7.3 Consider the speed control system given in Problem 6.2, where a, b, d are as-
sumed unknown.
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CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL
(a) Design an APPC law to achieve the following performance specifications:
(i) The time constant of the closed-loop system is less than 2 sec.
(ii) The steady state error is zero for any constant disturbance d
(b) Design an ALQC law such that
∞
J =
( y 2 + λu 2) dt
0
is minimized. Simulate the closed-loop ALQC system with a = 0 . 02 , b =
1 . 3 , d = 0 . 5 for different values of λ. Comment on your results.
7.4 Consider the following system:
ω 2
y =
n
u
s 2 + 2 ξωns + ω 2 n
where the parameter ωn (the natural frequency) is known, but the damping
ratio ξ is unknown. The performance specifications for the closed-loop system
are given in terms of the unit step response as follows: (a) the peak overshoot
is less than 5% and (b) the settling time is less than 2 sec.
(a) Design an estimation scheme to estimate ξ when ωn is known.
(b) Design an indirect APPC and analyze the stability properties of the
closed-loop system.
7.5 Consider the plant
s + b
y =
u
s( s + a)
(a) Design an adaptive law to generate ˆ a and ˆ b, the estimate of a and b,
respectively, on-line.
(b) Design an APPC scheme to stabilize the plant and regulate y to zero.
(c) Discuss the stabilizability condition ˆ a and ˆ b have to satisfy at each time
t.
(d) What additional assumptions you need to impose on the parameters a
and b so that the adaptive algorithm can be modified to guarantee the
stabilizability condition? Use these assumptions to propose a modified
APPC scheme.
7.6 Repeat Problem 7.3 using a hybrid APPC scheme.
7.7 Solve the MRAC problem given by Problem 6.10 in Chapter 6 using Remark
7.3.1 and an APPC scheme.
7.8 Use Remark 7.3.1 to verify that the MRC law of Section 6.3.2 shown in Figure
6.1 is a special case of the general PPC law given by equations (7.3.3), (7.3.6).
7.8. PROBLEMS
529
7.9 Establish the stability properties of the hybrid ALQ control scheme given in
Table 7.10.
7.10 Establish the stability properties of the hybrid APPC scheme of Table 7.8.
7.11 For n = 2 , q = 1, show that A( t) defined in (7.7.5) satisfies det( sI − A( t)) =
A∗Λ q, where A∗, Λ q are defined in Table 7.4.
Chapter 8
Robust Adaptive Laws
8.1
Introduction
The adaptive laws and control schemes developed and analyzed in Chapters 4
to 7 are based on a plant model that is free of noise, disturbances and
unmodeled dynamics. These schemes are to be implemented on actual plants
that most likely deviate from the plant models on which their design is based.
An actual plant may be infinite dimensional, nonlinear and its measured
input and output may be corrupted by noise and external disturbances.
The effect of the discrepancies between the plant model and the actual plant
on the stability and performance of the schemes of Chapters 4 to 7 may
not be known until these schemes are implemented on the actual plant. In
this chapter, we take an intermediate step, and examine the stability and
robustness of the schemes of Chapters 4 to 7 when applied to more complex
plant models that include a class of uncertainties and external disturbances
that are likely to be present in the actual plant.
The question of how well an adaptive scheme of the class developed in
Chapters 4 to 7 performs in the presence of plant model uncertainties and
bounded disturbances was raised in the late 1970s. It was shown, using
simple examples, that an adaptive scheme designed for a disturbance free
plant model may go unstable in the presence of small disturbances [48].
These examples demonstrated that the adaptive schemes of Chapters 4 to
7 are not robust with respect to external disturbances. This nonrobust
behavior of adaptive schemes became a controversial issue in the early 1980s
530
8.2. PLANT UNCERTAINTIES AND ROBUST CONTROL
531
when more examples of instabilities were published demonstrating lack of
robustness in the presence of unmodeled dynamics or bounded disturbances
[85, 197]. This motivated many researchers to study the mechanisms of
instabilities and find ways to counteract them. By the mid-1980s, several
new designs and modifications were proposed and analyzed leading to a body
of work known as robust adaptive control.
The purpose of this chapter is to analyze various instability mechanisms
that may arise when the schemes of Chapters 4 to 7 are applied to plant
models with uncertainties and propose ways to counteract them.
We start with Section 8.2 where we characterize various plant model un-
certainties to be used for testing the stability and robustness of the schemes
of Chapters 4 to 7. In Section 8.3, we analyze several instability mechanisms
exhibited by the schemes of Chapters 4 to 7, in the presence of external dis-
turbances or unmodeled dynamics. The understanding of these instability
phenomena helps the reader understand the various modifications presented
in the rest of the chapter. In Section 8.4, we use several techniques to modify
the adaptive schemes of Section 8.3 and establish robustness with respect to
bounded disturbances and unmodeled dynamics. The examples presented in
Sections 8.3 and 8.4 demonstrate that the cause of the nonrobust behavior
of the adaptive schemes is the adaptive law that makes the closed-loop plant
nonlinear and time varying. The remaining sections are devoted to the devel-
opment of adaptive laws that are robust with respect to a wide class of plant
model uncertainties. We refer to them as robust adaptive laws. These robust
adaptive laws are combined with control laws to generate robust adaptive
control schemes in Chapter 9.
8.2
Plant Uncertainties and Robust Control
The first task of a control engineer in designing a control system is to obtain
a mathematical model that describes the actual plant to be controlled. The
actual plant, however, may be too complex and its dynamics may not be
completely understood. Developing a mathematical model that describes
accurately the physical behavior of the plant over an operating range is a
challenging task. Even if a detailed mathematical model of the plant is
available, such a model may be of high order leading to a complex controller
whose implementation may be costly and whose operation may not be well
532
CHAPTER 8. ROBUST ADAPTIVE LAWS
understood. This makes the modeling task even more challenging because
the mathematical model of the plant is required to describe accurately the
plant as well as be simple enough from the control design point of view.
While a simple model leads to a simpler control design, such a design must
possess a sufficient degree of robustness with respect to the unmodeled plant
characteristics. To study and improve the robustness properties of control
designs, we need a characterization of the types of plant uncertainties that
are likely to be encountered in practice. Once the plant uncertainties are
characterized in some mathematical form, they can be used to analyze the
stability and performance properties of controllers designed using simplified
plant models but applied to plants with uncertainties.
8.2.1
Unstructured Uncertainties
Let us start with an example of the frequency response of a stable plant.
Such a response can be obtained in the form of a Bode diagram by exciting
the plant with a sinusoidal input at various frequencies and measuring its
steady state output response. A typical frequency response of an actual
stable plant with an output y may have the form shown in Figure 8.1.
It is clear that the data obtained for ω ≥ ωm are unreliable because
at high frequencies the measurements are corrupted by noise, unmodeled
high frequency dynamics, etc. For frequencies below ωm, the data are accu-
rate enough to be used for approximating the plant by a finite-order model.
An approximate model for the plant, whose frequency response is shown in
Figure 8.1, is a second-order transfer function G 0( s) with one stable zero
and two poles, which disregards the phenomena beyond, say ω ≥ ωm. The
modeling error resulting from inaccuracies in the zero-pole locations and
high frequency phenomena can be characterized by an upper bound in the
frequency domain.
Now let us use the above example to motivate the following relationships
between the actual transfer function of the plant denoted by G( s) and the
transfer function of the nominal or modeled part of the plant denoted by
G 0( s).
Definition 8.2.1 (Additive Perturbations) Suppose that G( s) and G 0( s)
are related by
G( s) = G 0( s) + ∆ a( s)
(8.2.1)
8.2. PLANT UNCERTAINTIES AND ROBUST CONTROL
533
5
0
-5
20log
y ( jω)
10
-10
-15
-20 -1
0
1
2
10
10
10
10
ω
ω
m
Figure 8.1 An example of a frequency response of a stable plant.
where ∆ a( s) is stable. Then ∆ a( s) is called an additive plant perturbation
or uncertainty. The structure of ∆ a( s) is usually unknown but ∆ a( s) is
assumed to satisfy an upper bound in the frequency domain, i.e.,
|∆ a( jω) | ≤ δa( ω) ∀ω
(8.2.2)
for some known function δa( ω). In view of (8.2.1) and (8.2.2) defines a family
of plants described by
Π a = {G | |G( jω) − G 0( jω) | ≤ δa( ω) }
(8.2.3)
The upper bound δa( ω) of ∆ a( jω) may be obtained from frequency re-
sponse experiments. In robust control [231], G 0( s) is known exactly and the
534
CHAPTER 8. ROBUST ADAPTIVE LAWS
uncertainties of the zeros and poles of G( s) are included in ∆ a( s). In adap-
tive control, the parameters of G 0( s) are unknown and therefore zero-pole
inaccuracies do not have to be included in ∆ a( s). Because the main topic
of the book is adaptive control, we adopt Definition 8.2.1, which requires
∆ a( s) to be stable.
Definition 8.2.2 (Multiplicative Perturbations) Let G( s) , G 0( s) be re-
lated by
G( s) = G 0( s)(1 + ∆ m( s))
(8.2.4)
where ∆ m( s) is stable. Then ∆ m( s) is called a multiplicative plant pertur-
bation or uncertainty.
In the case of multiplicative plant perturbations, ∆ m( s) may be constrained
to satisfy an upper bound in the frequency domain, i.e.,
|∆ m( jω) | ≤ δm( ω)
(8.2.5)
for some known δm( ω) which may be generated from frequency response
experiments. Equations (8.2.4) and (8.2.5) describe a family of plants given
by
|G( jω) − G
Π
0( jω) |
m =
G
≤ δ
|G
m( ω)
(8.2.6)
0( jω) |
For the same reason as in the additive perturbation case, we adopt Defini-
tion 8.2.2 which requires ∆ m( s) to be stable instead of the usual definition
in robust control where ∆ m( s) is allowed to be unstable for a certain family
of plants.
Definition 8.2.3 (Stable Factor Perturbations) Let G( s) , G 0( s) have
the following coprime factorizations [231]:
N
N
G( s) =
0( s) + ∆1( s) , G
0( s)
(8.2.7)
D
0( s) =
0( s) + ∆2( s)
D 0( s)
where N 0 and D 0 are proper stable rational transfer functions that are co-
prime, 1 and ∆1( s) and ∆2( s) are stable. Then ∆1( s) and ∆2( s) are called stable factor plant perturbations.
1Two proper transfer functions P ( s) , Q( s) are coprime if and only if they have no finite
common zeros in the closed right half s-plane and at least one of them has relative degree
zero [231].
8.2. PLANT UNCERTAINTIES AND ROBUST CONTROL
535
✲ ∆ a( s)
❄
y
✲
u
G
✲
+ +
❧
✲
0( s)
Σ
(i)
✲ ∆ m( s)
❄
y
✲
+ +
❧
u
Σ ✲ G
✲
0( s)
(ii)
✲ ∆1( s)
+ ❄
y
✲
u
N
✲ ❧✲
+
✲
0( s)
Σ
❧
Σ
+
− ✻
❧
+
Σ ✛
D
✛
0( s) − 1
✻
+
∆
✛
2( s)
(iii)
Figure 8.2 Block diagram representations of plant models with (i) addi-
tive, (ii) multiplicative, and (iii) stable factor perturbations.
Figure 8.2 shows a block diagram representation of the three types of plant
model uncertainties.
The perturbations ∆ a( s) , ∆ m( s) , ∆1( s), and ∆2( s) defined above with no additional restrictions are usually referred to as unstructured plant model
uncertainties.
536
CHAPTER 8. ROBUST ADAPTIVE LAWS
8.2.2
Structured Uncertainties: Singular Perturbations
In many applications, the plant perturbations may have a special form be-
cause they may originate from variations of physical parameters or arise
because of a deliberate reduction in the complexity of a higher order math-
ematical model of the plant. Such perturbations are usually referred to as
structured plant model perturbations.
The knowledge of the structure of plant model uncertainties can be ex-
ploited in many control problems to achieve better performance and obtain
less conservative results.
An important class of structured plant model perturbations that describe
a wide class of plant dynamic uncertainties, such as fast sensor and actuator
dynamics, is given by singular perturbation models [106].
For a SISO, LTI plant, the following singular perturbation model in the
state space form
˙ x = A 11 x + A 12 z + B 1 u,
x ∈ Rn
µ ˙ z = A 21 x + A 22 z + B 2 u,
z ∈ Rm
(8.2.8)
y = C 1 x + C 2 z
can be used to describe the slow (or dominant) and fast (or parasitic) phe-
nomena of the plant. The scalar µ represents all the small parameters such
as small time constants, small masses, etc., to be neglected. In most appli-
cations, the representation (8.2.8) with a single parameter µ can be achieved
by proper scaling as shown in [106]. All the matrices in (8.2.8) are assumed
to be constant and independent of µ. As explained in [106], this assumption
is for convenience only and leads to a minor loss of generality.
The two time scale property of (8.2.8) is evident if we use the change of
variables
zf = z + L( µ) x
(8.2.9)
where L( µ) is required to satisfy the algebraic equation
A 21 − A 22 L + µLA 11 − µLA 12 L = 0
(8.2.10)
to transform (8.2.8) into
˙ x = Asx + A 12 zf + B 1 u
µ ˙ zf = Af zf + Bsu
(8.2.11)
y = Cs x + C 2 zf
8.2. PLANT UNCERTAINTIES AND ROBUST CONTROL
537
where As = A 11 −A 12 L, Af = A 22 + µLA 12, Bs = B 2 + µLB 1, Cs = C 1 − C 2 L.
As shown in [106], if A 22 is nonsingular, then for all µ ∈ [0 , µ∗) and some
µ∗ > 0, a solution of the form
L = A− 1
22 A 21 + O( µ)
satisfying (8.2.10) exists. It is clear that for u = 0, i.e.,
˙ x = Asx + A 12 zf
(8.2.12)
µ ˙ zf = Af zf
the eigenvalues of (8.2.12) are equal to those of As and Af /µ, which, for
small µ and for Af nonsingular, are of O(1) and O(1 /µ), respectively2. The
smaller the value of µ, the wider the distance between the eigenvalues of As
and Af /µ, and the greater the separation of time scales. It is clear that if
Af is stable then the smaller the value of µ is, the faster the state variable
zf goes to zero. In the limit as µ → 0, zf converges instantaneously, i.e.,
infinitely fast to zero. Thus for small µ, the effect of stable fast dynamics is
reduced considerably after a very short time. Therefore, when A 22 is stable
(which for small µ implies that Af is stable), a reasonable approximation of
(8.2.8) is obtained by setting µ = 0, solving for z from the second equation
of (8.2.8) and substituting for its value in the first equation of (8.2.8), i.e.,
˙ x 0 = A 0 x 0 + B 0 u, x 0 ∈ Rn