Weiwen Dissertation
163
A COMPARATIVE STUDY OF NONLINEAR OBSERVERS AND CONTROL DESIGN STRATEGIES WEIWEN WANG Bachelor of Science in Electrical Engineering Guangxi University, P.R. China July 1994 Master of Science in Electrical Engineering Guangxi University, P.R. China July 1997 submitted in partial fulfillment of the requirements for the degree DOCTOR OF ENGINEERING at the CLEVELAND STATE UNIVERSITY December 2003
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Transcript of Weiwen Dissertation
A COMPARATIVE STUDY OF NONLINEAR OBSERVERS
AND CONTROL DESIGN STRATEGIES
WEIWEN WANG
Bachelor of Science in Electrical Engineering
Guangxi University, P.R. China
July 1994
Master of Science in Electrical Engineering
Guangxi University, P.R. China
July 1997
submitted in partial fulfillment of the requirements for the degree
DOCTOR OF ENGINEERING
at the
CLEVELAND STATE UNIVERSITY
December 2003
This dissertation has been approved
for the Department of ELECTRICAL AND COMPUTER ENGINEERING
and the College of Graduate Studies by
Dissertation Committee Chairperson, Dr. Zhiqiang Gao
Department/Date
Dr. James H. Burghart
Department/Date
Dr. Dan Simon
Department/Date
Dr. Toshinori Munakata
Department/Date
Dr. Anna V. Stankovic
Department/Date
DEDICATED
TO MY PARENTS
ACKNOWLEGEMENT
I would like to express my deepest appreciation to my advisor, Dr. Zhiqiang Gao,
for giving me such a challenging opportunity, for his inspiring direction for the research
and for his great support in all aspects of my study.
I wish to thank Professor Jingqing Han, Institute of System Science, Academia
Sinica for his invaluable advice on this research.
I would like to acknowledge the kind support and assistance of Professor Carl
Knospe, University of Virginia, who provide the simulation files and document of the
AMB benchmark problem.
I would like to thank Dr. James H. Burghart, Dr. Dan Simon, Dr. Anna V.
Stankovic and Dr. Toshinori Munakata, who are on my committee, for their time in
reviewing and evaluating this dissertation.
Thanks to Professor George Kramerich, for his encourage and help with my work.
I would also thank the members of Applied Control Research Lab for their help.
They include Shaohua Hu, Yi Hou, Shahid Parvez, Robert Miklosovic, Minshao Zhu,
Bosheng Sun and Jing Liu.
A special thank you goes to my family for their encouragement and support of my
studies.
v
A COMPARATIVE STUDY OF NONLINEAR OBSERVERS
AND CONTROL DESIGN STRATEGIES
WEIWEN WANG
ABSTRACT
Conventional observers are based on an accurate mathematical model of the plant,
which limits their applications in the real world where the accurate model may not be
available. The Extended State Observer (ESO) is a novel model-independent observer. It
has been successfully applied to many practical applications: such as hard disk computer
drives, web tension regulation, DC-DC converters, and industrial servo systems.
This research focuses on the comparative study of different observers and observer
based control strategies. The purpose is to find a set of practical solutions that make
control design and tuning easy and effective. Three observer methods, including the
ESO, the High Gain Observer (HGO), and the Sliding Mode Observer (SMO) are
investigated. The simulation and experimental results of motion systems show that ESO
has a better overall performance. Active disturbance rejection control (ADRC) is an
ESO-based control strategy. With a novel optimization tuning method, linear ADRC
achieves effective results when dealing with large inertia, friction variations and torque
disturbance. A simulation of an active, magnetic-bearing benchmark problem is used to
test the method. Based on the Popov criterion, a stability analysis of second order ADRC
is investigated.
Keywords: High Gain Observer, Sliding Mode Observer, Extended State
Observer, Active Disturbance Rejection Control, Motion Control, Popov Criterion.
vi
TABLE OF CONTENTS
Page
NOMENCLATURE............................................................................................. ix
LIST OF TABLES ............................................................................................... xi
LIST OF FIGURES .............................................................................................. v
CHAPTER
I. INTRODUCTION........................................................................................ 1
1.1 Background ......................................................................................... 2
1.2 Nonlinear Control Design Methods Review ....................................... 3
1.2.1 Nonlinear H∞ Method ................................................................... 4
1.2.2 Variable Structure, Sliding-Mode Control .................................... 7
1.2.3 Feedback Linearization ................................................................. 8
1.2.4 Adaptive Nonlinear Control .......................................................... 9
1.2.5 Recursive- Interlacing ................................................................... 9
1.3 Motivation of the Research ............................................................... 10
1.4 Problem Formulation......................................................................... 11
II. NONLINEAR OBSERVERS.................................................................... 13
2.1 Observer-based State Feedback Design ............................................ 14
2.2 High Gain Observers ......................................................................... 18
2.3 Sliding Mode Observers .................................................................... 21
vii
2.4 Nonlinear Extended State Observers................................................. 22
2.5 Active Disturbance Rejection Control (ADRC)................................ 26
2.6 Summary ........................................................................................... 28
III. COMPARISON OF ADVANCED STATE OBSERVER DESIGN
TECHNIQUES........................................................................................... 29
3.1 Simulation Results............................................................................. 30
3.1.1 Open-loop Comparison ............................................................... 30
3.1.2 Selection of Nonlinear Gains for NESO ..................................... 36
3.1.3 Closed-loop Comparison............................................................. 39
3.2 Hardware Test Results....................................................................... 41
3.3 Summary ........................................................................................... 46
IV. COMPARISON OF CONTROLLERS FOR MOTIONCONTROL
APPLICATIONS ....................................................................................... 47
4.1 Linear Active Disturbance Rejection Control Design Concepts....... 48
4.2 Loop Shaping Design Technique ...................................................... 51
4.3 Simulation Study and Comparison.................................................... 55
4.4 Experiment Study and Comparison................................................... 62
4.5 Summary ........................................................................................... 67
V. NOVEL SOLUTIONS FOR THE ACTIVE MAGNETIC BEARING
BENCHMARK PROBLEM ..................................................................... 68
5.1 Background ....................................................................................... 69
5.2 System Dynamics and Design Specifications ................................... 70
5.3 The Proposed Solutions....................Error! Bookmark not defined.3
viii
5.4 Simulation Study and Results............................................................ 78
5.5 Summary ........................................................................................... 87
VI. STABILITY ANALYSIS OF ADRC ....................................................... 88
6.1 Absolute Stability .............................................................................. 89
6.2 Stability Analysis of ADRC.............................................................. 92
VII. CONCLUDING REMARKS .................................................................. 96
REFERENCES.................................................................................................. 988
APPENDICES................................................................................................... 106
A. C Source Code used in Hardware Tests ........................................... 1077
B. Parameters of Active Magnetic Bearing Benchmark Problem....... 14848
ix
NOMENCLATURE
A System matrix of the state space control
B Input matrix of the state space control
C Output matrix of the state space control
L Observer gain matrix of the state space control
x State vector of the state space control
y Output vector of the state space control
u Control command
x Estimate state vector of the state space control
hi Observer gains of the high gain observer
ki Observer gains of the sliding mode observer
f (.) Dynamics of the plant and the disturbance
zi Estimate states of the ESO
w (t) External disturbance of the plant
J Inertia of the load
iβ Gains of the ESO
ig Nonlinear functions of ESO
0ω The observer bandwidth
cω The controller bandwidth
x
kp Proportional gain of the controller
kd Derivative gain of the controller
cω Closed loop natural frequency
r Reference input
ξ Damping ratio
Tmax Maximum torque
0δ Output error of the plant
xi
LIST OF TABLES
Table I Controller Parameters......................................................................................... 58
Table II Performances Comparison with ADRC, LADRC and LSC .............................. 66
Table III Simulation Comparison .................................................................................... 85
v
LIST OF FIGURES
Figure Page
1. The Sliding Condition .................................................................................................. 8
2. State-Feedback Design using an Observer................................................................. 14
3. Comparison of Linear and Nonlinear Gains .............................................................. 24
4. Structure of ADRC..................................................................................................... 26
5. Open Loop Simulation Model.................................................................................... 32
6. Estimation Error of the Nominal Plant....................................................................... 33
7. x3=a (t) and its Estimation z3 ..................................................................................... 34
8. Estimation Error of the Plant with 0.5N-m Coulomb Friction................................... 35
9. Disturbance and its Estimation................................................................................... 35
10. Estimated Error of the Plant with 100% Change of Inertia........................................ 36
11. Disturbance and its Estimation................................................................................... 36
12. Nonlinear Gain Functions .......................................................................................... 37
13. Estimated Error of the Plant with Initial Values ........................................................ 38
14. Observer-based State Feedback Control Configurations ........................................... 39
15. Nominal Responses of the Control Systems .............................................................. 40
16. Simulation Results with 0.5 N-m Coulomb Frictions ................................................ 41
17. Simulation Results with Sinusoid Disturbance .......................................................... 41
18. Experimental Setup .................................................................................................... 42
19. A Diagram for DC Brushless Servo System .............................................................. 42
vi
20. Responses to the Nominal Plant................................................................................. 44
21. Responses to Coulomb Friction ................................................................................. 45
22. Feedback Control Configuration................................................................................ 51
23. Loop Shaping ............................................................................................................. 53
24. The Simulink Diagram............................................................................................... 55
25. Position Error and Control Signal of ADRC System................................................. 59
26. Position Error and Control Signal of LADRC System .............................................. 59
28. Position Error and Control Signal of ADRC System with Inertia Change ................ 60
29. Position Error and Control Signal of LADRC System with Inertia Change.............. 60
30. Position Error and Control Signal of LSC System with Inertia Change.................... 61
31. Position Error and Control Signal of ADRC System with Torque Disturbance........ 61
32. Position Error and Control Signal of LADRC System with Torque Disturbance ..... 61
33. Position Error and Control Signal of LSC System with Torque Disturbance............ 62
34. Responses of the Nominal Plant ................................................................................ 63
35. Responses of the Plant with 30% Torque Disturbance at 4 second ........................... 64
36. Responses of the Plant with Friction.......................................................................... 65
37. Benchmark experiment (from [59]) ........................................................................... 70
38. Current as a function of flux and air gap length......................................................... 72
39. The AMB Simulation Model...................................................................................... 78
40. The Model of Leakage ............................................................................................... 78
41. The Model of Beam Dynamics .................................................................................. 79
42. Displacement Errors................................................................................................... 80
43. Voltages (u1&u2) and Flux ( 1φ & 2φ ) of PP ................................................................ 81
vii
44. Voltages (u1&u2) and Flux ( 1φ & 2φ ) of NDP............................................................. 81
45. Voltages (u1&u2) and Flux ( 1φ & 2φ ) of LADRC ....................................................... 82
46. Displacement Errors for the Plant Coil Resistance Increased by 25% ...................... 82
47. Voltages (u1&u2) and flux ( 1φ & 2φ ) of PP ................................................................. 83
48. Voltages (u1&u2) and Flux ( 1φ & 2φ ) of NDP............................................................. 83
49. Voltages (u1&u2) and Flux ( 1φ & 2φ ) of LADRC ....................................................... 84
50. Saturated High Gain Controller Responses (from [61]) ............................................ 86
51. Nonlinear Dynamics Placement Controller Responses.............................................. 87
52. The Problem of Lure and Postnikov .......................................................................... 90
53. Diagram of ADRC ..................................................................................................... 92
1
CHAPTER I
INTRODUCTION
Automatic control systems were first developed over two thousand years ago. The
primary motivation for feedback control in times of antiquity was the need for the
accurate determination of time. Around 270 B.C., the Greek Ktesibios invented a float
regulator for a water clock, which kept time by regulating the water level in a vessel [1].
In 1868 J.C. Maxwell explained the instabilities exhibited by the flyball governor using
differential equations to describe the control system. Since that time, control theory has
made significant strides.
2
1.1 Background
The period before 1868 is called “the prehistory of automatic control [2].” In 1877,
E. J. Routh provided a numerical technique for determining when a characteristic
equation has stable roots [Routh 1877], followed by A. M. Lyapunov’s study of the
stability of nonlinear differential equations using a generalized notion of energy in 1892
[Lyapunov 1892]. The period from 1868 to the early 1900s may be called “the primitive
period of automatic control” because of these discoveries.
In 1922 N. Minorsky introduced the Proportional-Integral-Derivative (PID)
controller. During 1930s and 1940s, frequency-domain approaches, the Nyquist stability
criterion, the Bode plot, and the root locus were used to analyze and design single-input
and single-output (SISO) linear systems. Simple compensators like lead-lag were
developed. The term “cybernetics” was coined by Norbert Weiner to refer to control
theory [Weiner 1949, Weiner 1961]. The period from 1900 to 1960 is often called“the
classical period,” and the period from 1960 through the present is “the modern period.”
In a series of famous papers, Kalman and his coauthors discussed the optimal
control of systems and provided the design equations for the linear quadratic regulator
(LQR) and Kalman filter. This approach was firmly rooted in time-domain and relied on
techniques from linear algebra. Today this branch of control still remains very active
under the name of “the linear system theory.” Digital control theory was formulated with
the advent of the microprocessor in 1969. In 1981, the H∞ control approach was proposed
by Zames. The first significant development in optimal control was the famous work of
Pontryagin, Boltyanskii, Gamkrelidze and Mishchenko in 1986.
3
Finally, everything narrowed down to nonlinear control. Zames, LaSalle, and
Willems contributed a significant amount of work on the theory of the stability of
nonlinear systems. [Zames 1961a, Zames 1961b, LaSalle 1968, and Willems1972].
Compared to classical control theory, modern control design is fundamentally a time-
domain technique. A state-space model of the system is required, which can represent a
multiple-input and multiple-output (MIMO) system as well as a SISO system.
1.2 Review of Nonlinear Control Design Methods
Today, many researchers and designers, from such broad areas as aircraft and
spacecraft control, robotics, process control, and biomedical engineering, have shown an
active interest in the development and applications of nonlinear control methodologies.
Linear control methods work well for the small range of the plant. But, when the required
operation range is large, a linear controller is likely to perform poorly because there is not
proper compensation for nonlinearities in the system.
When designing linear controllers, it is usually necessary to assume that the
parameters of the system model are reasonably well known. However, many control
problems involve uncertainties in the model parameters. A linear controller based on
inaccurate values of the model parameters may exhibit significant performance
degradation or even instability. Nonlinearities can be intentionally introduced into the
controller of a control system so that model uncertainties can be tolerated. Robust
controllers and adaptive controllers are two good examples.
4
The topic of nonlinear control design for large-range operation has attracted
particular attention because of the advent of microprocessors, which has made the
implementation of nonlinear controllers a relatively simple matter. Modern technology,
such as high-speed, high-accuracy robots and high-performance aircrafts, is also
demanding control systems with more stringent design specifications. Many new papers
and reports have recently been written on nonlinear control research and applications.
1.2.1 Nonlinear H∞ Method
The aim of the work performed in this area is to design a robust controller for
nonlinear systems using generalizations of the highly successful H∞ methods used in
linear systems theory [3, 4, 5, 6].
For the linear system:
( ) ( ) ( ) ( );
( ) ( ) ( ),
0;
.
u w
y yu
T
yu y
T
yu yu
x t Ax t B u t B w t
y t C x t D u t
where
D C
D D I
= + += +
=
=
&
(1.1)
u(t): input; y(t): output; w(t): disturbance.
The problem with H∞ full information control is finding a feedback controller, using the
state and disturbance that minimizes the closed-loop system infinity-norm.
2,[0, ]
,[0, ]
2,[0, ]
( )sup
( )
f
f
f
t
yw t
t
y tJ G
w t∞= =
5
when J< r, (r is called performance bound). A controller that satisfies this bound is called
a suboptimal controller. The following is the procedure of the Hamilton-Jacobi inequality
(HJI) equation-based design method.
The Hamiltonian Equations are described as
2
0
( , , ) ( ) ( ) ( ) ( ) 2 ( ){ ( ) ( ) ( ) ( )}
ft
T T T
u wJ u w p y t y t r w t w t p t Ax t B u t B w t x t dt= − + + + −∫ & .
( , , , , , ) 0J u w p u w pδ δ δ∇ =
( )
( )
x t
p t
⇒ =
&
&
2( ) ( )
( ) ( )
T T
u u w w
T T
y y
A B B r B B x t x tL
C C A p t p t
−
∞
− + = − −
(1.2)
with L∞ =2T T
u u w w
T T
y y
A B B r B B
C C A
− − + − −
11 12( )
21 22
( ) ( )
( ) ( )f f fL t t
f f
t t t te
t t t t
φ φφ φ
∞ − − − = − −
1
22 21( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )T T
u u f fu t B P t x t B t t t t x t K t x tφ φ−= − = − − = − (1.3)
The feedback gain exists for arbitrary time intervals, and there is a solution to the H∞
sub-optimal control problem only if the Hamiltonian matrix has no purely imaginary
eigenvalues. The matrix P(t) is the solution of a Riccati equation. It comes from the
Hamilton equation
( ) ( ) ( ) ( ) ( )p t P t x t P t x t= +&& & , ( ) ( ) ( ).p t P t x t= (1.4)
Putting (1.4) into (1.2) yields
2( ) ( ) ( ) ( )( ) ( )T T T T
y y u u w wP t P t A A P t C C P t B B r B B P t−= − − − + −&
6
For the nonlinear systems,
G: ( ) ( ) 1( ) 2( ) ,
( ) 1( ) ,
x f x f x g x w g x u
y h x h x u
= + ∆ + += +
& (1.5)
Assume f (0) =0, h (0) =0, K: y->u.
( )f x∆ is bounded. 1 ( )( ( ) 1( )) (0, )h x h x h x I′ =
Lemma If ( ( ), ( ))f x h x is zero-state detectable and there exists a positive – definite
function V(x) such that the following Hamilton-Jacobi inequality (HJI) holds:
2
1 1( ) ( ) ( ) 1( ) 1 ( ) ( ) ( ) 2( ) 2 ( ) ( ) '( ) ( ) 0
4 4x x x x x
V x f x V x g x g x V x V x g x g x V x h x h xr
′ ′′ ′+ − + ≤
then the following state feedback law
1( ) 2 '( )
2x
u x g x V′= −
solves the performance problem for the system (1.5).
The key task for nonlinear H∞ control is to obtain necessary and sufficient
conditions from appropriate nonlinear generalizations of the Riccati equations. These
involve partial differential equations and inequalities. Local linearization of (1.5) at x=0,
gives
(A, B, C)=(1 2
( (0), (0), (0), (0))df dg dg dh
dx dx dx dx
If the H∞ control problem for the linearized system is solvable, then the H∞ problem for
the nonlinear system is solvable locally near x=0. A controller can be constructed near
x=0 using V (x) which is locally smooth.
7
1.2.2 Variable Structure, Sliding-Mode Control
Variable structure control [7, 8] has been widely used over the last decade for the
control of uncertain systems because of its robustness to modeling uncertainties and
disturbances.
Consider the dynamic system:
( ) ( ) ( ; ) ( ; ) ( ) ( )nx t f X t b X t u t d t= + + (1.6)
where u(t) is scalar control input, x is the scalar output of interest, and
( 1)[ , ,..., ]n TX x x x −= & is the state. The function ( ; )f X t is not exactly known, but the extent
of imprecision f∆ on ( ; )f X t is upper bounded by a known continuous function of X
and t; similar to the control gain b(X; t). The disturbance d (t) is unknown but bounded in
absolute value by a known continuous function of time. The control problem is to get the
state X to track a specific state ( 1)
,[ ,..., ]n T
d d d dX x x x−= & in the presence of model
imprecision on f(X;t), b(X;t) and disturbance of d(t). To guarantee that this is achievable
using a finite control u, it must be assumed:
0
0t
X=
=% (1.7)
where ( 1): [ , ,..., ]n T
dX X X x x x
−= − = &% % % % . A time-varying sliding surface s(t) is defined in the
state-space nR as ( ; ) 0s x t =% with
1( ; ) ( ) , 0nds X t x
dtλ λ−= + >% % (1.8)
where λ is a positive constant. Given initial condition (1.7), the problem of tracking
dX X≡ is equivalent to that of remaining on the surface s(t) for all t>0; indeed s ≡ 0
represents a linear differential equation whose unique solution is 0X ≡% . Thus, the
8
problem of tracking the n-dimensional vector d
x can be reduced to that of keeping the
scalar quantity s at zero. A sufficient condition for such positive invariance of s (t) is to
choose the control law u(t) of (1.6) to satisfy
21( ; )
2
ds x t s
dtη≤ − (1.9)
where η is a positive constant. Inequality (1.9) constrains trajectories to point towards
the surface s (t) (Fig. 1), and is referred to as the sliding condition.
S (t)
Figure 1: The Sliding Condition
1.2.3 Feedback Linearization
Feedback linearization techniques [7, 8] can be viewed as ways of transforming
original system models into equivalent models of a simpler form. Feedback linearization
is based on the idea of transforming nonlinear dynamics into a linear form by using state
feedback. It has two approaches: input-state linearization for complete linearization and
input-output linearization for partial linearization. They can be used for stabilization and
tracking control problems, single-input or multiple-input systems, and a number of
9
practical nonlinear control problems, both as a system analysis tool and a controller
design tool.
1.2.4 Adaptive Nonlinear Control
Currently the most systematic methodology for adaptive nonlinear control design is
backstepping [9, 10, 11], which is mainly applicable to systems having a cascaded or
triangular structure. The central idea of the approach is to recursively design controllers
for subsystems in the structure and “step back” the feedback signals towards the control
input. This differs from the conventional feedback linearization in its ability to avoid
cancellation of useful nonlinearities in pursuing the objectives of stabilization and
tracking. In addition, by utilizing the control Lyapunov function, backstepping has the
flexibility to introduce appropriate dynamics in order to make the system behave in a
desirable manner. However, no analytical measure of performance or notions of
optimality have been shown to exist for controllers derived via backstepping approaches.
The two main methods for adaptive backstepping design are the turning function
design and the modular design. These assume that the full state of the system is available
for measurement. If it is not available, the observer backstepping method [12] is used.
1.2.5 Recursive- Interlacing
The recursive-interlacing procedure was proposed for the design of globally-
stabilizing controllers [13, 14, 15]. The procedure introduces new interlacing steps and
combines them with the existing recursive or backstepping design. It is not limited to
systems of any specific order and imposes no restrictions on either locations or types of
10
(uncertain) dynamics. Backstepping design is included as a special case, and no
assumption is needed on the stability of the nominal system or on the existence of
Lyapunov function. The nonlinear system under investigation contains uncertainties that
do not satisfy any of the previously developed conditions for matching. The recursive-
interlacing procedure is a fully-programmed integration of four components: recursion
(or backstepping), forward interlacing, backward interlacing, and concepts of admissible
fictitious controls and Lyapunov functions.
1.3 Motivation of the Research
From the literature review, it appears that several unresolved issues exist in the
current nonlinear control methods, including
• The need for a model: nonlinear ∞H and feedback linearization requires a model
of the plant to make a design;
• The complexity of design procedure: recursive-interlacing has four components
for its design, and the nonlinear ∞H method requires partial differential equations
and inequalities;
• The lack of robustness: feedback linearization cannot deal with the variety of
parameters;
• Scope of performance improvement: the sliding mode has chattering phenomena,
as well as steady-state error.
11
Most nonlinear controllers are state feedback controllers. When the state of the
system is unavailable or the sensor is expensive, observers must be used in estimation.
The motivation of this research is to find a set of observer based the control systems that
are relatively independent of the mathematical model, perform better, and are simple to
implement.
Since Minorsky invented PID in 1922, PID-based control technology still remains a
tool of choice in over ninety percent of industrial applications. The goal of this research is
to find a set of novel, practical tools that make control design and tuning easy and
effective.
1.4 Problem Formulation
The Extended State Observer (ESO), proposed by Professor Jingqing Han, can
estimate the state without a mathematical model of the system. It is a novel concept for
observer design, estimating not only the state, but also the internal and external
disturbances, thus making disturbance rejection control possible. The invention of ESO is
a revolutionary concept for control theory and application. It has several properties
including model-independence, active estimation, compensation for disturbances, simple
design, and strong robustness. This method has evolved as an important technique for the
state feedback control of nonlinear systems.
In this dissertation, the concepts of the ESO, the High Gain Observer (HGO), and
the Sliding Mode Observer (SMO) are introduced in Chapter II. Chapter III presents a
12
comparative study of these three observers, which is based on the robustness of the
performance with respect to the uncertainties of the plant and the observer tracking
errors, both at steady-state and during transients. The optimal tuning methods for linear
active disturbance rejection control (LADRC) and loop-shaping are tested, and along
with active disturbance rejection control (ADRC), are applied to a servo motor system in
Chapter IV. A design procedure for ESO-based control systems is described. The active
magnetic-bearing benchmark problem is solved using LADRC in Chapter V, and Chapter
VI discusses the stability analysis of ADRC.
13
CHAPTER II
NONLINEAR OBSERVERS
Observers provide state estimates of the plant to closed-loop control algorithms.
The control algorithm is designed in two parts: a “full-state feedback” part based on the
assumption that all of the state variables can be measured and an observer to estimate the
state of the process based on the observer output. In this chapter, the concepts of
observer-based state feedback design are introduced first. The high gain observers,
sliding mode observers and extended state observers are reviewed, followed by an
introduction to ADRC.
14
2.1 Observer-Based State Feedback Design
The concept of separating the state feedback control design into the full-state
feedback part and observer is known as the separation principle, which has rigorous
validity in linear systems, as well as a limited class of nonlinear systems. Figure 2
illustrates the state-feedback design using an observer.
B
L
A
C∫xu +
++
+−
To controller
yy
Estimated
outputPlant output
Estimated
output error
Figure 2: State-Feedback Design using an Observer
Consider a linear, continuous-time dynamic system
x Ax Bu
y Cx
= +=
& (2.1)
A full-order observer for the linear process is defined by
ˆ ˆ ˆ( )x Ax L y Cx Bu= + − +& (2.2)
The estimation error is
ˆe x x= − (2.3)
From (2.2) and (2.3) we have
ˆ( )e A LC e Ae= − =& (2.4)
15
The estimation error will converge to zero if A is a stability matrix. When A is constant,
its eigenvalues must be in the open left half-plane. This asymptotic state estimator is
known as the Luenberger observer [16]. Since the matrices A, B and C are defined by the
plant, the only freedom in the design of the observer is in the selection of the gain matrix
L. Optimization and pole placement are two standard design methods.
Since the observer given by (2.2) has the structure of a Kalman filter, its gain matrix
can be chosen as a Kalman filter gain matrix [5], i.e,
1L PC R
−′= (2.5)
where P is the covariance matrix of the estimation error and satisfies the matrix Riccati
equation
1P AP PA PC R CP Q
−′ ′= + − +& (2.6)
where R is a positive definite matrix and Q is a positive semi-definite matrix. In most
applications the steady-state covariance matrix is used in (2.5). This matrix is given by
setting P& in (2.6) to zero. The resulting equation is known as the algebraic Riccati
equation. Algorithms to solve the algebraic Riccati equation are included in popular
control system software packages such as MATLAB and CONTROL-C. In order to make
the gain matrix given by (2.5) and (2.6) to be genuinely optimum, the process noise and
the observation noise must be white, with the matrices Q and R as their spectral densities.
It is nearly impossible to determine these spectral density matrices in practical
applications. Hence, the matrices Q and R are best treated as design parameters that can
be varied to achieve overall system design objectives.
An alternative to solving the algebraic Riccati equation in order to obtain the
observer gain matrix is to select L to place the poles of the observer [17, 18, 19, 20], i.e.,
16
the eigenvalues of A in (2.4). From (2.4), the characteristic equation of the error is now
given by
det[ ( )] 0sI A LC− − = (2.7)
If L is chosen so that A-LC has stable and reasonably fast eigenvalues, e will then decay
to zero and remain there, independent of the known forcing function u (t), its effect on the
state x (t), and irrespective of the initial condition e (0). Therefore, ˆ( )x t will converge to
x (t), regardless of the value ˆ(0)x . Furthermore, the dynamics of the error can be chosen
for stability, as well as for speed, as opposed to the open-loop dynamics determined by A.
If we do not have an accurate model of the plant (A, B, C), the dynamics of the error are
no longer governed by (2.4). However, we can typically choose L so that the error system
is at least stable and the error remains acceptably small, even with (small) modeling
errors and disturbing inputs. It is important to emphasize that the nature of the plant and
that of the estimator are quite different. The selection of L can be approached in exactly
the same fashion that K is selected in the control-law design. If the desired location of the
estimator error poles is specified as
1 2, ,..., ,i n
s β β β=
then the desired estimator characteristic equation is
1 2( ) ( )( )...( )e n
s s s sα β β β= − − − (2.8)
We can solve for L by comparing coefficients in (2.7) and (2.8).
The observer is used mainly to estimate the state for the purpose of feedback control.
ˆu Gx= − (2.9)
where
x x e= − (2.10)
17
The closed-loop dynamics is given in part by
( )x Ax BG x e= − −& (2.11)
when a full-order observer is used
ˆ ( )e Ae A LC e= = −& (2.12)
Thus, the complete closed-loop dynamics is
0
x A BG BG x
e A LC e
− = −
&
& (2.13)
Suppose
0
A BG BGA
A LC
− = −
0sI A sI A BG sI A LC− = − + − + = (2.14)
The closed-loop eigenvalues are the eigenvalues of A-BG, the full-state feedback system;
and the eigenvalues of A-LC, the dynamics matrix of the observer. This is a statement of
the well-known separation principle, which permits one to design the observer and the
full-state feedback control independently, with the assurance that the poles of the closed-
loop dynamic system will be the poles selected for the full-state feedback system and
those selected for the observer.
18
2.2 High Gain Observers
The high gain observer has been used in the design of output feedback controllers
because of its ability to robustly estimate the unmeasured states and derivatives of the
output, while asymptotically attenuating disturbances [21, 22, 23, 24, 25, 26]. The
technique was first introduced by Esfandiari and Khalil and since then has been used in
approximately forty papers.
Consider the second-order nonlinear system:
1 2
2
1
( , )
x x
x x u
y x
== Φ
=
&
& (2.15)
Supposing that ( )u xγ= is a state feedback control that stabilizes the origin x = 0 of the
closed-loop system, the following observer is used
1 2 1 1
2 0 2 1
ˆ ˆ ˆ( )
ˆ ˆ ˆ( , ) ( )
x x h y x
x x u h y x
= + −
= Φ + −
&
& (2.16)
where, 0ˆ( , )x uΦ is a nominal model of the nonlinear function ( , )x uΦ . The estimation
error equations are
1 1 1 2
2 2 1 ( , )
x h x x
x h x x xδ= − +
= − +
&% % %
&% % %
(2.17)
where 0ˆ( , ) ( , ( )) ( , ( ))x x x x x xδ γ γ= Φ − Φ) )
%
1 1 1
2 2 2
ˆ
ˆ
x x xx
x x x
− = = −
%%
% (2.18)
19
As in any asymptotic observer, the observer gain 1
2
hH
h
=
is designed to achieve
asymptotic error convergence; that is, lim ( ) 0t
x t→∞ =% . In the absence of the disturbance
term ( , ),x xδ % asymptotic error convergence is achieved by designing the observer gain
such that the matrix
1
0
2
1
0
hA
h
− = −
(2.19)
is Hurwitz; that is, its eigenvalues have negative real parts. For this second-order system,
A0 is Hurwitz for any positive constants h1 and h2. In the presence ofδ , the observer gain
must be designed with the additional goal of rejecting the effect of the disturbance term
δ on the estimation error, x% . This is ideally achieved, for any disturbance termδ , if the
transfer function from δ to x% is identically zero. The observer gain can then be
designed 1 2 1h h>> >> , such that the transfer function 0H is arbitrarily close to zero.
0 2
11 2
11( )H s
s hs h s h
= ++ +
(2.20)
In particular, taking
1 21 2 2
,h hα αε ε
= = (2.21)
for some positive constant 1 2, andα α ε , with 1,ε << 0 0lim ( ) 0.H sε → =
The scaled estimation errors are defined as
11 2 2,
xxη η
ε= =%
% (2.22)
Then the newly defined variables satisfy the following equation
1 1 1 2εη α η η= − +&
20
2 2 1 ( , )x xεη α η εδ= − +& % (2.23)
This equation shows that reducing ε diminishes the effect of the disturbance termδ . It
also shows that, for smallε , the dynamics of the estimation error will be much faster than
the dynamics of x. But the change of variable (2.22) may cause the initial condition
1(0)η to be of order 1
( )Oε
, even when 1(0)x% is of order (1)O . With this initial condition,
the solution of (2.22) will contain a term of the form /1( ) at
eε
ε− for some a>0. While this
exponential mode decays rapidly, it exhibits an impulse-like behavior where the transient
peaks to 1
( )Oε
values before decaying toward zero. This behavior is known as the
peaking phenomenon, which is an intrinsic feature of any HGO design that rejects the
effect of the disturbance term δ in (2.17); that is, any design with 2 1 1.h h>> >> The peak
phenomenon could destabilize the closed-loop system, as the impulse-like behavior is
transmitted from the observer to the plant.
The HGO is basically an approximate differentiator, which can be easily seen in a
special case when the nominal function 0Φ is chosen to be zero; for which the observer
is linear. For the full-order observer (2.16) the transfer function from y to x)
is given by
1 22
2
1 2
1 ( / ) 1
( )
s
s ss s
εα ααε α ε α
+ → + +
as 0ε →
Realizing that the HGO is basically an approximate differentiator, the measurement
noise and unmodeled high frequency sensor dynamics will put a practical limit on how
small ε could be.
21
The combination of globally-bounded state feedback control with an HGO allows
for a separation approach where the state feedback control is designed to meet the design
objectives first. The HGO follows quickly enough to recover the performance achieved
under state feedback. Most papers that use an HGO incorporate this separation approach.
The HGO has recently been applied in many nonlinear systems. Henrik Rehbinder
and Xiaoming Hu [27] used it to estimate nonlinear pitch and roll for walking robots.
Seungrohk Oh and Hassan K.Khalil [28] discovered the use of nonlinear output-feedback
tracking using HGOs and variable structure control. E.S.Shin and K.W.Lee [29] designed
a robust output feedback control of robot manipulators. J. De Leon, K Busawon, and G.
Acosta [30] used the digital implementation of a HGO-based control for a rigid robot.
Nael H. El-Farra and Pamagiotis D. Christofides [31] designed an HGO with a bounded,
robust, optimal state feedback controller for a broad class of nonlinear processes with
uncertain variables and actuator constraints.
2.3 Sliding Mode Observers
Slotune, Hedrick, and Misawa [32] proposed the design of state observers using
sliding surfaces. Consider a second order system
1x f=&&
where f is a nonlinear, uncertain function of the state x. Based on the above discussion, an
observer structure of the following form is used
22
1 1 1 2 1 1
2 2 1 2 1
ˆ ˆ sgn( )
ˆ sgn( )
x x x k x
x x f k x
α
α
= − + −
= − + −
& % %
)& % %
(2.24)
where 1 1 1ˆx x x= −% , f
) is the estimated value of f, and the constants
iα ( 1,2)i = are chosen
as in a Luenberger observer in order to place the poles of the linearized system at desired
locations.
The error dynamics can be written:
1 1 1 2 1 1
2 2 1 2 1
sgn( )
sgn( )
x x x k x
x x f k x
αα
= − + −
= − + ∆ −
&% % % %
&% % %
The dynamic uncertainty f∆ is assumed to be explicitly bounded. The switching gain k1
is chosen as a bound on the steady-state estimation error on x2, and k2 is chosen to be
larger than the modeling errors.
2.4 Nonlinear Extended State Observers
The extended state observer is a novel observer for a class of uncertain systems,
proposed by Han [33, 34, 39].
Consider an th
n order nonlinear system described by
( ) ( 1)( , ,..., , ( )) ( )n ny f y y y w t bu t−= +& (2.25)
where f (.) is an uncertain function, ( )w t is the unknown external disturbance, u (t) is the
known control input, and y (t) is the measured output. The system is equivalent to
23
1 2
1
1
1
:
( )
n n
n
x x
x x bu
x a t
y x
+
+
= = + = =
&
&
&
1
( )n
i
i i
f fa t x w
x w=
∂ ∂= +∂ ∂∑ & & (2.26)
where f (.) has been regarded as an extended state 1nx − . Then the nonlinear continuous
observer is designed for system (2.25):
1
1 2 01 1
1 0
1 0 1 1
( )
( )
:
( )
( )
n n n n
n n n
e z y t
z z g e
z z g e bu
z g e
β
ββ
+
+ + +
= − = − = − +
= −
&
&
&
(2.27)
when t → ∞ , ( ) ( )i i
z t x t→ ( 1)i n∈ + , 1nz + is the estimate of the extended state f . The
observer (2.27) is the ESO for system (2.25). The observer can be given as:
1 2 01 1 1 1
1 0 1 0
1 1 1 1, 1
( ( ), , )
:
( ( ), , )
( ( ), )
n n n n n
n on n n
z z fal z y t
z z fal z y t b u
z fal z y t
β α δ
β α δβ α δ
−
+ + + +
= − − = − − + = − −
&
&
&
(2.28)
where b0 is the normal value of b, and fal (.) is defined as:
1
( ),
( , , ),
sign
fal
α
α
ε ε ε δε α δ ε ε δ
δ −
>=
≤
0δ > (2.29)
The nonlinear function in (2.29), which is used to make the observer more efficient,
was selected heuristically based on experimental results. Intuitively, it is a nonlinear gain
function where small errors correspond to higher gains, and large errors correspond to
smaller gains. When the error is small, it prevents excessive gain, which causes high
frequency chattering in some simulation studies. Figure 3 illustrates the difference
24
between the linear and nonlinear gain. If i
α are chosen as unity, then the observer is
equal to the well-known Luenberger observer. The ESO does not include a model, yet
can reconstruct states reliably for nonlinear plants. Because it does not use a model, it is
simpler and easier to construct, easier to implement due to its efficiency in many cases,
and freer from model uncertainties such as parameter variations and external
disturbances. Therefore, in this case, it is robust. If part of 1 2( , , , )f t x x w , says
1 1 2( , , , )f t x x w is known, and then it should be incorporated into the observer as
1 2 01 1 1 1
1 1 1 0 1 0
1 1 1 1 1 1, 1
( ( ), , )
:
( , ,... , ) ( ( ), , )
( , ,..., , ) ( ( ), )
n n n n n n
n n on n n
z z fal z y t
z z f t z z w fal z y t b u
z h t z z w fal z y t
β α χ
β α δβ α δ
−
+ + + +
= − − = + − − + = − −
&
&
&
(2.30)
where 1 1( , ,...., , )n
h t z z w = 1 1( , ,..., , )n
f t z z w& . This will make the observer more efficient.
The ESO is not only a state observer, but also a disturbance observer. Since Han’s
observer uses nonlinear functions, it was named Nonlinear Extended State Observer
(NESO) and can be applied for MIMO systems also.
y = x
y = f( x, αααα , δδδδ )
x
y
1
δδδδ
Figure 3: Comparison of Linear and Nonlinear Gains
25
For example, considering an n-dimensional p-input q-output state equation
( )x Ax B u w
y Cx
= + +=
& (2.31)
where w is disturbance, a NESO is defined as
ˆ ˆ ( )
ˆ ( )x x x
x Ax Bu F e
x F e
ββ
= + −
= −
&
& (2.32)
where
ˆ( ) : ( ) ( )e t x t x t= − ;
ˆx
x : Estimation of Extended States;
( )F e 1 1: ( ( , , ),......, ( , , ))n n
fal e fal eα δ α δ= , a 1n× vector;
1 1( ) : ( ( , , ),......, ( , , ))x n n x x
F e fal e fal eα δ α δ+ += , a 1n× vector.
For
ˆx
Bw x= (2.33)
If B has full column rank, the unique solution w can be obtained from the above equation
[35].
26
2.5 Active Disturbance Rejection Control
Based on ESO, ADRC is developed. Consider a second-order system:
1 2
2 1 2 0
1
( , , , )
x x
x f t x x w b u
y x
= = + =
&
& (2.34)
where 1 2( , , , )f t x x w is the uncertainty of the system. The uncertainty could be external
and internal disturbances, such as measurement noise, sensor noise, frictions, backlash,
and hysteresis, or the variation and uncertainties of the plant dynamics. The inputs of the
system are ,w u , the output is y , and 0b is the known constant. ADRC is applied to
counteract the uncertainty by the estimated signals [36]. The structure of the controller is
illustrated in the following figure.
Profile+_
+_
Non-linearCombination +_
PlantControlled
ExtendedState Observer
(ESO)
1/b0 b0
v(t) v2(t)
v1(t)
u0(t) u(t) y(t)
z3(t)
z2(t)
z1(t)
w(t)
Figure 4: Structure of ADRC
ADRC mainly consists of three parts: the ESO, the Profile Generator, and the
Nonlinear Combination. Using ESO given above to (2.34), we get the z1, z2, z3, which are
27
the estimated 1x , 2x and 1 2( , , , )f t x x w , respectively. The plant can now be dynamically
compensated by
0 3 0( ) ( ) ( ) /u t u t z t b= − (2.35)
Combining (2.34) with (2.35), we get
1 2
2 0 0
1
x x
x b u
y x
= = =
&
& (2.36)
The control problem of (2.34) is now simplified to a double integrator control problem
(2.36). Here the nonlinear combination takes the form:
0 ( ) ( , , ) ( , , )p p p p d d d d
u t k fal k falε α δ ε α δ= + (2.37)
where 1 1Pv zε = − , 2 2d
v zε = − are error and its differentiation, respectively. The
controller gains are p
k andd
k . The nonlinear controller performs better than the linear
controller [39], providing good gain scheduling and completely agreeing with the
intuition obtained from working with practical problems. As a matter of fact, many fuzzy
logic controllers exhibit these kinds of characteristics on their error surface. ADRC is
defined as the control method where the value of 1 2( , , , )f t x x w is estimated in real time
and compensated by the use of the control signal u . This control law can also be viewed
as an integrator-less PID because the dynamically compensated plant provides the
integral action.
28
2.6 Summary
Observer structures usually need to include a plant model in their equations which
inevitably accompanies some practical burdens. Without a model, observers cannot be
constructed: even if it is available, unless it is accurate enough, a reliable state
reconstruction could not be expected. Even when a model is accurate enough, the
observer could often become too complicated (because of model complexity) to have any
practical use, especially on a real-time basis. The advantages and benefits of an accurate
and efficient model cannot be emphasized too much, when it is available, unless such a
model is difficult to obtain. HGOs, SMOs, and ESOs are three kinds of observers
designed for the plant in the presence of disturbances, dynamic uncertainties, and
nonlinearities in practical applications.
29
CHAPTER III
COMPARISON OF ADVANCED STATE OBSERVER DESIGN TECHNIQUES
This chapter presents a comparison study of the characteristics and performances of
the HGO, SMO, and ESO. These observers were originally proposed to address the
dependence of the classical observers, such as the Kalman Filter and the Luenberger
observer, on the accurate mathematical representation of the plant. Several novel
nonlinear gain functions are proposed to address the difficulty of dealing with unknown
initial conditions. The simulations conducted give insight into observer behavior in both
open-loop and closed–loop scenarios. The experimental results are also provided to give
realism.
30
3.1 Simulation Results
A typical servo motor plant is made up of a motor, a servo drive amplifier, a system
of gears, and a load. This plant is used for software and hardware in the loop simulation.
A current drive is typically used in high performance servo motor systems to reduce the
order of the mathematic model of the servomotor by eliminating the effect of the
inductance. In motion control literature, a servo motor can be approximated as a linear,
time-invariant equation of the following form:
c b
y y uJ J
= +&& & (3.1)
where y is position, u is the motor current, J and, c are the total inertia and total viscous
friction reflected to the load, and b is the total hardware gain which typically includes the
gear ratio. The transfer function is
( )( )
bG s
s Js c=
+ (3.2)
3.1.1 Open-loop Comparison
The criteria for comparison is based on the robustness of the performance with
respect to the uncertainties of plant and the observer tracking errors, both at steady state
and during transients. Simulations are conducted to give insight into observer behavior in
both open-loop and closed-loop scenarios.
An industrial motion control test-bed made by ECP [38] is used for simulation. Its
linear model was derived as:
1.41 23.2y y u= − +&& & (3.3)
31
where y is the output position, and u is the control voltage sent to the power amplifier
that drives the motor. Initially, no friction, disturbance or backlash is intentionally added.
The model can be written as:
1 2
2 2
1
23.2 1.41
x x
x u x
y x
= = − =
&
& (3.4)
The HGO for the model (3.4) is
1 2 1 1
2 2 2 1
ˆ ˆ ˆ( )
ˆ ˆ ˆ23.2 1.41 ( )
x x h y x
x u x h y x
= + −
= − + −
&
& (3.5)
The observer gains are adjusted as
1 21 2 2
,h hγ γε ε
= = (3.6)
where 0 1ε< << .
The SMO is designed as
1 1 2 1
2 2 2 2
ˆ ˆ sgn( )
ˆ ˆ23.2 1.41 sgn( )
x e x k e
x e u x k e
γ
γ
= − + −
= − + − −
&
& (3.7)
The extended state-space model for (3.4) is
1 2
2 3
3
1
23.2
0
x x
x x u
x
y x
= = + = =
&
&
&
(3.8)
where 21.41x− is treated as 1 2( , , , )f t x x w , as well as an extended state, 3x .
32
The NESO is given as:
1 2 1 1
2 3 2 2
3 3 3
( )
( ) 23.2
( )
z z g e
z z g e u
z g e
ββ
β
= + = + + =
&
&
&
(3.9)
where e=y-z1. ig (.) 1,2,3i = are the same as (2.29). The Matlab/Simulink package from
Mathworks was used for the simulation in this research. The simulation model is shown
in Figure 5.
Figure 5: Open Loop Simulation Model
Pole placement is used to determine the position of the poles at –4.2, {γ1 , γ2 } in
HGO and SMO, and {βi, i = 1,2,3}in NESO. In addition, ε=0.2 is used for the HGO,
k1=0.5 and k2=15 are used for the SMO, αi = {1, .5, .25} and δ=10-3
are used for the
NESO. All three observers were implemented digitally with a sampling rate of 1 kHz.
The output measurement is corrupted by white noise to make the comparison realistic.
The quality of observers is measured by the speed and accuracy of the states of the
33
observer converging to those of the plant. To make the comparison fair, the parameters of
the observers are adjusted so that their sensitivities to measurement noise are roughly the
same. The exact outputs of y and y& are obtained directly from the simulation model of
the plant to calculate the state estimation error.
For open-loop tests, the input to the plant is a step function, and the observers are
evaluated according to their capability in tracking the step response. The tests were run
in three conditions:
• Nominal plant;
• Nominal plant plus coulomb friction;
• Nominal plant with 100% increase in inertia.
The same set of observer parameters are used in all simulations. Figure 6 illustrates the
position and velocity estimation errors for the nominal plant in terms of the tracking
errors for y and y& .
Figure 6: Estimation Error of the Nominal Plant
0 1 2 3 4 5- 0 . 0 2 5
- 0 . 0 2
- 0 . 0 1 5
- 0 . 0 1
- 0 . 0 0 5
0
0 . 0 0 5
0 . 0 1
0 . 0 1 5
T i m e ( s e c . )
Y-Z
1 e
stim
atio
n e
rro
r
N E S O
S M O
H G O
0 1 2 3 4 5- 0 . 4
- 0 . 3
- 0 . 2
- 0 . 1
0
0 . 1
0 . 2
0 . 3
T i m e ( s e c . )
Y2
-Z2
estim
atio
n e
rro
r
N E S O
S M O
H G O
34
All three observers perform well in steady-state and have roughly the same
accuracy and sensitivity to the noise. As expected, NESO takes longer to reach steady-
state, because it does not assume the knowledge of the plant dynamics. Interestingly, z3
converges to the unknown function f = -1.41 y& , as shown in Figure 7.
0 1 2 3 4 5- 2 5
- 2 0
- 1 5
- 1 0
- 5
0
5
T i m e ( s e c . )
Z 3 ( t )
a ( t )
Figure 7: x3=a (t) and its Estimation z3
Figure 8 illustrates the tracking errors for the plant with added coulomb friction.
Only the estimated errors of NESO are zero, clearly demonstrating that NESO is much
more robust than HGO and SMO in the presence of disturbance. Once again, z3
converges quickly and accurately to the combination of unknown dynamics and
disturbance, a(t) = ( , , )f y y w& , as shown in Figure 9. Figure 10 and 11 illustrate the
simulation results for the plant with a 100% increase of inertia. Compared with the other
two observers, the NESO is the quickest in following the states. Its performance is the
best overall, followed by SMO.
35
Figure 8: Estimation Error of the Plant with 0.5N-m Coulomb Friction
0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 4 . 5 5- 2 5
- 2 0
- 1 5
- 1 0
- 5
0
5
T i m e ( s e c . )
Co
ulo
mb
fri
ctio
n
Z 3 ( t )
a ( t )
Figure 9: Disturbance and its Estimation
0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 4 . 5 5- 0 . 0 7
- 0 . 0 6
- 0 . 0 5
- 0 . 0 4
- 0 . 0 3
- 0 . 0 2
- 0 . 0 1
0
0 . 0 1
0 . 0 2
T i m e ( s e c . )
Y-Z
1 e
stim
ati
on
err
or N E S O
S M O
H G O
0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 4 . 5 5- 1 . 8
- 1 . 6
- 1 . 4
- 1 . 2
- 1
- 0 . 8
- 0 . 6
- 0 . 4
- 0 . 2
0
0 . 2
T i m e ( s e c . )
Y2
-Z2
es
tim
ati
on
err
or
N E S O
S M O
H G O
36
0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 4 . 5 5- 0 . 0 6
- 0 . 0 5
- 0 . 0 4
- 0 . 0 3
- 0 . 0 2
- 0 . 0 1
0
0 . 0 1
0 . 0 2
T i m e ( s e c . )
Y1
-Z1
es
tim
atio
n e
rro
r
N E S O
S M O
H G O
Figure 10: Estimated Error of the Plant with 100% Change of Inertia
0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 4 . 5 5- 2 5
- 2 0
- 1 5
- 1 0
- 5
0
5
T i m e ( s e c . )
Dis
turb
an
ce
Z 3 ( t )
a ( t )
Figure 11: Disturbance and its Estimation
3.1.2 Selection of Nonlinear Gains for NESO
The nonlinear gain function for NESO was first proposed by J. Han [39]. The
research for this dissertation revealed that, for the plant with unknown initial conditions,
a new nonlinear function i
f (.), as shown in (3.10), could be used in NESO to avoid
significant transient estimation error:
0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 4 . 5 5- 0 . 0 6
- 0 . 0 5
- 0 . 0 4
- 0 . 0 3
- 0 . 0 2
- 0 . 0 1
0
0 . 0 1
0 . 0 2
T i m e ( s e c . )
Y1
-Z1
esti
ma
tio
n e
rro
r
N E S O
S M O
H G O
0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 4 . 5 5- 1 . 4
- 1 . 2
- 1
- 0 . 8
- 0 . 6
- 0 . 4
- 0 . 2
0
0 . 2
0 . 4
T i m e ( s e c . )
Y2
-Z2
es
tim
atio
n e
rro
rN E S O
S M O
H G O
37
2 1 2
1 2
1
( )*( )* ,( , , )
* , , 0
i i i
i i i
i
k e sign e k k ef e k k
k e e
δ δδ δ
+ − >= ≤ >
(3.10)
with 1 2, 0i ik k > . Furthermore, by choosing 0iα < in (2.29), the transient error was
significantly reduced. Three curves from (2.29) and (3.10) are shown in Figure 12 to
illustrate the differences. As in (2.29), δ defines the range of a high gain section where
the observer is very aggressive. This range is usually small.
δδδδ - δδδδ x
y( . )i
f
0
g(.),
-..-..
-.-.-
----
g(.),
0 1α< <0α <
Figure 12: Nonlinear Gain Functions
The following simulation tests were performed for the comparisons of three gain
functions for NESO as shown in Figure 12. The initial conditions are set as
(0) 10y = (rev) and (0) 0y =& . The observer poles used in simulation are placed at –8
rad/sec. The simulation results are shown in Figure 13.
The corresponding gain parameters that are used in simulation are:
(.)if :
11 12 13 1 2 324, 192, 512, 1,k k k δ δ δ= = = = = =
21 22 23200, 60, 130.k k k= = − = −
1(.)g : 11 12 1324, 192, 512,k k k= = = 1 2 3 1δ δ δ= = = ,
1 2 32, 4, 4.α α α= = − = −
2 (.)g : 11 12 1324, 192, 512,k k k= = = 1 2 3 1δ δ δ= = = ,
1 2 32, .2, .2.α α α= = =
38
Figure 13: Estimated Error of the Plant with Initial Values
The simulation results indicate clearly that the 1(.)g and the (.)if
function achieve the
best performance with the smallest tracking errors. The results also suggest that the
negative power should be used in (2.29) to counter unknown initial conditions.
0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6- 1
- 0 . 5
0
0 . 5
1
1 . 5
2
2 . 5
3
3 . 5
4
T i m e ( s e c . )
Y1
-Z1
es
tim
atio
n e
rro
r
f i ( . ) - . - . - . g 1 ( . ) g 2 ( . ) . . . . . .
0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 1 . 6 1 . 8 2- 1 4
- 1 2
- 1 0
- 8
- 6
- 4
- 2
0
2
T i m e ( s e c . )
Y2
-Z2
esti
ma
tio
n e
rro
r
f i ( . ) - . - . - . g 1 ( . ) g 2 ( . ) . . . . . .
0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 1 . 6 1 . 8 2- 3 5
- 3 0
- 2 5
- 2 0
- 1 5
- 1 0
- 5
0
5
T i m e ( s e c . )
Y3
-Z3
es
tim
atio
n e
rro
r
f i ( . ) - . - . - . g 1 ( . ) g 2 ( . ) . . . . . .
39
3.1.3 Closed-loop Comparison
Based on their open loop performance, NESO and SMO are evaluated in a closed-
loop feedback setting, such as the one in Figure 14 for NESO. The profile generator
provides the desired state trajectory in both y and y& , using an industry standard
trapezoidal profile. Based on the separation principle, the controller is designed
independently, assuming that all states are accessible in the control law.
Profile
Generator+_
PD +_ Plant
Extended
State Observer
(ESO)
1/b0 b0
r(t)
v1(t)
u0(t) u(t) y(t)
z3(t)
z2(t)
z1(t)
w(t)
Figure 14: Observer-based State Feedback Control Configurations
In the case of NESO, the extended state information, z3, which converges to x3=f
(y, y& , w), is used to compensate for the unknown f (y, y& , w). In particular, the control law
is
3 0
( ) /u z Ke b= − + (3.11)
where e=[v1-z1, -z2] T
and K is the state feedback gain that is equivalent to a proportional-
derivative (PD) controller design. Note that2
z− , instead of1 2( )v t z−& , is used to avoid
taking the derivative of the set point. It is also used to make the closed-loop transfer
function a pure second-order without a zero. Substituting (3.9) in (2.29),
3
( ( , , ) )y f y y w z Ke= − +&& & (3.12)
40
K can be determined via pole-placement. More details about this design strategy can be
found in [40-43].
For SMO-based state feedback design, only the position and velocity estimates, z1
and z2, are available and the corresponding PD design yields the following closed-loop
system,
0( , , ) 1.41y f y y w Ke y Ke= + = − +&& & & (3.13)
Note that the extended state, z3, is not available in SMO. The poles of the observers are
selected as –12rad/sec. For the controller design, the closed-loop poles are placed at –15
rad/sec. In addition, k1=0.5 and k2=15 are used for SMO, and αi = {1, 0.5, 0.25} and
δ=10-3
are used for NESO. For the sake of fairness, the same observer and control poles
are used for both NESO- and SMO-based designs. The main difference in design is that
the NESO-based method assumes no knowledge of f (y, y& , w).
Simulation results are shown in Figures 15, 16, and 17. Both control systems have
similar output responses for the nominal plant. However, as soon as unknown friction or
disturbances are introduced, the differences between NESO and SMO become apparent;
indicating that the NESO-based design has inherent robustness against the uncertainties.
By estimating f (y, y& , w), instead of modeling it, the controller becomes independent of it.
0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 4 . 5 5- 0 . 0 5
0
0 . 0 5
0 . 1
0 . 1 5
0 . 2
0 . 2 5
0 . 3
T i m e ( s e c . )
Po
sitio
n e
rro
r (r
ev.)
N E S O . . . . . . . S M O
Figure 15: Nominal Responses of the Control Systems
41
0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 4 . 5 5- 0 . 0 5
0
0 . 0 5
0 . 1
0 . 1 5
0 . 2
0 . 2 5
0 . 3
T i m e ( s e c . )
Po
sit
ion
err
or
(re
v.)
N E S O . . . . . . . S M O
Figure 16: Simulation Results with 0.5 N-m Coulomb Frictions
0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 4 . 5 5- 0 . 0 5
0
0 . 0 5
0 . 1
0 . 1 5
0 . 2
0 . 2 5
0 . 3
T i m e ( s e c . )
Po
sitio
n e
rro
r (r
ev
.)
N E S O . . . . . . . S M O
Figure 17: Simulation Results with Sinusoid Disturbance
3.2 Hardware Test Results
An industrial motion control test-bed is used to verify the above results and show
potential practical applications. The setup includes a PC-based control platform and a DC
brushless servo system, shown in Figures 18 and 19.
42
0-5V
Figure 18: Experimental Setup
PC-basedADRC
DACDC Singal
Conditioning
DC drive
(2 channels)
Electro-
mechanicalPlant
Encoders
Quadrature
CountingBoard
Disturbance
Signal
Figure 19: A Diagram for DC Brushless Servo System
The servo system includes a power amplifier, an encoder that provides the position
measurement, and two motors: one as an actuator, and the other as a disturbance source.
The inertia, friction and backlash are all adjustable, making it convenient to test the
control algorithms. The input to the plant is a voltage signal to the servo current
Plant DC drive
Signal
Generator
DC Signal Conditioning
LS7166
Data acquisition
board (Clock 8254,
DAC, RS232)
+/-
3.5V
43
amplifier. The amplifier input is rated for +/- 10 volts peak and +/- 3.5 volts safe. A safe
value is used because the servo drive for the plant is a dumb amplifier, meaning there is
no over-current protection. If the safe limit were not used, a control scheme would have
to be implemented to prevent damage to the drive. For details on the current drive, see
[38]. A Pentium 133 MHz PC running in DOS is programmed as the controller. It
contains a CIO-DAS16/F data acquisition card to read the position encoder output signal.
This card has two 12-bit analog output channels that are configured 0-5 volts. For details
on setting and programming this card, see [44]. Since the output from the data
acquisition card is 0-5 volts, and +/-3.5 volts is required by the servo amplifier, additional
signal conditioning circuit had to be added. Since the feedback from the control loop is a
quadrature encoder signal, a special quadrature encoder data acquisition card is used
(LS7166 by US Digital). This card can accept input frequencies from DC to 10 MHZ and
is capable of reading four encoders simultaneously. The sampling frequency is 1 kHz.
The output of the controller is limited to ± 3.5 V. The drive system has a dead zone of
± 0.5 V.
The plant is modeled approximately as it was in (3.1). Initially, no friction,
disturbance or backlash is intentionally added. To verify the effectiveness of the NESO,
the same controller poles are used for the NESO-based and SMO-based control systems.
The response of the nominal system is plotted in Figure 19. Figure 20 shows the response
of the plant having coulomb friction. The results indicate that the NESO-based controller
performs better in both steady state accuracy and transient response. A chattering
problem is also apparent in the SMO-based design.
44
In general, the hardware test results are consistent with those of the software
simulations.
Figure 20: Responses to the Nominal Plant
0 1 2 3 4 5 6 7 8 9 1 00
0 . 2
0 . 4
0 . 6
0 . 8
1
1 . 2
1 . 4
Po
sit
ion (
rev
.)
N E S O . . . . . . . S M O
0 1 2 3 4 5 6 7 8 9 1 0- 0 . 0 5
0
0 . 0 5
0 . 1
0 . 1 5
0 . 2
0 . 2 5
0 . 3
0 . 3 5
Po
sit
ion
err
or
(re
v.)
0 1 2 3 4 5 6 7 8 9 1 0- 2
- 1 . 5
- 1
- 0 . 5
0
0 . 5
1
1 . 5
2
2 . 5
3
T i m e ( s e c . )
Co
ntr
ol (v
olta
ge
)
45
Figure 21: Responses to Coulomb Friction
0 1 2 3 4 5 6 7 8 9 1 00
0 . 2
0 . 4
0 . 6
0 . 8
1
1 . 2
1 . 4
Po
sitio
n (
rev.)
N E S O . . . . . . S M O
0 1 2 3 4 5 6 7 8 9 1 0- 0 . 0 5
0
0 . 0 5
0 . 1
0 . 1 5
0 . 2
0 . 2 5
0 . 3
0 . 3 5
Po
sitio
n e
rro
r (r
ev
.)
0 1 2 3 4 5 6 7 8 9 1 0- 1 . 5
- 1
- 0 . 5
0
0 . 5
1
1 . 5
2
2 . 5
3
3 . 5
T i m e ( s e c . )
Co
ntr
ol (v
olta
ge
)
46
3.3 Summary
A comparison study of advanced observer designs, including NESO, HGO, and
SMO, was performed. A gain modification method is proposed for the NESO to deal with
the unknown initial conditions. Both software simulation and hardware tests are
performed. The following observations are made based on the results:
• As a state estimator, NESO performs better than HGO and SMO. The robustness of
the NESO to plant uncertainty and external disturbance is inherent in its structure.
The chattering problem is the main drawback of the SMO in practical applications.
• The simulation and experimental results seem to justify the design concepts of NESO.
By augmenting the plant and treating the unknown dynamics as an extended state, an
alternative design method for state observers and an alternative to system
identification were discovered. That is, instead of trying to find a mathematical
expression of the dynamics and disturbances, a state observer can be built to estimate
it and compensate for it in real time.
47
CHAPTER IV
COMPARISON OF CONTROLLERS FOR MOTION
CONTROL APPLICATIONS
ADRC is a NESO-based nonlinear control strategy. The tuning procedure is
complicated. For the sake of simplification and ease for practical application, LADRC is
investigated in this chapter.
A comparison study of performances and characteristics of ADRC, LADRC and
loop shaping design method is presented. Simulation and experimental results are shown,
and the new tuning method of LADRC is given.
48
4.1 Linear Active Disturbance Rejection Control Design Concepts
ADRC represents a control strategy that is rather independent of the mathematical
models of the plants, thus achieving inherent robustness. An overview of this design
philosophy and the associated algorithms was introduced in Chapter II. Although the
unique advantages of this method have been proven in several practical applications, the
tuning of the controller can be challenging because of the number of nonlinear functions.
Rewrite the plant in (2.34) in state space form as
1 2
2 3 0
3
1
x x
x x b u
x h
y x
= = + = =
&
&
& (4.1)
where 3
,x f h f= = & . It is equal to
x Ax Bu Eh
y Cx
= + +=
& (4.2)
where
[ ]0
0 1 0 0 0
0 0 1 , , 1 0 0 , 0
0 0 0 0 1
A B b C E
= = = =
The linear version of ESO (LESO) can be designed as
ˆ( )
ˆ
z Az Bu L y y
y Cz
= + + −=
& (4.3)
where1 2 3
[ ]T
L β β β= , the observer gain vector, which is obtained using pole placement.
Combining (4.2) and (4.3), the error equation can be written as:
49
e
e A e Eh= +& (4.4)
where
, 1,2,3i i i
e x z i= − =
1
2
3
1 0
0 1
0 0
eA A LC
βββ
− = − = − −
(4.5)
Obviously, the LESO is BIBO-stable if the roots of the characteristic polynomial of e
A
3 2
1 2 3( )s s s sλ β β β= + + + (4.6)
are all in the left half-plane and h is bounded. Suppose the observer poles are all placed at
oω−
3 2 3
1 2 3( ) ( )
os s s s sλ β β β ω= + + + = + (4.7)
then
2 3
1 2 33 , 3 ,o o oβ ω β ω β ω= = = (4.8)
The choice of o
ω could be a trade-off between how fast the observer tracks the states and
how sensitive it is to the sensor noises.
The control law for the plant (4.1) is given as a
3 0
0
z uu
b
− += (4.9)
Ignoring the inaccuracy of the observer,
3 0 0( )y f z u u= − + ≈&& (4.10)
which is a unit-gain double integrator that can be controlled with a PD controller
0 1 2( )p d
u k r z k z= − − (4.11)
50
where r is the set point. This results in a pure second-order closed-loop transfer function
of
2
1( )
d p
G ss k s k
=+ +
(4.12)
The gains can be simply selected as
22d c p ck and kξω ω= = (4.13)
where c
ω and ξ are the desired closed-loop natural frequency and damping ratio. ξ can
be simply chosen as one to avoid any oscillations, and c
ω is determined based on the
transient response requirements.
The LESO-based control scheme is referred to as LADRC [45] which uses linear
observer and linear feedback gains in place of the nonlinear ones in (2.29) and (2.37).
Optimization of LADRC
The goal of controller design is to achieve the maximum closed-loop bandwidth,
subject to design constraints. This is denoted as the parameter optimization. There are
only two parameters in LADRC: o
ω , which represents the bandwidth of the observer;
and c
ω , which represents the bandwidth of the controller. The following is the procedure
for determining these optimal parameters:
Step 1: Select an 3 5o c
ω ω≈ � , where c
ω is the bandwidth of a transient profile;
Step 2: Set c o
ω ω= and simulate the LADRC in a simulation;
51
Step3: Increase both until the noise level and/or oscillations in the control signal and
output exceed the tolerance;
Step4: Increase or decrease c
ω and o
ω individually, to make trade-offs among different
design considerations such as the maximum error during the transient period, the
disturbance attenuation, and the magnitude and smoothness of the controller.
4.2 Loop Shaping Design Technique
The loop shaping design method is a frequency response-based control design
technique, and the only one to comprehensively address multiple design concerns, such
as transient response, disturbance rejection, stability margins, and noise. As a concept and
design tool, loop-shaping has helped practicing engineers greatly improving PID loop
performance and stability margins. Consider the system described in Figure 22.
P(s)
Prefilter Controller Plant
( )c
G s ( )pG s
-
e(t)
Disturbance
d(t)
+Output
y(t)
n(t)
+
Sensor Noise
r(t)
Reference
Input
u(t)
Figure 22: Feedback Control Configuration
52
Design Specifications:
• Command following
( ) ( )( )
1 ( ) ( ) 1( ) 1 ( ) ( )
p c
p c
p c
G s G sY sG j G j
R s G s G sω ω= ≈ ⇒
+� (4.14)
• Disturbance rejection
( )( )
( ) ( ) 1( ) 1 ( ) ( )
p
p c
p c
G sY ssmall G j G j
D s G s G sω ω= ⇒
+� (4.15)
• Sensor noise reduction
( ) ( )( )
( ) ( ) 1( ) 1 ( ) ( )
p c
p c
p c
G s G sY ssmall G j G j
N s G s G sω ω= − ⇒
+� (4.16)
• Robust stability
( ) ( ) 1
( ) ( ) 11 ( ) ( ) ( )
p c
p c
p c
G j G jG j G j
G j G j lm j
ω ωω ω
ω ω ω< ⇒ ≤
+ (4.17)
• Stability (gain/phase) margin
One pole roll-off at the crossover frequency
Constraints on ( ) ( )p c
G j G jω ω :
Low Frequency: ( ) ( ) 1p c
G j G jω ω �
High Frequency: ( ) ( ) 1p c
G j G jω ω �
Crossover Frequency: good gain and phase margin
The loop shaping design process consists of two steps: 1) converting all design
specifications to loop gain constraints, as shown in Figure 23; and 2) finding a controller
( )c
G jω to meet the specifications.
53
cωω
LowFrequency
Constraints
HighFrequency
Constraints
dB )()( ωω jGjG cp
decdB /20−
1ω 2ω
Figure 23: Loop Shaping
Normally, the desired loop gain can be characterized as
1
2
1 1( ) ( ) ( )
1 1
m
p c n
c
sL s G S G s
ss s
ω
ω ω
+ = = + +
(4.18)
where c
ω is the bandwidth, and 1 2, , 0,c c
m andω ω ω ω< > ≥ 0n ≥ are selected to meet
constraints shown in Figure 23. Both m and n are integers. The default values for 1ω and
2ω are
1 /10c
ω ω= (4.19)
2 10 cω ω= (4.20)
which yield a phase margin of approximately ninety degrees. The controller can be
derived from (4.18)
11
2
1 1( ) ( )
1 1
m
c pn
c
sG s G s
ss s
ω
ω ω
−+ = + +
(4.21)
54
An additional constraint on n is that
1
2
1 1( )
1 1
pn
c
G ss s
ω ω
−
+ +
is proper. (4.22)
This design is valid only if the plant is minimum phase. For a non-minimum phase plant,
a minimum phase approximation of 1( )pG s− should be used instead. A compromise
between 1ω and the phase margin can be made by adjusting 1ω upwards, which will
increase the low frequency gains as the cost of reducing the phase margin. Similar
compromise can be made between the phase margin and 2ω .
Optimization of loop shaping controller
The controller used here has two parameters, p
k and c
ω . Usually, 0.1 0.7p
k ≈ � .
cω
can be optimized using the following steps:
Step1: Set an initial value of c
ω based on the bandwidth of required transient response;
Step2: Increase c
ω while performing tests on the simulator, until the control signal
becomes too noisy and/or too bumpy, or the output is oscillatory.
55
4.3 Simulation Study and Comparison
In order to accurately compare the three control algorithms, the simulation system
was set up for the servo motor model, shown in Figure 24. The controller blocks are s-
functions written in c-code.
y
u1
e
Zero-Order
Hold6
Zero-Order
Hold5
Zero-Order
Hold4
Zero-Order
Hold3
Zero-Order
Hold2
Zero-Order
Hold1
In1Out1Out2Out3
In1Out1Out2Out3
In1
Out1
Out2
Out3
Step2
Step1
Step
Out1
Out2
ProfileIn1
In2Out1
LOOP
In1
In2
Out1Out2Out3
LADRC1
In1In2In3
Out1
ADRC
u
Workspace2
e
Workspace1
y
Workspace
Figure 24: The Simulink Diagram
Rewrite the motion control model in (3.3) as below:
23.24( )
( 1.41)p
G ss s
=+
Selecting m=n=2, the loop shaping controller can be obtained from (4.21)
2
1
2
2
2
1
2
2
1 1 ( 1.41)( )
23.241 1
( ) ( 1.41)
23.24 ( 1)( 1)
c
c
c
s s sG s
ss s
s s
s ss
ω
ω ω
ω
ω ω
+ + = + +
+ +=+ +
(4.23)
56
In the simulations, ( )c
G s is digitized as ( )c
G z using the Tustin or Trapezoidal method at
a sampling time of 1ms.
2( 1)
( 1)
zs
T z
−=+
(4.24)
Assume that 1 2* , 10*c c
kpω ω ω ω= = ,( )
( )( )
c
u zG z
e z= , where ( )u z and ( )e z are the
control signal and the error signal. Combining (4.23) with (4.24), the digitized loop
shaping controller is calculated as
u[k] = (a12-a5*a11*u [k-4]-(a5*a10-8*a11)*u [k-3]-(a11*a4 -8*a10+a5*a9)*u [k-2]-
(a4* a10-8*a9)*u [k-1])/a4/a9.
where k = 0……. n, integer;
a1 = 4 *c
ω + 4 * kp *c
ω *c
ω * h + kp * kp *c
ω *c
ω *c
ω * h * h;
a2 = 2 * kp * kp * c
ω * c
ω * c
ω * h * h – 8 *c
ω ;
a3 = kp * kp * c
ω * c
ω * c
ω * h * h + 4 *c
ω – 4 * kp * h * c
ω *c
ω ;
a4 = 4 + 2 *c
ω * h;
a5 = 4 – 2 * h *c
ω ;
a6 = 200 * c
ω * c
ω * h + 141 * c
ω * c
ω * h * h;
a7 = 282 * c
ω * c
ω * h * h;
a8 =141 * c
ω * c
ω * h * h – 200 * c
ω * c
ω * h;
a9 = 4 + 40 * h * c
ω + 100 * c
ω * c
ω * h * h;
a10 =200 * c
ω * c
ω * h * h - 8;
a11=100 * c
ω * c
ω * h * h – 40 * h * c
ω + 4;
57
a12 = (a1 * a6 * e[k] + (a1*a7+a2*a6)*e [k-1] + (a1*a8+a2*a7+a3*a6) *e [k-2] + (a2*a8
+a3*a7)*e [k-3] +a3*a8*e [k-4])/23.24.
In addition, a profiler is added into the loop shaping control system to reduce the
overshoot.
1( 0)
11
c
k
sw k
>+
The ADRC controller is formulated as
ESO:
1 2 1 1 1 1
2 3 2 1 2 2 0
3 3 1 3 3
( , , )
( , , )
( , , )
z z fal z y
z z fal z y b u
z fal z y
β α δβ α δ
β α δ
= + − = + − + = −
&
&
&
where 2 3
1 2 33 , 3 ,o o oβ ω β ω β ω= = = ; 0 0;ω > 0; 0.i i
α δ> >
0 ( ) ( , , ) ( , , )p p p p d d d d
u t k fal k falε α δ ε α δ= +
0 3 0( ) ( ) ( ) /u t u t z t b= −
where 1 1Pv zε = −
, 2 2dv zε = − ; 0, 0; ,
p d a dδ δ α α> > are real.
The LADRC controller is designed as
LESO:
1 2 1 1
2 3 2 1 0
3 3 1
( )
( )
( )
z z z y
z z z y b u
z z y
ββ
β
= + − = + − + = −
&
&
&
where 2 3
1 2 33 , 3 ,o o oβ ω β ω β ω= = = , 0 0.ω >
Assume 1ζ = ,
2
0 2
1
2( ) ( )
16.5 16.5
( ) ( )
c cu e t z t
e t r z t
ω ω= −
= −
58
0 3 0( ) ( ) ( ) /u t u t z t b= −
The parameters for these three controllers are shown in Table I. The sample time h is
0.001 second. The simulation responses of the nominal plant are shown in Figures 25, 26,
and 27. The parameter tuning is based on the same noise sensitivities of the control
signals. Note that the output of the ADRC system has much less transient error than the
LADRC and loop shaping control (LSC) systems because the ADRC controller has two
desired input signals, r and the derivative of r. The setting time and control signals of the
three systems are very similar.
Table I: Controller Parameters
BLOCK PARAMETERS
ADRC
a=20, β1= 3a, β2=3a2;
β3=a3, α1=1.0, δ1=0.01;
α2=0.5, δ2=0.1, α3=0.25;
δ3=0.1, kp=0.1, αp=0.4;
δp=0.01, kd=10, αd=1.5;
δd=0.001, b0=25;
LADRC
a=40, β1 = 3a, β2=3a2;
β3=a3, 30
nω = , b0=25;
LSC kp=0.7, 20c
ω = , k=0.5;
59
0 1 2 3 4 5 6 7 8 9 1 0 - 0 . 0 5
0
0 . 0 5
0 . 1
0 . 1 5
0 . 2
Po
sitio
n e
rro
r (r
ev.)
0 1 2 3 4 5 6 7 8 9 1 0 - 2
- 1
0
1
2
3
T im e ( s e c . )
Co
ntr
ol (v
olta
ge
)
Figure 25: Position Error and Control Signal of ADRC System
0 1 2 3 4 5 6 7 8 9 1 0 - 0 . 0 5
0
0 . 0 5
0 . 1
0 . 1 5
0 . 2
Po
sitio
n e
rro
r (r
ev.)
0 1 2 3 4 5 6 7 8 9 1 0 - 2
- 1
0
1
2
3
T im e ( s e c . )
Co
ntr
ol (v
olta
ge
)
Figure 26: Position Error and Control Signal of LADRC System
0 1 2 3 4 5 6 7 8 9 1 0 - 0 . 0 5
0
0 . 0 5
0 . 1
0 . 1 5
0 . 2
Po
sitio
n e
rro
r (r
ev.)
0 1 2 3 4 5 6 7 8 9 1 0 - 2
- 1
0
1
2
3
T im e ( s e c . )
Co
ntr
ol (v
olta
ge
)
Figure 27: Position Error and Control Signal of LSC System
60
In order to test the robustness of the three control systems, the change of inertia and
torque disturbance are added into the plant separately. The parameters of the controllers
remain the same. Figures 28, 29, and 30 show the simulation responses of the plant with
100% inertia increased. The output of LSC system obviously has more overshoot than
the ADRC system. The setting time for the ADRC and LADRC systems is roughly the
same. Figures 31, 32, and 33 illustrate the responses of the plant with 20% torque
disturbance. Note that the system using LSC has the longest recovery time, followed by
ADRC.
0 1 2 3 4 5 6 7 8 9 1 0 - 0 . 0 5
0
0 . 0 5
0 . 1
0 . 1 5
0 . 2
Positio
n e
rror
(rev.)
0 1 2 3 4 5 6 7 8 9 1 0 - 4
- 2
0
2
4
T im e ( s e c . )
Contr
ol (v
oltage)
Figure 28: Position Error and Control Signal of ADRC System with Inertia Change
0 1 2 3 4 5 6 7 8 9 1 0 - 0 . 0 5
0
0 . 0 5
0 . 1
0 . 1 5
0 . 2
Po
sitio
n e
rro
r (r
ev.)
0 1 2 3 4 5 6 7 8 9 1 0 - 4
- 2
0
2
4
T im e ( s e c . )
Co
ntr
ol (v
olta
ge
)
Figure 29: Position Error and Control Signal of LADRC System with Inertia Change
61
0 1 2 3 4 5 6 7 8 9 1 0 - 0 . 0 5
0
0 . 0 5
0 . 1
0 . 1 5
0 . 2
Po
sitio
n e
rro
r (r
ev.)
0 1 2 3 4 5 6 7 8 9 1 0 - 4
- 2
0
2
4
T im e ( s e c . )
Co
ntr
ol (v
olta
ge
)
Figure 30: Position Error and Control Signal of LSC System with Inertia Change
4 5 6 7 8 9 1 0 - 8
- 6
- 4
- 2
0
2 x 1 0
- 3
Po
sitio
n e
rro
r (r
ev.)
0 1 2 3 4 5 6 7 8 9 1 0 - 2
- 1
0
1
2
3
T im e ( s e c . )
Co
ntr
ol (
volta
ge
)
Figure 31: Position Error and Control Signal of ADRC System with Torque Disturbance
4 5 6 7 8 9 1 0 - 8
- 6
- 4
- 2
0
2 x 1 0
- 3
Po
sitio
n e
rro
r (r
ev.)
0 1 2 3 4 5 6 7 8 9 1 0 - 2
- 1
0
1
2
3
T im e ( s e c . )
Co
ntr
ol (
volta
ge
)
Figure 32: Position Error and Control Signal of LADRC System with Torque Disturbance
62
4 5 6 7 8 9 1 0 - 8
- 6
- 4
- 2
0
2 x 1 0
- 3
Positi
on
err
or
(rev.)
0 1 2 3 4 5 6 7 8 9 1 0 - 2
- 1
0
1
2
3
T im e ( s e c . )
Co
ntr
ol (
voltag
e)
Figure 33: Position Error and Control Signal of LSC System with Torque Disturbance
4.4 Experimental Study and Comparison
The motion control setup in Chapter III is used for the hardware test. The
parameters of the controllers are the same as those that are used in simulations. The
results for the nominal plant are shown in Figure 34. Note that the system controlled by
LSC has overshoot. Again the ADRC system has the smallest transient error. The
following tests are performed individually in order to compare the robustness of the
controllers:
• Increase the load by 75% (replace two 0.5 Kg weights to the disc at radius of
6.6cm)
• Increase the friction significantly by adjusting the rubbing screw
• Introduce 30% torque disturbance using the disturbance motor
The performances are evaluated in Table II. Figure 35 shows the responses for the plant
with 30% torque disturbance at 4 seconds. Note that the output of LADRC system has the
63
shortest recovery time. Figure 36 illustrates the responses of the plant with friction. Note
that the performances of the LADRC and LSC systems are better than ADRC.
0 1 2 3 4 5 6 7 8 9 100
0.5
1
1.5P
os
itio
n (
rev
.)
0 1 2 3 4 5 6 7 8 9 10-0.2
0
0.2
Po
sit
ion
Err
or
(re
v.)
0 1 2 3 4 5 6 7 8 9 10-5
0
5
Time (sec.)
Co
ntr
ol
(vo
lta
ge
) ADRC LADRC -.-.-. LSC ......
Figure 34: Responses of the Nominal Plant
.
64
0 1 2 3 4 5 6 7 8 9 100
0.5
1
1.5
Po
sit
ion
(re
v.)
0 1 2 3 4 5 6 7 8 9 10-0.02
0
0.02
Po
sit
ion
err
or
(re
v.)
0 1 2 3 4 5 6 7 8 9 10-5
0
5
Time (sec.)
Co
ntr
ol
(vo
lta
ge
)
ADRC LADRC -.-.-. LSC ......
Figure 35: Responses of the Plant with 30% Torque Disturbance at 4 second
65
0 1 2 3 4 5 6 7 8 9 100
0.5
1
1.5
Po
sit
ion
(re
v.)
0 1 2 3 4 5 6 7 8 9 10-0.2
0
0.2
Po
sit
ion
err
or
(re
v.)
0 1 2 3 4 5 6 7 8 9 10-5
0
5
Time (sec.)
Co
ntr
ol
(vo
lta
ge
)
ADRC LADRC -.-.-. LSC ......
Figure 36: Responses of the Plant with Friction
66
Table II: Performances Comparison with ADRC, LADRC and LSC
Over-
Shoot (%)
Settling Time
(sec.)
Steady State
Error (rev.)
Nominal Case 0.00 0.60 84.889 10−×
Load Added 0.00 0.60 1.87 410−×
Friction Exerted 0.00 2.00 41 10−×
ADRC
30% of Tmax
Disturbance
0.00
2.00
84.889 10−×
Nominal Case 0.00 0.70 84.889 10−×
Load Added 0.00 0.70 84.889 10−×
Friction Exerted 0.00 0.80 41.5 10−×
LADRC
30% of Tmax
Disturbance
0.00
0.10
84.889 10−×
Nominal Case 3.00 0.70 56.254 10−×
Load Added 6.00 0.80 56.254 10−×
Friction Exerted 5.00 0.90 41 10−×
LSC
30% of Tmax
Disturbance
0.50
2.00
0.00
67
4.5 Summary
The simulation and experimental results presented above demonstrate that with the
optimal tuning method, the loop shaping control can be a viable solution for practical
applications. Since it is a model-based control method, compared with the LADRC and
the ADRC approaches, it has least disturbance rejection ability.
With the optimal tuning method, LADRC can outperform ADRC. The tuning for
the ADRC controller is done through trial-and-error. There is a lot of room for the
performance to be improved. If this problem could be solved, the ADRC system would
get better performance than the LADRC system.
68
CHAPTER V
NOVEL SOLUTIONS FOR THE ACTIVE MAGNETIC BEARING
BENCHMARK PROBLEM
Active magnetic bearings (AMBs) are bearings used to suspend rotor by magnetic
forces without any contact. They have many advantages over conventional bearings
because contact-free support eliminates mechanical friction, lubrication and wear
problems. In this chapter, based on a nonlinear model of an active magnetic bearing
system, nonlinear controllers are developed for the electromechanical system, using
nonlinear dynamics placement control scheme and LADRC approach. Simulation results
illustrate the performances of these methods.
69
5.1 Background
AMBs are currently used in many engineering applications (e.g., compressors,
milling spindles, flywheels, and blood pumps). Since they can be actively controlled, they
offer many other potential advantages, such as adjusting the stiffness of the suspending
load, eliminating vibration active damping, or providing an automatic balancing
capability.
There have been many reported methods for the design of AMB controllers. Most
of them are based on the linearized electromechanical system model such as
eigenstructure assignment, H ∞ control [46, 47], sliding mode control [48, 49], µ -
synthesis [50], Q -parameterization [51], and geometric approach [52]. The effective
range of these control systems is usually limited to a very narrow area near the
equilibrium point because the linearized model cannot express the exact behavior of the
system at any other operating points. In AMB systems, where the nonlinearity is
inherently strong, it may be more effective to design a control system based on the
nonlinear model rather than its linear approximation. Feedback linearization (FL) has
been widely accepted [53, 54]. However, FL sometimes cancels out the nonlinear terms
that should not be canceled and, consequently, requires unnecessary control efforts.
Backstepping has been applied to magnetic bearing systems [55-57], based on the
Lyapunov method, and it accounts for the nonlinearities of the AMB system.
Many current linear control design methods are restrictive because they require the
plant to be linear. Based on a nonlinear model of an AMB system, a nonlinear dynamics
placement (NDP) control scheme for the AMB benchmark system [58] is proposed. As
70
specified in [34, 42, and 43], nonlinear feedback mechanisms are, in general, superior to
the linear ones and should be explored. LADRC, is also proposed for this system
because of its strong disturbance rejection and simple tuning. The research is motivated
by the need to account for the periodic disturbance acting on the AMB system.
5.2 System Dynamics and Design Specifications
The AMB problem is briefly introduced in [58, 59]. The AMB benchmark system
consists of a rigid beam that is free to rotate about a pivot located at its center of mass
(see Figure 37) [59]. The beam’s angular position is controlled via two electromagnets
positioned at each end of the beam. A non-contacting displacement sensor measures the
rotation of the beam. An additional actuator, located at one of the ends of beam, is used to
apply a perturbation force that emulates a periodic, exogenous disturbance. This system
incorporates all of the typical nonlinearities of an AMB system. The plant is open-loop
unstable.
Figure 37: Benchmark experiment (from [59])
71
The system can be modeled by the differential equations (5.1) through (5.8) [58].
The parameters and the signals used for these equations are listed in Appendix B.
1 1 1
1( ( ) , ) ( )
Rh p g y g y sat u
N Nφ φ= − + + +& (5.1)
2 2 2
1( ( ) , ) ( )
Rh p g y g y sat u
N Nφ φ= − − − +& (5.2)
2
1 1[ ( ) ]f c p g y φ= + (5.3)
2
2 2[ ( ) ]f c p g y φ= − (5.4)
1 2( )L
f f dJ
θ = − −&& (5.5)
with system outputs given by
y Kθ= (5.6)
1 1( ( ) , )I h p g y g yφ= + + (5.7)
2 2( ( ) , )I h p g y g yφ= − − (5.8)
K converts the rotation of the beam to mils and K<0. The flux leakage function (.)p can
be described by the function1( ) ( )p x ax b
−= + , where the constants are given in Appendix B.
The nonlinear function h (.) is given as a look-up table in the simulation model in [58], as
shown in Figure 38.
72
Figure 38: Current as a function of flux and air gap length
The control objective is to regulate of the beam’s angular position θ, with minimum
employed magnetic flux 1φ and 2φ . The controller is designed to achieve the following
goals:
• High performance in terms of regulating the beam quickly and accurately;
• Better disturbance rejection;
• Strong robustness by increasing the tolerance of the control system to dynamic
variations such as coil resistant change;
• Simple design procedure and ease of implementation for real applications;
The main challenges in this problem include: (1) the system is highly nonlinear; (2)
there is a periodic disturbance torque, d, as shown in (5.5), with the magnitude limited to
less than 4 N-m and frequency between 10 and 50 Hz; and (3) the beam dynamics include
an unmodeled flexible mode.
73
5.3 The Proposed Solutions
The use of nonlinear gains has been contemplated by many researchers and reported
in literature; see for example [34, 42, 43]. Nonlinear feedback is known to have distinct
characteristics that can be explored in order to bring unique advantages to feedback
control. The nature of the AMB problems and the difficulty in its control design
motivates us to seek such a solution.
Note that the AMB problem described above is similar to a double integrator plant,
as seen in (5.5). Therefore, let’s consider a nominal double integrator plant
y w u= +&& (5.9)
where y is the output, u is the control signal, and w is the disturbance. With r as the
reference input, a typical linear control law is
1 2( )u k r y k y= − − & , 0i
k > i=1, 2 (5.10)
where k1 and k2 are determined to place the closed-loop poles at a predetermined location.
Such a control law, known as proportional-derivative (PD) controller, is widely used in
motion control industry. Difficulties may arise when there is a constant disturbance, w,
which induces a steady-state error. The common remedy is to add an integration term to
the controller, but the associated phase lag reduces the stability margin.
Instead of adding an integrator to address the steady-state error problem associated
with the PD controller in (5.10), an alternative method is to employ nonlinear gains,
instead of linear ones [34, 42, 43]. The main idea is to make the controller more sensitive
to small errors by increasing the gain in the small error region. As demonstrated, this
method not only relieves the controller of the integrator phase lag, but also results in
74
better disturbance rejection and transient response. That is, nonlinear dynamics can be
purposely placed in the closed-loop system to make it behave better. One such alternative
controller of (5.10) is
1 1 1 2 2 2( , , ) ( , , )u k fal e k fal yα δ α δ= − & (5.11)
where ik >0, e r y= − , 1, 2i = , (.)fal is a nonlinear function, defined as the same as
(2.29)
1
( ). ,( , , )
. ,
asign x x when x
fal xx when x
α
δα δ
δ δ−
>= <
Here, α ,δ are two parameters of the nonlinear function. For 1α= , it is equivalent to the
linear function of y x= . For 0 1,α< < , the nonlinear controller provides higher gain when
error is small and lower gain when error is large. The idea is that making the controller
more responsive to small error helps avoid the use of the integrator control. Here, δ is a
small positive number applied to create a small linear gain area around the origin to avoid
singularity. Combining (5.9) and (5.11), the closed-loop system is purposely made
nonlinear and is, therefore, called NDP.
The AMB problem is solved in two steps: (1) by treating the mechanical part of the
plant as a double integrator, an NDP design is carried out as discussed above; and (2) a
proportional controller is designed for the flux loop.
Combining (5.5) and (5.6) and assuming 0,d = the beam angular acceleration can
be written as
1 2( )
KLy K f f
Jθ= = −&&&& (5.12)
75
Breaking (5.12) up into two parts, the plant can be represented as
0y u=&& (5.13)
and
0 1 2( )
KLu f f
J= − (5.14)
Applying the NDP approach to (5.13), the control law is chosen as
0 1 1 1 1 2 2 2 2( , , ) ( , , )u k fal e k fal yα δ α δ= − & (5.15)
where e=r-y, and r is the desired position.
To reduce the noise sensitivity, an approximate differentiator, s/(τ s+1)2, is used in
place of the pure differentiation, s, whereτ is the filter time constant to be adjusted
according to the noise level in the control signal. All design parameters can be initially
selected from a linear design and the nonlinearity is gradually added into the simulation
for maximum performance and robustness.
Once u0 is obtained, 1f and 2f can be determined from (5.14). Since both 1f and
2f only take non-negative values, according to (5.3) and (5.4), the following scheme is
used to compute them:
0 1 2 0
0 1 0 2
0, 0, ;
0, , 0.
Ju f f u
K L
Ju f u f
K L
≥ = =
< = = (5.16)
This scheme switches between the two actuators and effectively utilizes the power
supply voltage 1( )sat u and
2( )sat u [61]. Next, the desired flux *φ is obtained by solving
(5.3) and (5.4):
76
1*
1
/
( )
f c
p g yφ =
+, 2*
2
/
( )
f c
p g yφ =
− (5.17)
and used as the set point for (5.1) and (5.2). Assuming the actuators are operating in
linear regions, (5.1) and (5.2) can be rewritten as
1 1 1
1RI u
N Nφ = − +& (5.18)
2 2 2
1RI u
N Nφ = − +& (5.19)
Because this power electronic component has a much higher bandwidth than its
mechanical counter part, a simple proportional control is usually sufficient. That is, the
control law that makes φ follow *φ is designed as:
*
1 3 1 1 1( )u Nk RIφ φ= − + (5.20)
*
2 4 2 2 2( )u Nk RIφ φ= − + (5.21)
where k3 and k4 are the feedback gains that can be determined using any linear design
method.
The key component here is to use a nonlinear feedback in (5.15) to make the
controller more sensitive to small errors and, therefore, aggressively regulate the beam
position in the presence of external disturbances. It also reduces or even eliminates the
need of an integral control, thus enhancing the stability margins of the closed-loop
control system. This approach has been successfully tested in similar motion control
applications, where it exhibits good disturbance rejection characteristics, and was tested
in simulation for the AMB problem.
77
Another solution is using the LADRC method, instead of NDP, in the first step. The
signal can be obtained from the LESO. Therefore, the differentiator is not necessary for
this method. The beam angular acceleration equation is
1 2( )KL
y K f f dJ
θ= = − −&&&&
It can be written as
1 2( )KL
y K f f d u uJ
θ= = − − − +&&&& (5.22)
Assume 1 2( )KL
f f f d uJ
= − − − , then
y f u= +&& (5.23)
where f is the uncertainty of the plant. u is the control signal. Applying the LADRC to
(5.22)
LESO:
1 2 1 1
2 3 2 1
3 3 1
( )
( )
( )
z z z y
z z z y u
z z y
ββ
β
= + − = + − + = −
&
&
&
The control law is
3 0u z u= − + , 0 1 2( )p d
u k r z k z= − − , (5.24)
where 2d c
k ξω= and 2
p ck ω= . The plant is changed into
0y u=&& .
The next steps are the same as (5.16) ~ (5.21).
78
5.4 Simulation Study and Results
The simulation model of AMB [58] is shown in Figure 39. Figures 40 and 41
present the models of leakage and beam dynamics.
5
ddy/dt
4
dy/dt
3
Current2
2
Current 1
1
Displacement
(mi ls)
Gap Flux 2
Gap 2Current2
flux to current 2
Saturation2
Saturation 1
Total Flux 2
gap 2
Gap Flux 2
Leakage 2
gap 1
Total Flux 1
Gap Flux 1
Leakage 1
1
s
1
s
disp
gap1 mils
gap 2 mils
Gap Posi tion
Flux2 Force2
Force calculation2
Flux1 Force1
Force calculation1
Gap 1
Gap Fiux 1
Current 1
Flux to Current 1
Current 2 Volt 2
Coil Resistance 2
Current 1 Volt 1
Coil Resistance 1
Force 1
Force 2
Out 1
Out 1 1
Out 1 2
Beam Dynamics
-K-
1
----------------
Number Turns 2
-K-
1
----------------
Number Turns 1
3
Disturbance
2
U2
1
U1
Figure 39: The AMB Simulation Model
1
Gap Flux 1
Gap1_mils in P1 Out
P1 Gain Calculation
Mult2
Total Flux 1
1
gap 1
1
P1 Out.026496
slope
Sum Product1
Product
1.0528
Intercept
1
Gap1_mils in
Phi_tot / Phi_gap Phi_gap / Phi_tot
Figure 40: The Model of Leakage
79
1
Out 1
-K-
acc
-5724
Radian to
Mils Gain
Mux
Mux
x' = Ax+Bu
y = Cx+Du
Flexible Modes
1
s
1
s
2
Force 2
1
Force 1
Figure 41: The Model of Beam Dynamics
The proposed control approaches are tested in simulation. The control laws (5.15),
(5.16), (5.20), (5.21), and (5.24) are implemented digitally with a sampling period of 10-4
seconds. In order to more accurately demonstrate the advantages of the NDP approach, a
linear Pole Placement (PP) method is induced to the simulation test. Instead of using the
nonlinear control law that was shown in (5.15), the PP method uses the linear PD control
scheme. A smooth profile is used in simulation as the desired trajectory of the output.
The torque disturbance is ( ) 4sin(40 )d t tπ= . The initial condition is (0) 10y = − mils and
(0) 2y = −& mils/sec. The controller parameters are chosen as follows:
NDP: 53 10p
k = × ,1
0.7α = , 1 0.05δ = , 55 10τ −= × ,
31.5 10d
k = × , 2
0.92α = , 2 1δ = , 3,4 1000k = .
PP: 5 5
3,42 10 , 800, 5 10 , 1000.p dk k kτ −= × = − = × =
LADRC: 3000, 300, 1,o c
ω ω ξ= = =
2 3
1 2 33* , 3* , .
o o oβ ω β ω β ω= = =
Figure 42 illustrates the steady-state displacement errors of the three systems. Note
that the NDP and LADRC systems have smaller steady-state errors than the PP system.
The LADRC system’s setting time is slightly longer than the NDP and the PP systems.
80
0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
Time(sec.)
Error
PP NDP LADRC
Figure 42: Displacement Errors
Figure 43, 44, and 45 show the voltages (u1&u2) and flux (1
φ &2
φ ) of the PP, the
NDP and the LADRC systems. It is clear that the fluxes that are employed are almost
equal. The controller robustness is evaluated by increasing the coil resistance by 25%
because the actual coil resistance varies with the coil temperature. Other parameters of
the controller and plant are unchanged. The results are shown in Figure 46, 47, 48, and
49. Again, the LADRC and the NPD systems have smaller steady-state error. The fluxes
employed by both systems are almost the same.
81
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-10
0
10
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-10
0
10
20
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20
0.5
1x 10
-4
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20
0.5
1x 10
-4
Time (sec.)
flux2
flux1
voltage2
voltage1
Figure 43: Voltages (u1&u2) and Flux (1
φ &2
φ ) of PP
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-10
0
10
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-10
0
10
20
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20
0.5
1x 10
-4
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20
0.5
1x 10
-4
Time (sec.)
flux2
flux1
voltage2
voltage1
Figure 44: Voltages (u1&u2) and Flux (1
φ &2
φ ) of NDP
82
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-10
0
10
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-10
0
10
20
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20
0.5
1x 10
-4
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20
0.5
1x 10
-4
Time (sec.)
flux2
flux1
voltage2
voltage1
Figure 45: Voltages (u1&u2) and Flux (1
φ &2
φ ) of LADRC
0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
Time(sec.)
Error
PP NDP LADRC
Figure 46: Displacement Errors for the Plant Coil Resistance Increased by 25%
83
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-10
0
10
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-10
0
10
20
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20
0.5
1x 10
-4
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20
0.5
1x 10
-4
Time (sec.)
flux1
flux2
voltage2
voltage1
Figure 47: Voltages (u1&u2) and flux (1
φ &2
φ ) of PP
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-10
0
10
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-10
0
10
20
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20
0.5
1x 10
-4
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20
0.5
1x 10
-4
Time (sec.)
flux2
flux1
voltage2
voltage1
Figure 48: Voltages (u1&u2) and Flux (1
φ &2
φ ) of NDP
84
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-10
0
10
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-10
0
10
20
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20
0.5
1x 10
-4
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20
0.5
1x 10
-4
Time (sec.)
flux2
flux1
voltage2
voltage1
Figure 49: Voltages (u1&u2) and Flux (1
φ &2
φ ) of LADRC
To further investigate the robustness of the controller, the following tests are
performed individually:
• Change the frequency of the disturbance to 50Hz;
• Reduce initial position to 5 mils.
The results are evaluated in terms of setting time and steady-state error, which are shown
in Table III. These results reveal that the proposed NDP method and LADRC have
distinct advantage concerning disturbance attenuation.
85
Table III: Simulation Comparison
Setting time
(2%vs initial
disp.)(sec)
Steady
State
Error (mils)
4sin(40 )d tπ= 0.0218 0.05
4sin(100 )d tπ= 0.0225 0.05
(0) 5 ,
(0) 2 / sec
m ils
m ils
θθ
=
=&
4sin(40 )d tπ=
0.0195 0.05 NDP
25% change
resistance
4sin(40 )d tπ=
0.0216 0.12
4sin(40 )d tπ= 0.023 0.16
4 sin(100 )d tπ= 0.0235 0.16
(0) 5 ,
(0) 2 / sec
mils
mils
θθ
=
=&
4sin(40 )d tπ=
0.021 0.16 PP
25% change
resistance
4sin(40 )d tπ=
N/A 0.35
4sin(40 )d tπ= 0.025 0.06
4sin(100 )d tπ= 0.025 0.1
(0) 5 ,
(0) 2 / sec
mils
mils
θθ
=
=&
4sin(40 )d tπ=
0.03 0.05
LADRC
25% change
resistance
4sin(40 )d tπ=
0.025 0.17
86
Finally, NDP is compared with the saturated high gain controller proposed in [61].
Figures 50 and 51 plot the responses of the two systems in the same scale. Due to the lack
of information, the simulation results in [61] were directly imported in Figure 50. It
shows that the steady-state error of a saturated high gain controller is 2.5mils, which is
much larger that the one of the NDP system. Furthermore, the control voltages used by
the NDP system are much smaller than those used by the PP system. The NDP system
has an overall better performance in terms of smoothness of the output, smaller steady-
state error, and less control actions required.
Figure 50: Saturated High Gain Controller Responses (from [61])
87
0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1- 1 5
- 1 0
- 5
0
5
y
0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1- 4 0
- 3 0
- 2 0
- 1 0
0
1 0
2 0
3 0
4 0
sa
t(u
1)
0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1- 4 0
- 3 0
- 2 0
- 1 0
0
1 0
2 0
3 0
4 0
T i m e ( s e c . )
sa
t(u
2)
Figure 51: Nonlinear Dynamics Placement Controller Responses
5.5 Summary
Two novel control methods are proposed to resolve the AMB problem, and
simulation results are very promising. For AMB systems, where the nonlinearity is
inherently strong, a control system based on the nonlinear model is more effective than
on the linear model. Furthermore, both controllers are practical and easily implemented
into a digital controller.
88
CHAPTER VI
STABILITY ANALYSIS OF ADRC
The ADRC design method introduced earlier is an effective alternative approach,
especially when other controllers cannot achieve the desired performance in terms of set
point change, large parameter drift, large inertia, friction variations, external torque
disturbance and unknown system model. Stability is a basic problem in control system
design. Based on Popov criteria, a stability analysis for the ADRC system is proposed in
this chapter.
89
6.1 Absolute Stability
Simulation and experimental results show that ADRC has many advantages, such as
faster reaction, smaller overshoot and strong robustness. It can estimate disturbance using
ESO and compensate the system in each sampling period. The dynamic compensation
reduces the system to approximately a double integrator which can be easily controlled
using a PD controller.
Although ADRC is used in many fields, the stability analysis has not been
completely resolved. For ADRC to be considered as a serious contender in industrial
control design, a measure of stability or a certain degree of safety must be provided.
ADRC can be viewed as a nonlinear controller. At present, the approaches that are used
to analyze the stability of nonlinear systems include describing function, phase plane
analysis, the extended circle criterion, and Lyapunov’s method. Lyapunov’s method is
probably the most widely used approach. The second method of Lyapunov determines the
stability of equilibrium in a system by nonlinear differential equations. It is especially
useful when the control systems are described by differential equations. On the other hand,
stability analysis in classical control theory has been centered on frequency-domain
methods, propelled by the beauty, practicality, and generality (for linear systems) of the
Nyquist criterion. Each of these approaches has its advantages. Frequency-domain
methods use very compact model specifications and are able to address robustness through
gain and phase margins. However, the generalization to nonlinear systems of frequency-
domain methods is awkward. Differential equation models have proved a much better way
of addressing nonlinearities overall.
90
Through the problem of determining conditions for the stability of systems with one
nonlinear element, control witnessed a symbiosis between stability theory of Lyapunov
methods and frequency-domain techniques. The catalyst in this development was Popov’s
stability criterion.
In 1944, Lure and Postnikov [63] began to study the stability of systems of the form
depicted in Figure 52. Such systems are assumed to be completely linearizable with the
exception of a single nonlinear element denoted by f (.). The operator W is assumed to be
linear, time-invariant (LTI), while the nonlinear characteristic f(.) satisfies the following
conditions.
Definition: Function Class {F}. A function is 0
( )f δ said
0( ) { }f Fδ ∈ if it is a real
continuous single-value scalar function that satisfies
0 0( ) 0.fδ δ ≥ (6.1)
where 0
( )f δ is constrained to fall within the first and third quadrants of the0
δ -0
( )f δ
plane.
W(s)
f(.)
input 0δ
0( )f δ
-
+
r=0
Figure 52: The Problem of Lure and Postnikov
Lure and Postnikov determined the conditions for constraints on the LTI plant that would
be sufficient to ensure the stability of the system for any nonlinear gain { }(.)f F∈ . The
resulting criterion could then be applied to any specific problem, where the nonlinear
function is any member of the class{ }F . This concept is an important philosophical basis
91
for many developments in stability theory. This problem of stability of a system with one
nonlinear element leads also to the Aizerman conjecture. This states that if there exist k1,
k2 such that 1 2
( )fk k
δδ
< < for all Rδ ∈ , and if the linear system obtained by replacing the
nonlinear element by a linear one with gain k, is asymptotically stable for all [ ]1 2,k k k∈ ,
then the nonlinear system is also asymptotically stable in the large. This is also called
absolute stability of the system shown in Figure 52.
Romanian scientist Popov proposed an absolute stability criterion for nonlinear
systems in 1959. It is a kind of absolute stability judge method in frequency field, similar
to the Nyquist method of linear systems. The problem of stability for the non-linear
system can only be solved when computing the frequent properties of a system’s linear
parts. This criterion can be formulated as follows:
Consider the feedback system described by (see Figure 52):
0 0; ; ( )
Tdx Ax bu c x u f
dtδ δ= + = = − (6.2)
with , , ,n n nA R b c R×∈ ∈ and :f R R→ a continuous nonlinearity with (0) 0f = . Denote
by W the transfer function of the linear part, i.e., 1( ) ( ) .TW s c Is A b−= − Assume
furthermore that A is a stability matrix. The matrix A does not need to have all of its
eigenvalues in the open left half of the complex plane. In fact, Popov assumes that A has
all of its eigenvalues in the open left half of the complex plane, except, possibly, one
eigenvalue at the origin.
Popov proves that the equilibrium trajectory 0x = is globally asymptotically stable if
the following conditions are satisfied:
92
1. 0 0
( ) 0fδ δ > for all 0
0 ,Rδ≠ ∈
2. There exists 0q ≥ such that Re(1 ) ( ) 0qj w jω ω+ ≥ for all .Rω ∈
This result is known as the Popov criterion. The following section will describe how to
analyze the stability of ADRC using the Popov criterion.
6.2 Stability Analysis of ADRC
The diagram of the ADRC system is shown in Figure 53.
Observer
f(e) Plant
r=0 e
-
y
1/b
0u u
-
linear part
Figure 53: Diagram of ADRC
The following assumptions are made:
1. Set point 0r = ;
2. Replacing the NESO of ADRC with the linear one (LESO);
3. Nonlinear feedback functions in (2.37) satisfy the formula
( ) 0, 0ef e e> ≠ .
where 0
( ) ( , , ) ( , , )p p p d d d
f e u k fal e k fal eα δ α δ= = + & .
93
For a first-order linear plant:
x ax bu
y x
= − +=
& (6.3)
with ,a b R+∈
The LESO for the first-order plant in (6.3) is
1 2 1
2 2 1 ,
o
o o
z z e bu
z e where e y z observer error
ββ
= + += = −
&
&
(6.4)
The control law is
20
,z
u ub
= − (6.5)
with 0
( ) ( ), (0) 0.u f y f y f= − = − = , ( ) 0yf y > , 0.r =
Putting (6.5) into (6.4) and (6.3),
0 2
1 1 0
2 2
o
o
x ax bu z
z e bu
z e
y x
ββ
= − + − = + = =
&
&
& (6.6)
[ ]
1 1 1 1 0
2 2 22
1
2
0 1
0
0 0
1 0 0
x a x b
z z b u
zz
x
y z
z
β ββ β
− − ⇒ = − + −
=
&
&
&
(6.7)
The transfer function of the linear part of the Figure 53 can be obtained as
21
1 1 2
2 2
1 1 2 1 1 2
( )( )
( ) ( )( ) [ ( ) ]
bb s
bs s bsG s C SI A Bs a s a s s a s a
ββ β ββ β β β β β
−+ + + += − = =
+ + + + + + + +.
94
det( ) 0I Aλ − = ⇒
1
2
2 1 1 2
2
3 1 1 2
0,
1 1 1( ) 4 ,
2 2 2
1 1 1( ) 4 .
2 2 2
a a
a a
λ
λ β β β
λ β β β
= = − − + − − = − − − − −
Since a,β1 and β2 are positive, Re(λ2) and Re(λ3) negative. Therefore the matrix
1 1
2 2
0 1
0
0
a
A β ββ β
− − = − −
is stable. Note that
[ ]2 2
1 2
22 2
1 2 1
( )Re ( )
( ) ( )
baG j
a a
ω β βωω β β β ω
+ −=− − + +
(6.8)
[ ]2 2 2 4 2
1 2 1 2 2
22 2
1 2 1
2( )
( ) ( )m
qb bq qb qa b q bq I G j
a a
β ω ω β ω β β βω ωω β β β ω
− + − − −=− − + +
(6.9)
Re(1 ) ( ) Re(1 )[Re ( ) ( ) ]
Re ( ) ( )
m
m
qj G j qj G j I G j i
G j q I G j
ω ω ω ω ωω ω ω
+ = + += −
(6.10)
Combining (6.8), (6.9) and (6.10),
[ ]2 2 2 2 2 4 2
1 2 1 2 1 2 2
22 2
1 2 1
( ) 2Re(1 ) ( )
( ) ( )
ba qb bq qb qa b q bqj G j
a a
ω β β β ω ω β ω β β βω ωω β β β ω
+ − + − + + ++ =− − + +
(6.11)
Obviously, 2 2
1 2( )aω β β− − + [ ]2
1( )aβ ω+ >0, for all Rω ∈ .
Let 22
aq
β= >0, the numerator of (6.11) becomes
2 2 242 2 1 1
1
2 22 2 2 2
ba qb a babba
β β ω βωββ β
− + + + (6.12)
95
Since the last three terms in (6.12) are all positive, i.e.
2 2
1
2
0,2
qbβ ωβ
≥4
2
02
abωβ
≥ and 2
1 0.2
a bβ >
Re(1 ) ( ) 0qj w jω ω+ ≥ for all ω ∈R if 2 21
2
ββ ≥ .
That is, according to the Popov criterion, for the first-order ADRC system, equilibrium
trajectory 0x = is globally asymptotically stable if the following conditions are satisfied:
1. The linear plant is asymptotically stable with 0;a >
2. The parameters of the ESO are positive, and 2 21
2
ββ ≥ .
96
CHAPTER VII
CONCLUDING REMARKS
A comparison study of three advanced observers, including ESO, HGO, and SMO,
is performed using simulations and hardware tests. The results show that ESO performs
better than HGO and SMO for the systems with uncertainties. The robustness of the
NESO is inherent in its structure.
ADRC is a NESO-based nonlinear control strategy, which can compensate for the
nonlinear dynamics in a model-independent way. It has proved to be a powerful and
effective controller. Due to the nonlinear functions used in ADRC, tuning the parameters
to achieve best performance is not straight-forward. LADRC was used to simplify the
design procedure of ADRC. In this dissertation, the effectiveness of LADRC is tested
using experiments on motion control of a DC brushless motor. The experimental results
reveal that LADRC offers the potential of improving the control precision and the quality
of the control signal for the actuators. A comparison of LADRC, ADRC, and loop
97
shaping design shows that with the optimal tuning method, LADRC can achieve high
performance.
The AMB problem is a good benchmark to test the control method because it is a
highly nonlinear plant with a periodic disturbance torque. Two methods were used for
dealing with this problem: NDP and LADRC. To show the advantages of the NDP
method, a traditional PP method is also used for comparison. Simulation results showed
that LADRC and NDP had better performance than the linear PP method. Their results
also exceed all other control schemes [53, 54, 56, 58].
Based on the assumption that the observer is linear, a stability analysis of second-
order ADRC is discussed by using the Popov criterion. The system equilibrium at 0x =
is shown to be globally asymptotically stable if the following conditions are satisfied: the
linear plant is asymptotically stable, and the parameters of observer satisfy the condition:
2 21 0
2
ββ ≥ > .
For future research, the design process of ADRC can be further investigated so that
the nonlinear gains can be easily tuned for optimal performance. The stability analysis of
high-order ADRC should also be explored.
98
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parameterization/ µ Control of an Electromagnetic Suspension System,” Proc. of the
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104
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106
APPENDICES
107
A. C Source Code used in Hardware Tests
/* LS7166 APPLICATION NOTE, EXAMPLE PROVIDED FOR REFERENCE BY */
/* U.S. Digital Corp (206) 260-7451 3-11-93 */
/* note: pin 3 and pin 4 are normally high, pin 18 to address 0 */
/* Borland C source code */
#include <stdio.h>
#include <dos.h>
#include <conio.h>
/* addresses */
#define BASE 0X340 /* base address of interface board */
#define DATA1 BASE+0 /* data register of LS7166 */
#define CONTROL1 BASE+1 /* control register of LS7166 */
#define LOAD BASE+8 /* a write toggles pin 3 of LS7166 */
/* LS7166 commands */
#define MASTER_RESET 0X20 /* master reset command */
#define INPUT_SETUP 0X68 /* setup counter input mode */
#define QUAD_X1 0XC1 /* quadrature multiplier to 1 */
#define QUAD_X2 0XC2 /* quadrature multiplier to 2 */
#define QUAD_X4 0XC3 /* quadrature multiplier to 4 */
108
#define ADDR_RESET 0X01 /* reset address pointer */
#define LATCH_CNTR 0X02 /* latch counter */
#define CNTR_RESET 0X04 /* reset counter */
#define PRESET_CTR 0X08 /* transfer preset to counter */
unsigned long cntr_ol=0x080000000; /* output of the CNTR */
unsigned char bwt_old; /* the last BWT flag */
unsigned char cyt_old; /* the last CYT flag */
void init_7166(void) { /* initialize the 7166 */
unsigned char status;
outportb(CONTROL1, MASTER_RESET);
outportb(CONTROL1, INPUT_SETUP);
outportb(CONTROL1, QUAD_X4);
outportb(CONTROL1, CNTR_RESET);
/* preset counter to 44 */
outportb(CONTROL1, ADDR_RESET); /* reset addr */
outportb(DATA1, 0x0); /* output preset value */
outportb(DATA1, 0x0); /* output preset value */
outportb(DATA1, 0x080); /* output preset value */
outportb(CONTROL1, PRESET_CTR); /* preset to counter */
109
status = inportb(CONTROL1);
bwt_old = status & 0x01;
cyt_old = status & 0x02;
}
void soft_latch(void) { /* latch position with software command */
outportb(CONTROL1, LATCH_CNTR); /* counter to latch */
}
void hard_latch(void) { /* latch position with low pulse on pin 3 */
outportb(LOAD, 0); /* toggle pin 3 of LS 7166 */
}
unsigned long read_7166(void) { /* read position of encoder */
unsigned char status;
unsigned long position;
soft_latch();
status = inportb(CONTROL1);
outportb(CONTROL1, ADDR_RESET); /* reset address pointer */
position = (unsigned long)inportb(DATA1); /* least significant byte */
position += (unsigned long)inportb(DATA1) << 8;
position += (unsigned long)inportb(DATA1) <<16; /* most significant byte */
cntr_ol = ( cntr_ol & 0x0ff000000)|position;
110
if ((status&0x01) != bwt_old) {
cntr_ol -= 0x01000000;
}
if ((status&0x02) != cyt_old) {
cntr_ol += 0x01000000;
}
bwt_old = status&0x01;
cyt_old = status&0x02;
return (cntr_ol);
}
/* Initialize the 8254 */
#include <stdio.h>
#include <dos.h>
#include <conio.h>
void init_8254(void) { /* initialize the 8254 */
outportb(0x43, 0xBA); /* config count down from FFFFh(65535) */
outportb(0x42, 0x0);
outportb(0x42, 0x0);
111
}
long read_8254(void) {
long t1,t2,t;
outportb(0x43, 0x80); /* latch new time of loop */
t1 = (long)inportb(0x42);
t2 = (long)inportb(0x42);
t=t1+t2*256;
return (t);
}
/* DAC program */
/* Borland C source code */
#include <stdio.h>
#include <dos.h>
#include <conio.h>
/* addresses */
#define DAC_BASE 0X300 /* base address of DAC board */
void DAC_out(double voltage){
112
double D;
int DH,DL;
if(voltage>5.0) voltage=5.0;
if(voltage<0.0) voltage=0.0;
D=4095*voltage/5.0;
DH=D/16;
DL=D-DH*16;
DL=DL*16;
outport(DAC_BASE+6, DL);
outport(DAC_BASE+7, DH);
}
void init_DAC(void) { /* initialize the DAC board */
DAC_out(2.5);
}
113
****************************************************************
Loop shaping Control for Servo Motor with trapezoid profile December 2002
****************************************************************
#include <stdio.h>
#include <math.h>
#include <stdlib.h>
#include <dos.h>
#include <conio.h>
#define pi 3.141592653589793
unsigned long read_7166();
long read_8254();
void main(){
double a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12;
double u=0,u_1=0,u_2=0,u_3=0,u_4=0; /* control signal */
double e=0, e_1=0, e_2=0, e_3=0, e_4=0; /* error signal */
double t=0,h, ysp=0; /* time and simulation stepsize */
double V_max, t5, v0, out; /* parameters of trapezoid */
/* parameters of controller */
double kp=0.7, wc=20;
114
double feedback = 0;
FILE *fp;
int i, I=10,k=0,n=15,first=1;
int K=1000;
char buff[10];
long t_previous,t_current,position;
int time_count;
double *Time;
double *Feedback;
double *Control_out;
double *error;
double *Control;
fp=fopen("loop_trz.dat","w");
if (fp==NULL) {printf("Can't creat the output file out.dat!\n");exit(1);}
printf("\n Input how many revolutions to go [default 1]: ");
gets(buff);
if (strlen(buff) == 0) v0=1;
else
v0 = atof(buff);
printf("\n Input the max speed (rad/sec) [default 14]: ");
gets(buff);
115
if (strlen(buff) == 0) V_max= 14.0;
else
V_max = atof(buff);
printf("\n Input the kp [default 0.7]: ");
gets(buff);
if (strlen(buff) == 0) kp = 0.7;
else
kp = atof(buff);
printf("kp=%f\n", kp);
printf("\n Input the wc (rad/sec) [default 20]: ");
gets(buff);
if (strlen(buff) == 0) wc = 20.0;
else
wc = atof(buff);
printf("wc = %f\n",wc);
Time = (double *)malloc(K*sizeof(double));
if (!Time) {printf(" No enough memory for Time Array!\n");
exit(1);}
Control_out=(double *)malloc(K*sizeof(double));
if (!Control_out) {printf(" No enough memory for Control_out Array!\n");
exit(1);}
116
Control = (double *)malloc(K*sizeof(double));
if (!Control) {printf(" No enough memory for Control Array!\n");
exit(1);}
Feedback=(double *)malloc(K*sizeof(double));
if (!Feedback) {printf(" No enough memory for Feedback Array!\n");
exit(1);}
error = (double *) malloc(K*sizeof(double));
if (!error) {printf(" No enough memory for error array!\n");
exit(1);}
printf("Program running...\n");
printf(" \n");
printf("Press any key to exit.\n");
init_DAC();
init_7166();
init_8254();
t5 = 3*pi*v0/V_max;
117
a1=4*wc+4*kp*wc*wc*h+kp*kp*wc*wc*wc*h*h;
a2=2*kp*kp*wc*wc*wc*h*h-8*wc;
a3=kp*kp*wc*wc*wc*h*h+4*wc-4*kp*h*wc*wc;
a4=4+2*wc*h;
a5=4-2*h*wc;
a6=200*wc*wc*h+141*wc*wc*h*h;
a7=282*wc*wc*h*h;
a8=141*wc*wc*h*h-200*wc*wc*h;
a9=4+40*h*wc+100*wc*wc*h*h;
a10=200*wc*wc*h*h-8;
a11=100*wc*wc*h*h-40*h*wc+4;
t_previous=read_8254(); /* right now, h=0.0, feedback=0.0 */
while(!kbhit()) {
for (i=0;i<I;i++) { /* every I steps output a data */
delay(n);
t_current=read_8254();
if (t_current>t_previous) t_previous+=65536;
time_count=(int)(t_previous-t_current);
h=(double)time_count/1193180.0;
if (first) h=0.001;
t_previous=t_current;
t=t+h;
118
position=read_7166()-(unsigned long)0x080800000;
feedback=(double)(position*2*pi/16000.0);
if (t<= t5/3) ysp = 3*V_max/t5/2*t*t;
else
if (t<= 2*t5/ 3) ysp = V_max * t5/6 + V_max * (t-t5/3);
else
if (t<=t5) ysp = V_max * t5/2
+ ((t-2*t5/3)*t5-(t*t-4*t5*t5/9)/2)*3*V_max/t5;
else
ysp = 2*V_max*t5/3;
e=ysp-feedback;
if (first) {
first=0;
continue;
}
a12=(a1*a6*e+(a1*a7+a2*a6)*e_1+(a1*a8+a2*a7+a3*a6)*e_2+(a2*a8+a3*a7)*e_3+a3
*a8*e_4)/23.24;
u=(a12-a5*a11*u_4-(a5*a10-8*a11)*u_3-(a11*a4-8*a10+a5*a9)*u_2-(a4*a10-
8*a9)*u_1)/a4/a9;
e_4=e_3;
119
e_3=e_2;
e_2=e_1;
e_1=e;
u_4=u_3;
u_3=u_2;
u_2=u_1;
u_1=u;
if (u > 3.5) u = 3.5;
if (u <-3.5) u =-3.5;
out=5.0*u/7.0+2.5;
DAC_out(out);
} /* for i loop */
if ( k < K ) {
Time[k]=t;
if ( k == 0) Control[k] = 0;
else Control[k] = (feedback - Feedback[k-1])/h/10;
/* Control[k]=u0; */
Control_out[k]=u;
Feedback[k]=feedback;
error[k] = e;
120
++k;
}
else
goto exit1;
} /* for k loop */
exit1:
init_DAC();
for(k=0;k<K;k++){
fprintf(fp,"%f %f %f %f %f\n",Time[k],Control[k],
Control_out[k],Feedback[k],error[k]);}
fclose(fp);
printf("Program completed !\n");
}
/* end of main */
121
*********************************************************************
ESO based state feedback control (LADRC) for Servo Motor using trapezoid profile,
August 2002
*********************************************************************
#include <stdio.h>
#include <math.h>
#include <stdlib.h>
#include <dos.h>
#include <conio.h>
#define pi 3.141592653589793
double fal(double x,double alpha,double delta);
unsigned long read_7166();
long read_8254();
void main(){
double z1=0,z2=0,z3=0; /* three states of ESO */
double v1,v2,v3; /* temple states of ESO */
double z4=0,z5=0;
double v4,v5;
double u0,u=0,out; /* control signal */
122
double t=0,h=0; /* time and simulation step size */
double beta_01=12, alpha_01=0.5, delta_01=0.001,
beta_02=10, alpha_02=1.0, delta_02=0.001,
beta_03=160, alpha_03=2.0, delta_03=0.001;
double r=20,rr=20,v0=0.125*2*pi;
double V_max, t1; /* parameters of trapezoid */
double K_P=1, K_D=1;
double b0=25;
double e,fe,fe1,fe2;
double e1=0,e2=0;
double feedback=0;
FILE *fp;
int i,I=10,k=0,n=15;
int K=1000;
char buff[10];
long t_previous,t_current,position;
int time_count;
double *Time;
double *Feedback;
double *Control_out;
double *error;
double *error_dot;
123
double *Control;
fp=fopen("parasw.dat","r");
if (fp==NULL) {printf("Can't open the parameters file paras.dat !\n");exit(1);}
fscanf(fp,"%lf %lf\n",&r,&v0);
fscanf(fp,"%lf\n",&K_P);
fscanf(fp,"%lf\n",&K_D);
fscanf(fp,"%lf %lf %lf\n",&beta_01,&alpha_01, &delta_01);
fscanf(fp,"%lf %lf %lf\n",&beta_02,&alpha_02, &delta_02);
fscanf(fp,"%lf %lf %lf\n",&beta_03,&alpha_03, &delta_03);
fscanf(fp,"%lf",&b0);
fscanf(fp,"%lf",&rr);
fclose(fp);
fp=fopen("eso_trz.dat","w");
if (fp==NULL) {printf("Can't creat the output file out.dat!\n");exit(1);}
printf("\n Input how many revolutions to go [default 1]: ");
gets(buff);
if (strlen(buff) == 0) v0=1;
else
v0 = atof(buff);
printf("\n Input the max speed (rad/sec) [default 14]: ");
gets(buff);
if (strlen(buff) == 0) V_max= 14.0;
124
else
V_max = atof(buff);
r=fabs(r);
if (r>rr) r=rr;
t1 = 3*pi*v0/V_max;
printf("Program running...\n");
printf(" \n");
printf("Press any key to exit.\n");
Time = (double *)malloc(K*sizeof(double));
if (!Time) {printf(" No enough memory for Time Array!\n");
exit(1);}
Control_out=(double *)malloc(K*sizeof(double));
if (!Control_out) {printf(" No enough memory for Control_out Array!\n");
exit(1);}
Control = (double *)malloc(K*sizeof(double));
if (!Control) {printf(" No enough memory for Control Array!\n");
exit(1);}
Feedback=(double *)malloc(K*sizeof(double));
if (!Feedback) {printf(" No enough memory for Feedback Array!\n");
exit(1);}
error = (double *) malloc(K*sizeof(double));
if (!error) {printf(" No enough memory for error array!\n");
exit(1);}
125
error_dot = (double *) malloc(K*sizeof(double));
if (!error_dot) { printf(" No enough memory for error_dot arry!\n");
exit(1);}
delay(n);
init_DAC();
init_7166();
init_8254();
t_previous=read_8254 (); /* right now, h=0.0, feedback=0.0 */
while(!kbhit()) {
for (i=0;i<I;i++) { /* every I steps output a data */
t=t+h;
/* ESO */
position=read_7166()-(unsigned long)0x080800000;
feedback=(double)(position*2*pi/16000.0);
e=z1-feedback;
fe=fal(e,alpha_01,delta_01);
v1=z1+h*(z2-beta_01*fe);
fe=fal(e,alpha_02,delta_02);
v2=z2+h*(z3-beta_02*fe+b0*u);
fe=fal(e,alpha_03,delta_03);
v3=z3-h*beta_03*fe;
if (t<= t1/3) v4 = 3*V_max/t1/2*t*t;
126
else
if (t<= 2*t1/3) v4 = V_max * t1/6 + V_max * (t-t1/3);
else
if (t<=t1) v4 = V_max * t1/2
+ ((t-2*t1/3)*t1-(t*t-4*t1*t1/9)/2)*3*V_max/t1;
else
v4 = 2*V_max*t1/3;
if (h == 0) v5 =0;
else
v5 = (v4 - z4)/h;
z4=v4; z5=v5;
e1=z4-z1;
e2=-z2;
u0=K_P*e1+K_D*e2;
u=u0-z3/b0;
if (u > 3.5) u = 3.5;
if (u <-3.5) u =-3.5;
out=5.0*u/7.0+2.5;
DAC_out(out);
delay(n);
127
t_current=read_8254();
if (t_current>t_previous) t_previous+=65536;
time_count=(int)(t_previous-t_current);
h=(double)time_count/1193180.0;
t_previous=t_current;
z1=v1;z2=v2;z3=v3;
} /* for i loop */
if ( k < K ) {
Time[k]=t;
if ( k == 0) Control[k] = 0;
else Control[k] = (feedback - Feedback[k-1])/h/10;
Control_out[k]=u;
Feedback[k]=feedback;
error[k] = z3;
error_dot[k] = z5;
++k;
}
else
goto exit1;
} /* for k loop */
exit1:
init_DAC();
128
for(k=0;k<K;k++){
fprintf(fp,"%f %f %f %f %f %f\n",Time[k],Control[k],
Control_out[k],Feedback[k],error[k],error_dot[k]);}
fclose(fp);
printf("Program completed !\n");
} /* end of main */
double fal(double x,double alpha,double delta) {
if(x>delta) return(pow(x,alpha));
else if(x<-delta) return(-pow(-x,alpha));
else return(x*pow(delta,alpha-1));
}
129
*************************************************
ADRC for Servo Motor using trapezoid profile, May 2000
*************************************************
#include <stdio.h>
#include <math.h>
#include <stdlib.h>
#include <dos.h>
#include <conio.h>
#define pi 3.141592653589793
double sature(double x, double delta);
double fal(double x,double alpha,double delta);
double fst2(double x1, double x2, double u0, double r, double h);
unsigned long read_7166();
long read_8254();
void main(){
double z1=0,z2=0,z3=0; /* three states of ESO */
double v1,v2,v3; /* temple states of ESO */
double z4=0,z5=0;
double v4,v5;
130
double u0,u=0,out; /* control signal */
double t=0,h=0; /* time and simulation step size */
double beta_01=12, alpha_01=0.5, delta_01=0.001, /* parameters of ESO */
beta_02=10, alpha_02=1.0, delta_02=0.001,
beta_03=160, alpha_03=2.0, delta_03=0.001;
double r=20,rr=20,v0=0.125*2*pi;
double V_max, t1; /* parameters of trapezoid */
double K_P=2, alpha_P=0.75, delta_P=0.001, /* parameters of Nonlinear Combination */
K_D=0.5, alpha_D=1.5, delta_D=0.001;
double b0=15.04;
double e,fe,fe1,fe2;
double e1=0,e2=0;
double feedback=0;
FILE *fp;
int i,I=10,k=0,n=15;
int K=1000;
char buff[10];
long t_previous,t_current,position;
int time_count;
double *Time;
double *Feedback;
131
double *Control_out;
double *error;
double *error_dot;
double *Control;
fp=fopen("paras1.dat","r");
if (fp==NULL) {printf("Can't open the parameters file paras.dat !\n");exit(1);}
fscanf(fp,"%lf %lf\n",&r,&v0);
fscanf(fp,"%lf %lf %lf\n",&K_P, &alpha_P, &delta_P);
fscanf(fp,"%lf %lf %lf\n",&K_D, &alpha_D, &delta_D);
fscanf(fp,"%lf %lf %lf\n",&beta_01,&alpha_01, &delta_01);
fscanf(fp,"%lf %lf %lf\n",&beta_02,&alpha_02, &delta_02);
fscanf(fp,"%lf %lf %lf\n",&beta_03,&alpha_03, &delta_03);
fscanf(fp,"%lf",&b0);
fscanf(fp,"%lf",&rr);
fclose(fp);
fp=fopen("adrc_trz.dat","w");
if (fp==NULL) {printf("Can't creat the output file out.dat!\n");exit(1);}
printf("\n Input how many revolutions to go [default 1]: ");
132
gets(buff);
if (strlen(buff) == 0) v0=1;
else
v0 = atof(buff);
printf("\n Input the max speed (rad/sec) [default 14]: ");
gets(buff);
if (strlen(buff) == 0) V_max= 14.0;
else
V_max = atof(buff);
r=fabs(r);
if (r>rr) r=rr;
t1 = 3*pi*v0/V_max;
printf("Program running...\n");
printf(" \n");
printf("Press any key to exit.\n");
Time = (double *)malloc(K*sizeof(double));
if (!Time) {printf(" No enough memory for Time Array!\n");
exit(1);}
133
Control_out=(double *)malloc(K*sizeof(double));
if (!Control_out) {printf(" No enough memory for Control_out Array!\n");
exit(1);}
Control = (double *)malloc(K*sizeof(double));
if (!Control) {printf(" No enough memory for Control Array!\n");
exit(1);}
Feedback=(double *)malloc(K*sizeof(double));
if (!Feedback) {printf(" No enough memory for Feedback Array!\n");
exit(1);}
error = (double *) malloc(K*sizeof(double));
if (!error) {printf(" No enough memory for error array!\n");
exit(1);}
error_dot = (double *) malloc(K*sizeof(double));
if (!error_dot) { printf(" No enough memory for error_dot arry!\n");
exit(1);}
delay(n);
init_DAC();
134
init_7166();
init_8254();
t_previous=read_8254(); /* right now, h=0.0, feedback=0.0 */
while(!kbhit()) {
for (i=0;i<I;i++) { /* every I steps output a data */
t=t+h;
/* ESO */
position=read_7166()-(unsigned long)0x080800000;
feedback=(double)(position*2*pi/16000.0);
e=z1-feedback;
fe=fal(e,alpha_01,delta_01);
v1=z1+h*(z2-beta_01*fe);
fe=fal(e,alpha_02,delta_02);
135
v2=z2+h*(z3-beta_02*fe+b0*u);
fe=fal(e,alpha_03,delta_03);
v3=z3-h*beta_03*fe;
if (t <= t1/3) v4 = 3*V_max/t1/2*t*t;
else
if (t<= 2*t1/3) v4 = V_max * t1/6 + V_max * (t-t1/3);
else
if (t<=t1) v4 = V_max * t1/2
+ ((t-2*t1/3)*t1-(t*t-4*t1*t1/9)/2)*3*V_max/t1;
else
v4 = 2*V_max*t1/3;
if (h == 0) v5 =0;
else
v5 = (v4 - z4)/h;
z4=v4; z5=v5;
e1=z4-z1;
e2=z5-z2;
fe1=fal(e1,alpha_P,delta_P);
136
fe2=fal(e2,alpha_D,delta_D);
u0=K_P*fe1+K_D*fe2;
u=u0-z3/b0;
if (u > 3.5) u = 3.5;
if (u <-3.5) u =-3.5;
out=5.0*u/7.0+2.5;
DAC_out(out);
delay(n);
t_current=read_8254();
if (t_current>t_previous) t_previous+=65536;
time_count=(int)(t_previous-t_current);
h=(double)time_count/1193180.0;
t_previous=t_current;
z1=v1;z2=v2;z3=v3;
} /* for i loop */
137
if ( k < K ) {
Time[k]=t;
if ( k == 0) Control[k] = 0;
else Control[k] = (feedback - Feedback[k-1])/h/10;
Control_out[k]=u;
Feedback[k]=feedback;
error[k] = z3;
error_dot[k] = z5;
++k;
}
else
goto exit1;
} /* for k loop */
exit1;
init_DAC();
for(k=0;k<K;k++){
fprintf(fp,"%f %f %f %f %f %f\n",Time[k],Control[k],
Control_out[k],Feedback[k],error[k],error_dot[k]);}
138
fclose(fp);
printf("Program completed !\n");
} /* end of main */
double sature(double x, double delta) {
if (x>delta) return(1.0);
else if (x<-delta) return(-1.0);
else return(x/delta);
}
double fal(double x,double alpha,double delta) {
if(x>delta) return(pow(x,alpha));
else if(x<-delta) return(-pow(-x,alpha));
else return(x*pow(delta,alpha-1));
}
double fst2(double x1, double x2, double u0, double r, double h)
{
double d,d0,y1,a0,a1;
139
d=r*h;
d0=d*h;
y1=x1-u0+h*x2;
a0=sqrt(d*d+8.0*r*fabs(y1));
if (y1>d0) a1=x2+(a0-d)/2.0;
else if (y1<-d0) a1=x2-(a0-d)/2.0;
else a1=x2+y1/h;
return(-r*sature(a1,d));
}
140
***********************************************************************
Slid Mode Observer based state feedback control for Servo Motor using trapezoid profile,
August 2002
************************************************************************
#include <stdio.h>
#include <math.h>
#include <stdlib.h>
#include <dos.h>
#include <conio.h>
#define pi 3.141592653589793
double fal(double x,double alpha,double delta);
unsigned long read_7166();
long read_8254();
void main(){
double z1=0,z2=0; /* two states of Slid */
double v1,v2; /* temple states of Slid */
double z4=0,z5=0;
double v4,v5;
double u=0,out; /* control signal */
double t=0,h=0; /* time and simulation step size */
141
/* parameters of Slid */
double beta_01=7, alpha_01= 0.5, beta_02= 7.8, alpha_02=15;
double r=20,rr=20,v0=0.125*2*pi;
double V_max, t1; /* parameters of trapezoid */
double K_P=1, K_D=4.5; /* parameters of state feedback */
double b0=25;
double e,fe,fe1,fe2;
double e1=0,e2=0;
double feedback=0;
FILE *fp;
int i,I=10,k=0,n=15;
int K=1000;
char buff[10];
long t_previous,t_current,position;
int time_count;
double *Time;
double *Feedback;
double *Control_out;
double *error;
double *error_dot;
double *Control;
fp=fopen("paraslid.dat","r");
142
if (fp==NULL) {printf("Can't open the parameters file paras.dat !\n");exit(1);}
fscanf(fp,"%lf %lf\n",&r,&v0);
fscanf(fp,"%lf\n",&K_P);
fscanf(fp,"%lf\n",&K_D);
fscanf(fp,"%lf %lf\n",&beta_01,&alpha_01);
fscanf(fp,"%lf %lf\n",&beta_02,&alpha_02);
fscanf(fp,"%lf",&b0);
fscanf(fp,"%lf",&rr);
fclose(fp);
printf(" r= %lf\n", r);
printf("v0= %lf\n", v0);
printf(" beta_01= %lf\n", beta_01);
printf("alpha_01= %lf\n", alpha_01);
printf("beta_02 = %lf\n", beta_02);
printf("alpha_02= %lf\n", alpha_02);
fp=fopen("slid_trz.dat","w");
if (fp==NULL) {printf("Can't creat the output file out.dat!\n");exit(1);}
printf("\n Input how many revolutions to go [default 1]: ");
gets(buff);
if (strlen(buff) == 0) v0=1;
143
else
v0 = atof(buff);
printf("\n Input the max speed (rad/sec) [default 14]: ");
gets(buff);
if (strlen(buff) == 0) V_max= 14.0;
else
V_max = atof(buff);
r=fabs(r);
if (r>rr) r=rr;
t1 = 3*pi*v0/V_max;
printf("Program running...\n");
printf(" \n");
printf("Press any key to exit.\n");
Time = (double *)malloc(K*sizeof(double));
if (!Time) {printf(" No enough memory for Time Array!\n");
exit(1);}
Control_out=(double *)malloc(K*sizeof(double));
if (!Control_out) {printf(" No enough memory for Control_out Array!\n");
exit(1);}
Control = (double *)malloc(K*sizeof(double));
144
if (!Control) {printf(" No enough memory for Control Array!\n");
exit(1);}
Feedback=(double *)malloc(K*sizeof(double));
if (!Feedback) {printf(" No enough memory for Feedback Array!\n");
exit(1);}
error = (double *) malloc(K*sizeof(double));
if (!error) {printf(" No enough memory for error array!\n");
exit(1);}
error_dot = (double *) malloc(K*sizeof(double));
if (!error_dot) { printf(" No enough memory for error_dot arry!\n");
exit(1);}
delay(n);
init_DAC();
init_7166();
init_8254();
t_previous=read_8254(); /* right now, h=0.0, feedback=0.0 */
while(!kbhit()) {
for (i=0;i<I;i++) { /* every I steps output a data */
t=t+h;
position=read_7166()-(unsigned long)0x080800000;
feedback=(double)(position*2*pi/16000.0);
e=z1-feedback;
145
fe=sgn(e);
v1=z1+h*(z2-beta_01*e-alpha_01*fe);
v2=z2+h*(-1.41*z2-beta_02*e+b0*u-alpha_02*fe);
if (t<= t1/3) v4 = 3*V_max/t1/2*t*t;
else
if (t<= 2*t1/3) v4 = V_max * t1/6 + V_max * (t-t1/3);
else
if (t<=t1) v4 = V_max * t1/2
+ ((t-2*t1/3)*t1-(t*t-4*t1*t1/9)/2)*3*V_max/t1;
else
v4 = 2*V_max*t1/3;
if (h == 0) v5 =0;
else
v5 = (v4 - z4)/h;
z4=v4; z5=v5;
e1=z4-z1;
e2=-z2;
u=K_P*e1+K_D*e2;
if (u > 3.5) u = 3.5;
if (u <-3.5) u =-3.5;
out=5.0*u/7.0+2.5;
146
DAC_out(out);
delay(n);
t_current=read_8254();
if (t_current>t_previous) t_previous+=65536;
time_count=(int)(t_previous-t_current);
h=(double)time_count/1193180.0;
t_previous=t_current;
z1=v1;z2=v2;
/*printf("position= %ld\n", position);*/
} /* for i loop */
if ( k < K ) {
Time[k]=t;
if ( k == 0) Control[k] = 0;
else Control[k] = (feedback - Feedback[k-1])/h/10;
Control_out[k]=u;
Feedback[k]=feedback;
++k;
}
else
goto exit1;
} /* for k loop */
exit1:
147
init_DAC();
for(k=0;k<K;k++){
fprintf(fp,"%f %f %f %f\n",Time[k],Control[k],
Control_out[k],Feedback[k]);}
fclose(fp);
printf("Program completed !\n");
} /* end of main */
int sgn(double x) {
if(x>0) return(1);
else if(x<0) return(-1);
else return(0);
}
148
B. Parameters of Active Magnetic Bearing Benchmark Problem
Signals
iφ
total magnetic flux produced by actuator
, 1, 2i i =
iu command voltage applied to actuator coil
if
force produced upon beam by actuator
, 1, 2i i =
θ rotation angle of beam, 0.00178θ ≤ radian
g nominal air gap length, 13 mils
y
displacement of beam’s end as measured
by sensor, 10y ≤ mils
iI current in coil of actuator , 1,2i i =
d disturbance torque applied to beam
Nonlinearities
(.)p ratio of air gap flux to total flux,
1( ) ( )p x ax b −= +
(.)h current as a function of gap flux and gap
length
Constants
J mass moment of inertia of beam,
149
0.00967J = kg-m2
L
distance from pivot to actuator,
0.145L = m
N
number of coil turns in each actuator,
321N =
R coil resistance, 1.55R ohms=
K
conversion factor (radians to mils)
5724 /K mils rad= −
sV power supply voltage, 30
sV = volts
a,b,c
magnetic constants,
90.0265, 1.053, 5.87 10a b c= = = ×
