p( s)
Reference signal
Qm( s) ym = 0
Assumptions
Same as in Table 9.5
Robust
Same as in Table 9.5 to generate ˆ
Zp( s, t) , ˆ
Rp( s, t)
adaptive law
and ˆ
A, ˆ
B
State observer
Same as in Table 9.6
Calculation
ˆ
Kc = λ− 1 ˆ
B P
of controller
ˆ
A P + P ˆ
A − 1 P ˆ
B ˆ
B P + CC = 0
λ
parameters
C = [1 , 0 , . . . , 0] , P = P
> 0
Control law
¯
up = − ˆ
Kc( t)ˆ e, up = Q 1 ¯ u
Q
p
m
Design variables
λ > 0 and Q 1 , Qm as in Table 9.6
Example 9.5.3 Consider the same plant as in Example 9.5.2, i.e.,
b
yp =
(1 + ∆
s + a
m( s)) up
and the same control objective that requires the output yp to track the constant
reference signal ym = 1. A robust ALQC scheme can be constructed using Table 9.7
as follows:
Adaptive Law
˙ θp = Pr[Γ( φ − σsθp)]
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CHAPTER 9. ROBUST ADAPTIVE CONTROL SCHEMES
z − θ
=
p φ , m 2 = 1 + n 2
m 2
s,
n 2 s = ms
˙
ms = −δ 0 ms + u 2 p + y 2 p, ms(0) = 0
s
1
z =
y
[ u
s + λ p, φ = s + λ p, −yp]
θp = [ˆ b, ˆ a]
The projection operator Pr[ ·] constraints ˆ b to satisfy |ˆ b( t) | ≥ b 0 ∀t ≥ 0 where b 0 is a known lower bound for |b| and σs is the switching σ-modification.
State Observer
˙ˆ e = ˆ
Aˆ
e + ˆ
B ¯
up − ˆ
Ko([1 0]ˆ e − e 1)
where
ˆ
−ˆ a 1
1
10 − ˆ a
A =
,
ˆ
B = ˆ b
,
ˆ
K
0
0
1
o =
25
Controller Parameters ˆ
Kc = λ− 1 ˆ
B P where P = P
> 0 is solved pointwise
in time using
ˆ
A P + P ˆ
A − P ˆ
Bλ− 1 ˆ
B P + CC = O, C = [1 0]
s + 1
Control Law
¯
up = − ˆ
Kc( t)ˆ e, up =
¯
u
s
p
9.6
Adaptive Control of LTV Plants
One of the main reasons for considering adaptive control in applications is to
compensate for large variations in the plant parameters. One can argue that
if the plant model is LTI with unknown parameters a sufficient number of
tests can be performed off-line to calculate these parameters with sufficient
accuracy and therefore, there is no need for adaptive control when the plant
model is LTI. One can also go further and argue that if some nominal values
of the plant model parameters are known, robust control may be adequate
as long as perturbations around these nominal values remain within certain
bounds. And again in this case there is no need for adaptive control. In
many applications, however, such as aerospace, process control, etc., LTI
plant models may not be good approximations of the plant due to drastic
changes in parameter values that may arise due to changes in operating
9.6. ADAPTIVE CONTROL OF LTV PLANTS
733
points, partial failure of certain components, wear and tear effects, etc. In
such applications, linear time varying (LTV) plant models of the form
˙ x = A( t) x + B( t) u
y = C ( t) x + D( t) u
(9.6.1)
where A, B, C, and D consist of unknown time varying elements, may be
necessary. Even though adaptive control was motivated for plants modeled
by (9.6.1), most of the work on adaptive control until the mid 80’s dealt
with LTI plants. For some time, adaptive control for LTV plants whose
parameters vary slowly with time was considered to be a trivial extension of
that for LTI plants. This consideration was based on the intuitive argument
that an adaptive system designed for an LTI plant should also work for a
linear plant whose parameters vary slowly with time. This argument was
later on proved to be invalid. In fact, attempts to apply adaptive control to
simple LTV plants led to similar unsolved stability and robustness problems
as in the case of LTI plants with modeling errors. No significant progress was
made towards the design and analysis of adaptive controllers for LTV plants
until the mid-1980s when some of the fundamental robustness problems of
adaptive control for LTI plants were resolved.
In the early attempts [4, 74], the notion of the PE property of certain
signals in the adaptive control loop was employed to guarantee the exponen-
tial stability of the unperturbed error system, which eventually led to the
local stability of the closed-loop time-varying (TV) plant. Elsewhere, the
restriction of the type of time variations of the plant parameters also led to
the conclusion that an adaptive controller could be used in the respective
environment. More specifically, in [31] the parameter variations were as-
sumed to be perturbations of some nominal constant parameters, which are
small in the norm and modeled as a martingale process with bounded covari-
ance. A treatment of the parameter variations as small TV perturbations,
in an L 2-gain sense, was also considered in [69] for a restrictive class of LTI-
nominal plants. Special models of parameter variations, such as exponential
or 1 /t-decaying or finite number of jump discontinuities, were considered in
[70, 138, 181]. The main characteristic of these early studies was that no
modifications to the adaptive laws were necessary due to either the use of
PE or the restriction of the parameter variations to a class that introduces
no persistent modeling error effects in the adaptive control scheme.
734
CHAPTER 9. ROBUST ADAPTIVE CONTROL SCHEMES
The adaptive control problem for general LTV plants was initially treated
as a robustness problem where the effects of slow parameter variations were
treated as unmodeled perturbations [9, 66, 145, 222]. The same robust adap-
tive control techniques used for LTI plants in the presence of bounded distur-
bances and unmodeled dynamics were shown to work for LTV plants when
their parameters are smooth and vary slowly with time or when they vary
discontinuously, i.e., experience large jumps in their values but the disconti-
nuities occur over large intervals of time [225].
The robust MRAC and APPC schemes presented in the previous sections
can be shown to work well with LTV plants whose parameters belong to the
class of smooth and slowly varying with respect to time or the class with
infrequent jumps in their values. The difficulty in analyzing these schemes
with LTV plants has to do with the representations of the plant model. In
the LTI case the transfer function and related input/output results are used
to design and analyze adaptive controllers. For an LTV plant, the notion
of a transfer function and of poles and zeros is no longer applicable which
makes it difficult to extend the results of the LTI case to the LTV one. This
difficulty was circumvented in [223, 224, 225, 226] by using the notion of
the polynomial differential operator and the polynomial integral operator to
describe an LTV plant such as (9.6.1) in an input-output form that resembles
a transfer function description.
The details of these mathematical preliminaries as well as the design
and analysis of adaptive controllers for LTV plants of the form (9.6.1) are
presented in a monograph [226] and in a series of papers [223, 224, 225].
Interested readers are referred to these papers for further information.
9.7
Adaptive Control for Multivariable Plants
The design of adaptive controllers for MIMO plant models is more complex
than in the SISO case. In the MIMO case we are no longer dealing with a
single transfer function but with a transfer matrix whose elements are trans-
fer functions describing the coupling between inputs and outputs. As in the
SISO case, the design of an adaptive controller for a MIMO plant can be
accomplished by combining a control law, that meets the control objective
when the plant parameters are known, with an adaptive law that generates
estimates for the unknown parameters. The design of the control and adap-
9.7. ADAPTIVE CONTROL OF MIMO PLANTS
735
tive law, however, requires the development of certain parameterizations of
the plant model that are more complex than those in the SISO case.
In this section we briefly describe several approaches that can be used to
design adaptive controllers for MIMO plants.
9.7.1
Decentralized Adaptive Control
Let us consider the MIMO plant model
y = H( s) u
(9.7.1)
where y ∈ RN , u ∈ RN and H( s) ∈ CN×N is the plant transfer matrix that
is assumed to be proper. Equation (9.7.1) may be also expressed as
yi = hii( s) ui +
hij( s) uj,
i = 1 , 2 , . . . , N
(9.7.2)
1 ≤j≤N
j= i
where hij( s), the elements of H( s), are transfer functions.
Another less obvious but more general decomposition of (9.7.1) is
yi = hii( s) ui +
( hij( s) uj + qij( s) yj) , i = 1 , 2 , . . . , N
(9.7.3)
1 ≤j≤N
j= i
for some different transfer functions hij( s) , qij( s) .
If the MIMO plant model (9.7.3) is such that the interconnecting or
coupling transfer functions hij( s) , qij( s) ( i = j) are stable and small in some sense, then they can be treated as modeling error terms in the control design.
This means that instead of designing an adaptive controller for the MIMO
plant (9.7.3), we can design N adaptive controllers for N SISO plant models
of the form
yi = hii( s) ui, i = 1 , 2 , . . . , N
(9.7.4)
If these adaptive controllers are designed based on robustness considerations,
then the effect of the small unmodeled interconnections present in the MIMO
plant will not destroy stability. This approach, known as decentralized adap-
tive control, has been pursued in [38, 59, 82, 185, 195].
The analysis of decentralized adaptive control designed for the plant
model (9.7.4) but applied to (9.7.3) follows directly from that of robust adap-
tive control for plants with unmodeled dynamics considered in the previous
sections and is left as an exercise for the reader.
736
CHAPTER 9. ROBUST ADAPTIVE CONTROL SCHEMES
9.7.2
The Command Generator Tracker Approach
The command generator tracker (CGT) theory was first proposed in [26] for
the model following problem with known parameters. An adaptive control
algorithm based on the CGT method was subsequently developed in [207,
208]. Extensions and further improvements of the adaptive CGT algorithms
for finite [18, 100] as well as for infinite [233, 234] dimensional plant models
followed the work of [207, 208].
The adaptive CGT algorithms are based on a plant/reference model
structural matching and on an SPR condition. They are developed using
the SPR-Lyapunov design approach.
The plant model under consideration in the CGT approach is in the state
space form
˙ x = Ax + Bu ,
x(0) = x 0
y = C x
(9.7.5)
where x ∈ Rn; y, u ∈ Rm; and A, B, and C are constant matrices of appro-
priate dimensions. The control objective is to choose u so that all signals in
the closed-loop plant are bounded and the plant output y tracks the output
ym of the reference model described by
˙ xm = Amxm + Bmr
xm(0) = xm 0
ym = Cmxm
(9.7.6)
where xm ∈ Rnm, r ∈ Rpm and ym ∈ Rm.
The only requirement on the reference model at this point is that its
output ym has the same dimension as the plant output. The dimension of
xm can be much lower than that of x.
Let us first consider the case where the plant parameters A, B, and C
are known and use the following assumption.
Assumption 1 (CGT matching condition) There exist matrices S∗ 11,
S∗ 12, S∗ 21, S∗ 22 such that the desired plant input u∗ that meets the control
objective satisfies
x∗
S∗ 11 S∗ 12
xm
(9.7.7)
u∗
=
S∗ 21 S∗ 22
r
9.7. ADAPTIVE CONTROL OF MIMO PLANTS
737
where x∗ is equal to x when u = u∗, i.e., x∗ satisfies
˙ x∗ = Ax∗ + Bu∗
y∗ = C x∗ = ym
Assumption 2 (Output stabilizability condition) There exists a ma-
trix G∗ such that A + BG∗C
is a stable matrix.
If assumptions 1, 2 are satisfied we can use the control law
u = G∗( y − ym) + S∗ 21 xm + S∗ 22 r
(9.7.8)
that yields the closed-loop plant
˙ x = Ax + BG∗e 1 + BS∗ 21 xm + BS∗ 22 r
where e 1 = y − ym. Let e = x − x∗ be the state error between the actual and
desired plant state. Note the e is not available for measurement and is used
only for analysis. The error e satisfies the equation
˙ e = ( A + BG∗C ) e
which implies that e and therefore e 1 are bounded and converge to zero
exponentially fast. If xm, r are also bounded then we can conclude that all
signals are bounded and the control law (9.7.8) meets the control objective
exactly.
If A, B, and C are unknown, the matrices G∗, S∗ 21, and S∗ 22 cannot be
calculated, and, therefore, (9.7.8) cannot be implemented. In this case we
use the control law
u = G( t)( y − ym) + S 21( t) xm + S 22( t) r
(9.7.9)
where G( t) , S 21( t) , S 22( t) are the estimates of G∗, S∗ 21 , S∗ 22 to be generated by an adaptive law. The adaptive law is developed using the SPR-Lyapunov
design approach and employs the following assumption.
Assumption 3 (SPR condition) There exists a matrix G∗ such that
C ( sI − A − BG∗C ) − 1 B is an SPR transfer matrix.
738
CHAPTER 9. ROBUST ADAPTIVE CONTROL SCHEMES
Assumption 3 puts a stronger condition on the output feedback gain
G∗, i.e., in addition to making A + BG∗C stable it should also make the
closed-loop plant transfer matrix SPR.
For the closed-loop plant (9.7.5), (9.7.9), the equation for e = x − x∗
becomes
˙ e = ( A + BG∗C ) e + B ˜
Ge 1 + B ˜
Sω
e 1 = C e
(9.7.10)
where
˜
G( t) = G( t) − G∗,
˜
S( t) = S( t) − S∗
S( t) = [ S 21( t) , S 22( t)] ,
S∗ = [ S∗ 21 , S∗ 22]
ω = [ xm, r ]
We propose the Lyapunov-like function
V = e P
˜
˜
ce + tr[ ˜
G Γ − 1
1 G] + tr[ ˜
S Γ − 1
2 S]
where Pc = Pc > 0 satisfies the equations of the matrix form of the LKY
Lemma (see Lemma 3.5.4) because of assumption 3, and Γ1 , Γ2 are symmet-
ric positive definite matrices.
Choosing
˙˜
G = ˙
G = −Γ1 e 1 e 1
˙˜
S = ˙
S = −Γ2 e 1 ω
(9.7.11)
it follows as in the SISO case that
˙
V = −e QQ e − νce Lce
where Lc = Lc > 0 and νc > 0 is a small constant and Q is a constant matrix.
Using similar arguments as in the SISO case we can establish that e, G, S
are bounded and e ∈ L 2. If in addition xm, r are bounded we can establish
that all signals are bounded and e, e 1 converge to zero as t → ∞. Therefore
the adaptive control law (9.7.9), (9.7.11) meets the control objective.
Due to the restrictive nature of Assumptions 1 to 3, the CGT approach
did not receive as much attention as other adaptive control methods. As
9.7. ADAPTIV