x(0) = 0
v = C x + Dvp
(9.8.3)
Because A is a stable matrix, we have eA( t−τ) ≤ λ 0 e−α 0( t−τ) for some λ 0 , α 0 > 0.
Applying Lemma 3.3.5 to (9.8.3) we obtain
v
( t−t 1)
t,t
≤ ce− δ 2
|x( t
(9.8.4)
1
1) | + H ( s) ∞δ vpt,t 1
and for H( s) strictly proper
|v( t) | ≤ ce− δ ( t−t
2
1) |x( t 1) | + H( s) 2 δ vp
(9.8.5)
t,t 1
for any 0 < δ < 2 α 0 and for some constant c ≥ 0. If H( s) is analytic in Re[ s] ≥
−δ 0 / 2 then A − δ 0 I is a stable matrix and from Lemma 3.3.3 we have
2
|x( t) | ≤ c vpt 2 δ ≤ cm( t) ∀t ≥ 0
0
Hence, |x( t 1) | ≤ cm( t 1), (i) and (ii) follow from (9.8.4) and (9.8.5), respectively.
When H( s) is not analytic in Re[ s] ≥ −δ 0 / 2 we use output injection to rewrite (9.8.3) as
˙ x = ( A − KC ) x + Bvp + KC x
or
˙ x = Acx + Bcvp + Kv
(9.8.6)
where Ac = A − KC , Bc = B − KD and K is chosen so that Ac − δ 0 I is a stable 2
matrix. The existence of such a K is guaranteed by the observability of ( C, A)[95].
Applying Lemma 3.3.3 (i) to (9.8.6), we obtain
|x( t) | ≤ c vpt 2 δ + c v
≤ cm( t) ∀t ≥ 0
(9.8.7)
0
t 2 δ 0
where the last inequality is established by using the assumption vt 2 δ , v
≤
0
pt 2 δ 0
cm( t) of the Lemma. Hence, x( t 1) ≤ cm( t 1) and (i), (ii) follow directly from
(9.8.4), (9.8.5).
✷
Instead of mf given by (9.8.1), let us consider the signal
m 2
2
2
f ( t) = e−δ( t−t 1) m 2( t
+ y
, t ≥ t
1
1) + upt,t
p
1 ≥ 0
(9.8.8)
1
t,t 1
The signal mf has very similar normalizing properties as m
1
f as indicated by the
following lemma:
Lemma 9.8.3 The signal mf given by (9.8.8) guarantees that
1
9.8. STABILITY PROOFS OF ROBUST MRAC SCHEMES
747
(i) ωi( t) /mf , i = 1 , 2 ; ω
/m
and n
∈ L
1
t,t 1
f 1
s/mf 1
∞
(ii) If θ ∈ L∞, then yp/mf , u
, ω/m , W ( s) ω/m
∈ L
/m ,
1
p/mf 1
f 1
f 1
∞ and
upt,t
f
1
1
˙ yp
/m
∈ L
t,t
f
∞
1
1
(iii) If θ, ˙ r ∈ L∞, then ˙ ωt,t /m ∈ L
1
f 1
∞
where ( ·) t,t
denotes the L
1
2 δ -norm defined over the interval [ t 1 , t] , t ≥ t 1 ≥ 0 ; δ is
any constant in the interval (0 , δ 0] and W ( s) is a proper transfer function, which is
analytic in Re[ s] ≥ −δ 0 / 2 .
Proof (i) We have ω 1 = α( s) u
y
is strictly
Λ( s) p, ω 2 = α( s)
Λ( s) p. Since each element of α( s)
Λ( s)
proper and analytic in Re[ s] ≥ − δ 0 and u
, y
≤ cm, it follows from
2
pt 2 δ 0
pt 2 δ 0
Lemma 9.8.2 that
|ω
( t−t 1)
1( t) |, ω 1
≤ ce− δ 2
m( t
≤ cm
t,t
1) + c up
f
1
t,t 1
1
|ω
( t−t 1)
2( t) |, ω 2
≤ ce− δ 2
m( t
≤ cm
t,t
1) + c yp
f
1
t,t 1
1
and, therefore, |ωi( t) |, ωi
, i = 1 , 2 are bounded from above by m . Because
t,t
f
1
1
ω
2
2
2
2
t,t
≤ ω
+ ω
+ y
+ c ≤ cm 2
1
1 t,t
2
p
1
t,t 1
t,t 1
f 1
therefore, ωt,t
is bounded from above by m . We have n 2
1
f 1
s = ms = m 2 − 1 and
n 2
2
2
s
= ms( t) = e−δ 0( t−t 1)( m 2( t 1) − 1) + up
+ y
t,t
p
1
2 δ 0
t,t 1 2 δ 0
≤ e−δ( t−t 1) m 2( t
2
2
1) + up
+ y
= m 2
t,t
p
1
t,t 1
f 1
for any given δ ∈ (0 , δ 0], and the proof of (i) is complete.
(ii) We have
yp = Wm( s) ρ∗ ˜
θ ω + ρ∗η + Wm( s) r
Because Wm( s) is strictly proper and analytic in Re[ s] ≥ −δ 0 / 2, and Lemma 9.8.1
together with ˜
θ ∈ L∞ imply that ρ∗ ˜
θ ωt 2 δ ≤ cm( t) ∀t ≥ 0, it follows from
0
Lemma 9.8.2 that
|y
( t−t 1)
p( t) |, yp
≤ ce− δ 2
m( t
+ c η
+ c, ∀t ≥ t
t,t
1) + c ωt,t
t,t
1 ≥ 0
1
1
1
Now η = ∆( s) up + dη where ∆( s) is strictly proper and a nalytic in Re[ s] ≥ − δ 0 , 2
dη ∈ L∞ and up
≤ m( t). Hence from Lemma 9.8.2 we have
t
2 δ 0
η
( t−t 1)
t,t
≤ ce− δ 2
m( t
+ c
1
1) + c upt,t 1
Because up = θ ω and θ ∈ L∞, we have up
≤ c ω
, and from part (i) we
t,t
t,t
1
1
have ω
( t−t
t,t
≤ cm . Therefore u
≤ cm , η
≤ ce− δ 2
1) m( t
+ c
1
f 1
pt,t
f
t,t
1)+ cmf
1
1
1
1
and
|y
( t−t 1)
p( t) |, yp
≤ ce− δ 2
m( t
+ c ≤ cm
t,t
1) + cmf
f
1
1
1
748
CHAPTER 9. ROBUST ADAPTIVE CONTROL SCHEMES
In a similar manner, we show
˙ yp
≤ cm . From |ω( t) | ≤ |ω
t,t
f
1( t) | + |ω 2( t) | +
1
1
|yp( t) | + c, it follows that |ω( t) | ≤ cmf . Because u
1
p = θ ω and θ ∈ L∞, it follows
directly that |up( t) | ≤ cmf .
1
Consider v = W ( s) ω where W ( s) is proper and analytic in Re[ s] ≥ −δ 0 / 2.
Because from Lemma 9.8.1 ωt 2 δ ≤ cm( t), it follows from Lemma 9.8.2 that
0
|v( t) | ≤ ce− δ ( t−t
2
1) m( t 1) + c ωt,t
≤ cm
1
f 1
(iii) We have ˙ ω = [ ˙ ω 1 , ˙ ω 2 , ˙ yp, ˙ r] , where ˙ ω 1 = sα( s) u y
Λ( s)
p, ˙
ω 2 = sα( s)
Λ( s)
p. Because
the elements of sα( s) are proper, it follows from the results of (i), (ii) that ˙ ω
≤
Λ( s)
it,t 1
cmf , i = 1 , 2 which together with ˙ r ∈ L
≤ cm
imply (iii).
✷
1
∞ and
˙ ypt,t
f
1
1
9.8.2
Proof of Theorem 9.3.2
We complete the proof of Theorem 9.3.2 in five steps outlined in Section 9.3.
Step1. Express the plant input and output in terms of the parameter error term
˜
θ ω. We use Figure 9.3 to express up, yp in terms of the parameter error ˜
θ and
modeling error input. We have
G
1
Λ − C∗
y
0Λ c∗
0
˜
1
p =
r +
θ ω +
η
(Λ − C∗
1
1 ) − G 0 D∗
1
c∗ 0
c∗ 0Λ
η 1 = ∆ m( s)( up + du) + du
Λ c∗
1
G
u
0
˜
0 D∗
1
p =
r +
θ ω +
η
(Λ − C∗
1
1 ) − G 0 D∗
1
c∗ 0
(Λ − C∗ 1) − G 0 D∗ 1
where C∗ 1( s) = θ∗ 1 α( s) , D∗ 1 = θ∗ 3Λ( s) + θ∗ 2 α( s). Using the matching equation G 0Λ c∗ 0
= W
(Λ − C∗
m
1 ) − G 0 D∗
1
we obtain
1
y
˜
p
= Wm r +
θ ω + η
c∗
y
0
1
u
˜
p
= G− 1
0 Wm
r +
θ ω + η
c∗
u
(9.8.9)
0
where
θ∗
Λ − θ∗
η
3 Λ + θ∗
2 α
1 α
u =
W
W
c∗
mη 1 ,
ηy =
mη 1
0Λ
c∗ 0Λ
9.8. STABILITY PROOFS OF ROBUST MRAC SCHEMES
749
Let us simplify the notation by denoting ( ·) t 2 δ with · . From (9.8.9) and
the stability of Wm, G− 1
0 Wm, we obtain
yp ≤ c + c ˜
θ ω + ηy ,
up ≤ c + c ˜
θ ω + ηu
(9.8.10)
for some δ > 0 by applying Lemma 3.3.2. Using the expressions for ηu, ηy, we have
Λ( s) − θ∗
η
1 α( s)
y
≤
W
c∗
m( s)∆ m( s) ∞δ up
+ cd 0
0Λ( s)
∞δ
θ∗
η
3 Λ( s) + θ∗
2 α( s)
u
≤
W
c∗
m( s)∆ m( s) ∞δ up
+ cd 0
0Λ( s)
∞δ
where d 0 is the upper bound for du and c ≥ 0 denotes any finite constant, which
implies that
ηy ≤ c∆ ∞mf + cd 0 ,
ηu ≤ c∆ ∞mf + cd 0
(9.8.11)
where ∆ ∞ = Wm( s)∆ m( s) ∞δ. From (9.8.10) and (9.8.11), it follows that the
fictitious normalizing signal m 2 = 1 + u 2 + y 2 satisfies
f
p
p
m 2 f ≤ c + c ˜ θ ω 2 + c∆2 ∞m 2 f