ROBUST HYSTERESIS COMPENSATION FOR NANOPOSITIONING

CONTROL

By

Yasir Khudhair Al-Nadawi

A DISSERTATION

Submitted to

Michigan State University

in partial fulﬁllment of the requirements

for the degree of

Electrical Engineering - Doctor of Philosophy

2021

ABSTRACT

ROBUST HYSTERESIS COMPENSATION FOR NANOPOSITIONING

CONTROL

By

Yasir Khudhair Al-Nadawi

Piezoelectric and other smart material-based actuators are widely used in micro- and nano po-

sitioning applications. However, the intrinsic hysteretic behavior of these actuators deteriorates

their tracking performance. This dissertation, composed of three parts, is focused on nonlinear

control methods for compensating the hysteresis and achieving high-precision control in the pres-

ence of model uncertainties. An inversion-based adaptive conditional servocompensator (ACS)

is ﬁrst proposed, where a nanopositioning system represented as a linear system preceded with a

hysteresis nonlinearity modeled with a Modiﬁed Prandtl-Ishlinskii (MPI) operator. With an ap-

proximate inverse MPI operator as a compensator, the resulting system takes a semi-afﬁne form.

The proposed controller consists of two parts, a continuously-implemented sliding mode control

(SMC) law followed by an ASC. The hysteresis inversion error is treated as a matched disturbance

and its analytical bound is used to minimize the conservativeness of the SMC design. Under mild

assumptions, the well-posedness and periodic stability of the closed-loop system are established.

The second part of the dissertation focuses on designing an inversion-free ACS to achieve

precise tracking control of systems with hysteresis, without requiring explicit inversion of the

hysteresis. To facilitate the control design, the MPI operator is rearranged into a form comprised

of three parts: a linear term, a nominal hysteretic term represented by a classical Prandtl-Ishlinskii

(PI) operator, and a hysteretic perturbation. The bound on the hysteretic perturbation is further

derived based on the parameter uncertainty of the MPI operator. To properly “cancel” the nominal

hysteresis effect without inversion, a technique involving a low-pass ﬁlter is introduced. It is shown

that, with persistent excitation, the closed-loop variables are ultimately bounded and the tracking

error approaches a neighborhood of zero, where the neighborhood can be made arbitrarily small

via the choice of the SMC boundary layer width parameter and the servocompensator order.

In the third part, an output feedback-based hysteresis compensation approach is used using

dynamic inversion and extended high-gain observers. With mild assumptions on the properties of

the hysteresis nonlinearity, the system can be represented as an uncertain, non-afﬁne, nonlinear

system containing a hysteretic perturbation. Dynamic inversion is used to deal with the non-afﬁne

input, uncertainties, and the hysteretic perturbation, where the latter two are estimated using an

extended high-gain observer. Analysis of the closed-loop system under output feedback shows

that the tracking error converges to a small neighborhood near the origin, which can be made

arbitrarily small via proper choice of time-scale parameters of dynamic inversion and the observer,

respectively.

The efﬁcacy of the three proposed controllers is veriﬁed experimentally on a commercial

nanopositioning device under different types of periodic reference inputs, via comparison with

multiple inversion-based and inversion-free approaches.

Copyright by
YASIR KHUDHAIR AL-NADAWI
2021

To all my beloved family

My parents, my wife, and my brothers

for their endless love, sacriﬁces,

prayers, support, and encouragement.

v

ACKNOWLEDGMENTS

In the name of Allah, the most merciful and the most beneﬁcent. All praise and thanks are due

to Allah the Almighty for providing me with the opportunity to undertake and complete this task.

I would like to acknowledge the invaluable help and guidance of my academic advisors, Prof.

Hassan K. Khalil and Prof. Xiaobo Tan. I am extremely fortunate to have had the opportunity

to work so closely with them over the past several years with such preeminent scholars as Prof.

Khalil and Prof. Tan. Their wisdom and vast, seemingly boundless knowledge of our ﬁeld was

the most precious and useful resources of all. They often exhibited great patience with me through

my learning process. For their unwavering support and encouragement, I shall be everlastingly

grateful. I would also like to thank my other committee members, Prof. Guoming Zhu and Assist.

Prof. Vaibhav Srivastava for their support and tremendously helpful feedback during my research

activities. I am so grateful to my colleagues and friends, particularly Dhrubajit Chowdhury, for

collaborating with me in one of the research projects of this dissertation and for all the valuable

discussions we had through our work together. Also, I would like to thank the National Science

Foundation (CMMI 1301243) for supporting my research.

I would not be at this momentous juncture of my life if not for my parents, Khudhair and

Thamera, to whom I owe everything. They both unfailingly led by example when it came to

teaching and encouraging me to always think for myself and others. They have always been there

for me to support and encourage me no matter how hard the situation is. With all they gave to me,

I will always be indebted to them for the rest of my life. And to my wife, Nawar, who since I met

her have provided me a boundless love, constant support, and encouragement that have made all

this possible. Finally, I am very thankful to my brothers; Ayad, Samer, Zeyad, and Ashraph for

their constant encouragement, love, and advices. I will be always grateful to them.

vi

TABLE OF CONTENTS

LIST OF TABLES .

.

LIST OF FIGURES .

.

.

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.

.

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.

.

.

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

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

.

ix

x

Chapter 1

Introduction .

1.1
.
1.2 Literature Survey .

.
.

.
.

.
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.
.

. .

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

Introduction and Literature Survey . . . . . . . . . . . . . . . . . . . . .
.
.
1.2.1 Hysteresis Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Inversion-based Control . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.2
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.3
Inversion-free Control
.
Inversion-based ACS Approach . . . . . . . . . . . . . . . . . . . . . . .
Inversion-free ACS Approach . . . . . . . . . . . . . . . . . . . . . . . .

1
1
2
2
2
5
6
7
1.3.1
1.3.2
8
1.3.3 Dynamic Inversion Based on Extended High-gain Observer Approach . . . 10

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

1.3 Overview of Contributions

Chapter 2

.

.

.
.

. .

Introduction .

2.1
.
2.2 Problem Formulation .

Inversion-based Hysteresis Compensation Using Adaptive Conditional
Servocompensator .
.
.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.1
System Model with Hystersis Nonlinearity . . . . . . . . . . . . . . . . . . 14
2.2.2 MPI Hysteresis Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
. . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.3
2.3 Adaptive Output Feedback Controller Design . . . . . . . . . . . . . . . . . . . . 25
2.3.1 Continuously-Implemented SMC Law Design . . . . . . . . . . . . . . . . 25
2.3.2 Adaptive Conditional Servocompensator Design . . . . . . . . . . . . . . . 28
2.3.3 Output-feedback Closed-loop System Dynamics . . . . . . . . . . . . . . . 32

Inversion of the MPI Operator

2.4 Well-Posedness and Periodic Stability of the Closed-Loop System With Hysteresis

Inversion Perturbations .
.

2.5 Experimental Results .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
.

Chapter 3

.
.

.
.
.

.
.
.

. .
.

Introduction .

3.1
3.2 System Model .

Inversion-free Hysteresis Compensation Via Adaptive Conditional Ser-
vomechanism .
. 49
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
.
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2.1 Transformation into The Semi-Afﬁne form . . . . . . . . . . . . . . . . . 52
. 57
3.3.1 Continuously-Implemented SMC Design . . . . . . . . . . . . . . . . . . 57
3.3.2 Adaptive Conditional Servocompensator Design . . . . . . . . . . . . . . . 59
3.4 Analysis of Output-Feedback Closed-loop System . . . . . . . . . . . . . . . . . . 63
3.4.1 Reaching Phase .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.4.2 Boundary Layer Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.3 Adaptive Conditional Servocompensator Design . . . . . . . . . . . . . . . . . .

vii

3.5 Application to Nanopositioning Control

. . . . . . . . . . . . . . . . . . . . . . . 70

Chapter 4

.

.
.
.

. .

Introduction .

Dynamic Inversion-based Hysteresis Compensation Using Extended High-
gain Observer
.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
.
4.1
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.2 Problem Formulation .
.
4.3 State Feedback Controller .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.4 Output Feedback Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.5 Simulation Results
.
4.6 Experimental Results .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Sinusoidal Reference Signals . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.6.1
4.6.2
. . . . . . . . . . . . . . . . . . . . . . . . . 110
Sawtooth Reference Signals
4.6.3 Multi-Harmonic Reference Signals . . . . . . . . . . . . . . . . . . . . . . 110

.
.

.
.

.

Chapter 5

.

. .

5.1 Conclusions .

Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . 114
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
Inversion-based ACS Approach . . . . . . . . . . . . . . . . . . . . . . . 114
5.1.1
Inversion-free ACS Approach . . . . . . . . . . . . . . . . . . . . . . . . 115
5.1.2
5.1.3 Dynamic Inversion Based on Extended High-gain Observer Approach . . . 116
5.2 Future Work Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

.

.

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.

.

APPENDICES .

.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.1 Proof of Theorem 2.1 .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.2 Proof of Theorem 3.1 .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.3 Proof of Theorem 3.2 .
A.4 Derivation of Inequality (A.16) . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . 120
. 120
. 124
. 129
. 133

.
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.

.

BIBLIOGRAPHY .

.

.

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.

.

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.

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.

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.

.

. . . . . . . . . . . . . . . . . . . 138

viii

LIST OF TABLES

Percentage mean tracking error (mean |e(t)|%) with respect to the ref-
erence maximum peak under sinusoidal reference input for the proposed
controller versus comparable methods.
Peak mean tracking error (max |e(t)|%) with respect to the reference max-
imum peak under sinusoidal reference input for the proposed controller
versus comparable methods.

. . . . . . . . . . . . . . . . . . . . . . . . 47

. . . . . . . . . . . . . . . . . . . 46

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Percentage Tracking errors for the proposed controller with respect to the
reference maximum peak under sawtooth and van der Pol desired refer-
ences.
Percentage Peak tracking error (max |e(t)|%) for the proposed controller versus
other controllers in percent of the reference maximum peak under sinusoidal
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
reference input.
Percentage Mean tracking error (mean |e(t)|%) for the proposed controller ver-
sus competing methods in percent of the reference maximum peak under sinu-
soidal reference input. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

. 73

.

Percentage Tracking errors for the proposed controller in percent of the
reference maximum peak under sawtooth and van der Pol reference inputs. 74
Simulation results: Mean and Max |e(t)|% in percent of the reference amplitude
in the state feedback case.

. . . . . . . . . . . . . . . . . . . . . . . . . . . 106

. . . . . . . . . . . . . . . . . . . 107

.

.

Simulation results: Mean and Max |e(t)|% in percent of the reference amplitude
in the output feedback case with µ = 0.1.
Experimental results: Comparison of the Mean |e(t)|% in tracking a sinusoidal
reference. .
Experimental results: Comparison of the Max |e(t)|% in tracking a sinusoidal
reference. .
Experimental results: Mean and Max |e(t)|% in percent of the reference ampli-
tude for sawtooth and Multi-Harmonic references.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

. . . . . . . . . . . . . . . 108

Table 2.1:

Table 2.2:

Table 2.3:

Table 3.1:

Table 3.2:

Table 3.3:

Table 4.1:

Table 4.2:

Table 4.3:

Table 4.4:

Table 4.5:

ix

LIST OF FIGURES

Figure 2.1:

Schematic of the class of systems considered, with linear dynamics pro-
ceeded by a hysteresis operator.

. . . . . . . . . . . . . . . . . . . . . . 16

Figure 2.2:

Figure 2.3:

Figure 2.4:

Figure 2.5:

Block diagram of the ﬁnite-dimensional MPI hysteresis operator. Tri-
angular blocks represent the weight gains of the PI and the dead-zone
operators, respectively.

. . . . . . . . . . . . . . . . . . . . . . . . . . . 17

The feedforward hysteresis inversion.
Characteristics of the backlash operator Pbi as compared to the stop op-
erator Psi.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

. . . . . . . . . . . . . . . . . . . 20

.

.

.

Experimental setup of the the nanopositioner system. (a) The complete
setup including the nanopositioner stage Nano-OP65, Nano-Drive power
ampliﬁer unit, and the dSPACE DS1104 data acquisition interface unit;
(b) Magniﬁed picture of the nanopositioner stage Nano-OP65.

. . . . . . 40

Figure 2.6: Measured displacement versus sawtooth desired reference with 100 Hz

frequency using the inversion-based ACS.

. . . . . . . . . . . . . . . . . 42

Figure 2.7:

Tracking error with a sawtooth reference with 100 Hz frequency.

. . . . . 42

Figure 2.8:

Frequency spectrum of the tracking error with a 100 Hz sawtooth refer-
ence in the boundary-layer phase. . . . . . . . . . . . . . . . . . . . . . . 43

Figure 2.9: Measured displacement versus van der Pol desired reference with 100 Hz

frequency using the inversion-based ACS.

. . . . . . . . . . . . . . . . . 43

Figure 2.10: Tracking error with a van der Pol reference with 100 Hz frequency. . . . . 44

Figure 2.11: Frequency spectrum of the tracking error with a 100 Hz van der Pol ref-

erence in the boundary-layer phase. . . . . . . . . . . . . . . . . . . . . . 44

Figure 3.1: Measured Bode plot of the nanopositioner system (in blue) and the output of the

identiﬁed high-frequency dynamics (in red).

. . . . . . . . . . . . . . . . . . 76

Figure 3.2:

Nanopositioner system output used in the identiﬁcation and the resulting model
output.

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

.

.

.

.

.

Figure 3.3: Measured hysteresis loops (in blue) compared to predicted hysteresis loops (in

red).

.

.

.

.

.

.

.

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

x

Figure 3.4:

Figure 3.5:

Figure 3.6:

Figure 3.7:

Figure 3.8:

Figure 3.9:

Experimental results of measured displacement on tracking a sawtooth reference
with 100 Hz frequency using the inversion-free ACS. . . . . . . . . . . . . . . 78

Experimental results of the tracking error with a sawtooth reference with 100 Hz
frequency.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

.

.

.

.

Experimental results of the tracking error with a 100 Hz sawtooth reference in
the boundary-layer phase.

. . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Experimental results of measured displacement on tracking a van der Pol refer-
ence with 100 Hz frequency using the inversion-free ACS.

. . . . . . . . . . . 79

Experimental results of the tracking error with a van der Pol reference with 100
Hz frequency. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

.

.

Experimental results of the tracking error with a 100 Hz van der Pol reference in
the boundary-layer phase.

. . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Figure 3.10: Tracking error signal frequency spectrum with a sawtooth reference with 100 Hz

frequency in boundary-layer phase.

. . . . . . . . . . . . . . . . . . . . . . 81

Figure 3.11: Tracking error signal frequency spectrum with a van der Pol reference with 100

Hz frequency in boundary-layer phase using the inversion-free ACS.

. . . . . . 81

Figure 4.1:

Illustration of a classical play operator. . . . . . . . . . . . . . . . . . . . 88

Figure 4.2:

Block diagram illustrating the proposed approach. Here EHGO stands
for extended high-gain observer.

. . . . . . . . . . . . . . . . . . . . . . 97

Figure 4.3:

Figure 4.4:

Figure 4.5:

Experimental results on tracking a 100 Hz sinusoidal reference signal. (a)
Positioner displacement; (b) tracking error; (c) frequency spectrum of the
tracking error.

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

.

Experimental results on tracking a 100 Hz sawtooth reference signal. (a)
Positioner displacement; (b) tracking error; (c) frequency spectrum of the
tracking error.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

.

Experimental results on tracking a reference signal generated via the multi-
harmonic signal with a primary frequency of 100 Hz. (a) Positioner dis-
placement; (b) tracking error; (c) frequency spectrum of the tracking error. 113

xi

Chapter 1

Introduction and Literature Survey

1.1

Introduction

Smart material-based actuators such as piezoelectric or magnetostrictive actuators are signiﬁcant in

high-precision applications due to their ability to achieve manipulation in micro/nano meter level

with relatively fast responses, their compact size, and large bandwidth of operation [1,2]. One im-

portant area where smart material-actuators are used is nanopositioning, which is of importance to

applications such as atomic force microscopy, scanning probe microscopy (SPM) [3,4], ultra-high

density data storage, micro/nanofabrication [5], micro-manipulators [6,7], ultra-precision grinding

operation [8], mechanical nanomanufacturing system for nanomilling [9], high-precision electro-

chemical etching-based micromachining [10], nanofabrication of materials [11], and investigation

of biological systems over scales ranging from single-molecules to whole cells [12, 13].

However, these types of actuators exhibit a severe phenomenon called hysteresis. Generally, a

system is said to exhibit hysteresis when a characteristic looping behavior of the input-output graph

is displayed. The relationship between the input and output cannot be represented in terms of a

single-valued function [14]. The presence of this non-smooth input nonlinearity causes deteriora-

tion in system performance and oscillations, poor tracking responses [2], or even instability [15].

Therefore, the necessity to compensate the hysteretic behavior of these actuators by feedback con-

trollers has became indispensable. There has been signiﬁcant research interest for hysteresis mod-

1

eling and control design techniques to mitigate its effect [16].

1.2 Literature Survey

1.2.1 Hysteresis Modeling

Hysteresis models can be mainly classiﬁed into physics-based and phenomenology-based models.

The physics-based models are system-dependent and focus on ﬁrst principles of physics, while

the phenomenology-based models mainly replicate the natural behavior of the system without giv-

ing insight into the physics of the system. Since they are not speciﬁc to a physical system, these

models are generally applicable for a large class of hysteretic systems, examples of these mathe-

matical models, also widely known as hysteresis operators. There are many models developed and

studied in literature; for example: Bouc-Wen model [17], the Duhem model [18,19], the Maxwell-

Based model [20], the Jiles-Atherton model [21], the Krasnosel’skii-Pokrovskii model [22,23], the

Preisach model [24,25], and the Prandtl-Ishlinskii (PI) model [26,27]. The later two approaches are

widely investigated in the literature, however; one disadvantage of the Preisach operator as com-

pared to the Prandtl-Ishliskii model is that it does not admit an analytical inverse model, which

poses a challenge in real-time implementation [28]. This makes the Prandtl-Ishlinskii operator

more applicable in real-time control systems due to its analytic inversion feature [29].

1.2.2 Inversion-based Control

One control methodology for systems with (input) hysteresis is to use a feedforward open-loop

scheme by placing an inverse compensator at the input side to mitigate the effect of the hystere-

sis [1, 20, 30]. Examples of published work under this approach; the work of [31–36]. One draw-

2

back of this approach is that it depends strongly on how accurate the hysteresis model is. Despite

the reasonable tracking performance achieved using the feedforward control methodology, it has

been shown that it is necessary to ensure system robustness against model uncertainties and exter-

nal disturbances [37]. Moreover, operators like the Preisach operator, does not admit an analytical

form of its inverse and therefore it needs to be constructed through computation-intensive tech-

niques [25]. For operators, like the ﬁnite-dimensional classical [31], modiﬁed [38], and general-

ized [39] PI operators, where analytical formulas for their inverses exist, the number of hysterons

needs to be sufﬁciently high in order to have good model accuracy, which entails more storage

space and computational effort. In addition, due to the multi-valued and non-smoothness feature

of hysteresis, these models show strong sensitivity of the model parameters to unknown measure-

ment errors in the model identiﬁcation [40]. Therefore, the robustness of the system against the

inversion error perturbations, any external disturbance, or noise inputs can not be guaranteed.

Consequently, a more effective approach is to integrate the inverse compensator within a

closed-loop system, with the latter designed to handle the residual tracking error by considering

bounded disturbances resulted from the hysteresis inversion error and other perturbations. Exam-

ples of work done along this line using a popular choice of feedback controller is the PID control;

see for the work explored in [1, 41, 42]. Although these types of classical controllers fail to track

at high-frequency references [43]. Adaptive control can be used to deal with the inversion error

which mainly arises due to uncertainty in the hysteresis parameters. Examples of adaptive control

designed to adapt the hysteresis inverse operators’ parameters [15, 44–47].

More advanced techniques like sliding-mode control are robust against disturbances and pa-
rameter uncertainties [48–50] have been used. An H∞ robust control method was proposed by Al

Janaideh et al. in [51], where the controller was designed based on the characterization of the out-

put of the inverse PI operator, which consists of a linear reversible term and a bounded nonlinear

3

term.

Internal model-based regulators present a promising approach. The internal model is designed

to model sinusoidal disturbances inputs, where the role of the internal model is to act as an external

generator that rejects these sinusoidal matched disturbances [52]. Example of this approach is the

work of [5], where the knowledge of the internal model frequencies was assumed in the internal

model design. An important problem is that in most cases, one does not have knowledge of the

frequencies of the exogenous signals generated by the exosystem. Therefore, adaptation of the

internal model for estimating those frequencies becomes a necessity to compensate these exosys-

tem uncertainties. This limitation motivates the work of Serrani et al. [53], where an adaptation

mechanism was proposed to estimate a ﬁnite number of frequencies. This approach was adopted

by Esbrook et al. [54] for the control of piezo-driven actuators. In their work, the hysteresis was

modeled by using the classical PI operator, which was compensated with a feedforward inverse

operator. The inversion error was treated as a periodic matched disturbance, and a servomecha-

nism was designed to accommodate a ﬁnite number of frequencies, which were estimated through

a direct adaptation approach. Despite the initial success reported in their paper, further experi-

ments conducted later showed that the adapted parameters failed to converge, which motivated the

authors to replace the direct adaptation servomechanism with an indirect adaptation scheme [55].

Many other methodologies have been investigated with the inversion-based framework in the

literature, including, for example, iterative control [56, 57], model reference adaptive inverse con-

trol [45], and hysteretic perturbation estimation [58–61]. However, the achieved tracking precision

will depend mainly on the smallness of the hysteresis inversion error, which is highly dependent

on the accuracy of the hysteresis model and its identiﬁed parameters.

4

1.2.3 Inversion-free Control

The high computational effort required to implement the inverse hysteresis models and the

inversion error resulted due to the mismatch between the hysteresis model and the actual hysteresis

motivates the development of the inversion-free approaches. In this class of control methods, either

the inversion is achieved implicitly rather than explicitly or the controller does not involve any

inversion at all.

In [40] an inversion-free adaptive control design is proposed based on the assumption that the

hysteresis nonlinearity is modeled as a Prandtl-Ishlinskii operator. The hysteretic input term is

expressed as a combination of a control linear-term and a hysteretic perturbation term assumed to

be bounded by a known constant. The hysteretic disturbance term is compensated by an adaptation

law that estimates the density function, which is treated as a parameter in this framework. A model

reference-based adaptive control approach is developed in [62] for a system that includes hysteresis

nonlinearity described as a Prandtl-Ishlinskii operator. In their work, the inversion of the hysteresis

operator is implemented implicitly to avoid certain difﬁculties in theoretical analysis.

In [63] a retrospective cost adaptive control (RCAC) is designed in discrete time for the tracking

control of a shape memory alloy (SMA) actuator. The SMA actuator is assumed to be modeled

by a Wiener model that has linear dynamics cascaded with a hysteresis nonlinearity modeled as

a generalized Prandtl-Ishlinskii operator. Their adaptation law ensures the compensation of the

hysteretic disturbance without the need to use an inverse operator. The work of [64] presents

an inversion-free state-feedback controller design for a minimum-phase system proceeded by a

Prandtl-Ishlinskii operator. A general controller structure with an integral term is assumed in this

work. The major contribution of this work is that the boundedness of the signals of the closed-loop

system is guaranteed if a speciﬁc linear matrix inequality (LMI) condition is satisﬁed.

5

A feedforward inversion-free hysteresis compensator was proposed in discrete time in [29]

based on the rate-dependent Prandtle-Ishlinskii (RDPI) hysteresis model without the need to con-

struct a rate dependent inverse PI operator. The compensator utilizes an arrangement of the (RDPI)

model into a linear reversible term and a rate-dependent hysteretic perturbation. A major assump-

tion about this work is the availability of a good (RDPI) hysteresis model, due to its open-loop

nature. This work was extended in [37], in which the feedforward scheme is being integrated

within a feedback control system framework that is designed based on the internal model control

(IMC) theory. The proposed controller was implemented on a piezoelectric micropositioner to test

the validity of the controller.

Another alternative approach towards inversion-free hysteresis compensation for linear systems

is by using disturbance observers; see [65–67]. This approach mainly relies on the decomposition

of the hysteresis operator into a linear part, where the control input appears linearly, and another

part treated as a hysteretic perturbation (disturbance). Disturbance observer-based control methods

involve estimating this hysteretic perturbation and canceling it by using its estimate in the feedback

control law.

1.3 Overview of Contributions

In this dissertation, three approaches for hysteresis compensation are presented, which are the

inversion-based adaptive conditional servocompensator, the inversion-free adaptive conditional

servocompensator, and the dynamics inversion based on extended high-gain observer approach.

The following subsections provide a detailed breakdown for each approach.

6

1.3.1 Inversion-based ACS Approach

In chapter 2, we focus on the design of an adaptive conditional servocompensator for precise

tracking control of a class of systems with hysteresis. The hysteresis is assumed to be modeled

by an MPI hysteresis operator [38], which outperforms the classical PI operator with its ability to

incorporate asymmetric hysteretic behavior. The chapter has the following major contributions:

1. The hysteresis inversion error is analyzed and an analytical bound on the inversion error

is derived and used in the sliding mode controller design,which results in less conservative

results as compared to the case when a constant bound is used. The controller based on

a constant bound results in larger and more aggressive control actions with larger tracking

errors.

2. An output feedback controller is designed using adaptive conditional servocompensator by

assuming that the residual disturbance due to imperfect hysteresis inversion is composed of a

ﬁnite number of unknown frequencies. The unmeasured states are estimated by a high-gain

observer.

3. Periodic stability analysis is conducted using contraction mapping arguments by following

the stability analysis framework introduced in [68]. This approach is useful in establishing

the periodic stability in a less conservative manner as compared to Lyapunov-based stability

arguments under the smallness of the hysteretic inversion perturbation. However, the main

challenge is that our proposed closed-loop control system does not ﬁt exactly the system

assumed in [68] due to the inclusion of non-smooth terms in our control law. We are able

to establish, under mild assumptions, that the closed-loop system solution will converge

exponentially to a unique periodic solution when the inversion error is sufﬁciently small

without complicating the control law with additional terms that might be computationally

heavy.

7

The proposed control approach is experimentally evaluated on a commercial piezoelectric

nanopositioner. It shows superior precision tracking performance as compared to for other control

approaches implemented on the same apparatus. The four control approaches considered for com-

parison are sliding mode control [48], Single Harmonic Servo-Compensator (SHSC) and Multi-

Harmonic Servo-Compensator (MHSC) [5], the dynamic inversion based on extended high-gain

observer [69], and a classical PI controller without hysteresis inversion.

1.3.2 Inversion-free ACS Approach

In chapter 3, an adaptive conditional servocompensator is designed without requiring the use of

an analytical inverse hysteresis operator in the loop. The hysteresis nonlinearity is assumed to

be modeled with Modiﬁed Prandtl-Ishlinskii (MPI) operator. The following contributions are in-

tended;

1. A reconstruction of the (MPI) hysteresis operator that transforms the input hysteric term

into a linear part in terms of the control input, a hysteretic term modeled with a clas-

sical (PI) operator parameterized with the nominal weights, and a hysteretic perturbation

term. This arrangement converts the system from a non-afﬁne form into a semi-afﬁne one.

This result is basically established in Preposition 3.1 based on an assumption of an addi-

tive form of parametric uncertainty for the hysteresis weights. In Proposition 3.2, it was

shown that the hysteretic bound obeys a linear growth bound that is not all of its terms are

necessarily small. This bound will be utilized in designing a less-conservative continuously-

implemented sliding-mode control law.

2. One difﬁculty arose in the design of the equivalent portion of sliding-mode control law due

to the dependence of the nominal PI operator term on the control input. This required the

8

solution of a very complicated nonlinear equation that has a PI operator in one of its terms

in terms of the equivalent control input. The complexity lies in both the theoretical analysis

and implementation, where this issue creates a logical-loop due to the dependence of the

instantaneous value of the control input on that particular nominal hysteric term. To solve

this issue, a low-pass ﬁlter is inserted in the loop to avoid certain complexities and to simplify

the realization of the controller without the need to solve the above mentioned complicated

equation.

3. The adaptive conditional servocompensator is then designed based on an assumed number of

frequencies to compensate the residual hysteretic disturbances in the boundary-layer phase.

One challenge on the analysis side is that the original approach of the adaptive conditional

servocompensator [70] does not accommodate hysteretic perturbations. Moreover, the theo-

retical frame work to prove stability of [68] cannot be used here, because the a major assump-

tion of smallness of the hysteretic perturbation term is not satisﬁed in this case. Therefore,

the stability analysis is established by extending the theoretical frame-work in [70] to han-

dle the hysteretic perturbation. These results are embodied in Theorem 3.1 and 3.2, where

in Theorem 3.1, we show that in the reaching phase, all the closed-loop system variables

are bounded and will converge to a positively-invariant set that is parameterized by cer-

tain parameters related to the ﬁlter time-constant, the boundary-layer width, and a high-gain

observer parameter. Theorem 3.2 establishes the ultimate boundedness of the closed-loop

system in boundary-layer stage under the hysteretic-perturbation-free case and when there

exist valid perturbations.

We validate our control design with experimental implementation on a commercially available

piezo-actuated nanopositioner. The obtained results show a signiﬁcant reduction of the tracking

9

error for all the tested references as compared to the results of three other approaches implemented

on the same experimental apparatus. The ﬁrst is a classical proportional-integral controller applied

directly without inversion in the loop. The second approach is the sliding mode control approach

[48], and the third approach is the servocompensator approach [5].

1.3.3 Dynamic Inversion Based on Extended High-gain Observer Approach

In chapter 4, we consider the tracking problem for an uncertain nonlinear single-input-single-

output system, preceded by an unknown hysteresis operator. The following points summarize the

contributions:

1. With a mild assumptions on the general properties of the hysteresis, we apply dynamic-

inversion [71], to effectively compute the hysteresis inverse without requiring an explicit

model of the hysteresis. In order to implement the dynamic inversion algorithm, an extended

high-gain observer [72] is used to estimate the hysteretic part, system states, and uncertain-

ties at a time scale faster than the plant dynamics and the dynamic inversion. While an

extended high-gain observer has been used with dynamic inversion in [73] for memory-less

non-afﬁne nonlinearities, the proposed approach is the ﬁrst attempt to deal with a nonafﬁne

hysteretic nonlinearity using dynamic inversion and an extended high-gain observer. For the

resulting three-time-scale closed-loop system, we show that the tracking error converges to

a small neighborhood near the origin, which can be made arbitrarily small via proper choice

of time-scale parameters of dynamic inversion and the observer, respectively.

2. The non-afﬁne hysteretic nonlinearity poses a theoretical challenge in the analysis. There-

fore, we have developed two propositions in the chapter (Propositions 4.1 and 4.2), which

prove certain properties of the hysteresis operator that are then used for the analysis of the

10

closed-loop system later in the chapter. The proposed algorithm presents several advan-

tages over other inversion-free approaches in literature. Firstly, the non-hysteretic compo-

nent is assumed to be general with mild assumptions, unlike the other inversion-free ap-

proaches [65, 74], which assume that the non-hysteretic component to be linear. Secondly,

the plant is considered to be nonlinear satisfying the minimum-phase assumption. Thirdly,

the complexity of the proposed controller is invariant with respect to the complexity of the

underlying hysteresis operator. This is in contrast to adaptive hysteresis inversion-based [15]

or pseudo inversion-based approaches [75], where the controller complexity is correlated

directly with the number of hysteretic elements in the model. The extended high-gain ob-

server is used to estimate the hysteretic perturbation and then cancel it. Compared to [76],

we have two major differences: i) we consider an uncertain nonlinear single-input-single-

output system with nonafﬁne hysteretic nonlinearity, which is more general; ii) we consider

rate-independent hysteresis effect which can be modeled by standard operators like the PI

and Preisach operators.

We compare the proposed method with the sliding mode controller (SMC) [48], single har-

monic servocompensator (SHSC) [5] and the multiple harmonic servocompensator (MSHC) [5],

all of which use an explicit hysteresis inversion operator. Selecting these robust control techniques

for comparison is mainly because of the following reasons: (i) The aforementioned approaches

are comparable to our approach in the sense that they are robust control algorithms. (ii) All these

approaches are implemented with an inverse operator inserted at the input side. Therefore, obtain-

ing comparable or better results shows the efﬁcacy of the proposed controller, which is simpler

in structure as compared to the compared approaches. (iii) The compared approaches were im-

plemented on the same apparatus under comparable experimental conditions, which makes the

comparison fair when evaluating the methods. The experiments on a nanopositioner conﬁrm the

11

effectiveness of the proposed control scheme and demonstrate comparable tracking performance

with the aforementioned algorithms. In particular, we have demonstrated tracking of a 100-Hz

sinusoidal reference signal with 0.2051 % mean error and 0.6666 % max error for a travel range of

40 µm, compared with 0.83 % mean and 2.75 % peak errors, respectively, for SMC.

I would like to mention that the work in chapter 4, is the result of the collaboration with my

colleague Dhrubajit Chowdhury and Prof. Xiaobo Tan. Chowdhury’s main contribution is in

preparing the theoretical part (in particular the Propositions 4.1 and 4.2), designing the controller,

and the stability analysis (Theorem 4.1 and Theorem 4.2) with my support throughout the steps

of the work. I took the lead in the simulation part and in the experimental implementation of the

controller with Chowdhury’s help in the controller tuning process.

12

Chapter 2

Inversion-based Hysteresis Compensation

Using Adaptive Conditional

Servocompensator

2.1

Introduction

In this chapter, we consider the problem of tracking periodic reference inputs for a class of

nonlinear systems comprised of linear dynamics preceded by a hysteresis operator. Following [70],

we design an adaptive conditional servocompensator for tracking a periodic reference. An MPI

operator [38] is used to capture the hysteresis nonlinearity with enhanced performance over the

classical PI operator. The main difference between this approach and the classical internal model

work in [53], is that the internal model adaptation is active only inside a small neighborhood of

a certain manifold. The controller design is done through a three-step process. First, we design

a continuously-implemented sliding mode controller, where both the hysteresis and the system

dynamics are assumed to be uncertain. We derive the upper bound of the hysteresis inversion error

and make use of this bound to design the sliding mode control to mitigate the conservativeness

of the SMC design. In the second step, we design a conditional servocompensator with adaptive

tuning of its coefﬁcients. Finally, a high-gain observer is introduced for estimating the unavailable

13

states.

Stability analysis is conducted for the closed-loop system to demonstrate semi-global error

regulation. First, the well-posedness of the closed-loop system in the presence of small hysteresis

inversion error is established using the contraction mapping principle. We then show that under the

inherited exponential stability properties from the hysteresis-free closed-loop system, the stability

of the closed-loop system with hysteretic perturbations can be established. In particular, we show

that, under proper conditions, the tracking error converges exponentially to a periodic solution in

the neighborhood of the origin. The proposed controller design is experimentally validated on a

commercial nanopositioner, where a series of experiments are conducted by applying sinusoidal

reference inputs.

The remaining sections are organized as follows. In Section 2.2, the problem formulation, the

system model with hysteresis, and the derivation of the analytical bound of the inversion error

are presented. Section 2.3 explains the adaptive conditional compensator design. In Section 2.4,

the periodic stability analysis for the closed-loop system with hysteresis perturbation is discussed.

Finally, experimental results are given in Section 2.5.

2.2 Problem Formulation

2.2.1 System Model with Hystersis Nonlinearity

Consider the following non-afﬁne system (depicted in Fig. 2.1) modeled with linear dynamics

preceded by hysteresis input nonlinearity,

˙x (t) = Fp (x (t)) + Bpuo (t)

y (t) = x1 (t) ,

14

(2.1)

where x ∈ Rn is the state vector, y is the measured output, and uo ∈ R is the output of the hysteresis

operator,

uo (t) = Fh [uin (·);νo] (t)

(2.2)

where Fh denotes the hysteresis operator, uin ∈ R is the system input, and νo represents the initial
memory of the hysteresis (we will elaborate more on the hysteresis model in the next section). The

function Fp and the vector Bp are deﬁned as follows:



x2

x3
...

xn



fn (x)

n×1

Fp =

, Bp =





0

0
...
...

b

n×1

(2.3)

where fn (x) = −a1x1− a2x2−···− anxn, ai’s and b > 0 are the system parameters. The objective
is to make the system output y (t) track a desired reference input yd (t), which is assumed to obey

the following assumption.

Assumption 2.1. The desired reference yd and its time-derivatives up to order n are piecewise
continuous in t, bounded for all t ≥ 0, and T-periodic (i.e., yd (t) = yd (t + T )) for some T ≥ 0.

The tracking error is deﬁned as

e1 (t) = y (t)− yd (t)

Using the above equation, we can obtain the error dynamics as follows:

˙e (t) = Fp (e (t)) + Bpuo (t) + Bdδd (t) ,

(2.4)

15

where Fp is deﬁned as in (2.3) with argument e instead of x, e = [e1,e2,··· ,en]T , ei = di−1e
i = 2,··· ,n, (·)T denotes the transpose, Bd = [0,··· ,0,1]T , and δd is deﬁned as:

dti−1 for

δd (t) = −a1yd − a2 ˙yd −···− an−1yn−1

d − anyn

d

Figure 2.1: Schematic of the class of systems considered, with linear dynamics proceeded by a
hysteresis operator.

2.2.2 MPI Hysteresis Operator

To model the hysteresis operator Fh in Eq. (2.2), we use the Modiﬁed Prandtl-Ishlinskii (MPI) op-

erator (illustrated in Fig. 2.2 below). This model was originally proposed by Kuhnen [38] in 2003

by combining the classical Prandtl-Ishlinskii (PI) hysteresis operator with another operator, which

is represented by a weighted superposition of one-sided deadzone nonlinearities. This modiﬁcation

makes the MPI operator capable of modeling asymmetric hysteretic characteristics as compared

to the classical PI operator. The PI operator is a weighted superposition of basic hysteresis units,

each of which is modeled with a backlash operator with a threshold rth. The backlash operator for
a continuous and monotone input v (t) for t ∈ [0,T ] is given as:

ubk (t) = max{min{v (t) + rth,ubk (0)},uin (t)− rth}

(2.5)

16

where ubk (0) is the initial state. In essence, the PI operator consists of weighted integral of a

continuum of backlash operators, which makes it an inﬁnite-dimensional operator [22]. Due to

practical consideration, a ﬁnite-dimensional PI operator is often considered, which is represented

by a weighted sum of a ﬁnite number of backlash operators. Accordingly, the following assumption

is made for the MPI operator.

Assumption 2.2. The hysteresis nonlinearity Fh is modeled with a ﬁnite-dimensional MPI opera-

tor, which consists of (q +1) backlash operators and (2l +1) one-sided deadzone operators, where
q and l ∈ N +.

Figure 2.2: Block diagram of the ﬁnite-dimensional MPI hysteresis operator. Triangular blocks
represent the weight gains of the PI and the dead-zone operators, respectively.

Based on Assumption 2.2, the output ub of the PI operator Fb under the input v is given by

ub (t) = Fb [v;xb (0)] (t) =

q
∑
i=0

θbixbi (t)

where xbi (t) represents the output of the ith backlash operator such that

xbi (t) = Pbi [v;xbi (0)] (t)

17

(2.6)

(2.7)

in which xbi (0) is the initial state, and θbi ≥ 0, is the weight of the ith backlash operator. Let

(cid:3) be the backlash radii vector,

θb =(cid:2)θb0,θb1,··· ,θbq

where r0 = 0 and ri ≥ 0 for i = 1,2,··· ,q. Deﬁne the following vectors:

(cid:3) be the weight vector and r =(cid:2)r0,r1,··· ,rq
Pb :=(cid:2)Pb0,Pb1,··· ,Pbq
(cid:3)T
xb :=(cid:2)xb0,xb1,··· ,xbq
(cid:3)T

xb (0) :=(cid:2)xb0 (0) ,xb1 (0) ,··· ,xbq (0)(cid:3)T

(2.8)

(2.9)

(2.10)

By using equations (2.8)-(2.10), we rewrite equation (2.6) in a compact form as follows:

ub (t) = θ T

b xb = θ T

b Pb [v;xb (0)]

As mentioned above, the deadzone operator contains a weighted superposition of one-sided dead-
zone functions. For each one-sided deadzone function Pdi with input ub (t) and a threshold dz, the
output is given as

Pdi (ub (t)) =

max{ub (t)− dz,0}
ub (t)
min{ub (t)− dz,0}

if di > 0

if di = 0

if di < 0

(2.11)



Let d := [d−l,··· ,d0,d1,··· ,dl]T and θd := [θd−l,··· ,θd0,θd1,··· ,θdl]T be the deadzone op-
erator thresholds and weights vectors, respectively. Deﬁne the deadzone operator vector Pd :=
[Pd−l,··· ,Pd0,Pd1,··· ,Pdl]T . The output uo (t) of the MPI operator Fh under the input v (t) with

18

an initial condition state xb (0) will be

uo (t) = Fh [v;xb (0)] (t) :=

(cid:16)

(cid:17)

= θ T

d Pd

θ T
b xb (t)

(cid:33)
θb jPb j [v;xb (0)] (t)

(cid:32) q

∑
j=0

θdiPdi

l
∑
i=l

(2.12)

Let W1,1

t

be the Banach space of the absolutely continuous function u : [0,t] → R equipped

with a standard norm (cid:107)·(cid:107)W1,1
1-norm of the ﬁrst order time-derivative of u such that;

t

, which is a combination of the 1-norm of the function u and the

(cid:107)u(cid:107)W1,1

t

= (cid:107)u(cid:107)1 +

(cid:107)u(cid:48) (τ)(cid:107)1dτ

(2.13)

(cid:90) t

0

where u(cid:48) (τ)) is the derivative of u with respect to τ. In Proposition 2.1 below, we establish the
local Lipschitz property for the MPI operator Fh for any input u ∈ W1,1

L ⊂ W1,1

t where

L = {u|u : R+ → R,u|[0,t] ∈ W1,1
W1,1

t

, ∀t ≥ 0}

(2.14)

Proposition 2.1. The MPI operator Fh in equation (2.12) is locally Lipschitz continuous with
constant Lh = ∑l−l |θdi| ∑q

j=0|θb j| with the following condition:

sup
τ∈R+

(cid:107)Fh[u1;xb (0)] (τ) − Fh[u2;xb (0)](τ)(cid:107)W1,1

τ

≤ Lh sup
τ∈R+

(cid:107)u1 (τ) − u2 (τ)(cid:107)W1,1

τ

(2.15)

where u1, u2 are two different inputs in the set W1,1
L .

Proof. Consider the PI operator Fb in Eq. (2.6), under inputs u1, u2, and the initial condition xb (0).
By utilizing the Lipschitz continuity property of the PI operator Fb inside W1,1
L

[45, 77], one can

19

derive the following inequality:

sup
τ∈R+

(cid:107)Fb[u1;xb (0)] (τ)−Fb[u2;xb (0)] (τ)(cid:107)W1,1

τ

≤ q
∑
i=0

|θbi| sup
τ∈R+

(cid:107)u1 (τ)−u2 (τ)(cid:107)W1,1

τ

(2.16)

Similarly, utilizing the the Lipschitz continuity property of the dead-zone operator Pd (2.11), we

can show

sup
τ∈R+

(cid:107)Fh (Fb[u1;xb (0)] (τ))− Fh (Fb[u2;xb (0)] (τ))(cid:107)W1,1

τ

≤ l
∑
i=−l

|θdi| sup
τ∈R+

(cid:107)Fb[u1;xb (0)] (τ)− Fb[u2;xb (0)] (τ)(cid:107)W1,1

τ

(2.17)

By combining the two inequalities (2.16) and (2.17), we get (2.15).

2.2.3 Inversion of the MPI Operator

Figure 2.3: The feedforward hysteresis inversion.

As shown in Fig. 2.3, the hysteresis inversion is achieved by cascading an inverse MPI operator

with the MPI hysteresis operator Fh. Let ˆFh denotes the approximated forward MPI operator

resulted from the model identiﬁcation.

Assumption 2.3. For the MPI operator (2.12), only the values of the radii vector r and the thresh-

olds vector d are known.

20

The above assumption implies that exact values of both the PI operator’s weight vector θb and

the deadzone operator’s weight vector θd are unknown. This assumption is common and it has

been used frequently in the literature (for example see references [38], [78], and [79]).

Based on Assumption 2.3, let ˆθb and ˆθd be the estimated values of the weight vectors θb and

θd, respectively. We denote the errors of these weight vectors as follows:

∆b = θb − ˆθb,
∆d = θd − ˆθd

(2.18)

where ∆b ∈ Rq+1 and ∆d ∈ R2l+1 are the weight error vectors of the weight vectors θb and θd,
respectively. Consider Fig. 2.3, and let F−1
denotes the inverse hysteresis operator and ν (t)
h
represents the generated output of the operator F−1
h

. The inverse operator F−1
h

is also an MPI

operator and it is deﬁned as:

(cid:34) l

[uin; ¯xb (0)] (t)
¯θbi ¯Pbi

(cid:35)
¯θd ji ¯Pd j (uin (t)); ¯xbi (0)

(t)

(2.19)

ν (t) = F−1
h
q
∑
i=0
¯Pb

= ¯θ T
b

:=

(cid:104) ¯θ T

d

∑
(cid:105)
j=l
¯Pd (uin (t)); ¯xb (0)

(t)

(cid:3)T and ¯Pd = [ ¯Pd−l,··· , ¯Pd−1, ¯Pd0,··· , ¯Pd1,··· , ¯Pdl]

where ¯Pbi and ¯Pdi are the individual inverse PI and deadzone operators, respectively, the vectors
T are the vector of the in-

¯Pb =(cid:2) ¯Pb0,··· , ¯Pbq
verse PI and deadzone operator vectors, respectively. The vectors ¯xb =(cid:2) ¯xb0, ¯xb1,··· , ¯xbq
(cid:3)T and
¯xb (0) =(cid:2) ¯xb0 (0) , ¯xb1 (0) ,··· , ¯xbq (0)(cid:3)T are the inverse PI operator state and its initial state, respec-
(cid:3)T . Let
respectively, where ¯θb =(cid:2) ¯θb0, ¯θb1,··· , ¯θbq
(cid:3)T be the thresholds’ vectors of the in-
¯r =(cid:2)¯r0, ¯r1,··· , ¯rq

(cid:3)T and ¯θd =(cid:2) ¯θd−l,··· , ¯θd1, ¯θd0, ¯θd1,··· , ¯θdl

(cid:3)T and ¯d =(cid:2) ¯d−l,··· , ¯d−1, ¯d0, ¯d1,··· , ¯dl

tively. The weights ¯θbi and ¯θd j are for the individual inverse backlash and deadzone operators,

21

verse PI and deadzone operators, respectively.

Remark 2.1. Due to space limitation, the procedure of how to calculate the inverse MPI operators

parameters’ vectors ¯θb, ¯θd, ¯r, and ¯d has been omitted. For more details about these calculations,

the readers may consult the reference [38].

In the subsequent steps, we show under sufﬁciently accurate estimation of the MPI operator

weights (θb and θd), we can make the inversion error arbitrarily small. Moreover, we drive an

analytical bound on the hysteresis inversion error, which will be used later to design the controller.

Let δinv denotes the hysteresis perturbation due to imperfect inversion, which can be expressed as:

The approximated MPI hysteresis operator can be expressed as;

δinv (t) = uo (t)− uin (t)

(cid:16) ˆθ T

(cid:17)
b Pb [ν;xb (0)]
(cid:105)

[uin; ¯xb (0)];xb (0)

ˆFh = ˆθ T

d Pd

h

(cid:104) ˆF−1
(cid:16)

(2.20)

(2.21)

Using (2.21) and by rewriting uin = ˆFh

, one can rewrite equation (2.20)

as:

δinv = Iinv [uin;xb (0)] (t)

= θ T

d Pd

θ T
b xb (t)

(cid:17)− ˆθ T

d Pd

(cid:16) ˆθ T

b xb (t)

(cid:17)

(2.22)

where Iinv denotes the operator resulted due to the inversion process. In the following proposition,
we will show that the output δinv of the operator Iinv obeys a growth condition, whose upper bound
is a linear function of the input uin and it can be used later to design a less conservative controller
as compared to the case when the inversion error is bounded by a constant such that |δinv| ≤ kδ ,
where kδ is some positive constant.

22

Figure 2.4: Characteristics of the backlash operator Pbi as compared to the stop operator Psi.

Proposition 2.2. Under Assumption 2.3, and if the operator Iinv is modeled as in equation (2.22),
then its output δinv (t) will satisfy the following condition:

where the constants ∆0 and ∆1 can be calculated from the following formulas:

|δinv| ≤ ∆0|uin (t)| + ∆1

(cid:1)(cid:32)
(cid:0)(cid:107) ˆθb(cid:107)∞ +(cid:107) ˆθd(cid:107)∞
(cid:1)(cid:32)
(cid:0)(cid:107) ˆθb(cid:107)∞ +(cid:107) ˆθd(cid:107)∞

∆0 = εhmax

∆1 = εhmax

(cid:33)
| ¯θd j|]
(cid:33)

q
∑
[
i=0

| ¯θbi|] [

l
∑
j=−l

q
∑
[
i=0

| ¯θbi|] (cid:107)¯r(cid:107)∞ +(cid:107)r(cid:107)∞

(2.23)

(2.24)

(2.25)

where εhmax = (q + 1) (2l + 1)max ((cid:107)∆b(cid:107)∞,(cid:107)∆d(cid:107)∞) is the maximum perturbation of the weight
vectors and (cid:107)·(cid:107)∞ is the inﬁnity norm.

Proof: The backlash operator Pbi with threshold ri can be represented using the stop operator

by the following formula [80]:

Psi [ ν (t);xbi (0)] (t) = ν (t)−Pbi [ν (t);xbi (0)] (t)

(2.26)

23

where Psi is the stop operator with threshold ri. As shown in Fig. 2.4, the stop operator is bounded
from above by its threshold ri such that

|Psi [ ν (t);xbi (0)] (t)| ≤ ri

Therefore, using (2.27), we can show that

|Pbi [ν (t);xbi (0)] (t)− ν (t)| ≤ ri

(2.27)

(2.28)

From the Lipschitz property of the deadzone nonlinearity, we can show that Pdi satisﬁes the fol-
lowing inequality:

|Pdi (ν1 (t))−Pdi (ν2 (t))| ≤ |ν1 (t)− ν2 (t)|

Considering equation (2.22), by adding and subtracting the term(cid:0) ˆθ T

and by using (2.18), we can rewrite (2.22) as follows:

δinv = ∆T

d Pd

(cid:20)
(cid:16)
(cid:17)
(cid:17)(cid:21)
b Pb [ν;xb (0)] (t)
Pd
θ T
b Pb [ν;xb (0)] (t)
−Pd

(cid:16) ˆθ T

+ ˆθ T
d

(cid:16)

(2.29)

d Pd

(cid:0)θ T
b Pb [ν (t);xb (0)] (t)(cid:1)(cid:1)
(cid:17)
b Pb [ν;xb (0)] (t)
θ T

(2.30)

By taking the absolute value of both sides of the above equation, using inequalities (2.28) and

(2.29), and applying the H ¨older(cid:48)s inequality, one can get;

(cid:19)

(2.31)

(cid:107) ˆθb(cid:107)∞ +(cid:107) ˆθd(cid:107)∞

(cid:1)(cid:18)

(cid:0)ν (t) +(cid:107)r(cid:107)∞

|δinv| ≤ εhmax

24

By taking the absolute value of (2.19), we get

(cid:34) q

ν (t) ≤

| ¯θbi|

∑
i=0

(cid:35)(cid:18)(cid:34) l

∑
j=−l

(cid:35)
| ¯θdi|

(cid:19)

|uin (t)| +(cid:107)¯r(cid:107)∞

(2.32)

(cid:4)

Finally, by inserting (2.32) back into (2.31) and arranging the terms, we can obtain (2.23).

The smallness of the inversion error bound depends directly on the maximum perturbation

εhmax, which depends on how accurate the MPI hysteresis model is. Therefore, the following

assumption is made to characterize the smallness of the inversion error in the closed-loop system.

Assumption 2.4. There exists a small positive constant εh ≤ εhmax such that the inversion error
(2.22) can be rewritten in the following form:

δinv = Ib [uin;xb (0)] (t) = εhWin (t)

(2.33)

where Win (t) = Winv[uin;Win(0)] (t) and the hysteresis operator Winv is the composite MPI oper-
ator due to the inversion.

2.3 Adaptive Output Feedback Controller Design

2.3.1 Continuously-Implemented SMC Law Design

By using (2.20), we can convert the nonafﬁne hysteretic error system (2.4) into the following semi-

afﬁne system:

˙e (t) = Fp (e (t)) + Bp (uin (t) + δinv (t)) + Bdδd (t) ,

(2.34)

The system is called semi-afﬁne because it has input uin appearing linearly and the perturbation

(inversion error δinv) is modeled with a nonlinear function of the input uin as in equation (2.22).

25

The ﬁrst step in designing the controller is to design a continuously-implemented sliding mode

control law. The following surface function is considered;

ξc = k1e1 + k2e2 +··· + kn−1en−1 + en

(2.35)

where the coefﬁcients k1,k2,··· ,kn−1 are positive and chosen to make the following polynomial

λ n + kn−1λ n−1 +··· + k1

Hurwitz. For design purposes, the control input uin is partitioned into two parts; the ﬁrst part is the

equivalent control uq and the second part is the switching control uw:

uin = uq + uw

(2.36)

Let Vc = 1

2ξ 2 be used as a Lyapunov function candidate. By using the control law (2.36) and
calculating the time-derivative of the Lyapunov function Vc, the equivalent control uq is designed

as:

(cid:2)− fn (e)− δe (e)− δd (t)(cid:3)

uq (t,e) =

1
b

where fn (e) = −a1e1 − a2e2 −···− anen and δe (e) = k1e2 +··· + kn−1en.

Let the switching control law uw be

uw = −βw (t,e)sat

(cid:19)

(cid:18)ξc

µ

(2.37)

(2.38)

where βw is the switching gain function, which will be designed shortly, sat (·) is a standard satu-

26

ration function, which is deﬁned as follows [81]:

 v,

sat (v) =

sign (v) ,

if |v| ≤ 1
if |v| > 1

(2.39)

The parameter µ > 0 is chosen small. To design the switching function βw, consider the error

bound inequality (2.23). By substituting the desired control law (2.36) back into the inequality

(2.23), we get

|δinv| ≤ ∆1 + ∆0|uw (t,e) + uq (t,e)| ≤ ∆1 + ∆0|uw (t,e)| + ∆0|uq (t,e)|

(2.40)

Assuming ∆0 < 1, the switching function is designed as

(cid:20)∆1 + ∆0|uq (t,e)|

1− ∆0

(cid:21)

+ Γ0

(2.41)

βw (t,e) = Γ1

where Γ0 > 0 and Γ1 > 1 are controller parameters. Inserting the control laws (2.36) and (2.37),
and utilizing the error bound inequality (2.40), we get

˙Vc < −b(cid:2)(cid:0)∆1 + ∆0|uq (t,e)|(cid:1) (Γ1 − 1) + Γ0Γ1]|ξc|

which implies that the control law (2.36) achieves a non-zero steady-state error. In other words, if µ
is chosen small enough, the closed-loop system trajectory will reach the boundary layer {|ξc| ≤ µ},

and will stay in there for all future time. The purpose of using the saturation function instead of the

signum function in the switching control law (2.38) is to avoid the chattering of the control action;
however, the drawback is that the error e will be O (µ) instead of zero. To mitigate the residual

error, we use the adaptive conditional servomechanism [70], [82], which will be discussed in the

27

next section.

For the output feedback case, we assume that only e1 is available for measurement. Therefore,

to reconstruct the unmeasured states, we design a high-gain observer with the following structure:

˙ˆe (t) = ˆfo (e) + ˆgo (e1 − ˆe1)

(2.42)

where

ˆfo =



ˆe2

ˆe3
...

ˆen

0



n×1



h1
ε
h2
ε2
...
hn
εn



n×1

,

ˆgo =

where ˆei is the estimated state of ei, ε is a small positive design parameter, and hi is the estimation
gain for ith state, where the gains are chosen such that the polynomial λ n +h1λ n−1 +···+hn−1λ +
hn is Hurwitz. Accordingly, The desired control law uin (2.36) and the surface function ξc (2.35)

are modiﬁed by replacing e by ˆe.

In order to avoid the effect of observer peaking, the observer states are saturated before before

being plugged into the control law. This remedy, suggested in [83], will make the control law

globally bounded in its arguments in the domain of interest.

2.3.2 Adaptive Conditional Servocompensator Design

In the boundary-layer phase, due to the hysteresis inversion error, there is a non-vanishing matched

perturbation δinv. In theory, the disturbance δinv (t) could contain an inﬁnite number of harmonics

of the reference signal frequency yd. However, for practical reasons and based on the adaptive

conditional servomechanism theory [70] and [82], we will assume a ﬁnite number of frequencies

28

to be estimated by the conditional servocompensator.

Assumption 2.5. The hysteresis inversion error δinv (t) is generated by an exogenous neutrally

stable linear dynamical system:

˙ω = Sω,

δinv (t) = Γω

(2.43)

where ω ∈ κ ⊂ Rm is the state vector of the exosystem, κ is a compact set, and the matrices S and

Γ are given by

S =

Γ =



(cid:20)



,

m×m

0

0
...

0

1

0

0

1

. . .

. . .

. . .

. . .

0

0

0
...

1

c0

c1

. . .

. . .

1 0 . . .

. . . 0

cm−1

(cid:21)

1×m

The above assumption means that the disturbance δinv is a linear combination of constant and

sinusoidal components. The word “conditional” means that the servocompensator is designed to
be active only when the sliding surface variable ξc enters the boundary layer {|ξc| ≤ µ}. Let ξ be

the new sliding surface variable, which has the servocompensator part, such that

ˆξ = KT

ϑ ϑ + ˆξc

(2.44)

where ϑ ∈ Rm is the conditional servocompensator state vector. The conditional servocompen-

sator dynamics are described as:

(cid:16) ˆξ /µ

(cid:17)

˙ϑ = Aϑ ϑ + µBϑ sat

29

(2.45)

Kϑ is a unique servomechanism gain vector, which will make the eigenvalues of matrix(cid:0)Aϑ + Bϑ KT

ϑ

(cid:1)

equal to the eigenvalues of S, and the pair (Aϑ ,Bϑ ) is chosen such that they are in the controllable

canonical form

Aϑ =



1

0

0

1

0

0
...

0 . . .
∗
∗

. . .
∗

. . . 0

. . . 0
...

0
∗

1
∗







0

0
...

0
∗

Bϑ =

where the matrix Aϑ is Hurwitz, therefore if ϑ (0) is O (µ), ϑ (t) will be O (µ) also.

m×m

m×1

To include the servomechanism in the control law, the switching control law uw (2.38) is mod-

iﬁed to

uw = −βw (t,e)sat

Let N be the solution of the following Sylvester equation

(cid:33)

(cid:32) ˆξ

µ

(cid:16)

(2.46)

(cid:17)

NS− SN = −P−1Bϑ

Γ + KT

ϑ PN

It has been established in [84] that the existence and uniqueness of the solution N is ensured if Aϑ

and S have no common eigenvalues. In addition to that, if the pair (Aϑ ,Bϑ ) is controllable and the

pair (S,Γ) is observable, the matrix N is non-singular [84]. The matrix P is a unique non-singular

similarity transformation matrix such that

P−1(cid:16)

(cid:17)

Aϑ + Bϑ KT
ϑ

P = S, P−1Bϑ = [0 0 . . . 0 1]T

If the eigenvalues of the exosystem (2.43) are unknown, which is due to the frequencies of the

30

inversion error δinv (t) being unknown, it is needed to manipulate Kϑ in (2.44) online with an
adaptation law. Let ˆKϑ be the manipulated servomechanism gain vector. The adaptation law is

designed to change the eigenvalues of the matrix(cid:0)Aϑ + Bϑ ˆKT

(cid:1) online to become equal to the

ϑ

eigenvalues of S. To lower the number of adapted variables, we use the partial adaptation as

suggested in [70], by deﬁning the following vectors

λϑ = Iϑ Kϑ

and

vϑ = Iϑ ϑ

where λϑ ∈ Rι, vϑ ∈ Rι, and ι ≤ m is the number of the adapted variables, Iϑ is (ι × m) matrix
with its rows are unit vectors, and vϑ represents the adaptive regressor vector. As we mentioned

above, the pair (Aϑ ,Bϑ ) is in the controllable canonical form, implying that the number of adap-

tation variables are

 m

2 ,
m−1
2 ,

ι =

if the number of frequencies is even

if the number of frequencies is odd

With the assumption that λϑ belongs to the convex hypercube κϑ = {λϑ|ai ≤ λϑi ≤ bi,1 ≤

i ≤ ι}, the adaptation law can be designed as

(cid:16) ˆξc,vϑ ,βw

(cid:17)
˙ˆλϑ = βϑ

& γϑ

= γϑ βwvϑ

µϑ sat

(cid:16) ˆξ , µ

(cid:17) Pr
(cid:18)

(cid:17)(cid:17)

(cid:16)

(cid:16) ˆξc,vϑ ,βw
(cid:17)(cid:19)
(cid:16) ˆξc/µϑ

γϑ

/µ2

(2.47)

where the function Pr (·) is a parameter projection operator that retains ˆλϑ in κδ ⊃ κϑ ∀t ≥ 0,
(cid:16)
where κδ = {λϑ|ai−δ ≤ λϑi ≤ bi +δ ,1 ≤ i ≤ ι}, and δ > 0. γϑ is the adaptation gain. The func-
tion

and 0 < µϑ < µ is the componentwise smooth projection Pr

(cid:16) ˆξc,vϑ ,βw

(cid:16) ˆξc/µϑ

(cid:17)(cid:17)

µϑ sat

(cid:16)

γϑ

(cid:17)(cid:17)

31

is deﬁned by

(cid:20)
Pr (γϑ (·))

(cid:21)

=

i

(cid:16) ˆξ , µ

(cid:17)

The function βϑ

(cid:33)
(cid:33)

γϑi

γϑi

bi−ˆλϑi

δ
−ai
ˆλϑi
δ



(cid:32)
(cid:32)

1 +

1 +

γϑi

if ˆλϑi > bi and γϑi > 0

if ˆλϑi < ai and γϑi < 0

Otherwise

is deﬁned as:

βϑ

(cid:17)

(cid:16) ˆξ , µ
(cid:16) ˆξ , µ



0

1
1− | ˆξ|−µ
µ

if | ˆξ| ≥ 2µ
if | ˆξ| ≤ µ
if µ < | ˆξ| < 2µ

is to keep ˆλϑ constant outside the boundary set {| ˆξ| ≤ 2µ}.

=

(cid:17)

The purpose of the function βϑ

2.3.3 Output-feedback Closed-loop System Dynamics

The closed-loop dynamics under output feedback are composed of the exosystem dynamics (2.43),

the conditional servomechanism (2.45), the semi-afﬁne error dynamics (2.34), the adaptation law

(2.47), the time-derivative of the modiﬁed sliding mode surface function (2.44) and the high-gain

observer dynamics (2.42). By inserting the output feedback control law (2.36) into equations (2.44)

and (2.42) and then doing algebraic manipulation, we can derive the closed-loop system in compact

form as follows;

˙X (t) = Fcl (t,X ,Win) = ¯Fcl (t,X ) + εhDinv (X ,Win)

(2.48)

32

where Fcl is the closed-loop system under hysteresis inversion, ¯Fcl is hysteresis-free closed-loop
system (δinv (t) = 0), Dinv (X ,Win) = BinvWin (t), and these functions are deﬁned as:



,

(cid:17)(cid:21)



µ

Sω

Aϑ ϑ + µBϑ sat

(cid:17)
(cid:16)ξ−L(ε)φ
ϑ ϑ(cid:1)
(cid:0)ξ − ˆKT
(cid:17)
(cid:16)ξ−L(ε)φ
(cid:16)ξ−L(ε)φ
∆d1 (·) + ∆d2 (·)− bβw (t, ˆe)sat
(cid:17)(cid:17)

− δe (e)− bβw (t, ˆe)sat

Aζ ζ + Bζ

µ

µ

(cid:17) Pr
(cid:17)

γϑ

(cid:16)

(cid:16) ˆξc,vϑ ,βw
and(cid:0)Aϕ ,Bϕ

¯Fcl (t,X ) =

ε−1Aφ φ + Bφ

(cid:20)
(cid:16) ˆξ , µ
Binv = [0,0,0,b,b,0]T ,
(cid:16)

βϑ

where X = [ω,ϑ ,ζ ,ξ ,φ ,λϑ ]T , the pairs
able canonical forms, respectively, and the matrices Aζ and Aϕ are Hurwitz. The state vector φ is

Aζ ,Bζ

(cid:1) are in controllable and observ-

the scaled estimation variable, which is deﬁned as:

ϕi =

1

εn−i (ei − ˆei)

The other terms and variables are deﬁned as follows:

∆d1 (·) = ˆKT

ϑ Aϑ ϑ + µ ˆKϑ Bϑ sat ((ξ − L(ε)ϕ) /µ) + ˙ˆλϑ vϑ ,

∆d2 (·) =

i=1
∑
n−1

kiei+1 − δe (e) ,

ζ T = [e1 e2 . . . en−1] ,

φ T = [φ1 φ2 . . . φn] ,

33

(cid:20)
k1εn−1 k2εn−2 k3εn−3 . . . kn−1ε 1

(cid:21)

,

L (ε) =

Notice that the closed-loop system dynamics (2.48) include some nonlinearities, for example, the

saturation function in the control law (2.38) and the projection function in the adaptation control

law (2.47). For the purpose of conducting the analysis in the next section, the following assumption

is needed. Moreover, Lemma 2.1 is needed to establish the exponential stability of the hysteresis-

free closed-loop system.

Assumption 2.6. If the desired reference input yd is T-periodic, then there exists a unique T-
periodic solution XT (t) for the hysteresis-free closed-loop system ¯Fcl (t,X ) in (2.48).

Lemma 2.1. (Exponential Stability of the Hysteresis-Free Closed-loop System) Suppose Assump-

tions 2.1-2.5 are satisﬁed. Consider the hysteresis-free closed-loop system

˙X (t) = ¯Fcl (t,X )

(2.49)

under ideal hysteresis inversion (δinv (t) = 0). Let Ω1 ⊂ R, Ω2 ⊂ Rn, Ω3 ⊂ Rn, and Ω4 ⊂ Rm be
compact sets. Let ϑ (0) = O (µ) and ν0 and ˆζ (0) be bounded. Let the switching control under the
hysteresis-free case are chosen as:

(cid:18) ξ

(cid:19)

uw = −βwmaxsat

µ

where βwmax = maxe∈Ω1, t≥0{βw (t,e)}. Then, there is µ∗ > 0, such that for every µ ∈ (0, µ∗],
there is ε∗ = ε∗ (µ) > 0 such that for every µ ∈ (0, µ∗] and ε ∈ (0,ε∗] and for all initial condi-
tions ξ (0) ,e (0) ∈ Ω1 × Ω2, ˆe (0) ∈ Ω3, ϑ (0) ∈ Ωm, ˆλϑ ∈ κϑ , if ¯vϑ is persistently exciting, then
the closed-loop variables vector X is bounded ∀t ≥ 0 and the hysteresis-free closed-loop system

34

¯Fcl (t,X ) has an exponentially stable equilibrium point at
, where
˜ϑ = ϑ − ¯ϑ, ˜λϑ = ˆλϑ −λϑ , ¯vϑ is the regression vector, and ¯ϑ is the servocompensator state vector
in the boundary-layer stage.

e = 0, ˜ϑ = 0, ˆe = 0, ˜λϑ = 0

(cid:16)

(cid:17)

The proof of this lemma is carried out by repeating the steps of the proof of Theorem 1 in [70]

for the hysteresis-free closed-loop system ¯Fcl (t,X ).

2.4 Well-Posedness and Periodic Stability of the Closed-Loop

System With Hysteresis Inversion Perturbations

Before proving the existence of an exponentially stable, periodic solution of the closed-loop system

dynamics under hysteresis inversion (2.48), we need to establish that the system (2.48) is well-

posed. By establishing well-posedness, we mean establishing the existence and uniqueness of the

solution of the closed-loop system (2.48).

Theorem 2.1. (Well-Posedness of the Hysteretic Closed-loop System) Consider the closed-loop
system under hysteretic inversion perturbation (2.48). Let ΩX ⊂ Rrc and ΩH ⊂ W1,1
t
sets and are deﬁned as:

be compact

ΩX := {X ∈ Rrc : (cid:107)X (t)−X (0)(cid:107)1 ≤ rX}

ΩH := {Win ∈ W1,1
t

: (cid:107)Win (t)−Win (0)(cid:107)W1,1

t

≤ rH}

where rc = 2n+2m+ι and rX , rH > 0. Under the piecewise continuity of the function Fcl (t,X ,Win)

35

in t and from its local Lipschitz property in arguments X and Win, such that the condition

(cid:16)
t,X1,Win1

(cid:17)−Fcl

(cid:16)
t,X2,Win2

(cid:17)(cid:107)1

(cid:107)Fcl

≤ Lχ(cid:107)X1 (t)−X2 (t)(cid:107)1 + LW(cid:107)Win1 (t)−Win2 (t)(cid:107)1

(2.50)

is satisﬁed for any X1, X1 ∈ ΩX and Win1, Win2 ∈ ΩH ∀t ∈ [0,tu], where tu > 0, then there exists
0 < tc < tu, such that the system (2.48) has a unique solution XT (t) for all X (0) ∈ ΩX and
Win (0) ∈ ΩH over the time interval [0,tc].

The proof of this theorem can be found in Appendix A.1. By establishing the well-posedness

of the hysteretic closed-loop system (2.48), we are now prepared to prove its periodic stability.

Deﬁne uT

in = αin (XT ) and vT = F−1

in will be
T -periodic and vT will also be T -periodic, but after some transient period of time. To prove the

h

(cid:2)uT
in; ¯xb (0)(cid:3) (t). Under Assumptions 2.1 - 2.6, uT

existence of an exponentially stable periodic solution of the hysteretic closed-loop system (2.48),

we need to establish the existence of a contraction property for the composite hysteresis operator
Win resulted due to the inversion. This property can be established for a T -periodic reference input
yd (t) if uin (t) and v (t) satisfy the following condition.

Assumption 2.7. For any absolute continuous function vc, deﬁne its oscillation within the time

interval [t1,t2] as

Assume

osc[0,T ]

&

where rmax = (cid:107)r(cid:107)∞ and ¯rmax = (cid:107)¯r(cid:107)∞.

osc[t1,t2][vc] =

sup

t1≤τ1,τ2≤t2

|vc (τ1)− vc (τ2)|

(cid:2)∑l−l θdiPdi

(cid:0)uT

in

(cid:1)(cid:3) > 2¯rmax

osc[T,2T ][vT ] > 2rmax

36

Theorem 2.2. (Periodic Stability of the Hysteretic Closed-loop System) Consider the hysteretic
closed-loop system (2.48). Let Assumptions 2.1 - 2.7 be satisﬁed and let ¯ΩX ⊂Rrc and ¯ΩH ⊂ W1,1
t
be compact sets. Assume ∆0 < 1, under the exponential stability of the hysteretic-free closed-loop
system (2.49), then there exists ε∗
h ≤ εhmax, such that for all the initial conditions X (0) ∈ ¯ΩX
and Win (0) ∈ ¯ΩH, the solution X (t) of the hysteretic closed-loop system (2.48) will converge
exponentially to a unique periodic solution XT (t).

Proof: Consider the ith backlash operator Pbi with radius ri in Eq. (2.7), for the input uin (t)
(0). It can be seen that the

with osc[0,T ] > 2ri and two different initial conditions X a
bi
backlash operator Pbi has the following property

(0) and X b
bi

(cid:2)uin;X a

bi (0)(cid:3) (t)−Pbi[uin;X b

|Pbi

bi (0)] (t)| = 0, ∀t > T

(2.51)

This mainly follows the essential properties of the backlash operator (2.7), where its output os-

cillation becomes independent of the initial condition once its input oscillation amplitude exceeds
2ri. Refer to the composite MPI operator Win in (2.33) resulting from the inversion process. This
operator is a composition of the inverse MPI operator F−1
(whose largest radius is ¯rmax) and the
h

MPI operator Fh (whose largest radius is rmax). Using the property (2.51) and Assumption 2.7, we
can show that the operator Win obeys the following contraction property:

|Win[uin;xa

inv (0)] (t)−Win[uin;xb

inv (0)] (t)| = 0 ∀t > 2T

(2.52)

where xa

inv (0) and xb

inv (0) are two different applicable initial inputs. Notice that even if Win oper-
ator includes deadzone nonlinearities, we still can establish the contraction property (2.52). This is

because the deadzone nonlinearity is memory-less and preserves the local Lipschitz property. Sim-

ilarly, by using the property (2.52) along with Assumption 2.7, we can show that the shift operator

37

Shεh deﬁned as:

Shεh : (X (0) ,Win (0)) → (X (2T ) ,Win (2T ))

(2.53)

has a contraction property for εh > 0. Since the MPI operator Win satisﬁes both the Volterra and
semi-group properties [14], also from Proposition 2.1, we can show that

|Win (t)| ≤ ¯∆0|X (t)| + ¯∆1

where ¯∆0 and ¯∆1 are positive constants and are function of the constants ∆0 and ∆1 of inequality

(2.23). From Lemma 2.1, under the persistency of excitation of the regressor vector ¯νϑ , we can
show that the hysteresis-free closed-loop system (2.49) is T -convergent about XT [68]. From
Theorem 2.1, we have established the existence and uniqueness (well-posedness) of the solution of

the hysteretic closed-loop system (2.48). Therefore, by following similar steps to those of Theorem

(2.1) of [68], we can establish that the solution of the hysteretic closed-loop system (2.48) will
(cid:4)

converge exponentially to a unique periodic solution when εh is sufﬁciently small.

In Theorem 2.2, we have established that the solution of the closed-loop system converges

exponentially to a periodic solution provided the inversion error is sufﬁciently small. Furthermore,

it can be shown that an ultimate bound on the tracking error can be reduced by reducing the

controller parameters µ and ε. The ultimate boundedness can be established by following similar

steps in the proofs of Theorems 1 and 2 of [85]. The ﬁrst step is to show that the closed-loop

systems variables in the reaching phase will converge exponentially to a positively-invariant set that

is parameterized by the parameters (µ and ε), which can shrink to zero set if these two parameters

are pushed to zero. The second step is to establish that the closed-loop system variables in the

boundary layer phase are ultimately bounded by a bound that depends on the controller parameters
(µ and ε) and the inversion error perturbation εhDinv (X ,Win).

38

2.5 Experimental Results

In this section, we examine the performance of the proposed control scheme by implementing

tracking experiments on a commercial piezo-actuated nanopositioner stage (Nano OP-56) shown in

Fig. 2.5. Experiments on this platform, manufactured by Mad City Labs Inc., provides a practical

importance to benchmark our controller in handling hysteretic disturbances. Position measurement
is provided by a built-in capacitive sensor. The travel range of the nanopositioner is ±65 µm. The

power ampliﬁer unit (Nano Drive, Mad City Labs, Inc.) drives the piezo actuator and has a gain

of 15. In the system setup, the manipulated control input is the one to the power ampliﬁer instead

of the actual voltage input to the piezo actuator. For real-time implementation, the controller is

deployed in a dSPACE (DS1104) platform using Matlab/simulink real-time coder tools.

The nanopositioner system is modeled with a second order nonlinear system in the form of

equation (2.1). The hysteresis part is modeled with an MPI operator with 8 play operators and 9

dead-zone units, and the weights and thresholds are identiﬁed as:

ˆθ T
b = [0.719,0.183,0.035,0.055,0.034,0.033,0.023,0.061] ,
rT = [0,0.33,0.66,1.00,1.33,1.66,2.00,2.33] ,

d =(cid:2)1.062,0.473,0.641,0.311,8.426,−0.636,−0.501,−0.614,−0.415(cid:3),

ˆθ T
dT = [−2.68,−1.97,−1.22,−0.42,0,0.32,1.02,1.76,2.57] ,

The linear part of the model is identiﬁed using frequency-based identiﬁcation methods and its
parameters are found to be a1 = 1.795× 108, a2 = 5696.88, and b = 1.063× 109. Notice that due
to the system high resonant frequency, the identiﬁed parameters are very large.

The high-gain observer (2.42) is implemented with the gains h1 and h2 being 3 and 20, respec-
tively. The parameter ε is taken as 0.0001. For the parameters of the switching function βw (·) in

39

(a)

(b)

Figure 2.5: Experimental setup of the the nanopositioner system. (a) The complete setup including
the nanopositioner stage Nano-OP65, Nano-Drive power ampliﬁer unit, and the dSPACE DS1104
data acquisition interface unit; (b) Magniﬁed picture of the nanopositioner stage Nano-OP65.

(2.46), we calculate ﬁrst the constants ∆0 and ∆1 of the bound (2.23) by using the formulas (2.24)

and (2.25). From the identiﬁed weights and thresholds of the MPI hysteresis model mentioned

above, we calculate the following:

(cid:107) ˆθb(cid:107)∞ = 0.719,

(cid:107) ˆθd(cid:107)∞ = 8.426,

and (cid:107)r(cid:107)∞ = 2.33,

40

Then we use the procedure provided in [38] to calculate the inverse MPI operator parameters, and

accordingly we ﬁnd;

| ¯θbi| = 1.9082,

7
∑
i=0

4
∑
j=−4

| ¯θd j| = 0.187,

and (cid:107)¯r(cid:107)∞ = 2.2379,

To ensure periodic stability, one major assumption of Theorem 2.2 is ∆0 < 1. To comply with this

assumption, let εhmax = 0.015, and then with formulas (2.24) and (2.25), the error bound (2.23)
constants can be computed as ∆0 = 0.9058 and ∆1 = 0.049. The rest of the switching function

(2.41) parameters are chosen as; Γ1 = Γ2 = 10. We choose the sliding function (2.35) constant

k1 = 5000 and the boundary layer width constant µ = 1000.

For the adaptive servocompensator, we assume that the residual disturbance in the boundary-

layer phase (due to hysteresis inversion) has only in its frequency spectrum the fundamental fre-

quency of the desired reference input. Moreover, we assume that there is an additional bias distur-

bance term alongside the periodic disturbance terms. As a result, the internal model will be a third

order model (namely a second order model augumented with an integrator state).

To ensure the robustness of the adaptation law (2.47) against noise inputs and the observer

peaking effect, the adaptation parameter λϑ is assumed to be retained in the following convex set;

κδ = {λϑ|− 8.1 ≤ λϑ ≤ 8.1}

The remaining adaptation law parameters are chosen as; γϑ = 2.68 × 105, and µϑ = 1. In our
experiments, we test the proposed controller using three types of desired reference signals. The

ﬁrst one is a sinusoidal input deﬁned as

yd (t) = 10 sin (2π f t) + 10 µm

41

Figure 2.6: Measured displacement versus sawtooth desired reference with 100 Hz frequency using
the inversion-based ACS.

Figure 2.7: Tracking error with a sawtooth reference with 100 Hz frequency.

with frequency f = 5, 25, 50, and 100 Hz. It is worth mentioning that we published part of our

evaluation results using sinusoidal reference in [86]. Therefore, we are not going to repeat these re-

42

00.0050.010.0150.020.0250.03Time (seconds)-505101520Displacement (m)Desired Reference yd(t)Measured Output y(t)012345678910Time (seconds)-4.5-4-3.5-3-2.5-2-1.5-1-0.500.5Tracking Error (m) 00.0050.010.0150.020.0250.03-4-2077.027.047.067.087.1-0.04-0.0200.020.04Figure 2.8: Frequency spectrum of the tracking error with a 100 Hz sawtooth reference in the
boundary-layer phase.

Figure 2.9: Measured displacement versus van der Pol desired reference with 100 Hz frequency
using the inversion-based ACS.

sults in this chapter. In Tables 2.1 and 2.2, we conduct a comparison between the achieved tracking

error accuracy of our proposed approach in the boundary-layer phase as compared to other control

43

01002003004005006007008009001000Frequency (Hz)0123456Absolute Magnitude of Tracking error (m)10-300.0050.010.0150.020.0250.03Time (seconds)-5051015202530Displacement (m)Desired Reference yd(t)Measured Output y(t)Figure 2.10: Tracking error with a van der Pol reference with 100 Hz frequency.

Figure 2.11: Frequency spectrum of the tracking error with a 100 Hz van der Pol reference in the
boundary-layer phase.

approaches proposed in previous projects implemented on the same experimental setup, including

a) the inversion-based sliding mode controller proposed in [48], which will be abbreviated as SMC,

44

012345678910Time (seconds)-30-25-20-15-10-505Tracking Error (m)00.0050.010.0150.020.0250.03-30-20-10077.027.047.067.087.1-0.0500.0501002003004005006007008009001000Frequency (Hz)00.0020.0040.0060.0080.010.0120.0140.016Absolute Magnitude of Tracking Error (m)b) the results of [5], in which both of Single Harmonic Servo-Compensator (SHSC) and Multi-

Harmonic Servo-Compensator (MHSC) are designed and implemented, c) the results obtained

from combining extended high-gain observer and the dynamic inversion approaches (EHGO-DI),

which appears in [69], and d) a classical PI controller without hysteresis inversion implemented

experimentally by the authors and its gains are chosen to yield the best possible performance.

In Table 2.1 and 2.2, we show the percentage of the maximum absolute tracking error and

the percentage of the mean absolute tracking error with respect to the maximum peak. It can be

noticed in both tables that the inversion-based adaptive conditional servocompensator approach

(Inv-B ACS) results have greatly outperformed the other ﬁve approaches in reducing the tracking

errors for all frequencies. The trend in both tables shows that the next best tracking results is the

extended high-gain observer combined with dynamics inversion (EHGO-DI) approach. Notice that

the mean percentage absolute error for the EHGO-DI approach is higher than the Inv-B ACS one

by almost 7 folds for the 100 Hz frequency.

The second round of experiments are done using a sawtooth desired reference with the same

frequencies used with the sinusoidal reference (frequencies 5, 25, 50, and 100 Hz). A second

order pre-ﬁlter is inserted to smooth out the reference signal to avoid spiking impulses at the signal

edges. The measured output displacement y (t) under the inversion-bases ACS control method

for the 100 Hz frequency reference case is shown in Figure 2.6. In Figure 2.7, the tracking error

e1 (t) is presented. The magniﬁed sub-ﬁgure to the left side shows the tracking error response

for the ﬁrst 0.03 seconds. Notice that the tracking error is converging quickly in around 0.003

seconds. Another magniﬁed sub-ﬁgure added to the right to show an instant of the tracking error

in the boundary-layer phase (7.0-7.1 seconds). Notice that the error is not increasing in the period

(0.03-10 seconds).

In Figure 2.8, we show the frequency spectral content of the tracking error in the boundary

45

layer phase (0.03-10 seconds) for the 100 Hz reference. It is noticed that we have 9 harmonics

shown in the spectrum with odd harmonics being relatively stronger than the even harmonics,

which are barely noticeable. This is mainly attributed to the hysteresis nonlinearity exciting the

odd harmonics more than the even ones. It can be seen that the ﬁrst harmonic (the fundamental)

has magnitude of less than 5.5 nm and the rest of harmonics have smaller magnitudes.

Another set of experiments are conducted using van der Pol oscillator output used as the desired

reference input to the system. In Figure 2.9, the measured output displacement y (t) is shown for

the case with the frequency of 100 Hz. In Figure 2.10, we demonstrate the tracking error e (t).

Similar to the sawtooth reference case, it can be noticed that the tracking error converges within

around 0.003 seconds. However, the tracking error magnitudes are a little bit more compared to

the sawtooth reference case. This can be seen clearly in Figure 2.11, in which we demonstrate the

spectrum content of the tracking error frequency. It can be seen that the fundamental harmonic has

magnitude of 14 nm as compared to less than 5.5 nm in the sawtooth reference case. We can see

in Figure 2.11 that we have 9 harmonics in the signal spectrum similar to the sawtooth reference,

but with higher amplitudes.
Table 2.1: Percentage mean tracking error (mean |e(t)|%) with respect to the reference maximum
peak under sinusoidal reference input for the proposed controller versus comparable methods.

freq. (Hz)

5
25
50
100

SMC
0.0595
0.3100
0.3300
0.4150

SHSC MHSC
0.1355
0.3245
0.1340
0.3535
0.3850
0.1420
0.1760
0.4075

EHGO-DI

0.0672
0.0665
0.0686
0.1026

PI

0.0736
0.0939
0.1498
0.2897

Inv-B ACS

0.0016
0.0041
0.0101
0.0148

Finally, in Table 2.3, we offer a comparison in absolute percentage tracking errors for the cases

of the sawtooth and van der Pol references’. It can be observed from the table that for all the

tested frequencies, the van der Pol reference case has higher tracking errors as compared to the

sawtooth case. For instance, in the 100 Hz frequency case, we can see that absolute percentage

46

Table 2.2: Peak mean tracking error (max |e(t)|%) with respect to the reference maximum peak
under sinusoidal reference input for the proposed controller versus comparable methods.

freq. (Hz)

5
25
50
100

SMC
0.4750
0.8500
1.1250
1.3750

SHSC MHSC
0.8600
0.4495
0.4405
0.9250
0.5050
0.9650
1.1900
0.7850

EHGO-DI

0.1153
0.1383
0.1821
0.3333

PI

0.7894
0.8422
1.0058
1.5185

Inv-B ACS

0.0083
0.0212
0.0465
0.0610

Table 2.3: Percentage Tracking errors for the proposed controller with respect to the reference
maximum peak under sawtooth and van der Pol desired references.

freq. (Hz)

5
25
50
100

Sawtooth

van der Pol

Mean |e(t)|% Max |e(t)|% Mean |e(t)|% Max |e(t)|%

0.0016
0.0054
0.0104
0.0226

0.0133
0.0481
0.1027
0.2584

0.0017
0.0075
0.0202
0.0707

0.0210
0.0560
0.1291
0.3695

mean error of the van der Pol reference case is larger by more than three times compared to the

sawtooth reference case. This is an indication that the van der Pol reference excites strongly the

odd harmonics as compared to the previous two references.

It is worth mentioning that we tried replacing the switching function designed based on the
analytical error bound by a constant gain taken as the maximum of the switching function βw (·)
for t ≥ 0. However, the results obtained from the ﬁxed switching gain have shown that the tracking

errors have increased with more aggressive control actions for all the considered references.

Remark 2.2.

• In the implementation of the adaptive servocompensator (2.45), to avoid the case when the

matrix S has very large eigenvalues, we utilized the technique suggested in [53] to scale

down the internal model matrices such that Aϑ = gϑ ¯Aϑ and Bϑ = gϑ ¯Bϑ , where ¯Aϑ is cho-
sen as a Hurwitz matrix with eigenvalues (-1,-1.5,-2) and the pair ( ¯Aϑ , ¯Bϑ ) is in controllable
canonical form, gϑ is the scaling factor and is chosen to be 2π × 600. This technique helps
in making the adaptation parameters in Kϑ reasonably small.

47

• In the adaptation law (2.47), We added the function

(cid:16)

(cid:16) ˆξc/µϑ

(cid:17)(cid:17)

µϑ sat

due to the bene-

ﬁt seen in the experimental implementation of the controller for enhancing the adaptation

performance; however, in the original approach of [70], it is not mandatory.

It’s worth

mentioning that this modiﬁcation has not changed the ﬁnal theoretical outcomes.

• In the controller evaluation, we have limited the desired reference frequency to maximum of

100 Hz. This is mainly to avoid causing excessive vibration, which could lead to potential

damage of the device, especially, when the highest harmonic frequency is very close to the

ﬁrst resonant frequency of the system.

48

Chapter 3

Inversion-free Hysteresis Compensation Via

Adaptive Conditional Servomechanism

3.1

Introduction

In this chapter, we propose an inversion-free approach to the control of systems with hysteresis,

removing the computational complexity in constructing an inverse compensator. The hysteresis

nonlinearity is modeled as a Modiﬁed Prandtl-Ishlinskii (MPI) operator. We utilize the properties

of the MPI hysteresis model to transform the system into a semi-afﬁne form, where one term has the

control input appearing linearly and the other term represents the hysteretic perturbation. The pro-

posed controller is designed based on an adaptive conditional servocompensator approach, which

is a continuously-implemented sliding mode control law powered with an adaptive servocompen-

sator. An analytical bound on the hysteretic perturbation is derived and used in the design of the

sliding mode control law. A low-pass ﬁlter is augmented with the control law, to avoid solving a

complicated equation involved. Our stability analysis shows that, under reasonable assumptions,

the boundedness of the closed-loop system trajectories is ensured.

The rest of the chapter is organized as follows. In Section 3.2, the system dynamics with MPI

operator is described, the transformation into a semi-afﬁne form is presented, and the derivation of

hysteretic bound is explained. Section 3.3 details the design of the adaptive conditional compen-

49

sator. Analysis of the output-feedback closed-loop system is carried out in Section 3.4. Finally, in

Section 3.5, we present the experimental validation of the proposed controller.

3.2 System Model

Consider the following dynamics, which are represented as a linear system preceded with a hys-

teresis nonlinearity modeled by an MPI operator:



˙xi = xi+1

1 ≤ i ≤ n− 1

˙xn = f (x) + bυ (t)

y = x1

(3.1)

where x ∈ Rn is the state vector, y is the output, the function f (x) = −a1x1 −···− an−1xn−1 −
anxn, ai and b are model parameters, b > 0, and υ ∈ R is the output of the hysteresis nonlinearity
modeled as follows:

υ (t) = Γhd [u;υ0] (t) = θ T

d Πd

θ T
h Πh [u;υ0] (t)

(t)

(3.2)

(cid:104)

(cid:105)

where u is the system input, Γhd, Πd, and Πh represent the MPI, the asymmetric dead-zone opera-
d = [θd1,θd2, . . . ,θdp]T is the dead-zone operator’s
tor, and the classical PI operator, respectively, θ T
weight vector, θ T
h = [θh0,θh1, . . . ,θhq]T represents the PI operator’s weights vector, υ0 represents
the initial state of the hysteresis, and (·)T represents the transpose. As shown in Eq. (3.2) above,

the MPI operator consists of the PI operator and the dead-zone operator [32]. The PI operator is

modeled as a sum of weighted backlash operators (also known as play operators), each of which is

50

represented as:

υhi (t) = πhi

(cid:104)
(cid:16)

(cid:105)
(cid:16)

u,υh0i

(t)

= max

min

u + ri,υh0i

(cid:17)

(cid:17)

,u− ri

(3.3)

for i ∈ [0,q], where υhi is the output of the ith backlash operator and ri is the corresponding
threshold. Let r = [r0,r1, . . . ,rq]T , with 0 = r0 ≤ r1 ≤ ··· ≤ rq. The PI operator is deﬁned as

υh (t) = θ T

h Πh [u;υh0] (t) =

q
∑
i=0

θhiπhi [u;υh0] (t)

(3.4)

The asymmetric dead-zone operator is a weighted sum of asymmetric dead-zone functions. The

output of each asymmetric dead-zone function under the input υh (t) can be represented as



υdi (t) = πdi (υh) =

υh (t)− di, υh (t) ≥ di
0,
υh (t)− di, υh (t) ≤ di

di < υh (t) < di

(3.5)

where di and di are respectively the corresponding positive and negative thresholds for i ∈ [1, p]
and πdi is the ith asymmetric dead-zone nonlinearity. Notice that the use of asymmetric dead-zone
functions is a modiﬁcation from the original model of [32], where one-sided dead-zone functions

are used. We had to do this modiﬁcation, which will reduce the ability of the MPI operator to rep-

resent the asymmetric hysteresis behavior, because of some difﬁculty in establishing the stability

of the closed-loop system later. Accordingly, the dead-zone operator is deﬁned as

υd (t) = θ T

d Πd [υh (t)] =

p
∑
i=1

θdiπdi (υh)

51

where Π T

d = [πd1, . . . ,πdp]T . Let dT = [−d1, . . . ,−d p]T and dT

= [d1, . . . ,d p]T .

The tracking error is deﬁned as

e1 = y− yr

(3.6)

where yr is the desired reference, which is assumed to satisfy the following condition.

Assumption 3.1. The reference signal yr (t) and its derivatives up to y(n)
r
uous in t and are bounded for all t ≥ 0.

(t) are piecewise contin-

Shifting the system dynamics using (3.6), the error dynamics are found to be

˙ei = ei+1

1 ≤ i ≤ n− 1

˙en = f (e) + ∆re f (t) + bυ (t)

(3.7)

where e ∈ Rn is the vector of error states, e = (e1,··· ,en)T , f (e) = −a1e1 −···− an−1en−1 −
anen, and ∆re f (t) = −a1yr − a2 ˙yr −···− an−1y(n−1)

− any(n)
r

r

.

3.2.1 Transformation into The Semi-Afﬁne form

Both the system model (3.1) and the error model (3.7) are non-afﬁne due to the hysteresis non-

linearity at the input. In the subsequent steps, we will utilize the properties of the MPI hysteresis

model to transform the error model into a semi-afﬁne form, where the input nonlinearity is parti-

tioned into two parts; the ﬁrst part is linear, while the second part consists of a nominal hysteretic

nonlinearity modeled as a classical PI operator and a hysteretic perturbation.

Assumption 3.2. The uncertainties of the MPI operator (3.2) are limited only to the PI operator’s

vector θh, and the dead-zone operator’s weight vector θd, which are represented as

θh = θ N

h + ∆h

52

(3.8)

θd = θ N

d + ∆d

(3.9)

where θ N

h and θ N

d are the nominal weight vectors for the PI operator and the dead-zone operator,

respectively, while ∆h and ∆d represent the respective uncertainties of the two operators.

Proposition 3.1. Under Assumption 3.2, the MPI operator (3.2) can be rewritten in the following

semi-afﬁne form

υ (t) = ΘN

d θ N
h0

u + ΘN

d Qh−0 (t) + ϕ∆ (t)

where θ N
h0

is the corresponding weight of the backlash operator πh0, ΘN

d = θ N T

d

(3.10)

1 = ∑p

i=1 θ N
di

, 1

is a column vector with all of its entries are equal to 1, and Qh−0 (t) is the output of the nominal
hysteresis nonlinearity modeled the the classical PI operator with the term πh0 excluded, and ϕ∆
is a hysteretic perturbation term.

Proof: Based on Assumption 3.2, by substituting (3.8) and (3.9) into Eq. (3.2), one obtains

(cid:104)

(cid:105)

(θ N T

h + ∆T

h )Πh [u;υo] (t)

υ (t) = (θ N T

d + ∆T

d )Πd

(t)

(3.11)

Using the fact that the asymmetric dead-zone nonlinearity can be represented by an asymmetric

saturation nonlinearity [87], we write

πdi [Q] = Q− σdi (Q)

(3.12)

where

σdi (Q) =



di

Q

di

53

if Q > di
di ≤ Q ≤ di
Q < di

Let

Qh (t)

de f
= θ N T

h Πh [u;υ0] (t)

Q∆ (t)

de f
= ∆T

h Πh [u;υ0] (t)

With Eqs. (3.13) and (3.14), Eq. (3.11) can be rewritten as

υ (t) = (θ N T

d + ∆T

d )Πd [Qh + Q∆] (t)

which, based on (3.12), can be expressed as

where σd =(cid:2)σd1, . . . ,σdp

where

υ (t) = (θ N T

d + ∆T

d ) [1 (Qh + Q∆)− σd (Qh + Q∆)]

(cid:3)T . Accordingly, we can rewrite (3.16) as

υ (t) = ϕN (t) + ϕ∆ (t)

ϕN (t) = ΘN

d Qh (t)

(3.13)

(3.14)

(3.15)

(3.16)

(3.17)

(3.18)

. Notice here that the

is the nominal part of the input hysteresis nonlinearity and ΘN

d = ∑p

i=1 θ N
di

nominal input nonlinearity is modeled with a classical PI operator. Other hysteretic perturbation

components are lumped in the term ϕ∆(t) as follows:

ϕ∆ (t) = θ N T

d

1 Q∆ (t) + ∆T

d 1 Qh (t) + ∆T

d 1 Q∆ (t)− (θ N T

d + ∆T

d ) σd (Qh (t) + Q∆ (t))

where the term ϕ∆ (t) will be treated as a matched disturbance input. By separating the backlash

54

operator with r0 = 0 (namely, the non-hysteretic term) from the rest of backlashes in the nominal

term ϕN in (3.18), one gets

where

(cid:104)

θ N
h1

,θ N
h2

, . . . ,θ N
hq

(cid:105)T

and θ N

h−0 =

Qh (t) = θ N

hou + Qh−0 (t)

Qh−0 (t) = θ N T

h−0 Πh−0 [u;υ0] (t)

is the vector of nominal weights for the PI operator excluding the

(trivial) backlash with zero threshold term. Consequently, the nominal term of the input nonlinear-

ity is expressed as

ϕN (t) = ΘN

d Qh−0 (t)

d θ N

hou (t) + ΘN

Using (3.17) and (3.19), we get (3.10).

(3.19)

(cid:3)

Accordingly, the error dynamics (3.7) can be transformed from a non-afﬁne form to a semi-

afﬁne form as follows 

˙ei = ei+1

1 ≤ i ≤ n− 1

+ b ϕ∆ (t)

˙en = f (e) + ∆re f (t) + bhu (t) + bd Qh−0 (t)

(3.20)

where bh = b ΘN

d θ N

ho is assumed to have a nonzero value and bd = b ΘN
d .

The error dynamics (3.20) will be used in the subsequent sections for the controller design,

observer design, and closed-loop analysis purposes. We can further derive a bound on the hysteretic

perturbation term φ∆.

Proposition 3.2. If Assumption 3.2 holds, then the hysteretic perturbation term ϕ∆ (t) in Proposi-

55

tion 3.1 is bounded by

where

|ϕ∆ (t)| ≤ Ko|u| + K1

Ko = ∆max

d (cid:107)∞ +(cid:107)θ N

(cid:104)(cid:107)θ N
(cid:104)(cid:107)θ N

(cid:105)

h (cid:107)∞ + ∆max
(cid:105)

K1 = Ko(cid:107)r(cid:107)∞ + p

d (cid:107)∞ +(cid:107)∆d(cid:107)∞

σdmax

∆max = p (q + 1)max ((cid:107)∆h(cid:107)∞,(cid:107)∆d(cid:107)∞)

(3.21)

(3.22)

(3.23)

(3.24)

where (cid:107)·(cid:107)∞ is the inﬁnity norm, which is the maximum of the absolute values of the vector entries

and σdmax = max(cid:0)(cid:107)d(cid:107)∞,(cid:107)d(cid:107)∞

(cid:1).

Proof: The backlash operator (3.3) can be represented using a stop operator in the following

way [80]:

Since the stop operator πsi

(cid:104)

(cid:104)

u;υh0i

(cid:105)

(t)

πsi [u;υ0] (t) = u (t)− πhi
(cid:105)

u;υh0i

(t) satisﬁes the inequality

(cid:104)

(cid:105)

|πsi

u;υh0i

(t)| ≤ ri

(cid:105)

(cid:104)

u;υh0i

each play operator πhi

|πhi

As a result, it can be easily seen that

(t) is bounded by

(cid:104)

(cid:105)

u;υh0i

(t)| ≤ |u (t)| + ri

|Qh (t)| ≤ (q + 1)(cid:107)θ N

h (cid:107)∞ (|u| +(cid:107)r(cid:107)∞)

56

(3.25)

(3.26)

(3.27)

|Q∆ (t)| ≤ (q + 1)(cid:107)∆h(cid:107)∞ (|u| +(cid:107)r(cid:107)∞)

By using H¨older’s inequality, one can see that

|(θ N T

d + ∆T

d )σd (Qh + Q∆)| ≤ p

(cid:104)(cid:107)θ N

d (cid:107)∞ +(cid:107)∆d(cid:107)∞

(cid:105)

σdmax

Utilizing the bounds (3.27), (3.28), and (3.29), we can derive (3.21).

(3.28)

(3.29)

(cid:3)

3.3 Adaptive Conditional Servocompensator Design

3.3.1 Continuously-Implemented SMC Design

The design of adaptive conditional servocompensator involves ﬁrst designing a continuously-

implemented SMC law [70]. The continuously-implemented SMC is designed by replacing the

discontinuous switching function with a saturation or sigmoidal function [88]. Consider the slid-

ing function

sc = k1e1 + k2e2 +··· + kn−1en−1 + en

Its time-derivative is found to be;

˙sc = f (e) + ∆re f (t) + ∆e (e) + bh u + bd Qh−0 (t) + b ϕ∆ (t)

where ∆e (e) = k1e2 +··· + kn−1en. Let the control input be

u = ueq + us

(3.30)

57

where ueq denotes the equivalent control term, while us denotes the switching control term. For

the control law design purposes, we use the following Lyapunov function candidate:

Vs =

1
2

s2
c

Using the control law (3.30), the time-derivative of Vs will be

(cid:20)

˙Vs = sc ˙sc = sc

f (e) + ∆re f (t) + ∆e (e) + bh ueq + bh us + bd Qh−0 (t) + b ϕ∆ (t)

(cid:21)

Note that, to design ueq, we need the existence of a feasible solution to the following equation:

f (e) + ∆re f (t) + ∆e (e) + bh ueq + bd Qh−0 (t) = 0

(3.31)

with respect to ueq. Since the hysteresis operator corresponding to Qh−0 does not contain the

non-hysteretic backlash and is thus not strictly increasing, it does not admit a unique inverse.

Furthermore, due to the implicit nature of (3.31), one cannot obtain a closed-form solution to

(3.31) even if the analytical inverse of Qh−0 were available. To address this challenge, we propose

the use of a low-pass ﬁlter which can be expressed as

τ ˙zh + zh = Qh−0 (t)

(3.32)

where τ is a design parameter chosen small enough. Now, the equivalent control ueq is designed

by using the output of the ﬁlter zh to mitigate the effect of the signal Qh−0 (t)

(cid:20)

ueq =

1
bh

(cid:21)

− f (e)− ∆re f (t)− ∆e (e)− bdzh

58

(3.33)

The switching control law is taken as

us = −βs (t,e,zh)sat (sc/µ)

(3.34)

where sat(·) represents the saturation function, and βs (,·,·) is the switching-gain function and it
will be deﬁned in Theorem 3.1 later. Here µ > 0 is a design parameter that represents the slope

of the saturation function, such that limµ→0 sat (sc/µ) = sgn (sc). If µ is chosen small enough

(smaller than τ as our analysis will show later), we can guarantee that the trajectory will be attracted

in ﬁnite time to a small neighborhood around the the surface sc = 0 , namely, the boundary layer
{|sc| ≤ µ}, and will stay conﬁned therein for all the future time. This choice of the switching
control law is to avoid the chatter of the control; however, the error e will be O (µ). To deal with

this error we use the adaptive conditional servomechanism [70], which is discussed next.

3.3.2 Adaptive Conditional Servocompensator Design

The control law (3.34) does not achieve zero tracking error because in the boundary-layer layer

phase, there is a non-vanishing matched disturbance

χ = µb ϕ∆ (t)

which (in theory) generates an inﬁnite number of harmonics of the reference signal frequency. To

accommodate the adaptive conditional servomechanism theory [70] and [82], we will consider a

ﬁnite number of frequencies (the frequencies themselves need not to be known; only their number

should be known).

Assumption 3.3. The approximate disturbance ¯χ (t) is generated by an exogenous neutrally stable

59

linear dynamical system:

˙ω = Sω,

¯χ = Γω

(3.35)

where ω ∈ ϖ is the state vector of the approximated exosystem, and ϖ ⊂ Rm is a compact set,

(cid:20)

, Γ =

(cid:21)

1 0 . . .

. . . 0

1×m



S =

0

0
...

0

1

0

0

1

. . .

. . .

. . .

. . .

0

0

0
...

1

c0

c1

. . .

. . .

cm−1



m×m

The internal model (i.e., the servocompensator) is designed to be active only when the sliding

surface trajectory enters the boundary layer in the following way;

˙η = Aηη + µBηsat (s/µ)

(3.36)

where η ∈ Rm is the conditional servocompensator state vector. The matrix Aη is chosen to be

Hurwitz, and the pair (Aη ,Bη) is in the controllable canonical form. Let s be the new surface

variable, which includes the servomechanism,

s = KT

η η + sc

where Kη is a unique vector that assigns the eigenvalues of

(cid:16)

Aη + BηKT
η

(cid:17)

at the eigenvalues of S.

Because Aη is Hurwitz, if η (0) is O (µ), η (t) will be O (µ). The switching control law (3.34) is

modiﬁed to

us = −βs (t,e,zh)sat (s/µ)

(3.37)

60

In this work, we consider the output feedback case, where only e1 is available for measurement.

Therefore, we utilize a high-gain observer to reconstruct the remaining error states. Similar to [86],

we design a high-gain observer with a linear structure:

˙ˆei = ˆei+1 +

gi
εi (e1 − ˆe1)

˙ˆen =

gn
εn (e1 − ˆe1)

1 ≤ i ≤ n− 1

(3.38)

where ˆei is the estimate of ei, ε (cid:28) 1 is a positive design parameter, and gi is the corresponding
estimation gain, where all the gains are chosen such that the polynomial λ n + g1λ n−1 + ··· +
gn−1λ +gn is Hurwitz. The control laws (3.33), (3.37), and the conditional compensator dynamics

(3.36) are modiﬁed by replacing e and s by ˆe and ˆs, respectively, where

ˆs = KT

η η + ˆsc

(3.39)

It is worth mentioning that in order to prevent the observer peaking, the estimated observer states

are saturated before being fed into the control law, which makes the control law globally bounded

in its arguments in the domain of interest. The work of [84] ensures the existence and uniqueness
of the solution M of the following Sylvester equation MS− SM = −P−1Bη
and S have no common eigenvalues. Moreover, the matrix M is non-singular because the pair

(cid:1) is controllable and the pair (S,Γ) is observable [84]. Note that here P is a unique non-

(cid:0)Aη ,Bη

Γ + KT

η PM

, if Aη

(cid:16)

(cid:17)

singular similarity transformation matrix such that

P−1(cid:16)

(cid:17)

Aη + BηKT
η

P = S, P−1Bη = [0 0 . . . 0 1]T

(3.40)

Now, suppose that the frequencies of the approximated disturbance signal ¯χ are unknown, which

61

means that the eigenvalues of the exosystem (3.35) are unknown. An adaptation mechanism is

implemented by replacing Kη in (3.39) by ˆKη, and then an adaptive law is designed to adjust

(cid:16)

(cid:17)

the eigenvalues of the matrix

Aη + Bη ˆKT
η

online to become equal to the eigenvalues of the

exosystem matrix S. In case of having a partial adaptation, we deﬁne the following ι-dimensional

vectors

λη = ZηKη

and

v = Zηη

where ι ≤ m is the number of the adapted variables, Zη is (ι × m) matrix with its rows are unit

vectors, and v represents the regressor vector. Since the pair (cid:0)Aη ,Bη

(cid:1) is chosen to be in the

controllable canonical form, it can be seen that the number of adaptation variables ι will be m/2 if
the number of frequencies is even, or (m− 1)/2 if the number of frequencies is odd. Suppose that
λη belongs to the convex hypercube ϖη = {λη|ai ≤ ληi ≤ bi,1 ≤ i ≤ ι}. As we will show later in
Section 3.4.2, the adaptation law can be designed as

˙ˆλη = B ( ˆs, µ) P (γ ( ˆsc,v,βs))

(3.41)

where the function P (·) is a parameter projection operator that retains ˆλη in ϖδ for all t ≥ 0.
Moreover, deﬁne the set ϖδ = {λη|ai−δ ≤ ληi ≤ bi +δ ,1 ≤ i ≤ ι}, and δ > 0. The function γ is
deﬁned as

γ ( ˆsc,v,βs) = gaβsv Pr ( ˆsc) /µ2

where ga is the adaptation gain. The componentwise smooth projection P (γ ( ˆsc,v,βs)) is deﬁned

62

by

(cid:20)

P (γ ( ˆsc,v,βs))

(cid:21)

=

i

The function Pr ( ˆsc) = µηsat(cid:0) ˆsc/µη

γi



1 +

1 +

(cid:19)
(cid:19)

bi−ˆληi
ˆληi−ai

δ

(cid:18)
(cid:18)
(cid:1) is added as an additional optional step that we ﬁnd later

if ˆληi > bi and γi > 0
if ˆληi < ai and γi < 0

Otherwise

γi

γi

δ

beneﬁcial in the controller implementation, where 0 < µη < µ. The above projection ensures that
ˆλη ∈ ϖη . The function B (s, µ) is deﬁned as



B (s, µ) =

0

1
1− |s|−µ
µ

if |s| ≥ 2µ
if |s| ≤ µ
if µ < |s| < 2µ

The purpose of the function B ( ˆs, µ) is to keep ˆλη constant outside the boundary set {|s| ≤ 2µ}.

3.4 Analysis of Output-Feedback Closed-loop System

3.4.1 Reaching Phase

In this phase, the closed-loop system under output feedback consists of the exosystem dynamics

(3.35), the conditional servomechanism (3.36), the semi-afﬁne error dynamics (3.20), and the high-
gain observer dynamics (3.38). By introducing the change of variables ˜zh = zh − Qh−o (t), the
shifted ﬁlter dynamics equation is obtained, and by considering the sliding surface function (3.39)

under output feedback, we can obtain the following three time-scale singularly perturbed closed-

63

loop system;

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

˙ω = Sω
˙η = Aηη + µBηsat ((s− N (ε)ϕ) /µ)
˙ζ = Aζ ζ + Bζ
˙s = −bd ˜zh + ∆1 (·) + ∆2 (·)− bh.ψ (η, ˆe,zh, µ) + ψz (·)

s− ˆKT
η η

(cid:17)

(cid:16)

τ ˙˜zh = −˜zh − τ∆q (·) + ψq (·)

(cid:2)− bd ˜zh − ∆e (e) + ∆2 (·)

ε ˙ϕ = Aϕϕ + ε Bϕ

− bh.ψ (η, ˆe,zh, µ) + ψz (·)(cid:3)

˙ˆλη = B ( ˆs, µ) .P (γ ( ˆsc,v,βs))

(3.42)

(cid:16)

Aζ ,Bζ

(cid:17)

and(cid:0)Aϕ ,Bϕ

(cid:1) are

where the matrices Aζ and Aϕ are Hurwitz. Furthermore, the pairs

in controllable and observable canonical forms, respectively. The scaled estimation dynamics are

given by

ϕi =

1

εn−i (ei − ˆei)

Notice here if ϕ = 0, the closed-loop equations will coincide with the corresponding equations

under state-feedback. The terms

∆1 (·) = ˆKT

η Aηη + µ ˆKηBηsat ((s− N(ε)ϕ) /µ) + ˙ˆληv

∆2 (·) = bϕ∆ (t)

ψz (·) = bdψq (·) + b ( ˆϕ∆ (t)− ϕ∆ (t))

ψq (·) = ˆQh−o (t)− Qh−o (t)

64

(cid:0) ˙ueq + ˙us

(cid:1)

∆q (·) =

∂ Qh−o

∂ u

ζ T = [e1 e2 . . . en−1]

ϕT = [ϕ1 ϕ2 . . . ϕn]

(cid:20)
k1εn−1 k2εn−2 k3εn−3 . . . kn−1ε 1

(cid:21)

N (ε) =

where ˆϕ∆ and ˆQh−o are the corresponding terms under the output-feedback case. Finally,

ψ (η,e,zh, µ) = βs (t,e,zh)sat (s/µ)

Since the matrices Aη, Aζ , and Aϕ are Hurwitz, we deﬁne the following Lyapunov functions;

Vη

de f
= ηT Pηη, Vζ

de f
= ζ T Pζ ζ , Vϕ

de f
= ϕT Pϕϕ,

de f
=

Vs

1
2

s2, and Vq

de f
=

1
2

˜z2
h

where the symmetric positive deﬁnite matrices Pη, Pζ , and Pϕ are the solutions of the following
Lyapunov equations: PηAη + AT

ηPη = −I, and Pζ Aζ + AT

ζ Pζ = −I, PϕAϕ + AT

ϕPϕ = −I.

re-

Ωc

spectively. Given any positive constants c > µ and r > τ, we deﬁne the following compact sets:
= {η : Vη ≤ ρη µ2},
de f
= {ϕ : Vϕ ≤ ρϕε2}, where the param-
de f

= {(ζ ,s) : Vζ ≤ ρζ c2,|s|≤ c}, Ωµ
= {˜zh : Vq ≤ ρqr2}, Ψτ
de f

= {(ζ ,s) : Vζ ≤ ρζ µ2,|s|≤ µ}, Ωη

= {˜zh : Vq ≤ ρqτ2}, and Ψε
de f

de f

de f

Σr

eters ρζ , ρη, ρq, and ρϕ will be deﬁned in the proof of Theorem 3.1.

Theorem 3.1. Suppose that Assumptions (3.1)-(3.3) hold and consider the closed-loop system
(3.42). Let Ω1, Ω2 be compact sets and be in the interior of Ωc and Σr, respectively, η (0) ∈ Ωη,

65

and let υo and ˆζ (0) be bounded. Let the switching-gain function βs (t,e,zh) be chosen as

(cid:16)

(cid:17)(cid:17)

(cid:16)

Ko/

βs (t,e,zh) =

K1 + Ko|ueq|
ho (1− ¯Ko)
ΘN
d θ N

+ γo

(3.43)

d θ N
ΘN
ho

and γo > 0. Assume ¯Ko < 1. Then there is τ∗ > 0, such that for each
with ¯Ko =
τ ∈ (0,τ∗], there is µ∗ = µ∗ (τ) > 0, such that for each µ ∈ (0, µ∗], there is ε∗ = ε∗ (τ, µ) > 0, such
that for every τ ∈ (0,τ∗], µ ∈ (0, µ∗], and ε ∈ (0,ε∗] and for all initial conditions (ζ (0) ,s (0)) ∈
Ω1, ˜zh (0) ∈ Ω2, η (0) = O (µ), and ˆλη ∈ ϖη, all the state variables are bounded and converge to
a positively-invariant set ϒτ,µ,ε = Ψτ × Ωµ × Ψε in ﬁnite time.

The proof of this theorem is deferred to Appendix A.2.

3.4.2 Boundary Layer Phase

In this section, we will analyze the closed-loop system stability after the trajectory enters the
set ϒτ,µ,ε. Inside this set, the function sat ((s− N (ε)ϕ) /µ) = s− N (ε)ϕ/µ. We introduce the

following change of variables;

where

ϑ =

(η − ¯η)

µ

+ Bης

(cid:90) en

dy

0

bhβs (t,e1, . . . ,en−1,y,zh)

ς =

(3.44)

and ¯η is the conditional servocompensator state in the boundary-layer phase under the case when

the internal model is not perturbed (Eq. (3.35) of Assumption 3.3). Under the internal-model-
perturbation-free case, the switching control law us uses a ﬁxed-gain ¯βsmax instead of the variable

66

gain function βs (t, ˆe,zh), where ¯βsmax is a positive constant such that inside the set ϒτ,µ,ε, the
KT
η ¯η
switching-gain satisﬁes |βs (t, ˆe,zh)| ≤ ¯βsmax, and consequently, ¯χ = − ¯βsmax
µ . The main rea-
son of this modiﬁcation is because the switching function βs (t, ˆe,zh) is not smooth. Therefore, the
change of variables (3.44) cannot be used to obtain the dynamics ˙ϑ. In the forthcoming equations,

we rewrite the term

ˆKT
η η = KT

η η + ˜λ T
η v

(3.45)

where ˜λη = ˆλη − λη. Accordingly, using the equations (3.44)-(3.45), the closed-loop dynamics

can be written in four time-scales as follows;

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

˙ω = Soω

˙ϑ = Aηϑ + Fϑ (ζ , ˜zh,sc) + bϑ (ϕ) /µ + ∆ϑ ( ¯η,ζ ,zh, ¯χ, ˜χ, µ)
˙ζ = Aζ ζ + Bζ sc
τ ˙˜zh = −˜zh − τ∆q (·) + ψq (·)
µ ˙sc = −bhβs (t, ˆe,zh)sc − bhβs (t, ˆe,zh) ˜λ T

η v + µFsc (ϑ ,ζ , ˜zh,sc)

(3.46)

+ bsc (ϕ) + µ∆sc ( ¯η,ζ ,zh, ¯χ, ˜χ, µ) + µψz (·)

(cid:2)− bhβs (t, ˆe,zh)sc/µ − bhβs (t, ˆe,zh) ˜λ T

ε ˙ϕ = Aϕϕ + εBϕ

+ Fsc (ϑ ,ζ , ˜zh,sc) + bsc (ϕ) /µ + ∆sc ( ¯η,ζ ,zh, ¯χ, ˜χ, µ) + ψz (·)(cid:3)

η v/µ

˙˜λη = γβs (t, ˆe,zh)H.v.sc/µ2 + Fλ (ϑ ,ϕ) /µ2

where the functions bϑ , bsc, Fϑ , Fsc, and Fλ are locally Lipschitz functions and satisfy

bϑ (0) = bsc (0) = Fϑ (0,0,0) = Fsc (0,0,0,0) = Fλ (0,0) = 0

67

The internal model perturbation term

˜χ ( ¯η,e, µ) = b (ϕ∆ (u)− ¯χ)

is generated due to the effect of the approximation of the internal model. The ι × ι matrix H is a

diagonal matrix, whose entries could be 1, 1 +

bi−ˆληi

δ

, or 1 +

ˆληi−ai

δ

.

Remark 3.1. Inside the set ϒµ,τ, the following arguments are applied:

1. From the analysis of Section 3.4.1, the control law (3.30) is bounded due to the boundedness

of its arguments and because of the smooth projection of the adaptation law (3.41), which

implies the boundedness of the term ϕ∆ (u). Moreover, ¯η is of order of µ, therefore it can be

easily shown that

| ˜χ ( ¯η,e, µ)| ≤ δh

where δh is a positive constant dependent on µ. Consequently, we can show that

|∆ϑ (·)| ≤ δϑo,

and

|∆sc (·)| ≤ δsco

(3.47)

Moreover, the following terms are found to satisfy the following inequalities;

| ˙ueq| ≤ ¯keq,

|∂ Qh−o
∂ u

| ≤ ¯kq,

and

| ˙βs (t, ˆe,zh)| ≤ ¯ks

where ¯ks =

Ko.¯keq
d θN
ΘN
h0

(1− ¯Ko)

. Therefore, we can show that

| ˙us| ≤ ¯ks
µ

68

Consequently, the term

(cid:16)¯keq +

(cid:17)

.

¯ks
µ

where δz = ¯kq

|∆q (·)| ≤ δz

2. By the local Lipschitz properties, the following inequalities are satisﬁed

ψz (·)| ≤ ¯lz|ϕ(cid:107),

and

|ψq (·)| ≤ ¯lq|ϕ(cid:107)

(3.48)

| 1
bh

where ¯lz and ¯lq are positive constants.

Let ¯v = Zη ¯η be the regressor vector inside the set ϒµ,τ. if ¯v is not persistently excited (i.e. Zη

is not of full rank), the next lemma, which is adapted from [70], is needed.

Lemma 3.1. For every initial condition ω (0), there exists a non-singular matrix Eη, which is

dependent on ω (0), such that

Eη ¯v =

 ,

 ¯va

0

(3.49)

where ¯va ∈ Rιa is the persistently exciting vector with ιa ≤ ι. Under the fully persistently exciting
case (ι = ιa), Eη will be equal to the identity matrix and ¯v = ¯va. If ¯v = 0, equation (3.49) holds

with ¯va = 0.

Theorem 3.2. Consider the closed-loop dynamics (3.46). Under the assumptions of Theorem 3.1,

and inside the set ϒτ,µ,ε, if the regressor vector of the conditional servocompensator state ¯v is
persistently exciting, then there exists τ(cid:63) > 0, such that for each τ ∈ (0,τ(cid:63)], there is µ(cid:63) = µ(cid:63) (τ) >
0, such that for each µ ∈ (0, µ(cid:63)], there is ε(cid:63) = ε(cid:63) (τ, µ) > 0, such that, for every τ ∈ (0,τ(cid:63)],
µ ∈ (0, µ(cid:63)], and ε ∈ (0,ε(cid:63)],

69

τ, τ
µ

(cid:17)
, the tracking error e (t) = O(cid:0)¯λ(cid:1).

(cid:16)
(cid:3), the tracking error will be O (δh).

• For the internal-model-perturbation case (| ˜χ| ≤ δh): there exists δ (cid:63)

h such that for every

• For the internal-model-perturbation-free case ( ˜χ = 0): if τ is chosen arbitrarily small, then

for ¯λ = max

δh ∈(cid:0)0,δ (cid:63)

h

The proof of the theorem is provided in Appendix A.3.

3.5 Application to Nanopositioning Control

In this section, the proposed control approach and the theoretical results are conﬁrmed by conduct-

ing a series of tracking experiments with different types of reference inputs. For that purpose, we

use the experimental setup shown in Figure 2.5. The nanopositioner dynamics are modeled in the

form of equation (3.1) with n = 2. The parameters of the fast dynamics part of the model are iden-

tiﬁed using a sinusoidal input with a small amplitude with frequency range from 1 Hz to 5 KHz.
Accordingly, the parameters of the fast dynamics are identiﬁed as a1 = 1.838× 108, a2 = 4622,
and b = 1.0688× 109. The measured Bode plot of the nanopositioner system output is shown in

Figure 3.1 along with that of the identiﬁed system dynamics. The slow dynamics part, which is

represented by the MPI hysteresis operator, is identiﬁed by injecting a sinusoidal input with grad-

ually decreased amplitude to exhibit the inner loops. The identiﬁed MPI model is constructed with

8 play operators and 9 deadzone elements, and the identiﬁed weights and thresholds vectors are

found to be

θ T
h = [0.001,0.7301,0.1293,0.0969,0.0695,0.0324,0.0329,0.0871] ,
rT = [0,0.0460,0.4654,1.1071,1.8248,2.5574,5.0429,3.4279] ,

70

d =(cid:2)− 14.775,−8.287,−1.1174,−0.9753,−0.9403,0.3892,0.4419,

θ T

14.2164,19.1489(cid:3),

dT = [−7.4049,0,−8,−5,−7,−10,−9,−2,−1] ,
dT

= [1.4344,0,5.3450,4.0961,2.6166,9.3856,7.4485,1.4278,0.4928] ,

To validate the overall model, we cascade the MPI hysteresis model with the fast dynamics transfer

function. Notice that the fast dynamics transfer function has a non-unity dc gain. Since we model

the slow dynamics using the MPI operator model, the fast dynamics transfer function dc gain is

changed to unity when the model is implemented. In Figures 3.2, we show the system output

versus the overall identiﬁed system output using the same input reference used for identiﬁcation.

While in Figure 3.3, we plot the measured hysteresis loops as compared to the predicted ones.

As it can be seen in both ﬁgures, the identiﬁed system can capture the system output with small

prediction error. Moreover, we repeat the same experiment by changing the frequency of the input

signal with a range up to 100 Hz, and as expected the prediction error increased as we increased

the frequency of the input; however, the identiﬁed model is still able to capture the output of the

system reasonably well.

To reconstruct the unmeasured states, the high-gain observer (3.38) is implemented with the

parameters ε, the gains g1 and g2 being 0.0001, 3 and 20, respectively. The low-pass ﬁlter (3.32)

is implemented with τ = 2000. The parameters of the switching function βs (Eq. (3.43)) are
calculated after calculating the bound (3.21). We calculate σdmax = 10 and, with ¯Ko = 0.97, then
the parameter K1 is found to be 217.5. For the switching control part, the following parameters are

used: µ = 1000 and k1 = 5000. The parameter γo is chosen as 10.

The conditional servocompensator dynamics (3.36) is designed as a third-order system. This

is based on the assumption that the frequency spectrum of the boundary-layer disturbance (χ)

71

includes only the fundamental frequency of the reference input. In Addition to that, an integral state

is augmented with the second-order internal model dynamics to compensate for any disturbance

bias.

To robustify the adaptation law against noisy inputs, the adaptation parameter λη is assumed

to be conﬁned to the following convex set;

ϖη = {λη|− 8 ≤ λη ≤ 8}

The adaptation gain γ, the parameters δ , and µη are chosen as 2.7× 104, 0.1, and 1, respectively.

The experiments are focused on three types of reference signals. The ﬁrst signal is a sinusoidal

input deﬁned as yr = 10 sin (2π f t) + 10 µm with frequency f = 5, 25, 50, and 100 Hz. Notice

that the controller evaluation is limited to reference frequencies with range up to 100 Hz, to avoid

excessive vibration and potential damage to the nanopostioner when the highest harmonic fre-

quency is getting close to the ﬁrst resonant frequency of the system. In Table 3.1 and 3.2, we show

a comparative analysis for the tracking error of our proposed approach with experimental results

of different control approaches taken from previous projects on the same device. The considered

approaches for comparison are: 1) the work of [48] in which a sliding mode controller (SMC) was

designed, 2) the results of the Single Harmonic Servo-Compensator (SHSC) and Multi-Harmonic

Servo-Compensator (MHSC) demonstrated in [5], 3) a classical PI controller implemented and

tuned to best possible performance without utilizing any feedforward inversion. The percentage

of the maximum tracking error with respect to the maximum peak is shown in Table 3.1, while in

Table 3.2 the percentage of the mean tracking error with respect to the maximum peak is presented.

In both tables, it can be seen that the inversion-free adaptive conditional servocompensator based

approach (Inv-F ACS) results are superior to the other four approaches in reducing the tracking

72

error very signiﬁcantly.

We further test the controller by using a sawtooth reference input with frequencies 5, 25, 50, and

100 Hz. To avoid spiking responses of the closed-loop system, we insert a second order pre-ﬁlter to

smooth out the signal edges. In Figure 3.4, the measured displacement of the inversion-free control

schema under the 100 Hz frequency reference case is shown. The tracking error is shown in Figure

3.5 for the period 0-0.03 second. It can be noticed in both Figure 3.4 and 3.5 that the tracking error

undershoots to around -3.5 µm and then overshoot to less than 1 µm before converging quickly in

less than 0.005 seconds. The tracking error of the inversion-free controller in the boundary-layer

phase (0.01-10 seconds) are shown in Figure 3.6. Notice that the tracking error are not increasing

in this period.

Another set of experiments are conducted with a periodic reference generated by the van der

Pol oscillator with the same frequency ranges used with the previous two reference inputs. See

Figures 3.7, 3.8, and 3.9 for the tracking performance for the 100 Hz case.
Table 3.1: Percentage Peak tracking error (max |e(t)|%) for the proposed controller versus other controllers
in percent of the reference maximum peak under sinusoidal reference input.

Hz
5
25
50
100

SMC
0.4750
0.8500
1.1250
1.3750

SHSC MHSC
0.4495
0.8600
0.4405
0.9250
0.5050
0.9650
1.1900
0.7850

PI

0.7894
0.8422
1.0058
1.5185

Inv-F ACS

0.0105
0.0165
0.0272
0.0414

Table 3.2: Percentage Mean tracking error (mean |e(t)|%) for the proposed controller versus competing
methods in percent of the reference maximum peak under sinusoidal reference input.

Hz
5
25
50
100

SMC
0.0595
0.3100
0.3300
0.4150

SHSC MHSC
0.3245
0.1355
0.1340
0.3535
0.1420
0.3850
0.4075
0.1760

PI

0.0736
0.0939
0.1498
0.2897

Inv-F ACS

0.0007
0.0021
0.0039
0.0082

To give a further insight, the frequency content spectrum of the tracking error for both cases

73

Table 3.3: Percentage Tracking errors for the proposed controller in percent of the reference max-
imum peak under sawtooth and van der Pol reference inputs.

Hz
5
25
50
100

Sawtooth

van der Pol

Max |e(t)|% Mean |e(t)|% Max |e(t)|% Mean |e(t)|%

0.0070
0.0292
0.0414
0.1529

0.0007
0.0030
0.0053
0.0111

0.0071
0.0195
0.0281
0.1157

0.0005
0.0024
0.0045
0.0100

of the sawtooth and the van der Pole reference inputs with 100 Hz frequency are given in Figures

3.10 and 3.11, respectively. Two observations can be drawn. Firstly, the harmonics excited for the

sawtooth case include 4 odd harmonics, while in the van der Pol case we have 5 odd harmonics.

Secondly, it is noted in the sawtooth case that most of the residual error comes from the ﬁrst and

third harmonic, while in the van der Pol case, most of the residual error comes from the third har-

monic as compared to the ﬁrst harmonic amplitude, which is reduced to less than 1 nm. In Table

3.3, the percentage tracking errors are presented for both the sawtooth and the van der Pol cases for

the considered frequency range. The results show that the van der Pol case shows slightly better

performance than the sawtooth case for all frequencies.

Remark 3.2. It can be noticed that the controller parameters µ, τ, and k1 are chosen relatively

large. This is mainly attributed to the large parameters of the nanopositioner system’s model. This

can be explained by appropriately scaling the state and time variables of the semi-afﬁne error

dynamics (3.20). Let a1 = ω2

n and a2 = 2ξωn, where ωn is the nanpositioner’s natural frequency

and ξ is the damping coefﬁcient. To scale the system, we apply the following change of variables:

z1 (ts) = e1 (t) ,

z2 (ts) =

e2 (t)
ωn

,

ts = ωnt

74

which transforms the error system (3.20) into

 ˙z1 = z2

˙z2 = −z1 − 2ξ z2 +

(cid:2)∆re f + bd + Qh−0 + b ϕ∆

(cid:3)

bh
ω2
n

u +

1
ω2
n

where ˙zi denotes the derivative of zi with respect to ts. The switching component of the continuously-

implemented sliding mode control law (3.34) for the transformed system is given by

(cid:18) ¯k1z1 + z2

(cid:19)

¯µ

sat

= sat

(cid:18) ¯k1e1 + e2/ωn

(cid:19)

¯µ

(3.50)

where ¯µ is a small parameter. Comparing the switching component of (3.50) with

(cid:18)k1e1 + e2

(cid:19)

µ

sat

in our controller, we can see that

k1 = ¯k1ωn,

µ = ¯µωn

In other words, the non-scaled surface function sc will be of order of ωn = 1.3557× 104, which

was observed in the experiments. Inside the boundary layer, we have

˙e1 = −k1e1 + sc

and |sc| ≤ µ. Let Ve1 = 1
2e2

1. Then the time-derivative of Ve1 satisﬁes

˙Ve1 = −k1e2

1 + e1sc ≤ −k1e2

1 +|e1|µ ≤ −k1
2

1, ∀ |e1| ≥ 2µ
e2
k1

75

Therefore, the trajectory reaches the set {|e1|≤ 2µ
, |sc|≤ µ} in ﬁnite time. Hence e1 = O (µ/k1) =
k1
O (0.2). Adding the conditional adaptation will make the error much smaller. Our theory requires

choosing τ > µ, so we picked τ = 2000.

Figure 3.1: Measured Bode plot of the nanopositioner system (in blue) and the output of the identiﬁed
high-frequency dynamics (in red).

76

5505005000Frequency (Hz)0102030Magnitude (dB)MeasuredEstimated5505005000Frequency (Hz)-400-2000Phase (deg)MeasuredEstimatedFigure 3.2: Nanopositioner system output used in the identiﬁcation and the resulting model output.

Figure 3.3: Measured hysteresis loops (in blue) compared to predicted hysteresis loops (in red).

77

050100150200Time (sec)-100102030405060Output Displacement (m) MeasuredPredicted0123456Input (volts)01020304050Hysteresis Output (m) MeasuredPredictedFigure 3.4: Experimental results of measured displacement on tracking a sawtooth reference with 100 Hz
frequency using the inversion-free ACS.

Figure 3.5: Experimental results of the tracking error with a sawtooth reference with 100 Hz frequency.

78

00.0050.010.0150.020.0250.03Time (seconds)-505101520Displacement (m)Desired ReferenceOutput Displacement00.0050.010.0150.020.0250.03Time (seconds)-4-3-2-101Tracking Error (m)Figure 3.6: Experimental results of the tracking error with a 100 Hz sawtooth reference in the boundary-
layer phase.

Figure 3.7: Experimental results of measured displacement on tracking a van der Pol reference with 100
Hz frequency using the inversion-free ACS.

79

0246810Time (seconds)-0.03-0.02-0.0100.010.020.03Tracking Error (m)00.0050.010.0150.020.0250.03Time (seconds)-5051015202530Displacement (m)Desired ReferenceOutput DisplacementFigure 3.8: Experimental results of the tracking error with a van der Pol reference with 100 Hz frequency.

Figure 3.9: Experimental results of the tracking error with a 100 Hz van der Pol reference in the boundary-
layer phase.

80

00.0050.010.0150.020.0250.03Time (seconds)-30-25-20-15-10-50510Tracking Displacement (m)0246810Time (seconds)-0.03-0.02-0.0100.010.020.03Tracking Displacement (m)Figure 3.10: Tracking error signal frequency spectrum with a sawtooth reference with 100 Hz frequency
in boundary-layer phase.

Figure 3.11: Tracking error signal frequency spectrum with a van der Pol reference with 100 Hz frequency
in boundary-layer phase using the inversion-free ACS.

81

01002003004005006007008009001000Frequency (Hz)00.511.52Absolute Amplitude (m)10-302004006008001000Frequency (Hz)00.511.52Absolute Magnitude of e1 (m)10-3Chapter 4

Dynamic Inversion-based Hysteresis

Compensation Using Extended High-gain

Observer

4.1

Introduction

In this chapter, we consider the tracking problem for an uncertain nonlinear single-input-single-

output system, preceded by an unknown hysteresis operator. With a mild assumption on the non-

hysteretic component, the system preceded by hysteresis can be treated as an uncertain non-afﬁne

nonlinear system. We then use dynamic-inversion [71], to deal with the nonafﬁne nonlinearity as

it computes the inverse in a fast time-scale compared to the plant dynamics. In order to implement

the dynamic inversion algorithm, the knowledge of the right hand side of the last state derivative

of the plant is required which includes the hysteretic part, and uncertainties. To overcome this,

we use extended high-gain observers [72], to estimate the hysteretic part, system states, and un-

certainties in the fastest time-scale. Therefore, the closed-loop system using the output feedback

controller results in a three-time-scale structure, where the extended high-gain observer estimates

unmeasured states, uncertainties and hysteretic perturbation in the fastest time-scale, the dynamic

inversion is used in the intermediate time scale, while the plant dynamics evolves in the slowest

82

time scale.

The remainder of this chapter is organized as follows. In Section 4.2, we introduce the system

class and assumptions regarding the hysteresis operator and the nonlinear plant. We provide Propo-

sitions 4.1 and 4.2, which are then used for the analysis of the closed-loop system. In Section 4.3

the state feedback controller design is ﬁrst described, in preparation for the discussion of the out-

put feedback controller. The analysis of the state feedback case determines the saturation level that

will be used in output feedback to deal with the observer peaking. Moreover, the state feedback

controller is designed ﬁrst to shape the transient response of the closed-loop system. The transient

response under output feedback is shaped by the performance recovery property, which is achieved

by bringing the trajectories under output feedback arbitrarily close to the trajectories under state

feedback for sufﬁciently small observer parameter. In Section 4.4, the output feedback control

scheme is described and the resulting closed-loop system is analyzed utilizing the assumptions

and propositions from Section 4.2 and the saturation level from Section 4.3. Finally, simulation

results are presented in Section 4.5, followed by experimental validation and comparison results in

Section 4.6.

4.2 Problem Formulation
We consider the tracking problem for a single-input-single-output system deﬁned in the normal

form, with a hysteretic operator at the input side:

˙η = f0(η,ξ ,w),

˙ξ = Aξ + B[ f (η,ξ ,w) + ϒ(t)],

ϒ(t) = Γ(cid:2)u(τ)|t

(cid:3) (t),

τ=0,H0

y = ξ1,

83

(4.1a)

(4.1b)

(4.1c)

(4.1d)

where A ∈ Rρ×ρ and B ∈ Rρ×1 are deﬁned as

0 1

0 0
...
...

0

1
...

. . . 0

. . . 0
...
...

0 0 . . .

0

1

0 0 . . .

. . . 0



A =





,



0

0
...

0

1

, B =

η ∈ Rn−ρ, ξ = col(ξ1, . . . ,ξρ ) ∈ Rρ are the state variables, u ∈ R is the control input, and y ∈ R
is the output of the system, Γ[·,·] is a hysteresis operator, ϒ ∈ R is the output of the hysteresis
operator, which depends on the history of the input u(τ)|t
τ=0 and the initial internal state H0 of the
operator Γ, w ∈ Rl is an exogenous input and ρ is the relative degree of the system. The normal
form is deﬁned for η ∈ Dη ⊂ Rn−ρ and ξ ∈ Dξ ⊂ Rρ for some domains Dη and Dξ . The system
is required to satisfy the following assumptions.

Assumption 4.1. For any given t, w(t) and ˙w(t) are bounded, and w(t) belongs to a known com-
pact set W ⊂ Rl.

Assumption 4.2. The function f is continuously differentiable with locally Lipschitz derivatives
and f0 is locally Lipschitz for all (η,ξ ,w) ∈ Dη × Dξ ×W .

Assumption 4.3. There exist a continuously differentiable function V0(η), class K∞ functions Ψ1,
Ψ2, and K function Ψ3, such that

Ψ1(||η||) ≤ V0(η) ≤ Ψ2(||η||)

∂V0
∂η

f0(η,ξ ,w) ≤ 0, for ||η|| ≥ Ψ3(||ξ|| +||w||)

84

for all (η,ξ ,w) ∈ Dη × Dξ ×W .
Assumption 4.3 implies that the internal dynamics of the system is regionally input-to-state stable.

Assumption 4.4. The hysteresis operator Γ can be decomposed as

Γ(cid:2)u(τ)|t

τ=0,H0

(cid:3) (t) = g(u(t)) + ˜Γ(cid:2)u(τ)|t

τ=0,H0

(cid:3) (t)

for some non-hysteretic function g(·) and remaining hysteretic term ˜Γ, and

(i) g(·) is continuously differentiable and satisﬁes

β1 ≤ dg(u)
du

≤ β2

where β1 and β2 are positive constants with β2 ≥ β1.

(ii) ˜Γ is a monotone hysteresis operator in the sense that

0 ≤ (u1(t)− u2(t))( ˜Γ[u1(τ)|t

τ=0,H0](t)− ˜Γ[u2(τ)|t

τ=0,H0](t)) ≤ k(u1(t)− u2(t))2

(4.3)

(4.4)

(4.5)

τ=0 and u2(τ)|t

where u1(τ)|t
u1(0) = u2(0), and k is a positive constant.

τ=0 are both monotonically increasing (or decreasing) inputs with
τ=0,H0](t) is piecewise con-
tinuously differentiable with respect to u(t) for a monotonic function u; i.e., any segment of the

In addition, ˜Γ[u(τ)|t

hysteresis graph ( ˜Γ[u,H0] vs. u) has a piecewise continuous slope function.

Remark 4.1. Assumption 4.4 is satisﬁed by a wide class of hysteresis operators, including the

classical PI operator, the generalized PI operator, and the Preisach operator. For example, for a

classical PI operator with N + 1 plays, the hysteresis output is given by

(cid:2)u(τ)|t

τ=0,H0

(cid:3) (t)

ϒ(t) = θ0u(t) +

N
∑
i=1

θiΓri

85

(4.6)

where θi is the weight associated with play Γri with threshold ri, θ0 is the weight associated with

the play with threshold ri = 0 (the memory-less term), and H0 represents the vector of initial values
of all plays. If θ0 > 0 and θi ≥ 0 for i = 1,2, . . . ,N, the classical PI model satisﬁes all conditions
in Assumption 4.4. Similarly, a generalized PI model satisﬁes Assumption 4.4 if the non-hysteretic

component satisﬁes (4) and all generalized plays have non-negative weights.

Remark 4.2. Assumption 4.4 is also satisﬁed by Preisach-like operators (where the basic building

block is a variant of the Preisach hysteron), such as the Krasnosel’skii-Pokrovskii (KP) operator.

This operator has been used for hysteresis modeling, e.g., shape memory alloy (SMA) actuators

[23]. The output of a KP model is given by

ϒ(t) = g(u) + Γ[u(τ)t

τ=0,H0](t)

where g(u) satisﬁes (4) and Γ[·] consists of KP hysterons which are the same as that of a play
operator, when it is unsaturated, namely, when its output is within the interval (-1, 1).

Proposition 4.1. Consider a hysteresis operator Γ satisfying Assumption 4.4. Then, for a contin-

uous monotonic input u,

0 ≤ ∂ ˜Γ[u(τ)|t
∂ u(t)

τ=0,H0](t)

≤ k

(4.7)

and the derivative exists except possibly for a discrete number of points.

Proof: Without loss of generality, consider a continuous monotonically increasing u. Deﬁne
u1, such that u1(τ) = u(τ) for τ ∈ [0,t], and u1(τ) = u(t) + ∆u(τ) for τ > t, where ∆u(t) = 0 and
∆u(τ) is a continuous, strictly increasing function for τ ≥ t. Deﬁne u2 such that u2(τ) = u(τ) for
τ ∈ [0,t], and u2(τ) ≡ u(t), for τ > t. Pick some τ = t1 > t. It can be easily veriﬁed that

∂ ˜Γ[u(τ)|t

τ=0,H0](t)
∂ u(t)

= lim
t1→t

∆Γ(t1)
∆u(t1)

86

(4.8)

where

∆Γ(t1)

(cid:52)
= ˜Γ[u1(τ)|t1

τ=0,H0](t1)− ˜Γ[u2(τ)|t1

τ=0,H0](t1)

(4.9)

Eq. (4.7) then follows by utilizing (5) in (4.8) and (4.9). And the derivative thus deﬁned exists

except possibly for a discrete number of points due to the piece-wise continuity of the derivative

as assumed in Assumption 4.4.

(cid:3)

Remark 4.3. From Assumption 4.4, for a monotonic input u(t) we have the piece-wise continuity of
the derivative of ˜Γ[u(τ)|t
there will be only a ﬁnite number of jumps of the derivative of ˜Γ[u(τ)|t

τ=0,H0](t) with respect to u(t). Therefore, for any compact time interval,
τ=0,H0](t). Moreover, these

jumps will be bounded and this follows from (4.5).

Remark 4.4. The classical PI operator satisﬁes Proposition 4.1, with the hysteretic part given by

˜Γ[u(τ)|t

τ=0,H0] =

N
∑
i=1

θiΓri

(cid:2)u(τ)|t

τ=0,H0

(cid:3) (t)

from which we have

∂ ˜Γ[u(τ)|t
∂ u(t)

τ=0,H0]

=

N
∑
i=1

θi

∂

∂ u(t)

(cid:2)u(τ)|t

τ=0,H0

(cid:3) (t)

Γri

(cid:2)u(τ)|t

(cid:3) (t) = 0 or 1 depending on whether

∂

(cid:2)u(τ)|t

As illustrated in Fig. 4.1,

(cid:3) (t) is in the play or linear region and is discontinuous only when the operator

τ=0,H0

∂ u(t)

Γri

τ=0,H0

Γri
operates at the intersection of the play and linear regions. Therefore, it follows that

0 ≤ ∂ ˜Γ[u(τ)|t
∂ u(t)

τ=0,H0]

≤ N
∑
i=1

θi

87

Figure 4.1: Illustration of a classical play operator.

Proposition 4.2. Consider a hysteresis operator Γ that satisﬁes Assumption 4.4. Then,

(i) Γ admits a unique (right) inverse, denoted as Γ−1, in the sense that, Γ ◦ Γ−1 = I (iden-

tity), i.e., Γ[Γ−1[ϒ(τ)|t
ϒ, where H−

τ=0,H−

0 ],H0](t) = ϒ(t), for any continuous and monotone function

0 is the corresponding initial condition of Γ−1.

(ii) The inverse hysteresis operator Γ−1 satisﬁes

≤ ∂ Γ−1[ϒ(τ)|t
∂ϒ(t)

τ=0,H−

0 ](t)

1

β2 + k

≤ 1
β1

(4.10)

Proof: By differentiating Γ[u(τ)|t

τ=0,H0](t) with respect to u(t), we have

∂Γ[u(τ)|t

τ=0,H0](t)
∂ u(t)

=

∂ g(u(t))
∂ u(t)

+

∂ ˜Γ[u(τ)|t

τ=0,H0](t)
∂ u(t)

and from Eq. (4.4) and Proposition 4.1, we get

β1 ≤ ∂ Γ[u(τ)|t

τ=0,H0](t)

∂ u(t)

88

≤ β2 + k

(4.11)

For part (i), the existence of a unique inverse is guaranteed by the property that the operator Γ,

per Eq. (4.11), is strictly monotonically increasing; namely, the output is strictly increasing (resp.,

decreasing) with a strictly increasing (resp., decreasing) input.

For Part (ii), Eq. (4.11) shows that the slope of any segment on the hysteresis graph (ϒ vs. u)

for Γ is bounded between two positive constants, β1 and β2 + k. Since the graph of the inverse
operator Γ−1 is obtained by swapping the horizontal axis and the vertical axis of the graph for the

forward hysteresis operator Γ, one can immediately conclude that the slope of any segment of the

inverse hysteresis graph (u vs. ϒ) is the reciprocal of the slope of the corresponding segment on
the forward hysteresis graph. Eq. (4.10) then follows from (4.11). (cid:3)

Remark 4.5. By breaking ϒ into piecewise monotone segments, one can easily see that Proposition

4.2 holds for any continuous function ϒ.

Remark 4.6. Eq. (4.10) shows that the slope of any segment on the hysteresis graph (u vs. ϒ) for
Γ−1 is bounded between two positive constants, 1/(β2 + k) and 1/β1. From Assumption 4.4, since
any segment of the hysteresis graph ( ˜Γ[u,H0] vs. u) has a piece-wise continuous slope function for

a monotonic input u, it follows that for a monotonic input ϒ(t), we have the piece-wise continuity

of the derivative of the inverse hysteresis operator. Therefore, for any compact time interval, there

will be only a ﬁnite number of jumps of the derivative of the inverse hysteresis operator. Moreover,

these jumps will be bounded and this follows from (4.10).

Assumption 4.5. The reference signal r(t) and its derivatives up to r(ρ+1)(t) are bounded for all
t ≥ 0. Moreover, R(t) = col(r(t),r(1)(t), . . . ,r(ρ−1)(t)) ∈ Dξ for all t ≥ 0.

89

The change of variables

e1 = ξ1 − r(t), e2 = ξ2 − r(1)(t), . . . , eρ = ξρ − r(ρ−1)(t)

(4.12)

transforms the system (4.1.a-4.1.d) into the form

˙η = f0(η,e +R,w),
(cid:16)
η,e,R,w,r(ρ)(cid:17)

˙e = Ae + B

(cid:110)

F

e1 = y− r,

(cid:111)

+ Γ[u](t)

,

(4.13a)

(4.13b)

(4.13c)

(cid:16)
η,e,R,w,r(ρ)(cid:17)

where F

= f (η,e+R,w)−r(ρ) and e = col(e1,e2, . . . ,eρ ). For ease of notation,
we have dropped the initial condition from the hysteresis operator argument, and for simplicity,

we write the hysteresis output as Γ [u] (t). Similarly, we drop t from the deﬁnition of the reference

signal and its derivatives.

4.3 State Feedback Controller

Dynamic inversion is used for control of nonlinear non-afﬁne-control systems [71]. It is useful for

systems where the inverse of the input nonlinearity does not have a closed-form solution. In this

chapter the controller u needs to be designed such that the closed-loop system behaves as

˙e = (A− BK)e

(4.14)

where K is chosen such that the matrix (A− BK) is Hurwitz. Let Ve(e) = eT Pe, where P is the
solution to the Lyapunov equation P(A − BK) + (A − BK)T P = −Q, for some positive-deﬁnite

90

symmetric matrix Q. Let c be a positive constant and deﬁne Ωc = {e : Ve(e) ≤ c}. Choose c > 0
such that for every e ∈ Ωc, ξ = e +R ∈ Dξ and therefore {ξ : Ve(e) ≤ c} ⊂ Dξ . Next we deﬁne the
set Ω0 = {η : V0(η) ≤ c0}, which is compact and contained in Dη, where c0 is chosen such that
||w||, and κ3 = max
c0 ≥ Ψ2(Ψ3(κ1(c) + κ2 + κ3)), where κ1(c) = max
||R(t)||.
e∈Ωc
t
From Assumption 4.3, the set Ω0 × Ωc is positively invariant with respect to the system

||e||, κ2 = max
w∈W

˙η = f0(η,e +R,w),

˙e = (A− BK)e

because on the boundary {η : V0(η) = c0}

Ψ2(||η||) ≥ c0 ≥ Ψ2(Ψ3(κ1(c) + κ2 + κ3))
=⇒ ||η|| ≥ Ψ3(κ1(c) + κ2 + κ3)
=⇒ ||η|| ≥ Ψ3(||ξ|| +||w||) =⇒ ˙V0 ≤ 0

and on the boundary {e : Ve(e) = c}, ˙Ve < 0.

In this section we design the state feedback controller and we assume that we have perfect
knowledge of the function f (·), states (η,ξ ), exogenous input w, the reference signal and its

derivatives, and the hysteresis output Γ [u] (t). The dynamic inversion algorithm is then given by

µ ˙u = −(cid:104)

F

η,e,R,w,r(ρ)(cid:17)
(cid:16)

(cid:105)

+ Γ[u](t) + Ke

where µ is a small positive constant. The quasi-steady state is obtained by setting µ = 0,

f (η,e +R,w) + Γ [u] (t)− r(ρ) = −Ke

91

(4.15)

(4.16)

|r(ρ+1)(t)|.
From Assumption 4.5, r(ρ) and r(ρ+1) are bounded. Let d1 = max
t
Deﬁne the sets R0 = {R : ||R|| ≤ κ3}, R1 = {r(ρ) : |r(ρ)| ≤ d1}, and R2 = {r(ρ+1) : |r(ρ+1)| ≤ d2}.
From Proposition 4.2, if (η,e,R,w,r(ρ)) ∈ Ω0 × Ωc × R0 ×W × R1 for each τ ∈ [0,t], Eq. (16)
has a unique solution given by

|r(ρ)(t)| and d2 = max
t

u∗(t) = φ [χ](t)

= Γ−1(cid:2)χ(τ)|t

(cid:52)

τ=0

(cid:3) (t)

where

χ(τ) = − f (η(τ),e(τ) +R(τ),w(τ)) + r(ρ)(τ)− Ke(τ),

and Γ−1 is the inverse of the hysteresis operator Γ. Furthermore, from Assumptions 4.1, 4.2, and

4.5, χ is continuously differentiable with respect to t.

The change of variables

z(t) = u(t)− φ [χ] (t)

transforms system (4.13.b) and (4.15) into

˙e = (A− BK)e + B{Γ [z + φ ]− Γ [φ ]} ,

µ ˙z = − (Γ [z + φ ]− Γ [φ ])− µ ˙φ ,

(cid:16)
Γ−1(cid:2)χ(τ)|t

τ=0

(cid:3)(cid:17)

˙χ(t).

˙φ =

∂

∂ χ(t)

(4.17a)

(4.17b)

where

Note that ˙χ(t) is bounded for all
(η,e,R,w,r(ρ),r(ρ+1)) ∈ Ω0 × Ωc × R0 × W × R1 × R2. From Proposition Proposition 4.2, it
is bounded for all (η,e,R,w,r(ρ)) ∈ Ω0 × Ωc × R0 ×W ×
follows that

(cid:16)
Γ−1(cid:2)χ(τ)|t

(cid:3)(cid:17)

τ=0

∂

∂ χ(t)

92

R1, from which we can conclude that ˙φ is bounded for all (η,e,R,w,r(ρ),r(ρ+1)) ∈ Ω0 × Ωc ×
R0 ×W × R1 × R2.

The right-hand side of (4.17.b) has a ﬁnite number of jumps on any compact time interval and

this follows from Proposition 4.2. The solution of (4.17.b) is therefore studied in the sense of

Caratheodory solution [89], which is absolutely continuous curves which satisﬁes

(cid:90) t

0

z(t) = z(0) +

1
µ

E(τ)dτ

where the integral is the Lebesgue integral and E(τ) = − (Γ [z(τ) + φ (τ)]− Γ [φ (τ)])− µ ˙φ (τ).

Let |u| ≤ q0, where q0 is a positive constant, and we deﬁne the compact set Ωz = {z : |z| ≤ q},

where

q ≥ q0 +

max

(η,e,R,w,r(ρ))∈Ω0×Ωc×R0×W×R1

|φ|.

Theorem 4.1. Consider the closed-loop system formed of the plant (4.13) and the dynamic inver-

sion algorithm (4.15). Suppose that Assumptions 4.1-4.5 are satisﬁed and (η(0),e(0)) belongs to
a known compact subset in the interior of Ω0 × Ωc, |z(0)| < q. Then there exists µ∗ > 0 such that
for all µ ∈ (0, µ∗), the trajectories of the closed-loop system are bounded for all t ≥ 0, and

limsup
t→∞

e(t) = O(µ),

limsup
t→∞

z(t) = O(µ).

(4.18)

Proof: Consider the Lyapunov functions Ve = eT Pe and Vz = 1

2z2. Taking the time derivative

of Vz along (4.17.b) yields

µ ˙Vz = −z (Γ [z + φ ] (t)− Γ [φ ] (t))− µz ˙φ

93

The term z (Γ [z + φ ] (t)− Γ [φ ] (t)) can be written as

z (Γ[z + φ ](t)− Γ[φ ](t)) = z [g(z + φ )− g(φ )] + z(cid:2) ˜Γ[z + φ ](t)− ˜Γ[φ ](t)(cid:3)

The term z [g(z + φ )− g(φ )] can be written as

z [g(z + φ )− g(φ )] = (z + φ − φ )(g(z + φ )− g(φ )),

and from Eq. (4.4), using the mean value theorem [81], we have

β1z2 ≤ (z + φ − φ ) [g(z + φ )− g(φ )] ≤ β2z2.

From Assumption 4.4, the hysteresis operator ˜Γ [(·)] is monotone and

z(cid:2) ˜Γ[z + φ ](t)− ˜Γ[φ ](t)(cid:3) = (z + φ − φ )(cid:2) ˜Γ[z + φ ](t)− ˜Γ[φ ](t)(cid:3)

from which we have

z (Γ [z + φ ] (t)− Γ [φ ] (t)) ≥ β1z2

Over the compact set Ω0×Ωc× R0×W × R1× R2, since ˙φ is bounded, we have | ˙φ| ≤ b, for some
positive constant b independent of µ. Therefore, we have

µ ˙Vz ≤ −β1z2 + µ|z|| ˙φ|

=⇒ ˙Vz ≤ − β1
2µ

z2,

for |z| ≥ 2µb
β1

Take b1 = 2b/β1. There exists µ1 > 0, such that for all µ ∈ (0, µ1), the set {z : |z| ≤ b1µ} is

94

positively invariant, and for sufﬁciently small µ, the set {z : |z| ≤ b1µ} is in the interior of Ωz.

Taking the time derivative of Ve along (4.17.a) yields

˙Ve = −eT Qe + 2eT PB (Γ [z + φ ] (t)− Γ [φ ] (t))

≤ −λmin(Q)||e||2 + 2˜k||e||·||PB||·|z|

where ˜k = k + β2 from (4.3), (4.4) and (4.5), λmax(Q) and λmin(Q) denote the maximum and
minimum eigenvalues of the matrix Q, respectively. And for z ∈ {z : |z| ≤ b1µ}, we have

˙Ve ≤ −λmin(Q)||e||2 + 2˜kb1µ||PB||·||e||

≤ −λmin(Q)

2

||e||2,

for ||e|| ≥ 4˜kb1µ||PB||
λmin(Q)

Take b2 =

16˜k2b2

1||PB||2λmax(P)
(λmin(Q))2

, from which we have

˙Ve ≤ −λmin(Q)

2

||e||2,

for Ve ≥ b2µ2

Therefore, we can choose µ2 > 0 such that for all µ ∈ (0, µ2), the set {e : Ve ≤ b2µ2} is positively
invariant and for sufﬁciently small µ, the set {e : Ve ≤ b2µ2} is in the interior of {e : Ve ≤ c},
from which we can conclude that the set Ω0 × Ωc × Ωz is positively invariant. Finally by taking
µ∗ = min{µ1, µ2}, (4.18) follows.

(cid:3)

Remark 4.7. Under Assumptions 4.1-4.3, the stability result will be regional. However, if all the
assumptions hold globally, i.e. Dη = Rn−ρ, Dξ = Rρ, W = Rl and ˙η = f0(η,ξ ,w) is input-to-
state stable, then the constants c0 and c, can be chosen arbitrarily large, and any compact set of

95

Rn−ρ ×Rρ can be put in the interior of {η : V0(η) ≤ c0}×{ξ : Ve(e) ≤ c}. Then, in this case the
stability result will be semi-global.

4.4 Output Feedback Controller

The implementation of the dynamic inversion algorithm in the previous section requires the knowl-
edge of the function F(η,e,R,w,r(ρ)) and the hysteresis output Γ[u](t), which is typically not

available for use in practice. Therefore, we propose an output feedback controller by combining

an extended high-gain observer with the dynamic inversion algorithm to compensate for the hys-

teresis present in the system. High-gain observers are used to robustly estimate the states of a

system while extended high-gain observers are used for estimating an unknown input to the sys-

tem without requiring the input to be slowly varying [72]. Fig. 4.2 illustrates the proposed control

approach.

The extended high-gain observer provides an estimate of the combined value at time t for the

function F and the hysteresis output Γ, which is then used in the dynamic inversion algorithm. The

output feedback controller has two parameters µ and ε, which are small and determine the time

scales of the dynamic inversion algorithm and the observer, respectively. Typically the observer

dynamics should be faster than the dynamic inversion, so that the estimates can converge to the

true values rapidly, and the dynamic inversion should be faster than the plant dynamics so that the

control input converges to the desired control quickly.

We set σ (t) = F(η(t),e(t),R,w(t),r(ρ)(t)) + Γ[u](t). If σ and e were available for feedback,

the dynamic inversion algorithm (4.15) would be given by

µ ˙u = −[σ + Ke]

96

Figure 4.2: Block diagram illustrating the proposed approach. Here EHGO stands for extended
high-gain observer.

Since σ and e are not available, we use an extended high-gain observer to get the estimates ( ˆe, ˆσ )

of (e,σ ). The extended high-gain observer is constructed as

αi
εi (e1 − ˆe1), 1 ≤ i ≤ ρ − 1
˙ˆei = ˆei+1 +
αρ
ερ (e1 − ˆe1)
˙ˆeρ = ˆσ +
αρ+1
ερ+1 (e1 − ˆe1)

˙ˆσ =

where α1 to αρ+1 are chosen such that the polynomial

sρ+1 + α1sρ + . . . + αρ+1

97

(4.19a)

(4.19b)

(4.19c)

(4.20)

is Hurwitz, and ε > 0 is a small parameter. For sufﬁciently small ε, the estimation error decays

to zero much faster than the system dynamics. High-gain observer suffers from the peaking phe-

nomenon [82], where the estimation error peaks to the order of O(1/ερ ) during the transient period

and then it decays quickly to the order of O(ε). In feedback control the peaking is overcome by

designing the control to be a globally bounded function of the estimates ( ˆe, ˆσ ). Let

|Ke|,

M > max
e∈Ωc

N >

|F(η,e,R,w,r(ρ)) + Γ[z + φ ]|.

max

(η,e,w)∈Ω0×Ωc×W,
R∈R0, r(ρ)∈R1,

z∈Ωz

The estimates are saturated as

(K ˆe)s = Msat

(cid:18)K ˆe

(cid:19)

M

,

ˆσs = Nsat

(cid:18) ˆσ

(cid:19)

N

where sat(y) = min{|y|,1} sign(y) for y ∈ R. The dynamic inversion algorithm with the saturated

estimates is given by

µ ˙u = −[ ˆσs + (K ˆe)s]

(4.21)

Theorem 4.2. Consider the closed-loop system formed of the plant (4.13), the extended high-gain

observer (4.19), and the dynamic inversion algorithm (4.21). Suppose that the Assumptions 4.1-

4.5 are satisﬁed, α1 to αρ+1 are chosen such that the polynomial (4.20) is Hurwitz, (η(0),e(0))
belongs to a known compact subset in the interior of Ω0 × Ωc, |z(0)| < q, and the initial states of
the observer belong to a compact subset of Rρ+1. Then there exists λ∗
1 > 0, such that for ε and µ
with max {µ,ε/µ} ≤ λ∗
1 , the trajectories of the closed-loop system are bounded for all t ≥ 0, and

98

there exists T∗(λ∗

1 ) > 0 such that

˙e = (A− BK)e + O(λ∗

1 ), ∀ t ≥ T∗(λ∗
1 ).

(4.22)

Moreover given any Ξ > 0, there exists λ∗

2 > 0, such that for ε and µ with max {µ,ε/µ} ≤ λ∗

2

||e(t)− er(t)|| ≤ Ξ, ∀ t ≥ 0.

(4.23)

where er(t) is the solution of the target system (4.14), with er(0) = e(0).

Proof: Deﬁne the change of variables

ζi = (ei − ˆei)/ερ+1−i

for 1 ≤ i ≤ ρ,

ζρ+1 = σ − ˆσ = f (η,e +R,w) + Γ[u](t)− r(ρ) − ˆσ

The fast variable dynamics is given by

ε ˙ζi = −αiζ1 + ζi+1 for 1 ≤ i ≤ ρ − 1,
ε ˙ζρ = −αρζ1 + ζρ+1,

ε ˙ζρ+1 = −αρ+1ζ1 + ε∆0 + (ε/µ)∆1,

(4.25a)

(4.25b)

(4.25c)

where

∆0 =

∂σ
∂η

f0 +

∂σ
∂ e [(A− BK)e + B (Γ[z + φ ](t)− Γ[φ ](t))] +

∂σ
∂ w

˙w +

∂σ
∂R ˙R− r(ρ+1)

99

∆1 = − ∂σ
∂ u(t)

[ ˆσs − σ + (K ˆe)s − Ke + Γ[z + φ ](t)− Γ[φ ](t)]

where the functions f0, F are written without their arguments. Their complete forms are f0 =
f0(η,e +R,w) and F = F(η,e,R,w,r(ρ)). The term

can be written as

∂σ
∂ u(t)

∂σ
∂ u(t)

=

∂ Γ[u(τ)|t
∂ u(t)

τ=0](t)

=

∂ g(u(t))
∂ u(t)

+

∂ ˜Γ[u(τ)|t
∂ u(t)

τ=0](t)

Using Proposition 4.1, for all (η,e,w,R,r(ρ),r(ρ+1),z) ∈ Ω0 × Ωc ×W × R0 × R1 × R2 × Ωz, the
term

is bounded. The closed-loop system is represented by

∂σ
∂ u(t)

˙η = f0(η,e +R,w),
˙e = (A− BK)e + B[Γ[z + φ ](t)− Γ[φ ](t)],

µ ˙z = − (Γ[z + φ ](t)− Γ[φ ](t))− µ ˙φ + ∆2,

ε ˙ζ = Λζ + ε ¯B∆0 + (ε/µ) ¯B∆1,

(4.26a)

(4.26b)

(4.26c)

(4.26d)

where

∆2 = − [ ˆσs − σ + (K ˆe)s − Ke] , ζ = col(ζ1, . . . ,ζρ+1),

Λ =

, ¯B =

.



0

1

0
...

1
...

−α1
−α2
...
−αρ
0 . . .
−αρ+1 0 . . .



. . . 0

. . . 0
...
...

0

1

. . . 0

100





0

0
...

0

1

(ρ+1)×1

The solution of (4.26) is studied in the sense of Caratheodory solution. The matrix Λ is Hurwitz by
construction. Since the initial states (η(0),e(0),z(0)) lie in the interior of Ω0 × Ωc × Ωz and the
right-hand-side functions of Eq. (4.26.a)-(4.26.c) are bounded uniformly in ε, there is time ¯T > 0
such that (η(t),e(t),z(t)) ∈ Ω0 × Ωc × Ωz for all t ∈ [0, ¯T ]. During this time period the terms ∆0,
∆1, ∆2 are bounded by

|∆0| ≤ k1,

|∆1| ≤ k2, |∆2| ≤ k3

where k1, k2, k3 are positive constants independent of µ and ε. This follows from the continuous

differentiability of f , local Lipschitz property of f0, boundedness of

∂σ
∂ u(t)

, and global bounded-

ness of ˆσs and (K ˆe)s.

(cid:26) ε

(cid:27)

µ , µ

We deﬁne λ = max

and show that there exist positive constants ˜λ3, γ1, and γ2, such
that the set Ω0 × Ωc × {z : |z| ≤ γ1λ} × {ζ : ζ T P0ζ ≤ γ2(ε/µ)2} is positively invariant for all
λ ≤ ˜λ3, where P0 = PT
0 > 0 is the solution to the Lyapunov equation P0Λ + ΛT P0 = −I. We
construct the Lyapunov function Vζ = ζ T P0ζ and take its time derivative along Eq. (26d), which
yields

ε ˙Vζ = −ζ T ζ + 2εζ T P0 ¯B∆0 + 2(ε/µ)ζ T P0 ¯B∆1
ε ˙Vζ ≤ −||ζ||2 + 2k1ε||P0 ¯B||·||ζ|| + 2(ε/µ)k2||P0 ¯B||·||ζ||

(cid:18)

(cid:19)

≤ −||ζ||2 + 2ε||P0 ¯B||

from which we have

k1 +

k2
µ

||ζ||

ε ˙Vζ ≤ −1
2

||ζ||2, for ||ζ|| ≥ 4ε
µ

||P0 ¯B||(k1 + k2)

101

By taking γ2 = 16||P0 ¯B||2λmax(P0)(k1 + k2)2, where λmax(P0) is the maximum eigenvalue of P0,
one gets

ε ˙Vζ ≤ −1
2

||ζ||2, for Vζ ≥ γ2(ε/µ)2

Therefore, on the boundary {ζ : Vζ = γ2(ε/µ)2}, we have ˙Vζ < 0. Following the high gain ob-
server theory [73], one can show that initially ζ (0) could be outside the set {ζ : Vζ ≤ γ2(ε/µ)2},
but it converges towards the set within time T1(ε), where lim
T1(ε) = 0. For sufﬁciently small ε,
ε→0
T1(ε) < (1/2) ¯T . Therefore, ζ (t) enters the set {ζ : ζ T P0ζ ≤ γ2(ε/µ)2}, while (η(t),e(t),z(t))
remain in the interior of Ω0×Ωc×Ωz. When ζ (t) enters the set {ζ : ζ T P0ζ ≤ γ2(ε/µ)2} we have

K ˆe = Ke + O(λ ),

ˆσ = σ + O(λ )

Therefore, for sufﬁciently small λ , we have K ˆe < M and ˆσ < N which implies that the saturation

in the estimates is no longer active, hence

K ˆes = K ˆe, ˆσs = ˆσ

Next we analyze the system (4.26.c) for t ≥ T1(ε). Note that z(t) ∈ Ωz during the time period
z2 and take its time derivative along Eq.
[0,T1(ε)]. We construct the Lyapunov function Vz =

1
2

(4.26.c), from which we have

µ ˙Vz = −z{Γ[z + φ ](t)− Γ[φ ](t)} + z∆2 − µz ˙φ

The term ∆2 after time t ≥ T1(ε) is bounded by ||∆2|| ≤ k4||ζ||, where k4 is a positive constant

102

independent of λ , from which we have

µ ˙Vz ≤ −β1z2 +

(cid:20) ε

µ

k4

(cid:114) γ2

λmin(P0)

(cid:21)

+ µ| ˙φ|

|z|

≤ −β1z2 + λ k5|z|

where λmin(P0) is the minimum eigenvalue of P0 and k5 is a positive constant independent of λ .

Therefore, we have

Taking γ1 = 2k5/β1 ensures that

µ ˙Vz ≤ −β1
2

|z|2,

for |z| ≥ 2k5λ
β1

µ ˙Vz ≤ −β1
2

|z|2,

for |z| ≥ γ1λ

Therefore, for all λ ≤ ˜λ1, where ˜λ1 is a small positive constant chosen such that the set {z :
|z| ≤ γ1λ} is in the interior of Ωz. Similarly, following [73], one can show that initially z(0)
could be outside the set {z : |z| ≤ γ1λ}, but it converges towards the set within time T2(µ), where
T2(µ) = 0. For sufﬁciently small µ, the system trajectory enters the set Ω0 × Ωc ×{z : |z| ≤
lim
µ→0
γ1λ}×{ζ : ζ T P0ζ ≤ γ2(ε/µ)2} for t ≥ T1(ε) + T2(µ).

Next we analyze (4.26.b) for t ≥ T1(ε) + T2(µ). Note that (η(t),e(t)) ∈ Ω0 × Ωc during the
time period [0,T1(ε) + T2(µ)]. Taking the time derivative of Ve along Eq. (4.26.b), which satisﬁes

˙Ve =− eT Qe + 2eT PB{Γ[z + φ ](t)− Γ[φ ](t)}

≤− λmin(Q)||e||2 + 2˜k||e||·|z|·||PB||
≤− λmin(Q)||e||2 + 2λ k6||e||, ∀ |z| ≤ γ1λ

103

where ˜k = k + β2 using (3), (4), and (5), k6 = γ1k||PB|| and λmin(Q) is the minimum eigenvalue
of Q, which implies

˙Ve ≤ −λmin(Q)

2

||e||2 for ||e|| ≥ 4λ k6
λmin(Q)

Therefore, for all λ ≤ ˜λ2, where ˜λ2 is a small positive constant, ˙Ve < 0 on the boundary {e :
Ve(e) = c}. Taking λ ≤ ˜λ3 = min{˜λ1, ˜λ2}, ensures that the set Ω0 × Ωc ×{z : |z| ≤ γ1λ}×{ζ :
ζ T P0ζ ≤ γ2(ε/µ)2} is positively invariant and enters it within time T1(ε) + T2(µ). Therefore, for
λ ≤ λ∗

1 = min{˜λ1, ˜λ2, ˜λ3}, there is time T∗(λ∗
Next we show (4.23). Since the right-hand-side functions of Eq. (4.14) and Eq. (4.26.b) are

1 ), such that for all t ≥ T∗(λ∗

1 ), (4.22) follows.

bounded, uniformly in ε and µ, and e(0) = er(0), it follows that

e(t)− er(t) = O(T1(ε) + T2(µ)), for 0 ≤ t ≤ T1(ε) + T2(µ)

Hence, e(T1(ε) +T2(µ))−er(T1(ε) +T2(µ)) = O(T1(ε) +T2(µ)), and for t ≥ T1(ε) +T2(µ), e(t)
satisﬁes

˙e = (A− BK)e + O(λ ).

Therefore, by continuous dependence of the solutions of differential equations and parameters [81,
Theorem 9.1] and the exponential stability of the origin of ˙e = (A− BK)e, we conclude that

e(t)− er(t) = O(λ ) + O(T1(ε) + T2(µ)), t ≥ 0

Hence, given Ξ > 0, there exists λ∗

2 > 0, such that for all λ ∈ (0,λ∗

2 ), (4.23) follows.

104

4.5 Simulation Results

In this section, a simulation study is conducted to examine the performance of the proposed con-

troller and to highlight the impact of the dynamic inversion parameter µ and the high-gain ob-

server parameter ε on the smallness of the tracking error. For simulation purposes, we consider the

model for a piezo-actuated nanopositioner, which was identiﬁed experimentally and used for the

controller design in [54]. Its worth mentioning that the same setup is used later for experimental

veriﬁcation of the proposed controller presented in next section. The model is composed of two

cascaded blocks. The ﬁrst block represents the input hysteresis nonlinearity, modeled by a Modi-

ﬁed Prandtl-Ishlinskii (MPI) operator consisting of 8 play operators and 9 deadzone operators. The

other block represents the vibrational dynamics, represented as a linear state-space system. The

augmented system can be represented as

˙ξ = Aξ ξ + Bξ

(cid:34)

Γξ [u;νo] (t) + dξ (t)

(cid:21)

(4.27)

where ξ ∈ R2 represents the displacement and velocity of the nanopositioning stage, Γξ is the MPI
operator with νo being its initial state, dξ ∈ R is the input disturbance, and the matrices



Aξ =

0

1

1.795× 108 5696.88

 and Bξ =





0

1.063× 109

It should noted that the model (4.27) is a special case of the general model (4.1) in Section 4.2.

Throughout the simulation studies, the observer parameters are taken as α1 = 3, α2 = 3, and α3 =

1, while the controller gain is taken as K = [2 1]. Moreover, it was assumed that the nanopositioner

is initially at a stationary state with position zero at the starting point. A sinusoidal reference is

105

Table 4.1: Simulation results: Mean and Max |e(t)|% in percent of the reference amplitude in the state
feedback case.

µ
0.1
0.01
0.001
0.0001

Mean |e(t)|% Max |e(t)|%

0.0584
0.0526
0.0119
0.0012

0.0956
0.0895
0.0219
0.0022

used with frequency of 100 Hz and amplitude of 10 µm. The input disturbance signal dξ consists

of the 3rd, 5th, and 7th harmonics of the reference signal frequency with amplitudes equal to 1,

0.2, and 0.1, respectively.

Table 4.1 shows the simulation results for the state feedback case, where all system state vari-

ables and the hysteretic output are measured. Both the mean and maximum values of the tracking

error, normalized by the amplitude of the reference signal (and thus expressed as %), are shown in

Table 4.1, for a range of dynamic inversion parameter µ. As observed clearly, both the mean and

maximum errors are reduced signiﬁcantly when the value of the parameter µ is decreased.

Table 4.2 shows the simulation results for the output feedback case, where the value of the

parameter µ is ﬁxed at 0.1, while the extended high-gain observer parameter ε is chosen at a few

different values as shown in the table. Note that both the mean and maximum tracking errors are

reduced dramatically when the value of ε is reduced from 0.01 to 0.001. However, reducing the

value of ε to lower values only results in slight decrease in the tracking errors, which inch towards

the tracking error performance in the state-feedback case for µ = 0.1.

The above simulation results support the major theoretical ﬁndings as summarized in both

Theorem 4.1 and 4.2; in particular, the tracking error will be reduced with a decreasing µ under

state feedback, and with a decreasing ratio of ε

µ under output feedback case, respectively.

106

Table 4.2: Simulation results: Mean and Max |e(t)|% in percent of the reference amplitude in the output
feedback case with µ = 0.1.

ε
0.01
0.001
0.0001
0.00001

Mean |e(t)|% Max |e(t)|%

0.2202
0.0609
0.0607
0.0602

0.4512
0.0958
0.0954
0.0953

4.6 Experimental Results

To further verify the theoretical results, a series of tracking experiments are conducted on a com-

mercial nanopositioner (Nano OP-65, from Mad City Labs). The piezo-actuated positioner has

a built-in capacitive sensor that measures its displacement. The data acquisition and control are

realized through dSPACE (DS1104) as shown in Fig. 2.5. Three types of reference signals are

used in the experiments, including sinusoidal signals, sawtooth signals, and multi-harmonic sig-

nals. The observer and the controller parameters used in the experiments are α1 = 3, α2 = 3,
α3 = 1, µ = 0.01, ε = 6.6667× 10−6, and K = [2,1.3398× 104]. To mitigate the effect of noise
ampliﬁcation due to the use of high-gains in the observer, a low-pass ﬁlter with 1 KHz bandwidth

is inserted at the measurement side.

4.6.1 Sinusoidal Reference Signals

In the experiments we have used four different sinusoidal signals with frequencies of 5, 25, 50,

100 Hz, respectively. The experimental results for the highest reference frequency, i.e., 100 Hz,

are shown in Fig. 4.4. As can be seen in Fig. 4.4(a), the measured position output converges to the

reference signal at roughly 0.009 seconds, which is shown more clearly in Fig. 4.4(b). Moreover,

it can be seen in the magniﬁed part of Fig. 4.4(b) that the tracking error at the steady state is within
±0.055 µm. In Fig. 4.4(c), the frequency content of the tracking error signal e1 (t) is shown. In

107

Table 4.3: Experimental results: Comparison of the Mean |e(t)|% in tracking a sinusoidal reference.

Frequency

5 Hz
25 Hz
50 Hz
100 Hz

EHGO-DI

0.1343
0.1330
0.1371
0.2051

SMC
0.1190
0.6200
0.6600
0.8300

SHSC MHSC
0.2710
0.6490
0.7070
0.2680
0.2840
0.7700
0.8150
0.3520

PI

0.5141
0.5540
0.6286
0.8242

Table 4.4: Experimental results: Comparison of the Max |e(t)|% in tracking a sinusoidal reference.

Frequency

5 Hz
25 Hz
50 Hz
100 Hz

EHGO-DI

0.2306
0.2765
0.3642
0.6666

SMC
0.9500
1.7000
2.2500
2.7500

SHSC MHSC
1.7200
0.8990
0.8810
1.8500
1.0100
1.9300
2.3800
1.5700

PI

2.3477
2.4854
3.0326
3.8560

addition to the fundamental frequency at 100 Hz, the third is also relatively pronounced. Note that

most of the tracking error magnitude is due to ﬁrst harmonic, which its absolute amplitude is less

than 0.03 µm.

Tables 4.3 and 4.4 further provide a comparison of the tracking error performance between

the proposed controller (EHGO-DI) and several competing controllers, including a sliding mode

controller (SMC) with hysteresis inversion in [48], a single-harmonic servocompensator (SHSC)

[5], a multiple-harmonic servocompensator (MHSC) [5], and a proportional integral controller

implemented solely without using feedforward inversion, for the mean error and the maximum

error, respectively. Note that hysteresis inversion is used in both SHSC and MHSC [5]. From both

tables, it can be seen that, overall the proposed controller outperforms all the other approaches,

Table 4.5: Experimental results: Mean and Max |e(t)|% in percent of the reference amplitude for sawtooth
and Multi-Harmonic references.

Frequency

5 Hz
25 Hz
50 Hz
100 Hz

Mean |e(t)|% Max |e(t)|% Mean |e(t)|% Max |e(t)|%

Multi-Harmonic

sawtooth

0.1354
0.1358
0.1401
0.1893

0.2355
0.3422
0.5388
0.8905

108

0.1389
0.1397
0.1496
0.2486

0.2440
0.3063
0.4671
0.9744

(a)

(b)

(c)

Figure 4.3: Experimental results on tracking a 100 Hz sinusoidal reference signal. (a) Positioner
displacement; (b) tracking error; (c) frequency spectrum of the tracking error.

109

00.0050.010.0150.020.0250.03Time (s)-50510152025Displacement (m)ReferenceOutput00.0050.010.0150.020.0250.03Time (s)-25-20-15-10-50510Tracking Error (m)0.0120.0140.0160.0180.020.0220.0240.0260.0280.03-0.0500.0502004006008001000Frequecny (Hz)00.0050.010.0150.020.0250.03Absolute Magnitude of e1 (m)despite that it does not use hysteresis modeling and inversion as the other methods do. In Table

4.3, it can noticed that our proportional integral controller is comparable to the SHSC controller

and it does slightly better than the SMC approach. However, the PI controller’s maximum absolute

error comes higher than all other approaches.

4.6.2 Sawtooth Reference Signals

The next set of experiments are implemented with sawtooth reference signals with frequencies of

5, 25, 50, and 100 Hz, respectively. To avoid impulsive behavior, a second-order linear ﬁlter is

utilized to smooth out the signal edges. Fig. 4.5(a) and 4.5(b) show the time trajectories of the

positioner displacement and the tracking error for the 100 Hz case, where it can be seen that the

displacement converges to the reference signal at roughly 0.003 seconds. Note that the tracking
error is conﬁned to around ±0.05 µm at the steady state. It can be observed in Fig. 4.5(c), that the

error signal contains more harmonic terms than the case of tracking a sinusoidal reference. The

ﬁrst part of Table 4.5 shows the mean and maximum tracking errors over the considered range

of frequencies. Notice that the growth of tracking error with the increase of frequency is largely

consistent with the results in the sinusoidal case.

4.6.3 Multi-Harmonic Reference Signals

In the ﬁnal set of experiments, the tracking of a multi-harmonic signal with ﬁrst and second har-

monics is implemented. Similar to the previous experiments, we consider a range of primary

frequencies, 5,25,50,100 Hz. As seen from Fig. 4.6(a) and 4.6(b), the positioner output converges

to the reference signal in roughly 0.006 seconds. The tracking error at the steady state is larger than

the two previous cases (sinusoidal and sawtooth references). Fig. 4.6(c) also shows the amplitudes

110

(a)

(b)

(c)

Figure 4.4: Experimental results on tracking a 100 Hz sawtooth reference signal. (a) Positioner
displacement; (b) tracking error; (c) frequency spectrum of the tracking error.

111

00.0050.010.0150.020.0250.03Time (s)-50510152025Displacement (m)ReferenceOutput00.0050.010.0150.020.0250.03Time (s)-5-4-3-2-101Tracking Error (/mum)0.0050.010.0150.020.0250.03-0.0500.0502004006008001000Frequency (Hz)00.0050.010.0150.02Absolute Magnitude of e1 (m)of the 1st and 2nd harmonic components of the error. It can be noticed that the absolute amplitude

of the 2nd harmonic is larger than the 1st one. The second half of Table 4.5 shows quantitative

results on the tracking error for all frequencies. Overall the results are satisfactory; the tracking

errors are slightly larger comparing to the smoothed sawtooth case. This can be attributed to the

pronounced 2nd harmonic component in the reference signal.

112

(a)

(b)

(c)

Figure 4.5: Experimental results on tracking a reference signal generated via the multi-harmonic
signal with a primary frequency of 100 Hz. (a) Positioner displacement; (b) tracking error; (c)
frequency spectrum of the tracking error.

113

00.0050.010.0150.020.0250.03Time (s)-50510152025Displacement (m)ReferenceOutput00.0050.010.0150.020.0250.03Time (s)-20-15-10-50510Tracking Error (m)0.010.0150.020.0250.03-0.1-0.0500.0502004006008001000Frequency (Hz)00.0050.010.0150.020.0250.030.035Absolute Magnitude of e1 (m)Chapter 5

Conclusions and Future Work

5.1 Conclusions

5.1.1 Inversion-based ACS Approach

We ﬁrst focused on designing an “inversion-based” adaptive conditional servocompensator closed-

loop system for a class of hysteretic systems. Under this control law, the closed-loop system will

have behavior over two different stages. The ﬁrst one is a reaching phase, where the controller is

a continuously-implemented sliding mode control law. In this stage, we have designed the control

law to accommodate hysteric inversion error perturbations by deriving an analytical bound on the

these perturbations. The switching part of the control law is designed using the analytical bound

to reduce the conservativeness as compared to the case when a constant bound is used. The second

stage starts when the sliding variable enters the boundary layer and stays therein forever. In this

case, the adaptation law is activated “conditionally” to handle the residual hysteretic perturbations.

Our stability analysis embodied in Theorem 2.1 and 2.2 establishes well-posedness and periodic

stability for the closed-loop system. The theoretical framework used to prove the periodic stability

is originally presented in [68]; however, new results were developed in this work since our closed-

loop system does not ﬁt the assumptions of [68] completely due to non-smooth terms in both

the control and the adaptation laws. Next, we validated the inversion-based experimentally. The

results conﬁrmed its superiority as compared with other control algorithms implemented on the

114

same device.

5.1.2 Inversion-free ACS Approach

We next considered designing an “inversion-free” adaptive conditional servocompensator to mit-

igate the effect of hysteresis in systems preceded by an MPI operator. Under the assumption that

only the weights of the operator are uncertain, we proved that the MPI operator can be repre-

sented in a semi-afﬁne form. This transformation converts the MPI operator into a sum of a linear

part, a hysteretic term modeled with a classical PI operator with nominal weights, and a hys-

teretic perturbation term. Then, we established that the hysteretic perturbation term obeys a linear

growth bound, which is utilized in designing less conservative continuously-implemented sliding-

mode control law as compared to the case when a general constant bound is used. Similar to the

inversion-based control algorithm, the ﬁrst part of the controller is a continuously-implemented

sliding mode controller, where its switching part is designed with a variable-gain function utilizing

a derived analytical bound on the hysteretic perturbation to reduce conservativeness.

Next, we introduced a low-pass ﬁlter technique to address a challenge in solving for the equiva-

lent control term. The validity of the approach was justiﬁed by analysis. It should be noted that, the

same issue appeared in the work of [40], where the authors approximated an integral term of the

PI operator with a summation over a time interval divided by a number of time steps, but without

incorporating the approximation error in the analysis. It is worth mentioning that both converting

the system to the semi-afﬁne form and designing the low-pass ﬁlter are crucial to producing a less

conservative sliding mode control law, as they allow one to isolate the nominal hysteretic part from

the hysteric perturbation terms. In particular, this makes it plausible to assume ¯Ko < 1. The second

part of the control law is the adaptive conditional servomechanism, which is designed based on

the assumption that the disturbance is generated by neutrally stable exogenous system. Due to this

115

approximation, an internal model perturbation term is generated to accommodate the residual er-

ror. Next we focused on conducting the stability analysis of the inversion-free closed-loop system

without the requirement of the smallness of the hysteric perturbation, as required in the theoreti-

cal framework introduced in [68]. Then we proved the ultimate boundedness of the closed-loop

systems’ variables and the tracking error was shown to be bounded by a small bound.

We next demonstrated the efﬁcacy of the inversion-free control algorithm through experimental

study by implementing the proposed control algorithm on a commercial nanopositioner and com-

pared its performance with other competing schemes. The proposed inversion-free method showed

superior performance in tracking different types of periodic desired references.

5.1.3 Dynamic Inversion Based on Extended High-gain Observer Approach

Finally, we presented a novel hysteresis compensation algorithm using dynamic inversion and

extended high-gain observer. While many of the state-of-the-art methods for hysteresis compensa-

tion require an explicit model of the hysteresis and the construction of (often times) computation-

intensive inverse hysteresis model, the proposed approach does not require either. In addition,

the controller is robust with respect to the system uncertainties and does not require the exact

knowledge of the plant parameters. This is because such uncertainties can be combined with the

hysteresis output in the signal σ, which is estimated by the extended high-gain observer. The mild

assumption on the hysteresis nonlinearity is easily satisﬁed by popular hysteresis operators like

the classical PI operator, the generalized PI operator and the Preisach operator. We extended the

dynamic inversion theory to the case of hysteretic systems with rigorous analysis. Then we further

validated the proposed approach with both simulation and extensive experimental evaluation.

116

5.2 Future Work Recommendations

There are still remaining ideas to be explored along the lines of the approaches introduced in this

dissertation. One potential direction is to explore the inversion-based and the inversion-free ap-

proaches with different hysteresis models like the Preisach model, or the Krasnosel’skii-Pokrovskii

model. One limitation of the proposed inversion-free control algorithm in Chapter 3 is the restric-

tive assumption that reduces the ability of the MPI operator to represent asymmetric hysteresis

behavior. A potential future work is to try to relax that assumption, which will widen the scope of

the proposed controller to more general applications that suffer from asymmetric hysteresis.

In both Chapters 2 and 3, we designed the output feedback inversion-based and inversion-free

control algorithms using standard high-gain observers [81] to reconstruct the unmeasured states.

However, one challenge in numerical implementation of the observer is that the observer gain is

(cid:17)

(cid:16) 1
internal states are O(cid:16) 1

εn

O

(cid:17)

εn−1

, where n is the system model order. During the peaking (transient) phase, the observer

. One recommended future research is to consider the cascade high-

gain observer scheme, which was introduced in [90] by Khalil in 2017. The cascade high-gain

observer is designed by cascading lower-dimensional observers with saturation functions inserted

in between them. One advantage of following such approach is that the observer gain in the cascade

observer is are limited to be O(cid:16) 1

(cid:17)

and the observer internal states peak will be O(cid:16) 1

(cid:17)

also during

ε

ε

the transient period.

Another suggested approach is motivated by the work of Astolﬁ et al. [91]. In their work, they

also proposed a low-power cascade high-gain observer that preserves the main feature of standard

high-gain observers in terms of arbitrarily fast convergence of the estimation error to zero, while

overcoming their main drawbacks, namely the “peaking phenomenon” during the transient and the

numerical implementation issue deriving from the high-gain parameter that is powered up to the

117

order of the system. The proposed cascade observer has superior features in terms of sensitivity of

the estimation error to high-frequency measurement noise when compared with standard high-gain

observers. The main difference of cascade observer structure of the work of [91] as compared to

Khalil’s cascade observer [90] is that ﬁrst one is built using second-order observers, while the later

is built using ﬁrst-order observers. It is worth mentioning that in Khalil’s cascade observer [90], the

effect of measurement noise on a feedback control system is comparable to its effect on a system

that uses the standard observer.

Another recommendation is to consider robust design of nonlinear internal models without

adaptation proposed by Isidori et al.

in 2012 [92]. Unlike the approaches used in Chapters 2

and 3, which rely on conventional adaptation schemes to estimate the frequency of the exogenous

signals, their proposed approach utilizes regression-like arguments to derive a nonlinear internal

model that is able to offset the presence of an unknown number of harmonic exogenous inputs of

uncertain amplitude, phase and frequency. However, one challenge that should be resolved is that

the work in [92] does not consider the hysteretic disturbances in the system. Therefore, further

efforts should be spent to extend this approach to include hysteretic perturbations.

118

APPENDICES

119

A.1 Proof of Theorem 2.1

Deﬁne the following integration operator:

X (t) := C[X ] (t)
(cid:90) t

= X o +

0

Fcl (s,X ,Winv[uin;Win(0)] (s))ds

(A.1)

where X o = X (0) and the mapping C[X ] (t) : ΩX × ΩH → ΩX × ΩH. Due to the equivalence of
solutions of both (2.48), and (A.1), we will continue the proof using (A.1). Let us deﬁne a closed
set S ⊂ [0,t]×Rrc × W1,1
t

,

S := {t ∈ [0,tc], X ∈ ΩX , and Win ∈ ΩH}

where tc > 0 will be calculated shortly. The ﬁrst step in order to establish that C[X ] (t) is a con-
traction over S is to show that C[X ] (t) maps S into itself. To do this, we rewrite (A.1) by adding
and subtracting the term Fcl

C[X ] (t)−X o =

(cid:0)s,X o,Winv[uo
in;Win(0)] (s)(cid:1) inside the integral,
(cid:20)
(cid:90) t
Fcl (s,X ,Winv[uin;Win(0)] (s))
(cid:0)s,X o,Winv[uo
(cid:0)s,X o,Winv[uo
(cid:0)t,X ,Winv[uo

in;Win(0)] (s)(cid:1)
in;Win(0)] (s)(cid:1)(cid:21)
in;Win(0)] (t)(cid:1) is bounded over the interval

0
+Fcl
−Fcl

(A.2)

ds

in is the control input evaluated at t = 0. From the local Lipschitz continuity properties

where uo
of the function Fcl (·), the function Fcl
t ∈ [0,tu], and for all X ,Win ∈ S, where the ﬁnal value of tc will be picked later no greater than tu.

120

Hence, deﬁne

(cid:0)t,X o,Winv[uo

in;Win(0)] (t)(cid:1)(cid:107)1

b f

(cid:52)
= max
t∈[0,tu]
X ,Win∈S

(cid:107)Fcl

From the Lipschitz properties of the function Fcl (·) and in light of the deﬁnition of the standard
norm (2.13), we can show that the ﬁrst two terms of the integrand in (A.2) satisfy the following

inequality:

(cid:107)Fcl (s,X ,Winv[uin;Win(0)] (s))−Fcl
Lx1(cid:107)X −X o(cid:107)|W1,1

(cid:0)s,X o,Winv[uo
+ Lh1(cid:107)Winv [uin;Win(0)] (s)−Winv[uo

in;Win(0)] (s)(cid:1)(cid:107)1 ≤
in;Win(0)] (s)(cid:107)W1,1

t

t

(A.3)

where Lx1 and Lh1 are the corresponding Lipschitz constants dependent on X o, rX , and rH. By
using the Lipschitz property of the MPI operator Winv [·], proven in Proposition 2.1, we have

(cid:107)Winv [uin;Win(0)] (s)−Winv[uo

in;Win(0)] (s)(cid:107)W1,1

t

≤ Lh2(cid:107)uin − uo

in(cid:107)W1,1

t

Let uin (t) = αin (.), where αin (.) is a piecewise continuous function in t, then inside the set S, we
can derive

(cid:107)uin − uo

in(cid:107)W1,1
≤

t

(cid:107)u(cid:48)
in (s)− uo(cid:48)

(cid:90) t
(cid:13)(cid:13)(cid:13)(cid:13)∂αin (X )

∂X

0

=

(cid:90) t

0

(cid:90) t

0

∂X X (cid:48)(cid:13)(cid:13)(cid:13)(cid:13)1
(cid:13)(cid:13)(cid:13)(cid:13)∂αin (X )

c1(cid:107)X (cid:48) (s)(cid:107)1ds

(A.4)

(A.5)

0

in (s)(cid:107)1ds ≤
(cid:90) t
(cid:107)X (cid:48)(cid:107)1 ≤
≤ c1rX

(cid:13)(cid:13)(cid:13)(cid:13)1
(cid:13)(cid:13)(cid:13)(cid:13)∂αin(X )

(cid:13)(cid:13)(cid:13)(cid:13)1

∂X

c1 = maxX∈S

where

Notice that the norm (cid:107)·(cid:107)1 in (A.5) is an induced norm. Now taking the 1-norm of both sides of
(A.2), and utilizing the local Lipschitz properties of Fcl (·) along with inequalities (A.3)- (A.4),

121

we get

(cid:107)C[X ] (t)−X o(cid:107)1 ≤ (cid:107)C[X ] (t)−X o(cid:107)Wt1,1 ≤

(cid:13)(cid:13)Winv[uin;Win(0)] (s)−Winv[uo

+ Lh1

(cid:90) t

(cid:20)
Lx1(cid:107)X (s)−X o (s)(cid:107)Wt1,1
ds ≤ tcbq

in;Win(0)] (s)(cid:13)(cid:13)Wt1,1 + b f

(cid:21)

0

(A.6)

where bq = Lx1rX +Lh1Lh2c1rX +b f . From the above inequality, we have established the bound-
edness of the function Fcl (s,X ,Winv[uin;Win(0)] (s)). Hence, by choosing tc ≤ rX
bq , we can ensure
that C[X ] (t) : S → S.
The next step is to show that the mapping C[X ] (t), with careful selection of tc, is a contraction
mapping over S. Let X1 and X2 ∈ S, and consider the norm

(cid:107)C[X1] (t)−C[X2] (t)(cid:107)1 =

−Fcl

(cid:17)

(cid:16)
0
s,X2,Winv[u2

(cid:90) t

(cid:13)(cid:13)(cid:13)(cid:13)Fcl
(cid:13)(cid:13)Winv[u1

+ Lh1

(cid:16)
s,X1,Winv[u1
ds ≤

in;Win(0)] (s)

(cid:17)(cid:13)(cid:13)(cid:13)(cid:13)1

(cid:20)
(cid:90) t
Lx1(cid:107)X1 (s)−X2 (s)(cid:107)Wt1,1
in;Win(0)] (s)(cid:13)(cid:13)Wt1,1
(cid:2)Lx1(cid:107)X1 (s)−X2 (s)(cid:107)Wt1,1 + Lh1Lh2(cid:107)u1

in;Win(0)] (s)
0
in;Win(0)] (s)−Winv[u2
≤ tc

in − u2

(cid:21)

ds

in(cid:107)Wt1,1

(cid:3) (A.7)

Similar to the steps that lead to (A.4), we have

(cid:107)u1

in (s)− u2

in (s)(cid:107)W1,1

t

(cid:90) t

0

=

in (s)− u2(cid:48)
(cid:107)u1(cid:48)
(cid:20)
in (s)(cid:107)1ds
(cid:90) t
(cid:107)∂αin (X1 (s))
1(s)− ∂αin (X2 (s))
X (cid:48)
≤
(cid:19)
(cid:18)∂αin(X1(s))

∂X

∂X

0

∂X

X (cid:48)
2(s)

X (cid:48)
2(s)(cid:107)1ds

(A.8)

By adding and subtracting the term

, inside the norm of the above integral,

122

we have

2 (S)(cid:3)
(cid:20)
(cid:90) t
c1(cid:107)X (cid:48)
2 (s)(cid:107)W1,1
(cid:21)
ds ≤ bρ(cid:107)X1 (s)−X2(cid:107)W1,1

1 (s)−X (cid:48)

t

t

(A.9)

(cid:107)u1

(cid:34)
in (s)(cid:107)W1,1
in (s)− u2
≤
∂αin (X1 (s))

t

−

(cid:90) t
− ∂αin (X2 (s))

(cid:13)(cid:13)(cid:13)(cid:13)∂αin (X (s))
(cid:21)
X (cid:48)
2(s)

(cid:2)X (cid:48)
1 (S)−X (cid:48)
ds ≤

∂X

0

(cid:13)(cid:13)(cid:13)(cid:13)1

∂X
∂X
+ Lαin(cid:107)X2(cid:107)W1,1
maxX∈S(cid:107)X1 (s)−X2(cid:107)W1,1

0

t

t

where bρ = c1 + c2Lαin, c2 = maxX∈S
function ∂αin

, and Lαin is the the local Lipschitz constant of the
∂X , which is dependent on the constants rX and tc. By combining inequalities (A.8)

t

(cid:107)X2(cid:107)W1,1

and (A.9), with inequality (A.7), we get

(cid:107)C[X1] (t)−C[X2] (t)(cid:107)W1,1

t

≤ tcρX(cid:107)X1 (t)−X2 (t)(cid:107)W1,1

t

(A.10)

where ρX = Lx1 + Lh1Lh2bρ. Therefore, by taking tc ≤ ρc

ρX , for any 0 ≤ ρc ≤ 1, we have

(cid:107)C[X1] (t)−C[X2] (t)(cid:107)W1,1

t

≤ ρc(cid:107)X1 (t)−X2 (t)(cid:107)W1,1

t

which implies that C[X ] (t) is a contraction mapping over the set S. Combining all the previous

analysis, and by using the contraction mapping Theorem (B.1) of [81], we conclude that if

tc ≤ min{tu,

rX
bq

,

ρc
ρX

}

then there is a unique solution X (t) ∈ S that satisﬁes (A.1), for all t ≤ tc. With that, we have
established the uniqueness of the solution of (A.1) in the set S. We need to show the uniqueness
. We can prove that by showing that for any Xo ∈ ΩX , the solution cannot
of the solution in W1,1

t

123

leave the set ΩX . To see this, let the time tp be such that the solution X(cid:0)tp

(cid:1) leaves the boundary

of the set ΩX . By following similar steps that led to (A.6), we get

rX = (cid:107)X(cid:0)tp

(cid:1)−Xo(cid:107)1 ≤ tpbq ⇒ tp ≥ rX

⇒ tp ≥ tc

bq

which means that X (t) cannot leave the set ΩX for all time t ≤ tc, and this implies that any
solution X (t) ∈ W1,1
lies in S, from which the uniqueness of the solution is established in the
t
(cid:4)
space W1,1

.

t

A.2 Proof of Theorem 3.1

The proof is done in three steps.

Step 1: By using the appropriate Lyapunov functions of the last 5 variables of the closed-loop
system (3.42), we show that there exist ρϕ > 0, τ∗

1 (τ) > 0, and ε∗

1 > 0, µ∗
(cid:3), the compact set ϒr,c,ε = Σr ×Ωc×Ψε is positively-

1 (τ, µ) > 0 such that for

each τ ∈(cid:0)0,τ∗

(cid:3), µ ∈(cid:0)0, µ∗

(cid:3), and ε ∈(cid:0)0,ε∗

1

1

1

invariant. This is done by showing that the time-derivatives of the Lyapunov functions are negative

on the boundaries of this set.

To achieve this step, ﬁrst by using the following inequality

˙Vη ≤ −(cid:107)η(cid:107)2 + 2µ(cid:107)η(cid:107)(cid:107)PηBη(cid:107)
(cid:0)Pη
(cid:113)
it can be shown that for the set Ωη with ρη = 4(cid:107)PηBη(cid:107)2λmax
Vη = ρη µ2.
parameter projection, we have (cid:107) ˆKη(cid:107) ≤ bmax = max
η∈Ωη

Inside Ωη, we have (cid:107)η(cid:107) ≤ µ ¯ρη, where ¯ρη =

(cid:1), ˙Vη ≤ 0 on the boundary
(cid:1). Because of the
(cid:0)Pη

ρη /λmin

(bi + δ ) ,1 ≤ i ≤ ι. Therefore, inside the set

124

Ωη ×{|s| ≤ c}, we have the inequality

˙Vζ ≤ −(cid:107)ζ(cid:107)2 + 2(cid:107)ζ(cid:107)(cid:107)Pζ Bζ(cid:107)(cid:0)c + µ ¯ρη .bmax

(cid:1)

if µ ≤ c/(cid:0) ¯ρη .bmax

(cid:1), then we have

˙Vζ ≤ −(cid:107)ζ(cid:107)2 + 4(cid:107)ζ(cid:107)(cid:107)Pζ Bζ(cid:107)c
(cid:17)(cid:17)

(cid:16)
ρζ = 16(cid:107)Pζ Bζ(cid:107)2λmax

(cid:16)

Pζ

Let Ωζ = {Vζ ≤ ρζ c2}, where
. The above inequality shows that
˙Vζ ≤ 0 on the boundary Vζ = ρζ c2. Inside the set Ωη ×Ωζ ×{|s| ≤ c}, we have (cid:107)ζ(cid:107) ≤ c ¯ρζ , where
¯ρζ =

. Hence, by using

ρζ /λmin

(cid:114)

(cid:17)

(cid:16)

Pζ

e = ¯Kζ ζ + Bϕ

(cid:0)s− ˆKηη(cid:1)
(cid:20)

where

 I
it can be veriﬁed, if µ ≤ c/(cid:0) ¯ρη .bmax

¯Kζ =

 , Kζ =
(cid:1), then (cid:107)e(cid:107) ≤ ρec, where ρe =

k1 k2

. . .

−Kζ

kn−1

(cid:16)

(cid:21)

(cid:17)

¯ρζ + 2

. The time-derivative

of the Lyapunov function Vϕ is given by

(cid:2)− bd ˜zh − ∆e + (e)∆2

ε ˙Vϕ = −ϕT ϕ + 2εϕT PϕBϕ

− bh.ψ (η, ˆe,zh, µ) + ψz (·)(cid:3)

The function ψ (η, ˆe,zh, µ) is globally bounded in ˆζ for all (η,ζ ,s, ˜zh) ∈ Ωη × Ωζ ×{|s| ≤ c}×
Σr. Then there exists a positive constant lϕ such that |− bd ˜zh − ∆e + (e)∆2 − bh.ψ (η, ˆe,zh, µ) +

125

ψz (·)| ≤ lϕ. Therefore, the time-derivative of Vϕ satisﬁes the following inequality

ε ˙Vϕ ≤ −(cid:107)ϕ(cid:107)2 + 2ε(cid:107)PϕBϕ(cid:107)lϕ .(cid:107)ϕ(cid:107) ≤ −1
2

(cid:107)ϕ(cid:107)2

(A.11)

∀(cid:107)ϕ(cid:107) ≥ 4ε(cid:107)PϕBϕ(cid:107)lϕ. It follows that the above inequality is satisﬁed for all Vϕ ≥ ρϕε2, and ˙Vϕ ≤ 0

on the boundary Vϕ = ρϕε2, where ρϕ = 16(cid:0)(cid:107)PϕBϕ(cid:107)lϕ

(cid:1)2 λmax

(cid:0)Pϕ

(cid:1).

Consider the ﬁlter dynamics equation. Inside the set {Ωη × Ωζ ×{|s| ≤ c}× Σr × Ψε}, the term
∆q (·) will be bounded due to boundedness of the state variables in the domain of interest. There-

fore, we have

| ˙ueq| ≤ keq

and

|∂ Qh−o
∂ u

| ≤ kq

where keq and kq are positive constants, independent of τ. Moreover, it can be established that

| ˙βs (t, ˆe,zh)| ≤

Ko
ho (1− ¯Ko)

ΘN
d θ N

In addition, due to the Lipschitz properties of the term ψq in its arguments and since (cid:107)ϕ(cid:107) ≤ ¯ρϕε
(where ¯ρϕ =

(cid:1)), one has

(cid:0)Pϕ

ρϕ /λmin

(cid:113)

|ψq| ≤ lq(cid:107)ζ − ˆζ(cid:107) ≤ lq(cid:107)ϕ(cid:107) ≤ lq ¯ρϕε

(A.12)

for some positive constant lq independent of τ. Therefore, if ε ≤ kz
lq ¯ρϕ

, where

(cid:32)

(cid:33)

kz = kqkeq

1 +

Ko
ho (1− ¯Ko)

ΘN
d θ N

126

then, the time-derivative of Vq satisﬁes the following inequality;

τ ˙Vq ≤ −|˜zh|2 + τ(cid:0)kz + lq ¯ρϕε(cid:1)|˜zh| ≤ −1

|˜zh|2

(A.13)

2

for all |˜zh| ≥ 4τkz. This implies that the ˙Vq is negative on the boundary of the set Στ, where
ρq = 4k2
z . Consider now the surface dynamics ˙s. Obtain the time-derivative of the Vs and add and

subtract the terms bh.ψ (η,e,zh, µ) and (βs (t, ˆe,zh) .sat (s/µ)) inside the brackets. Then it can be

shown, by utilizing the Lipschitz properties of the these terms, that if ε is small enough, we have

|ψ (η,e,zh, µ)− ψ (η, ˆe,zh, µ)| ≤(cid:16)

(cid:17)(cid:107)ζ − ˆζ(cid:107) ≤(cid:16)

(cid:17)(cid:107)ϕ(cid:107)

lβ + ls/µ

lβ + ls/µ

(A.14)

Similar to inequality (A.12), we can show that

|ψz| ≤ lz(cid:107)ζ − ˆζ(cid:107) ≤ lz(cid:107)ϕ(cid:107) ≤ lz ¯ρϕε

where lz is a positive constant and independent of τ and ε. Inside the set {Ωη × Ωζ × Ψτ × Ψε},
consider the term ∆1 (·). Since (cid:107)η(cid:107) ≤ µ ¯ρη and because of the parameter projection used in the
adaptation law, we can conclude that (cid:107)˙ˆλη(cid:107)≤ βsmax and (cid:107)v(cid:107)≤ µ ¯ρη, where βsmax = max
Consequently, we can show |∆1 (·)| ≤ µρ∆, where ρ∆ = ¯ρη (bmax + βsmax) +bmax +(cid:107)Bη(cid:107). By us-
ing (cid:107)ϕ(cid:107) ≤ ¯ρϕε, and inequalities (A.12), (A.14), utilizing the inequality |∆1 (·)| ≤ µρ∆, and the
switching gain function (3.43), the time derivative of the surface s will satisfy the following in-

{t≥0} (|βs (t, ˆe,zh)|).

equality

(cid:18)

˙Vs ≤ b.|s|

− ¯γo +

τkzz
b +

µρ∆
b

(cid:16)

(cid:16)

+ ε.

d .θ N
ΘN
h0

lβ + ls/µ

(cid:17)

¯ρϕ + lz ¯ρϕ /b

(cid:17)(cid:19)

127

where ¯γo = γoΘN

d θ N
h0

(1− ¯Ko). If

τ ≤ ¯γo.b
3kzz

, µ ≤ ¯γo.b
3.ρ∆

and

(cid:16)

ΘN
d .θ N
h0

(cid:16)

¯γo.b
lβ + ls/µ

(cid:17)

ε ≤

3.

(cid:17)

¯ρϕ + lz ¯ρϕ /b

we can show ˙Vs ≤ −b. ¯γo.|s|, which proves that ˙Vs ≤ 0 on the boundary |s| = c. The above analysis
shows that the setϒr,c,ε is positively-invariant and by choosing c and r large enough, we can include

any compact set of (ζ ,s) in the interior of ϒr,c,ε. This completes the proof of the ﬁrst step.

Step 2: Let a, b, r1, and r2 be positive constants such that a < b < c and r1 < r2 < r. Show that
for any bounded ˆe (0) and for any (η (0) ,ζ (0) ,s (0)) ∈ Ωa and ˜zh (0,0,0) ∈ Σr1, where Ωa and
2 (τ, µ) > 0 such that for every ε ∈(cid:0)0,ε∗
Σr1 are compact sets in the interior of the compact sets Ωb and Σr2, respectively, and there exists
ε∗
show that for any (η (0) ,ζ (0) ,s (0)) ∈ Ωb and ˜zh (0,0,0) ∈ Σr2, there exists τ∗
(cid:3) and for every ε ∈(cid:0)0,ε∗
every τ2 ∈(cid:0)0,τ∗

(cid:3) the trajectory enters the set ϒb,r2 in ﬁnite time. Then
(cid:3) the trajectory enters the set ϒc,r in ﬁnite time.

2 > 0 such that for

2

2

2

(cid:16)(cid:16) s−N(ε)ϕ

(cid:17)(cid:17)

Since for (η,ζ ,s) ∈ Ωη × Ωb and ˜zh ∈ Σr2, the term sat
formally in ˆζ , it follows that the right-hand side of the slow dynamics of the closed-loop system

is globally-bounded uni-

µ

(3.42) is globally-bounded uniformally in ε. Consequently, there exists a time To, independent of
ε, such that for all (η,ζ ,s) ∈ Ωb and ˜zh ∈ Σr2 for 0 ≤ t ≤ To, using the inequality (A.11), we can
show that

˙Vϕ ≤ −

1
2ελmax

(cid:1)Vϕ

(cid:0)Pϕ

The above inequality shows that for any bounded ˆe (0), there exists a time T1 (ε) < To, where

T1 (ε) = 0, during which ϕ enters Σε in ﬁnite time. Therefore, choosing ε small enough to

lim
ε→0

128

make (T1 (ε) < To) guarantees that the trajectories will enter ϒb,r2 in ﬁnite time. Consider the
shifted ﬁlter dynamics, due to the boundedness of the term ∆q, then there exists a time Tf > To,
independent of τ and ε, such that for all (η,ζ ,s) ∈ Ωc and ˜zh ∈ Σr, for 0 ≤ t ≤ Tf , we can show
using inequality (A.13) that

˙Vq ≤ −1
τ

Vq

Therefore, there exists a time T2 (τ) < Tf , where lim
τ→0

enters the set Στ in ﬁnite time. Consequently, choosing τ small enough to make(cid:0)T2 (τ) ≤ Tf

T2 (τ) = 0. During the interval (0,T2], ˜zh

(cid:1) will

ensure that all the trajectories enter to the set ϒc,r in ﬁnite time. This completes the second step of

3 (τ, µ) > 0, such that for all ε ∈(cid:0)0,ε∗

the proof.
Step 3: Show that there exists ε∗
trajectory in ϒr,c,ε enters a small positively-invariant set ϒτ,µ,ε = Ψτ × Ωµ × Ψε, which vanishes
to the origin as τ, µ and ε tend to zero.

(cid:3), such that for every

3

To prove this statement, similar arguments used to show that ϒr,c,ε is positively-invariant can be

repeated to show that any trajectories started inside ϒr,c,ε will enter a positively-invariant set ϒτ,µ,ε
(cid:3)

and stay therein for all the future time. With this step, the proof is concluded.

A.3 Proof of Theorem 3.2

Consider the following composite Lyapunov Function candidate;

Va = ϑ T Pηϑ + κζ ζ T Pζ ζ +

1
2.bh

s2
c +

1
2

˜z2
h +

(cid:18) µ

(cid:19)

2.γ

˜λ T ˜λ

(A.15)

Consider each term of (A.15) separately. Due to the Lipschitz properties of the functions bϑ , bsc,

Fϑ , Fsc, Fλ , ψz, and ψq and due to the boundedness of the terms ∆ϑ , ∆sc, ∆q, and ˜χ as indicated

129

in remark (3.1) above, the time-derivative of Va satisﬁes the following inequality

˙Va ≤ −XT QaX +

(cid:18)
c11 +
|˜zh|.(cid:107)ϕ(cid:107)− βs (t, ˆe,zh) .sc.˜λ T
η v

(cid:107)ϑ(cid:107).(cid:107)ϕ(cid:107) +

c4
µ

+

(cid:19)

c12
µ
µ
γ

+

c13
τ

µ
+ δϑ .(cid:107)ϑ(cid:107) + δsc.|sc| + δz.|˜zh|

.|sc|.(cid:107)ϕ(cid:107)

λ T
η

˙˜λη

(A.16)

where δϑ = 2∗(cid:107)Pη(cid:107).δϑo, δsc = 1
bh

.δsco, the matrix



Qa =

1
−c1
− (c3 + c8) −(cid:16)
−c2
(cid:20)

−c1
κζ

0

c9 + κζ c5

−c2
0

1
τ

(cid:17) −c10
(cid:21)T



(cid:17)

−(cid:16)
− (c3 + c8)
c9 + κζ c5
(cid:16) c6
−c10
µ − c7

(cid:17)

(cid:107)ϑ(cid:107) (cid:107)ζ(cid:107) (cid:107)˜zh|

and the composite vector X =

.The constants ci, 1 ≤ i ≤ 13 are pos-
itive and independent of τ and µ. In order to make the matrix Qa positive deﬁnite, choose κζ
large enough to make the 2× 2 principal minor positive, and then choose τ small enough to make
the 3 × 3 principal minor positive, then choose µ small enough to make the determinant of the

|Sc|

matrix Qa positive. The detailed derivation of inequality (A.16) is presented in Appendix A.4. By

applying the adaptation rule under output-feedback (the last equation of the closed-loop dynamics

(3.46)), and considering the case when ¯v is not fully persistently exciting, we use Lemma 3.1 along

with the change of coordinates (3.44). Therefore, the inequality (A.16) can be rewritten as

˙Va ≤ −α1(cid:107)X(cid:107)2 + α2(cid:107)X(cid:107).(cid:107)ϕ(cid:107) + α3(cid:107)˜ληa(cid:107).(cid:107)ϕ(cid:107)

+(cid:0)δϑ + δsc + δz

(cid:1).(cid:107)X(cid:107)

130

where α1, α2, and α3 are positive constants. α2 is dependent on τ, µ, and (cid:107)˜ληb(cid:107), while α3 is
˜ληb are the adapted parameters corresponding to the excited and
dependent on µ only, and ˜ληa
non-excited modes, respectively. Next, consider the subsystem formed of the ˙sc and ˙˜ληa equations,

which is found to be ˙sc

˙˜ληa

 =

+

µ βs (t, ˆe, ˜zh) −bh
γ bh
µ βs (t, ˆe, ˜zh) ¯vT
a

0

µ βs (t, ˆe, ˜zh) ¯vT
a

 −bh
 g1 (·)

g2 (·)





 sc

˜ληa



(A.17)

where

g1 (·) = −bhβs (t, ˆe, ˜zh) ˜λ T
ηa
− bhβs (t, ˆe, ˜zh) ˜λ T
ηb

(cid:2)ϑa − Bηaς(cid:3)
(cid:105)
(cid:104)
ϑb − Bηbς

+ Fsc (ϑ ,ζ , ˜zh,sc) + bsc (ϕ) /µ + ∆sc ( ¯η,ζ ,zh, ¯χ, ˜χ, µ)
+ ψz (·)
bh
µ

a .sc + βs (t, ˆe, ˜zh)Na.Hv(cid:2)sc

βs (t, ˆe, ˜zh) ¯vT

g2 (·) = −γ

− N (ε)ϕ(cid:3)/µ2

Notice here that (cid:107)˜ληb(cid:107) is treated as a bounded time-varying disturbance due to the smooth projec-
tion. Therefore, the functions g1 (·) and g2 (·) in (A.17) vanish at ϑ = 0, ζ = 0, sc = 0, ϕ = 0,
˜ληa = 0, ˜χ = 0 regardless of the value of (cid:107)˜ληb(cid:107). Therefore, according to Section 13.4 of [93], the
origin of the homogeneous part of (A.17) has an exponentially stable equilibrium point at sc = 0,
˜ληa = 0. By using the converse Lyapunov function (Theorem 4.14 of [81]), there is a Lyapunov

131

function Vb whose time-derivative along the system (A.17) satisﬁes the following inequality

˙Vb ≤− α4|sc|2 − α5(cid:107)˜ληa(cid:107)2 + α6(cid:107)X(cid:107)2 + α7(cid:107)X(cid:107)(cid:107)ϕ(cid:107)
+ α8(cid:107)X(cid:107)(cid:107)˜ληa(cid:107) + α9(cid:107)˜ληa(cid:107)(cid:107)ϕ(cid:107) + α10δsc(cid:107)X(cid:107)

where the positive constants αi, 4 ≤ i ≤ 10, may depend on µ but are independent of ε. Consider
now the estimation error dynamics ( ˙ϕ equation). It can be shown that the time-derivative of the

Lyapunov function Vϕ satisﬁes the following inequality;

˙Vϕ ≤ −1
ε

(cid:107)ϕ(cid:107)2 + α11(cid:107)ϕ(cid:107)2 + α12(cid:107)˜ληa(cid:107)(cid:107)ϕ(cid:107) + α13(cid:107)X(cid:107)(cid:107)ϕ(cid:107) + α14δsc(cid:107)ϕ(cid:107)

where αi, 11 ≤ i ≤ 14, are positive constants and are independent of ε. Finally, consider the
composite Lyapunov function candidate for the closed-loop system: Vcl = κVa +Vb +Vϕ, where

κ is a positive constant. It can be shown that the time-derivative of Vcl satisﬁes the following

inequality;

(cid:20)

where Y =

˙Vcl ≤ −Y T QclY + δχ(cid:107)Y(cid:107)
(cid:21)T

and the matrix

(cid:107)X(cid:107) (cid:107)˜ληa(cid:107) (cid:107)ϕ(cid:107)



Qcl =

κα1 − α15

−α8

−α8
α5

− (κα2 + α16) − (κα3 + α17)

− (κα2 + α16)
(cid:17)
− (κα3 + α17)

(cid:16) 1
ε − α11

(A.18)



where α15 = α6 − α4, α16 = α7 + α13, and α17 = α9 + α12. In order to make the matrix Qcl
positive deﬁnite, choose κ large enough to make the 2× 2 principal minor positive, then choose ε

small enough to make Qcl positive deﬁnite. If the second term of inequality (A.18) vanishes, by

132

Theorem 8.4 of [81], we can show that lim
t→∞

(cid:107)Y(cid:107) = 0. Accordingly, we can show that lim
t→∞

Also under persistency of excitation condition and by Lemma 3.1 we have lim
t→∞

(cid:107)e(cid:107) = 0.
˜λη = ˜ληa = 0. If

the second term does not vanish, the above inequality can be rewritten in the following way;

˙Vcl ≤ −θ1Vcl + δχθ2

(cid:112)Vcl

for some positive constants θ1 and θ2. Let W =

√
V , then the above inequality will became

˙W ≤ −θ3W + θ4δχ

where θ3 and θ4 are positive constants. This inequality shows that the trajectory will converge

exponentially to a positively invariant set, in which all the variables are ultimately bounded by a
(cid:3)

constant depends on µ in ﬁnite time, by which the proof is completed.

A.4 Derivation of Inequality (A.16)

To construct the quadratic form of inequality (A.16), we need to obtain the time-derivative of each

term of the composite Lyapunov function (A.15). Consider the ﬁrst term, let Vϑ = ϑ T Pηϑ, then

the time-derivative of Vϑ is

(cid:20)
˙Vϑ = −ϑ T ϑ + 2ϑ T Pη .

Fϑ (ζ , ˜zh,sc) + bϑ (ϕ) /µ

(cid:21)

(A.19)

+ ∆ϑ ( ¯η,ζ ,zh, ¯χ, ˜χ, µ)

133

By looking inside the brackets, consider ﬁrst the function F (·). By adding and subtracting appro-

priate terms and after rearranging we obtain

Fϑ (ζ , ˜zh,sc) = Fϑ (ζ , ˜zh,sc)− Fϑ (0, ˜zh,sc)
+ Fϑ (0, ˜zh,sc)− Fϑ (0,0,sc)
+ Fϑ (0,0,sc)− Fϑ (0,0,0)

+ Fϑ (0,0,0)

Based on the local Lipschitz properties of the function Fϑ , the following inequality yields

|Fϑ (ζ , ˜zh,sc)| ≤ l f1(cid:107)ζ(cid:107) + l f2|˜zh| + l f3|sc|

where l f1, l f2, and l f3 are positive constants. The other functions bϑ , bsc, Fsc, and Fλ will be
treated similarly to obtain their bounds. For the second term inside the bracket, we can show that

|bϑ (ϕ) /µ| ≤ l f4
µ

(cid:107)ϕ(cid:107)

Based on inequalities (3.47-3.48) in remark (3.1) and from equation (A.19), we can show that

˙Vϑ ≤ −(cid:107)ϑ(cid:107)2 + c1(cid:107)ϑ(cid:107).(cid:107)ζ(cid:107) + c2(cid:107)ϑ(cid:107).|˜zh| + c3(cid:107)ϑ(cid:107).|sc| +

(cid:107)ϑ(cid:107).(cid:107)ϕ(cid:107) + δϑ(cid:107)ϑ(cid:107)

c4
µ

(A.20)

where c1 = 2l f1.(cid:107)Pη(cid:107), c2 = 2l f2.(cid:107)Pη(cid:107), c3 = 2l f3.(cid:107)Pη(cid:107), c4 = 2l f4.(cid:107)Pη(cid:107), and δϑ = 2δϑo.(cid:107)Pη(cid:107).
Consider the second term of (A.15). Let Vζζ = κζ ζ T Pζ ζ , then its time-derivative is

(cid:104)−ζ T ζ + 2ζ T .Pζ Bζ sc

(cid:105)

˙Vζζ = κζ

134

It can be easily seen that ˙Vζζ satisﬁes

| ˙Vζζ| ≤ −κζ(cid:107)ζ(cid:107)2 + κζ c5(cid:107)ζ(cid:107).|sc|

(A.21)

where c5 = 2(cid:107)Pζ Bζ(cid:107). Now, consider the third term of (A.15) and let Vsc = 1
2.bh
the time-derivative will result in

s2
c. Then taking

˙Vsc = −βs (t, ˆe,zh)s2

c/µ − βs (t, ˆe,zh) .sc ˜λ T

η v/µ

+ Fsc (ϑ ,ζ , ˜zh,sc) .sc/bh + bsc (ϕ) .sc/ (µbh)
+ ∆sc ( ¯η,ζ ,zh, ¯χ, ˜χ, µ) .sc/bh + ψz (·) .sc/bh

(A.22)

Since the functions Fsc and bsc are locally Lipschitz, we can derive the following two inequalities

accordingly;

|Fsc (ϑ ,ζ , ˜zh,sc) /bh| ≤ ls1(cid:107)ϑ(cid:107) + ls2(cid:107)ζ(cid:107) + ls3|˜zh| + ls4|sc||bsc (ϕ) / (µbh)| ≤ ls5
µ

(A.23)

where lsi,1 ≤ i ≤ 5 are positive constants. By using the inequalities provided in remark (3.1),
and inequality (A.23), we can show that the time-derivative of Vs (A.22) satisﬁes the following

inequality

˙Vsc ≤ −

(cid:19)

(cid:18)c6
µ
+δsc|sc|− βs (t, ˆe, ˜zh)sc.˜λ T
η v

− c7

.|sc|2 + c8|sc|.(cid:107)ϑ(cid:107) + c9|sc|.(cid:107)ζ(cid:107) + c10|sc|.|˜zh| +

µ

(cid:18)

c11 +

(cid:19)

c12
µ

|sc|.(cid:107)ϕ(cid:107)

(A.24)

135

where c6 = ¯βsmax, c7 = ls4, c8 = ls1, c9 = ls2, c10 = ls3, c11 = ¯lz, c12 = ls5, and δsc = 1
bh
Finally, consider the forth term of (A.15), and let Vzz = 1
h. Then the time-derivative of Vzz is

2 ˜z2

.δsco.

˙Vzz = − ˜z2
h
τ

− ∆q (·) ˜zh +

ψq (·) ˜zh

τ

By utilizing the inequalities provided in remark (3.1), we can obtain

˙Vzz ≤ −1
τ

|zh|2 +

c13
τ

|˜zh|.(cid:107)ϕ(cid:107) + δz|˜zh|

(A.25)

where c13 = ¯lq. By combining the inequalities (A.20), (A.21), (A.24), and (A.25) and obtaining

the derivative of the ﬁfth term of the equation (A.15) with respect to time, we can derive inequality

(A.16).

136

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