707
Using Swapping Lemma A.1 we have
Wm( s)˜
θ 0 ω 0 = ˜ θ 0 φ 0 + Wc( s)( Wb( s) ω 0 )˙˜ θ 0
where the elements of Wc, Wb are strictly proper transfer functions with the
same poles as Wm( s). From the expression of the normalized estimation
error in (9.4.30), we have
m 2 = −˜
θ 0 φ 0 − dη
where dη = Wm( s) d 1. Therefore, m 2 = −˜
θ 0 φ 0 − Wm( s) d 1 leading to the
tracking error equation
1
e 1 =
− m 2 + W
c∗
c( Wbω 0 ) ˙˜
θ 0 + Wmua
0
Because , Wc, Wb, ω 0 , ˙˜
θ 0 are known, the input ua is to be chosen to reduce
the effect of ˜
θ 0 , ˙˜
θ 0 on e 1.
Let us choose
ua = −Q( s)[ − m 2 + Wc( s)( Wb( s) ω 0 )˙˜ θ 0] = −Q( s) Wm( s)(˜ θ 0 ω 0 + d 1) (9.4.36)
where Q( s) is a proper stable transfer function to be designed. With this
choice of ua, we have
1
e 1 =
[(1 − W
c∗
m( s) Q( s)) Wm( s)( ˜
θ 0 ω 0 + d 1)]
0
which implies that
e 1 t ∞ ≤ hm 1( (˜ θ 0 ω 0) t ∞ + d 1 t ∞)
where hm( t) is the impulse response of (1 − Wm( s) Q( s)) Wm( s).
As in Method 1 if we choose Q( s) as
W − 1
Q( s) =
m ( s)
(9.4.37)
( τ s + 1) n∗
we can establish that hm 1 → 0 as τ → 0.
708
CHAPTER 9. ROBUST ADAPTIVE CONTROL SCHEMES
With (9.4.37) the modified control law (9.4.29), (9.4.36) becomes
W − 1
u
m ( s)
p = θ 0 ω 0 + c∗ 0 r −
( − m 2 + W
( τ s + 1) n∗
c( s)( Wb( s) ω 0 ) ˙˜
θ 0)
(9.4.38)
where Wc, Wb can be calculated from the Swapping Lemma A.1 and ˙˜
θ 0 = ˙ θ 0
is available from the adaptive law.
Theorem 9.4.2 The modified MRAC scheme described by (9.4.30), (9.4.38)
with τ ∈ (0 , 1 /δ 0) guarantees the following properties when applied to the
plant (9.4.27):
(i) All signals are bounded.
(ii) The tracking error e 1 satisfies
lim sup |e 1( τ 0) | ≤ τ( c + d 0)
t→∞ τ 0 ≥t
where d 0 is an upper bound for the disturbance du and c ≥ 0 is a finite
constant independent of τ .
(iii) When du = 0 we have |e 1( t) | → 0 as t → ∞.
The proof of Theorem 9.4.2 follows from the proofs of the robust MRAC
schemes presented in Section 9.3 and is given in [39].
In [39], the robustness properties of the modified MRAC scheme are
analyzed by applying it to the plant
Z( s)
yp = kp
(1 + µ∆
R( s)
m( s)) up
where µ∆ m( s) is a multiplicative perturbation and µ > 0. It is established
that for τ ∈ ( τmin, 1 ) where 0 < τ
, there exists a µ∗( τ
δ
min < 1
min) > 0 such
0
δ 0
that all signals are bounded and
lim sup |e 1( τ 0) | ≤ τc + µc
t→∞ τ 0 ≥t
where c ≥ 0 is a constant independent of τ, µ. The function µ∗( τmin) is
such that as τmin → 0 , µ∗( τmin) → 0 demonstrating that for a given size
of unmodeled dynamics characterized by the value of µ∗, τ cannot be made
arbitrarily small.
9.5. ROBUST APPC SCHEMES
709
Remark 9.4.1 The modified MRAC schemes proposed above are based on
the assumption that the high frequency gain kp is known. The case of
unknown kp is not as straightforward. It is analyzed in [37] under the
assumption that a lower and an upper bound for kp is known a priori.
Remark 9.4.2 The performance of MRAC that includes transient as well
as steady-state behavior is a challenging problem especially in the
presence of modeling errors. The effect of the various design param-
eters, such as adaptive gains and filters, on the performance and ro-
bustness of MRAC is not easy to quantify and is unknown in gen-
eral. Tuning of some of the design parameters for improved perfor-
mance is found to be essential even in computer simulations, let alone
real-time implementations, especially for high order plants. For addi-
tional results on the performance of MRAC, the reader is referred to
[37, 119, 148, 182, 211, 227, 241].
9.5
Robust APPC Schemes
In Chapter 7, we designed and analyzed a wide class of APPC schemes for
a plant that is assumed to be finite dimensional, LTI, free of any noise and
external disturbances and whose transfer function satisfies assumptions P1
to P3.
In this section, we consider APPC schemes that are designed for a sim-
plified plant model but are applied to a higher-order plant with unmodeled
dynamics and bounded disturbances. In particular, we consider the higher
order plant model which we refer to it as the actual plant
yp = G 0( s)[1 + ∆ m( s)]( up + du)
(9.5.1)
where G 0( s) satisfies P1 to P3 given in Chapter 7, ∆ m( s) is an unknown
multiplicative uncertainty, du is a bounded input disturbance and the overall
plant transfer function G( s) = G 0( s)(1+∆ m( s)) is strictly proper. We design
APPC schemes for the lower-order plant model
yp = G 0( s) up
(9.5.2)
but apply and analyze them for the higher order plant model (9.5.1). The ef-
fect of perturbation ∆ m and disturbance du on the stability and performance
of the APPC schemes is investigated in the following sections.
710
CHAPTER 9. ROBUST ADAPTIVE CONTROL SCHEMES
Plant
du ✲ ∆ m( s)
y
❄
+
❄
+
m
−e
✲
+ ❧ 1
up
yp
Σ ✲ C( s)
✲ ❧
✲ ❧
Σ
Σ ✲ G
✲
0( s)
−
+
+
✻
Figure 9.8
Closed-loop PPC schemes with unmodeled dynamics and
bounded disturbances.
We first consider the nonadaptive case where G 0( s) is known exactly.
9.5.1
PPC: Known Parameters
Let us consider the control laws of Section 7.3.3 that are designed for the
simplified plant model (9.5.2) with known plant parameters and apply them
to the higher order plant (9.5.1). The block diagram of the closed-loop plant
is shown in Figure 9.8 where C( s) is the transfer function of the controller.
The expression for C( s) for each of the PPC laws developed in Chapter 7 is
given as follows.
For the control law in (7.3.6) and (7.3.11) of Section 7.3.2 which is based
on the polynomial approach, i.e.,
QmLup = P ( ym − yp) ,
LQmRp + P Zp = A∗
(9.5.3)
where Qm( s) is the internal model of the reference signal ym, i.e., Qm( s) ym =
0, we have
P ( s)
C( s) =
(9.5.4)
Qm( s) L( s)
The control law (7.3.19), (7.3.20) of Section 7.3.3 based on the state-variable
approach, i.e.,
˙ˆ e = Aˆ e + B¯ up − Ko( C ˆ e − e 1)
Q
¯
u
1
p
= −Kcˆ e,
up =
¯
u
Q
p
(9.5.5)
m
9.5. ROBUST APPC SCHEMES
711
where Kc satisfies (7.3.21) and Ko satisfies (7.3.22), can be put in the form
of Figure 9.8 as follows:
We have
ˆ
e( s) = ( sI − A + KoC ) − 1( B¯ up + Koe 1)
and
¯
up = −Kc( sI − A + KoC ) − 1( B¯ up + Koe 1)
i.e.,
K
¯
u
c( sI − A + KoC ) − 1 Ko
p = −
e
1 + K
1
c( sI − A + KoC ) − 1 B
and, therefore,
K
Q
C( s) =
c( sI − A + KoC ) − 1 Ko
1( s)
(9.5.6)
(1 + Kc( sI − A + KoC ) − 1 B) Qm( s)
For the LQ control of Section 7.3.4, the same control law (9.5.5) is used,
but Kc is calculated by solving the algebraic equation
A P + P A − P Bλ− 1 B P + CC = O,
Kc = λ− 1 B P
(9.5.7)
Therefore, the expression for the transfer function C( s) is exactly the same
as (9.5.6) except that (9.5.7) should be used to calculate Kc.
We express the closed-loop PPC plant into the general feedback form
discussed in Section 3.6 to obtain
C
−CG
u
p
y
= 1 + CG
1 + CG
m
(9.5.8)
y
CG
G
p
du
1 + CG
1 + CG
where G = G 0(1 + ∆ m) is the overall transfer function and C is different
for different pole placement schemes. The stability properties of (9.5.8) are
given by the following theorem:
Theorem 9.5.1 The closed-loop plant described by (9.5.8) is internally sta-
ble provided
T 0( s)∆ m( s) ∞ < 1
where T 0( s) = CG 0 is the complementary sensitivity function of the nomi-
1+ CG 0
nal plant. Furthermore, the tracking error e 1 converges exponentially to the
residual set
De = {e 1 ||e 1 | ≤ cd 0 }
(9.5.9)
712
CHAPTER 9. ROBUST ADAPTIVE CONTROL SCHEMES
where d 0 is an upper bound for |du| and c > 0 is a constant.
Proof The proof of the first part of Theorem 9.5.1 follows immediately from equa-
tion (9.5.8) and the small gain theorem by expressing the characteristic equation of
(9.5.8) as
CG
1 +
0
∆
1 + CG
m = 0
0
To establish (9.5.9), we use (9.5.8) to write
CG
G
1
G
e 1 =
− 1 y
d
y
d
1 + CG
m + 1 + CG u = − 1 + CG m + 1 + CG u
It follows from (9.5.4), (9.5.6) that the controller C( s) is of the form C( s) = C 0( s) Qm( s)
for some C 0( s). Therefore
Q
GQ
e
m
m
1 = −
y
d
Q
m +
u
m + C 0 G
Qm + C 0 G
where G = G 0(1 + ∆ m). Since Qmym = 0 and the closed-loop plant is internally
stable due to T 0( s)∆ m( s) ∞ < 1, we have
(1 + ∆
e
m) G 0 Qm
1 =
d
Q
u + t
(9.5.10)
m + C 0 G 0 + ∆ mC 0 G 0
where t is an exponentially decaying to zero term. Therefore, (9.5.9) is implied by
(9.5.10) and the internal stability of the closed-loop plant.
✷
It should be pointed out that the tracking error at steady state is not
affected by ym despite the presence of the unmodeled dynamics. That is, if
du ≡ 0 and ∆ m = 0, we still have e 1( t) → 0 as t → ∞ provided the closed-
loop plant is internally stable. This is due to the incorporation of the internal
model of ym in the control law. If Qm( s) contains the internal model of du
as a factor, i.e., Qm( s) = Qd( s) ¯
Qm( s) where Qd( s) du = 0 and ¯
Qm( s) ym = 0,
then it follows from (9.5.10) that e 1 = t, i.e., the tracking error converges
to zero exponentially despite the presence of the input disturbance. The
internal model of du can be constructed if we know the frequencies of du.
For example, if du is a slowly varying signal of unknown magnitude we could
choose Qd( s) = s.
The robustness and performance properties of the PPC schemes given by
Theorem 9.5.1 are based on the assumption that the parameters of the mod-
eled part of the plant, i.e., the coefficients of G 0( s) are known exactly. When
9.5. ROBUST APPC SCHEMES
713
the coefficients of G 0( s) are unknown, the PPC laws (9.5.3) and (9.5.5) are
combined with adaptive laws that provide on-line estimates for the unknown
parameters leading to a wide class of APPC schemes. The design of these
APPC schemes so that their robustness and performance properties are as
close as possible to those described by Theorem 9.5.1 for the known param-
eter case is a challenging problem in robust adaptive control and is treated
in the following sections.
9.5.2
Robust Adaptive Laws for APPC Schemes
The adaptive control schemes of Chapter 7 can meet the control objective
for the ideal plant (9.5.2) but not for the actual plant (9.5.1) where ∆ m( s) =
0 , du = 0. The presence of ∆ m and/or du may easily lead to various types
of instability. As in the case of MRAC, instabilities can be counteracted
and robustness properties improved if instead of the adaptive laws used in
Chapter 7, we use robust adaptive laws to update or estimate the unknown
parameters.
The robust adaptive laws to be combined with PPC laws developed for
the known parameter case are generated by first expressing the unknown
parameters of the modeled part of the plant in the form of the parametric
models considered in Chapter 8 and then applying the results of Chapter 8
directly.
We start by writing the plant equation (9.5.1) as
Rpyp = Zp(1 + ∆ m)( up + du)
(9.5.11)
where Zp = θ∗b αn− 1( s) , Rp = sn + θ∗a αn− 1( s); θ∗b, θ∗a are the coefficient vectors of Zp, Rp respectively and αn− 1( s) = [ sn− 1 , sn− 2 , . . . , s, 1] . Filtering each side of (9.5.11) with
1
, where Λ