say, the state transition matrix Φ( t, τ ) of A( t) satisfies Φ( t, τ ) ≤ k 1 e−k 2( t−τ)
∀t ≥ τ and t, τ ∈ I 1. This property of A( t) (when m 2 grows large) is used in
Step 3 to contradict the hypothesis that m 2 could grow unbounded and conclude
boundedness.
Let us start by assuming that m 2 grows unbounded. Because all the elements
of the state x are the outputs of strictly proper transfer functions with the same
poles as the roots of Λ( s) and inputs up, yp (see Section 7.7.1) and the roots of Λ( s)
are located in Re[ s] < −δ 0 / 2, it follows from Lemma 3.3.2 that x ∈ L
m
∞. Because
¯
ym, m ∈ L∞, it follows from (9.9.1) that yp , up ∈ L
m
m
∞. Because u 2
p, y 2
p are bounded
from above by m 2, it follows from the equation for m 2 that m 2 cannot grow faster
than an exponential, i.e., m 2( t) ≤ ek 1( t−t 0) m 2( t 0) , ∀t ≥ t 0 ≥ 0 for some k > 0.
Because m 2( t) is assumed to grow unbounded, we can find a t 0 > ¯
α > 0 for any
arbitrary constant ¯
α > t 2 − t 0 such that m 2( t 2) > ¯
αek 1 ¯ α. We have
¯
αek 1 ¯ α < m 2( t 2) ≤ ek 1( t 2 −t 0) m 2( t 0)
which implies that
ln m 2( t 0) > ln ¯
α + k 1[¯
α − ( t 2 − t 0)]
Because ¯
α > t 2 − t 0 and t 0 ∈ ( t 2 − ¯
α, t 2), it follows that
m 2( t 0) > ¯
α,
∀t 0 ∈ ( t 2 − ¯
α, t 2)
Let t 1 = sup τ≤t {arg( m 2( τ) = ¯ α) }. Then, m 2( t
2
1) = ¯
α and m 2( t) ≥ ¯
α, ∀t ∈ [ t 1 , t 2)
where t 1 ≤ t 2 − ¯
α, i.e., t 2 − t 1 ≥ ¯
α. Let us now consider the behavior of the
homogeneous part of (9.9.1), i.e.,
˙
Y = A( t) Y
(9.9.3)
over the interval I 1 = [ t 1 , t 2) for which m 2( t) ≥ ¯
α and t 2 − t 1 ≥ ¯
α where ¯
α > 0 is an
arbitrary constant. Because det( sI − A( t)) = A∗( s), i.e., A( t) is a pointwise stable matrix, the Lyapunov equation
A ( t) P ( t) + P ( t) A( t) = −I
(9.9.4)
9.9. STABILITY PROOFS OF ROBUST APPC SCHEMES
761
has the solution P ( t) = P ( t) > 0 for each t ∈ I 1. If we consider the Lyapunov function
V ( t) = Y ( t) P ( t) Y ( t)
then along the trajectory of (9.9.3), we have
˙
V = −Y Y + Y
˙
P Y ≤ −Y Y +
˙
P ( t) Y Y
As in the proof of Theorem 3.4.11, we can use (9.9.4) and the boundedness of P, A
to establish that
˙
P ( t) ≤ c ˙
A( t) . Because λ 1 Y Y ≤ V ≤ λ 2 Y Y for some
0 < λ 1 < λ 2, it follows that
˙
V ≤ −( λ− 1
˙
2
− cλ− 1
1
A( t) ) V
i.e.,
t
˙
V ( t) ≤ e−
( λ− 1 −cλ− 1 A( s) ) ds
τ
2
1
V ( τ )
∀ t ≥ τ ≥ 0. For the interval I 1 = [ t 1 , t 2), we have m 2( t) ≥ ¯
α and, therefore,
t
˙
d 2
A( τ ) dτ ≤ c(∆2
0
2 + f 0 +
)( t − τ ) + c
τ
¯
α
and therefore,
V ( t) ≤ e−λ 0( t−τ) V ( τ ) ,
∀t, τ ∈ [ t 1 , t 2)
(9.9.5)
and t ≥ τ provided
d 2
c( f
0
0 + ∆2
2 +
) < λ
¯
α
0
(9.9.6)
where λ 0 = λ− 1
2
. From (9.9.5) we have
2
λ 1 Y ( t) Y ( t) ≤ Y ( t) P Y ( t) ≤ e−λ 0( t−τ) λ 2 Y ( τ) Y ( τ) which implies that
λ
|Y ( t) | ≤
2 e−λ 0( t−τ) |Y ( τ) |, ∀t, τ ∈ [ t
λ
1 , t 2)
1
which, in turn, implies that the transition matrix Φ( t, τ ) of (9.9.3) satisfies
Φ( t, τ ) ≤ β 0 e−α 0( t−τ) , ∀t, τ ∈ [ t 1 , t 2)
(9.9.7)
where β 0 =
λ 2 , α
. Condition (9.9.6) can now be satisfied by choosing
λ
0 = λ 0
1
2
¯
α large enough and by requiring ∆2 , f 0 to be smaller than some constant, i.e.,
c( f 0 + ∆22) < λ− 1
2
, say. In the next step we use (9.9.7) and continue our argument
4
over the interval I 1 in order to establish boundedness by contradiction.
762
CHAPTER 9. ROBUST ADAPTIVE CONTROL SCHEMES
Step 3. Boundedness using the B-G Lemma and contradiction. Let us apply
Lemma 3.3.6 to (9.9.1) for t ∈ [ t 1 , t 2). We have
xt,t
≤ ce−δ/ 2( t−t 1) |x( t
+ c
1
1) | + c ( m 2) t,t 1
where ( ·) t,t
denotes the L
1
2 δ norm
( ·) t,t 1 2 δ defined over the interval [ t 1 , t), for
any 0 < δ < δ 1 < 2 α 0. Because x ∈ L
m
∞ it follows that
xt,t
≤ ce−δ/ 2( t−t 1) m( t
+ c
1
1) + c ( m 2) t,t 1
Because ypt,t , u
≤ c x
+ c ( m 2)
+ c, it follows that
1
pt,t 1
t,t 1
t,t 1
ypt,t , u
≤ ce−δ/ 2( t−t 1) m( t
+ c
1
pt,t 1
1) + c ( m 2) t,t 1
Now m 2( t) = 1 + ms( t) and
m
2
2
s( t) = e−δ 0( t−t 1) ms( t 1) + ypt,t
+ u
1
2 δ
pt,t
0
1
2 δ 0
Because ( ·) t,t
≤ ( ·)
for δ ≤ δ
1
2 δ 0
t,t 1
0, it follows that
m 2( t) = 1 + m
2
2
s( t) ≤ 1 + e−δ 0( t−t 1) m 2( t 1) + ypt,t
+ u
∀t ≥ t
1
pt,t 1
1
Substituting for the bound for ypt,t , u
we obtain
1
pt,t 1
m 2( t) ≤ ce−δ( t−t 1) m 2( t
2
1) + c ( m 2) t,t
+ c ∀t ≥ t
1
1 ≥ 0
or
t
m 2( t) ≤ c + ce−δ( t−t 1) m 2( t 1) + c
e−δ( t−τ) 2 m 2 m 2( τ ) dτ
(9.9.8)
t 1
Applying B-G Lemma III we obtain
t
t
c
2 m 2 dτ
t 2
m 2( t) ≤ c(1 + m 2( t
m 2 dτ
1)) e−δ( t−t 1) e
t 1
+ cδ
e−δ( t−s) ec s
ds, ∀t ≥ t 1
t 1
For t, s ∈ [ t 1 , t 2) we have
t
d 2
c
2 m 2 dτ ≤ c ∆2
0
2 + f 0 +
( t − s) + c
s
¯
α
By choosing ¯
α large enough so that c d 20 < δ and by requiring
¯
α
4
δ
c(∆22 + f 0) < 4
we have
t 2
m 2( t
( t 2 −t 1)
( t 2 −s)
2)
≤ c(1 + m 2( t 1)) e− δ 2
+ cδ
e− δ 2
ds
t 1
≤ c(1 + m 2( t
( t 2 −t 1)
1)) e− δ 2
+ c
9.9. STABILITY PROOFS OF ROBUST APPC SCHEMES
763
Because t 2 − t 1 ≥ ¯
α, m 2( t 1) = ¯
α and m 2( t 2) > ¯
α, we have
¯
α < m 2( t 2) ≤ c(1 + ¯
α) e− δ¯ α
2
+ c
Therefore, we can choose ¯
α large enough so that m 2( t 2) < ¯
α which contradicts the
hypothesis that m 2( t 2) > ¯
α. Therefore, m ∈ L∞ which implies that x, up, yp ∈ L∞.
The condition for robust stability is, therefore,
λ− 1 δ
c( f
2
0 + ∆2
2) < min {
, } = δ∗
2
4
for some finite constant c > 0.
Step 4. Establish bounds for the tracking error. A bound for the tracking
error e 1 is obtained by expressing e 1 in terms of signals that are guaranteed by the
adaptive law to be of the order of the modeling error in m.s.s. The tracking error
equation has exactly the same form as in the ideal case in Section 7.7.1 and is given
by
Λ( s) sn− 1 Q
Λ( s) α
e
m( s)
n− 2( s)
1 =
m 2 +
v
A∗( s)
A∗( s)
0
(see equation (7.7.21)) where v 0 is the output of proper stable transfer functions
whose inputs are elements of ˙ θp multiplied by bounded signals. Because ˙ θp, m 2 ∈
S( η 2 + f
≤ c(∆2
m 2
0) and η 2
m 2
2 + d 2
0), due to m ∈ L∞, it follows from Corollary 3.3.3
that e 1 ∈ S(∆22 + d 20 + f 0) and the proof is complete.
9.9.2
Proof of Theorem 9.5.3
We use the same steps as in the proof of Example 9.5.2.
Step 1. Develop the state error equations for the closed-loop plant. We start
with the plant equation
Rpyp = Zp(1 + ∆ m)( up + du)
Operating with Qm( s) on each side, we obtain
Q 1( s)
Q
Q
Q
R
m
m
m
p
y
e
[∆
Q
p = Rp
1 = Zp ¯
up + Zp
m( up + du) + du]
1
Q 1
Q 1
i.e.,
Z
Z
e
pQ 1
pQm
1 =
¯
u
[∆
R
p +
m( up + du) + du]
pQm
RpQm
Because Zp ∆
R
m is strictly proper, we can find an arbitrary polynomial Λ( s) whose
p
roots are in Re[ s] < −δ 0 / 2 and has the same degree as Rp, i.e., n and express e 1 as Z
Λ Q
e
pQ 1
m
1 =
¯
u
η
(9.9.9)
R
p +
pQm
RpQm
764
CHAPTER 9. ROBUST ADAPTIVE CONTROL SCHEMES
where
Z
η =
p [∆
Λ
m( up + du) + du]
We express (9.9.9) in the following canonical state-space form
˙ e = Ae + B ¯
up + B 1 η
e 1 = C e + d 1 η
(9.9.10)
where C ( sI − A) − 1 B = ZpQ 1 , C ( sI − A) − 1 B
with
R