 06
3.5178e − 04

2.7411e − 06

6.1678e − 07 −8.6683e − 05
3.4392e − 04
1.2268e − 04
4.405e − 05 −3.2221e − 04 −4.8872e − 04
4.0479e − 02 −2.4278e − 01
43.08e − 01

8.7408e − 05 −3.5454e − 04
1.2295e − 04 −4.9701e − 04 −2.2854e − 01





(B.15)

(B.16)



G2
3 =



G2A
0 =

100




G2A
1 =

G2A
2 =

3.6796e − 05 −1.7858e − 06 −2.4327e − 06 −2.4595e − 06
7.6751e − 05

1.2054e − 05 −1.3257e − 05

1.206e − 05 −5.7651e − 06 −1.3122e − 07

−5.7205e − 06
−3.7252e − 07 −1.5181e − 06
2.9831e − 06 −3.3113e − 06 −1.3209e − 05
−7.6883e − 07
−9.0988e − 05

4.1088e − 06
3.7241e − 04

2.7185e − 06

7.9486e − 07 −9.5248e − 05
3.912e − 04
1.5131e − 04
4.405e − 05 −3.1524e − 04 −6.3172e − 04
4.1129e − 02 −1.6123e − 01
37.591e − 01

−3.966e − 04 −1.7717e − 01

8.0012e − 05 −3.2804e − 04
9.6802e − 05

3.6872e − 05 −1.7061e − 06 −2.5486e − 06 −3.9118e − 06
7.9304e − 05

1.2038e − 05 −1.3235e − 05

1.2067e − 05 −5.7914e − 06 −1.4607e − 07
−5.7468e − 06
−3.8481e − 07 −1.4533e − 06
3.0026e − 06 −3.4104e − 06 −1.3189e − 05
−1.183e − 06
−8.8986e − 05

3.3699e − 06
3.5128e − 04

2.7412e − 06

6.7547e − 07 −8.7462e − 05
3.4587e − 04
1.1497e − 04
4.405e − 05 −3.2238e − 04 −4.5687e − 04
4.0815e − 02 −2.4882e − 01
43.775e − 01

8.7379e − 05 −3.5453e − 04
1.2343e − 04 −4.9817e − 04 −2.2927e − 01




(B.17)

(B.18)

101

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