ROBUST CONTROLLERS FOR SYSTEMS
WITH UNCERTAIN DYNAMICS
by
Van Thu Nguyen
B.S., Purdue University, 1984
A thesis submitted to the
faculty of the Graduate School of the
University of Colorado in partial fulfillment
of the requirements for the degree of
Master of Science
Department of Electrical Engineering
1992
This thesis for the Master of Science
degree by
Van Thu Nguyen
has been approved for the
Department of
Electrical Engineering
by
Edward T. Wall
Date
or/o^/^2.
Nguyen, Van Thu (M.S., Electrical Engineering)
Robust Controllers for Systems with Uncertain Dynamics
Thesis directed by Associate Professor Jan T. Bialasiewicz
This thesis introduces two robust controllers which deal with a class of systems
with uncertain dynamics or parameter variations. The first one is a Sliding Mode
Controller based on variable structure theory. A simple approach is presented. The
performance of the controller was evaluated and shown to be robust to certain
parameter variations in the model. The second one is a Time Delay Controller. The
control's system structure and design procedure for a linear timeinvariant and single
input singleoutput system are presented. A combination of Nyquist's criterion and
Kharitonov's theorem is used to determine the stability of systems whose
coefficients are known to vary within certain defined intervals. The approach and the
usefulness of the technique were illustrated and evaluated through simulations. The
simulation results indicate that the control method shows an excellent model
following and robustness properties to uncertain dynamics.
The form and content of this abstract are approved. We recommend its publication.
Edward T. Wall
CONTENTS
Chapter
1. Introduction.........................................................1
1.1 Overview....................................................1
1.2 Thesis Outline..............................................3
1.3 Notation....................................................4
2. Case study: A flexible onelink robotic arm........................ 6
2.1 Modelling Aspects.........................................6
2.2 Model Order Reduction......................................10
2.3 Numerical Example..........................................13
3. Sliding Mode Control................................................16
3.1 Sliding Mode Concept...................................... 16
3.2 Existence and Reachability of Sliding Mode.................19
3.3 Advantages and Disadvantages of Sliding Mode Control.......20
3.4 Chattering Elimination.....................................21
3.5 Control Law............................................... 23
3.6 Application to Flexible Arm................................28
3.7 Performance of Sliding Mode Control....................... 31
3.7.1 Robustness of Control............................... 33
3.7.2 SMC with Boundary Layer..............................36
3.7.3 Effect of Discontinuous Function.................... 38
3.8 Conclusions................................................40
4. Time Delay Control
41
4.1 Derivation of the Control Law............................41
4.2 Time Delay Control for LTISISO Systems..................47
4.3 Design Procedure.........................................52
4.4 Stability Analysis.......................................53
4.5 Implementation and Evaluation............................62
4.6 Application to the Case Study........................... 66
4.7 Conclusions..............................................76
Appendix
A.........................................................77
B.........................................................82
C.........................................................83
References.........................................................85
v
7
8
9
15
16
17
18
21
22
31
34
36
39
50
50
50
50
51
54
FIGURES
Experimental setup configuration...................
PoleZero location of the original system..........
Frequency response of the orginal system...........
Frequency response of the reduced system...........
Variable structure control system..................
Phaseplane of system in Fig. 3.1 .................
Sliding mode control trajectory....................
Boundary layer concept ............................
Modified control with boundary layer ..............
Performance of sliding mode control ...............
Robust Control Test ...............................
SMC with boundary layer............................
SMC with discontinuous function....................
Block diagram of TDC for LTISISO plant............
Reduced block diagram #1 of TDC for a LTISISO plant
Reduced block diagram #2 of TDC for a LTISISO plant
Root locus of TDC for a LTISISO plant example.....
Reduced block diagram #3 of TDC for a LTISISO.....
Nyquist diagram of a pure time delay system .........
4.7 Nyquist plot of time delay..................................................56
4.8 Nyquist diagrams of G(s) and F(s) ..........................................57
4.9 G(s) does not intercect the unit circle ....................................58
4.10 G(s) intersects the unit circle ........................................... 59
4.11 Nyquist plot of first order system example..................................64
4.12 A net rotation to the unit circle with k > 1................................65
4.13 There is no net rotation with k < 1........................................65
4.14 Nyquist plot of G(s) for nominal plant......................................67
4.15 Bode plot of G(s) for nominal plant ........................................68
4.16 Position response of nominal plant and ref. model (5ms) ....................69
4.17 Error in position of nominal plant..........................................69
4.18 Control effort of nominal plant.............................................70
4.19 Position response of nominal plant at critical time delay...................70
4.20 Error of plant at critical time delay ......................................71
4.21 Control effort of plant at critical time delay............................. 71
4.22 Position response with time delay larger than critical time delay.........72
4.23 Nyquist plot for 10 percent parameter variations ...........................73
4.24 Bode plots of Remin Immin for 10 percent parameter variations...........73
4.25 Bode plots of Remin Immax for 10 percent parameter variations...........74
4.26 Bode plots of Remax Immm for 10 percent parameter variations...........74
4.27 Bode plots of Remax Immax for 10 percent parameter variations...........75
vii
TABLES
Table
2.1 Model Parameters for First Three Flexible Modes...................... 8
2.2 Poles and Zeros of the original system.................................9
2.3 Poles and Zeros of the reduced model system ......................... 14
3.1 Parameters and Error Bounds .........................................29
4.1 Calculations for first order system ..................................65
4.2 Calculations for analysis with 10 percent parameter variations .......75
viii
CHAPTER 1
INTRODUCTION
1.1 OVERVIEW
In designing a controller for a real system, the mathematical model of the system
is the first consideration. Quite often such models contain uncertain parameters due to
the system operating in environments where unpredictable large system parameter
variations and unexpected disturbances are possible. Robot manipulators, underwater
vehicles, and autonomous systems are a few examples. Thus, a fixedgain controller
will be inadequate to achieve satisfactory performance in the entire range over which
the characteristics of the system may vary.
In some situations one would like to design a controller which works for a range of
some parameter value. For example, in designing a controller for a robotic manipulator
which is required to move payloads of different masses from one point to another, it is
more desirable to have a single controller which works for a range of payload masses
than to change the controller or adjust some controller parameter for each mass. This
problem is the same as that of designing a controller for an uncertain system.
This thesis presents two control methods, Sliding Mode Control (SMC) [1]
and Time Delay Control (TDC) [2] which can be applied for such systems.
(1) Sliding Mode Control : This method of control belongs to a wider class
commonly referred to as variablestructure controls (VSC) which derive their
name from their capability to modify the structure of the system. Variable
structure control systems have been studied in great detail in the Soviet Union
where a substantial part of the sliding control theory was developed.
The main feature of sliding mode control is to let a sliding mode occur on a
predetermined switching surface. While in sliding mode, the system will be
forced to "slide" along or in the vicinity of the switching surface. The system
is then robust and insensitive to the interactions.
The sliding mode control, however, has several drawbacks which limit the
application of the method in practice due to discontinuous high frequency
chattering. To overcome this disadvantage, continuous control laws [3],[4] and
refined switching surface [5] have been proposed.
(2) Time Delay Control: This method of control depends on the direct estimation of
a function representing the effect of uncertainties. This is accomplished by using
a time delay. The controller uses the gathered information to cancel the unknown
dynamics and the unexpected disturbances simultaneously, and then inserts the
desired dynamics into the plant. The TDC uses the past observation of the system
response and control input to directly modify the control actions rather than
adjusting the controller gains or identifying system.
These two methods are studied and applied to the endpoint control of a flexible
robotic arm considered as a case study in this thesis.
2
1.2 THESIS OUTLINE
For convenience in evaluating the performance of each controller, this paper starts
with an introduction of a flexible singlelink robotic arm in chapter 2. The model
parameters and the transfer function are based on the experimental arm of Cannon and
Schmitz [6]. Since the arm is a highorder system and is unstable, a model order
reduction method of Meyer [7] is applied to obtain a stable low order system. This
reduced order system is used as the case study to evaluate the performance of the
proposed controllers.
Chapter 3 begins with an introduction of the concept of slidingmode control
based on the switching logic in Variable Structure Control. General necessary and
sufficient conditions for the existence of sliding regime have been presented. A
description of simple slidingmode control approach of Paul A. Zaffiro is presented
[1], The controller is applied to the case study and the performance of the controller
respect to parameter variations is evaluated through a series of simulations followed
by discussions and conclusions.
Chapter 4 presents a Time Delay Controller of Komal YoucelToumi and Osamu
Ito [2], The control problem is defined and a control law for linear timeinvariant and
singleinput singleoutput systems is derived. A method of analysis combining the
Nyquist criterion and Kharitonov's polynomials is presented to determine the
stability of systems whose coefficients are known to vary within certain defined
intervals. A simple example is analyzed to illustrate the approach and the usefulness of
the technique. The effectiveness of the controller is evaluated by applying to the tip
position control of a flexible onelink arm through simulations.
3
1.3 NOTATION
A System matrix
B1 Inverse of matrix B
B+ Speuso matrix of B
E Young's modulus of elasticity (N/m2)
e Error state vector
I Cross sectional areamoment of inertia (m4)
Ih Hub inertia (kgm2)
lb Moment of inertia of the beam (kgm2)
IT Total inertia of the arm (1^ + 1^)
I Identity matrix
L Length of the arm (m)
0 Joint angle
S(x,t) Function representing a switching plane.
S(x,t) First derivative of function representing a
switching plane,
sat Saturation function
sgn Signum function
s Laplace operator
T Torque applied by motor (N.m)
t Time
T Time delay
^crit Critical time delay
4
4>i Eigen function of i* mode
Ci Damping ratio of mode i
P Linear density (kg/m)
The generalized coordinate of the system
0)i Natural frequency of mode i
r Reference signal
r Reference vector
X State vector
T X Vector transpose
8 Belong to
Rn nvector space real number
R Positive real number
u Vector input
CE Closedloop characteristic equation
SMC Sliding mode control
VSC Variable structure control
I Grammians
i Hankie singular values
wc Controllability Grammians
W0 Observability Grammians
Minimum value
+ Maximum value
5
CHAPTER 2
CASE STUDY: A Flexible OneLink Robotic Arm
The use of lightweight flexible links in robot manipulators is desirable from the
point of view of achieving robot arms which are faster and less expensive than the
typical robot arms in use today. These arms generally have highly vibratory poles
and low damping factors. Thus accurate position control becomes more difficult.
In this chapter, the flexible singlelink robotic arm of Cannon and Schmitz is
presented and is used to investigate the performance of the proposed controllers.
2.1 MODELLING ASPECTS
The experiment fexible singlearm of Cannon and Schmitz has been described in
detail in [6]. It was constructed from two thin aluminum beams with a series of
flexible bridges, made of alminum and beryllium copper, which act as pinned joints
for horizontal bending while maintaining vertical and torsional stiffness. The arm is
shown in Fig. 2.1.
The modelling of this particular structure has been presented in many works.
The transfer function of the system is derived in [6] and presented as
Y(s)
T(s)
L_
ITs2
i(L>aÂ£(0)
s^+ 2C,i cOjS + coi
(2.1)
where Y and T are the endpoint position and torque input, respectively.
positionsensitive device
Fig. 2.1 The experimental setup configuration
From (2.1) it can be seen that the model consist of an infinite number of modes.
In practice the controller design must choose the main dominant modes to design
control law. In this paper, only the rigid mode and the first three vibrational modes
of the arm are considered for the controller. In [6] it has been shown that the
parameters of the system are uncertain and may change with time.
The model parameters of the experimental flexible arm are given in Table 2.1. The
eighthorder model describing the flexible beam can be represented as:
G(S) = = (szQ .. .(sz6)
T(s) (spj). . (spfe)
(2.2)
7
Table 2.1
Model parameters for first three flexible modes
Mode Number PinnedFree Frequency Fj(Hz) Model Damping Actuator Model Gain d<]) /dx(0)(rad) Tipsensor Model Gain 0 (1) (m)
0* 0 0 1.0 1.12
1 1.88 0.015 2.97 1.1
2 3.44 0.02 3.0 0.09
3 7.70 0.02 1.25 1.21
* Rigid body.
where the poles and zeros of the eighthorder system are given in Table 2.2. The pole
zero location of the nominal system can be seen in Fig. 2.2.
Fig. 2.2. PoleZero Location of the original system
8
PHASE ANGLE (DEGREES) MAGNITUDE (DB)
Table 2.2
Poles and zeros of the original system
Poles Zeros
0 0 .0282 + 1.8798i .0600 + 3.4393i .1540 t7.6959i 1.7070 1.6552 3.2094 +4.257 li 3.3717 +4.0864i
The frequency response of the arm is shown in Fig 2.3
102 101 100 10i 102 103
Fig. 2.3 Frequency response of the original system.
9
2.2 MODEL ORDER REDUCTION
The singlelink flexible arm in Section 2.1 is assumed to consist of the regid body
mode and the first three vibratory modes. This results in a high order system. Control
for such a model would require a high order controller which may not be feasible to
implement in practice. This section presents a Fractional Balanced Reduction
technique which applies to unstable systems. Their resulting loworder models are
stable and cognizant of the full order models. Hence, a controller designed for the
reduced model will not only stabilize the original plant but the resulting closed loop
performance will also match closely the performance of the reduced order model plus
controller.
Fractional Balanced Reduction is an extention of the balancing technique of
Moore [8] to unstable plants without requiring the division of the plants into stable
and completely unstable parts. The technique is based on transforming the state to a
coordinate system in which the controllability and observability Grammians are equal
and diagonal, and then deleting those states which are only weakly controllable and
observable.
The reduced order models possess the following properties which are useful in
the design of reduced order controllers:
(i) The reduced order models preserve stability, controllability, observability
and minimality.
(ii) The steady state values of the reduced model and the original system are
identical.
(iii) Error bound of the reduced order model
10
o[GGco) Gr (jco)] <2 Zai Vw
i=r+l
where o(.) denotes the largest singular value of a matrix, Gr (jw) is the transfer function
of the reduced model.
The algorithm of the Fractional balanced Reduction method can be summarized
in six steps as follows:
Step 1: Create
G(s) = C(sl A)'1 B where {A, B, Â£) are minimal state space
realizations of the system
A minimal realization is the realization of a model that has the redundant or
unnecessary states eliminated.
Step 2: Solve the algebraic Riccati equation (ARE)
P A + ATP PBBTP + CT(T = 0
with P is the unique positive definite solution to the Algebraic Riccati
Equation. Define
A = A + BK
where
K = BtP
The gain parameter K is chosen so that the close loop system matrix A has
eigenvalues in the lefthalf plane (LHP).
11
Step 3: Balance the stable, minimal state space triple { A, B, (*2)} to obtain a
1 K J
new balanced state space triple { A,B,(C)} and Qrammian Z .
Z =diag(ai, o7,.., on),
where Oj > 0 are the Hankel singular values.
The balance realization of the system then can be obtained by using a linear
state transformation matrix T to transform the given system to balanced
form with equal and diagonal Grammians
T1 WcT'T= TtW0 T = Z = diag( G;)
where and WÂ£, are controllability and observability Grammians,
respectively.
p
Step 4: Partition {A, B,( K)} as:
and
r n r m
All A 12 > B = r Bi
A 21 A 22 n r B2
r n r
C P c, c2
K m Ki k2
where
n is order of the full system
r is order of the reduced system
p is number of output
m is number of input
12
and partition the Gramians as:
r n r
Si 0
0 Z2
whereZi = diag(o1; o2,..,or)
and Z2 = diag( Gr+1 o+2 ,.., on )
 Q
Step 5: Take { An, Bj ( ^ )} as a minimal, stable, rthorder ROM (reduced
p
order model) for { A B, (^)}.
Step 6: Let An= AnBjKj
Then { An, Bj Cj } is the rthorder FBR reducedorder model for G(s)
r
Z =
n r
2.3 NUMERICAL EXAMPLE
The flexible arm with the poles and zeros given in Table 2.2 is used in this
example. Following the procedure outlined in the previous section, we obtain the
Hankel singular values (see Appendix A) in nonascending order are:
CT1 2 a3 4 CT5 6 7 8
.6124 .5035 .3501 .3016 .2051 .0023 .0015 .0003
Note that c% thru are very small which suggest that a fifth order model is
reasonable for the system. The reduced order model can be written as:
13
(2.3)
^/x 14 s2 26.6 s + 1105.5
Gr(s) = ,:;
s5+ 5s4+ 74.7s3+ 305s2 + 607.5s + 2747.5
and the poles and zeros of the reduced system are given in Table 2.3.
We find (from Table 2.2 and 2.3) that the poles and zeros of the original system are
not simply related to those of the reduced system. Although, the original system is
strictly proper, the reduced system is not so. However,
8
a[G(jco) Gr(jco)] < 2 Z a. < 0.0082
i=6
Table 2.3
Poles and zeros of the reduced model system
Poles Zeros
3.5706 0.2940 + 7.90851 0.4207 + 2.5087i 15.8919 0.9459 + 2.0422i
14
PHASE (DEGREE) MAGNITUDE (DB)
The frequency response of the reduced system is shown in Fig. 2.4
40 r
20
0
20 :
40L
m
N;:
102 10i 100 10i
FREQUENCY (HZ)
102
103
200
100
0
100
200
102 10i 100 10i 102 103
FREQUENCY (HZ)
Fig. 2.4 Frequency response of the reduced system
15
CHAPTER 3
Sliding Mode Control
In this chaper a simple sliding mode control technique is presented. The robustness
property respect to uncertain parameters, the boundary concept to eleminate chattering
and the effect of discontinuous function in control law are evaluated.
3.1 SLIDING MODE CONCEPT
Sliding mode control (SMC) is classified as a variable structure control (VSC).
This method of control is currently receiving a growing interest in controlling
nonlinear systems and systems with uncertain parameters. The basis characteristic of
VSC is its ability to modify the dynamics of a given system. One such VSC is a
control law defined by a rapid switching between two values which may be constant
or variable.
With this type of control, the states of the system are attracted to a hyperplane
commonly referred to as the switching surface or the sliding surface, where the states
remain thereafter. This surface is considered as a desired trajectory and is composed
from the parts of different structures. This concept is illustrated by the following
example of Utkin [9] shown in Fig. 3.1.
Fig. 3.1 Variable structure control system
Consider a secondorder sytem in Fig. 3.1. It has a switch that can be used to
connect the system in the negativefeedback mode or the positivefeedback mode. To
configure a negativefeedback structure, the switch is thrown to position A. The
system response for this negativefeedback configuration is unstable and its phase
plane portrait is illustrated in Fig. 3.2(a).
On the other hand, when the switch in Fig. 3.1 is thrown to position B, a positive
feedback system can be obtained. The system response for this configuration is
displayed in Fig. 3.2(b), and it clearly shows the system is unstable.
(a)
x2=xi
(b)
Fig. 3.2 Phaseplane for the system in Fig. 3.1: (a) Negative feedback;
(b) positive fedback.
17
Note that the system was unstable for either configuration. However, if a
switching law is defined such as
where
u
if S > 0
if S < 0
(3.1)
S = CX!+ x2 = 0,
(3.2)
with c > 0 to force the system switches between the negative and positive
configurations, then a stable and robust response can be obtained. Its phase plane
portrait is shown in Fig. 3.3, where the phase trajectories are directed towards the
switching line S = cxj + x2 and once on this line the states must remain on it. The
motion along a line which is not a trajectory of any of the structures is called in
sliding mode.
Fig. 3.3. Slidingmode control trajectory of the system in Fig. 3.1
18
3.2 EXISTENCE AND REACHABILITY OF SLIDING MODE
The existence of a sliding mode is very critical in designing a siliding mode
controller. It demands that upon intersection of the switching surface, the
trajectories must remain there indefinitely. This description only applies to the ideal
case. In practice, due to switching delays, ideal switching does not occur and the
trajectories can only be guaranteed to lie within an neigborhood of the surface. Thus,
the existence of a sliding mode is related to the stability of the system.
The property of reachability signifies that the trajectories converge to the sliding
surface for all initial conditions in a domain of attraction Q, for all t > 0. The
sufficient conditions guaranteeing the existence and reachability of the sliding mode
based on Lyapunov's stability theorems are presented as follows:
Consider the general singleinput non linear system
x(t) = f(t, x, u) (3.3)
where
x(t) 8 Rn and u(t) 8 Rn
The switching surface is in general a Rn 1 submanifold of Rn. Let the
switching be represented by the equation
S(x) = 0 (3.4)
Referring to the Lyapunov stability methods (Ukin [9]), the following Lyapunov
function candidate is considered
V{S(x)} =yS2(x)>0 (3.5)
19
Theorem 1 (Ukin [9]). A sliding mode exists if
dV(S(x))
dt
= SS <0
(3.6)
about S = 0.
Theorem 2 (Ukin [9]). The subspace for which the switching surface S = 0
may be attained is given by
Q = {x : S(x)S(x) <0 for all t > 0 }
The second method of Lyapunov states that the system (3.3) is asymptotically
stable for all initial conditions x(t ) e Q, if S(^)S(x) < 0 evaluated along a solution x
for all x e Q.
3.3 ADVANTAGES AND DISADVANTAGES OF SLIDING MODE CONTROL
An important advantage of sliding mode control is that the nonlinear model is
used directly to calculate the controller and a linearization is not neccessary. Also,
the order of a system may be reduced and the structure of a nonlinear system may be
transformed to that of a linear system. Probably the most significant advantage is
the robustness. When in sliding mode, the system is robust against external
disturbances and plant parameters. Thus, the sliding mode controller has capability
to reduce large overshoots which occur at the instant of step change of parameters.
20
The disadvantage of this particular type of control is chattering phenomenon
caused by discontinuous function in the control law. In general, chattering is highly
undesirable in engineering applications since it implies extremly high control activity
and further may excite highfrequency dynamics neglected in the course of modelling.
This disadvantage becomes apparent during the physical implementation since the
switching frequency is finite.
3.4 CHATTERING ELIMINATION
The chattering problem can be remedied by a smoothing approximation in a
boundary layer near the switching surface as shown in Fig. 3.4. This concept was
first proposed by Slotine and Sastry [10] and is achieved by replacing the sgn function
with the following saturation function.
sgn(S)=
{+1 if S >0
1 ifS <0
*
sat(S/d>) =
!
sgn(S),
s/o,
if ISI >0
if d> < S < d>
21
where O (O > 0) is the boundary layer thickness. The boundary concept is realized
by the modifed control law
r asgn(S), ISI >
l^S, ISI <
(3.7)
The modified control has the form of a linear function with saturation as seen in
Fig. 3.5.
Fig. 3.5 Modified control with boundary layer
The parameter d> determines the "width" of the boundary layer and when tends
zero the control tends to the ideal discontinous control. As d> increases, the control
becomes more and more smooth but at the same time, the advantanges of ideal
sliding mode control diminish. Thus, there is a tradeoff between the benefits gained
by using layer and the degrations of performance which result.
22
3.5 CONTROL LAW
Consider a singleinput singleoutput (SISO) system in the controllable
canonical form
Xi = x2
x2 = x3
xn = f(x,t) + b(x,t).u(t)+ d(t) (3.8)
and
y(t) = c(x,t) (39)
where xT= [ x3, x2.xn], u(t) is the scalar control input, d(t) is a scalar disturbance
and y(t) is the scalar output of interest. In general f(x,t) and c(x,t) do not need to be
known exactly, but they must be bounded by some known continuous function of x.
Our goal is to find a contol law that will make the output follow a desired trajectory
in the presence of model and disturbance uncertainties.
If any desired output, yd, is chosen, the control problem is then to get the output
to track the desired output or to get the error between the output and the desired
output to go to zero. Let the output tracking error be define as
e(t) = y(t) yd(t)
(3.10)
We now define a time varying surface S(x,t) in the state space Rn, which presents
the desired dynamics defined by the scalar equation
23
S(x,t): S(x,t) = 0
(3.11)
In general, the sliding surface is a linear combination of the tracking error and its
derivatives:
S(x,t):= (JjL+ j^eCt) A>0 (3.12)
= X,i e(t) + A,2e(t) + ... + ^ (3.13)
dtr_1
The constants A,; in Eq. (3.13) must be chosen so that S(x,t) = 0 yields acceptable
error transients and r is the number of times Eq. (3.12) must be differentiated until
the control, u(t), appears.
The control problem of tracking the desired output is equivalent to remaining on
the sliding surface S(x,t) for all time t > 0. This can be accomplished by choosing
S(x,t) as a Lyapunov function candidate and choosing the control u(t) such that
2 "dt ^2
To insure that S(x,t) will go to zero in a finite time which we can control, we
redefine (3.14) such as
~2 "dtS 2(x,t) ^ ^ ISI, rl>0 (315)
or
S (x,t) = T).sgn(S) (3.16)
where r\ is a positive constant. Inequality of Eq. (3.15) constraints trajectories to
point toward the surface S(x,t) and is referred to as sliding condition.
24
The idea behind equations (3.14) and (3.15) is to choose a wellbehaved function
of the tracking error according (3.14), and select a feedback control law, u(t), in Eq.
(3.8) such that S2satisfies equation (3.15) despite the presence of modeling errors and
parameter uncertainties. Further, if ylt=o f ya lt=o the sliding surface S(x,t) will still be
reached in finite time while equation (3.14) guarantees that e(t) ^ 0 as t >oo.
The control law can be derived by differentiating equation (3.13) and substituting
into equation (3.16) yields the control law which is given by
S = Xi e(t) + X2e(t) + ... + ^ = ri.sgn(S)
dt1
(3.17)
or
S = *4 e(t) + X2e(t) + ..+XT.i + % %d = Ti.sgn(S)
dtrl dtr dt1
(3.18)
Assume that the output must be diffentiated r times until the control, u(t), appears
then dry j dtr can be written in the form
dry
dt
7 = g(x,t) + h(x,t).u(t) + d(t)
(3.19)
Inserting dry/dtr in Eq. (3.19) into Eq. (3.18) gives:
e(t) + ?i2e(t) + .. +A.ri
dtrl
g(x,t) + h(x,t).u(t) + d(t) ^rd = ti.sgn(S)
(3.20)
25
From Eq. (3.20) the control law u(t) then has the form
u(t) = .7u +ri.sgn(S))
h(x,t)
(3.21)
where u is
(3.22)
The control law given in equation (3.21) assumes that the system model is perfectly
known. Of course, this is never the case. In fact, modeling errors are an inevitable in
control work. They include everything from modelled dynamics to variations in
physical constants. To account modeling errors and uncertainties into control problem,
the control law in equation (3.21) must be modified.
Suppose that errors are assumed to exist in both g(x,t) and h(x,t) in equation (3.19).
Let g(x,t) be of the form
general, the modelling errors are not known, but their magnitudes are upper bounded
as
g(x,t) = g(x,t) + 5g(x,t)
(3.23)
where g(x,t) is the modelled system dynamics and 5g(x,t) is the modelling error. In
I 5g(x,t)l < y (x,t)
(3.24)
If h(x,t) can be bounded as
tvnin< h(x,t) < b,
'max
(3.25)
26
then Slotine [4] has shown the control law can be defined as
u(t) =
7 (k.sat(S/0) + ft)
h
(3.26)
where
u' A.^ e(t) + + A,
dr1e(t)
rl
dt
rl
+ g(x,t) 
dryd
dtr
(3.27)
and
K = p(Ti + Y) +(pl)lul
(3.28)
6 =
bm in
(3.29)
P =
j bmax
V b^T
(3.30)
The timevarying gain, K, can be thought of as a modified (and timevarying) T.
It guarantees that the control is robust to modelling errors in g(x,t) and h(x,t) not in
excess of the bounds defined by equations (3.24) and ( 3.25).
The control parameters r, A, and O are chosen by the designer. The parameter r\
insures that equation (3.16) is satisfied and that the system converges to desired
output in a finite time. The choice of r\ depends on the plant is modelled. If the plant
is modelled poorly, r\ must be large in order to guarantee that the system is robust to
the modelled errors.
27
3.6 APPLICATION TO FLEXIBLE ARM
The reduced model developed in Sec. 2.3. is used in this application. The
equations of control system are written in a controllable canonical form as:
Xj = x2
x5 = a0 Xj aj X2 a2 x3 a3 x4 a4 x5 + u
and
(3.31)
y = Cj Xj+ c2x2+ c3 x3
where x =[xj, x2, .. x5]T, y is the tip position and u is the motor torque. The
parameters given in Table 2.2 are considered as the nominal parameters of the plant.
For the purpose of this study, the model will be assumed to be accurate within
10% of the nominal plant. Thus, the upper and lower bounds on the parameter
errors can be determined. The nominal parameters and their bounds are shown in
Table 3.1.
The control objective is to maintain the armtip y at a desired position which is
position step input yd. The error between the actual tip position and the desired
position is defined as
e(t) = y(t) yd(t) (3.32)
28
Table 3.1
Parameters and Error Bounds
Param %err Nom + 
ao 10 2747.5 3022.25 2472.75
at 10 607.5 668.25 546.8
a2 10 305 335.5 274.5
a3 10 74.7 82.1 67.0
a4 10 5 5.5 4.5
Cl 10 1105.5 1216.05 994.95
92 10 26.6 23.9 .29.3
C3 10 14 15.4 12.6
Using equation (3.12), the sliding surface S(x,t) is defined as
S(x,t) = ^(t) + e(t) + e(t)
(3.33)
or
S(x,t) = A.i(y yd) + (y ya) + (y y'd)
(3.34)
Differentiating (3.34) gives
S(x,t) =Aiy + y + y"
(3.35)
since the derivatives of step are zero for a regulator. Differentiating y in Eq. (3.31)
three times gives
Cjalx2' c3a2X3+ (C1 c3a3)x4
+ (qz C3 a4 )X5 + C3 u
(3.36)
29
and thus
g(x,t) =c3aQx1 Cja^ c3a2x3+ (q C3a3)x4
+ (C2 c3a4)x5
Using in Table 3.1 as the control gain of the input u, the bounds and bmi,
found to be
bmax = 15.4
b = 12.6
mm
and from equations (3.27) and (3.28) we have
h ^ bmaxbmin = 13.93
P = '\j bmax/bmin = 1.1
Using the nominal values in Table 3.land inserting into Eq. (3.17) gives:
g(x,t) = 20258XJ 8505x2 4270.4x3 965.3x486.6x5
The control law used for simulation is
u(0 =  ( K.sat(S/
where
fi = 13.93
and
u = X,iy (t) + y(t) + g(x,t)
(3.37)
are
(3.38)
(3.39)
(3.40)
(3.41)
(3.42)
(3.43)
(3.44)
30
3.7 PERFORMANCE OF SLIDING MODE CONTROL
In this section the results of the simulation of control law based on slidingmode
control are presented and evaluated. After some iteration, the control parameters were
chosen as given in Table 3.7.1. From Table 3.7.1, the value of fc is:
K = 1490+. Liu'I (3.45)
A
Using the value of K, Eq. (3.44) and the control law of Eq. (3.34), the simulation
results are shown in Figs. 3.6. Fig 3.6(a) shows the trajectories of the reference model
and the tip position. The error signal and the control effort are shown in Figs 3.6(b)
and 3.6(c).
Table 3.7.1 Control Parameters
?ll o T\ y
10 25 55 1300
Fig 3.6(a)
31
CONTROL EFFORT (N.m)
Fig. 3.6(b)
Fig. 3.6(c)
00
Z
0
1 o
D
u.
2
t
2 3
SECONDS
Fig. 3.6(d)
32
Fig. 3.6(e)
Fig. 3. 6 Performance of sliding mode control, (a) Tip and desired position,
(b) error signal, (c) control effort, (d) function S. (e) function SS.
3.7.1 Robustness of Control
The next test for the controller is its robustness to parameter variations and
unmodeled dynamics. Slidingmode controllers are expected to outperform more
than conventional feedback schemes with respect to their robustness. This is true
for the following two reasons. First, choosing T, then sufficiently large will
guarantee that the system approaches the sliding surface. Secondly, the controller
is lower order and is not an exact representation of the actual physical process. It
is naturally more robust than the original higher dimensionality system. In order to
evaluate this property, the amplitude of control parameter r is increased as given in
Table 3. 7.2 The simulation results are shown in Figs. 3.7.
33
CONTROL EFFORT (N.m) TIP POSITION & REF. MODEL (m)
Table 3.7.2 Control Parameters
^1 o T\ Y
10 25 110 1300
1.2
1
0.8
0.6
0.4
0.2
0 1 2 3
u SECONDS
Fig. 3.7(a)
2^::
1.5
1
0.5
0
0.5
1 2 3 4 5
SECONDS
Fig. 3.7(b)
34
Fig. 3.7(c)
300
200
Fig. 3.7(d)
Fig. 3.7 Robust Control Test. (a) Trajectories of armtip and reference,
(b) Control Effort (c) Function S. (d) Function SS.
Fig 3.7(a) shows that the arm tip tracks the reference model very well, except
a vibration appears at the output. This vibration dues to a large increase in the
magnitude of the control input that provokes the manifestation of the physical
phenomenon known as chattering. This chattering is shown clearly in Figs 3.7(c)
and 3.7(d).
35
3.7.2 SMC with Boundary Layer
The chattering effect may be eliminated by the introduction of the boundary layer
which is characterised by control parameter d>. In the previous section, the magnilitude
r] was increased to illustrate the robustness of the controller, thus, the chattering affected
the control input. In order to eliminate this problem, the boundary layer needs to be
modified. Table 3.7.3 presents the control parameters with the boundary layer is increased
to 40 and the simulation results are shown in Fig. 3.8.
A comparison of Fig. 3. 7 and Fig. 3.8 reveals a considerable difference in the
form of the actual control for the two choice of . Indeed, with = 40 the control is
much smoother. It is clear that finding the optimal value of O which gives an acceptable
form of the control, and at the same time is robust, needs a series of experiments.
Table 3.7.3 Control Parameters
X\ o T1 y
10 40 110 1300
Fig. 3.8(a)
36
CONTROL EFFORT (N.m)
300
200
100
0
100
200
300
0
2 3
SECONDS
Fig. 3.8(b)
4
5
Fig. 3.8(d)
Fig. 3.8 SMC with boundary layer. (a) Trajectories of armtip and reference,
(b) control effort (c) function S. (d) function SS.
37
3.7.3 Effect of Discontinuous Function
As mentioned earlier, the main disadvantage of slidingmode control is chattering
phenonmenon with very high frequency due to the sgn(S) function in the controller.
In practice, the physical system may not tolerate such very high frequency at the input
to generate control. To see this disadvantage, a simulation is carried out by deleting
the boundary layer in the last simulation with the control parameters given in
Table 3.7.4 The simulation results are shown in Fig. 3.9. Fig. 3.9(a) shows that
the tracking is not satisfied. This implies the fact that the trajectories of the system
only lie within a neighborhood of the switching surface as shown in Fig. 3.9(c). Fig.
3.9(b) shows that the control input is excited to a very high magnitude.
Table 3.7.4 Control Parameters
X,i T1 Y
10 0 110 1300
38
5000
5000
2 SECONDS^
Fig. 3.9(b)
100
50
2 0
50
100
200
2 ^
SECONDS
Fig. 3.9(c)
2 3
SECONDS
Fig. 3.9(d)
Fig. 3.9 SMC and discontinous function, (a) Trajectories of armtip and reference,
(b) control effort (c) function S. (d) function SS.
39
3.8 CONCLUSIONS
A slidingmode control technique based on variable structure theory was presented
and applied to the tip control of a flexible robotic arm, an uncertain linear SISO system.
The controller demonstrated the robustness property with respect to uncertainties in the
presence of the modelling errors. This was illustrated by changing the model error for
which SMC was found to be good tracking accuracy. The approach also shows a
reduced computational requirement for meeting robustness and performance
requirements.
40
CHAPTER 4
Time Delay Control
This chapter introduces another control method, Time Delay Control. A control
law for linear uncertain SISO system is discussed. A stability analysis procedure
combined from the Nyquist criterion and Kharitonov's theorem is used to determine
the stability of systems whose coefficients are known to vary within defined intervals.
The effectiveness of the TDC system is evaluated through simulations.
4.1 DERIVATON OF THE CONTROL LAW
Consider a nonlinear system described by the following dynamic equation
x = f(x,t) + h(x,t) + B(x,t)u + d(t) (4.1)
where
x is an nxl plant state vector,
u is an rxl control vector,
B(x,t) is an nxr control distribution matrix with rank r,
f(x,t) is nxl vector representing known part of the plant dynamics,
h(x,t) is nxl vector representing unknown part of the plant dynamics,
d(t) is an unknown disturbance vector, and
t time.
Note that the above expression is common in many applications. For example,
in robot manipulators, h(x,t) can correspond to nonlinear torques caused by Coriolis
and centrifugal effects or nonlinearities such as dry fiction at each joint. The vector
d(t) represents any kind of disturbance such as external torques.
The reference model that generates the desired trajectory as a linear time
invariant system is defined by:
(4.2)
where
xm is an nxl state vector,
Am is an run constant stable system matrix,
Bm is the nxr constant command distribution matrix, and
r is an rxl command vector.
The error vector, e, is defined as the difference between the plant state vector and
the reference model state vectors
The control objective is to force the error to vanish with a desired dynamics:
where Ae is an nxn error system matrix, which defines some desired dynamics. By
combining equations (4.1) through (4.5) and adding Amx + AmX, an equation that
governs the error dynamics is obtained as:
(4.3)
and the time rate of change of the error is
(4.4)
e = Aee
(4.5)
e = Ame +{f h d+ Amx + Bmr Bu }
(4.6)
42
If it is possible to determine a control u of equation (4.6) such that the following
equation is always met
Ke = f hd + An^c + BmrBu (4.7)
then, substitution of Eq. (4.6) into Eq. (4.5) leads to a desired dynamics
e = (Am+K)e = A,e (4.8)
where K is an nxn error feedback matrix and the error system matrix Aecan be
arbitrarily determined through a proper choice of the error feedback gain matrix K.
The control signal u, that satisfies equation (4.7), must then be selected in order
to obtain a desired error dynamics. However, equation (4.7) cannot always be
satisfied because the number of signals control is generally smaller than the number
of states. Thus, the best approximate solution of the equation is adopted to determine
the control u:
u = B+{ f h d + Amx + Bmr Ke ) (4.9)
where B+= (BTB)'1 BT and is known as a pseudoinverse matrix. Note that the matrix
B B is an rxr nonsingular matrix since B is of rank r.
By substituting Eq. (4.9) into Eq. (4.1) and through some algebraic manipulations
given in Appendix B, the error dynamics is obtained
e = (Am+ K)e + {I BB+}{fh d + Aji+ B^Ke} (4.10)
43
In order to obtain the desired error dynamics given by Eq. (4.8), the following
structural constraint must be met so that the error vanishes as time goes to infinity.
{I BB+}{ f h d + Amx +Bmr Ke} = 0 (4.11)
The structural constraint of the equation (4.11) is always satisfied if B is nxn and
B 1 exists since I BB + = I BB 1 is a zero matrix. It is shown in reference [2], this
condition is always satisfied for systems expressed in canonical form (see Appendix C).
Since the matrix I BB+ has rank nr, Eq. (4.11) effectively consists of nr constraint
equations. So, the structural constraint simply indicates that r inputs can really control
only r states and the rest nr states should automatically be controlled under the
contraint.
The control u that will force the plant to follow the reference model in the presence
of unknown dynamics h(x,t) and unexpected disturbance d(t) which are in the right hand
side of Eq. (4.8). The two terms appear as a sum and their effect can be determined
form the plant dynamics equation (4.1):
h(x,t) + d(t) = x(t) f(x,t) B(x,t)u (412)
The estimate of the effect of the term h(x,t) + d(t) can be obtained by evaluating the
function h(x,t) + d(t) at the present time and at time t X in the past for a small time
delay X,
fi(x,t) + d(t) = h(x,t X) + d(t X)
(4.13)
44
Combining Eqs. (4.12) and (4.13), the effect of the function h(x,t) + d(t) is
estimated by:
fi(x,t) + d(t) = x(t T) f(x,t X) B(x,t T)u(t X) (4.14)
The TDC control law is then obtained by substituting Eq. (4.14) into Eq. (4.9)
and is given by
u(t) = B+(t){ f(t) x(t X) + f(t X)
+B(t X)u(t X) + Amx(t) + Bmr(t) Ke(t)} (415)
This controller observes the states and the inputs of system at time t X, one step
into the past, and determines the control action that should be commanded at time t.
Note that the controller identifies the unknown dynamics by evaluating a function
directly every X seconds. Therefore, any changes in dynamics will be detected within
the time period X.
As mentioned earlier, a class of systems that satisfy the structural constraint are
those expressed in canonical form. Thus, the Time Delay Control law for this specific
class of systems with n states and r inputs is derived if each term of equation (4.1) can
be partitioned as follows:
x =
f(x,t)
fr(x>0
h (x,t) =
0
hr(x,t)
45
B (x,t) =
(4.16)
0
Br (X,t)
d(t) =
0
dr(t)
X
where xq and 0 are (nr)xl vectors, x s =[x j,.., xn] is also an (nr)xl vector, x
h r and dr are rxl vectors, and Br is an rxr nonsingular matrix.
The reference model and an error feedback gain matrix can also be partitioned in
a similar fashion:
Ollq " 0 0
Amr ; Bm = Bmr ; k = Kr
mr _
(4.17)
where, in this case, 0 is an (nr)x (nr), (nr)xr or (nr)xn matrix, respectively and Iq is
an (nr)x(nr) matrix, Amr is an rxn matrix, Bmr is an rxr matrix and K ris an rxn
matrix. For the specific canonical form considered, B+ is given by
b* = (btb)'bt= {[Â£] [i;]} LoT;Br]
= (Br Br )_1 [0 :Bj]=B\\0 :Ir] = [0 :B't) (4.18)
Substituting Eqs. (4.16), (4.17) and (4.18) into Eq. (4.15), the control action now
reduces to
u(t) = B'1 (t){ fr(t) x r(t X) + f r(t X)
+ Br (t X)u(t X) + A mix(t) + Bmrr(t) K re(t)} (4.19)
46
4.2 TIME DELAY CONTROL FOR LTISISO SYSTEMS
Consider a linear timeinvariant and singleinput singleoutput plant with system
matrices, A and B, are unknown
x = Ax + Bu (4.20)
T
where x = [xj, x2,..., xn] The matrices A and B are written in controllable
canonical form and are partitioned
Ar= [^q ,aj,.. ,an_1],
Br=b (4.21)
where aQ, a},... an.j and b are unknown constants
The reference model and error feedback gain matrix K can be chosen based on Eq.
(4.17) in order to satisfy the structural constraint.
x = Amx +^r
ym = xm.i (4.22)
where
xm xm,2 Xm,rJ
^mr [VO "am,l am,nl
Bmr = bm, and
Kr = [k0,k ........kn.J (4.23)
For this case fr(x,t) = 0 and hr (x,t) = Ar. Substituting Eqs. (4. 20 ) through (4.23)
into Eq. (4. 19), the TDC control law can be obtained as follows
u(t) = u(t T) + (l/b){ x,, (t X) arn nxn (t)
 amJx (t) + bm r(t) + kn.1(xm n xn ) + .. + k0(xmil Xj)} (4.24)
47
If the output of the plant is designated as y = , the transfer function relating y
to the input u is
GP(s) =
Y(s)
U(s) P(s)
(4.25)
where Y(s) and U(s) are the Laplace transforms of y(t) and u(t), respectively and the
polynomial P(s) is
/ v n n1
P(s) = s + ani s + ... + a0
(4.26)
The transfer function of the reference model is written in the same way
GJs) =
Ym(s) bn
R(s) Pm(s)
where
n1
Pm (s) = s + am n.lS + ... + am 0
(4.27)
(4.28)
Substituting Eqs. (4.25) and (4.27 ) into the Laplace transform of Eq. (4.24), gives
U(s) =
Pk(s)Xn(s) + bmR(s)^k(s)Y(s)
b(l e^s)
(4.29)
where
Pk(s) = knis"1 + + k1s + k0
(4.30)
and
Pmk(s) =eXs sn+ (amin4+ k^Os11'1 + . + (am 0+ k0)
(4.31)
48
Inserting Eq. (4.25) in to Eq. (4.29) gives
Y(s) =
Pk(s)Ym(s) + bmR(s)
P(s)(lexs) + Pmk(s)
(4.32)
By combining Eq. (4.27) and Eq. (4.32), the closedloop transfer function is obtained
Y(s) b^ / (Pm(s) + Pk(s) \
R(s) Pm(s)\i eTs)P(s)+ Pmk(s) ' (433^
The closedloop characteristic equation is then
Pm (s) [ (1 e Xs )P(S) + Pmk (s)] =0 (4.34)
The main feature of a TDC system can be interpreted as one where the command
r is prefiltered by the reference model into ym and then the error ym y is forced to zero
by the high gain integrator 1/Ts with pole/zero cancellation. This feature of the TDC
system in frequency domain can be described by analyzing the LTISISO systems
defined above.
Suppose that the desired error dynamics is governed by
e Am e
which simplifes the algebra by making all the error feedback gains to zero, a block
diagram for the whole TDC system can be described as shown in Fig. 4.1.
49
Time Delay Controller
Fig. 4.1 Block diagram of TDC for a LTISISO plant
For a small time delay X, the following approximation holds
xn (t X) = x n (t),
the block diagram of Fig. 4.1 can reduced to the one of Fig. 4.2 and then to the one of
Fig. 4.3.
Time Delay Controller
Ref. model ' piant
Fig. 4.2 Reduced block diagram #1 of TDC for a LTISISO plant
50
model I 1
Fig. 4.3 Reduced block diagram #2 of TDC for a LTISISO plant
A typical root locus of the closed loop part of the system inside the dotted line of Fig.
4.3 is shown in Fig. 4.4
Fig. 4.4 Root locus of TDC for LTISISO plant example
51
If the effective controller gain 1/T is sufficiently large, one closedloop pole goes
to negative infinity whereas all other poles approach the fixed closed loop zeros.
Therefore, the pole/zero cancellation occurs, which makes the whole closed loop part
look like a first order system with a large bandwidth. The resultant block diagram
is shown in Fig. 4.5.
r bm ym 1 y >
Ri(s) 4 > 5 s + 1
Fig. 4.5 Reduced block diagram #3 of TDC for a LTISISO
4.3 DESIGN PROCEDURE
In the control law of Eq. (4.19), design parameters which can be chosen are: error
feedback gains ko through kn l and time delay I, which is related to model following
performance. Once the error system martix of Eq. (4.8) is obtained, its characteristic
equation can be calculated as
sI1+ (Wi + kni)s1'1 + + (am,0+ k0 ) = 0
The parameters of the above equation are combined from the parameters of model
reference and error feedback gains, k0 through k^, which can be chosen to give the
desired poles of error dynamics. The choice of time delay X however, must belong
to the stability of the system as shown in Eq. (4.33). The choice of time delay and
stability analysis will be discussed in the next section.
52
4.4 STABILITY ANALYSIS
As shown in the last section, the closedloop characteristic equation (CE) of the
time delay system is:
Pm(s)[(leTs)P(s) + P,k(s)]=0 (435)
This CE involves the stable characteristic polynomial of the reference model Pm(s),
the delay term e'^s and the plant characteristic polynomial P(s). The polynomial
Pmk(s), given by Eq. (4.31), is made up from coefficients in the reference model
and feedback gain matrix, and the delay term. The locations of the roots of this CE
determine the stability of the continuous time closedloop system.
The formulation of the delay term in Eq. (4.35) complicates the stability analysis.
For example, the RouthHurwitz criterion cannot be used with systems of this type,
since the system characteristic equation is not a polynomial in s. On the other hand,
the effect of the delay term in a feedback closedloop system is to add a phase lag to
the system and makes the system less stable. Fig. 4.6 illustrates the effect of the time
delay in an pure time delay system. Note that the Nyquist diagram is rotated closer
to the 1 + jO point and the system becomes unstable.
In this section, the stability of systems where the TDC is applied to singleinput
singleoutput, linear timeinvariant plant whose coefficients are known to vary within
certain defined interval is examined. To deal with those systems, a method based
on the Nyquist criterion and four Kharitonov polynomials will be used to determine
53
Fig. 4. 6. Nyquist diagram of a pure time delay system
the stability of the closedloop system from its CE. Equivalently, the purpose is to
determine the largest possible of time delay X (X > 0 ) for which the CE has no zeros
in the closedright half of the complex plane.
First of all, the CE may be written in the form
G(s) F(s) = 0 (4.36)
where G(s) and F(s) are some functions of the variable s. The poles of Eq. (4.36) are
those values of s for which G(s) and/or F(s) approach infinity. Usually, the model's
characteristic equation Pm(s) in Eq. (4.35) is chosen to be stable, thus the
characteristic equation to examine is:
54
(1 e'Cs)P(s) + ^k(s) = 0
(4.37)
The following manipulations are performed to put the characteristic equation given
by Eq. (4.37) into the form of Eq. (4.36).
Expanding Eq. (4.37) gives
P(s) P(s)eTs + sn e 'Xs + (amji + kn4 )sn4+ ... + (a^ kQ) = 0 (4.38)
Define the new polynomial
Pmk(s) = (am i>1+ kn] )sn_1+ .. + (am 0+ kg ) (4.39)
The characteristic equation, Eq. (4.37), can now be written as
P(s) + Pmk(s) eXs(sn + P(s)) = 0
or
/ P(s) + Pmk(s) \ .(eTs)=o
k sn+ P(s) '
Adding and subtracting sn from the numerator allows
(4.40)
/ sn + Pmk(s)
' sn+ P(s)
+1 ) (e"Xs) = 0
(4.41)
55
Let us define
and
G(s)= (
Sn + Pmk(s)
sn+ P(s)
F(s) = eXs
(4.42)
(4.43)
The characteristic equation, given by Eq. (4.37) is now in the desired form of Eq.
(4.36),G(s)(e'Xs) = 0.
The frequency transfer function of e'Xs is
F(j(o) = ej)X
F(jco) = 1 / mX
(4.44)
It has a magnitude of unity and a negative angle whose magnitude increases directly in
proportion to frequency. The Nyquist diagram is a simply unit circle center at origin
and is traced indefinitely, as shown in Fig. 4.7.
Re
Fig. 4.7 Nyquist plot of time delay F(s)
56
The G(s) term is now a ratio of polynomials with no delay terms. Plotted against
e~^s is the rest of the system, the G(s) term. The resulting Nyquist plot is a simple
curve and is used to determine the stability of the system. In this problem, the
contour followed by s is the Nyquist path which encompasses the entire right half
plane. Fig. 4.8 shows the diagrams of G(s) and F(s) when s follows the Nyquist path
the splane.
Fig. 4.8 Nyquist diagrams of G(s) and F(s)
Two cases can arise when analyzing the plots, either of G(s) plot intersects the
unit circle or none. If G(s) intersects the unit circle the further analysis of the plot is
done to determine the critical time delay for stability, Tcrjt, based on the gain and
phase margin (IG(jco) = II). If there is no intersection the stability can be
immediately determined by the Nyquist criterion.
57
There are two possibilities that G(s) does not intersect the unit circle. Fig. 4.9(a)
shows that there is no net rotation and the system is stable for any time delay X. The
delay X has no effect on stability. Fig. 4.9(b) shows that there is a net rotation that is
G(s) encircles the unit circle. Thus, the CE has a zero in the right half plane and the
system is unstable for any X.
Fig. 4.9 G(s) does not intersect the unit circle.
When the diagram of G(s) intersects the unit circle, the procedure is to evaluate
the intersections to determine the critical delay time Xcril .
Fig. 4.10 shows the case when G(s) and the unit circle reach the intersection point
with the same frequency co. This happens when X = Xcrit, there are zeros of the
characteristic equation (poles of the system) on the jco axis. The time delay Xcrit is
determined as follows:
a (rad)
co (rad/sec)
(4. 45)
58
where a is the phase angle and to is frequency at IG(jco) I = 1 (0 in decibels). Note
that Tcrit is considered as the upper limit of the time delay that the system remains
stable.
Fig. 4.10 G(s) intersects the unit circle.
Now we will use the idea of Kharitonov [11] for stability of plants with varying
parameters within certain ranges. Kharitonov showed that a given polynomial K(s)
with parameter variation s will be stable for any combination of parameter within a
permissible range if and only if the four Kharitonov polynomials are stable:
Ki(s) = %' + qjs + cÂ£s2+ q+s3+ q^s4 + q's5+ q+s6 + ;
K2(s) = qj + q+s + qjs2 + q's3+ cÂ£s4 + q+s5 + qgs6 + ; (4.46)
Kg(s) = cÂ£ + qjs + c^s2+ q's3+ q+s4 + qs5+ q+s6 +;
K4(s) = % + qjs + qj s2 + q+s3+ q;s4 + q+s5+ q'6s6 +;
where K(s) is a strictly Hurwitz polynomial
K(s) = % + qj s + % s2 + . + qn sn (4 47)
59
with the coefficient bounds qr and qj" ; i = 0,1, , n.
More precisely, for s = jco and co Â£ R, we introduce the following notation for the
real and imaginary parts of the Kharitonov polynomials:
Re"(co) = Re Kj (jco);
Re+ (co) = Re K2 (jco).
y
Im'(co) = Im K3(jco).
y
Im+ (co) = Im K4O00)
To illustrate, for n = 5, we obtain
Re(co) = q^ q2co2+ q^co4;
Re(co) = qQ q2'co2+ qljco4;
Im" (co) = qj +q3 co3 + q^ co5;
Im+ (co) = q" q^co3 + qljco5;
(4.48)
(4.49)
Using four above Kharitonov's polynomials, we find four bounding curves in the
Nyquist plot of G(s) and show that if each of these curves yields a stable system,
then every system in the family is also stable. Here we assume the plant parameters
vary in certain ranges, a; Â£ [a^, a+] and define a as a family of individual a;.
60
We define the four bounding functions as:
G(jcd)
G(jw),
G(jco)
mm ,min = RefCtjCo)] + jmin Im[G(j
Vaea Vaea
mm .max = +^^[000))]
= max Re[G(jco)] + jmin Im[G(jco)]
max, mm yaÂ£a Vaea
G(jco) = max Re[G(jco)] + jmax Im[G(jco)]
m8X. max w _ \_/ , _
Vaea Vaea
(4.50)
(4.51)
(4.52)
(4.53)
To facilitate in calculating the four functions (4.50) (4.53), we define
G(s) = G*(s)+ 1
(4.54)
where
G*(s)
Sn + Pmk(s)
 sn + P(S)
(4.55)
Substituting s = jco into G*(s) gives a complex number of the form
G*(ju) = + (456)
Y + jP
For a given w we need to find the maximum and the minimum real and imaginary
parts of G*as the plant parameters vary in their intervals.
We note that a and <> are known and constant since they are part of the desired
polynomial (controller). The a parameters enter the denominator of G*(a) vary and
lead to ye [y~, Y+J and P Â£ [ (3\ P+] where
61
(4.57)
y~ = a"0 + a^co4 ...
y+= aj a"2o? + a} to4 ... (4.58)
(3 = a"Jco a^co3 + ajco5 ... (4.59)
P+= a^co a3co3 + a'jW5 .. . (4.60)
After finding the minimum and maximum real and imaginary parts of G*(a), one (1) is
added to the real parts of G*(a) to get G(a).
4.5 IMPLEMENTATION AND EVALUATION
The following example will illustrate the method using a first order system with
known b. A reference model and a plant are defined as
with
x
xm+r>
 ax + u,
a e l 1 2]
For this system, P(s) = s + a0, aQe [1 2], b = 1, and Pmk(s) = (1+k) where k is the
error feedback gain. Using Eq. (4.42), G(s) becomes
0(s) = _id!_tkL + 1
a0
(4.61)
First, using k = 0 and substituting s = jco into Eq. (4.59), the parameters of G*(jco) of
62
Eq. (4.56) are:
a = 1,
= CO,
Y"= ao = 1,
Y+= aj= 2, and
P" = = 0.
With
G(jco) = 1 + j[ + 1
ao
the extrema in Eqs. (4.50) (4.53) are given by
min Re[G(j(o)] = 1 1 + 1=0
max Re[G(ja))] = 1 2  + 1 =0.5
min lm[G(jco)] = CO 1 = CO
max Im[G(jco)] = CO 2 0.5co
The four bounding functions are
G(jco) . . = 0+jco, G(jco) . = 0+j(0.5co)
'mm,nun J M 'min.max J
G(jco)mv = 0.5 + jco, G(jco) =0.5 +j(0.5co)
V) 'max,mm J ^ 'max,max J
(4.62)
(4.63)
(4.64)
where co is positive. For the part of the Nyquist plot when s is off the imaginary axis,
s = ReJa and a varies from JL to JL in the limit.
2 2
63
G(Re*a) =
+ i _ Re*
ao
for R >. and a0 > 0. The four resulting Nyquist diagrams are plotted along with
the unit circle in Fig. 4.11. Note that there are two pairs of overlapping diagrams.
Fig. 4.11 Nyquist plot of first order system example
When the diagram of G(s) intersects the unit circle, the procedure is to evaluate the
intersections to define the critical time delay XCrit The Xcrjtof the four plots of Eqs.
(4.63)(4.64) are calculated and summarized in Table 4.1. From Table 4.1,
G(ico) is the most restrictive so if X < 0.582 seconds the TDC with k = 0 will
^ max,max
stablize every first order plant with a e [ 1 ,2],
G(s), Eq. (4.60), of this example reveals that for k > 1 anf k < 1 the four plots of
the bounding G(joi) do not intersect the unit circle. In theses cases the time delay can
have no effect on stability. With k > 1 there is a net rotation to the unit circle, the
system is unstable for any X, Fig. 4. 12.
64
Table 4.1
Calculations for first order ystem
Function (0 at which IG(jco)l = 1 phase, 0, at which IG(jco)l = 1 T 0C cnt CO
G(>)min,min 1 7t 2 1.571
G(jw) min,max 2 n 2 0.785
G(jco) max,min .85 n 3 1.232
G(jco) v 'max.max 1.73 n 3 0.582
65
With k < 1 there is no net rotation to the unit circle, the system is stable for any time
delay T, Fig. 4.13.
4.6 APPLICATION TO THE CASE STUDY
The reduced order sytem with the parameters and the error bounds given in Table 3.1
is used in this study. After a canonical form is developed and partitioned in the same
form of Eq. (4.21), the transfer function of the nominal plant model is
Y(s) =_________________1105.5____________________
T(s) s5+ 5s4 + 74.7s3 + 305s + 607.5s2 + 2747.5 (465)
where Y(s) is tip position and T(s) is torque input.
Using Butterworth standard form for zero steadystate steperror systems, the model
reference is defined to have natural frequency to = 5 rad/s and bm= 3125 giving the
model reference transfer function
^Vn ________________________bhi________________________
Pm(s) s5+ 3.16cos4 + 5.3o)2s3+ 5.5cd3s2+ 3.2cd4s + co5
______________________3125_______________________
s5+ 15.8s4 + 132.5s3 + 687.5s2 + 2000s + 3125
Suppose that the feedback gains are all set to zero, so Pmk(s) in Eq. (4.39) is
f^k(s) = 15.8s4+ 132.5s3 + 687.5s2+ 2000s + 3125 (4.66)
The Nominal Plant:
First the nominal plant is analyzed. For this case b = 1105.5 given in Table 3.1 and
66
the nominal plant transfer function P(s) from Eq. (4.65) is
P(s) = S5+ 5s4 + 74.7s3 + 305s2+ 607.5s + 2747.5
G(s) in Eq. (4.42) is now specified:
x s5+ 15.8s4 + 132.5s3 + 687.5s2 + 2000s +3125
G(s) = + 1
5s4 + 74.7s3 + 305s2 + 607.5s + 2747.5
Substituting s = jco into (4.68) gives
G(jw)= jco5 + 15.8co4j!32.5co3 687.5co2+j2000co 3125 + +
5co4 j74.7co3 305co2+ j607.5co + 2747.5
Using Eq. (4.69) to plot the Nyquist diagram for G(jco). Fig. 4.14 shows that the
Nyquist plot of G(s) in the intersection of the unit circle.
(4.67)
(4.68)
(4.69)
67
Since G(jco) and the unit circle have intersection points, a further analysis is
neccessary to find the critical time delay. The required information is the frequency
and the phase at which the Nyquist plot of G(jco) intersects the unit circle. This
information is gotten from the Bode plots of G(jco) given in Eq. (4.69) shown in Fig.
4.15.
CQ
Q
W
Q
3
H
50
0
o
<
s 50
100
101
102
103
FREQUENCY
FREQUENCY
Fig. 4.15 Bode plot of G(s) for nominal plant
It was found that when IG(jco)l = 0 in decibels, co = 130 rad/s and a = 120 deg = 2.10
rad. For intersection, the unit circle must reach a phase angle of 2.10 rad at the same
frequency co = 130 rad/s. The phase of the unit circle is coT = 2.10 so
X . =^10 = 0.016 seconds
cnt 130
Simulations are created to verify the analysis and the performance of the controller.
68
Figs 4.16, 4.17 and 4.18 show the performance controller with X = 5 ms. These are
included to demonstrate the controller performance in a stable situation. Fig 4.16
indicates that the plant tracks the model very well.
Fig. 4.16 Position response (m) of nominal plant and reference model
with X = 5ms
Fig. 4.17 Error (m) in position with X = 5 ms
69
Fig. 4.18 Control effort (Nm) of nominal plant with X = 5 ms
The second set of simulation, Figs 4.19, 4.20 and 4.21, shows the response with
X > X it= 16 ms. The system is marginally stable, as predicted in the analysis.
Fig. 4.19 Position reponse (m) of nominal plant and reference model
with X = 16 ms (X = Xcrjt )
70
Fig. 4.20 Error (m) in position response with X = 16 ms (x = Xcrjt )
Fig. 4.21 Control effort (Nm) of nominal plant with X = 16 ms (X = Xcrjt )
A simulation with X > Xcritis included to illustrate the analysis. Fig. 4.22 shows
that the system is unstable.
71
Fig. 4.22 Position response of the nominal plant and reference model
with X = 20 ms (T > Tcrjt)
Plant with 10 Percent Parameter variations:
We now consider the plant parameters are known to be within 10 percent of the
values of the nominal plant. They may vary within the ranges given in Table 3.1. The
control parameter b varies in the range of [ 1216.05, 9994.95]. Using Eq. (4.49), the
four Kharitonov's polynomials of system are:
y= 2472.75 335.5m2 + 4.5m4
Y+ = 3022.25 274.95m + 5.5m4
pT = 546.7 82. 17m3 + ,9m5 ,,
p (4.70)
(3+ = 668.25 64.23m3 + 1.1m5
From Eq. (4.71), the four bounding equations in the form of Eqs. (4.50) (4.51) are
obtained. A short program is developed to calculate four bounding functions and the
resulting Nyquist plots are shown in Fig 4.23 with the Bode plots shows in Figs. 4.24
4.27.
72
The calculations of X rit from each of the four bounding functions are given in Table 4.2.
Fig. 4.23 Nyquist plots for 10 percent parameter variations
50
1801;: :i
100 101 102 103
Fig. 4.24 Bode plots of Remmlmmin for plant with 10 percent
parameter variations
73
MAGNITUDE PHASE MAGNITUDE
101
102
103
50
0..........!..
50
100 101 102 103
Fig. 4.25 Bode plots of Remm Immax for plant with 10 percent
parameter variations
FREQUENCY
Fig. 4.26 Bode plots of Remax Irnmin for plant with 10 percent
parameter variations
74
50
2
501; ;:
100 101 102 103
FREQUENCY
0
a.
1801______:__; ______________:__: :::::::______:: : : :
100 101 102 103
Fig. 4.27 Bode plots of Remax Immax for plant with 10 percent
parameter variations
From Table 4.2, the plant with 10 percent parameter variations^ varied between
0.009 and 0.020 seconds. For the nominal plant the requirement is X ri = 0.016 s, for
10 percent parameter variations the requirement is X = 0.009 s.
Table 4.2
Calculations for analysis with 10 percent parameter variations
Function co at which IG(jco)l = 1 phase, a, at which IG(jco)l = 1 xi ~Â£
130 2.618 0.020
G(jfi>) min,max 157 1.700 0.011
G(jco) max,mm 120 1.920 0.016
G(jco) v max.max 166 1.745 0.009
75
4.7 CONCLUSIONS
A Time Delay Control (TDC) method was presented and the control law for linear
timeinvariant (LTI) and singleinput singleoutput plant were discussed. It was shown
that the choice of error feedback gain matrix and the reference model is restricted
depending on the size of a control distribution matrix. Hence, the simplified TDC
control law given for systems in a canonical form.
The ideas of Nyquist and Kharitonov were used as a graphic procedure to determine
the stability of systems whose coefficients are known to vary within certain defined
intervals. The condition depends on maps of the Nyquist paths and stability depends
on four functions each yielding a stable system.
The controller was applied to the tip control of a flexible arm and the performance of
the controller was evaluated through simulations. The simulation results have indicated
that the controller shows a good model following which only depends on adjusting the
time delay. The robustness of the controller can be verified against the control algorithm.
That is the controller uses time delay to estimate the uncertain dynamics of systems and
then directly modifies the control action rather than adjusting the controller gains or
identifying system parameters thereby leading to a model independent controller.
76
APPENDIX A
This appendix describles the calculations which lead to the Hankel singular values
and then the reduced order model of the flexible arm.
Step 1. First of all, the minimal state space realizations of the full order system are
derived as follows:
1.0e+03 *
0.0000 0.0008 0.0012 0.8597 0.0073 2.3819 0.0318 0.7957
0.0010 0.0000 0.0000 0.0070 0.0001 0.0193 0.0003 0.0065
0.0000 0.0010 0.0001 0.0418 0.0004 0.1157 0.0015 0.0387
0.0000 0.0000 0.0010 0.0003 0.0000 0.0009 0.0000 0.0003
0.0000 0.0000 0.0000 0.0111 0.0001 0.0331 0.0004 0.0111
0.0000 0.0000 0.0000 0.0000 0.0010 0.0001 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0019 0.0000 0.0006
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0010 0.0000
0.9987 0.0081 0.0485 0.0004 0.0140 0.0000 0.0004 0.0000
1.0e+03 *
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 5.9486
Step 2.
Define R = 1, Q = diag(l 1 1 1 1 1 1 31762000) and use LQR (linear quadratic
regulator design) method, the unique positive definite matrix P is obtained:
77
1.0e+07 0.0005 0.0007 0.0002 0.0026 0.0354 0.0478 0.0523 0.0246
0.0007 0.0012 0.0005 0.0042 0.0550 0.0810 0.0851 0.0410
0.0002 0.0005 0.0012 0.0006 0.0167 0.0371 0.0493 0.0354
0.0026 0.0042 0.0006 0.0247 0.1938 0.2652 0.2241 0.0401
0.0354 0.0550 0.0167 0.1938 2.7783 38016 4.1993 2.0368
0.0478 0.0810 0.0371 0.2652 3.8016 5.6379 6.0928 3.1461
0.0523 0.0854 0.0493 0.2241 4.1993 6.0926 7.1959 4.3043
0.0246 0.0410 0.0354 0.0401 2.0368 3.1461 4.3043 4.2243
The parameter gain Kis defined as K = B1^ P,
K
1.0e + 03 *
0.1168 0.1465 0.1306 0.8526 8.64139.4667 9.2753 2.6219
A =
1.0e+04
0.0117
0.0002
0.0006
0.0000
0.0002
0.0000
0.0000
0.0000
0.0147
0.0001
0.0008
0.0000
0.0000
0.0000
0.0000
0.0000
0.0129
0.0001
0.0006
0.0001
0.0002
0.0000
0.0000
0.0000
$ = A + BK
0.1711 0.8623 1.1836 0.9295 0.1823
0.0014 0.0070 0.0096 0.0075 0.0015
0.0083 0.0419 0.0575 0.0052 0.0089
0.0001 0.0003 0.0005 0.0004 0.0001
0.0023 0.0121 0.0166 0.0131 0.0026
0.0000 0.0001 0.0006 0.0000 0.0000
0.0000 0.0004 0.0000 0.0004 0.0000
0.0000 0.0000 0.0000 0.0001 0.0000
and
1.0e+03 *
0.0000
0.1168
0.0000
0.1465
0.0000
0.1306
0.0000 0.0000 0.0000 0.0000 5.9486
0.8526 8.6413 9.4667 9.2753 2.6219
78
Step 3
Using minimal statespace triple { A, B, (C )} and balancing method, the new
K
__ Q
balanced state space triple {A, B, ( k) and its diagonal Grammians Z are obtained:
1.0e+04 *
0.5124 5.9624 0.5362 0.3775 5.4758 0.9977 0.9901 0.2848
5.9696 0.2502 1.4781 0.2045 2.1101 1.5192 0.6761 0.3230
0.6548 1.4842 0.8425 1.0350 2.2873 1.0099 0.8157 0.2893
0.4321 0.2311 1.0484 0.3736 0.6304 0.5041 0.2979 0.1257
4.4650 1.2153 1.5492 0.3114 2.1630 3.7695 1.3228 0.7277
0.6678 1.3794 0.5146 0.2666 1.0705 1.7004 3.3237 0.5315
0.2010 0.2946 0.2604 0.0803 1.1438 1.3712 0.8009 1.5464
0.1394 0.2585 0.0847 0.0402 0.1139 0.4155 0.4931 0.1366
0.6725
0.4428
0.6001
0.2402
1.5412
1.7020
1.2651
0.5349
and
I = .6124 .5035 .3501 .3016 .2051 .0023 .0015 0.0003
Note that a6 g7 and o8 are very small. It seems reasonable to eliminate the last three
states.
r
Step 4. Partition {A, B, (^)} as:
0.5583 0.4096 0.5851 0.2402 0.9025 0.7979 0.4184 0.3199
0.3747 0.1682 0.1331 0.0003 1.2493 1.5033 1.1393 0.5163
79
1.0e+04 *
0.5124 5.9624 0.5362 0.3775 5.4758 0.9977 0.9901 0.2848
5.9696 0.2502 1.4781 0.2045 2.1101 1.5192 0.6761 0.3230
0.6548 1.4842 0.8425 1.0350 2.2873 1.0099 0.8157 0.2893
0.4321 0.2311 1.0484 0.3736 0.6304 0.5041 0.2979 0.1257
4.4650 1.2153 1.5492 0.3114 2.1630 3.7695 1.3228 0.7277
0.6678 1.3794 0.5146 0.2666 1.0705 1.7004 3.3237 0.5315
0.2010 0.2946 0.2604 0.0803 1.1438 1.3712 0.8009 1.5464
0.1394 0.2585 0.0847 0.0402 0.1139 0.4155 0.4931 0.1366
0.6725 0.4428 0.6001 0.2402 1.5412
1.7020 1.2651 0.5349
Step 5.
1.0e+04
0.5124
5.9696
0.6548
0.4321
4.4650
5.9624
0.2502
1.4842
0.2311
1.2153
0.5362
1.4781
0.8425
1.0484
1.5492
0.3775
0.2045
1.0350
0.3736
0.3114
5.4758
2.1101
2.2873
0.6304
2.1630
B,
0.6725
0.4428
0.6001
0.2402
1.5412
Ci
Ki =
0.5583 0.4096 0.5851 0.2402 0.9025
0.3747 0.1682 0.1331 0.0003 1.2493
80
Step 6.
An An"
0.5124 5.9624
5.9696 0.2562
0.6541 1.4842
0.4321 0.2311
4.4651 1.2153
0.5326 0.3775 5.4757
1.4781 0.2545 2.1100
0.8425 1.7350 2.1872
1.0484 0.7736 0.6304
1.5492 0.3114 0.2628
The transfer function of the reduced order model is calculated as
Gr (s) = QUIAjj^Bj
___________14 s226.6 s + 1105.5______________
s5+ 5s4+ 74.7s3+ 305s2 + 607.5s + 2747.5
81
APPENDIX B
Algebraic Manipulation Leading to Equation (4.10)
Substituting Eq. (4.9) into Eq. (4.1) to obtain
x = f + h + BB+{ f h d + Amx+ Bnf Ke) + d
Subtracting Eq. (A.l) from Eq. (4.2),
xm x = Amxm BB+A,^ + BB+Ke + {I BB +)( f h d + Bf}
Adding A x + A x (= 0) to the righthand side of Eq. (A.2),
xmx = Amx Amx +A^BB+Amx
+BB+Ke+ { IBB+H fhd + B^r)
= Am(xmx) + Ke +
{ I BB+ }{ f h d + A mx + Bmr Ke }
By using the definition of the error in Eq. (4.4), Eq. (4.10) is obtained:
e = (Am+ K)e +{ I BB^ { f h d + Amx + Bmr Ke)
(B.l)
(B.2)
(B.3)
(4.10)
82
APPENDIX C
Canonical System and the Structural Constraint
From equation (4.16)
(BTB) 1
= (BrTBr j = B'r1(BrT)1
Then,
I BB+=I B (BTB )''1Bt
b;1(bt)1[o bJ]
= i 
0
I
[0
I]
I 0
0 0
From equation (4.16),
 f h d Aj^x + Bmr Ke
(C.l)
(C.2)
(C.3)
XS [ 0 Iq]X
 fr hrd Amrx + Bmir K re
0
fr hrd Amix +Bmrr Kre
83
Substituting equations (1.2) and (1.3) into equations (4.11), the structural constraint
becomes:
{ I BB+ } {f h d  + B^e }
I 0 0
0 0 fr hrd Amrx + Bmrr
0
(C.4)
Thus, the structural constraint is always met for the system expressed in canonical
form.
84
REFERENCES
[1] Zaffiro, P.A./'Design and Control of a Heating System for Thermoplastic
Composite Tape," Master's Thesis, MIT, 1988.
[2] YoucefToumi, K. and Ito, O.," A Time Delay Controller for Systems with
Unknown Dynamics, Journal of Dynamic Systems, Measurement, and
Control, March 1990, Vol. 112, pp. 133 142.
[3] Cho, D. and Hedrick, J. K., "Sliding Mode Control of Automotive Power
Trains Uncertain Actuator Models," American Control Conference,
November 1987, Vol. 2 of 3.
[4] Slotine, J.J. and Coetsee, J. A., "Adaptive Sliding Controller Synthesis for
Nonlinear Systems," Int. J. Control, 1986, Vol. 43, No. 6, pp 16311651.
[5] Tan, H.S. and Chin, J.K., "Vehicle Traction Control: Variable Structure
Control Approach," Journal of Dynamics Systems, Measurement, and
Control, July 1989, Vol. 113.
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