ym generated by the reference model
Z
y
m( s)
m = km
r
Rm( s)
where Rm( s) has the same degree as Rp( s). Examine stability when Zp( s) is
Hurwitz and when it is not. Comment on your results.
6.6 Consider the third order plant
k
y
p( s + b 0)
p =
u
s 3 + a
p
2 s 2 + a 1 s + a 0
where ai, i = 0 , 1 , 2; b 0 , kp are constants and b 0 > 0. The transfer function of the reference model is given by
1
ym =
r
( s + 1)2
(a) Assuming that ai, b 0, and kp are known, design an MRC law that guar-
antees closed-loop stability and yp → ym as t → ∞ for any bounded
reference input r.
(b) Repeat (a) when ai, b 0, and kp are unknown and kp > 0.
(c) If in (b) a 2 = 0 , a 1 = 0 , a 0 = 1 are known but kp, b 0 are unknown,
indicate the simplification that results in the control law.
6.7 Show that the MRC law given by (6.3.29) in Remark 6.3.6 meets the MRC
objective for the plant given by (6.3.1).
6.8 Show that the MRC law given by (6.3.27) or (6.3.28) in Remark 6.3.5 meets the
MRC objective for the plant (6.3.1) for any given nonzero initial conditions.
6.9. PROBLEMS
431
6.9 Repeat the proof of Theorem 6.4.2 by using a minimal state space representa-
tion of the error system e 1 = Wm( s) ( s + p 0) ρ∗ ˜
θ φ as explained in Remark
6.4.5.
6.10 Consider the third-order plant
1
yp =
u
s 3 + a
p
2 s 2 + a 1 s + a 0
where ai, i = 0 , 1 , 2 are unknown constants and the reference model
2
ym =
r
( s + 1)( s + 2)( s + 2 . 5)
(a) Design a direct MRAC law with unnormalized adaptive law so that all
signals in the closed-loop plant are bounded and yp( t) → ym( t) as t → ∞
for any bounded reference input r.
(b) Simulate your scheme by assuming the following values for the plant
parameters
a 0 = 1 , a 1 = − 1 . 5 , a 2 = 0
and examine the effect of your choice of r on parameter convergence.
6.11 Design and analyze a direct MRAC with normalized adaptive law for the
plant
b
yp =
u
s + a p
where b > 0 . 5 , a are unknown constants. The reference model is given by
3
ym =
r
s + 3
(a) Design a direct MRAC scheme based on the gradient algorithm
(b) Repeat (a) for a least-squares algorithm
(c) Repeat (a) using the SPR-Lyapunov design approach with normalization
(d) Simulate your design in (c) with and without normalization. For simu-
lations use b = 1 . 2 , a = − 1 and r a signal of your choice.
6.12 Repeat Problem 6.11(a), for a hybrid MRAC scheme. Simulate your scheme
using the values of b = 1 . 2 , a = − 1 and r a signal of your choice.
6.13 Consider the plant
( s + b
y
0)
p =
u
( s + a)2 p
where b 0 > 0 . 2 and a are unknown constants. The reference model is given
by
1
ym =
r
s + 1
432
CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL
(a) Design a direct MRAC scheme with unnormalized adaptive law
(b) Repeat (a) with a normalized adaptive law
(c) Design an indirect MRAC scheme with an unnormalized adaptive law
(d) Repeat (c) with a normalized adaptive law
(e) Simulate one direct and one indirect MRAC scheme of your choice from
(a) to (d) and compare their performance when b 0 = 2 , a = − 5 , a nd
r = 2 + sin 0 . 8 t. Comment.
6.14 Consider the control law (6.3.27) for the plant (6.3.1) where θ∗i, i = 1 , 2 , 3 , c∗ 0
are the desired controller parameters. Design and analyze a direct MRAC
scheme based on this control law.
6.15 Consider the control law given by equation (6.3.29) in Remark 6.3.6 designed
for the plant (6.3.1). Design and analyze a direct MRAC scheme based on
this control law.
6.16 Consider the SISO plant
k
y
pZp( s)
p =
u
R
p
p( s)
where Zp( s) , Rp( s) are monic, Zp( s) is Hurwitz and the relative degree
n∗ = 1. The order n of Rp is unknown. Show that the adaptive control law
up = − θyp sgn( kp) , ˙ θ = y 2 p
guarantees signal boundedness and convergence of yp to zero for any finite n.
(Hint: The plant may be represented as
˙ x 1 = A 11 x 1 + A 12 yp,
x 1 ∈ Rn− 1
˙ yp = A 21 x 1 + a 0 yp + kpup
where A 11 is stable.)
6.17 Following the same procedure used in Section 6.6.2, derive an indirect MRAC
scheme using unnormalized adaptive laws for a plant with n∗ = 2.
6.18 Let ω ∈ L∞ be PE and e ∈ S( µ)
L∞ where µ ≥ 0. Let ωµ = ω + e. Show
that there exists a µ∗ > 0 such that for any µ ∈ [0 , µ∗), ωµ is PE.
6.19 Consider the MRAC problem of Section 6.4.1. It has been shown (see Remark
6.4.3) that the nonzero initial condition appears in the error equation as
e 1 = Wm( s) ρ∗( up − θ∗ ω) + Cc ( sI − Ac) − 1 e(0)
6.9. PROBLEMS
433
Show that the same stability results as in the case of e(0) = 0 can be estab-
lished when e(0) = 0 by using the new Lyapunov-like function
e P
˜
θ Γ − 1 ˜
θ
V =
ce +
|ρ∗| + βe
2
2
0 P 0 e 0
where e 0 is the zero-input response, i.e.,
˙ e 0 = Ace 0 ,
e 0(0) = e(0)
P 0 satisfies Ac P 0 + P 0 Ac = −I and β > 0 is an arbitrary positive constant.
Chapter 7
Adaptive Pole Placement
Control
7.1
Introduction
In Chapter 6 we considered the design of a wide class of MRAC schemes for
LTI plants with stable zeros. The assumption that the plant is minimum
phase, i.e., it has stable zeros, is rather restrictive in many applications. For
example, the approximation of time delays often encountered in chemical
and other industrial processes leads to plant models with unstable zeros. As
we discussed in Chapter 6, the minimum phase assumption is a consequence
of the MRC objective that requires cancellation of the plant zeros in an
effort to make the closed-loop plant transfer function equal to that of the
reference model. The same assumption is also used to express the desired
controller parameters in the form of a linear or bilinear parametric model,
and is, therefore, crucial for parameter estimation and the stability of the
overall adaptive control scheme.
Another class of control schemes that is popular in the known parameter
case are those that change the poles of the plant and do not involve plant
zero-pole cancellations. These schemes are referred to as pole placement
schemes and are applicable to both minimum and nonminimum phase LTI
plants. The combination of a pole placement control law with a parameter
estimator or an adaptive law leads to an adaptive pole placement control
(APPC) scheme that can be used to control a wide class of LTI plants with
434
7.1. INTRODUCTION
435
unknown parameters.
As in the MRAC case, the APPC schemes may be divided into two
classes: The indirect APPC schemes where the adaptive law generates on-
line estimates of the coefficients of the plant transfer function that are then
used to calculate the parameters of the pole placement control law by solving
a certain algebraic equation; and the direct APPC where the parameters of
the pole placement control law are generated directly by an adaptive law
without any intermediate calculations that involve estimates of the plant
parameters.
The direct APPC schemes are restricted to scalar plants and to spe-
cial classes of plants where the desired parameters of the pole placement
controller can be expressed in the form of the linear or bilinear parametric
models. Efforts to develop direct APPC schemes for a general class of LTI
plants led to APPC schemes where both the controller and plant parameters
are estimated on-line simultaneously [49, 112], leading to a rather complex
adaptive control scheme.
The indirect APPC schemes, on the other hand, are easy to design and
are applicable to a wide class of LTI plants that are not required to be
minimum phase or stable. The main drawback of indirect APPC is the
possible loss of stabilizability of the estimated plant based on which the
calculation of the controller parameters is performed. This drawback can
be eliminated by modifying the indirect APPC schemes at the expense of
adding more complexity. Because of its flexibility in choosing the controller
design methodology (state feedback, compensator design, linear quadratic,
etc.) and adaptive law (least squares, gradient, or SPR-Lyapunov type),
indirect APPC is the most general class of adaptive control schemes. This
class also includes indirect MRAC as a special case where some of the poles
of the plant are assigned to be equal to the zeros of the plant to facilitate
the required zero-pole cancellation for transfer function matching. Indirect
APPC schemes have also been known as self-tuning regulators in the litera-
ture of adaptive control to distinguish them from direct MRAC schemes.
The chapter is organized as follows: In Section 7.2 we use several ex-
amples to illustrate the design and analysis of APPC. These examples are
used to motivate the more complicated designs in the general case treated in
the rest of the chapter. In Section 7.3, we define the pole placement control
(PPC) problem for a general class of SISO, LTI plants and solve it for the
436
CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL
case of known plant parameters using several different control laws. The
significance of Section 7.3 is that it provides several pole placement control
laws to be used together with adaptive laws to form APPC schemes. The
design and analysis of indirect APPC schemes for a general class of SISO,
LTI plants is presented in Section 7.4. Section 7.5 is devoted to the design
and analysis of hybrid APPC schemes where the estimated plant parameters
are updated at distinct points in time. The problem of stabilizability of the
estimated plant model at each instant of time is treated in Section 7.6. A
simple example is first used to illustrate the possible loss of stabilizability
and a modified indirect APPC scheme is then proposed and analyzed. The
modified scheme is guaranteed to meet the control objective and is therefore
not affected by the possible loss of stabilizability during parameter estima-
tion. Section 7.7 is devoted to stability proofs of the various theorems given
in previous sections.
7.2
Simple APPC Schemes
In this section we use several examples to illustrate the design and analysis of
simple APPC schemes. The important features and characteristics of these
schemes are used to motivate and understand the more complicated ones to
be introduced in the sections to follow.
7.2.1
Scalar Example: Adaptive Regulation
Consider the scalar plant
˙ y = ay + bu
(7.2.1)
where a and b are unknown constants, and the sign of b is known. The control
objective is to choose u so that the closed-loop pole is placed at −am, where
am > 0 is a given constant, y and u are bounded, and y( t) converges to zero
as t → ∞.
If a and b were known and b = 0 then the control law
u = −ky + r
(7.2.2)
a + a
k =
m
(7.2.3)
b
7.2. SIMPLE APPC SCHEMES
437
where r is a reference input, would lead to the closed-loop plant
˙ y = −amy + br
(7.2.4)
i.e., the control law described by (7.2.2) and (7.2.3) changes the pole of the
plant from a to −am but preserves the zero structure. This is in contrast to
MRC, where the zeros of the plant are canceled and replaced with new ones.
It is clear from (7.2.4) that the pole placement law (7.2.2) and (7.2.3) with
r=0 meets the control objective exactly.
Let us now consider the case where a and b are unknown. As in the
MRAC case, we use the certainty equivalence approach to form adaptive
pole placement control schemes as follows: We use the same control law as
in (7.2.2) but replace the unknown controller parameter k with its on-line
estimate ˆ
k. The estimate ˆ
k may be generated in two different ways: The
direct one where ˆ
k is generated by an adaptive law and the indirect one
where ˆ
k is calculated from
ˆ
ˆ a + a
k =
m
(7.2.5)
ˆ b
provided ˆ b = 0 where ˆ a and ˆ b are the on-line estimates of a and b, respec-
tively. We consider each design approach separately.
Direct Adaptive Regulation In this case the time-varying gain ˆ
k in the
control law
u = −ˆ
ky + r,
r = 0
(7.2.6)
is updated directly by an adaptive law. The adaptive law is developed as
follows: We add and subtract the desired control input, u∗ = −ky + r with
k = a+ am in the plant equation, i.e.,
b
˙ y = ay + bu∗ − bu∗ + bu = −amy − b(ˆ k − k) y + br
to obtain, for r = 0, the error equation
˙ y = −amy − b˜ ky
(7.2.7)
where ˜
k = ˆ
k − k, that relates the parameter error term b˜
ky and regulation
error y through the SPR transfer function
1
. As shown in Chapter 4,
s+ am
438
CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL
(7.2.7) is in the appropriate form for applying the Lyapunov design approach.
Choosing
y 2
˜
k 2 | b |
V =
+
2
2 γ
it follows that for
˙ˆ k = γy 2sgn( b)
(7.2.8)
we have
˙
V = −amy 2 ≤ 0
which implies that y, ˜
k, u ∈ L∞ and y ∈ L 2. From (7.2.7) and y, ˜ k ∈ L∞ we
have ˙ y ∈ L∞; therefore, using Lemma 3.2.5 we obtain y( t) → 0 as t → ∞.
In summary, the direct APPC scheme in (7.2.6) and (7.2.8) guaran-
tees signal boundedness and regulation of the plant state y( t) to zero. The
scheme, however, does not guarantee that the closed-loop pole of the plant
is placed at −am even asymptotically with time. To achieve such a pole
placement result, we need to show that ˆ
k → a+ am as t → ∞. For parameter
b
convergence, however, y is required to be PE which is in conflict with the
objective of regulating y to zero. The conflict between parameter identifica-
tion and regulation or control is well known in adaptive control and cannot
be avoided in general.
Indirect Adaptive Regulation
In this approach, the gain ˆ
k( t) in the
control law
u = −ˆ
k( t) y + r,
r = 0
(7.2.9)
is calculated by using the algebraic equation
ˆ
ˆ a + a
k =
m
(7.2.10)
ˆ b
with ˆ b = 0 and the on-line estimates ˆ a and ˆ b of the plant parameters a and
b, respectively. The adaptive law for generating ˆ a and ˆ b is constructed by
using the techniques of Chapter 4 as follows:
We construct the series-parallel model
˙ ym = −am( ym − y) + ˆ ay + ˆ bu
(7.2.11)
7.2. SIMPLE APPC SCHEMES
439
then subtract (7.2.11) from the plant equation (7.2.1) to obtain the error
equation
˙ e = −ame − ˜ ay − ˜ bu
(7.2.12)
where e = y − ym, ˜ a = ˆ a − a, ˜ b = ˆ b − b. Using the Lyapunov-like function
e 2
˜ a 2
˜ b 2
V =
+
+
2
2 γ 1
2 γ 2
fo