ENHANCING HIGH-GAIN-OBSERVER PERFORMANCE IN THE PRESENCE
OF MEASUREMENT NOISE
By
Alexis A. Ball

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Electrical Engineering
2011

ABSTRACT
ENHANCING HIGH-GAIN-OBSERVER PERFORMANCE IN THE
PRESENCE OF MEASUREMENT NOISE
By
Alexis A. Ball
High-gain observers are a prevalent and an important topic in state estimation
and output feedback control of nonlinear systems. In the absence of measurement
noise, this technique robustly estimates the derivatives of the output while achieving
fast convergence. Moreover, for a sufficiently fast observer and a globally bounded
controller, the high-gain observer is able to recover the system performance achieved
under state feedback control.
However, in the presence of measurement noise, a tradeoff exists between the measurement noise sensitivity and the speed of state reconstruction. As the observer gain
is increased, the bandwidth of the observer is extended. As the bandwidth increases,
the high-gain observer asymptotically approaches the behavior of a differentiator,
exacerbating the presence of measurement noise.
This dissertation addresses the challenging performance issues that arise when
implementing high-gain observers in the presence of measurement noise. In particular, we focus on the tradeoff between fast state reconstruction, minimizing the bound
on the steady-state estimation error, and rejecting the model uncertainty. The observer design and analysis is approached through three major thrust areas: observer
structure, tracking performance and filtering.

To my family; for their endless
support and encouragement.

iii

ACKNOWLEDGMENTS
I am indebted to my advisor, Professor Hassan Khalil, for molding me into the
researcher I am today. Professor Khalil’s expectations for excellence are uncompromising, and his patience boundless. I feel incredibly fortunate to have matured
scholarly under his tutelage. Thank you to my committee members Professor Ranjan
Mukherjee, Professor Xiaobo Tan and Professor Ning Xi for their insightful comments
that improved the quality of this dissertation.
The many thought provoking discussions that occurred with my colleagues at other
institutions would not have been possible without the support of the National Science
Foundation, the Alliance for Graduate Education and the Professoriate, the Graduate
School at Michigan State University, the College of Engineering at Michigan State
University and Dr. Barbara O’Kelly. I simply cannot thank Dr. O’Kelly enough for
coming to my rescue on several occasions.
I would like to thank my colleagues and friends for the helpful discussions, free
food and distractions. Most of all, thank you to my parents for believing in me and
my abilities. Their exaggerations of my superhuman research prowess stimulates me
to be that mythical heroine. Last, but certainly not least, thank you to my beloved
Zahar who inspires me to pursue the impossible.

iv

TABLE OF CONTENTS

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction . . . . . . .
1.1 Feedback Control . . . .
1.2 High-Gain Observers and
1.3 Organization . . . . . .

vii

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Measurement Noise
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2 Nonlinear-Gain High-Gain Observers . . .
2.1 Introduction . . . . . . . . . . . . . . . .
2.2 Problem Formulation and System Description
2.3 Observer Dynamics . . . . . . . . . . . . .
2.4 Closed-Loop System Analysis . . . . . . . .
2.5 Simulation: Field Controlled DC Motor . . .
2.6 Conclusions . . . . . . . . . . . . . . . .

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12
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49

3 High-Gain-Observer Tracking Performance
surement Noise . . . . . . . . . . . . .
3.1 Introduction . . . . . . . . . . . . . . .
3.2 Motivation . . . . . . . . . . . . . . . .
3.3 Tracking Performance . . . . . . . . . .
3.4 Problem Formulation . . . . . . . . . . .
3.5 Linear Systems Exploration . . . . . . . .
3.6 Nonlinear Systems Extension . . . . . . .
3.7 Conclusions . . . . . . . . . . . . . . .

in the Presence of Mea. . . . . . . . . . . .
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4 Enhancing High-Gain Observer Performance with Wavelet Denoising . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 A Wavelet Introduction . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 The Anatomy of a Wavelet Transform:
Continuous-Time . . . . . . . . . . . . . . . . . . . . . .
4.1.2 The Anatomy of a Wavelet Transform: Discrete-Time . . . .
4.2 Denoising: Offline . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Wavelet Type . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Wavelet Transform Levels . . . . . . . . . . . . . . . . . .
4.2.3 Thresholding Scheme . . . . . . . . . . . . . . . . . . . .
4.3 Denoising: Real-time . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Thresholding Scheme . . . . . . . . . . . . . . . . . . . .
4.3.3 Windowing . . . . . . . . . . . . . . . . . . . . . . . . .
v

50
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51
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60
66
78
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93
93

4.4

4.5
4.6

Example . . . . . . . . . .
4.4.1 Simulation . . . . .
4.4.2 Altering the Wavelet
4.4.3 Levels . . . . . . . .
4.4.4 Thresholding Logic .
4.4.5 Windowing . . . . .
Lowpass Filters . . . . . . .
Conclusions . . . . . . . .

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94
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5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
A Nonnegative Impulse Response

. . . . . . . . . . . . . . . . . 112

B A Block-Diagonal Form for Linear Systems

. . . . . . . . . . . 118

C Decomposition of Nonlinear Singularly Perturbed Systems
References

. . . 121

. . . . . . . . . . . . . . . . . . . . . . . . . . . 127

vi

LIST OF FIGURES

2.1

Plot of the two-piece nonlinear-gain function. . . . . . . . . . . . .

19

2.2

Plot of the two-piece nonlinear-gain function compared with the lineargain (g1 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

2.3

Plot of the three-piece nonlinear-gain function. . . . . . . . . . . .

21

2.4

Velocity reference trajectory

(r). . . . . . . . . . . . . . . . . . .
˙

42

2.5

Transient response of the error x2 − x2 vs. time for a (a) Two-Piece
ˆ
Nonlinear, (b) Switched, (c) Linear ε1 and (d) Linear ε2 gain high-gain
observer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

Transient response of the error x2 − x2 vs. time for a (a) Two-Piece
ˆ
Nonlinear, (b) Three-Piece Nonlinear, (c) Linear ε1 and (d) Linear ε2
gain high-gain observer. . . . . . . . . . . . . . . . . . . . . . .

45

Steady-state response of the error x2 − x2 vs. time for a (a) Twoˆ
Piece Nonlinear, (b) Switched, (c) Linear ε1 and (d) Linear ε2 gain
high-gain observer. . . . . . . . . . . . . . . . . . . . . . . . . .

46

State-state response of the error x2 − x2 vs. time for a (a) Two-Piece
ˆ
Nonlinear, (b) Three-Piece Nonlinear, (c) Linear ε1 and (d) Linear ε2
gain high-gain observer. . . . . . . . . . . . . . . . . . . . . . .

46

Transient response of the tracking error x2 − r vs. time for a
˙
(a) Two-Piece Nonlinear, (b) Switched, (c) Linear ε1 and (d) Linear
ε2 gain high-gain observer. . . . . . . . . . . . . . . . . . . . . .

47

2.10 Transient response of the tracking error x2 − r vs. time for a (a)
˙
Two-Piece Nonlinear, (b) Three-Piece Nonlinear, (c) Linear ε1 and
(d) Linear ε2 gain high-gain observer. . . . . . . . . . . . . . . . .

47

2.6

2.7

2.8

2.9

vii

2.11 Steady-state response of the tracking error x2 − r vs. time for a
˙
(a) Two-Piece Nonlinear, (b) Switched, (c) Linear ε1 and (d) Linear
ε2 gain high-gain observer. . . . . . . . . . . . . . . . . . . . . .

48

2.12 Steady-state response of the tracking error x2 − r vs. time for a
˙
(a) Two-Piece Nonlinear, (b) Three-Piece Nonlinear, (c) Linear ε1 and
(d) Linear ε2 gain high-gain observer. . . . . . . . . . . . . . . . .

48

Steady-state response of the error x2 − x2 vs. time for a high-gain
ˆ
observer with (a) ε = 0.001 and (b) ε = 0.0005. . . . . . . . . .

53

− x2 vs. time for a
ˆ
ε = 0.0005. . . . .

53

4.1

Diagram of a discrete wavelet transform implementation. . . . . . .

88

4.2

Potential hard thresholding function. . . . . . . . . . . . . . . . .

90

4.3

Potential soft thresholding function.

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

90

4.4

Comparison of the trajectories under (a) continuous-time, (b) sampleddata, (c) continuous-time and (d) sampled-data output feedback. . .

96

Comparison of the trajectories in steady-state (a) without wavelet denoising, (b) with wavelet denoising, (c) without wavelet denoising and
(d) with wavelet denoising. . . . . . . . . . . . . . . . . . . . . .

97

Transient performance comparison of the nonlinear-gain high-gain observer (a) without a wavelet pre-filter and (b) with a wavelet pre-filter.

98

3.1

3.2

4.5

4.6

Steady-state response of the tracking error x2
high-gain observer with (a) ε = 0.001 and (b)

4.7

Steady-state performance comparison of the nonlinear-gain high-gain
observer (a) without a wavelet pre-filter and (b) with a wavelet pre-filter. 99

4.8

Denoising performance in steady-state with the (a) Daubechies 1,
(b) Daubechies 4, (c) Daubechies 10 and (d) Daubechies 20 wavelets.

4.9

100

Denoising performance in steady-state with the (a) Daubechies 2,
(b) Coiflets 2 and (c) Symlets 2 wavelets. . . . . . . . . . . . . . . 101

4.10 Denoising performance in steady-state with Haar (a) level 1, (b) level
2, (c) level 3 and (d) level 4 wavelet transforms. . . . . . . . . . . . 102
4.11 Denoising performance in steady-state with (a) soft thresholding and
(b) hard thresholding. . . . . . . . . . . . . . . . . . . . . . . . 103
viii

4.12 Steady-state performance comparison of a (a) Haar wavelet denoising
scheme and (b) Butterworth filter. . . . . . . . . . . . . . . . . . 105
4.13 Transient performance comparison of a (a) Haar wavelet denoising
scheme and (b) Butterworth filter. . . . . . . . . . . . . . . . . . 105

ix

Chapter 1
Introduction
High-gain observers are an important tool in state estimation and output feedback
control. Some of the earlier research performed in the spirit of high-gain observers
can be viewed in [23] and [22]; see also [37], [44] and [31] for recent results. Yet, even
in early works such as [22], it was noted that noise in the system sensors can cause a
noticeable (and undesirable) effect on the system dynamics. Thus, the focus of this
dissertation is to analyze and address the issues associated with high-gain observer
performance degradation in the presence of measurement noise.

1.1

Feedback Control

Before describing the high-gain observer form, it is important to lay the foundation for
the types of systems that are considered in this body of work. Namely, the nonlinear
1

system

z = ψ(x, z, ς, u)
˙
x = Ax + Bφ(x, z, ς, u)
˙

(1.2)

y = Cx

(1.3)

w = Θ(x, z, ς) ,
where

(1.1)

(1.4)

z ∈ Rl and x ∈ Rn are the system states, y ∈ R and w ∈ Rs are

the measured outputs,

u ∈ R is the control input and ς(t) ∈ Rp represents the

exogenous signals. To create a more realistic problem, the function

φ(x, z, ς, u) is

assumed to be unknown; however, this does not excluded problems where the system
model may be known. The system matrices take the form



0

0

.
A = .
.

0

0



 
1 ··· ··· 0
0

 
0 
0 1 · · · 0

 
... . , B = .
.
.
.
.

 
0 
· · · · 0 1

 
0 ··· ··· 0
1

and

C = 1 0 ··· ··· 0 ,
where

A ∈ Rn×n , B ∈ Rn×1 and C ∈ R1×n . Logically, the system is assumed

to have dimension

n ≥ 2. In the event that the dimension condition is not met, the

construction of an observer is unnecessary; the measured output

y is simply used.

The structure for the model (1.1)-(1.4) includes mechanical systems, electromechanical systems and systems that can be placed in the normal form satisfying the
conditions of input-output linearization [30]. Some examples of mechanical and electromechanical systems where the displacements are measured, but not their deriva2

tives can be found in [3, 28, 33] for an induction motor, a rotational/translational
actuator and a smart material application, respectively. Furthermore, the additional
measurement

w may not be needed in every model. In this case, (1.4) can be re-

moved from the system representation. Yet, many models utilize the extra output.
For instance, consider a system in which the dynamics are extended by adding integrators; see [30]. Furthermore, the classic example of the magnetically suspended ball
is modeled such that the ball position and current are measurement outputs. The
position fits the chain of integrators form (in
state variable

x), whereas the current becomes the

w. Hence, many relevant and interesting systems are encapsulated in

the types of models of interest in this dissertation.

Assumption 1.1:
•

ς(t) is continuously differential and bounded;

•

ς(t) ∈ D ⊂ Rp , where D is compact;

•

φ, ψ , and Θ are locally Lipschitz in their arguments, uniformly in ς , over the
domain of interest; that is, for each compact subset of (x, z, u) in the domain of
interest, the functions satisfy the Lipschitz inequality with a Lipschitz constant
independent of

ς for all ς ∈ D.

The static state feedback controller takes the following form

u = γ(x, w, ς)

(1.5)

and is designed to meet the desired performance objectives. In practice, the controller
(1.5) cannot be implemented as written. Recall that only the first state is accessible.
Therefore, the controller cannot require values for any state beyond

x1 and remain

useful. Given the full state measurement is not available, an alternative method
3

is necessary to obtain the desired state information. One option is to construct an
observer that will estimate the system states from the available measurement y , which
leads to a dynamic output feedback controller of the form

u = γ(ˆ, w, ς) ,
x
where the state

(1.6)

x is replaced by the estimate x. For the class of systems defined by
ˆ

(1.1)-(1.4), we consider the high-gain observer

˙
x = Aˆ + Bφ0 (ˆ, w, ς, u) + h(y − x1 ) .
ˆ
x
x
ˆ

(1.7)

Typically, the gain function is defined as

h(y − x1 ) = H(y − x1 ) ,
ˆ
ˆ
where

H=
The

αn T
α1 α2
··· ··· n
.
ε ε2
ε

(1.8)

(1.9)

αi ’s are designed such that the roots of
sn + α1 sn−1 + · · · + αn−1 s + αn = 0

have negative real parts. The function

(1.10)

φ0 is locally Lipschitz and a known nominal

model of φ, which initially appears in (1.2). However, this is not the end of the story.
In general, the separation principle does not hold uniformly for nonlinear systems;
namely, maintaining the ability to design the state feedback controller independently
from the observer, with the result being a stable closed-loop system. The first separation principle for high-gain observers was reported in [50], followed by a more
comprehensive theorem in [6]. The stipulation is that the state feedback controller
4

be globally bounded and the observer parameter

ε be chosen sufficiently small. If

the state feedback controller is designed to achieve global stabilization, then it can
be shown that the output feedback controller utilizing the high-gain observer in (1.7)
realizes semiglobal stabilization.
Another important detail is the nature of the stability properties. Let the closedloop system (1.1)-(1.4) under the state feedback controller (1.5) be denoted as

χ = fr (χ, ς) ,
˙

(1.11)

where

χ=

x
z

∈ RN and fr (χ, ς) =

Ax + Bφ(x, z, ς, u)

.

ψ(x, z, ς, u)

Instead of stabilizing an equilibrium point of the system, the problem is posed as
rendering a certain compact set positively invariant and asymptotically attractive.
The allure of formulating the results in this fashion, is that the separation principle is
no longer limited to stabilization of an equilibrium point [5]. Some examples include
servomechanisms [25, 29] and finite time convergence to a set [20]. The regulation
problem for servomechanisms, for example, requires that the trajectories of the system
reach an invariant manifold where the tracking error is zero, usually called the zeroerror manifold. Instead of regulating the system about an equilibrium point, we desire
to study the dynamics on the manifold. Therefore, the set that we wish to render
positively invariant and asymptotically attractive is the zero-error manifold.
As stated in [4], uniform asymptotic stability with respect to a set
in ς , stipulates the following:
• Uniform Stability - For each

A, uniformly

ǫ > 0 there is a δ = δ(ǫ) such that

|χ(t0 )|A ≤ δ ⇒ |χ(t)|A < ǫ, ∀ t ≥ t0 ≥ 0, ∀ ς(t) ∈ D.
5

• Uniform Attraction - There is a constant
and for each

c > 0, independent of t0 and ς(t),

ǫ > 0 there is T = T (ǫ) such that

|χ(t)|A < ǫ, ∀ t ≥ t0 + T , ∀ |χ(t0 )|A < c, ∀ ς(t) ∈ D.
The expression |χ|A

= inf ν∈A χ−ν is the distance with respect to A. Further-

more, the system is said to be globally uniformly asymptotically stable with respect
to

A, if the uniform stability property holds with a class K∞ function δ , and the

uniform attraction property holds for any

r > 0 with T = T (ǫ, r). By extend-

ing the definition of stability to a compact positively invariant set instead of just an
equilibrium point, a wider variety of problem formulations can be encapsulated in
the above setup. Clearly, we can still address stabilization of the origin by defining

A = 0. However, we can just as easily structure the control objective as a regulation or tracking problem. For additional examples of control problems that can be
formulated as stabilization with respect to a set, see [4].

Assumption 1.2:

• The closed-loop system (1.11) is globally uniformly asymptotically stable with
respect to a compact positively invariant set

•

A, uniformly in ς ;

φ(x, z, ς, u) is zero in A, uniformly in ς .

To uncover some of the interesting properties inherent to high-gain observers,
consider the scaled estimation error

x −x
ˆ
ηi = i n−i i .
ε
6

(1.12)

For a second-order system the scaled estimation errors are

x1 − x1
ˆ
ε
η2 = x 2 − x 2 ,
ˆ
η1 =

(1.13)
(1.14)

which satisfy the singularly perturbed equation

εη1 = −α1 η1 + η2
˙
εη2 = −α2 η1 + εδ(x, z, ς, u) ,
˙
where

(1.15)
(1.16)

δ = φ − φ0 . As the value of ε is decreased, the effect of δ in (1.16) is

diminished. Hence, high-gain observers have the ability to reject the error due to
modeling uncertainty as

ε approaches zero. The smaller the value of ε, the faster

the time-scale of the observer relative to the plant (or system in

x). This difference

in time-scale leads to the possibility of peaking behavior. If there is any difference
in the initial conditions between the state
condition of η1 will be

x1 and the estimate x1 , then the initial
ˆ

O(1/ε). This peaking phenomenon is a by-product of the

observer gain structure in (1.9) and leads to a term of the form

a
exp(−at/ε)
ε
in the transient response of the solution to (1.15)-(1.16), where

(1.17)

a > 0. Given the

term will decay rapidly, the effects will be seen primarily in the transient response.
However, as

ε tends to zero, (1.17) approaches the behavior of an impulse function,

where its amplitude peaks at a value

O(1/ε). Before the impulse-like behavior of

the term can subside, this exponential mode has the ability to not only induce an unacceptable transient response, but destabilize the closed-loop nonlinear system. The
destabilizing effect of peaking interacting with the nonlinear feedback control was
7

first observed in [21]. Furthermore, the solution of saturating the control and/or the
state estimates outside a compact region of interest to achieve a globally bounded con-

ˆ
troller, and designing the nominal function φ to be globally bounded in the estimates,
protects the system plant from the destabilizing behavior during the peaking period.
Notice that the peaking period shrinks to zero as

ε tends to zero. Moreover, the

system trajectories under output feedback come arbitrarily close to the trajectories
under state feedback as the value of

ε approaches zero. This amounts to recovering

the system performance in addition to the stability properties of the system under
state feedback control; see [6].

1.2

High-Gain Observers and Measurement Noise

Return to the singularly perturbed representation of the derivative of the scaled estimation errors in (1.15)-(1.16). By adding noise into the system measurement such
that

y = x1 + v
becomes the output and

v is the additive measurement noise, (1.15)-(1.16) is altered

in the following manner

εη1 = −α1 η1 + η2 − (α1 /ε)v
˙

(1.18)

εη2 = −α2 η1 + εδ(x, z, ς, u) − (α2 /ε)v ,
˙

(1.19)

where reducing the amount of error in the estimation is no longer as simple as decreasing ε. Unlike the system without measurement noise there exists a tradeoff between
the steady-state errors due to the model uncertainty, captured in the function δ , and
the measurement noise v . In [2], the norm on the state vector
8

η and, ultimately, the

estimation error satisfies the inequality

µ
x(t) − x(t) ≤ c1 ε + c2 n−1 , ∀t ≥ T
ˆ
ε

(1.20)

for an n-dimensional system with positive constants c1 , c2 and T . The measurement
noise

v is assumed to be bounded by the positive constant µ. Furthermore, another

tradeoff exists between the speed of state recovery and the accuracy of that estimate
in steady-state. Moreover, it is crucial that the observer be sufficiently faster than
the dynamics of the plant in order to ensure recovery of the state feedback controller
performance. Therefore, choosing smaller values of ε results in better rejection of the
modeling uncertainty, faster reconstruction of the system states and recovery of the
performance under state feedback control. However, the presence of measurement
noise prevents

ε from being chosen arbitrarily small. Hence, the work in this disser-

tation seeks to further analyze the effects of measurement noise, while quantifying
and reducing the manifestations of the tradeoff in the system states.

1.3

Organization

The purpose of this dissertation is to tackle the challenging performance issues that
arise when implementing high-gain observers in the presence of measurement noise.
Thus, the work herein approaches observer design and analysis in the presence of
measurement noise through three major thrust areas: observer structure, tracking
performance and filtering. The divisions addressing those areas are briefly summarized below.
The first attempt at minimizing the tradeoffs present in the high-gain observer
are done through manipulating the observer gain structure. The gain is designed
as a function of the estimation error in the first state. Therefore, the gain function
9

responds to the value of this estimation error, such that the observer experiences a
larger gain during the transient period and a lower gain afterwards. By designing
the function in this fashion, the closed-loop system is able to obtain reasonably fast
state estimation and attenuate a larger portion of the measurement noise in steadystate. One may interpret this result as minimizing the classic tradeoff of speed versus
accuracy present in observer design.
However, the effect of measurement noise on the tracking error is less significant
than on the estimation error. Simulation observations suggest that pushing the observer gain too large can noticeably compromise the estimation error and, ultimately,
the system performance. Yet, such issues are not as apparent for control problems
that are formulated in the tracking framework. The effect that the measurement
noise has on the system tracking error is analyzed for linear systems and a class of
nonlinear systems.
In addition to augmenting the observer form with nonlinearities, the observer performance can potentially be improved by filtering out the measurement noise before
feeding the output to the observer. Typically, a lowpass filter is used to remove the
noise from signals in the feedback loop. However, depending on the order of the filter, unacceptable phase lag can be introduced with the potential for destabilizing the
system. In the interest of providing an alternative to the classic lowpass filter, the
feasibility of wavelets for denoising is studied in Chapter 4. The idea of using wavelets
to remove noise is not a new one, however, incorporating them into the feedback loop
for online denoising is a recent development. Historically, all of the signal is available
when the denoising algorithm is applied. Naturally, all of the past and present signal
values will not be available in the feedback loop. This is just one of the complications
introduced by attempting to denoise the output signal online. Overall, the investigation is carried out using wavelets to design various pre-filters, while comparing the
results to the lowpass filter. The final chapter speculates on possible future work and
10

provides some concluding remarks.

11

Chapter 2
Nonlinear-Gain High-Gain
Observers
2.1

Introduction

High-gain observers have developed into an important topic in state estimation and
output feedback control of nonlinear systems, beginning with papers such as [21]
and [22]. In the absence of measurement noise, this technique robustly estimates
the derivatives of the output while achieving fast convergence [21]. Moreover, for
a sufficiently high observer gain and a globally bounded controller, the high-gain
observer is able to recover the system performance achieved with the state feedback
control. Refer to [31] for a survey on high-gain observers.
However, observer theory reveals that a tradeoff exists between the measurement
noise sensitivity and the speed of state reconstruction [38]. As the observer gain is
increased, the bandwidth of the observer is extended. As the bandwidth increases
the high-gain observer asymptotically approaches the behavior of a differentiator,
exacerbating the presence of measurement noise. The authors of [43], in the context of discrete-time models, exploited this knowledge by designing a switched filter
12

composed of two linear filters (one for the transient response and the other for the
steady-state response); the value of the estimation error determines which filter is
active. The idea is to use a large filter gain (increasing the filter bandwidth) during the transient behavior to elicit a fast recovery of the state estimates. The filter
with the smaller gain is active once the estimation error has reached a steady-state
threshold, reducing the filter bandwidth and preventing a large magnification of the
measurement noise. In [52], the authors seek to minimize the effect quantization error
has on shaft encoder measurements by introducing a dead-zone nonlinearity into the
state estimation scheme. The dead-zone nonlinearity is used to alternate “smoothly”
between varying filter bandwidths to initially achieve fast state estimation and, ultimately, minimize the quantization error.
Recently, others have addressed the issue of measurement noise and observers
in [8, 45, 47]. The work in [47] investigates a high-gain observer with a sign-indefinite
gain adaptation for systems with potentially nonlocal Lipschitz functions and noisy
output. However, this approach often leads to highly oscillatory, although bounded,
behavior in the state estimates. In [8], the authors propose an extended Kalman filter
that utilizes an adaptive high-gain parameter to achieve noise rejection and global
convergence of the estimated state when careful tuning is exercised. More generally,
the work in [45] analyzes observers with improved transient performance for both
linear and nonlinear systems; the effects of measurement noise are briefly considered
for one high-gain observer design.
The effect of measurement noise in high-gain observers has been studied in [2,53].
It is shown in [2] that the steady-state estimation error has a component due to
modeling uncertainty, which can be attenuated by increasing the gain. Furthermore,
the error has a component due to measurement noise that is amplified by increasing
the gain. This tradeoff constrains the observer gain, which reduces the observer’s
ability to quickly reconstruct the states. In [2], the authors construct an observer
13

to diminish the manifestations of the tradeoff in the system states. A switched-gain
observer is proposed in [2] to force a large gain during the transient period for fast
state reconstruction, and allow for a smaller gain once the states are satisfactorily
estimated to reduce the effect of noise on the steady-state performance. However, a
number of complications are generally associated with a switched system. The time
in which the gains are switched, trigger threshold, and system peaking are all design
issues that must be addressed. Both from an analysis and design/implementation
prospective, using a switched observer can be tedious.

The purpose of this chapter is to construct high-gain observers containing a nonlinear gain that takes the form of a piecewise linear function with two or three distinct linear regions. The regions are chosen to correspond to the desired transient
and steady-state responses, respectively. By constructing the observer gain in this
manner, we can achieve fast state estimation and reduced steady-state error. Furthermore, the observer is devised such that the behavior of the innovation process can
be controlled separately from the other estimation errors. This is accomplished by
assigning one fast eigenvalue with the remaining eigenvalues chosen relatively slow.
Without this key step, the stability analysis for the proposed observers is unattainable. The analysis focuses on the proposed high-gain observer and the closed-loop
system dynamics. The discussion concludes with a simulation comparing the system
performance under the linear, nonlinear, and switched high-gain observer designs.
In particular, it is demonstrated that the nonlinear-gain observers are sufficient in
obtaining the desired estimation error dynamics, while reducing the implementation
complexity necessary with the switched gain observer.
14

2.2

Problem Formulation and System Description

Consider the nonlinear system

z = ψ(x, z, ς, u)
˙
x = Ax + Bφ(x, z, ς, u)
˙

(2.2)

y = Cx + v

(2.3)

w = Θ(x, z, ς) ,
where

(2.1)

(2.4)

z ∈ Rl and x ∈ Rn are the system states, y ∈ R and w ∈ Rs are the

measured outputs,
signals and v(t)

u ∈ R is the control input, ς(t) ∈ Rp represents the exogenous

∈ R is the measurement noise. The function φ(x, z, ς, u) may not

be known. We do not explicitly define noise in the output

w, given the purpose of

this work is to study how measurement noise directly enters the high-gain observer
from y , and mitigate the resulting effects. The triple (A,
of

B , C ) represents a chain

n integrators, where it is assumed that n ≥ 2. Possible sources for the model

(2.1)-(2.4) include mechanical systems, electromechanical systems and systems that
can be placed in the normal form satisfying the conditions of input-output linearization.

Assumption 2.1:
•

ς(t) is continuously differential and bounded;

•

ς(t) ∈ D ⊂ Rp , where D is compact;

•

v(t) is a measurable function of t and bounded, where the bound is defined as
|v(t)| ≤ µ;

•

φ, ψ , and Θ are locally Lipschitz in their arguments, uniformly in ς , over the
15

domain of interest; that is, for each compact subset of (x, z, u) in the domain of
interest, the functions satisfy the Lipschitz inequality with a Lipschitz constant
independent of

ς for all ς ∈ D.

The state feedback controller takes the following form

˙
θ = Γ(θ, x, w, ς)

(2.5)

u = γ(θ, x, w, ς)

(2.6)

and meets the requirements listed in Assumption 2.2.

Assumption 2.2:
•

Γ and γ are locally Lipschitz functions in their arguments, uniformly in ς , over
the domain of interest;

•

Γ and γ are globally bounded functions of x. The necessity for this assumption
is detailed in Chapter 1.

Let the closed-loop system (2.1)-(2.4) under the state feedback controller (2.5)-(2.6)
be denoted as

χ = fr (χ, ς) ,
˙
where

 
x
 
χ =  z  ∈ RN
 
θ


Ax + Bφ(x, z, ς, γ)


.
and fr (χ, ς) = 
ψ(x, z, ς, γ)


Γ(θ, x, w, ς)


16

(2.7)

Assumption 2.3:
• The closed-loop system (2.7) is globally uniformly asymptotically stable with
respect to a compact positively invariant set
•

A, uniformly in ς ;

φ(x, z, ς, γ) is zero in A, uniformly in ς .

Instead of stabilizing an equilibrium point of the system, the problem is posed as
rendering a certain compact set positively invariant and asymptotically attractive.
The allure of formulating the results in this fashion, is that the separation principle
is no longer limited to stabilization of an equilibrium point [5]; further details are
provided in Chapter 1. As stated in [4], uniform asymptotic stability with respect to
a set

A, uniformly in ς , stipulates the following:

• Uniform Stability - For each

ǫ > 0 there is a δ = δ(ǫ) such that

|χ(t0 )|A ≤ δ ⇒ |χ(t)|A < ǫ, ∀ t ≥ t0 ≥ 0, ∀ ς(t) ∈ D.
• Uniform Attraction - There is a constant
and for each

c > 0, independent of t0 and ς(t),

ǫ > 0 there is T = T (ǫ) such that

|χ(t)|A < ǫ, ∀ t ≥ t0 + T , ∀ |χ(t0 )|A < c, ∀ ς(t) ∈ D.
The expression |χ|A

= inf ν∈A χ−ν is the distance with respect to A. Further-

more, the system is said to be globally uniformly asymptotically stable with respect
to

A, if the uniform stability property holds with a class K∞ function δ , and the

uniform attraction property holds for any

r > 0 with T = T (ǫ, r). By extend-

ing the definition of stability to a compact positively invariant set instead of just an
equilibrium point, a wider variety of problem formulations can be encapsulated in
the above setup. Clearly, we can still address stabilization of the origin by defining
17

A = 0. However, we can just as easily structure the control objective as a regulation
or tracking problem. For examples of control problems that can be formulated as
stabilization with respect to a set, see [4] and Chapter 1.

2.3

Observer Dynamics

The intuition behind the nonlinear observer gain is the following:
• Achieve the desired (fast) state reconstruction with ε1 without sacrificing the
steady-state performance;
• Reduce the steady-state estimation error with ε2 while maintaining an acceptable rate of convergence in the estimates.
For a visual of the two-piece nonlinear-gain, see Figure 2.1. However, when examining the two-piece structure, it appears as if the slope (through the origin) g1 is not
equivalent to the slope g1 in the linear-gain observer, as shown in Figure 2.2. Hence,
it may be prudent to also investigate an observer constructed with a three-piece nonlinear gain; see Figure 2.3.

18

Figure 2.1: Plot of the two-piece nonlinear-gain function.

19

Figure 2.2: Plot of the two-piece nonlinear-gain function compared with the lineargain (g1 ).

20

Figure 2.3: Plot of the three-piece nonlinear-gain function.

The high-gain observer is defined as

˙
x = Aˆ + Bφ0 (ˆ, w, ς, u) + h(y − x1 ) ,
ˆ
x
x
ˆ

(2.8)

where the nonlinear gain is
i
i
i
ˆ
hi (y − x1 ) = αi g1 (y − x1 ) + d(g2 − g1 )sat
ˆ
21

y − x1
ˆ
d

(2.9)

for the two-piece structure and
i
i
ˆ
hi (y − x1 ) = αi g1 (y − x1 ) + d(g2 − gci )sat
ˆ

+

i
d2 (gci − g1 )sat

y − x1
ˆ
d
y − x1
ˆ
d2

(2.10)

for the three-piece version. The representations of the piecewise linear functions
shown in (2.9) and (2.10) are derived using Proposition 1 of [24], which states that
a piecewise linear function can be written as a summation of a linear function with
multiple saturation functions. The function “sat” denotes the saturation function
defined as

sat(e)

=

The expression for gci is given as


e,

if

sign(e),

if

|e| ≤ 1

.

(2.11)

|e| > 1

i
i
d2 g1 − d1 g2
,
gci =
d2 − d
where the observer gains are defined as

g1 = 1/ε1 and g2 = 1/ε2 , where ε1 < ε2
and chosen to correspond to the desired transient and steady-state responses, respectively. Both ε1 and ε2 are small positive parameters. The parameter d is defined such
that the observer gain is the smaller value g2 for
The function

φ0 is a nominal model of φ.

22

|x1 − x1 | ≤ d and d2 > d > µ.
ˆ

Assumption 2.4:

φ0 is locally Lipschitz in its arguments, uniformly in ς , over the domain of

•

interest;

φ0 is globally bounded in x and zero in A.

•
The

αi ’s are designed such that the roots of
sn + α1 sn−1 + · · · + αn−1 s + αn = 0

(2.12)

are real and negative, with one fast root and (n − 1) slow real roots. In this case,
(2.12) is written as

(sn−1 + β1 sn−2 + · · · + βn−2 s + βn−1 )(s + λ) = 0 ,
where the first polynomial is Hurwitz with
(2.12) to (2.13), it can be seen that
and

(2.13)

O(1) real roots and λ ≫ 1. Relating

α1 = λ + β1 , αi = βi−1 λ + βi ∀ 1 < i < n

αn = βn−1 λ. The output feedback controller is obtained by replacing x in

(2.5)-(2.6) with x.
ˆ

2.4

Closed-Loop System Analysis

For the closed-loop system analysis, the observer dynamics are replaced by the equivalent dynamics of the scaled estimation error

η = D(ε1 )(x − x) ,
ˆ
23

(2.14)

where

n−1
D(ε1 ) = diag[1, ε1 , · · · , ε1 ]. The closed-loop system under the output

feedback controller can be written as

χ = f (χ, ς, D−1 (ε1 )η)
˙


−1 (ε )η, w, ε))
Ax + Bφ(x, z, ς, γ(θ, x − D
1



=
ψ(x, z, ς, γ(θ, x − D−1 (ε1 )η, w, ε))


Γ(θ, x − D−1 (ε1 )η, w, ε)
¯
ε1 η = A0 η + B0 v + εn Bδ(χ, ς, w, D−1 (ε1 )η) + hδ1 (η1 + v) ,
˙
1

(2.15)

(2.16)

where



−α1
1 ··· ···

 −α2
0
1 ···


.
...
.
A0 = 
.

 −α
0
n−1 · · · · · ·

−αn · · · · · · · · ·
¯
hi = α i 1 −





−α1


 −α2
0


.
. 
.
.  , B0 = 

.
.


 −α
1
n−1


−αn
0
0

ε1
ε2







,




i

and

δ1 (η1 + v) =



sat



η1 +v
d

dd2
d2 −d

sat

d,
η1 +v
d

two-piece gain

− sat

η1 +v
d2

,

three-piece gain

where the value of δ1 depends on the form of the nonlinear gain. Yet, regardless
the structure of the gain, |δ1 | ≤ d. The function δ(χ, ς, w, D −1 (ε1 )η) is de24

fined as

φ(x, z, ς, γ(θ, x, w, ς)) − φ0 (ˆ, w, ς, γ(θ, x, w, ς)), and the matrix A0
ˆ
x
ˆ

is Hurwitz. The equations (2.15)-(2.16) resemble a model appearing in the standard
singularly perturbed form, as shown in [35]. The primary difference between this
system and the standard form is the presence of the negative powers of ε1 in the
term D −1 (ε1 )η . However, δ is a globally bounded function in x, implying that it is
ˆ
also globally bounded in D −1 (ε1 )η . This property allows us to extend the analysis
associated with standard singularly perturbed systems to the case involving (2.15)(2.16). The slow dynamics of (2.15) can be approximated by defining ε1 /ε2
setting ε2

εf ,

= 0 and keeping εf = 0, which yields η = 0. This reduces (2.15) to the

closed-loop system (2.7) under the state feedback controller (2.5)-(2.6). Moreover,
the system is globally uniformly asymptotically stable with respect to the compact
positively invariant set

A. Then, according to a converse Lyapunov theorem in [39],

there exists a smooth Lyapunov function V (χ), two class K∞ functions U1 and U2 ,
and a class

K function U3 such that

U1 ( χ A ) ≤ V (χ) ≤ U2 ( χ A )
∂V
f (χ, ς, 0) ≤ −U3 ( χ A )
∂χ
for all

(2.17)
(2.18)

ς ∈ D.

Theorem 2.1: Let Assumptions 2.1 through 2.4 hold and consider the closed-loop
system (2.7) with the observer (2.8). Moreover, let
and

M be any compact set in RN

N be any compact subset of Rn , where χ(t0 ) ∈ M and x(t0 ) ∈ N . Then,
ˆ

given the positive constant εf = ε1 /ε2 < 1, there is a positive constant
that for λ > λ∗ , the following properties hold:
• There exist positive constants
constant

λ∗ such

µ∗ and ca such that for µ < µ∗ there is a

εa = εa (µ) > ca µ1/n with limµ→0 εa (µ) = ε∗ > 0, such
2
25

that for each ε2
bounded for all

∈ (ca µ1/n , εa ] the trajectories of the closed-loop system are

t ≥ 0.

µ∗ > 0 and a class K function ρ1 such that for every µ < µ∗
2
2
and every Υ1 > ρ1 (µ), there are constants TΥ = TΥ (Υ1 ) ≥ 0 and εb =

• There exists

εb (µ, Υ1 ) > ca µ1/n , with limµ→0 εb (µ, Υ1 ) = ε∗ (Υ1 ) > 0, such that
b
for each ε2 ∈ (ca µ1/n , εb ]
max{|χ(t)|A , x(t) − x(t) } ≤ Υ1 , ∀t ≥ TΥ .
ˆ

(2.19)

µ∗ > 0 and a class K function ρ2 such that for every µ < µ∗
3
3
and every Υ2 > ρ2 (µ), there is a constant εc = εc (µ, Υ2 ) > ca µ1/n , with

• There exist

limµ→0 εc (µ, Υ2 ) = ε∗ (Υ2 ) > 0, such that for each ε2 ∈ (ca µ1/n , εc ]
c
χ(t) − χr (t) ≤ Υ2 , ∀t ≥ t0
where

(2.20)

χr (t) is the solution of (2.7) with χr (t0 ) = χ(t0 ).

The last two bullet items are similar to Theorem 1 of [2].

Remark 2.1: The procedure for choosing the design parameters ε1 , ε2 and

d is

fairly simple. Initially, the observer gain ε1 should be chosen to correspond to the
desired transient behavior in the estimation error dynamics; the approximate results
can be seen by running the chosen value in a linear observer. Without violating the
condition on the ratio of the ε’s, the value of the parameter ε2 is set to achieve the
desired steady-state estimation error dynamics. Finally, the value of
chosen as small as possible, while respecting the lower bound d
measurement noise. By choosing the value of

d should be

> µ, imposed by the

d small, the nonlinear observer is able

to maximize the use of ε1 during the transient phase. In the case of the three-piece
26

nonlinear gain, the additional parameter d2 should be chosen as close as possible to d.

Proof: In order to place the set

M in the interior of Ωc = {V (χ) ≤ c} ⊂ RN ,

c > maxχ∈M V (χ). The set Ωc is compact for any choice of c. We have
already established that δ is a globally bounded function in D −1 (ε1 )η . Therefore,

choose

there is a constant
and

η ∈ Rn .

Lδ > 0, independent of ε1 , such that δ ≤ Lδ for all χ ∈ Ωc

Consider the fast equation (2.16) for

χ ∈ Ωc . This equation possesses both

slow and fast variables due to the choice of the eigenvalues shown in (2.13). To
transform (2.16) into the singularly perturbed form,

A0 and B0 are represented as

A0 = A01 λ + A02 and B0 = B01 λ + B02 . The procedure from [35] is used with
the change of coordinates

ζ
η1

= Tη ,

(2.21)

where



−β1

1

0

··· 0





 −β2 0 1 · · · 0


Y
 .
... .
.
.
T =
= .
.


Z
−β

 n−1 0 · · · · · · 1
1
0 ··· ··· 0
and

T −1 = M N with M ∈ Rn×(n−1) and Y ∈ R(n−1)×n . Applying this

change of coordinates to (2.16) yields

¯
˙
ε1 ζ = Y A02 M ζ + Y B02 v + εn Y Bδ + Y hδ1
1
27

(2.22)

ε
ε1 η1 = −λη1 + ζ1 − (λ + β1 )v + (λ + β1 ) 1 − 1
˙
ε2

δ1 ,

(2.23)

where


ε1

 (λ + β1 ) 1 − ε
2




2 
−β
1 ··· 0 
ε

 .1

 (λβ1 + β2 ) 1 − 1
... . 
¯
 = λ ε1 a + b,
.
ε2
Yh= .
.
.


ε2


.
.


.
−βn−1 0 · · · 1 

n


ε1
λβn−1 1 −
ε2










a=




β

ε
β1 1 − 1
ε2
2
ε1
β2 1 −
ε2

n−1



.
.
.

1−

ε1
ε2

n−1









,






ε
ε
2
 −β1 1 − 1 + β2 1 − 1

ε2
ε2


ε
ε

−β1 β2 1 − 1 + β3 1 − 1
b=
ε2
ε2


.
.

.


ε
−βn−1 β1 1 − 1
ε2



2

3







,






Y A01 = 0, Y B01 = 0, ZA01 M = 0, ZA01 N = −1,
ZA02 M = 1 0 · · · 0 ∈ R1×(n−1) ,
28

A02 N = 0, and Y A02 M is by design a Hurwitz matrix. The solution matrix
P to the Lyapunov equation P Y A02 M + (Y A02 M )T P = −I is symmetric
and positive definite. Let the Lyapunov function candidate for (2.22) be chosen as

W1 = ζ T P ζ . It can be shown that
2
1
˙
ζ 2+
ζ ( P Y B02 µ + εn P Y B Lδ
W1 ≤ −
1
ε1
ε1
ε1
+ P
λ a + b
ε2

(2.24)

d).

It follows that

˙
W1 ≤ −
for

1
2ε1 P

W 1 , ∀ W 1 ≥ LW

LW = 4 P Y B02 µ + εn 4 P Y B Lδ
1
ε1
λ a + b
+4 P
ε2

Thus, the

(2.25)

(2.26)

d.

ζ states are bounded. The bound (2.26) is reached within the interval

[t0 , t0 + T1 (ε1 )], where

T1 (ε1 ) = 4 P ε1 ln
and kw1

σ1 =

2
kw 1

σ1 εn
1

→ 0 as ε1 → 0

(2.27)

> 0 is a constant independent of ε1 and ε2 , and
P P Y B Lδ .

Meanwhile, the next step is to show that the trajectories of (2.23) reach a positively
invariant strip defined as

{|x1 − x1 | ≤ L} ,
ˆ
where

(2.28)

1 2
0 < L < d. Using the Lyapunov function W2 = η1 , it can be shown that
2
29

1
1
−
η1 (η1 + v − δ1 )
ε1 ε2
(2.29)
1
1
1
1
−
δ − β v + ζ1 .
+ η1 β1
ε1 ε2 1 ε1 1
ε1

λ
˙
W2 = − η1 (η1 + v) − λ
ε2

It follows from d

> |v| that sign(η1 ) = sign(η1 + v) whenever |η1 + v| ≥ d and
−λ

1
1
−
ε1 ε2

η1 (η1 + v − δ1 ) ≤ 0.

To ensure that the condition in (2.28) is satisfied, we require a tighter bound on the
state ζ1 than what is provided in (2.25). Thus, the ultimate bound on |ζ1 | is denoted
as

ε
c0 + k1 µ + k2 εn Lδ + k3 1 λd + k4 d ,
1
ε2

(2.30)

where c0 , a constant due to initial conditions, can be made arbitrarily small for

t

large enough; k1 , k2 , k3 , and k4 are positive constants. Using the ultimate bound
in (2.30), after t0

+ T1 (ε1 ),
(1 − θ)λ
˙
|η1 |2 , ∀ |η1 | ≥ U
W2 ≤ −
ε2

for

U=

µ ε2
+
θ ε1

c1 µ + c2 d + c0 + k2 εn Lδ
1
λθ

ε
= β1 + k1 , c2 = β1 1 − 1
ε2
θ ∈ (0, 1) will be chosen in a later step.
where c1

(2.31)

k d
+ 3 ,
θ

(2.32)

+ k4 and ε1 ≤ 1. The parameter

To ensure that (2.28) implies

|x1 − x1 + v| ≤ d ,
ˆ
30

(2.33)

d is chosen as d > L + µ. In order for the strip (2.28) to be positively invariant, we
require the choice of L to be greater than U . This requirement leads to the inequality

U < L < d − µ. We rewrite the foregoing inequality as U + µ < d, and revisit U

defined in (2.32). Recall that the terms in U containing µ in the numerator and/or λ
in the denominator can be made sufficiently small. However, the third term, k3 d/θ ,
does not immediately appear to be necessarily small. To ensure that the inequality

U + µ < d is satisfied, the constant k3 needs to be handled with care. In particular,
we need k3 /θ

< 1 to avoid violating the foregoing inequality. According to Lemma

1 of [53], the constant k3 arises from the response of the system

ε
˙
¯
¯
ε1 ζ = Y A02 M ζ + aλ 1
ε2
and is given by

k3 =
for

∞
0

¯
|CeY A02 M t a|dt

(2.34)

¯
C = [1 0 · · · 0]. Define
¯
G(s) = C(sI − Y A02 M )−1 a .

Then, by selecting the poles and zeros of
m
¯

G(s) = K
i=1
where

(2.35)

G(s) to be real and distinct such that

s + zi
¯
s + pi
¯




n
¯
j=m+1
¯



1 
,
s + pi
¯

m ≤ n, zi > pi for i = 1, ...m, the impulse response of (2.34) is nonnega¯
¯ ¯
¯
¯

tive; see [26] for the proof. This assumption can always be satisfied by an appropriate

31

choice of β1 ,

· · · , βn−1 and ε1 /ε2 ; see Appendix A for examples. Hence,
∞

k3 =

¯
CeY A02 M t adt = G(s)

0

(2.36)
s=0

where solving for k3 leads to



−β1

1

0

... 0





 −β2 0 1 . . . 0


 .
... . ,
.
.
Y A02 M =  .
.


−β
0 . . . . . . 1
 n−2

−βn−1 0 . . . . . . 0

¯
C(sI − Y A02 M )−1 a =
and k3

=1−

ε1
ε2

ε n−1
ε
β1 1 − ε1 sn−2 + · · · + βn−1 1 − ε1
2

2

sn−1 + β1 sn−2 + · · · + βn−1

n−1
. Therefore,

n−1
c1 µ + c2 d + c0 + k2 εn Lδ (1 − εf )d
µ
1
U +µ= +µ+
+
.
θ
εf λθ
θ
Recall that

µ must be small enough and λ chosen large enough to ensure that U +

µ < d. Let
µ∗ =
1
λ∗

n−1
(1 − k)εf d
n−1
2 − kεf

,

c 1 µ∗ + c 2 d + k 2 L δ
1
= n
εf [(1 − k)d + kµ∗ ] − 2εf µ∗
1
1
32

µ ∈ (0, µ∗ ), λ > λ∗ and k ∈ (0, 1). Substituting the upper
1
∗ and the lower bound λ∗ into the expression for U + µ leads to
bound µ1
and require that

n−1
(1 − εf )d
µ∗
1 + µ∗ + 1 [εn−1 (1 − k)d + εn−1 kµ∗ − 2µ∗ ] +
U +µ <
1
1
1
f
θ
θ f
θ
1
n−1
n−1
n−1
= [(1 + θ + εf k − 2)µ∗ + (εf (1 − k) + 1 − εf )d]
1
θ
1
n−1
n−1
= [(1 + θ + εf k − 2)µ∗ + (1 − εf k)d].
1
θ

By choosing

n−1
θ = 1 − kεf and ε1 ≤ 1, the condition U + µ < d is satisfied.

We will now show that all trajectories reach the strip (2.28) in finite time. The
following inequality originates from (2.31)

2kεn
f

1
W 2 , ∀ W 2 ≥ L2 .
ε1
2

(2.37)

1
Σ2 = {W2 ≤ L2 } = {|η1 | ≤ L}
2

(2.38)

˙
W2 ≤ −λ
Therefore, the set

is positively invariant. If η1 (t0 ) is outside of

Σ2 , then from (2.37)

W2 (η1 (t)) ≤ W2 (η1 (t0 ))exp −λ

2kεn
f
ε1

(t − t0 ) .

From the scaling equation (2.14), it can be seen that whenever
bounded, there exists a constant

(2.39)

x(t0 ) and x(t0 ) are
ˆ

kw2 > 0, independent of ε1 and ε2 , such that

2
W2 (η1 (t0 )) ≤ kw2 . From (2.38) and (2.39), it can be seen that η1 reaches the set

Σ2 within the time interval [t0 , t0 + T2 (ε1 )], where
33





 k 
ε1
 w2 
ln 
T2 (ε1 ) =
 → 0 as ε1 → 0.
λkεn  1 
f
L
2

(2.40)

At this point, η1 is inside the strip (2.28) and cannot leave the strip for all future
time. Inside the strip, the parameter ε2 is driving the dynamics of the high-gain
observer, not ε1 . Therefore, it is appropriate to alter the scaling equation (2.14) to
obtain

ξ = D(ε2 )(x − x) ,
ˆ
where

(2.41)

n−1
D(ε2 ) = diag[1, ε2 , · · · , ε2 ]. Then, the error dynamics become

˙
ε2 ξ = A0 ξ + εn Bδ + B0 v ,
2

(2.42)

which is valid for trajectories inside the strip. Take the Lyapunov function candidate
as W3

= ξ T Sξ , where S is the positive definite symmetric solution to the Lyapunov

equation

SA0 + AT S = −I . Then, it can be shown that
0
˙
W3 ≤ −

where σ2

1
2ε2 S

W3 , ∀ W3 ≥ (σ2 εn + σ3 µ)2 ,
2

S and σ3 = 4 SB0

= 4 SB Lδ

S . Therefore, the set

Σ3 = {W3 ≤ (σ2 εn + σ3 µ)2 }
2
is positively invariant. For

(2.43)

(2.44)

ξ(t0 ) outside Σ3 , it can be seen from (2.43) that

W3 (ξ(t)) ≤ W3 (ξ(t0 ))exp

34

−(t − t0 )
2ε2 S

.

(2.45)

From the scaling equation in (2.41), for bounded

x(t0 ), x(t0 ) and ε2 , there exists a
ˆ

2
> 0, independent of ε2 , such that W3 (ξ(t0 )) ≤ kw3 . It follows from

constant kw3

(2.44) and (2.45) that ξ reaches the set Σ3 within the time interval [t0 , t0 +T3 (ε2 )],
where

T3 (ε2 ) = 4ε2 S ln

kw3
σ2 εn
2

→ 0 as ε2 → 0.

Σ3

Inside the set

µ
x(t) − x(t) = D−1 (ε2 )ξ(t) ≤ ε2 γ1 + n−1 γ2
ˆ
ε2
for γ1

= σ2 /

λmin (S) and γ2 = σ3 /

Σ is reached within the time interval

Fr (ε2 , µ)

(2.47)

λmin (S). Therefore, all of the trajec-

tories are traveling towards the positively invariant set
the set

(2.46)

Σ = {Σ2 ∩ Σ3 }. Moreover,

T (ε1 , ε2 ) = T1 (ε1 ) + T2 (ε1 ) + T3 (ε2 ) → 0 as ε2 → 0 ,

(2.48)

ε1 < ε2 ,
meaning that decreasing ε2 will eventually correspond to a decrease in ε1 . Hence,
only the reduction of ε2 is explicitly listed in (2.48).
where the times are defined in (2.27), (2.40) and (2.46). Recall that

We will now address the remaining arguments (including the slow vector

χ) for

the boundness of all trajectories, ultimate boundness where the trajectories come
close to the set

A × {x − x = 0} and closeness of trajectories. A similar approach
ˆ

for the slow states
For

χ can be found in [2].

(χ, η) ∈ Ωc × Σ the vector χ can be represented as
χ = f (χ, ς, D−1 (ε2 )ξ).
˙

The function

(2.49)

f is globally bounded in D−1 (ε2 )ξ , because it is globally bounded in
35

x and the term D−1 (ε2 )ξ results from substituting x − D−1 (ε2 )ξ for x. Hence,
ˆ
ˆ

there exists a positive constant kf , independent of ε2 , such that

f (χ, ς, D−1 (ε2 )ξ) ≤ kf .

(2.50)

The following lemma originally appeared in [2] and is included here for convenience;
see [2] for the proof.

Lemma 1: The function

Fr (ε2 , µ) has the following properties for ε2 > 0 and

µ ≥ 0. First, Fr (ε2 , µ) has a global minimum at
ε2 = [(n − 1)γ2 µ/γ1 ](1/n)

ca µ1/n

and
n−1
min Fr (ε2 , µ) = (c1 ca + γ2 /ca )µ1/n

ε2 >0
For ε2

ka µ1/n .

> ca µ1/n , Fr (ε2 , µ) is a strictly increasing function of ε2 and Fr (ε2 , µ) ≤

kb ε2 , where kb = γ1 + γ2 /cn . Then, given kr > 0, for every µ ∈ [0, (kr /ka )n )
a
there exist εm = εm (µ, kr ) ≥ 0 and εM = εM (µ, kr ) > ca µ1/n , with
εm ≤ min{ca µ1/n , (µγ2 n/kr )1/(n−1) }
and

lim εM (µ, kr ) = kr /γ1 ,

µ→0
such that

With

Fr (ε2 , µ) ≤ kr for all ε2 ∈ (εm , εM ].

η = 0, the output feedback expression in (2.15) reduces to the state feed-

back representation shown in (2.7). Then, for ε2
36

∈ (ca µ1/n , ε∗ ) and µ < µ∗ ,
2
1

there is a positive constant

L1 , independent of ε2 , such that the following Lipschitz

condition is satisfied

f (χ, ς, D−1 (ε2 )ξ) − f (χ, ς, 0) ≤ L1 D−1 (ε2 )ξ
for all (χ, η)

(2.51)

∈ Ωc × Σ, where the lower bound on ε2 is chosen according to Lemma

1. Taking the derivative of the smooth Lyapunov function

V (χ) yields

∂V
∂V
∂V
˙
f (χ, ς, D−1 (ε2 )ξ) +
f (χ, ς, 0) −
f (χ, ς, 0)
V =
∂χ
∂χ
∂χ
∂V
∂V
=
f (χ, ς, 0) +
f (χ, ς, D−1 (ε2 )ξ) − f (χ, ς, 0)
∂χ
∂χ
∂V
≤ −U3 (χ) +
f (χ, ς, D−1 (ε2 )ξ) − f (χ, ς, 0) .
∂χ
Let

L2 be an upper bound on ∂V /∂χ over Ωc . Then,
˙
V ≤ −U3 (χ) + L2 L1 x − x
ˆ

−U3 (χ) + Lr Fr (ε2 , µ)

(2.52)

(χ, η) ∈ Ωc × Σ. Let L3 = (1/Lr ) minχ∈∂Ωc U3 (χ) and apply
Lemma 1 with kr = L3 and set µ∗ = (kr /ka )n . Then, for µ < µ∗ and
4
4
˙
ε2 ∈ (ca µ1/n , εM ], we have V ≤ 0 for all (χ, η) ∈ ∂Ωc × Σ. From the

for all

previous analysis, we have that
set

˙
W3 ≤ 0 for all (χ, η) ∈ Ωc × ∂Σ. Therefore, the

Ωc × Σ is positively invariant, which implies that the set Ωc × Σ is positively

invariant.
Given

χ(t0 ) is in the interior of Ωc , we have
t

χ(t) − χ(t0 ) =
Considering that

f (χ(τ ), u(τ ))dτ .
t0

f is continuous and its arguments are bounded, f (χ, u) ≤ k1
37

for all

χ(t) ∈ Ωc , where k1 is independent of ε2 . Hence,
χ(t) − χ(t0 ) ≤ k1 (t − t0 )

as long as

(2.53)

χ(t) ∈ Ωc . Therefore, there exists a finite time T ∗ , independent of

ε2 , such that χ(t) ∈ Ωc for all t ∈ [t0 , t0 + T ∗ ]. The previous analysis showed
that η enters the set Σ during the finite time period [t0 , t0 + T (ε1 , ε2 )], where
T (ε1 , ε2 ) → 0 as ε2 → 0. Thus, there exists εu such that for all 0 < ε2 ≤ εu ,
T (ε1 , ε2 ) ≤ T ∗ . Thus, choosing µ∗ small enough that ca (µ∗ )1/n < εu and
5
5

µ∗ = min{µ∗ , µ∗ , µ∗ } and εa = min{εM , εu }, we conclude that for
1 4 5
0 ≤ µ < µ∗ and ca µ1/n < ε2 ≤ εa , the trajectory (χ, η) enters the set Ωc × Ω

setting

during the finite time period

[t0 , t0 + T (ε1 , ε2 )] and stays there for the remainder

of time. Prior to entering the set,

χ(t) and η(t) are bounded by (2.53) and (2.45),

respectively. Therefore, the closed-loop trajectories are bounded.
Ultimate Boundness: The proof for ultimate boundness utilizes the dynamics of
the

ξ vector inside the strip (2.28).

Apply Lemma 1 to the expressions (2.47) and (2.52) by setting kr = τ , where
τ ∈ (ka µ1/n , L3 ] and set εb = min{εa , εM }; the equations (2.47) and (2.52)
are satisfied for all t ≥ t0 + T ∗ . Thus, for ε2 ∈ (ca µ1/n , εb ] the function

Fr (ε2 , µ) ≤ τ . Then, from (2.52)

1
1
˙
V ≤ −U3 (χ) + Lr τ = − U3 (χ) − U3 (χ) + Lr τ .
2
2
This results in

1
˙
V ≤ − U3 (χ)
2
for all

χ ∈ {U3 (χ) ≤ 2Lr τ }. Given U3 (χ) is positive definite and continuous,
/

there is a positive constant τ ∗

< L3 such that the set U3 (χ) ≤ 2Lr τ is compact
38

τ ≤ τ ∗ . Let co (τ ) = maxU3 ≤2Lr τ {V (χ)}, then co (τ ) is a nondecreasing
function that tends to zero as τ → 0. Choose a class K function ϕ(τ ) such that
for

ϕ ≥ co (τ ). Then, for ϕ(τ ) ≤ V (χ) ≤ c

1
˙
V ≤ − U3 (χ).
2
Therefore, there exists a time

Tϕ = Tϕ (τ ) ≥ 0 such that
˙
V (χ(t)) ≤ ϕ(τ )

for all

t ≥ t0 + T ∗ + Tϕ (τ ). It follows from (2.17) that
−1
|χ(t)|A ≤ U1 (ϕ(τ ))

for all

ρa (τ )

t ≥ t0 + T ∗ + Tϕ (τ ). Moreover,
max{|χ(t)|A , x(t) − x(t) } ≤ max{τ, ρa (τ )}
ˆ

ρb (τ )

where ρb is a class K∞ function. Define ρ1 as ρ1 (µ) = ρb (ka µ1/n ) with µ <
µ∗ = min{µ∗ , (τ ∗ /ka )n }, where ρ1 is a class K∞ . Given Υ1 > ρ1 (µ), take
2
τ = min{τ ∗ , ρ−1 (Υ1 )} and set TΥ = t0 +T ∗ +Tϕ (τ ) to achieve the inequality
b
in (2.19).

Closeness of Trajectories: Using the fact that the closed-loop system under state
feedback is uniformly asymptotically stable with respect to the set
equality in (2.19), given

A and the in-

Υ2 > 2ρ1 (µ), there exists a time TΥ2 = TΥ2 (Υ2 ) > 0,

independent of ε2 , such that

|χ(t)|A ≤
39

Υ2
2

(2.54)

and

|χr (t)|A ≤
for all

Υ2
2

(2.55)

t ≥ TΥ2 for ε2 ∈ (ca µ1/n , εb ]. Then,
χ(t) − χr (t) ≤ χ(t) − x + χr (t) − x

(2.56)

t ≥ TΥ2 and x ∈ A. Then, we take the infimum of the right-hand side of
(2.56), over all x ∈ A, and substitute the values in (2.54) and (2.55). This results
for all

in the following

χ(t) − χr (t) ≤ |χ(t)|A + |χr (t)|A ≤ Υ2
for all

(2.57)

t ≥ TΥ2 . Furthermore, we can see from (2.53) that
χ(t) − χ(t0 ) ≤ k1 (t − t0 )

(2.58)

χr (t) − χ(t0 ) ≤ k1 (t − t0 )

(2.59)

and similarly

for all

t ∈ [t0 , t0 + T (ε1 , ε2 )]. Therefore,
χ(t) − χr (t) ≤ 2k1 T (ε1 , ε2 )

for all

(2.60)

t ∈ [t0 , t0 + T (ε1 , ε2 )]. Viewing the closed-loop system under output feed-

back as a perturbation of the closed-loop system under state feedback and applying
Theorem 3.4 of [30] over the interval

[t0 + T (ε1 , ε2 ), TΥ2 ] gives the following

χ(t) − χr (t) ≤ 2k1 c3 T (ε1 , ε2 ) + c4 Fr (ε2 , µ)
40

(2.61)

for some constants c3

≥ 1 and c4 > 0, independent from ε1 and ε2 . It can be

seen from (2.60) and (2.61) that the time for which (2.61) is valid can be simplified
to

[t0 , TΥ2 ]. Furthermore, it can be verified that T (ε1 , ε2 ) is a class K function of

¯
ε2 for ε2 ≤ (1/e)(kw3 /σ2 )1/n
ε2 . The inequality ca (µ∗ )1/n < ε2 can be
¯
6
satisfied for a sufficiently small µ∗ . Then, for each µ < µ∗ the following statement
6
6
holds

min

ε2 ∈(ca µ1/n ,¯2 ]
ε

{2k1 c3 T (ε1 , ε2 ) + c4 Fr (ε2 , µ)}
= 2k1 c3 T (εf ca µ1/n , ca µ1/n ) + c4 ka µ1/n

(2.62)

ρ3 (µ).
Moreover, it can be shown that ρ3 (µ) is a class K function. For each Υ2 > ρ3 (µ),
there exists εc = εc (µ, Υ2 ) > ca µ1/n with the limµ→0 εc (µ, Υ2 ) = ε∗ (Υ2 ) >
¯
¯
¯
¯c
0 such that for all ε2 ∈ (ca µ1/n , εc ]
¯

2k1 c3 T (ε1 , ε2 ) + c4 Fr (ε2 , µ) ≤ Υ2

(2.63)

is satisfied. The inequality in (2.20) can be found by taking

¯ ¯
ρ2 (µ) = max{2ρ1 (µ), ρ3 (µ)}, µ∗ = min{µ∗ , µ∗ } and εc = min{εb , εc , ε2 },
2 6
3
in combination with (2.57), (2.61) with the simplified time and (2.63).

2.5

Simulation: Field Controlled DC Motor

The system under consideration is a field controlled DC motor [30], where it is desired
that the shaft angular velocity track a reference signal as shown in Figure 2.4. The
41

system is represented as

x1 = x2
˙
x2 = φ(x, z, u)
˙

(2.65)

z = ψ(x, z, u)
˙

(2.66)

y = x1 + v

(2.67)

w=z,
where

(2.64)

(2.68)

x1 is the rotor position, x2 the rotor angular velocity and z the armature

current. Notice that the available measurements are the rotor position and the current. Moreover, the current measurement is not an output resulting from the chain
of integrators.

Velocity Reference

20
10
0
−10
−20
0

2

4

Time

6

Figure 2.4: Velocity reference trajectory

8

10

(r).
˙

u. The
10
controller expression is u =
(0.11ˆ2 + r − 100(y − r) − 20(ˆ2 − r)), where
x
¨
x
˙
w
feedback linearization is applied. The functions above are defined as φ(x, z, u) =
The field current is used as the source of control and is denoted by

−0.1x2 + 0.1zu and ψ(x, z, u) = −2z − 0.2x2 u + 200. The estimate x2
ˆ
42

is saturated outside [-100, 100]. The saturation values are chosen such that the
saturation is never active when the system is under state feedback control. The
nominal value for φ used in the observer is φ0 (ˆ, w, u)
x

= −0.11x2 + 0.1wu. The

state z is measured and need not be estimated; thus, the observer is second-order. The
gains for the observers are chosen as ε1
parameters are chosen as

= 0.0005 and ε2 = 0.01. The remaining

α1 = 71 and α2 = 70. The initial conditions are set at

x1 (0) = x2 (0) = x2 (0) = 0, x1 (0) = 0.02 and z(0) = 100 to match values
ˆ
ˆ
consistent with the physical system. The initial conditions are deliberately chosen to
be unequal to ensure peaking in the transient response of the system, lending itself to
a more realistic scenario. The measurement noise

v is generated using the Simulink

block “Uniform Random Number”, where the magnitude is limited to [-0.0016, 0.0016]
and the sampling time is set at 0.0008 seconds. The noise magnitude is based on a
1000 c/r encoder. The value of

d for the switched observer threshold is 0.005. The

parameters for the nonlinear-gain observers are d

= 0.0035 and d2 = 0.05. For the

switched observer, it was shown in [2] that if the system switches before the transient
response of the estimates of the higher order derivatives has subsided and entered
a positively invariant set, the other transients can take
and subsequently cause the value of

(y − x1 ) out of the strip
ˆ

ε to switch. If this occurs, the system could be

susceptible to multiple switching until all of the trajectories recover from peaking.
Thus, the switched observer requires the additional component of a switching timer,
based on the peaking period, that prevents the observer from switching before the
trajectories of the estimation error have reached a positively invariant set. The delay
timer is set for 0.15 seconds; details on how to choose this value can be found in [2].
Figure 2.5 shows the transient response of the error

x2 − x2 . As expected, the
ˆ

switched observer captures the behavior of the linear observer with the parameter ε1 .
Unlike the switched observer, the two-piece nonlinear-gain observer does not wait for

the transients to subside in both states before entering the strip. Therefore, the
43

system utilizing the nonlinear-gain observer does not perfectly mimic the transient
response of the system with the linear observer shown in Figure 2.5(c). However, the
nonlinear-gain observer is able to recover the performance of the observer in Figure
2.5(d) faster than the switched observer. As a result, the presence of noise is more
noticeable in the estimation error generated with the switched observer than with
the nonlinear-gain observer. Figure 2.6 compares the transient performance of the
three-piece nonlinear-gain observer with the two-piece and two linear-gain high-gain
observers. Clearly, there is no appreciable difference in the estimation error x2 − x2
ˆ
between the two nonlinear-gain observers.

20

10
Estimation Error

20

10

0

0

0

0.05

0.1
(a)

0.15

0.2

0

20

0.15

0.2

0.05

0.1
(d)

0.15

0.2

10

0

0.1
(b)

20

10

0.05

0

0

0.05

0.1
(c)

0.15

0.2

0
Time

Figure 2.5: Transient response of the error x2 − x2 vs. time for a (a) Two-Piece
ˆ
Nonlinear, (b) Switched, (c) Linear ε1 and (d) Linear ε2 gain high-gain observer.
In Figures 2.7 and 2.8, the estimation error steady-state behavior is practically
identical for all observers shown, with the exception of the linear ε1 observer.
Figure 2.9 shows the tracking error

x2 − r during the transient response of
˙

the observer dynamics. The transient response resulting from the system using the
nonlinear-gain observers is faster than both linear observers. As shown in Figures
44

20

10
Estimation Error

20

10

0

0

0

0.05

0.1
(a)

0.15

0.2

0

20

0.15

0.2

0.05

0.1
(d)

0.15

0.2

10

0

0.1
(b)

20

10

0.05

0

0

0.05

0.1
(c)

0.15

0.2

0
Time

Figure 2.6: Transient response of the error x2 − x2 vs. time for a (a) Two-Piece
ˆ
Nonlinear, (b) Three-Piece Nonlinear, (c) Linear ε1 and (d) Linear ε2 gain high-gain
observer.
2.11 and 2.12, the steady-state response of the tracking errors are nearly identical for
four out of the five observers. The nonlinear gain observers exhibit similar system
behavior, as reported in Figure 2.10 and Figure 2.12.
Overall, the nonlinear-gain observers are able to achieve better system performance than the two linear-gain observers, while bypassing the complications typically
associated with switching. In general, the two-piece observer is able to perform just
as well as an observer with three distinct gain regions.

45

5

0
Estimation Error

5

0

−5
0

−5
0

5
(a)

10

5

10

5
(d)

10

5

0

5
(b)

0

−5
0

5
(c)

−5
10
0
Time

Figure 2.7: Steady-state response of the error x2 − x2 vs. time for a (a) Two-Piece
ˆ
Nonlinear, (b) Switched, (c) Linear ε1 and (d) Linear ε2 gain high-gain observer.

5

0
Estimation Error

5

0

−5
0

−5
0

5
(a)

10

5

10

5
(d)

10

5

0

5
(b)

0

−5
0

5
(c)

−5
10
0
Time

Figure 2.8: State-state response of the error x2 − x2 vs. time for a (a) Two-Piece
ˆ
Nonlinear, (b) Three-Piece Nonlinear, (c) Linear ε1 and (d) Linear ε2 gain high-gain
observer.

46

30

20

20

10
Tracking Error

30

10

0

0

−10
0

0.05

0.1
(a)

0.15

−10
0

0.2

30

0.2

0.05

0.1
(d)

0.15

0.2

10

0

0.15

20

10

0.1
(b)

30

20

0.05

0

−10
0

0.05

0.1
(c)

0.15

0.2

−10
0
Time

Figure 2.9: Transient response of the tracking error x2 − r vs. time for a
˙
(a) Two-Piece Nonlinear, (b) Switched, (c) Linear ε1 and (d) Linear ε2 gain high-gain
observer.

30

20

20

10
Tracking Error

30

10

0

0

−10
0

0.05

0.1
(a)

0.15

−10
0

0.2

30

0.2

0.05

0.1
(d)

0.15

0.2

10

0

0.15

20

10

0.1
(b)

30

20

0.05

0

−10
0

0.05

0.1
(c)

0.15

0.2

−10
0
Time

Figure 2.10: Transient response of the tracking error x2 − r vs. time for a (a) Two˙
Piece Nonlinear, (b) Three-Piece Nonlinear, (c) Linear ε1 and (d) Linear ε2 gain
high-gain observer.
47

0.05

0
Tracking Error

0.05

0

−0.05
0

5
(a)

−0.05
0

10

0.05

0.05

0

5
(b)

10

5
(d)

10

0

−0.05
0

5
(c)

10

−0.05
0
Time

Figure 2.11: Steady-state response of the tracking error x2 − r vs. time for a
˙
(a) Two-Piece Nonlinear, (b) Switched, (c) Linear ε1 and (d) Linear ε2 gain high-gain
observer.

0.05

0
Tracking Error

0.05

0

−0.05
0

5
(a)

−0.05
0

10

0.05

10

5
(d)

10

0.05

0

5
(b)

0

−0.05
0

5
(c)

10

−0.05
0
Time

Figure 2.12: Steady-state response of the tracking error x2 − r vs. time for a
˙
(a) Two-Piece Nonlinear, (b) Three-Piece Nonlinear, (c) Linear ε1 and (d) Linear ε2
gain high-gain observer.
48

2.6

Conclusions

When high-gain observers are utilized in the presence of measurement noise, there
exists a tradeoff between fast state reconstruction and a reasonable state estimation
error. The nonlinear-gain high-gain observers adequately captured the transient and
steady-state performance seen in comparable linear-gain observers. Specifically, the
nonlinearity was chosen to have a higher observer gain during the transient period
and a lower gain afterwards, thus overcoming the tradeoff between fast state reconstruction and measurement noise attenuation. It appears as if altering the number
of piecewise linear regions in the nonlinear gain function do not have an appreciable
effect on the system performance, at least for the types of systems considered. Moreover, it was assumed that all assumptions hold globally, allowing the use of a converse
Lyapunov theorem of [39] in the proof. By a slight modification of the proof, it is
also possible to require the assumptions to hold only in a given region of the state
space and invoke the converse Lyapunov theorem of [7], which is a regional version
of the theorem of [39]. In particular, the first derivative of the exogenous signals, ς ,
˙
is required to be bounded.

49

Chapter 3
High-Gain-Observer Tracking
Performance in the Presence of
Measurement Noise
3.1

Introduction

Prior to this chapter, a significant amount of emphasis has been placed on the performance of the estimation error in the presence of measurement noise. Likewise, the
literature on high-gain observers subjected to measurement noise primarily focuses
on qualifying the system performance and quantifying potential bounds on the estimation error. However, there are many practical problems that seek to achieve more
than just stabilization; namely, path following and other goals that can be worked
into a tracking structure. This is not to suggest that the estimation error becomes a
moot point, however, given the observer estimates will be used in the feedback control. Yet, a greater insight into the challenges associated with measurement noise can
be gained by investigating the coupling between the observer design choices and the
effects on the tracking performance. For instance, simulation studies have suggested
50

that the effect of measurement noise on the tracking error is significantly less than
the effect manifested in the estimation error. If this can be shown mathematically,
even for just a special class of systems, designers may be able to achieve additional
leeway in constructing the high-gain observer and enhanced performance.
In order to explicitly show the importance and prevalence of this topic, Section
3.2 begins the discussion with a nonlinear example. Section 3.3 poses the questions
motivated by the previous section. Next, the class of systems investigated are defined
in Section 3.4. Before delving into the complexity that arises when dealing with
nonlinear systems, Section 3.5 investigates the effect of measurement noise on the
tracking error from a linear systems perspective. The result obtained from the linear
system analysis is extended to a class of nonlinear systems in Section 3.6, utilizing
the framework provided by ordinary differential equations and the unique properties
of multi-time scale systems. Ultimately, the tracking error is shown to be uniformly
bounded in the observer parameter ε.

3.2

Motivation

Consider the example found in [30]

x1 = x2
˙

(3.1)

x2 = x3 + u
˙
2

(3.2)

y = x1 + v ,
where the

(3.3)

xi ’s are the the system states, y the output, u the control and v the

measurement noise. The control objective is to have the state

x1 track a sinusoid

with an amplitude of 0.1 and a frequency of 0.3 rad/s. Using standard feedback
linearization techniques, the state feedback control is chosen as
51

u = −x3 − (x1 −
2

r) − (x2 − r) + r. However, only the first state is available for use in the controller.
˙
¨
Therefore, the output feedback control is constructed by replacing the system state

x2 with the estimate obtained from the high-gain observer defined as
2
˙
ˆ
x1 = x2 + (y − x1 )
ˆ
ˆ
ε
1
˙
x2 = 2 (y − x1 ) .
ˆ
ˆ
ε

(3.4)
(3.5)

Given the choice of output feedback, in order to prevent peaking in the plant during
the transient period, the controller is saturated outside [-1, 1]. The bounds on the
controller are chosen such that the saturation is not active under state feedback

x1 (0) = 0.1 and x2 (0) = x1 (0) =
ˆ
x2 (0) = 0. Note that x1 (0) and x1 (0) are deliberately chosen to be unequal
ˆ
ˆ
control. The initial conditions are set at

to depict a more realistic scenario. Additionally, when the initial conditions for the
system and observer differ, peaking is induced and appears in the transient response.
The measurement noise

v is generated using the Simulink block “Uniform Random

Number”, where the magnitude is limited to [-0.00011, 0.00011] and the sampling time
is set at 0.00005 seconds. In order to compare the effect the observer parameter
has on the estimation and tracking error, two separate trials are run with ε
and

ε

= 0.001

ε = 0.0005.

Figure 3.1 shows the steady-state response of the estimation error x2 − x2 for the
ˆ
linear observers. In particular, as the value of the observer parameter

ε is decreased,

the magnitude of the error significantly increases. However, the error in tracking
the reference signal, displayed in Figure 3.2, shows no appreciable change as
decreased. In fact, for values of ε

ε is

∈ [0.0005, 0.01], the steady-state response of the

tracking error is restricted to the range [−0.000029, 0.00011]. Hence, the tracking

error is uniformly bounded in ε. The same phenomenon is exhibited in linear systems
with measurement noise, and will be investigated in a subsequent section.
52

0.1

0.05

0.05

0

0

−0.05

Estimation Error

0.1

−0.05

−0.1
0

5

10
(a)

15

20

−0.1
0
Time

Figure 3.1: Steady-state response of the error x2
observer with (a) ε = 0.001 and (b) ε = 0.0005.

20

− x2 vs. time for a high-gain
ˆ

0

−0.05

15

0.05

0

10
(b)

0.1

0.05
Tracking Error

0.1

5

−0.05

−0.1
0

5

10
(a)

15

20

−0.1
0
Time

5

Figure 3.2: Steady-state response of the tracking error x2 −
high-gain observer with (a) ε = 0.001 and (b) ε = 0.0005.

53

10
(b)

15

20

x2 vs. time for a
ˆ

3.3

Tracking Performance

This section prepares the reader for the class of systems investigated, and poses the
questions motivated by Section 3.2. Referring back to the example in Section 3.2, the
simulation results suggest that the tracking error,

ε. Let’s investigate this claim further.

x1 − r, is uniformly bounded in

Unlike nonlinear representations, the transfer functions of a linear system can
reveal how the measurement noise impacts the tracking error. Before delving into the
details of a time domain analysis, consider the third-order system

x1 = x2
˙

(3.6)

x2 = x3
˙

(3.7)

x3 = a1 x1 + a2 x2 + a3 x3 + bu
˙

(3.8)

y = x1 + v ,
where the xi ’s are the system states,

(3.9)

y the output and u the control. The variable v

is the measurement noise. Tracking can be achieved by the state feedback controller

1
u = [−k1 (x1 − r) − k2 (x2 − r(1) ) − k3 (x3 − r(2) ) + r(3) ] ,
b
where

(3.10)

r(t) and r(j) (t) are the tracking signal and j th derivative of tracking signal,

respectively. The coefficients k1 , k2 , and k3 are chosen such that

s3 + (k3 − a3 )s2 + (k2 − a2 )s + (k1 − a1 )

(3.11)

is Hurwitz. The state estimates for the output feedback control are generated with
54

the linear high-gain observer

α
˙
ˆ
x1 = x2 + 1 (y − x1 )
ˆ
ˆ
ε
α2
˙
ˆ
x2 = x3 + 2 (y − x1 )
ˆ
ˆ
ε
α3
˙
x3 = 3 (y − x1 ) ,
ˆ
ˆ
ε
where the

(3.12)
(3.13)
(3.14)

αi ’s are designed such that
s 3 + α 1 s2 + α 2 s + α 3

(3.15)

is Hurwitz. The output feedback control is constructed by substituting the state
estimates, generated by (3.12)-(3.14), for the states

x that appear in (3.10). In this

discussion the control is not saturated, as typically employed to avoid peaking in the
plant during the transient response. In the case of tracking performance, the analysis
is concerned with the system behavior in steady-state, where the saturation is not
active.

Define the change of variables

e1 = x1 − r

e2 = x2 − r(1)

e3 = x3 − r(2)

for the tracking error.
55

(3.16)
(3.17)
(3.18)

The transfer functions from the noise to the tracking errors are

E
=
V

E1
V
E 
 2
V 
E3
V









H
 1
=  H2 
 
H3

,

(3.19)

where

∆
H1 = − 2
∆1
H2 = sH1
H 3 = s2 H 1
and

∆1 = ε3 s6 + (ε2 α1 − ε3 a3 )s5

+ (εα2 − ε2 a3 α1 − ε3 a2 )s4

+ (α3 − εa3 α2 − ε2 a2 α1 − a1 ε3 )s3

+ (−a1 ε2 α1 − a3 α3 − εa2 α2 + α1 ε2 k1 + α2 k2 ε + α3 k3 )s2
+ (−a2 α3 − a1 εα2 + α2 k1 ε + α3 k2 )s + (α3 k1 − a1 α3 )
∆2 = (α1 ε2 k1 + α2 k2 ε + α3 k3 )s2 + (α2 k1 ε + α3 k2 )s + α3 k1 .
The transfer functions H1 and H2 are two-frequency-scale transfer functions according to the definition of [40]. Therefore, their H∞ norms are of order O(1), i.e. they
are bounded uniformly in ε. The transfer function
because setting

H3 is not two-frequency-scale,

ε = 0 results in an improper transfer function. However, H3 can be

written as

1¯
H3 = H3 ,
ε
56

(3.20)

where

¯
¯
H3 is a two-frequency-scale transfer function. Hence, the H∞ norm of H3 is

O(1). This shows that the H∞ norm of H3 is O(1/ε). Furthermore, for any noise
that is an

O(µ).

L2 signal of order O(µ), the output will also be an L2 signal of order

Can a similar result be shown without the aid of frequency domain tools for linear
and nonlinear systems? What happens to the other states in the state vector
system of dimension

3.4

e for a

n? The next two sections seek to answer these questions.

Problem Formulation

Consider the class of nonlinear systems that can be represented in the form

x = Ax + B[bx (x) + au]
˙
y = Cx + v(t) ,
where x

(3.22)

∈ Rn is the system states, y ∈ R the measured output, u ∈ R the control

input and
and

(3.21)

v(t) ∈ R the measurement noise. The matrices A ∈ Rn×n , B ∈ Rn

C ∈ R1×n are defined as



0

0

.
A = .
.

0

0

 
0
1 ··· ··· 0
 

 0
0
1 · · · 0
 

... . , B = .
.
.
.
.
 

 0
· · · · · · 0 1
 

1
··· ··· ··· 0


and

C = 1 0 ··· ··· 0 ,
57

where it is assumed that n

≥ 2, which implies that the relative degree of the system

is greater than or equal to two. If this condition is not met, the construction of an
observer is unnecessary; the measured output
may not be known, and
measurable function,

y is simply used. The function bx (x)

a > 0. The measurement noise is assumed to be a bounded

|v(t)| ≤ µ. Given the purpose of this chapter is to investigate

the effects of measurement noise on the high-gain observer manifested in the tracking
error, the following change of variables is defined


x − r(t)
i
ei =
x − r(i−1) (t)
i

where

for
for

i=1

,

(3.23)

1<i≤n

r(t) and r(j) (t) are the tracking signal and its j th derivative. Then, ap-

plying the change of variables in (3.23) to the system (3.21)-(3.22) results in the
representation

e = Ae + B[b(t, e) + au]
˙
y = Ce + v(t) ,
where

(3.24)
(3.25)

b(t, e) replaces bx (x).

The state feedback controller takes the following form

u = −Ke ,

(3.26)

where the values of the vector K are chosen such that the origin of closed-loop system
is asymptotically stable. However, this does not exclude the possibility of employing a
nonlinear control technique. For instance, a continuously-implemented sliding mode
control is one type of nonlinear control scheme that will fit the controller form in
58

(3.26). For instance, the control

u = −β sat
takes the form

k1 e1 + k2 e2 + · · · + kn−1 en−1 + en
ρ

u = −Ke inside the boundary layer. It is appropriate to consider

the controller form inside the boundary layer, given the focus is on the steady-state
behavior of the tracking error.
The high-gain observer used to estimate the derivatives of the tracking error is
defined as

˙
e = Aˆ + H(y − C e)
ˆ
e
ˆ

(3.27)

and the gain is given by

α1 α2
αn
··· n
ε ε2
ε

HT =

.

Consider the scaled estimation error

e −e
ˆ
zi = i n−i i , 1 ≤ i ≤ n
ε

(3.28)

that when differentiated becomes

1
εz = A0 z + εBδ − B0 n−1 v ,
˙
ε
where



−α1

1

··· ··· 0



(3.29)



α1







 α2 
 −α2
0
1 · · · 0




 .
... . , B =  . 
. 
.
.
A0 =  .
 .
.

 0 


α
−α
· · · · · · 0 1

 n−1 
 n−1
αn
−αn · · · · · · · · · 0
59

(3.30)

and

δ = b(t, e) + au.

The output feedback controller takes the form

u = −K e = −Ke + KDz ,
ˆ
where

(3.31)

D = diag[εn−1 , εn−2 , · · · , 1]. Then, the closed-loop system under the

output feedback control (3.31) can be written in the following form

e = Ae + B[γ(t, e) + aKDz]
˙
1
εz = A0 z + εB[γ(t, e) + aKDz] − n−1 B0 v ,
˙
ε
where

(3.32)
(3.33)

γ(t, e) = b(t, e) − aKe.

To examine the dynamics of the “slow” system (3.32) separately from the dynamics
of the “fast” system (3.33), a decoupled form is necessary. Given the noise enters z in
˙
(3.33), and z enters e in (3.32), z should be removed from (3.32). If z is not removed
˙
from e, the noise entering
˙

z will lead to a more conservative bound on e. The next
˙

two sections address this concern as the bound on the tracking error is constructed.

3.5

Linear Systems Exploration

To adapt the general form given in (3.32)-(3.33) to a class of linear systems, we
require that b(t, e)

= b1 (t) + θT e be linear in e and γ(t, e) = b1 (t) + γ e, where
¯

γ = θT − aK . In preparation for the change of variables that will decouple the
¯
differential equations, write (3.32)-(3.33) as

e = A11 e + A12 z + Bb1 (t)
˙
1
εz = A21 e + A22 z + εBb1 (t) − n−1 B0 v ,
˙
ε
60

(3.34)
(3.35)

where



A11





= A + B¯ = 
γ




0

A21





= εB¯ = ε 
γ




1

0

···

···

0

0

...

.
.
.

0

.
.
.

1

···

θ1 − ak1 θ2 − ak2 · · · · · · θn − akn





= BaKD = 





0

···

0

0



A12

1



0
0

0
···

0

···

εn−1 ak1 εn−2 ak2 · · ·
0

0

0

.
.
.

···

··· ···

··· ···

···

0

0




0 

. 
. ,
.

0
0 

· · · akn

0


0


.
.
 and
.


0

θn − akn

··· ···

.
.
.

0

0

··· ···

0





,




θ1 − ak1 θ2 − ak2 · · · · · ·


−α1
1
0 ···
0



−α2
0
1 ···
0 




.
.
...
.
.
A22 = A0 + εBaKD = 
.
.
.


 −α
···
··· 0
1 
n−1


−αn + εn ak1 εn−1 ak2 · · · · · · −εakn

In order to remove the “fast” dynamics from the “slow” system, and vis versa, the
singularly perturbed system (3.34)-(3.35) is transformed into a block-diagonal form
provided in [35]; see Appendix B for further details. The change of variables, that

61

will provide the desired setup is

ψ = e − εM φ
εφ = εz + εL(ε)e
and exist for matrices

(3.36)
(3.37)

M and L that satisfy the linear algebraic equations

0 = ε(A11 − A12 L)M − M (A22 + εLA12 ) + A12

(3.38)

0 = A21 − A22 L + εL(A11 − A12 L),

(3.39)

respectively. Alternatively, (3.38)-(3.39) can be written as

0 = ε(A + BθT − BaK − BaKDL)M
− M (A0 + εBaKD + εLBaKD)

(3.40)

+ BaKD
0 = −εBaK−(A0 +εBaKD)L+εL(A+BθT −BaK−BaKDL), (3.41)
and is the form that will be used for all subsequent derivations. Rewrite (3.40) as

M A0 = BaKD + ε(A + BθT − BaK − BaKDL)M
− εM (I + L)BaKD
and bring

A0 to the right-hand side

M = BaKDA−1 + ε(A + BθT − BaK − BaKDL)M A−1
0
0
− εM (I + L)BaKDA−1 .
0

62

(3.42)

Hence, for sufficiently small ε, (3.42) is a contraction mapping. Therefore, we can
solve for

M using successive approximations [30]. Define

Mk+1 = BaKDA−1 + ε(A + BθT − BaK − BaKDL)Mk A−1
0
0
− εMk (I + L)BaKDA−1 .
0

According to [35], after

(3.43)

k iterations the exact solution M is approximated to within

O(εk ) error. Differentiating (3.36) yields
˙
˙
ψ = e − M (εφ)
˙
1
= (· · · ) − n−1 M B0 v .
ε
Therefore, we need

M B0 = O(εn−1 ) to eliminate the negative powers of ε from

˙
ψ . Note the following properties



0 0 ··· ···

1
− αn



α1





−1





  
 1 0 · · · · · · − α1   α 2   0 
αn  

  
α2   . 
−1 B = 
.  =  . ,
A0 0 0 1 0 · · · − α   .
 . 
.
n 
.
  . 
.
...
.
 α
  . 
.
.
.
  n−1   . 
α
0 0 0
1 − n−1
αn
0
αn


0



 
−1
 

−2 B = 
A0 0  0 
 . 
 . 
 . 
0

63

and



As the matrix

εn−1

··· ··· 0

0
εn−2


 0

−1 B =  .
DA0 0  .
.

 0

0




· · · 0

.
...
..
.

· · · ε 0

··· 0 1
0

···
···

A0 is raised to progressively higher powers and multiplied by B0 , the

only nonzero element in the product moves down the vector.
Now, show that

M B0 = O(εn−1 ) with the following argument:

M0 = (· · · )DA−1
0

M1 = (· · · )DA−1 + ε(· · · )DA−2
0
0

M2 = (· · · )DA−1 + ε(· · · )DA−2 + ε2 (· · · )DA−3
0
0
0

M3 = (· · · )DA−1 + ε(· · · )DA−2 + ε2 (· · · )DA−3 + ε3 (· · · )DA−4
0
0
0
0
−(n−1)

Mn−2 = (· · · )DA−1 + ε(· · · )DA−2 + · · · + εn−2 (· · · )DA0
0
0
+ εn−1 (· · · ) .

Then,

M B0 = (· · · )DA−1 B0 + ε(· · · )DA−2 B0 + · · ·
0
0
−(n−1)

+ εn−2 (· · · )DA0

B0 + εn−1 (· · · )B0

= Mn−2 B0 + O(εn−1 )

and

DA−i B0 = O(εn−i )
0
implying that

M B0 = O(εn−1 ) .
This shows that

ψ is O(µ).
64

(3.44)

Alternatively, we have

M0 = B(· · · )
M1 = B(· · · ) + εAB(· · · )

M2 = B(· · · ) + εAB(· · · ) + ε2 A2 B(· · · )

Mn−2 = B(· · · ) + εAB(· · · ) + · · · + εn−2 An−2 B(· · · )
+ εn−1 (· · · ) ,

where

 
 
 
0
0
0
.
.
.
.
.
.
.
.
.
 
  2
 
B = 0 , AB = 0 , A B = 1
 
 
 
 0
1 
 0
 
 
 
1
0
0

 
0
 
1 
 
n−2 B =   .
and A
0 
.
.
.
0

Then,

M = Mn−2 + O(εn−1 )
and

As a consequence of linearity,



O(εn−1 )





O(εn−2 )




.
.
M =

.


 O(ε) 


O(1)
φ=O

µ
εn−1
65

.

.

(3.45)

(3.46)

Then, combining (3.45) and (3.46) results in



O(εµ)

 O(µ)


µ
εM φ =  O

ε

.
.

.

µ
O n−2
ε
Given












.

(3.47)

ψ is O(µ), we can see that both e1 and e2 are O(µ), while the other com-

ponents of

e are of the order O(µ/εi ) with increasing powers of i.

Theorem 3.1: Consider the closed-loop system in (3.32)-(3.33), where
assumed to be linear in e. Then, for a linear system of dimension

b(t, e) is

n, the tracking

error and subsequent derivatives satisfy

|ei (t)| = O(µ), ∀i = 1, 2
µ
|ei (t)| = O i−2 , ∀2 < i ≤ n
ε
for

(3.48)
(3.49)

v ≤ µ.

3.6

Nonlinear Systems Extension

To isolate the dynamics of the tracking error, the decomposition method proposed in
[49] is utilized to eliminate the fast states, z , from the slow equation (3.32). In general,
the decomposition is valid for singularly perturbed differential systems. The goal of
the change of variables is to transform the system (3.32)-(3.33) into the following
66

form

˙
ψ = F (t, ψ, ε)
˙
εφ = G(t, ψ, φ, ε) .

(3.50)
(3.51)

The change of variables that achieves the transformation (3.50)-(3.51) is found by
setting

v = 0 in (3.32)-(3.33) and following the method of [49]; see Appendix C.

Note that the change of variables is actually applied to (3.32)-(3.33) for

v = 0. Let

f (t, e, z, ε) = Ae + B[γ(t, e) + aKDz]

(3.52)

g(t, e, z, ε) = A0 z + εB[γ(t, e) + aKDz] .

(3.53)

The integral manifold is defined as

z = h(t, e, ε); see Appendix C for a definition

of an integral manifold. A valid expression for

h(t, e, ε) is found by satisfying the

equation

ε

∂h
∂h
+ ε f (t, e, h, ε) = g(t, e, h, ε)
∂t
∂e

or, equivalently,

ε

∂h ∂h
+ε [Ae+Bγ(t, e)+BaKDh] = A0 h+εB[γ(t, e)+aKDh] . (3.54)
∂t
∂e

Furthermore, the function

h can be represented as a Taylor series in ε, i.e. h =

h0 (t, e) + εh1 (t, e) + · · · . The expressions for the hi ’s can be found by substituting the expansion for h into the partial differential equation (3.54). Matching the
coefficients of like powers of ε, it can be shown that the first set of coefficients are

h0 = 0
h1 = −A−1 Bγ(t, e) ,
0
67

where
Let

Ei is defined as an n × n diagonal matrix, whose ith diagonal element is 1.

φ = z − h(t, e, ε) and w = e − ψ . Then,

˙
ψ = Aψ + B[γ(t, ψ) + aKDh(t, ψ, ε)]

(3.55)

f (t, ψ, h(t, ψ, ε), ε)
w = f (t, ψ + w, φ + h(t, ψ + w, ε), ε) − f (t, ψ, h(t, ψ, ε), ε)
˙
= Aw + B[γ(t, ψ + w) − γ(t, ψ)

(3.56)

+ aKD[φ + h(t, ψ + w, ε) − h(t, ψ, ε)]]
f1 (t, ψ, w, φ, ε)
˙
εφ = g(t, ψ + w, φ + h(t, ψ + w, ε), ε)
− g(t, ψ + w, h(t, ψ + w, ε), ε)

∂h(t, ψ + w, ε)
[f (t, ψ + w, φ + h(t, ψ + w, ε), ε)
∂e
−f (t, ψ + w, h(t, ψ + w, ε), ε)]

−ε

= A0 + εBaKD − ε

∂h
BaKD φ
∂e

Z(t, ψ, w, φ, ε) .
Define

F (t, ψ, ε) = f (t, ψ, h(t, ψ, ε), ε)
= Aψ + Bγ(t, ψ) + BaKDh(t, ψ, ε)
G(t, ψ, φ, ε) = Z(t, ψ, εH(t, ψ, φ, ε), φ, ε)
= A0 + εBaKD + ε

68

∂h(t, ψ + εH, ε)
BaKD φ
∂e

(3.57)

introduced in (3.50)-(3.51). According to [49],

0=ε

H(t, ψ, φ, ε) satisfies the equation

∂H
∂H
∂H
+ε
F (t, ψ, ε) +
Z(t, ψ, εH, φ, ε)
∂t
∂ψ
∂φ

(3.58)

− f1 (t, ψ, εH, φ, ε)
H = H1 + εH2 + ε2 H3 + · · · , where Hi =
Hi (t, ψ, φ). Expanding the function f1 as a power series in ε results in

and has the asymptotic expansion

1
0
f1 (t, ψ, εH, φ, ε) = f1 (t, ψ, φ) + εf1 (t, ψ, φ) + · · · .
0
Solving for the coefficient f1 yields the expression
0
f1 (t, ψ, φ) = f1 (t, ψ, 0, φ, 0) = aBKEn φ .
1
Solving for the coefficient f1

∂f1 (t, ψ, 0, φ, 0)
∂f (t, ψ, 0, φ, 0)
H1 + 1
∂w
∂ε
∂γ(t, ψ)
∂h(t, ψ, 0)
= A+B
H1
+ BaKEn
∂e
∂e

1
f1 (t, ψ, φ) =

+ BaKEn−1 φ .
However, consider that

h(t, e, ε) = εh1 (t, e) + ε2 h2 (t, e) + · · ·
∂h(t, e, ε)
∂h
∂h
= ε 1 + ε2 2 + · · ·
∂e
∂e
∂e
∂h(t, e, 0)
=0.
∂e

69

(3.59)

1
Further simplifying the right-hand side of f1
1
f1 (t, ψ, φ) = A + B

∂γ(t, ψ)
H1 + BaKEn−1 φ .
∂e

(3.60)

2
Next, the coefficient f1 requires the following calculations
2
f1 (t, ψ, φ)

1 d2 f1 (t, ψ, εH(t, ψ, φ, ε), φ, ε)
=
2
dε2
ε=0
= AH2 + BaKEn−2 φ
1 d2
+ B 2 [γ(t, ψ + εH) + aKDh(t, ψ + εH, ε)
2 dε
−aKDh(t, ψ, ε)]

Realize that

h(t, ψ + εH, ε) − h(t, ψ, ε) = ε[h1 (t, ψ + εH) − h1 (t, ψ)]

+ ε2 [h2 (t, ψ + εH) − h2 (t, ψ)] + · · ·

is a valid expansion, where

h0 = 0. Then,

h1 (t, ψ + εH) = h0 (t, ψ + εH) + εh1 (t, ψ + εH) + · · ·
1
1
= h1 (t, ψ) +

∂h1 (t, ψ)
H1 + · · ·
∂e

leading to the conclusion that

h(t, ψ + εH, ε) − h(t, ψ, ε) = ε2

70

∂h1 (t, ψ)
H1 + · · · .
∂e

With a few more calculations

dγ(t, ψ + εH) dγ(t, ψ + εH1 + ε2 H2 + · · · )
=
dε
dε
∂γ(t, ψ + εH1 + ε2 H2 + · · · )
(H1 + 2εH2 + · · · )
=
∂e
and

d2 γ
∂γ(t, ψ)
T
H2 ,
= H1 Γh (t, ψ)H1 + 2
∂e
dε2 ε=0
where Γh is the Hessian matrix of γ with respect to e. Ultimately,
2
f1 (t, ψ, φ) = AH2 + BaKEn−2 φ

∂γ(t, e)
∂h (t, ψ)
+B
H2 + aKEn 1
H1
∂e
∂e

(3.61)
.

The remaining terms in the asymptotic expansion of f1 can be found in a similar
manner.

Return to (3.58), substitute in the expansion for

H(t, ψ, φ, ε) and solve for the

Hi ’s by matching the coefficients of like powers of ε. To obtain the expression for
H1 , gather the coefficients of (3.58) that do not contain ε and set that equation to
zero. For the remaining

Hi ’s, gather the coefficients that contain εi−1 and set the

resulting equation to zero to solve for

0=
where the coefficients without
for

Hi . For H1

∂H1
A φ − BaKEn φ ,
∂φ 0
ε are included, and the equation set to zero. Solving

H1
H1 = BaKEn A−1 φ .
0
71

(3.62)

For

H2
∂H2
A0 φ − ABaKEn A−1 φ
0
∂φ
∂γ(t, ψ)
BaKEn A−1 φ
−B
0
∂e
− BaKEn−1 φ ,

0 = BaKEn A−1 BaKEn φ +
0

where the coefficients of

ε are included and the equation set to zero. Solving for H2
∂γ(t, ψ)
BaKEn A−1
0
∂e
+ BaKEn−1

H2 = [ABaKEn A−1 + B
0

(3.63)

− BaKEn A−1 BaKEn ]A−1 φ .
0
0
The coefficients of ε2 set to zero are

0=

∂H2 ∂H2
+
[Aψ + Bγ(t, ψ)]
∂t
∂ψ
∂H3
∂H2
+
A0 φ +
BaKEn φ
∂φ
∂φ
∂H1
∂h (t, ψ)
+
BaKEn φ
BaKEn−1 + 1
∂φ
∂e
− AH2 − BAKEn−2 φ

∂γ(t, ψ)
1 T
H1 Γh (t, ψ)H1 +
H2
2
∂e
∂h (t, ψ)
H1 ,
+aKEn 1
∂e

−B

where

∂
∂γ(t, ψ)
∂H2
=
BaKEn A−2 φ ,
B
0
∂ψ
∂ψ
∂e

72

(3.64)

∂γ
∂γ
,
B=
∂e
∂en
∂ ∂γ(t, ψ)
∂H2
aKEn A−2 φ
=B
0
∂t
∂t
∂en
and

aKEn A−2 φ is a scalar. Substituting in the values above into (3.64) reveals
0

that all terms, except one, have the vector

φ only on the very right most side of the

expression. The term that is the exception is

1
− BφT A−T En K T aB T Γh (t, ψ)BaKEn A−1 φ ,
0
0
2
where the vector

(3.65)

φ appears in two locations. Aside from (3.65), the vector φ always

appears paired with

aKEn A−2 and becomes a scalar quantity; dividing both sides
0

of the equation by the scalar quantity eliminates

φ from all terms except (3.65). To

solve for H3 , the integral, with respect to φ, is taken of (3.64). However, the solution
supplied by the term in (3.65) is unclear. To eliminate (3.65) from (3.64), notice that

BT Γ
and require that the function

∂ 2 γ(t, ψ)
h (t, ψ)B =
∂e2
n

b be linear in en . This requirement implies that
∂ 2b
=0.
∂e2
n

Then, the term (3.65) is zero. For higher-order systems, more complicated terms like
(3.65) appear. However, there is no obvious way in which to eliminate their presence,
and solve for the remaining
focus on the case for

Hi ’s. Therefore, the remainder of this discussion will

n = 3, where H3 will not appear in the expansion of H and

no additional restrictions are placed on the system structure.

73

According to [49],

e = ψ + εH(t, ψ, φ, ε)

(3.66)

¯
z = φ + εh(t, ψ + εH(t, ψ, φ, ε), ε)

(3.67)

exists for choices of H and h that satisfy the partial differential equations (3.58) and
(3.54), respectively. Moreover, for (3.32)-(3.33), the right-hand side of (3.67) can be

¯
εh instead of h, because h0 = 0. Hence, the inverse transformation
must also take the form of an O(ε) perturbation, namely
written with

ψ = e + εQ(t, e, z, ε)

(3.68)

φ = z + εq(t, e, z, ε) .

(3.69)

Taking the derivative of (3.68) and (3.69) yields

∂Q
∂Q
∂Q
˙
+ε
e+
˙
z
˙
ψ =e+ε
˙
∂t
∂e
∂z
= F (t, ψ, ε)
∂q
∂q
∂q
+ ε2 e + ε z
˙
˙
∂t
∂e
∂z
= G(t, ψ, φ, ε) ,

(3.70)

˙
εφ = εz + ε2
˙

where

v=0

(3.71)

v = 0 in the expressions for e and z in (3.32) and (3.33), respectively. For
˙
˙
1 ∂Q
˙
B v
ψ = F (t, ψ, ε) − n−1
∂z 0
ε

(3.72)

∂q
1
˙
εφ = G(t, ψ, φ, ε) − n−1 I +
B0 v
∂z
ε

(3.73)

which reveals that the fast states are eliminated from the slow equation, and the

74

presence of measurement noise adds an additional term multiplied by the noise.

For a system with dimension

n = 3,

H = H1 + εH2 + ε2 HR ,
where

(3.74)

HR = HR (t, ψ, φ, ε) and is O(1). The change of variables in (3.66)-(3.67)

can be written as

e = ψ + εH1 + ε2 H2 + ε3 HR
= ψ + εBaKE3 A−1 φ
0
+ ε2 ABaKE3 A−1 + B
0

∂γ(t, ψ)
BaKE3 A−1
0
∂e

(3.75)

+BaKE2 − BaKE3 A−1 BaKE3 A−1 φ
0
0
+ ε3 (· · · )
¯
z = φ + εh(t, ψ + εH(t, ψ, φ, ε), ε)
= φ + εh1 (t, ψ + εH) + ε2 h2 (t, ψ + εH) + ε3 (· · · )
= φ + εh1 (t, ψ) + ε2
Applying (3.76) to (3.33), for

(3.76)

∂h1 (t, ψ)
H1 + ε2 h2 (t, ψ) + ε3 (· · · ) .
∂e

v = 0, results in

1
εz = A0 z + εB[γ(t, e) + aKDz] − 2 B0 v
˙
ε
∂h1 ∂h1 ˙
˙
ψ
+
= εφ + ε2
∂t
∂ψ
∂h
˙
+ ε2 1 BaKE3 A−1 (εφ) + ε3 (· · · )
0
∂e
75

(3.77)

and solving for

εφ yields
˙
εφ = − I

where

P = ε2

∂h (t, ψ)
BaKE
+ ε2 1
∂e

−1
3 A0

−1

P ,

(3.78)

∂h1 ∂h1 ˙
ψ − A0 z − εB [γ(t, e) + aKDz]
+
∂t
∂ψ
1
+ 2 B0 v + ε3 (· · · ) .
ε

The term of interest contains the measurement noise, and can be represented as

−I + ε2

∂h1 (t, ψ)
1
BaKE3 A−1 + ε3 (· · · ) 2 B0 v ,
0
∂e
ε

where the matrix identity
that

(3.79)

(I + L)−1 = I − L + L2 − L3 + · · · is used. Notice

E3 A−1 B0 = [0 0 · · · 0]T . Hence, (3.79) reduces to
0
1
− 2 B0 v + ε(· · · )v
ε

showing that

φ=O
Move

µ
εn−1

=O

µ
ε2

.

(3.80)

˙
ψ to the left-hand side and take the derivative of (3.75). Then, according to

(3.72),

1
˙
ψ = F (t, ψ, ε)+ 2 BaKE3 A−1 B0 v
0
ε
1
∂γ(t, ψ)
BaKE3 A−1
ABaKE3 A−1 + B
+
0
0
ε
∂e
+BaKE2 − BaKE3 A−1 BaKE3 A−1 B0 v
0
0

1
+ ε2 (· · · + 2 B0 v + · · · ) .
ε
76

(3.81)

˙
The first two terms in (3.81) with εφ are eliminated, due to the property E3 A−1 B0
0

=

E3 A−2 B0 = E2 A−1 B0 = 0. However, the ε2 term does not possess this prop0
0
erty and remains present in (3.81). Given the origin of the closed-loop system in
(3.32)-(3.33) is designed to be exponentially stable and the last term in (3.81) is

O(µ), ψ = O(µ). Repeating the argument from Section 3.5,

 
 
0
0
 
 
B = 0 , AB = 1
 
 
0
1

 
1
 
and A2 B = 0
 
0

showing that raising the A matrix to a power determines which terms of (3.75) enter
each component of the

e vector.

Proposition 3.1: For a closed-loop nonlinear system of the form (3.32)-(3.33), with
bounded measurement noise

v ≤ µ, the tracking error, first derivative of the track-

ing error and second derivative of the tracking error are of the following orders of
magnitude

e1 = ψ1 + ε3 (· · · ) = O(µ)

(3.82)

e2 = ψ2 + ε2 (· · · )φ + ε3 (· · · ) = O(µ)
µ
e3 = ψ3 + ε(· · · )φ + ε3 (· · · ) = O
,
ε

(3.83)
(3.84)

where the expression in (3.75) is used.

Moreover, the relationship shown in (3.48)-(3.49) for the linear system explored
in Section 3.5 holds for a nonlinear system of dimension
77

n = 3.

3.7

Conclusions

When high-gain observers are employed in the presence of measurement noise, there
exists a tradeoff between fast state reconstruction and a reasonable state estimation
error. However, this sort of compromise does not exist when the primary interest
is in the system tracking error. It was argued by constructing the system transfer
functions from the noise to the tracking error and its derivatives, that the error and
its first derivative are bounded uniformly in ε. Using singular perturbation analysis,
the results were extended to a class of linear systems of dimension

n. After the

tracking error and its first derivative, all remaining derivatives of the tracking error
are inversely proportional to increasing powers of ε. Subsequently, a similar result
was derived for a class of nonlinear systems using the special features of singularly
perturbed systems, further generalizing the results reported for linear systems. Due
to the form of the nonlinearity, the result for nonlinear systems is restricted to a
third-order system.
Although it has been shown that the tracking error is more immune to the effects of
measurement noise than the estimation error for both the linear and nonlinear forms
considered, this does not mean ε can be made arbitrarily small. It is important to keep
in mind that the estimates of the states will still be used in the controller. However,
the control may be able to tolerate a larger amount of error than the state estimates,
providing some additional flexibility in choosing the value of
focus.

78

ε when tracking is the

Chapter 4
Enhancing High-Gain Observer
Performance with Wavelet
Denoising
Using wavelets to extract various signal components for compression, feature detection
and denoising is a common approach in offline signal processing; see [17, 18, 36, 42]
and the references therein. Recently, the techniques for signal denoising offline have
been extended to produce acceptable results for online systems, [10, 34].
The goal of this chapter is to explore the role of wavelet denoising in improving
the performance of high-gain observers subjected to measurement noise. In order
to provide a basis for understanding the complexity of wavelets, the chapter begins
with a brief introduction in Section 4.1. Section 4.2 is intended to familiarize the
reader with the issues commonly addressed in designing a denoising algorithm with
wavelets. The discussion in Section 4.2 is extended to include the intricacies associated
with a real-time wavelet filter in Section 4.3. A simulation investigating the design
parameters, their varying success in denoising the measurement and a comparison
with the traditional lowpass filter is provided in Section 4.4. The last section addresses
79

directions for future work.

4.1

A Wavelet Introduction

Often times, we ignore the existence of noise in our systems to live in an idealized
world where control algorithms are simplified, proofs are elegant and assumptions
are abundant. However, a system with no noise is hardly a realistic scenario. When
noise is accounted for, it is generally assumed to have some special characteristics
that allow us to apply a particular type of filter (i.e. Kalman, lowpass, bandpass,
etc.). Yet, what if the noise is not so easily compartmentalized? The noise may
not be localized within a particular bandwidth, or may occupy the same frequency
space as the signal. When this occurs, traditional denoising techniques only utilizing
the temporal or frequency data cannot successfully remove the presence of noise
without causing attenuation of the desired signal. Moreover, if an algorithm could
localize in time and frequency where the noise occurs, a more thorough removal of
the undesirable signal components can be accomplished. Frequently, the noise is not
obviously periodic, and a non-stationary approach is necessary. This leads to the
introduction of wavelets.
Wavelets, like Fourier analysis, provide another domain in which to analyze signals. In Fourier analysis, a Fourier transform is used to decompose the signal into its
frequency components. Similarly, a wavelet can also be used to transform a signal into
the frequency domain. The primary difference is that a wavelet transform provides
the concentration of frequencies at each time instant. To follow the development of
the wavelet transform from the Fourier perspective, see [42]. Another mathematically
formal history of wavelets can be found in [14]. Some concerns that should be addressed when approaching the task of denoising with wavelets include how the noise
enters the system (additive, multiplicative), the noise profile (smooth, erratic, impul80

sive, etc.) and the signal to noise ratio (SNR), see [54] and [9]. Moreover, it is not
necessarily trivial to find the optimal wavelet and denoising technique pair; however,
it can be done for certain classes of problems, for instance [36].

4.1.1

The Anatomy of a Wavelet Transform:
Continuous-Time

A wavelet system is composed of two components: the wavelet function, known as
the mother wavelet and the scaling function. The scaling function is used to shift,
compress and stretch the mother wavelet. In particular, the wavelet expansion is
used to transform the signal into a domain where both time and frequency are localized. Moreover, the representation of the signal is compressed into a few wavelet
(expansion) coefficients. Then, these altered versions of the mother wavelet paired
with the transform coefficients can be used to reconstruct the original signal. This
idea is analogous to the the Fourier series using sinusoids to build a desired signal.
Unlike the expansion set (sinusoids) for the Fourier series, there is no definite form for
the mother wavelet function. In fact, there is an infinite number of choices available.
This chapter will focus on the most commonly used wavelets to cleanly capture
the signal multi-scale behavior; Haar, Daubechies, Coiflets and Symlets. The Haar
wavelet is part of the well-known and commonly used Daubechies family and is equivalent to the Daubechies order 1 transform. Some properties beyond simplicity, in the
case of the Haar wavelet, that make these wavelets prime candidates for signal denoising, is that all are compactly supported, orthogonal and have discrete implementations. Moreover, the transforms not only conserve the signal energy, but compress
it into a small number of coefficients; this is an important characteristic in the application of denoising. Furthermore, the least amount of asymmetry appears in the
Symlet wavelets. The discrete wavelet transform contains the mother wavelet
81

Ψj,k (t) = 2j/2 Ψ(2j t − k)

(4.1)

Φj,k (t) = 2j/2 Φ(2j t − k) ,

(4.2)

and the scaling function

where

j ∈ Z and k ∈ Z. The variables j and k are used to manipulate the

mother wavelet. From a practical standpoint, the value of
of the signal (as interpreted by the scaling function). For

j controls the resolution

j > 0 the scaling function

becomes narrower; this translates to capturing smaller details of the signal under the
transform. Conversely, for

j < 0 the information provided by the wavelet transform

is coarser. The notion of multiple scales is at the heart of what makes wavelets
so attractive for denoising and compression applications. The fact that the signal
components can be separated, and represented on different time scales, allows us to
single out the various frequencies occurring at different times. The variable

k simply

shifts the functions to cover the entire signal space. Specifically, the Haar mother
wavelet and the scaling functions are, respectively,

and


 1, 0 ≤ t < 1



2

1
Ψ(t) = −1, ≤ t < 1

2



0, else

Φ(t) =


 1, 0 ≤ t < 1
0, else

(4.3)

.

(4.4)

A closed-form expression is not available for all wavelet families, or necessarily as
82

easy to express in a clear fashion. For illustration and ease of understanding, the
expression for the Haar wavelet is provided.
Hence, the above functions can be used to form an orthonormal and compact
support basis for the signal of interest. The original signal

f (t) can be represented

as

f (t) =

∞

aj0 (k)Φj0 ,k (t) +

k=−∞

∞

∞

dj (k)Ψj,k (t) ,

(4.5)

k=−∞ j=j0

where the integer j0 dictates the coarsest scale whose space is spanned by the scaling
function; in general, the choice of j0 depends on the signal itself and the desired

f (t) always refers to the output signal y

resolution. In the context of this work,

corrupted by measurement noise. The approximation or average (low frequency)
coefficients are

aj0 (k) =

∞
−∞

f (t)Φj0 ,k (t)dt

(4.6)

and the fluctuation or detail (high frequency) coefficients are

dj (k) =

∞
−∞

f (t)Ψj,k (t)dt .

(4.7)

It should be noted that the above notation is intended for a discrete wavelet, even
though the function itself is piecewise continuous. In general, discrete wavelets are not
actually discrete in terms of the time variable. Rather, the dilation and translation
effects are discrete. The analysis and synthesis equations listed above assume that
the signal is infinite, and that all calculations are performed offline. Naturally, a finite
and truly discrete model is necessary to implement this in real-time and digitally.
83

4.1.2

The Anatomy of a Wavelet Transform: Discrete-Time

We will now proceed to discuss the discrete-time wavelet implementation, focusing on
the Haar wavelet. However, the process for Daubechies order 4 wavelet is similar. As
the above discussion implies, the information contained in the signal is encapsulated
by the wavelet transform coefficients (4.6) and (4.7), and the basis provided by the
functions shown in (4.1) and (4.2). The discrete-time representations for the Haar
wavelet approximation and detail coefficients are, respectively,

a(n) =

ˆ
ˆ
f (2n − 1) + f (2n)
√
2

(4.8)

and

ˆ
ˆ
f (2n − 1) − f (2n)
√
,
d(n) =
2
where

(4.9)

N
ˆ
f is the sampled signal with a sampling period of T , and n ∈ [1, ] where
2

N is the total number of samples. The inverse mapping is


 a(n) + d(n)

√

, ∀n odd
2
ˆ(n) =
f
−
 a(n)√ d(n) , ∀n even


2

.

(4.10)

At this point, we could easily take (4.8) through (4.10) and construct the necessary
matrices to transform the desired signal into the time-frequency (“wavelet”) domain,
and back into the time-domain. However, the expressions for the approximation and
detail coefficients given in (4.8) and (4.9) are generally not that simplistic nor informative. We seek to divide the coefficients into separate frequency scales, where
the signal is separated into its frequency subbands via a filter bank. The perspective gained from multiresolution analysis, via filter banks, will subsequently provide
84

invaluable insight when choosing a denoising scheme.
Typically, filter banks are chosen to implement discrete wavelet transforms, because they provide a relatively fast computation time (O(N ) complexity), reduced
data storage and convenient signal processing interpretation. With filter banks in
mind, the average coefficients are now represented as a convolution

ˆ l[2n]
a(n) = f ∗ ¯

(4.11)

between the signal of interest and the lowpass filter ¯ The detail coefficients
l.

ˆ ¯
d(n) = f ∗ h[2n]
are found via a convolution between the signal and the highpass filter

(4.12)

¯
h. Moreover,

both sets of coefficients are down-sampled by 2, given the process of filtering with both
a lowpass and highpass filter creates twice the information necessary to reconstruct
the original signal. These analysis filters are wavelet transform dependent, and are
constructed using the mother wavelet (4.1) and scaling (4.2) functions defined above.
Specifically, the lowpass filter in (4.11) is calculated from

Φ(t) =
m
where

√
l(m) 2Φ(2t − m) ,

(4.13)

¯ = l[−n], and the highpass filter in (4.12) is calculated from
l[n]
Ψ(t) =
m

√
¯
h(m) 2Φ(2t − m) ,

(4.14)

√

√
√ √
2 2
2 2
¯
where h[n] = h[n]. For a Haar filter l = [
,
] and h = [− ,
].
2 2
2 2
Thus, the implementation of the discrete wavelet transform can be viewed as nothing
more than FIR filter design, generally referred to as conjugate mirror filters. The
85

reconstructed signal is given by

ˆ
f (n) = a ∗ ˜ + d ∗ h[n] .
ˇ l[n] ˇ ˜
The coefficients (4.11) and (4.12) are up-sampled by 2, resulting in

(4.15)

ˇ
a and d, reˇ

spectively. The details concerning the construction of lowpass and highpass synthesis

˜
filters ˜ and h, along with the necessary and sufficient conditions on the analysis and
l
synthesis filter pairs, is available in [42]. For this discussion,√ synthesis filters for
the
√
√ √
a Haar transform are

˜
˜ = [ 2 , 2 ] and h = [ 2 , − 2 ]. These filter pairs
l
2 2
2
2

allow for a perfect reconstruction of the original signal, barring any intentional manipulation of the wavelet coefficients in between applying the analysis and synthesis
filters.
Lifting is another widely used method. Lifting seeks to improve the wavelet properties, in the context of perfect reconstruction filters (the methodology used above). It
can be used to both design the form of the wavelets, in addition to implementing the
discrete wavelet transform. Lifting tends to generate filter implementations that result in a faster runtime, when compared to other approaches. In theory, every perfect
reconstructible filter bank can be expressed in terms of lifting, [27,42]. In the context
of this work, we did not employ lifting techniques, given perfect reconstruction filters
yielded satisfactory results.

4.2

Denoising: Offline

The material covered thus far is a small portion of the entire picture. Unfortunately,
the above ideas in isolation do not guarantee a successful implementation of a realtime discrete-time wavelet transform. Thus, before tackling the construction of a realtime filter, we first address the components associated with wavelet denoising that
are necessary for both offline and real-time denoising applications. In this section,
86

we will temporarily neglect the issues that arise when implementing the filter in realtime. Details not covered here on wavelet denoising, and wavelet packet denoising,
are discussed in [15].

4.2.1

Wavelet Type

The first step in developing a wavelet denoising algorithm, is to settle on a wavelet
type. There are a plethora of mother wavelets and scaling functions available; in some
cases, you may even want to create your own. The more common wavelet families
include Daubechies, Symlets, Coiflets, Meyer and other biorthogonal functions, to
name a few. For a more lengthy discussion on the various wavelet families available
and their numerous properties, see [9, 14, 42]. Although we have focused on the first
three transform families in this work, that does not preclude the inclusion of another
type of wavelet. In fact, a different choice may achieve a smaller bound on the error in
steady-state than what can be achieved with the wavelets presented. However, there
will most likely be a tradeoff in the complexity. The authors of [11, 41, 51] suggest a
number of tools that can help determine the best wavelet for a given system. However,
only [41] and [11] also consider whether or not the choice will result in effective
denoising.
The type of wavelet function is heavily dependent on the type of noise in the
system, and potentially the signal itself. If the goal is to remove short and sporadic
noise, a wavelet function that captures this type of behavior is more appropriate. For
noise that is relatively smooth and omnipresent, a continuous and sufficiently smooth
wavelet is better suited, [9] and [54].
87

4.2.2

Wavelet Transform Levels

The number of levels is the number of times the signal is concurrently subjected to the
wavelet transform. For instance, recall the filter bank implementation of the discrete
wavelet transform. Initially, the output is simultaneously passed through a lowpass
(L) and highpass (H) filter, then down-sampled. This generates the level-1 coefficients. If the approximation coefficients (from the lowpass filter) are subsequently
passed through another set of low and highpass filters, followed by down-sampling,
the resultant would be the level-2 coefficients. We could repeat these steps for a
N-level wavelet transform, where N is the number of cascaded filter banks. This
technique is displayed in Figure 4.1.

Figure 4.1: Diagram of a discrete wavelet transform implementation.
Each subsequent level results in a more detailed scale of the signal behavior. The
first level is always the coarsest approximation of the signal. The coefficients from
the remaining levels are intended to add finer detail to the signal reconstruction. It
is worth noting, that different signal characteristics will appear under different scales
88

of the transform. Thus, the idea of multiresolution is to capture the majority of the
noise within a finite number of scales. Hence, more scales does not always mean better
results. In fact, the fewer number of coefficients necessary to capture the majority
of the signal energy, the better the chance of removing the noise without eliminating
significant features of the desired signal. The goal is to isolate the signal energy in
a limited number of (relatively) large coefficients, while eliminating the remaining
coefficients that are likely to contain the noise.

4.2.3

Thresholding Scheme

The basic premise behind thresholding, involves eliminating wavelet coefficients containing noise; analogously, keeping coefficients with a high percentage of the signal
energy. There are numerous ways in which to accomplish this task; some complex,
others remarkably simple. The first attempt at formalizing the wavelet coefficient
thresholding for removal of additive noise from deterministic signals was recorded
in [18].
Two of the most common (and simple) thresholding schemes are referred to as
soft and hard thresholding; incidentally, terms that tend to have loose definitions.
For a precise definition and interpretation of soft thresholding, see [16]. Both soft
and hard thresholding eliminate coefficients that are below a chosen threshold. In
hard thresholding, all values above the threshold are kept. Figure 4.2 is an example
of a hard thresholding function, where

x is the coefficient input and y the adjusted

coefficient after thresholding is applied. In soft thresholding, the coefficients at or
above the threshold value are altered (typically decreased) in some predefined fashion
(i.e. dead-zone nonlinearity). Figure 4.3 is an example of a soft thresholding function,
where

x is the coefficient input and y the adjusted coefficient after thresholding is

applied.

Hard thresholding is a poor choice when the signal and noise coefficients

are close in magnitude. Hard thresholding is not continuous, so it exaggerates even
89

Figure 4.2: Potential hard thresholding function.

Figure 4.3: Potential soft thresholding function.

90

the slightest difference in coefficients near the threshold value. This can cause undesired discontinuities/artifacts in the reconstructed signal. For a smoother transition
between the various signal transform components, soft thresholding is a valid choice.
This form of thresholding is not sensitive to coefficient values near the thresholding
value. However, soft thresholding decreases coefficients that contain a large percent
of the signal energy, regardless the proximity to the threshold. Meaning that no
coefficient remains unchanged after leaving the thresholding stage.
Another important detail of thresholding is the cutoff value. The cutoff value determines which coefficients are left untouched, and the others that will be diminished
or removed. Common ways to determine an appropriate thresholding cutoff include
Stein’s Unbiased Risk, the universal threshold and minimax thresholding. In this
work, comparable values were generated with all three methods. Hence, the remaining discussions (unless otherwise specified) will focus on the minimax principle, given
this is an approach commonly used in statistics when designing optimal estimators, in
the mean square error sense. Essentially, the minimax principle is used to predict the
likelihood that a coefficient contains mostly noise or signal information; a minimum
bound is found, and the threshold is chosen to respect that bound.

4.3

Denoising: Real-time

This section builds on the knowledge presented for offline denoising by adapting the
algorithms for real-time implementation.

4.3.1

Delay

One of the key obstacles in real-time implementation is time delay. In [12], the authors
investigate the use of wavelet denoising to attenuate the rotor vibration caused by
step changes in sinusoidal forcing for a flexible rotor-magnetic bearing system. They
91

observed that the system states experience lag that originates between the signal
measurement and the generation of the wavelet coefficients (prior to entering the
controller). One suggestion to reduce the delay is to choose wavelets that are more
asymmetric (relative to the Daubechies family). Given Daubechies wavelets tend to
be localized around the center of their time duration, the denoising performance is
reduced. A wavelet transform that is concentrated more towards the current point in
time will produce a more immediate response in the coefficients, and hence allow a
higher bandwidth of disturbance attenuation.
Another method to reduce delay is restricting the wavelet transform to the negative axis [10]. This allows the wavelet transform to only operate on past values
of the signal. In utilizing an average-interpolation method, the authors discovered
that taking estimates of the signal near the current time, and not precisely at the
time of interest, resulted in a more fruitful noise removal. However, an additional
amount of delay is introduced into the system that is proportional to the distance
of the measurement taken from the current estimate. The authors were also able
to increase the signal to noise ratio by using a redundant transform algorithm that
rendered the wavelet transitionally invariant; the coefficients remain the same regardless of the point in time the calculation is made. This is accomplished though cycle
spinning [17], which averages the coefficient results found at various time shifts of the
signal.
In [1], the authors used wavelet filtering in an adaptive controller for a structural
system, where the denoising does not take place in the feedback loop; thus, delay is
not a primary concern. In [19], wavelets are used to remove noise from process data
resulting from slowly-varying systems. In particular, they use wavelets to denoise
signals from a pilot-scale distillation column, where the process signals are updated
every 5 seconds. Although this algorithm is in real-time, the time scale is fairly slow.
Not surprisingly, increasing the wavelet levels can lead to an increase in the system
92

latency [34]. This is particularly the case when using filter banks that operate on the
same time-scale as the system itself, and/or when the discrete wavelet transform is
viewed as its own dynamic system [48].

4.3.2

Thresholding Scheme

If the bound on the measurement noise is unknown, the technique of coefficient thresholding, or wavelet shrinkage, can still be successful in eliminating noise. The central
idea is to avoid setting a constant threshold value a priori, and allow the evolution of
the system to define the threshold value. In this case, a larger window size (without
introducing a detrimental amount of delay) would probably yield better results. We
have not explored such methods in this work, given the level of complexity for those
algorithms and the potential of the excessive time-delay destabilizing the closed-loop
system.

4.3.3

Windowing

Given the denoising scheme is implemented online, we do not have access to the
entire signal at once. Thus, we must choose to view and manipulate only a portion
of that signal at anytime. This notion will be referred to as windowing. The size
of the window is limited by the design choices made in the previous sections. The
number of samples chosen dictates how high of a resolution the signal coefficients can
be (level), and what kind of thresholding scheme is appropriate. Moreover, if data
size is an issue (which it typically is in hardware implementation), the number of
samples necessary to complete a satisfactory denoising should be minimized.
One windowing approach taken in [46], suggests that the window should increase
as the number of data points increases; the interpretation is that by the time the last
data point is reached, the entire data set should be encapsulated in the window.
93

4.4

Example

This section investigates the performance of high-gain observers utilizing wavelet
denoising techniques.

4.4.1

Simulation

Consider the following simple pendulum stabilization problem

x1 = x2
˙

(4.16)

x2 = − sin (x1 + θr ) + u ,
˙

(4.17)

where the pendulum arm is to be regulated at a constant reference signal of θr , by
the control

u. The state variable x1 is defined as the difference between the actual

angular position, θ , and the desired angular position, θr ;

x2 is the velocity of the

pendulum arm. The state feedback controller

u = sin(x1 + θr ) − 29x1 − 10x2
is used to linearize the system and assign the closed-loop eigenvalues at

(4.18)

−5 ± 2j .

To obtain a globally bounded control, we saturate x1 at ±1.2 and x2 at ±1.7. The
bounds on the controller are chosen such that the saturation is never active when the
system is under state feedback control. Assuming the only measurement is
linear high-gain observer will take the following form
94

x1 , the

2
ˆ
x1 = x2 + (y − x1 )
˙
ˆ
ε
1
x2 = 2 (y − x1 ) ,
˙
ˆ
ε

(4.19)
(4.20)

y is corrupted by a bounded noise v , i.e. y = x1 + v ; the
1
1
observer eigenvalues are assigned to − and − . In the interest of exploring a more
ε
ε

where the measurement

realistic approach, this setup will be studied in the context of a sampled-data output
feedback controller. The actual implementation of the control algorithm is in discretetime, modeled after the problem explored in [32]. The control is sampled using a
zero-order-hold, where the control in (4.18) is held constant in-between the uniformly
spaced sampling points. The continuous-time observer (4.19)-(4.20) is discretized
using the Bilinear-Transformation method. The sampling period is chosen according
to the guidelines presented in [13]. Namely, the sampling period

T is designed as

T = αε, where 0 < r1 < α < r2 < ∞ for some positive constants r1 and r2 ,
independent of ε. For this example, α = 1 and T = ε.
The discrete-time observer is taken as

q(k + 1) = Ad q(k) + Bd y(k)
x(k) = D−1 [Cd q(k) + Dd y(k)] ,
ˆ

where

Ad =

1 −1 4
2 5
2 2 1
1 5
, Bd =
, Cd =
and Dd =
.
9 −4 7
9 2
9 −1 4
9 2
95

(4.21)
(4.22)

The sampled-data output feedback control is given by

u(k) = sin (ˆ1 (k) + θr ) − 29 sat1.2 (ˆ1 (k)) − 10 sat1.7 (ˆ2 (k)) ,
x
x
x
where satk (z)

= min {|z|,k}sign(z). The initial states are taken as x1 (0) = −1,

x2 (0) = 0, q1 (0) = 0 and q2 (0) = 0.

The noise is generated using the “Uniform Random Number” block in Simulink,
where the bound is selected as

±0.01 with a sample time of 0.001 seconds. The

high-gain observer parameter, ε, is set to

0.02. Figure 4.4 shows the system states

for both the continuous and sampled-data output feedback controllers.

0

0

x

1

0.5

x1

0.5

−0.5
−1
0

−0.5
5
(a)

−1
0

10

2

0
−5
0

10

5
(d)

10

5

x

x

2

5

5
(b)

5
(c)

10

0

−5
0
Time

Figure 4.4: Comparison of the trajectories under (a) continuous-time, (b) sampleddata, (c) continuous-time and (d) sampled-data output feedback.

The difference in the signal magnitudes at steady-state is due to aliasing introduced via the discretization. Hence, an anti-aliasing filter (i.e. Butterworth order
8 and cutoff frequency of 25Hz) can be used to remove the aliasing introduced by
sampling the control. As shown in [32], the pre-filter will reduced some of the noise.
96

The goal is to further reduce the noise level via wavelet denoising.
Consider the addition of a wavelet filter after the zero-order-hold sampling of y ,
but before

y is injected into the high-gain observer. The filter is constructed from

a Haar wavelet level 2 transform and performed on 4 samples (window length) at a
time. A soft threshold with a dead-zone nonlinearity is used. The threshold value
is chosen as the maximum value of the noise wavelet coefficients, and is the same
for all levels of the transform. In Figure 4.5, it is evident that the filter is indirectly
improving the performance of the state variables by eliminating a significant amount
of the remaining noise from the signal injected into the high-gain observer.
−3

5 x 10
1

1

5 x 10

(a)

9

−5
7

10

8

(b)

9

10

9

10

0.05

0

−0.05
7

x

x

0.05

2

8

2

−5
7

0

x

x

0

−3

8

(c)

9

10

0

−0.05
7
Time

8

(d)

Figure 4.5: Comparison of the trajectories in steady-state (a) without wavelet denoising, (b) with wavelet denoising, (c) without wavelet denoising and (d) with wavelet
denoising.

The nonlinear-gain observer developed in Chapter 2 was designed to attenuate the
measurement noise in steady-state, while maintaining fast state reconstruction in the
transient period. The notion of using a wavelet pre-filter is intended to remove some
of the noise from the output before entering the high-gain observer. However, these
97

two techniques are not mutually exclusive. The wavelet scheme can be used to filter
out additional noise from the measurement

y before the signal reaches the high-gain

observer with the nonlinear-gain. In fact, using these methods in a cascade fashion
can result in the ability to choose smaller values of ε, consequently increasing the
speed of the estimator. Consider the nonlinear-gain high-gain observer introduced
in Chapter 2, where ε1

= 0.001, ε2 = 0.02 and d = 0.2. Figure 4.6 shows the

transient performance of the state x2 , where the waveforms for the system with and
without the wavelet pre-filter look identical. The steady-state value of

x2 for the

system utilizing the nonlinear-gain observer with the wavelet pre-filter is significantly
reduced when compared to the system without the benefit of the pre-filter; see Figure
4.7. Clearly, the denoising algorithm further reduces the noise in the steady-state
trajectories.

3

2

2

1

1

0

0

−1

x

4

3

2

4

−1

−2
0

1

(a)

2

3

−2
0
Time

1

(b)

2

3

Figure 4.6: Transient performance comparison of the nonlinear-gain high-gain observer (a) without a wavelet pre-filter and (b) with a wavelet pre-filter.

98

0.04

0.02

0.02

x2

0.04

0

0

−0.02

−0.02

−0.04
8

8.5

9
(a)

9.5

10

−0.04
8
Time

8.5

9
(b)

9.5

10

Figure 4.7: Steady-state performance comparison of the nonlinear-gain high-gain observer (a) without a wavelet pre-filter and (b) with a wavelet pre-filter.

4.4.2

Altering the Wavelet

Suppose we chose a different type of wavelet; perhaps a smoother set of functions,
while leaving the level at 2, window size at 4 and thresholding soft. Figure 4.8, where
the wavelet filter is active, shows that the steady-state values of the state x2 contain
more noise as the order of the Daubechies wavelet increases. For this example, the
smoother the wavelet the poorer the noise approximation and subsequent removal.
Most likely, the decreased performance is due to the increasing length of support
as the order of the wavelet increases. In other words, more data (larger window)
may be required to properly utilize the higher-order wavelets. We have limited this
comparison to the Daubechies family, given the Haar transform is the lowest order
transform available in the Daubechies subset of functions.
Consider that the signal at steady-state is presumably constant (or almost constant) for a stabilization problem. An interpretation for the successful denoising with
this algorithm is that the shape of the Haar function can be used to approximate
99

0.019

0.019

0

0

4

x

2

−0.019
2

6
(a)

8

10

−0.019
2

0.019

6
(b)

8

10

4

6
(d)

8

10

0.019

0

4

0

−0.019
2

4

6
(c)

8

10

−0.019
2
Time

Figure 4.8: Denoising performance in steady-state with the (a) Daubechies 1,
(b) Daubechies 4, (c) Daubechies 10 and (d) Daubechies 20 wavelets.

the signal on multiple scales, and separate it from the varying (non-constant) noise
signal. In this case, the noise profile greatly differs from that of the signal, and the
wavelet is able to distinguish that difference. Hence, the signal characteristics will
still be clustered into a few large value wavelet coefficients, whereas the noise will
occupy a larger number of small detail coefficients. This discovery is welcomed, given
the Haar wavelet is one of the simplest to manipulate and implement, as previously
noted.
However, we would be remiss to not compare the denoising performance of wavelets
outside of the Daubechies family. Figure 4.9 provides a comparison of the steadystate denoising performance for order 2 Daubechies, Coiflets and Symlets wavelet.
The level is set at 1, window size 10 and thresholding soft; the Matlab “thselect”
function is used with the “minimax” option to determine the cutoff value. Clearly, all
three families produce similar denoising results. Comparing Figure 4.8 with Figure
4.9, suggests that the ad-hoc method of determining the cutoff threshold may be more
100

0.05
0

x2

−0.05
8

8.5

9
(a)

9.5

10

0.05

0.05

0

0

−0.05
8

8.5

9
(b)

9.5

10

−0.05
8
Time

8.5

9
(c)

9.5

10

Figure 4.9: Denoising performance in steady-state with the (a) Daubechies 2,
(b) Coiflets 2 and (c) Symlets 2 wavelets.
effective than the minimax method for smaller windows.

4.4.3

Levels

Although level 1 and level 2 transforms were chosen, we could have chosen a setup
that would provide us with a larger number of detail coefficients. Yet, in the case
of this feedback control stabilization problem, more detail is not more accuracy. In
fact, Figure 4.10 shows that as additional levels are added, the level of noise in
the state

x2 increases to more than twofold the amount seen with a level 1 Haar

transform; the window size is 16 and the thresholding type is a dead-zone nonlinearity
for the simulation considered. This result is partially a by-product from the way the
thresholding value was chosen. The bound on the measurement noise was known a
priori, implying that a bound on the wavelet coefficients is also known. The bound on
the coefficients was used as the threshold in this example, meaning that all wavelet
coefficients at or above that threshold will be eliminated before reconstructing the
101

0.05

0.05

0

0

8

x

2

−0.05
7

(a)

9

10

−0.05
7

0.05

(b)

9

10

9

10

0.05

0

8

0

−0.05
7

8

(c)

9

10

−0.05
7
Time

8

(d)

Figure 4.10: Denoising performance in steady-state with Haar (a) level 1, (b) level 2,
(c) level 3 and (d) level 4 wavelet transforms.
signal. If this approach is taken with a low level wavelet transform (i.e. 2 or lower),
the measurement signal can be recovered with minimal noise. However, even using an
online algorithm to determine the cutoff value still results in a decreased performance
as the level increases, although not as significantly as in the fixed threshold case; in
this work the minimax method is used to determine a cutoff value online.
We did not observe any obvious delay effects irrespective of the level. Unlike
the papers previously mentioned, the wavelet transforms in the denoising algorithms
discussed in this work were implemented as algebraic calculations.

4.4.4

Thresholding Logic

For stabilization problems with bounded measurement noise, soft thresholding reduces
the ultimate bound on the estimation error more so than hard thresholding; this
is in the context of choosing the threshold value a priori from the bound on the
measurement error.

Figure 4.11 provides a comparison between a hard and soft
102

x

2

0.05

0.05

0

−0.05
7

0

8

(a)

9

10

−0.05
7
Time

8

(b)

9

10

Figure 4.11: Denoising performance in steady-state with (a) soft thresholding and
(b) hard thresholding.
thresholding approach for a Haar, window size 4, 1 level denoising scheme. Clearly,
the bound on the error in

x2 is larger for the hard thresholding scheme. Thus, soft

thresholding does a better job of preserving the signal, and eliminating the noise.

4.4.5

Windowing

As an example, the minimum number of samples (per window) for a n-level Haar
transform is 2n . If more sophisticated (online) methods are used to determine the
threshold value, a larger window size can be required to achieve an accurate estimate
of the noise mean and variance; otherwise large portions of the signal could be removed
in error. This is not something previously addressed, given the measurement noise is
a bounded uniformly distributed value, where the bound is known a priori.
Given that past values are used to construct the sample set, the window must
utilize some set of initial conditions. These values could be significantly different
from the true values, and will remain in the window until pushed out. After
103

(k − 1)

samples, where

k is the sample size, the window will be populated with the true

system values. If too large of a window is chosen, latency may be introduced into the
system. For the system discussed here, there is no appreciable difference in system
performance for a larger versus a smaller window size (in the range of 2 samples
to 16 samples). This appears to be the case, given all algorithms are implemented
algebraically.

4.5

Lowpass Filters

Lowpass filters are a simple and common way to eliminate high-frequency noise.
However, if the frequency band of the signal overlaps that of the noise, eliminating
the noise can also remove a significant portion of the signal we wish to preserve.
Moreover, to produce a smoother denoised signal, we may want to increase the order
of the filter. However, the higher the order the greater the phase lag introduced
into the feedback loop. Thus, wavelets should have the potential to surpass lowpass
filters in noise removable; otherwise, the additional complexity cannot be justified.
However, this is not to suggest that all wavelet denoising schemes will outperform
the classic lowpass filter. In fact, some schemes shown in this chapter do not remove
more noise than a first-order lowpass filter. Thus, the many factors discussed in
this chapter must be carefully considered when constructing an appropriate wavelet
denoising scheme.
Consider an order eight digital Butterworth lowpass filter with a normalized cutoff
frequency of

0.9. This filter is realized by entering the order and normalized cutoff

frequency in the Matlab function “butter”. Next, a wavelet denoising scheme is
constructed for a level 2, window size 4, soft thresholding setup with an ad-hoc cutoff
value. For the sake of argument the simplest waveform, the Haar wavelet, is chosen.
Figure 4.12 shows the denoising performance for the Butterworth filter and the Haar
104

x2

0.05

0.05

0

−0.05
8

0

8.5

9
(a)

9.5

10

−0.05
8
Time

8.5

9
(b)

9.5

10

Figure 4.12: Steady-state performance comparison of a (a) Haar wavelet denoising
scheme and (b) Butterworth filter.

4

2

2

0

0

−2

−2

−4

x

6

4

2

6

−4

−6
0

1

(a)

2

3

−6
0
Time

1

(b)

2

3

Figure 4.13: Transient performance comparison of a (a) Haar wavelet denoising
scheme and (b) Butterworth filter.

105

wavelet denoising scheme for the steady-state signal of x2 . Clearly, the Haar wavelet
is able to remove more of the noise from the system state than the Butterworth filter.
Furthermore, Figure 4.13 shows that the high-gain observer utilizing the wavelet
denoising scheme results in the system state x2 having a slightly faster settling time
than the observer using the Butterworth filter.

4.6

Conclusions

This chapter provided a simulation-based feasibility study on wavelet denoising to
reduce the amount of measurement noise entering the high-gain observer. Logically,
the wavelet design can greatly affect the system performance. We found that for
denoising schemes with a relatively small window (small data set), an increase in
the wavelet order leads to an increase in the amount of noise in the system states.
To alleviate this deficiency, while maintaining the same wavelet function, a larger
window size can be chosen. However, if there is a limit on the amount of data
that can be stored, a different family of wavelets with a smaller support length is a
viable option. Moreover, if data storage is a primary concern, extra attention will
have to be taken in choosing the proper wavelet. It was shown that fewer wavelet
coefficients are necessary to reconstruct the output signal for a wavelet that quickly
captures the behavior of the noise. Moreover, excessive levels can lead to a decrease
in performance, given the signal energy is being distributed across a larger number
of coefficients that will likely be effected by the thresholding scheme. For the type of
system and bounded measurement noise investigated in this work, the way in which
we arrived at the cutoff value for the thresholding scheme appeared to have little to
no effect on the denoising outcome. However, the soft thresholding function resulted
in smoother and less noisy signals than the hard thresholding approach.
The ample degrees of freedom in wavelet design allow this denoising approach to be
106

extremely versatile. However, it is this flexibility that makes the problem incredibly
challenging. Overall, the simplistic Haar transform was shown to provide superior
steady-state and transient performance when compared to a lowpass Butterworth
filter, for additive bounded measurement noise.

107

Chapter 5
Conclusions
It is a well-known fact that high-gain observers are susceptible to measurement noise.
In particular, we discussed the tradeoff between fast state reconstruction, minimizing
the bound on the steady-state estimation error and rejecting the model uncertainty.
Hence, the focus of this dissertation has been to address issues concerned with observer
design and analysis in the presence of measurement noise through three major thrust
areas: observer structure, tracking performance and filtering.
Initially, a nonlinear-gain high-gain observer was constructed to capture the transient and steady-state performance present in comparable linear-gain observers. Specifically, the nonlinearity was chosen to have a higher observer gain during the transient
period and a lower gain afterwards. Throughout the course of the investigation,
we considered altering the number of piecewise linear regions in the nonlinear-gain
function. However, we concluded that a two-piece function produced satisfactory
results, suggesting that altering the number of piecewise linear regions is generally
unnecessary. This approach allowed us to reduce the tradeoff between fast state reconstruction and measurement noise attenuation. The stability of the closed-loop
system was proven for the case where all assumptions hold globally. Although, it
was noted in Chapter 2 that a slight modification would specialize the proof for a
108

regional version of the theorem provided. Through the simulation results, we showed
that the new gain structure successfully addresses the criticism encountered when
implementing high-gain observers in systems with measurement noise.
The compromise between the transient and steady-state performance obvious in
the estimation error is not apparent in the system tracking error. Motivating this notion with a nonlinear example, it became apparent that the tracking error is uniformly
bounded in

ε for systems without zero dynamics. Before attempting to confirm this

phenomenon rigorously, it was argued by constructing the system transfer functions
from the noise to the tracking error and its derivatives, that the error and its first
derivative are bounded uniformly in

ε for a third-order nonlinear system. Using sin-

gular perturbation analysis, the results were extended to a class of linear systems
of dimension

n. In particular, aside from the tracking error and its first derivative,

all remaining derivatives of the tracking error are inversely proportional to increasing
powers of ε. Subsequently, a similar result was derived for a class of nonlinear systems
using the special features of singularly perturbed systems, further generalizing the results reported for linear systems. Due to the form of the nonlinearity, the result for
nonlinear systems was restricted to a third-order system. Although it has been shown
that the tracking error is more immune to the effects of measurement noise than the
estimation error for both the linear and nonlinear forms considered, this does not
mean that

ε can be made arbitrarily small. It is important to keep in mind that the

estimates of the states will still be used in the controller. However, the control may
not be sensitive to a slight increase in the state estimation error, meaning we acquire
some additional flexibility in choosing the value of

ε when the tracking performance

is the primary focus. This work can be extended to included nonlinear systems with
dimension greater than three, additional control structures and the inclusion of zero
dynamics.
In order to maximize the performance possible with the newly developed nonlinear109

gain high-gain observer, we performed a simulation-based feasibility study on wavelet
denoising. The idea was to reduce the amount of measurement noise entering the
high-gain observer from the on-set. We found that for denoising schemes with a relatively small window (small data set), an increase in the wavelet order leads to an
increase in the amount of noise in the system states. To alleviate this deficiency,
while maintaining the same wavelet function, a larger window size can be chosen.
However, if there is a limit on the amount of data that can be stored, a different family of wavelets with a smaller support length was a viable option. Moreover, if data
storage is a primary concern, extra attention will have to be taken in choosing the
proper wavelet. It was shown that for a wavelet that quickly captures the behavior
of the noisy signal, fewer wavelet coefficients are necessary to reconstruct the system
output. Interestingly, iterating the wavelet transform excessively lead to a decrease
in performance, because the signal energy was distributed across a larger number of
coefficients that were reduced in the thresholding phase. Overall, the simplistic Haar
transform was shown to provide superior steady-state and transient performance when
compared to a lowpass Butterworth filter, for additive bounded measurement noise.
The ample degrees of freedom in wavelet design allow this denoising approach to be
extremely versatile. However, it is this flexibility that makes the problem incredibly
challenging. There are many possibilities for future work. Namely, generalizing the
results in Chapter 4 to a broader class of nonlinear systems, finding the ideal construction for an online wavelet denoising scheme and extending the work to include
the noise statistics in the filter design.

110

APPENDICES

111

Appendix A

Nonnegative Impulse Response

Consider the transfer function

G(s) =

ε
β1 1 − 1
ε2

sn−2

n−1
.

sn−1 + β1 sn−2 + · · · + βn−1

The poles and zeros of

(A.1)

G(s) can always be chosen real and distinct such that
m
¯

G(s) = K
i=1
for an appropriate choice of β1 ,

i = 1, ...m.
¯

1−

+ · · · + βn−1

ε1
ε2

s + zi
¯
s + pi
¯




n
¯
j=m+1
¯



1 
s + pi
¯

(A.2)

· · · , βn−1 and ε1 /ε2 , where m ≤ n, zi > pi for
¯
¯ ¯
¯

112

Example 1: System with relative degree 3 (i.e. n = 3)

For

n = 3, (A.1) becomes

G(s) =

ε
β1 1 − 1
ε2

+ β2

2

ε1
ε2

1−

.

s2 + β1 s + β2

(A.3)

By matching like terms of (A.3) to (A.2) we see that

(s + p1 )(s + p2 ) = s2 + β1 s + β2

(A.4)

p2 > p1 > 0

(A.5)

β1 = p1 + p2

(A.6)

β2 = p1 p2 .

(A.7)

meaning

Then,

β
ε
s+ 2 1+
β1 1 − 1
ε2
β1
G(s) =
(s + p1 )(s + p2 )
ε
β1 1 − 1 (s + z)
ε2
,
=
(s + p1 )(s + p2 )
where

β2
1+
β1
p1 p2
=
p1 + p2

z =

113

ε1
ε2
ε
1+ 1
ε2

.

ε1
ε2
(A.8)

Suppose ε1 /ε2

≥ p1 /p2 . Then,
z≥

Therefore, choose p1
With these choices,

p1 p2
p1 + p2

p
1+ 1
p2

= p1 .

(A.9)

= µ, p2 = 1, ε1 /ε2 = µ < 1 and z =

G(s) can be written as

µ
(1 + µ).
1+µ

(1 + µ)(1 − µ)
s+1
1 − µ2
,
=
s+1

G(s) =

(A.10)

where the condition in (A.2) is satisfied, and the poles and zeros of

G(s) are real

and distinct.

Example 2: System with relative degree 4 (i.e. n = 4)

For

n = 4, (A.1) becomes

G(s) =

ε
β1 1 − 1
ε2

s2

Choose the poles of (A.11) as

+ β2

1−

ε1
ε2

2

s + β3

1−

ε1
ε2

s3 + β1 s2 + β2 s + β3

3
.
(A.11)

−µ2 p, −µp and −p. The denominator of (A.11) can

be written as

(s + µ2 p)(s + µp)(s + p) = s3 + (1 + µ + µ2 )ps2
+ (1 + µ + µ2 )µp2 s + µ3 p3 ,
114

(A.12)

where

µ < 1. From (A.12), we can see that β1 = (1 + µ + µ2 )p, β2 = (1 + µ +

µ2 )µp2 and β3 = µ3 p3 . Focusing on the numerator,
ε
β1 1 − 1
ε2

s2

+ β2

1−

2

ε1
ε2

ε
1+ 1
ε2

β
s2 + 2
β1

ε
1+ 1 +
ε2

ε1
ε2

3

s

s + β3

ε
= β1 1 − 1
ε2
β
+ 3
β1

1−

ε1
ε2

=
(A.13)

2
,

where

β2
β1
β3
β1
Take

ε
1+ 1 +
ε2

ε
1+ 1
ε2
ε1
ε2

ε
= µp 1 + 1
ε2
µ3 p 3

2

=

ε
1+ 1 +
ε2

ε1
ε2

2
.

(1 + µ + µ2 )p

p = 1 and ε1 /ε2 = µ. Then, (A.13) becomes
(1 + µ + µ2 )(1 − µ)(s + µ)(s + µ2 ) .

Hence,

(A.14)

G(s) can be written as
(1 + µ + µ2 )(1 − µ)(s + µ)(s + µ2 ) 1 − µ3
,
G(s) =
=
s+1
(s + 1)(s + µ)(s + µ2 )

where the condition in (A.2) is satisfied, and the poles and zeros of
and distinct.

115

(A.15)

G(s) are real

Example 3: System with relative degree 5 (i.e. n = 5)
For

n = 5, (A.1) becomes

G(s) =

ε
β1 1 − 1
ε2

+ β2

1−

ε1
ε2

2

s2

s4 + β1 s3 + β2 s2 + β3 s + β4
β3

+

s3

1−

ε1
ε2

3

s + β4

1−

ε1
ε2

s4 + β1 s3 + β2 s2 + β3 s + β4

Choose the poles of (A.16) as

(A.16)

4
.

−µ3 , −µ2 , −µ, −1 and set ε1 /ε2 = µ. The

denominator of (A.16) can be written as

(s + µ3 )(s + µ2 )(s + µ)(s + 1)
= s4 + (1 + µ)(1 + µ2 )s3
+ (µ5

+ µ2 (1 + µ)2

+ µ)s2

(A.17)

+ µ3 (1 + µ)(1 + µ2 )s + µ6 .
From (A.17), we can see that β1

= (1+µ)(1+µ2 ), β2 = (µ5 +µ2 (1+µ)2 +µ),

β3 = µ3 (1+µ)(1+µ2 ) and β4 = µ6 . The numerator in (A.16) can be represented
as

β
β
β1 (1 − µ) s3 + 2 (1 + µ)s2 + 3 (1 + µ + µ2 )s
β1
β1
β
+ 4 (µ3 + µ2 + µ + 1) ,
β1

(A.18)

where substituting in the values for the βi ’s in the bracketed term results in

s3 + µ(µ2 + µ + 1)s2 + µ3 (µ2 + µ + 1)s + µ6 .
116

(A.19)

Thus,

G(s) is
β1 (1 − µ)(s + µ)(s + µ2 )(s + µ3 ) 1 − µ4
,
=
G(s) =
s+1
(s + 1)(s + µ)(s + µ2 )(s + µ3 )

where the condition in (A.2) is satisfied, and the poles and zeros of
and distinct.

117

(A.20)

G(s) are real

Appendix B

A Block-Diagonal Form for Linear
Systems

The ideas in this section are borrowed from [35]. Consider the following linear twotime-scale system

x = A11 x + A12 z
˙
εz = A21 x + A22 z
˙
for

(B.1)
(B.2)

0 < ε < 1. Before transforming (B.1)-(B.1) into a block-diagonal form, we first

seek to bring the system into a block-triangular form. In particular, the change of
variables

η(t) = z(t) + L(ε)x(t)

(B.3)

will bring the system into what is known as actuator form. Essentially, actuator form
means that the states of the “slow” equation (B.1) are removed from the dynamics of
the “fast” equation (B.2). This similarity transform will bring the system (B.1)-(B.2)
118

into the form

x(t)
˙
εη(t)
˙

=

A11 − A12 L
R(L, ε)

A12

x(t)

A22 + εLA12

η(t)

,

(B.4)

R(L, ε) must be zero for the state x to be removed from the

where the matrix

η equation. In order for R(L, ε) to be zero, the matrix L(ε) should satisfy the
˙
algebraic equation

R(L, ε) = A21 − A22 L + εLA11 − εLA12 L = 0 .

(B.5)

The system (B.1)-(B.2) is partially decoupled, providing a separate fast subsystem

εη(t) = (A22 + εLA12 )η(t) ,
˙
where

(B.6)

x does not appear.

However, another change of variables is necessary to achieve a complete separation
of the fast and slow states of the system (B.1)-(B.2), leading to a block-diagonal form.
Applying the change of variables

ζ(t) = x(t) − εM η(t)

(B.7)

results in

˙
ζ(t)
εη(t)
˙
where the matrix

=

A11 − A12 L
0

S(M, ε)

ζ(t)

A22 + εLA12

η(t)

,

(B.8)

M is required to satisfy the linear algebraic equation

S(M, ε) = ε(A11 − A12 L)M − M (A22 + εLA12 ) + A12 = 0 .
119

(B.9)

The exact slow system is given by

˙
ζ = (A11 − A12 L)ζ(t) .

(B.10)

Therefore, the system (B.1)-(B.2) assumes the block-diagonal form

˙
ζ(t)
εη(t)
˙

=

A11 − A12 L
0

0

ζ(t)

A22 + εLA12

η(t)

(B.11)

and is completely decoupled. Moreover, (B.11) has a unique solution for sufficiently
small ε.

120

Appendix C

Decomposition of Nonlinear
Singularly Perturbed Systems

The block-diagonal form for linear systems presented in Appendix B completely decouples the system dynamics of the “fast” and “slow” states. An analogous blockdiagonal form does not exist for nonlinear singularly perturbed systems. However,
there is a nonstandard change of variables that can accomplish partial decoupling of
the states. The decomposition method presented in [49] begins by removing the slow
input from the fast equation, resulting in an upper triangular form. When an additional change of variables is applied to remove the fast input from the slow equation,
some of the slow input is reintroduced into the fast equation; this results in a lower
triangular form. Ultimately, the complete transformation is able to eliminate the fast
input from the slow equation, but not the slow input from the fast equation. This
section contains additional details concerning the decomposition process.

For convenience, when referring to the original work, the naming conventions
121

in [49] are adopted here. Consider the following singularly perturbed system

x = f (t, x, y, ε)
˙
εy = g(t, x, y, ε)
˙
for

(C.1)
(C.2)

0 < ε < 1. The goal of this transform is to bring the system (C.1)-(C.2) into a

block-triangular form. In particular, the transform should reduce the system to

u = F (t, u, ε)
˙
εv = G(t, u, v, ε) ,
˙
where the “fast” input

(C.3)
(C.4)

v is eliminated from the “slow” equation (C.3). The change

of variables that will take the system into the desired form is

x = u + εH(t, u, v, ε)

(C.5)

y = v + h(t, x, ε) = v + h(t, u + εH(t, u, v, ε), ε) .

(C.6)

Under the assumptions listed in [49], the system (C.1)-(C.2) has an integral manifold
defined as

y = h(t, x, ε). Those assumptions for t ∈ R and x ∈ Rn are:

• The function

g(t, x, y, ε) in (C.2) evaluated at ε = 0 is zero, and has the

isolated solution

y = h0 (t, x);

f , g and h0 are twice continuously differentiable for |y −
h0 (t, x)| ≤ ρ and 0 ≤ ε ≤ ε0 ;

• The functions

• The eigenvalues of

∂g
(t, x, h0 (t, x), 0)
∂y

are negative.
122

In Chapter 3, the system (3.32)-(3.33) satisfies all of the above conditions.

Definition: Integral Manifold
Consider the differential equation

x = X(t, x)
˙
where

x, X ∈ Rn . The set S ⊂ R × Rn is said to be an integral manifold if for

(t0 , x0 ) ∈ S , the solution (t, x(t)), x(t0 ) = x0 is in S for t ∈ R.

Referring back to the system of interest (C.1)-(C.2), the dynamics on the manifold
can be described by the differential equation

x = f (t, x, h(t, x, ε), ε) ,
˙
where the state

(C.7)

y is replaced by the continuously differentiable function h. For

convenience when solving for

h, the asymptotic expansion

h = h0 (t, x) + εh1 (t, x) + ε2 h2 (t, x) + · · ·
is defined, where h(t, x, 0)

(C.8)

= h0 . In order for the transform in (C.5)-(C.6) to exist,

h must satisfy the partial differential equation
ε

∂h
∂h
+ ε f (t, x, h, ε) = g(t, x, h, ε) .
∂t
∂x

The coefficients of the expansion of
of

(C.9)

h in (C.8) can be found by matching like powers

ε in (C.9). Define the variables z = y − h(t, x, ε) and w = x − u. Then,

123

consider the functions

f1 = f (t, u + w, z + h(t, u + w, ε), ε) − f (t, u, h(t, u, ε), ε)

(C.10)

Z(t, u, w, z, ε) = g(t, u + w, z + h(t, u + w, ε), ε)
− g(t, u + w, h(t, u + w, ε), ε)
∂h
− ε (t, u + w, ε)[f (t, u + w, z + h(t, u + w, ε), ε)
∂x
− f (t, u + w, h(t, u + w, ε), ε)]

(C.11)

for the auxiliary differential system

u = f (t, u, h(t, u, ε), ε)
˙

(C.12)

w = f1 (t, u, w, z, ε)
˙

(C.13)

εz = Z(t, u, w, z, ε) .
˙

(C.14)

The system (C.12)-(C.14) has the integral manifold
function

w = εH(t, u, z, ε), where the

H satisfies the partial differential equation
ε

∂H
∂H
∂H
+ε
F (t, u, ε) +
Z(t, u, εH, v, ε)
∂t
∂u
∂v
= f1 (t, u, εH, v, ε) ,

where

F (t, u, ε) = f (t, u, h(t, u, ε), ε)
and

G(t, u, v, ε) = Z(t, u, εH(t, u, v, ε), v, ε) .

124

(C.15)

Often times, the function

H can be found as an asymptotic expansion of

εH = εH1 (t, u, v) + ε2 H2 (t, u, v) + ε3 H3 (t, u, v) + · · ·

(C.16)

from the expression in (C.15) by matching like coefficients in ε. Therefore, if both

h and H exist and satisfy the partial differential equations in (C.9) and (C.15),
respectively, (C.1)-(C.2) can be transformed into the system representation in (C.3)(C.4) using the change of variables in (C.5)-(C.6).

125

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