g 2( τ ) dτ ≤ c∆2
s
2( t−s)+ c, it follows that for c∆2
2 < δ, we have mf ∈ L∞.
From mf ∈ L∞, we have x, up, yp, m ∈ L∞ and therefore all signals are bounded.
The condition on ∆2 for signal boundedness is summarized as follows:
√
c∆2 < min[ δ, c∆ ∗] , 0 < δ < min[ δ 0 , 2 k 2]
where as indicated before ∆ ∗ > 0 is the bound for ∆2 for A( t) to be u.a.s.
Step 4. Establish tracking error bounds. As in the ideal case considered in
Example 7.4.1 (Step 4), we can establish by following exactly the same procedure
that
s( s + λ)
s + λ
1
˙
1
e
ˆ
1 =
m 2 −
( ˙ˆ a
y
b
u
( s + 1)2
( s + 1)2
s + λ p − s + λ p)
(9.5.24)
˙
This equation is exactly the same as in the ideal case except that m, ˙ˆ a, ˆ b ∈ S(∆22)
instead of being in L 2. Because yp, up, m ∈ L∞, it follows that
1
˙
1
m 2 , ( ˙ˆ a
y
ˆ b
u
s + λ p − s + λ p) ∈ S(∆22)
Therefore by writing s( s+ λ) = 1 + ( λ− 2) s− 1 , using m 2 ∈ S(∆2
( s+1)2
( s+1)2
2) and applying
Corollary 3.3.3 to (9.5.24), we obtain
t
e 21 dτ ≤ c∆22 t + c
(9.5.25)
0
which implies that the mean value of e 21 is of the order of the modeling error char-
acterized by ∆2.
Let us now simulate the APPC scheme (9.5.18), (9.5.19) applied to the plant
given by (9.5.17). For simulation purposes, we use a = − 1 , b = 1 and ∆ m( s)= − 2 µs .
1+ µs
The plant output response yp versus t is shown in Figure 9.9 for different values
of µ that indicate the size of ∆ m( s). As µ increases from 0, the response of yp
deteriorates and for µ = 0 . 28, the closed loop becomes unstable.
Remark 9.5.1 The calculation of the maximum size of unmodeled dynam-
ics characterized by ∆2 for robust stability is tedious and involves sev-
eral conservative steps. The most complicated step is the calculation
of ∆ ∗ using the proof of Theorem 3.4.11 and the rate of decay of the
state transition matrix of A( t). In addition, these calculations involve
the knowledge or bounds of the unknown parameters. The robustness
results obtained are therefore more qualitative than quantitative.
9.5. ROBUST APPC SCHEMES
721
2.4
2.2
2
µ=0.0
1.8
µ=0.05
1.6
µ=0.1
µ=0.2
1.4
output y
1.2
1
0.8
0.6
0.40
2
4
6
8
10
12
14
16
18
20
time (sec)
Figure 9.9 Plant output response for the APPC scheme of Example 9.5.1
for different values of µ.
Remark 9.5.2 In the above example the use of projection guarantees that
|b( t) | ≥ b 0 > 0 ∀t ≥ 0 where b 0 is a known lower bound for |b| and therefore stabilizability of the estimated plant is assured. As we showed
in Chapter 7, the problem of stabilizability becomes more difficult to
handle in the higher order case since no procedure is yet available
for the development of convex parameter sets where stabilizability is
guaranteed.
Let us now extend the results of Example 9.5.1 to higher order plants.
We consider the APPC scheme of Table 9.5 that is designed based on the
plant model (9.5.2) and applied to the actual plant (9.5.1).
Theorem 9.5.2 Assume that the estimated polynomials ˆ
Rp( s, t) , ˆ
Zp( s, t) of
the plant model are such that ˆ
RpQm, ˆ
Zp are strongly coprime at each time t.
There exists a δ∗ > 0 such that if
Z
c( f
p( s)
0 + ∆2
2) < δ∗,
where ∆2 =
∆
Λ
m( s)
p( s)
2 δ 0
722
CHAPTER 9. ROBUST ADAPTIVE CONTROL SCHEMES
then the APPC schemes of Table 9.5 guarantee that all signals are bounded
and the tracking error e 1 satisfies
t e 21 dτ ≤ c(∆22 + d 20 + f 0) t + c, ∀t ≥ 0
0
where f 0 = 0 in the case of switching-σ and projection and f 0 > 0 in the
case of fixed-σ( f 0 = σ) , dead zone ( f 0 = g 0) and -modification ( f 0 = ν 0) and d 0 is an upper bound for |du|.
The proof of Theorem 9.5.2 for du = 0 follows directly from the analysis
of Example 9.5.1 and the proof for the ideal case in Chapter 7. When du = 0
the proof involves a contradiction argument similar to that in the MRAC
case. The details of the proof are presented in Section 9.9.1.
Remark 9.5.3 As discussed in Chapter 7, the assumption that the esti-
mated time varying polynomials ˆ
RpQm, ˆ
Zp are strongly coprime can-
not be guaranteed by the adaptive law. The modifications discussed
in Chapter 7 could be used to relax this assumption without changing
the qualitative nature of the results of Theorem 9.5.2.
9.5.4
Robust APPC: State Feedback Law
A robust APPC scheme based on a state feedback law can be formed by
combining the PPC law (9.5.5) with a robust adaptive law as shown in
Table 9.6. The design of the APPC scheme is based on the plant model
(9.5.2) but is applied to the actual plant (9.5.1). We first demonstrate the
design and analysis of the robust APPC scheme using the following example.
Example 9.5.2 Let us consider the same plant as in Example 9.5.1,
b
yp =
(1 + ∆
s + a
m( s)) up
(9.5.26)
which for control design purposes is modeled as
b
yp =
u
s + a p
where a, b are unknown and ∆ m( s) is a multiplicative plant uncertainty that is
analytic in Re[ s] ≥ −δ 0 / 2 for some known δ 0 > 0. The control objective is to
9.5. ROBUST APPC SCHEMES
723
Table 9.6 Robust APPC scheme: state feedback law
Actual plant
yp = Zp (1 + ∆
R
m)( up + du)
p
Plant model
yp = Zp u
R
p
p
Reference
Qm( s) ym = 0
signal
Assumptions
Same as in Table 9.5
Robust
Same as in Table 9.5; it generates the estimates
adaptive law
ˆ
Zp( s, t) , ˆ
Rp( s, t)
˙ˆ e = ˆ
Aˆ
e + ˆ
B ¯
up − ˆ
Ko( t)( C ˆ e − e 1) , ˆ e ∈ Rn+ q
In+ q− 1
ˆ
A =
−θ 1( t) − − −− , ˆ
B = θ 2( t) , C =[1 , 0 , . . . , 0]
0
ˆ
RpQm = sn+ q + θ 1 ( t) αn+ q− 1( s)
State observer
ˆ
ZpQ 1 = θ 2 ( t) αn+ q− 1( s)
ˆ
Ko = a∗ − θ 1 , A∗o( s) = sn+ q + a∗ αn+ q− 1( s)
e 1 = yp − ym
Calculation of
Solve ˆ
Kc( t) from
controller
det( sI − ˆ
A + ˆ
B ˆ
Kc) = A∗c( s)
parameters
at each time t
Control law
¯
up = − ˆ
Kc( t)ˆ e, up = Q 1 ¯ u
Q
p
m
Q 1( s) , A∗o( s) , A∗c( s) monic of degree q, n + q, n + q, Design
respectively, with roots in Re[ s] < −δ 0 / 2; A∗c( s)
variables
has Q 1( s) as a factor; Qm( s) as in Table 9.5.
724
CHAPTER 9. ROBUST ADAPTIVE CONTROL SCHEMES
stabilize the actual plant and force yp to track the reference signal ym = 1 as close
as possible. As in Example 9.5.1, the parametric model for the actual plant is given
by
z = θ∗p φ + η
(9.5.27)
where z = s y
[ u
∆
s+ λ p, φ =
1
s+ λ
p, −yp] , θ∗
p = [ b, a] , η =
b
s+ λ
mu and λ > 0 is chosen
to satisfy λ > δ 0 / 2. A robust adaptive law based on (9.5.27) is
˙ θp = Pr[Γ( φ − σsθp)]
(9.5.28)
where θp = [ˆ b, ˆ a] ; Pr( ·) is the projection operator that guarantees that |ˆ b( t) | ≥
b 0 > 0 ∀t ≥ 0 where b 0 is a known lower bound for |b|. The other signals used in (9.5.28) are defined as
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
(9.5.29)
and σs is the switching σ-modification. Because ym = 1 we have Qm( s) = s and
select Q 1( s) = s + 1, i.e., we assume that δ 0 < 2. Choosing ˆ
Ko = [10 , 25] − [ˆ a, 0] ,
the state-observer is given by
˙
−ˆ a 1
1
−ˆ a + 10
ˆ
e =
ˆ
e +
ˆ b¯ u
([1 , 0]ˆ
e − e
0
0
1
p −
25
1)
(9.5.30)
where the poles of the observer, i.e., the eigenvalues of ˆ
A − ˆ
K 0 C, are chosen as
the roots of Ao( s) = s 2 + 10 s + 25 = ( s + 5)2. The closed-loop poles chosen as
the roots of A∗c( s) = ( s + 1)2 are used to calculate the controller parameter gain
ˆ
Kc( t) = [ˆ k 1 , ˆ k 2] using
det( sI − ˆ
A + ˆ
B ˆ
Kc) = ( s + 1)2
which gives
ˆ
1 − ˆ a
1
k 1 =
, ˆ
k
ˆ
2 =
b
ˆ b
The control law is given by
s + 1
¯
up = −[ˆ k 1 , ˆ k 2]ˆ e, up =
¯
u
s
p
(9.5.31)
The closed-loop plant is described by equations (9.5.26) to (9.5.31) and analyzed
by using the following steps.
Step 1. Develop the state error equations for the closed-loop plant. We start
with the plant equation
( s + a) yp = b(1 + ∆ m( s)) up
9.5. ROBUST APPC SCHEMES
725
which we rewrite as
( s + a) syp = b(1 + ∆ m( s)) sup
Using syp = se 1 and filtering each side with
1
= 1 , we obtain
Q 1( s)
s+1
s
s
( s + a)
e
u
s + 1 1 = b(1 + ∆ m)¯
up, ¯ up = s + 1 p
which implies that
b( s + 1)
e 1 =
(1 + ∆
s( s + a)
m)¯
up
(9.5.32)
We consider the following state representation of (9.5.32)
−a 1
1
λ − a + 1
˙ e =
e +
b¯
u
η
0
0
1
p +
λ
b
e 1 = [1 , 0] e + η, η =
∆
s + λ m( s) up
(9.5.33)
Let eo = e − ˆ e be the observation error. It follows from (9.5.30), (9.5.33) that eo
satisfies
− 10 1
1
1
λ − 9
˙ e
˜
o =
e
˜ ae
b¯
u
η
(9.5.34)
− 25 0
o +
0
1 −
1
p +
λ − 25
The plant output is related to ˆ
e, eo, e 1 as follows:
yp = C eo + C ˆ e + η + ym, e 1 = yp − ym
(9.5.35)
where C
= [1 , 0]. A relationship between up and ˆ e, eo, η that involves stable
transfer functions is developed by using the identity
s( s + a)
(2 − a) s + 1
+
= 1
(9.5.36)
( s + 1)2
( s + 1)2
developed in Section 7.4.3 (see (7.4.37)) under the assumption that b, s( s + a) are
coprime, i.e., b = 0. From (9.5.36), we have
s( s + a)
(2 − a) s + 1
up =
u
u
( s + 1)2 p +
( s + 1)2
p
Using sup = ( s+1)¯ up = −( s+1) ˆ
Kcˆ e and up = s+ a y
b
p −∆ mup in the above equation,
we obtain
s + a
[(2 − a) s + 1]( s + a)
[(2 − a) s + 1]( s + λ)
up = −
[ˆ
k