SAMPLE AND HOLD FUNCTIONS AND NON-MINIMUM-PHASE SYSTEMS
By
Yingxu Wang

A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Engineering Mechanics – Master of Science
2014

ABSTRACT
SAMPLE AND HOLD FUNCTIONS AND NON-MINIMUM-PHASE SYSTEMS
By
Yingxu Wang
With the same initial condition, a continuous-time system and the corresponding DE
system will have the same system responses at sampling times due to switched input and zeroorder hold input, respectively. A previous work had been done to show that using the Square
Pulse Sample and Hold Function (SPSHF) as the switched input, the corresponding Discrete
Equivalent (DE) system of a continuous-time non-minimum phase (NMP) system could have
minimum phase (MP) behavior. In this thesis, the switched input in the definition of DE system
was expended with another two sample and hold functions, Forward Triangle Sample and Hold
Function (FTSHF) and Backward Triangle Sample and Hold Function (BTSHF). The DE
systems and its Discrete Time (DT) systems equations were developed according to different
sample and hold functions. A simulation case study was proceeded to indicate the feasible
regions for selecting the sample and hold parameters for each switched input, which would lead
the resulting DE systems MP. The results of simulation case study implied that it was possible to
have an MPDT system with a smaller sampling period than discretizing the system directly using
zero-order hold (ZOH) method. In order to study the robustness of each sample and hold
function, the q-Markov cover system identification with Pseudo Random Binary Signal (PRBS)
was then studied for the purpose of doing controller-in-the-loop (CIL) simulation. The CIL
simulation used the dSPACE auto-box to simulate the system dynamics. A resistant-capacitor
(RC) filter was used to simulate actuator dynamics using different capacitors. At last the
characteristics of the three sample and hold functions were compared.

ACKNOWLEDGEMENTS

Two years had passed as if it was only two days. I can still remember the day that I
knocked at the door of Dr. Zhu’s office, he welcomed me with a worm smile that made me feel
like he was only an old friend but not a professor. Actually we did become friends. I really
appreciated the wisdom choice I made to be Dr. Zhu’s student. He treated his students as his
children, he thought what we thought and he felt what we felt. With his generous and patient
guidance, I finished my master thesis research. He also led me into the whole new area of control
by inspiring me to be confidence. He was always saying that “control is not a course or degree, it
is more about art, we can achieve the same goal in a thousand ways, and maybe none of them is
wrong. What we do is to enjoy the process.” This was what I learned during my way to
graduation and I really appreciated that. To be his student is my honor, my luck, and also my
destiny. Here, all I want to say is thank you.
I would also like to thank my committee members: Dr. Ranjan Mukherjee, who provided
the fundamental reference and help. He treated me as his buddy and supported me in both my
study and my life; Dr. Jongeun Choi, my first professor, who gave me the introduction of control,
and is always willing to help. Thank you for your help and support.
Without my friends in the lab, I could hardly imagine how hard my research will be. My
appreciation is also for you: Shupeng Zhang, Jie Yang, Tao Zeng, Hailiang Zheng, and Jerry
Yang.
Also, I would like to thank my girlfriend Changchang Yao for everything she did. She
pointed a way when I feel it was dark everywhere. She made me to believe I could do everything,
not for anyone, but for myself.
iii

Last but not least, to my parents: it was them, who supported me to finish my four years
collage and my two years graduate school. They know nothing about control, they did not even
know what mechanical engineering is about, but they could understand whether I’m suffering or
happy. They supported me without saying a word, but I could understand.
Actually I could never finish my acknowledgement in such few words. I simply wish
everyone healthy and enjoy their lives.

iv

TABLE OF CONTENTS

LIST OF TABLES ........................................................................................................................ vii
LIST OF FIGURES .................................................................................................................... viii
CHAPTER 1 INTRODUCTION .....................................................................................................1
1.1 Background and Motivation ...................................................................................................1
1.2 Research Overview ................................................................................................................2
1.3 Organization ...........................................................................................................................4
CHAPTER 2 DE SYSTEMS WITH THREE SAMPLE AND HOLD FUNCTIONS ...................5
2.1 DE System and the SPSHF ....................................................................................................5
2.2 The FTSHF and the Associate DE System ............................................................................7
2.3 The BTSHF and the Associate DE System ............................................................................8
2.4 The DT Systems ...................................................................................................................10
CHAPTER 3 A SIMULATION CASE STUDY ..........................................................................12
3.1 DT System with the SPSHF .................................................................................................13
3.2 DT System with the FTSHF .................................................................................................14
3.3 DT System with the BTSHF ................................................................................................16
3.4 Analysis ................................................................................................................................17
CHAPTER 4 Q-MARKOV COVER IDENTIFICATION USING PRBS INPUT ......................20
4.1 Introduction ..........................................................................................................................20
4.2 PRBS Used for Q-Markov Cover.........................................................................................22
4.3 System Identification............................................................................................................25
4.3.1 System Identification Steps.......................................................................................25
4.3.2 System Identification Parameter Determination .......................................................27
4.3.3 Identification Result of the SPSHF Case ..................................................................30
4.3.4 Identification Result of the FTSHF Case ..................................................................31
4.3.5 Identification Result of the BTSHF Case .................................................................32
4.4 DE System GUI for finding the DE systems ........................................................................33
4.4.1 Introduction ...............................................................................................................33
4.4.2 Procedure ..................................................................................................................34
4.4.3 Benefits .....................................................................................................................35
4.4.4 Future Work of the DE GUI .....................................................................................36
CHAPTER 5 CIL SIMULATION USING A DSPACE AUTO-BOX .........................................38
5.1 Introduction ..........................................................................................................................38
5.2 RC Filter ...............................................................................................................................42
v

5.3 CIL Result ............................................................................................................................44
5.4 Analysis ................................................................................................................................46
CHAPTER 6 CONCLUSIONS ....................................................................................................48
BIBLIOGRAPHY .........................................................................................................................50

vi

LIST OF TABLES

TABLE 1. Coefficients for maximum sequences .........................................................................23
TABLE 2. The pole and zero absolute value of the SPSHF-DT case ..........................................31
TABLE 3. The pole and zero absolute value of the FTSHF-DT case ..........................................32
TABLE 4. The pole and zero absolute value of the BTSHF-DT case ..........................................33
TABLE 5. Time constant VS. pole locations ...............................................................................43

vii

LIST OF FIGURES

Figure 1. The flowchart of MPDT system achieving process ........................................................3
Figure 2. Continuous input, ZOSHF and three switched sample and hold functions .....................5
Figure 3. A continuous-time system match with its DE system at sampling times .........................7
Figure 4. Maximum zero of the SPSHF-DT system VS. sample and hold parameters .................13
Figure 5. Pole zero plot of the SPSHF-DT system ........................................................................14
Figure 6. Maximum zero of the FTSHF-DT system VS. sample and hold parameters .................15
Figure 7. Pole zero plot of the FTSHF-DT system ........................................................................16
Figure 8. Maximum zero of the BTSHF-DT system VS. sample and hold parameters ................16
Figure 9. Pole zero plot of the BTSHF-DT system .......................................................................17
Figure 10. Regions of all the sample and hold functions for selecting the switch parameters so
that the resulting DT system is MP .............................................................................18
Figure 11. The layout of system identification using PRBS input ...............................................21
Figure 12. Several periods of PRBS signal with the order of 11 ...................................................24
Figure 13. The modified PRBS signal ..........................................................................................24
Figure 14. System response due to different inputs ......................................................................25
Figure 15. Simulation layout..........................................................................................................28
Figure 16. System Identification GUI ...........................................................................................29
Figure 17. The pole-zero plot of SPSHF-DT, identified and NMP systems .................................30
Figure 18. The pole-zero plot of FTSHF-DT, identified and NMP systems .................................31
Figure 19. The pole-zero plot of BTSHF-DT, identified and NMP systems .................................32
viii

Figure 20. The GUI for finding the MPDT system ......................................................................37
Figure 21. The flowchart of CIL Simulation ................................................................................39
Figure 22. dSPACE setup .............................................................................................................40
Figure 23. dSPACE control desk layout .......................................................................................41
Figure 24. The modified PRBS input signal ..................................................................................44
Figure 25. The zero moving trend of the identified system with SPSHF .....................................45
Figure 26. The zero moving trend of the identified system with FTSHF .....................................45
Figure 27. The zero moving trend of the identified system with BTSHF ....................................46
Figure 28. Overall zero moving trends of the three sample and hold functions ...........................47

ix

CHAPTER 1
INTRODUCTION

1.1 Background and Motivation
From transfer function point of view, in continuous-time domain, a NMP system is a plant
that has at least one zero locates in the right hand plan (RHP). And in discrete-time domain, a
NMP system has at least one zero locates outside of the unit circle. NMP system could be seen
everywhere in real life, including aircraft, hydraulic turbines, automotive continuously variable
transmissions, and so on [1]. People are always trying to avoid NMP characteristics when
considering a control method because of the problems caused by the unstable zeroes. Including
slow step response with large undershoot and an additional phase shift and excessive phase lag
with no attenuation at high frequencies [2]. During a control process, when a PI or PID controller
is applied to a NMP system, when necessary, a proportional gain may be very large, from the
aspect of Root Locus, when the proportional gain increases, the closed loop poles will move
toward the open loop zeroes, which may change the stability of the NMP system. So when a
NMP system is in the loop, prevent high gain control is applied, which leads to poor close-loop
performance [1].
In order to control NMP systems, several methods are studied [2]-[10] before. For example,
the fuzzy control was studied in [2] and [3] to eliminate the undershot in order to change the
transient response. In [4] an artificial neural network was studied to achieve improving the NMP
system performance. The genetic algorithm was used for off-line training in the artificial neural
network. In [5] a new method called direct adaptive control structure came up for controlling the
NMP systems. According to this result, the direct adaptive control method was also developed

1

for not necessarily MP systems based on simultaneous identification when using the input and
output of a prediction models [6]. Another method called variable structure system [7] was used
for controlling a NMP system by assigning the pole to achieve model reference control. The last
one, which is more close to the area researched in this thesis, an alternative approach presented
in [8], shows the study of DE system characteristics with a sample and hold function, different
from the zero order holding one. It specifically introduced the SPSHF. It had been validated that
a proper choice of sampling period and pulse duty cycle, it could make the DE system of a NMP
system become MP. In this thesis, a stable NMP system was studied as a numerical example.
Also in [9], it is found that a continuous time NMP system could be MP when discretizing the
system directly with zero-order hold method with a certain sampling rate. The theory from this
thesis will make the largest sampling rate for directly discretizing with zero-order hold even
larger.

1.2 Research Overview
In [8], the SPSHF was defined as the switched input for achieving the DE analytical state
space matrix equations. In this thesis, the switched input definition is expended with another two
sample and hold functions: FTSHF and BTSHF, and the corresponding DE equations are derived
in the same way as in [8].
When a different sample and hold function method is applied, the corresponding DE system
has the same poles, but the zeroes locate differently. Also, for the same sample and hold function
method, when the sample and hold parameters change, the zeroes moves accordingly. According
to this, it is possible to have the DE and DT of a NMP system to be MP. In the following
flowchart (figure 1), the process of obtaining a MPDT system of a NMP is shown.

2

DT System

DE System

Sample Data
System

SISO NMP
Plant

Holding
Functions

Continuous equivalent system with switched input
that matches output at sampling times

Discretized system MP or NMP

MP in discrete-time domain?
Figure 1. The flowchart of MPDT system achieving process

This method has its advantage of convenience which is no more need to study any methods of
controlling a NMP system with trying to avoid the negative effects of the system, but find the
MP DE system of the NMP system to determine the switched input, instead of studying the NMP
system itself.
However, this method has its disadvantage. According to different requirements, when the
duty cycle reduce, a large actuator with faster response time is needed, which means that the
energy cost may increase. As the limitation of actuator, the selection of the sample and hold
function duty cycle cannot be arbitrary small.

3

The main contribution of this thesis is to study the switched input of a NMP system. By
adjusting the sample and hold parameters, the NMP system could have MP characteristics.

1.3 Organization
The paper is organized as follows. Chapter II showed how the DE analytical state space
matrix equations with three sample and hold functions were driven. In chapter III, the properties
of the MPDE systems were studied, and also the DT systems. A numerical example showed how
to select sample and hold parameters for different sample and hold functions. In the next chapter,
system identification was studied as a tool for of doing controller-in-the-loop (CIL) simulation.
And also a built GUI was showed in order to make the calculation process simpler. Chapter V
was testing the robustness of this theory with a CIL real-time simulation. dSPACE auto-box was
used to be the hardware in the loop. The last chapter added some conclusions.

4

CHAPTER 2
DE SYSTEMS WITH THREE SAMPLE AND HOLD FUNCTIONS

2.1 DE System and the SPSHF
Consider the continuous time input u (t ) shown in Figure 2. (a). The input can be discretized
using Zero-Order Sample and Hold Function (ZOSHF) with the sampling period of δ and
represented by uˆ (t ) shown in Figure 2. (b). Base on the ZOSHF, the switched input is the SFSHF,
shown in Figure 2. (c).

u (t )

t0

t1

t2

(a)

tk

tk  

uZ (t )

t0

t1

t2

(b)

tk

tk  

u (t )

t0

t1

t2

(c)

tk
tk  p

tk  

u (t )

t0

t1

t2

(d)

tk
u (t )

t0

t1

t2

(e)

tk

Figure 2. Continuous input, ZOSHF and three switched sample and hold functions
5

The switched input u (t ) described in Figure 2. (c) is SFSHF and defined as

u (t ), if tk  t  tk  p
u (t )   Z k
if tk  p  t  tk  
0,

(1)

k  0, 1, 2, ...
In which uZ (tk ) is the ZOH value of u (t k ) over the interval [tk , tk   ] (shown in Figure 2. (b)),
and   (0, ) is the sample period of the input pulse, p  (0,1] is the duty cycle. Consider the
linear system in (2).

(2)

With the switched input in (1), the state variables at sampling times are defined as:
p

x(tk  p )  e Ap x(tk )   e A( p  ) Bu (tk   ) d
0

 e
x(tk 1 )

Ap

x(tk )  A1 (e Ap  I ) Bu (tk )

 e A(  p ) x(tk  p )
 e A x(tk )  A1 (e A  e A(1 p ) ) Bu (tk )

(3)

According to [10], the definition of DE system is defined as:
Definition 1 (DE): a continuous time system:

(4)

which has the state variables at sampling times tk   are given by


x (tk 1 )  e A x (tk )   e A (  ) B uZ (tk   )d
0

A

1

 e x (tk )  A (e

6

A

 I ) B u (tk )

(5)

is a DE system of system (2), if the state variables of (5) match with those in (3) at sampling
times tk , k  0, 1, 2, ... . This leads the state space matrices equation of DE system with the
SPSHF:
A 
B
C

A
1

  I  e  A   I  e  Ap  B
 C

(6)

The continuous time system with square pulse switched input and the corresponding DE system
are shown in Figure 3. The Matching behavior is clearly showed as well.
0.2
DE System
NMP System

0

-0.2

-0.4

-0.6

-0.8

-1

-1.2

0

1

2

3

4

5

6

7

8

9

10

Figure 3. A continuous-time system match with its DE system at sampling times

2.2 The FTSHF and the Associated DE System

7

With the same concept, the switched input definition is changed from the SPSHF to the
FTSHF and shown in Figure 2. (d). and defined is defined as:

 uZ (t )(tk  t )
 uZ (t ), if tk  t  tk  p

p
u (t )  
0,
if tk  p  t  tk  


(7)

k  0, 1, 2, ...
In where   (0, ) is the sampling period, p  (0,1] is the duty cycle of the switched input.
For the FTSHF, it is the same as SPSHF case: equaling the state variables of the original
system (2) and the state variables of the DE system with FTSHF in (8):

x(tk  p )

p

 e Ap xk   e A( p  ) Bu (tk   ) d
0

 e Ap xk  2 A1 (
x(tk 1 )

1 1 1 1 Ap
A 
A e  e Ap ) B
p
p

 e A( ( p )) x(tk  p )
 e A xk  2 A1 (

(8)

1 1 A(1 p )
1 1 A
A e

A e  e A ) B
p
p

The state space equation of DE system with the FTSHF is achieved:

A  A
B 

I  e 
 A

1

 1 1  Ap

I   I  B
 p A  e



(9)

C  C

2.3 The BTSHF and the Associated DE System
The last sample and hold function is the BTSHF which is shown in Fig. 2 (c) with dotted lines
and defined as:
8

 (t  tk )
uZ (tk ), if tk  t  tk  p

u (t )   p
 0,
if tk  p  t  tk  


(10)

k  0, 1, 2, ...
The process of achieving DE system with the BTSHF is the same as the FTSHF, substitute
(10) into (4), and the state variables are shown in (11):
p

x(tk  p )  e A(tk  p tk ) xk   e A( p  ) BuZ (tk   ) d
0

 e A( p ) xk 
x(ti1 )  e

A (  p )

2
2 2
2 2 Ap
p A1 B 
A B
A e B
p
p
p

(11)

x(tk  p )

 e A xk  2 A1 (

1
1 1 A(1 p )
1 1 A
p e A(1 p ) 
A e

A e )B
p
p
p

The DE system equation with the BTSHF is obtained by equaling (11) and (5):

A 
B
C

A

 1 1

 
A  I  e  Ap   e  Ap 
 p

 C

1

I  e  B
 A

(12)

From the equations of the DE systems (6), (9), and (12), DE system is related to the switched
input with its sample and hold parameters. And it is obvious that the input u (t ) is canceled out,
which means that the characteristics of the DE system only dependent on the sample and hold
parameters. The first assumption is that the original system state space matrices are minimal
realization from the system transfer function. According to this assumption and the equations of
DE systems, the poles of the DE system and the original system should be identical because

A  A . And the zeroes are different because B  B . And the B matrix of the DE system is

9

varying with the two sample and hold parameters (δ and p). As a result, it is possible that if a
specific combination of sample and hold parameters are chosen, the DE system could be MP.

2.4 The DT Systems
DT systems are achieved by discretizing the DE systems with zero order hold with the same
sampling period as the switched inputs. Apparently, the DT systems, DE systems, and the
original system will all have the same state variables at sampling times due to the same sample
and hold parameters and sample and hold function. The state space form of the DT system is
given by:

x(tk   )  Ad x(tk )  Bd u (tk ),

x(t0 )  x0

 Cd x(tk )

y(tk )

(13)

For the SPSHF, the DT system is calculated is referred as the SFPSH-DT system, and is
given by:
Ad

 e A

Bd

 M p A1  e A  I  I  e  A 

Cd

 C

1

I  e

 Ap

B

(14)

for the FTSHF-DT system:
Ad
Bd
Cd

 e A
1 

e Ap  I
 M t A1  e A  I  I  e  A   A1
 IB
p


 C

and the BTSHF-DT system:

10

(15)

Ad

 e A

Bd

 Mt A

Cd

 C

1

e

A

 I  I  e

 A



1

 1 I  e  Ap

 e  Ap  B
A
p



(16)

After discretization, the DT system with any sample and hold function described in this thesis
has the same pole locations. However the zeroes could be quite different since the control input
̅d of the DT system is different.
matrix B

11

CHAPTER 3
A SIMULATION CASE STUDY

An example of a continuous-time, SISO, NMP system with the form of transfer function
shown as below

G(s) 

(s 5)(s2  1)
(s 2)(s 1)(s2  2s 5)

(17)

To calculate the DE and the DT systems with state variables, the state space form of the
system has to be achieved. After minimal realization, the state space form of the system is

0
1
A
0

0
C  [0

0 0 10 
 5

1
0 0 19 
B 
 5
1 0 13

 
0 1 5 
1
0 0 1]
D  0

(18)

During the simulation, δ is bounded by 0 and 2. Since p is representing the duty cycle in
percentage, it is bounded by 0 and 1. In order to determine the feasible δ and p combination, 3-D
plots in Figure 4, Figure 5, and Figure 8 of the DT systems with the three sample and hold
functions are plotted. Z-axis is representing the max absolute value of the DT system, and X-Y
plane is representing the sample and hold parameters. As a result, the DT system is MP when the
max absolute zero is smaller than 1, hence the feasible regions for selecting the sample and hold
parameters so that the resulting DT system is MP is the part under the z = 1 plane.

12

3.1 DT System with the SPSHF
The 3-D plot of the SPAHF-DT system max zero distribution according to the sample and
hold parameters is shown in Figure 4:

Figure 4. Maximum zero of the SPSHF-DT system VS. sample and hold parameters
In this 3-D plot, the feasible region for selecting the sample and hold parameters is within the red
region. With the sample and hold parameter selected randomly, δ = 0.5, p = 0.4, the
corresponding pole-zero plot is:

13

The Zplane of DT System with SPHF
1

0.8

DT zero
DT pole
Unit Circle
NMP zero
NMP pole

0.6

Imaginary Part

0.4

0.2

0

-0.2

-0.4

-0.6

-0.8

-1
-2

-1.5

-1

-0.5
Real Part

0

0.5

1

Figure 5 Pole zero plot of the SPSHF-DT system
The red dots represent the NMP system zeroes and poles, and the blue dots represent the DT
system zeroes and poles. With this combination of the sample and hold parameters, the SPSHFDT system is MP.

3.2 DT System with the FTSHF
The 3-D plot of the FTSHF case is shown in Figure 6:

14

Figure 6. Maximum zero of the FTSHF-DT system VS. sample and hold parameters
It is obvious that the 3-D plot of the FTSHF-DT system is quite different from SPSHF-DT
system. These two plots has different feasible regions for selecting the sample and hold
parameters. With the sample and hold parameter selected randomly but inside the red region, δ =
0.5, p = 0.3, the corresponding pole-zero plot is:

15

The Zplane of DT System with FTHF
1

0.8

DT zero
DT pole
Unit Circle
NMP zero
NMP pole

0.6

Imaginary Part

0.4

0.2

0

-0.2

-0.4

-0.6

-0.8

-1
-2

-1.5

-1

-0.5
Real Part

0

0.5

1

Figure 7. Pole zero plot of the FTSHF-DT system

3.3 DT System with the BTSHF
The 3-D plot of the BTSHF-DT system is shown in Figure 8:

Figure 8. Maximum zero of the BTSHF-DT system VS. sample and hold parameters

16

With the sample and hold parameter picked within the red region randomly of δ = 0.5, p = 0.3,
the corresponding pole-zero plot of the BTSHF-DT system is shown in Figure 9:
The Zplane of DT System with BTHF
1

0.8

DT zero
DT pole
Unit Circle
NMP zero
NMP pole

0.6

Imaginary Part

0.4

0.2

0

-0.2

-0.4

-0.6

-0.8

-1
-2

-1.5

-1

-0.5
Real Part

0

0.5

1

Figure 9. Pole zero plot of the BTSHF-DT system

3.4 Analysis
In order to make it clearer, all the feasible regions of the three sample and hold functions are
plotted separately and overlapped in Figure 10. It has already been proved that if a combination
of the sample and hold parameters are selected within the feasible region, the resulting DT
system is MP. In Fig. 10. It depicts the difference between those three regions for each of the
sample and hold functions. Also, there are two blue straight dash-dotted lines lies horizontally
and vertically representing the fixed sampling rate and duty cycle.

17

1
11
0.8
1.00.8
0.81
0.6
0.80.8
0.6
0.4
0.60.6
0.4
0.4
0.2
0.2
0.4
0.4
0.2
0
00
0.20.20 0
0
00
10

p

(a)

MPDT

p

p

0.8

0.80.81

0.4

0.6

0.8

1

1.2

1.4

1.6

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0.8
0.8

11

1.2
1.2

1.4
1.4

1.6
1.6

0.2

1

1
1.0
0.8
0.8
0.81
0.8
0.6
0.6
0.8
0.6
0.6
0.4
0.6
0.4
0.4
0.2
0.4
0.20
0.2
00
0.2 0
00
0
01
0
1
1.0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

1.8
MPDT
1.8

1.8

2

2

2
2

(b)

MPDT
0.2
0.2

0.4
0.4

0.6
0.6

MPDT
1.8
1.8

22

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

(c)

0.61

0.60.6
0.8

MPDT

0.8
0.4

0.4
0.6
0.4

0.6
0.2

0.4
0.20.2

MPDT

0.40

00.20 0

p

0.2 0
1.00010
0
0.8
0.81

(d)

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0.2
0.2

0.4
0.4

0.6
0.6

0.8
0.8

1
1

1.2
1.2

1.4
1.4

1.6
1.6

1.8
1.8

2
2

0.8
0.8

1
1.0

1.2
1.2

1.4
1.4

1.6
1.6

1.8
1.8

22

1.2

1.4

1.6

1.8

2

0.679
0.8

0.61

1
0.6
0.4
0.8
0.8
0.354
0.2
0.4
0.6
0.2
0.6
0.2
0.4
00 00
0.4

0.485

0.20
0.2
0

0.2 0.3 0.32
0.40.42 0.55
0.60.6
0.2

0.2

0.4

0.6

0.8

00 0
0



1



δ

0.2
0.4
0.6
0.8
1.2
1.4
1.6
1.8
0.4
11
1.2
1.4
1.6
1.8
22
Figure 10. Regions of all the0.2
sample
and0.6
hold0.8
functions
for
selecting
the switch
parameters
so


that the resulting DT system is MP
From the result in [9], with ZOH method, the NMP system can also be discretized to MP
when the sampling period is sufficiently large. For this system, the zero-order discretizing
sampling period varies from 0.60s to 1.76s shown in Figure 10 (a) with the top boundary of the
shaded region. Note that this is a special case of the SPSHF at p is 1.

18

When p is at its minimum, which is 0.01 in here, the other sample and hold parameter δ are all
from 0.2112s, which is the smallest sampling period. This is because when the duty cycle is
extremely small, all of the three sample and hold functions converge to a narrow pulse.
From feedback control point of view, the smaller sampling period, the better. For a constant
duty cycle, which is 80% in Figure 10 (d), the FTSHF-DT system has the smallest sampling
period of 0.3232s, and the BTSHF has the largest sampling period, which is 0.5455s.
From the aspect of energy consumption for actuator, the lower the duty cycle is, the larger
magnitude it will be, the more energy it will cost for the actuator to generate the signal. From Fig
10, it is not hard to see that, for a constant sampling period, which is 0.3s, the FTSHF-DT system
has the largest duty cycle of 67.9%, then is the SPSHF-DT system, with 48.5% and the BTSHFDT system has the smallest duty cycle, which is 35.4%. That means the FTSHF input can obtain
a higher duty cycle so a lower energy is needed when comparing to the SPSHF and BTSHF
cases. From the sample and hold parameter point of view, the FTSHF has the best characteristics
among these three sample and hold functions.
No matter which sample and hold function is in use, the p value cannot reach 0. When p ≈ 0,
in each sample period, the input signal has to carry the whole information required in the whole
sample period in a significant small time period, which is not be able to actuate.

19

CHAPTER 4
Q-MARKOV COVER IDENTIFICATION USING PRBS INPUT

4.1 Introduction
System identification is a widely used method in modern control. It uses statistical methods
to build mathematical models of dynamical systems from measured data. The measured data
includes inputs and output (system response). According to the system behavior, system
identification considers the original system as a black box, and builds a new system which would
have the same response due to the same input. If the input changes due to a noise signal, the
system identification can generate a new system due to this changed input signal. Hence, the
system identification is used for testing the robustness of this theory. In the next chapter, a CIL
simulation with different error is done. After collecting the input and output data, the
corresponding DT system is generated by system identification. The robustness is tested by
plotting the pole-zero plot to see if the DT system is still MP after adding the error generator. If
there is no error generated in the CIL, according to definition 1, the mathematical dynamical
model generated by system identification should be a DT system corresponding to the original
system, and will have exactly the same poles with the original system. The process and principle
of system identification is shown in Fig. 11.

20

0.5

1

0.8

0.4
0.6

0.4

0.3

0.2

0

(s  5)(s  1)
G (s) 
(s  2)(s  1)(s 2  2s  5)
2

-0.2

-0.4

-0.6

-0.8

-1
0

1

2

3

4

5

6

7

8

9

10

PRBS with Selected
Sample and Hold
Functions

0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5

NMP System

Collected at sample
period of δ

Q-Markov Covariance
Equivalent Realization
Identification

0

1000

2000

3000

4000

5000

6000

7000

8000

System Response
Collected at sample
period of δ

Minimum Phase DT System

Figure 11. The layout of system identification using PRBS input
Usually, the q-Markov Covariance equivalent Realization method used for identification
required white or pulse response ([11] and [12]). But neither of them is a perfect signal.
PRBS is a signal that can be easily generated by shift register circuit. For doing system
identification, according to different systems, the order, sampling period and magnitude can be
adjusted to have the system well excited. The original q-Markov Cover method for identification
required white noise or pulse response [13], which are not perfect input signals. For the white
noise, because it is an unrealizable ideal, no signal generators can generate the white noise
exactly, hence the identification error is formed at the very beginning of the process.
Furthermore it has another huge defect is that it conflicts with saturations on the physical
actuators because of its unbounded property. For the pulse response, it is not suitable for all
systems because of the limited energy. Consequently, the noise in the process could cover the
pulse response for some specific situations. After all of these considerations, the PRBS was used
to replace the previous input signals [13].

21

4.2 PRBS Signal
The most commonly used PRBS is based on maximum length sequence (called m-sequence).
n
This type of PRBS is with the length of m  2  1 , in which n represents PRBS order. Let z 1 be
1
delay operator and pˆ (z ) be the polynomials [13]

pˆ( z 1 )  an z n1 

 a2 z 1  a1

(19)

where a i is either a > 0 or -a, and for any number , the calculation of a  a obeys,

1  1  1  1  1, 1  1  1  1  1

(20)

, n ) so that pˆ (z 1 ) is

In TABLE 1 [11], lists the selection of coefficients a i ( i  1, 2,

obtained to be a maximum sequence. Table 1 is just a part of the whole table in Ref [11], in
which the order of greater than 12 and smaller than 34 can be found. In here, order 11 is good
enough for doing system identification with this system. The PRBS signal can be generated by
the equation shown below:
uˆ[(k  1)Tp ]  pˆ ( z 1 )uˆ(kTp ), k  0, 1, 2,

where uˆ (0)  1 and uˆ ( 1Ts )  uˆ ( 2Ts ) 

,

(21)

 uˆ ( nTs )  0 , in where Ts is the PRBS generating

rate.
For q-Markov Cover use, the PRBS need to be inversed, and the inversed PRBS is calculated as

 a k even
u(k )  s(k )  [a  2 a uˆ (k )], s(k )  
a k odd
in which u(k ) has a period of 2m and u(k )  u(k  m) . The mean of the inverse PRBS is
22

(22)

1 2m1
mu  E2m (kTp ) 
 u(kTp )  0
2m i 0

(23)

TABLE 1. Coefficients for maximum sequences
Order of Polynomial

Period of the sequence m

n

Non-negative coefficients a i

2

3

a1 , a 2

3

7

a 2 , a3

4

15

a3 , a 4

5

31

a 3 , a5

6

63

a5 , a 6

7

127

a 4 , a7

8

255

a2 , a3 , a4 , a8

9

511

a5 , a 9

10

1023

a7 , a10

11

2047

a9 , a11

Since the inversed PRBS signal is the only thing used for q-Markov Cover system
identification, in the following of this thesis, “PRBS” stands for inversed PRBS. A part of the
PRBS signal with the order of 11 is shown in Figure 12:

23

1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0

1

2

3

4

5

6

7

8

9

10

Figure 12. Several periods of PRBS signal with the order of 11
In Figure 13, it is the modified PRBS signal, which is combing PRBS and the ZOSHF and
the three switched sample and hold functions. The first plot is the PRBS signal combined with
ZOSHF. The second plot is the PRBS combined with SPSHF, the third is combined with FTSHF,
and the last one is combined with BTSHF.
1
ZOHF

0
-1
0

1

2

3

4

5

6

7

2

8
SPHF

0
-2
0

1

2

3

4

5

6

7

8

5
FTHF

0
-5
0

1

2

3

4

5

6

7

8

5
BTHF

0
-5
0

1

2

3

4

5

6

Figure 13. The modified PRBS signal

24

7

8

The sample and hold parameters are selected according to the former section to make sure that
the resulting DT system is MP. Due to the different input signals, the NMP system response are
different and shown in Figure 14.
0.5
ZOHF

0
-0.5

0

1

2

3

4

5

6

7

0.5
0
-0.5

8
SPHF

0

1

2

3

4

5

6

7

0.5
0
-0.5

8
FTHF

0

1

2

3

4

5

6

7

0.5
0
-0.5

8
BTHF

0

1

2

3

4

5

6

7

8

Figure 14 System response due to different inputs

4.3 System Identification
In this thesis, the system identification is done by using a Matlab GUI file [14]. The PRBS
GUI was developed by Dr. Guoming Zhu for multi-rate PRBS q-Markov Cover. The number of
Markov parameters besides the order of the identified system model can be adjusted by using the
system identification GUI.

4.3.1 System Identification Steps
There are three steps of doing the system identification with PRBS and its parameter
selection:

25

Step1: Design the PRBS system identification
To design the experiment, according to parameter m, the magnitude and the order of the
PRBS signals need to be determined first. Usually, the magnitude of PRBS is determined by
choosing the most feasible number so that the ratio of system output signal to noise signal is
maximized. The order of PRBS is chosen based on covering the dominated frequencies of the
identified system such that the identified system could be excited by PRBS signal. Also, the
output sample period T need to be determined secondly, which is basing on the actuator response
to PRBS signal.
Step 2: Run the system identification test and collect data
After all the necessary parameters are determined, the next step is to run and collect both
input and output (system response) data. In order to achieve an accurate system response, a
steady state response is required, which means that for a certain system, the tenth cycle or more
could be used for data collection.
Step 3: Run the GUI to do the system identification
Convert all the data collected in the last section to a Matlab data file. The PRBS system
identification GUI can be used to obtain a linearized system model. During the use of GUI, the
output weighting matrix Q, q of Markov and covariance parameter are need to be determined.
The parameters are determined based on different identification systems. The weighting matrix is
used for balancing the output response scale for different output channels. The q of Markov is
used for adjusting the identification result by telling how well the identified model response
matches the original system response.

26

4.3.2 System Identification Parameter Determination
The order of the PRBS signal is set to be 11, so that the length is enough to excite the system
to do the system identification. Since the PRBS signal will combined with the sample and hold
functions, and the magnitude is set to be 1. With this two parameters (order and magnitude), the
PRBS can be generated by a Matlab m file. With the signal, a Simulink file could be established.
In the Simulink, the sample period of PRBS has to match with the sample and hold function’s
sample period, which is set to 0.5s for each sample and hold function. The duty cycle of each
sample and hold function is set the same as the p in last chapter. Since the bandwidth of the
system is 21.23Hz, according to Shannon-Hartley’s theorem [15], the smallest simulation step
size is determined to be 0.016s. So in here, during the simulation, the simulation step size is set
to be 0.001s.
The following figure is the Simulink layout for achieving the system output by inputting a
PRBS

signal.

27

Figure 15. Simulation layout

28

Figure 16. System Identification GUI

29

After running the Simulink, the output data including the modified input signal and the
system response signal are collected both at sampling period of 0.5s, which is the same as the
switched input sampling periods, and imported to the system identification GUI. In the GUI, the
number of Markov parameter is set to be 30 and the order of the identified model is 4 for a better
identification result. The identified system is then achieved.

4.3.3 Identified Result of the SPSHF Case
Since the input and output data were collected at the fixed sampling time of 0.5s, the
identified system was in discrete time domain. According to Figure 3 and Figure 9, the sample
and hold parameters were set to δ = 0.5, p = 0.4, because in Figure 3, this set of sample and hold
parameters made the maximum absolute zero of the SPSHF-DT system under the plane of z = 1,
so that the DT system was MP. The system identification result is shown in Figure 1.
The pole-zero plot of DT and ID system with SPHF
ID-zero
ID-pole
1
DE-zero
DE-pole
NMP zero
NMP pole

Imaginary Part

0.5

0

-0.5

-1

-1.5

-1

-0.5

0

0.5

1

Real Part

Figure 17. The pole-zero plot of SPSHF-DT, identified and NMP systems
In the following table, the numerical poles and zeroes of Figure 17 are listed.
30

TABLE 2. The pole and zero absolute value of the SPSHF-DT case
IDENTIFIED POLE

DT POLE

IDENTIFIED ZERO

DT ZERO

0.6069

0.6065

0.9456

0.9458

0.6069

0.6065

0.9456

0.9458

0.6117

0.6065

0.4501

0.4455

0.3539

0.3679

4.3.4 Identified Result of the FTSHF Case
The sample and hold parameters were selected to be δ = 0.5, p = 0.3 for the FTSHF case. The
result of system identification is shown in Figure 18.
The pole-zero plot of DT and ID system with FTHF
ID-zero
ID-pole
1
DE-zero
DE-pole
NMP zero
NMP pole

Imaginary Part

0.5

0

-0.5

-1

-1.5

-1

-0.5

0

0.5

1

Real Part

Figure 18. The pole-zero plot of FTSHF-DT, Identified and NMP
systems
In the following table, the numerical poles and zeroes of Figure 18 are listed.

31

TABLE 3. The pole and zero absolute value of the FTSHF-DT case
IDENTIFIED POLE DT POLE

IDENTIFIED ZERO

DT ZERO

0.6081

0.6065

0.9528

0.9537

0.6081

0.6065

0.9528

0.9537

0.6209

0.6065

0.2082

0.1899

0.3247

0.3679

4.3.5 Identified Result of the BTSHF Case
By applying the same process, the system identification result of BTSHF with

  0.5, p  0.3 is in Figure 19 and Table 4.
The pole-zero plot of DT and ID system with BTHF
ID-zero
ID-pole
1
DE-zero
DE-pole
NMP zero
NMP pole

Imaginary Part

0.5

0

-0.5

-1

-1.5

-1

-0.5

0

0.5

1

Real Part

Figure 19. The pole-zero plot of BTSHF-DT, identified and NMP systems
And the poles and zeroes are shown in the following table:

32

TABLE 4. The pole and zero absolute value of the BTSHF-DT case
IDENTIFIED POLE

DT POLE

IDENTIFIED ZERO

DT ZERO

0.6074

0.6065

0.9407

0.9425

0.6074

0.6065

0.9407

0.9425

0.6161

0.6065

0.3080

0.2789

0.3415

0.3679

From the three figures and the tables list in this section, the poles and the zeroes of the
identified system generated by the system identification match with the calculated DT system
perfectly with the same sample and hold parameters, which means that the system identification
can identify the system precisely and it is good for doing CIL simulation.

4.4 DE System GUI for Finding the MPDT Systems
4.4.1 Introduction
In order to make the calculation process faster and more convenient, a GUI was made for
finding the DT systems according to SPSHF, FTSHF and BTSHF. 3-D plots and 2-D plots are
made for observing the feasible regions for selecting sample and hold parameters so that the
resulting DT system is MP.
The DE GUI layout is shown in Figure 20. It consists five buttons, five figures and two tables.
The buttons are “Load System”, “RUN”, “Save”, “Parameter Selection” and “Achieve MP”. The
figures are “original pole-zero plot”, “holding function plot”,”3D plot of the holding parameter
33

VS. system zeroes”, “Feasible parameter area plot”, and “DT pole-zero plot”. The two tables are
used for displaying the poles and zeroes of the corresponding DT system.
4.4.2 Procedure
The “Load System” button is used for loading an external transfer function with the form of a
Matlab data file into the GUI system. The transfer function could be either NMP or MP, the GUI
will tell which sort of system it is and respond correctly.
After input the system transfer function, a sample and hold method should be chosen by
using the pop up menu. Three sample and hold function options are listed in the pop up menu,
which are “Pulse”, “Forward Triangle” and “Backward Triangle”. Once a sample and hold
method is selected, the general picture of the sample and hold function will appear in the figure
on the up right corner.
Then, the “RUN” button should be clicked for plotting the 3D plot of the sample and hold
parameter VS. system zeroes. After clicking, it will take a while for Matlab calculating the
matrix and plot it out. The 3D plot is the first overlook of choosing the sample and hold
parameters.
In order to make it clearer, the “Parameter Selection” button is used for plotting the feasible
sample and hold parameter area (green portion). By using the two slid bar (horizontal one is used
for choosing delta, and the vertical one is for choosing p), the feasible sample and hold
parameters are selected. When the bar moves, the corresponding parameter value will appear in
the text box in the middle. The cross section of those two red lines represents the sample and
hold parameter combination. So if the cross section is inside the green area, the outcome DT

34

system of the GUI will be MP. The desired sample and hold parameter could also be selected by
inputting directly into the edit box, which will also affect the slider bar position.
“Achieve MP” button is used for plotting the pole-zero plot of the achieved DT system.
Once it is clicked, the pole-zero plot will appear in the “DT pole-zero plot” figure. At the same
time, the poles and zeroes of the DT system will appear in those two tables also with their
absolute value. By looking at this figure or the tables, the characteristic of the DT system could
be observed.
At last, all the results including the original system, original pole and zero, sample and hold
method, sample and hold parameters, and the DE system will be saved automatically by clicking
the “Save” button. All the result will saved in a single Matlab data file in the same directory with
the loaded system file.

4.4.3 Benefits
This GUI has five benefits:
1. It becomes easier to identify a system is a MP or NMP. Simply input the system into the
GUI, and look at the first figure –“original pole-zero plot”.
2. It becomes faster to tell if a system has its MPDT or not by looking at the 3D plot or the
“Feasible parameter area plot”. If there is a certain area under the z = 0 plan, it means that
the system has its MPDT system.
3. It is much faster to find the DT system
4. The choice of finding DT system is visual and adjustable

35

4.4.4 Future work of the DE GUI
1. The higher the order of the system is, the longer time the calculation needs. Due to the
problem of long time calculation, the Matlab code could be optimized to achieve the
results in a shorter time.
2. The color of the outlook could be optimized in order to make the result clearer.
3. A new portion could be developed to show the transfer function of the DT system.
4. The boundary of the feasible area should be plotted in order to show the area clearer.

36

Figure 20. The GUI for finding the MPDT system

37

CHAPTER 5
CIL SIMULATION USING A DSPACE AUTO-BOX

5.1 Introduction
In this thesis, the dSPACE auto-box and a RC filter were connected to the system to perform
as real-time simulation in the CIL system. dSPACE was responsible for all the calculation
processes including the system dynamics and the modified PRBS input signal. Also, in order to
test the robustness of this theory, an adjustable RC filter was connected with the dSPACE autobox to simulate actuator dynamics using different capacitors. The system Simulink model and
the signal generating file were built in the main CPU board in the dSPACE. The control signal
was sampled and held in dSPACE, output by the D/A board, filtered by the circuit and input to
dSPACE by A/D board.
In Figure 21 and Figure 22, the flowchart and the actual layout of the whole process are
showed. After the CIL simulation was conducted, the modified PRBS input and the
corresponding system output data were collected both at the sampling period of δ (0.5s). With
the collected data, the system identification was conducted and form the corresponding identified
systems. Comparing the different identified systems due to the change of capacitors of the RC
filter with the MPDT system, the results of robustness test could be observed.

38

A/D

D/A
DSPACE 1005 CPU

RC Filter

Unideal actuator dynamics
Figure 21. The flowchart of CIL Simulation

39

Oscilloscope

dSPACE

Power

RC Filter

Figure 22. dSPACE setup

40

Figure 23. dSPACE control desk layout

41

5.2 RC Filter
As shown in figure 21, the RC filter was a simple resistance-capacitor circuit. To be an
adjustable RC filter, the resistance was fixed at 1kΩ, and the capacitors used in this experiment
were 1 µF, 3.3 µF, 10 µF, 47 µF, and 100 µF. The system was simulated using each value of the
capacitor to have different system responses.
During the simulation, the capacitor would discharge its stored energy through the resistor.
By applying Kirchhoff’s voltage equation [16], yielding(24).
V1 ( s )  ( R 

1
) I ( s)
Cs

(24)

1
)
Cs

(25)

And the output voltage is

V2 ( s )  I ( s )(

Therefore, solving (24) for I ( s) and substituting into(25):

V2 ( s ) 

(1 / Cs)V1 ( s)
R  1 / Cs

(26)

Then the transfer function is obtained:

G( s ) 

V2 ( s)
1
1


V1 ( s) RCs  1  s  1

(27)

 is the time constant of the network. With different capacitance, the time constant and the pole
of equation (32) in continuous time domain are shown in Table 5.

42

TABLE 5. Time constant VS. pole locations
TIME CONSTANT (S)

POLE

0.001

-1000

0.0033

-303

0.01

-100

0.047

-21.27

0.1

-10

The poles of the original system in the numerical simulation section were 2 , 1 , 1  2i ,

1  2i , the RC transfer function poles were too small when comparing with the original system
poles. As a result, the RC filter poles were ignored as a part of the transfer function due to the
0.001 simulation step length, instead, it was only been considered as an error, or noise. The
modified PRBS signal changed due to different capacitors are shown in Figure 24.

43

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

100

200

300

400

500

2

700

800

900

0

1000

1
0.5

500

550

600

650

700

750

800

850

900

950

0

100

200

300

400

500

600

2

w/o dSPACE
0 µf
1 µf
3.3 µf

1.5

0
450

600

1
0.5
0
450

800

900

1000

w/o dSPACE
10 µf
47 µf
100 µf

1.5

1000

700

500

550

600

650

700

750

800

850

900

950

1000

500

550

600

650

700

750

800

850

900

950

1000

PRBS ReTri Input from 0uf to 3.3 uf
2

2

1.5

1.5

1

1

0.5

0.5

0
450

500

550

600

650

700

750

800

850

900

950

0
450

1000

Figure 24. The modified PRBS input signal
From figure 24, it was obvious that when the capacitance was increasing, the modified input
becomes smoother and the magnitudes were smaller.
5.3 CIL Result
Due to the change of the input signal, the system response will definitely change. As a result
of observation, as the capacitor was increasing, the outcome of the system identification was
changing by moving the locations of the zeroes towards to a NMP direction (outside of the unit
circle). The simulation results of SPSHF-DT, FTSHF-DT and BTSHF-DT systems are shown in
Figure 25, Figure 26, and Figure 27.

44

Zero Movement
DE zero

1

ID-zero 0uf

0.8

ID-zero 1uf

0.6
ID-zero 3.3uf

0.4

Imaginary Part

ID-zero 10uf

0.2
ID-zero 47uf

0
ID-zero 100uf

-0.2
NMP zero
NMP pole

Imaginary Part

-0.4
-0.6

0.01
-0.8
-0.010
-1.8

-1.6

-1.4

-1.2

-1

-1

-0.8

-0.6

Real Part
-1.5

-1

-0.5

0
Real Part

0.5

1

1.5

Figure 25. The zero moving trend of the identified system with SPSHF
Zero Movement
DE zero

1

ID-zero 0uf

0.8
ID-zero 1uf

0.6
ID-zero 3.3uf

Imaginary Part

0.4

ID-zero 10uf

0.2

ID-zero 47uf

ID-zero 100uf

0

NMP zero

-0.2

NMP pole

Imaginary Part

-0.4

-0.6

0.02
-0.8
0
-0.02
-1.8

-1.6

-1.4

-1.2

-1
Real Part

-1
-1.5

-1

-0.5

-0.8

0

-0.6

0.5

-0.4

-0.2

1

Real Part

Figure 26. The zero moving trend of the identified system with FTSHF

45

1.5

Zero Movement
DE zero

1

ID-zero 0uf

0.8
ID-zero 1uf

0.6
ID-zero 3.3uf

0.4

Imaginary Part

ID-zero 10uf

0.2

ID-zero 47uf

ID-zero 100uf

0

NMP zero

-0.2

NMP pole

Imaginary Part

-0.4

-0.6

0.02
0 -0.8
-0.02
-1.8

-1.6

-1.4

-1.2

-1
Real Part

-1
-1.5

-1

-0.5

-0.8

0

-0.6

0.5

-0.4

1

-0.2

1.5

Real Part

Figure 27. The zero moving trend of the identified system with BTSHF

5.4 Analysis

From the CIL result figures, it is clear that for all of the three sample and hold functions, the
negative zero moving trend was the most significant and they would become smaller than -1 at a
certain capacitance value. In this experiment, when the capacitance reached 100 µF, only the
negative zero of the SPSHF became smaller than -1.
For the purpose of comparing the robustness of all the sample and hold functions, the zero
moving trends of the three sample and hold functions are plotted in Figure 28. The vertical value
of the plot is the difference between the new identified systems negative zero and the original
MPDT negative zero.

46

Zero Movement for All Three Inputs
0.7
Zero Shiftment for Pulse Input
Zero Shiftment for Triangle Input
Zero Shiftment for Reversed Triangle Input

0.6

0.5

0.4

0.3

0.2

0.1

0

0

20

40

60

80

100

Figure 28. Overall zero moving trends of the three sample and hold functions

From Figure 28, only the SPSHF-DT system become NMP when the capacitance of the RC
filter reached 100 µF. The moving trends of the other two sample and hold method were smaller
than the SPSHF. The smallest was the FTSHF, which means that the FTSHF method had the best
robustness.

47

CHAPTER 6
CONSLUSIONS

A new way of studying a NMP system was discovered by a previous work, which was
focusing on finding the DE system of a NMP system. The DE system with a ZOSHF input would
have exactly the same response with a NMP system with a switched input at sampling times. The
switched input used here was defined as SPSHF and the DE system state space matrices equation
was developed by equaling the sate variables of the DE system and the NMP system at sampling
times. And it was also proved that with a proper choice of the sample and hold parameters, the
corresponding DE system could be MP.
In this thesis, the definition of the switched input was expanded with another two sample
and hold functions, which were FTSHF and BTSHF. Followed the same steps of finding the DE
system in the previous work, the DE system state space equations of the FTSHF case and
BTSHF case were developed. A New definition called DT system was introduced by discretizing
the DE systems. For each case, the feasible combinations of the sample and hold parameters for
achieving MP characteristics were studied. 3D figures were plotted for indicating the relationship
between the sample and hold parameters and the maximum absolute zero of the corresponding
DT systems. When comparing the feasible regions for selecting the sample and hold parameters,
the FTSHF had the better choices (larger region) of a smaller sampling period and a larger duty
cycle. Also comparing with the zero-order discretizing, smaller sampling period was achieved
for the MP DT system.
The robustness study was completed using CIL simulations with the help of system
identification. After doing the CIL simulation, the results showed that the SPSHF has the worst
48

robustness with increased actuator modeling error; while FTSHF provided the best robustness
among the three sample and hold functions.
As a summary, the best sample and hold function is the FTSHF due to its robustness
property with the largest feasible sample and hold parameter region.

49

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50

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