RESET CONTROL SYSTEMS: STABILITY, PERFORMANCE ANDAPPLICATION

A Ph.D. Dissertation Presented

by

Qian Chen

Submitted to the Graduate School ofUniversity of Massachusetts Amherst

in partial fulfillment of the requirements for the degree of

DOCTOR OF PHILOSOPHY

September 2000

Mechanical Engineering

c° Copyright by Qian Chen 2000All Rights Reserved

RESET CONTROL SYSTEMS: STABILITY, PERFORMANCE ANDAPPLICATION

A Dissertation Presented

by

Qian Chen

Approved as to style and content by:

Yossi Chait, Chair

C. V. Hollot, Co-Chair

Theodore E. Djaferis, Member

Robert Gao, Member

Corrodo Poli, Department HeadMechanical Engineering

Dedicated to my parents and my wife

ACKNOWLEDGMENT

I really appreciate my advisors, Professors Yossi Chait and Christopher V. Hol-

lot, for their thoughtful guidance and contribution to this work. I also appreci-

ate their kindly helps, in both academic research and many other aspects, during

my three years of study here. My fruitful learning and achievement are based on

their patient guidance, encouragement and support. I would like to thank Professor

Theodore Djaferis and Professor Robert Gao for serving in my committee. I also

learned much while taking their courses and want to thank them for their instruc-

tions.

Thanks are also due to the other members in our research group of reset control,

Orhan Beker and Dr. Yuan Zheng. Their research are very helpful to this work and

Orhan Beker also provided many beneficial discussions and suggestions. This work

is supported by National Scientific Foundation and this support is appreciated.

I would like to thank all my colleagues in DACS Lab, UMASS, Amherst for

their kindly help during my three years of research and study. Especially I need to

thank Varodom Toochinda and Prof. Jinfu Liu for their help.

Finally, I am indebted to my wife for her support and encouragement. Without

her support, this work would not have been possible. I would like to devote this

dissertation to my parents and my wife.

v

vi

ABSTRACT

RESET CONTROL SYSTEMS: STABILITY, PERFORMANCE ANDAPPLICATION

SEPTEMBER 2000

QIAN CHEN, B.S., EAST CHINA NORMAL UNIVERSITY

M.S., EAST CHINA NORMAL UNIVERSITY

Ph.D., UNIVERSITY OF MASSACHUSETTS AMHERST

Directed by: Professors Yossi Chait and C. V. Hollot

Linear time invariant (LTI) control design is the most widely applied control

design technique. But it has inherent limitation. In LTI feedback control systems,

high-frequency loop gain is linked to both low-frequency loop gain and stability

margins through Bode’s gain-phase relationship. This linkage results in trade-offs

among competing performance specifications which LTI control design can not pro-

vide remedy. Reset control design is a great candidate of improving this limitation

in LTI control design. The basic idea of reset control is to reset the state of a linear

controller to zero whenever its input meets a threshold. Such a reset controller is

introduced in feedback control systems with the aim of providing better trade-offs

between competing specifications than could be achieved using linear controllers.

There have been some successful experimental applications of reset control tech-

nique. These examples demonstrate that reset control has the potential benefit of

improving the inherent performance trade-offs in LTI control, while inherits the ad-

vantages of LTI control design. However, during the past three decades, there is a

lack of theoretical results for reset control systems. For example, none of the exper-

imental reset control applications can provide formal proof of stability and theoretic

analysis of performance. Dedicated to solving this problem, this dissertation per-

forms a complete theoretic analysis of the stability and time-domain performance

vii

of reset control systems. A complete set of theoretic results on the stability and

time-domain performance are developed. These results build a solid theoretic base

for the further and wider application of reset control system. Also, reset control

design technique is applied successfully for the speed control of a rotational mechan-

ical system. This experiment demonstrates the benefit of reset control design over

LTI control design and the effectiveness of the theoretic results developed in this

dissertation.

viii

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vii

TABLE OF CONTENTS

Page

ACKNOWLEDGMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

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

CHAPTER

1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Description of Reset Control System . . . . . . . . . . . . . . . . . . . 6

1.3 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Dissertation Contributions . . . . . . . . . . . . . . . . . . . . . . . . 12

1.5 Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2. STABILITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 BIBO Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Robustness to Implementation Errors . . . . . . . . . . . . . . . . . . 23

2.4 Steady-State Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

viii

3. PERFORMANCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2 Steady-State Performance . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3 Transient Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.4 Input Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4. RESET CONTROL SYSTEMS WITH SECOND-ORDER PLANTS . . . . 47

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.2 BIBO Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.3 Zero Input Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.4 Step Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.5 A Specific Class of Reset Control Systems . . . . . . . . . . . . . . . . 59

4.5.1 Asymptotic and BIBO Stability . . . . . . . . . . . . . . . . . 59

4.5.2 Step Response . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.6 Demonstration of Transient Response Analysis . . . . . . . . . . . . . 61

5. EXPERIMENTAL APPLICATION . . . . . . . . . . . . . . . . . . . . . . 65

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.2 System Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.3 Performance Specifications and LTI control design . . . . . . . . . . . 70

5.4 Reset Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

ix

5.4.1 Design of Linear Controller C(s) . . . . . . . . . . . . . . . . . 79

5.4.2 Design of Reset Element . . . . . . . . . . . . . . . . . . . . . 81

5.4.3 Stability and Steady-State Behavior Analysis . . . . . . . . . . 84

5.4.4 Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . 87

5.5 Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6. CONCLUSIONS AND FUTURE WORK . . . . . . . . . . . . . . . . . . . 101

APPENDIX

A. Proof of |p11(τ )| < 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

B. Example of designed controller with µ-analysis toolbox . . . . . . . . . . . 106

C. A minimal realization of L(s) . . . . . . . . . . . . . . . . . . . . . . . . . . 107

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

x

LIST OF FIGURES

1.1 Typical LTI feedback control system . . . . . . . . . . . . . . . . . . . . 1

1.2 Illustration of “cost of feedback” . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Describing function of Clegg-integrator . . . . . . . . . . . . . . . . . . . 4

1.4 Describing function of FORE (b=25) and Bode-plot of 1/(s+25) . . . . . 5

1.5 Block diagram of reset control system . . . . . . . . . . . . . . . . . . . . 6

1.6 Comparison of step responses for the reset control system (solid)and its base linear system (dot) . . . . . . . . . . . . . . . . . . . . . . . 8

1.7 Comparison of responses to 2 rad/sec sinusoidal sensor noise forthe reset control system (solid) and its base linear system (dot) . . . . . 9

1.8 Comparison of response to 0.2 rad/sec sinusoid disturbance forthe reset control system (solid) and its base-linear system (dot) . . . . . 9

2.1 Schematic of the tape-speed control system . . . . . . . . . . . . . . . . . 30

2.2 The plot of Re[h(jω)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3 The plot of Re[h(jω)] with the selection b=22, 100 and 200 respectively. 32

2.4 Zero input response with sensor noise . . . . . . . . . . . . . . . . . . . . 33

3.1 Step response with sensor noise and step disturbance . . . . . . . . . . . 38

3.2 Comparison of the step responses of reset control systm and itsbase-linear system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

xi

3.3 Demonstration of “input decomposition” principle for reset con-trol system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.1 Zero input response of second-order reset contorl system . . . . . . . . . 54

4.2 Step response of second-order reset system . . . . . . . . . . . . . . . . . 56

4.3 A class of reset control systems . . . . . . . . . . . . . . . . . . . . . . . 59

4.4 Analysis of step response of second-order reset control system . . . . . . 63

4.5 Relationship between maximum overshoot and M . . . . . . . . . . . . . 64

4.6 Step response of redesigned reset control system . . . . . . . . . . . . . . 64

5.1 Picture of the rotational flexible mechanical system . . . . . . . . . . . . 65

5.2 Setup of rotational flexible mechanical system . . . . . . . . . . . . . . . 66

5.3 Block diagram of the servo system . . . . . . . . . . . . . . . . . . . . . 66

5.4 Comparison of the measured and identified frequency responses ofthe plant P(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.5 Block diagram of the feed-back control system . . . . . . . . . . . . . . . 70

5.6 Loop-shaping of LTI design 1 on Nichols chart . . . . . . . . . . . . . . . 73

5.7 Loop-shaping of LTI design 2 on Nichols chart . . . . . . . . . . . . . . . 75

5.8 Bode plots of LTI design 1 and 2 . . . . . . . . . . . . . . . . . . . . . . 76

5.9 Comparison of step response for LTI design 1 and 2 . . . . . . . . . . . . 76

5.10 Comparison of sinusoid sensor noise response for LTI design 1 and 2 . . . 77

xii

5.11 Comparison of disturbance response for LTI design 1 and 2 . . . . . . . . 77

5.12 Block diagram of reset control system design . . . . . . . . . . . . . . . . 78

5.13 The design table for FORE’s pole b . . . . . . . . . . . . . . . . . . . . . 82

5.14 Reset control design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.15 Plot of Re[h(jω)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.16 Plot of Re[h(jω)] with the selection of b=10, 30 and 100 respectively . . 85

5.17 Step response of reset control system with sensor noise and disturbance. 86

5.18 Step response of reset control design . . . . . . . . . . . . . . . . . . . . 87

5.19 5 Hz sinusoid sensor noise response of reset control design . . . . . . . . 88

5.20 0.5 Hz sinusoid disturbance of reset control design . . . . . . . . . . . . . 88

5.21 Comparison of step responses of reset control design and LTI de-sign 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.22 Comaprison of the responses of reset control design and LTI design1 to 5 Hz sinusoid sensor noise . . . . . . . . . . . . . . . . . . . . . . . . 90

5.23 Comparison of step responses of reset control design and LTI de-sign 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.24 Comparison of the responses of reset control design and LTI design2 to 5 Hz sinusoid sensor noise . . . . . . . . . . . . . . . . . . . . . . . . 91

5.25 Comparison of the output power spectrums of reset control designand LTI design 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

xiii

5.26 Comparison of the output power spectrums of reset control designand LTI design 2 to whitenoise input . . . . . . . . . . . . . . . . . . . . 93

5.27 Step response of LTI design 1 . . . . . . . . . . . . . . . . . . . . . . . . 95

5.28 Step response of LTI design 2 . . . . . . . . . . . . . . . . . . . . . . . . 95

5.29 Step response of reset control design . . . . . . . . . . . . . . . . . . . . 96

5.30 The response of LTI design 1 with reference and sinusoid sensor noise . . 96

5.31 The response of LTI design 2 with reference and sinusoid sensor noise . . 97

5.32 The response of reset control design with reference and sinusoidsensor noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.33 The response of LTI design 1 with step reference and sinusoid disturbance98

5.34 The response of LTI design 2 with step reference and sinusoid disturbance98

5.35 The response of reset control design with step reference and sinu-soid disturbance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.36 The response of reset control design with step reference, sensornoise and disturbance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.1 Comparison of step response of new reset system, reset controlsystem and base-linear system . . . . . . . . . . . . . . . . . . . . . . . . 102

xiv

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xv

CHAPTER 1

INTRODUCTION

1.1 Background

Linear time-invariant (LTI) feedback control design is the most mature and

widely applied control design technique. But LTI control design has its inherent

limitation. In LTI feedback control systems, high-frequency loop gain is related to

both low-frequency specifications and stability margins through Bode’s gain-phase

relationship. This relationship (referred to as the “cost of feedback” [17]) results in

trade-offs among competing performance specifications which are unavoidable in LTI

control. To further illustrate this problem, consider the typical LTI control system

as shown in Figure 1.1, where C(s) and P (s) are the transfer functions of controller

and plant respectively and signals r(t), y(t), n(t) and d(t) represent reference input,

output, sensor noise and disturbance respectively. The following are typical genetic

specifications for control design.

P(s)C(s)

n

yrd

_

Figure 1.1 Typical LTI feedback control system

1

1. In the low-frequency band, we require the sensitive function,

S(jω) =1

1+ |P (jω)C(jω)| ,

to be small, |S(jω)| < 1, for disturbance rejection. This requires that the open-loop

magnitude |P (jω)C(jω)| be high in low-frequency band.

2. In high-frequency band, we require the complementary sensitive function,

T (jω) =|P (jω)C(jω)|

1+ |P (jω)C(jω)| ,

to be small, |T (jω)| < 1, for sensor noise suppression. This requires that

|P (jω)C(jω)| be small in high-frequency band.

3. Enough stability margin.

The following example clearly explains how the trade-offs among these per-

formance specifications happen. Figure 1.2 shows the magnitude and phase of the

open-loop transfer functions P (jω)C(jω) of two LTI control designs. To further sup-

press the high-frequency sensor noise of linear design 1, we decrease its high-frequency

loop gain and get linear design 2. According to Bode’s gain-phase relationship, linear

design 2 will have larger phase lag in the cross-over frequency range. So the cost of

linear design 2 is that it has a smaller phase margin than linear design 1 as indicated

in Figure 1.2. This trade-off is unavoidable in LTI control.

This limitation has motivated researchers to consider various forms of nonlin-

ear control. A straightforward approach of improving the inherent limitation in LTI

design is to adopt nonlinear non-smooth filters which are able to circumvents the fre-

quency domain restriction of LTI filters, governed by Bode’s gain-phase relationship

2

100 101 102-20

0

20

40

Mag

(db)

100 101 102-250

-200

-150

-100

-50

Phas

e (d

egre

e)

Frequency (rad/sec)

linear design 1 linear design 2

phase margin 1

phase margin 2

Figure 1.2 Illustration of “cost of feedback”

[44], [45]. The design and analysis of such kind of these nonlinear filters are normally

based on the describing function approximation [14], [16]. These nonlinear filters are

targeted to have a describing function approximations with much smaller phase lag

and same magnitude slope characteristics compared to the corresponding LTI filters.

From the above illustration of the “cost of feedback” problem, we can see that it

is possible to improve the performance trade-offs by using these nonlinear filters in

place of the corresponding linear filters. Such nonlinear filters started with Lewis

servo [46] and the Kalman nonlinear gain element [47]. Other nonlinear non-smooth

filters include driven limiter [48] and the split-path-nonlinear filter [49]. Another

interesting example is the resetting virtual absorbers for vibration control by Bupp

and Bernstein [22]. Our interest is on the nonlinear integrator which is introduced

by Clegg in 1958 [1]. It is basically a linear integrator whose state resets to zero

3

10 -1 10 0 10 1 10 2-40

-20

0

20

40

Mag

(db)

10 -1 10 0 10 1 10 2

-80

-60

-40

-20

0

Phas

e (d

egre

e)

Figure 1.3 Describing function of Clegg-integrator

when its input crosses zero. This so-called Clegg-integrator can be described with

the following impulsive differential equations [18]:

xc(t) = u(t); u(t) 6= 0 (1.1)

xc(t+) = 0; u(t) = 0,

where u(t) and xc(t) denote the input and state of Clegg-integrator respectively.

Clegg-integrator was shown to have a describing function with similar magnitude

to the frequency response of a linear integrator but with 51.9◦ less phase lag (see

Figure 1.3). This feature makes it a great candidate for improving the performance

trade-offs in LTI control design.

The Clegg integrator was later generalized by Krishnan and Horowitz to a so-

called first-order reset element (FORE) [2]. FORE simply extended the Clegg concept

4

10-1 100 101 102-45

-40

-35

-30

-25

Mag

(db)

FORELinear

10-1 100 101 102-80

-60

-40

-20

0

Phas

e (d

egre

e)

Frequency (rad/sec)

Figure 1.4 Describing function of FORE (b=25) and Bode-plot of 1/(s+25)

to a lag filter 1/(s+ b), where b is called FORE’s pole. It can be described with the

impulsive differential equations:

xc(t) = −bxc(t) + u(t); u(t) 6= 0 (1.2)

xc(t+) = 0; u(t) = 0,

where u(t) and xc(t) denote the input and state of FORE respectively. The describing

function of FORE when b = 25 is shown in Figure 1.4 compared to the frequency

response of the linear filter 1/(s + 25). It shows that they have similar magnitude

while FORE has much less phase lag than the linear filter in high-frequency. The

implication is that we may improve performance trade-offs by using FORE in place

of corresponding linear filter.

5

1.2 Description of Reset Control System

This dissertation is focused on the reset control system which was introduced

as a possible means to mitigate this limitation. The basic idea of reset control is to

reset the state of a linear controller to zero whenever its input meets a threshold.

Typical reset controllers include the Clegg-integrator [1] and first-order reset element

(FORE) [3] introduced in preceding section. The structure of the reset control system

under consideration is shown in Figure 1.5, where signals r(t), y(t), e(t), n(t) and

d(t) represent reference input, output, error signal, sensor noise and disturbance

respectively and L(s) is the transfer function of the linear loop which includes both

the linear controller C(s) and the plant P (s), i.e. L(s) = C(s)P (s). In the absence

of resetting, the FORE behaves as the linear filter 1/(s + b). In this case, we refer

to the resulting linear, closed-loop system as the base linear system.

The FORE element can be described by the first-order impulsive differential

equation [18]:

xc(t) = −bxc(t) + e(t); e(t) 6= 0 (1.3)

xc(t+) = 0; e(t) = 0

L(s) FORE

n

yerd

_

Figure 1.5 Block diagram of reset control system

6

where xc(t) is the state of FORE and e(t) = r(t) − n(t) − d(t) − y(t) is the error.

The time instants when e(t) = 0 are called reset times and the set of reset times I is

defined as

I = {ti | e(ti) = 0, ti > ti−1, i = 1, 2, ...},

where ti is called the i th reset time. The transfer function L(s) denotes the linear

loop which can be described by the state equations

xp(t) = Axp(t) +Bxc(t) (1.4)

y(t) = Cxp(t) + d(t)

where xp(t) ∈ <n is the plant states and {A,B,C} is a minimal realization of L(s).

From (1.3) and (1.4) the reset control system in Figure 1.5 can then be described by

the following impulse differential equation [18]

xp(t) = Axp(t) +Bxc(t)

xc(t) = −Cxp(t)− bxc(t) + w(t); t /∈ I (1.5)

xc(t+) = 0; t ∈ I

where w(t) = r(t)− n(t)− d(t).

In the case that there is no reset actions, reset control system (2.1) reduces to

the following linear system which is referred as its base linear system.· .xpl (t).xcl (t)

¸= Acl

·xpl(t)xcl(t)

¸+

·0w(t)

¸, (1.6)·

xpl(0)xcl(0)

¸=

·xp(0)xc(0)

¸,

where

Acl =

·A −B−C −b

¸.

7

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

t (second)

y

Figure 1.6 Comparison of step responses for the reset control system (solid) and itsbase linear system (dot)

1.3 Motivation

Despite of its simplicity, the above reset control system was shown to have the

potential benefit of providing improved trade-offs in feedback control. The describing

functions of Clegg integrator and FORE give a reasonable explanation of the source

of potential benefit. The potential benefit of reset control has been exemplified by

the earlier work in [1]-[4]. Recently, the benefit of reset control was also confirmed

experimentally in [5] and [6] where a FORE was used in the design of a tape-speed

control system. In Chapter 5 of this dissertation, the experiment of control system

design for a rotational mechanical system also verifies the benefit of reset control

design. As an illustration, we now repeat a simple example originally presented in

[10]. In this example, L(s) is

L(s) =s+ 1

s(s+ 0.2)

8

0 10 20 30 40 50 60-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

t (second)

y

Figure 1.7 Comparison of responses to 2 rad/sec sinusoidal sensor noise for the resetcontrol system (solid) and its base linear system (dot)

60 80 100 120 140 160 180 200 220 240 260-0.06

-0.04

-0.02

0

0.02

0.04

0.06

t (seconds)

y

Figure 1.8 Comparison of response to 0.2 rad/sec sinusoid disturbance for the resetcontrol system (solid) and its base-linear system (dot)

9

and FORE’s pole is b = 1. Figure 1.6 shows that this reset control system achieves a

40% reduction in step response overshoot compared to its base linear system. Figure

1.7 and 1.8 shows that their responses to 2 rad/sec sinusoidal sensor noise and 0.2

rad/sec sinusoidal disturbance are similar. The implication is that compared to its

base linear system, a reset control system appears to have much less overshoot in

its step response while maintaining similar high-frequency sensor-noise suppression

and low-frequency disturbance rejection. Since overshoot is closely related to stability

margin, a smaller overshoot generally implies a larger stability margin. In LTI design,

a larger stability margin is normally achieved in the cost of sacrificing its performance

of sensor-noise suppression or disturbance rejection. This is not seen in reset control

which implies its potential of improving the performance trade-offs in LTI design.

Despite of this potential benefit, there is a lack of theoretic results in stability

and performance analysis of reset control system. The analysis in [1] was limited

to the describing function of Clegg integrator, and [2] and [3] provided some design

guidelines without formal proof. [8] and [9] developed results for asymptotic stability

limited to some special reset control systems with second-order plants. Although

[4] and [5] developed some general stability results, they are too conservative to

be applicable. For example, although the reset control design in [6] shows very

good performance, there is no formal proof of stability and steady-state behavior.

Dedicated to solving this problem, this dissertation builds the theoretical foundations

of stability and time-domain performance for reset control. It is believed that these

results will further enable the application of reset control in engineering applications.

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In this dissertation, we are faced with the following major challenges in the research

of reset control. To date there are no results available.

1. Theoretic analysis of stability under arbitrary bounded input.

2. Theoretic analysis of steady-state behaviors such as zero steady-state error.

3. Theoretic analysis of performance.

We theoretically establish basic properties of reset control systems including

stability, transient and steady-state performance and apply these results to some

experimental setups of reset control design. Some of the results have appeared in

[10]-[12].

Finally, we remark that reset control resembles a number of popular nonlinear

control strategies including relay control ([36], [37]), variable structure and sliding

mode control ([38], [39], [42], [43]) and switching control ([28], [40], [41]). A common

feature to all of these is the use of a switching surface to trigger change in the

control signal. The difference in reset control is that the same control law is used on

both sides of the switching surface. A change takes place on a fixed surface wherein

the controller states are reset to zero. So, compared to these popular nonlinear

control strategies, reset control may be easier to design and simpler to implement.

There is no need to design switching surface and different control laws. As we will

demonstrate later, reset control design is as easy as linear control design. Since reset

action can be modeled as the injection of judiciously-timed, state-dependent impulses

into an otherwise linear feedback system, reset control system is alike with system

with impulse effect ([18], [19], [29]) or system with jump [34]. This analogy is evident

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when reset control systems were modeled by impulsive differential equations; e.g.,

see [18]. However, for our specific case of reset control system, the existing results

for impulse differential equations are too conservative to be applicable. For example,

with these results we are not able to provide stability guarantee for the experimental

reset control system in [6].

1.4 Dissertation Contributions

This dissertation has made the following contributions:

1. Developed a sufficient condition which guaranteed BIBO stability of reset

control system even in the face of implementation errors. Furthermore, this condition

was demonstrated to be non-conservative.

2. Developed a general result regarding the steady-state behavior of reset control

system. Additional results such as asymptotic stability and zero steady-state errors

were derived as the specific cases of this result.

3. Developed a so-called “input decomposition” principle for reset control sys-

tems which simplified the analysis of system response.

4. Characterized the maximum overshoot, rise time and settling time for step

response. For a specific class of reset control systems of reset control systems with

second-order plants, we explicitly computed the values of maximum overshoot, rise

time and settling time.

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5. For the reset control systems with second-order plants, we developed a suffi-

cient and necessary condition for asymptotic stability and a sufficient condition for

BIBO stability.

6. We applied reset control design technique for the speed control of a rotational

mechanical system and performed theoretic analysis of this reset control system.

1.5 Dissertation Outline

The structure of this dissertation is as follows: Chapter 2 discusses stability

of reset control systems. A sufficient condition is developed which guarantees the

BIBO stability of reset control system even in the face of implementation errors. As

a by-product of this result, a general result regarding the steady-state behavior of

reset control system is derived. Additional results such as asymptotic stability and

zero steady-state errors are established as the specific cases of this result. Chapter 3

focuses on the performance issue. We give a result which guarantees zero steady-state

error for reset control systems. Also, the so-called “input decomposition” principle

for reset control systems is developed. In Chapter 4 we study the reset control sys-

tems with second-order plants. A sufficient and necessary condition for asymptotic

stability and a sufficient condition for BIBO stability are developed. The step re-

sponse is analyzed and the values of maximum overshoot, rise time and settling time

are explicitly computed for a specific class of reset control systems. In Chapter 5,

reset control design technique is applied for the speed control of a rotational flexible

mechanical system. We also perform theoretic analysis of stability and steady-state

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behaviors for this reset control system. Finally, in Chapter 6, conclusions and future

work are presented.

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