1 ( τ ) dτ
t
t
1
t+ T
k
≥
( q w)2 dτ − 4 T − ¯
k
2
3
(4.8.19)
t
a 2
Because w is PE, i.e.,
t+ T 0
( q w)2 dτ ≥ α 0 T 0
t
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CHAPTER 4. ON-LINE PARAMETER ESTIMATION
for some T 0 , α 0 > 0 and ∀t ≥ 0, we can divide the interval [ t, t + T ] into subintervals of length T 0 and write
t+ T
n 0
t+ iT 0
( q w)2 dτ ≥
( q w)2 dτ ≥ n 0 α 0 T 0
t
i=1
t+( i− 1) T 0
where n 0 is the largest integer that satisfies n 0 ≤ T . From the definition of n
T
0, we
0
have n 0 ≥ T − 1; therefore, we can establish the following inequality
T 0
t+ T
T
( q w)2 dτ ≥ α 0
− 1 T 0 = α 0( T − T 0)
(4.8.20)
t
T 0
Because the above analysis holds for any T > 0, we assume that T > T 0. Using
(4.8.19) and (4.8.20) in (4.8.18), we have
t+ T
1
α
k
( q w
0
4
3)2 dτ
≥
( T − T 0) −
T − ¯
k 3 − k 2
t
k 1
2
a 2
1
α
k
α
=
0 − 4 T −
0 T 0 + ¯ k
k
3 + k 2
(4.8.21)
1
2
a 2
2
Since (4.8.21) holds for any a, T > 0 and α 0 , k 4 are independent of a, we can first
choose a to satisfy
α 0
k
α
− 4 ≥ 0
2
a 2
4
and then fix T so that
α 0
α
T −
0 T 0 + ¯ k
4
2
3 + k 2
> β 1 k 1 T
for a fixed β 1 > 0. It follows that
t+ T
( q w 3( τ))2 dτ ≥ β 1 T > 0
t
and the lower bound in (4.8.17) is established.
✷
Lemma 4.8.4 Consider the system
˙
Y 1 = AcY 1 − Bcφ Y 2
˙
Y 2 = 0
(4.8.22)
y 0 = Cc Y 1
where Ac is a stable matrix, ( Cc, Ac) is observable, and φ ∈ L∞. If φf defined as
φf = Cc ( sI − Ac) − 1 Bcφ
4.8. PARAMETER CONVERGENCE PROOFS
233
satisfies
1
t+ T 0
α 1 I ≤
φ
T
f ( τ ) φf ( τ ) dτ ≤ α 2 I,
∀t ≥ 0
(4.8.23)
0
t
for some constants α 1 , α 2 , T 0 > 0 , then (4.8.22) is UCO.
Proof The UCO of (4.8.22) follows if we establish that the observability grammian
N ( t, t + T ) of (4.8.22) defined as
t+ T
N ( t, t + T ) =
Φ ( τ, t) CC Φ( τ, t) dτ
t
where C = [ Cc 0] satisfies
βI ≥ N ( t, t + T ) ≥ αI
for some constant α, β > 0, where Φ( t, t 0) is the state transition matrix of (4.8.22).
The upper bound βI follows from the boundedness of Φ( t, t 0) that is implied by
φ ∈ L∞ and the fact that Ac is a stable matrix. The lower bound will follow if we
establish the following inequality:
t+ T
y 20( τ) dτ ≥ α |Y 1( t) | 2 + |Y 2 | 2
t
where Y 2 is independent of t due to ˙ Y 2 = 0. From (4.8.22), we can write
τ
y 0( τ) = Cc Y 1( τ) = Cc eAc( τ−t) Y 1( t) −
Cc eAc( τ−σ) Bcφ ( σ) dσY 2
t
= y 1( τ) + y 2( τ)
for all τ ≥ t, where y 1( τ)= Cc eAc( τ−t) Y 1( t) , y 2( τ)= − τ C
t
c eAc( τ −σ) Bcφ ( σ) dσY 2.
Using the inequalities ( x + y)2 ≥ x 2 − y 2 and ( x + y)2 ≥ y 2 − x 2 with x = y 2
2
1 , y = y 2
over the intervals [ t, t + T ], [ t + T , t + T ], respectively, we have
t+ T
t+ T
y 2
t+ T
y 2
1 ( τ )
0 ( τ ) dτ
≥
dτ −
y 22( τ) dτ
t
t
2
t
t+ T y 2
t+ T
+
2 ( τ ) dτ −
y 21( τ) dτ
(4.8.24)
t+ T
2
t+ T
for any 0 < T < T . We now evaluate each term on the right-hand side of (4.8.24).
Because Ac is a stable matrix, it follows that
|y 1( τ) | ≤ k 1 e−γ 1( τ−t) |Y 1( t) |
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CHAPTER 4. ON-LINE PARAMETER ESTIMATION
for some k 1 , γ 1 > 0, and, therefore,
t+ T
k 2
y 2
1
1 ( τ ) dτ ≤
e− 2 γ 1 T |Y 1( t) | 2
(4.8.25)
t+ T
2 γ 1
On the other hand, since ( Cc, Ac) is observable, we have
t+ T
eA ( t−τ)
c
CcCc eAc( t−τ) dτ ≥ k 2 I
t
for any T > T 1 and some constants k 2 , T 1 > 0. Hence,
t+ T
y 21( τ) dτ ≥ k 2 |Y 1( t) | 2
(4.8.26)
t
Using y 2( τ) = −φ ( τ ) Y
f
2 and the fact that
t+ T
α 2 n 1 T 0 I ≥
φf φf dτ ≥ n 0 α 1 T 0 I
t+ T
where n 0 , n 1 is the largest and smallest integer respectively that satisfy
T − T
n 0 ≤
≤ n
T
1
0
i.e., n 0 ≥ T−T − 1, n
+ 1, we can establish the following inequalities sat-
T
1 ≤ T −T
0
T 0
isfied by y 2:
t+ T
T
y 22( τ) dτ ≤ α 2 T 0
+ 1 |Y 2 | 2
t
T 0
t+ T
T − T
y 22( τ) dτ ≥ α 1 T 0
− 1 |Y 2 | 2
(4.8.27)
t+ T
T 0
Using (4.8.25), (4.8.26), (4.8.27) in (4.8.24), we have
t+ T
k
k 2
y 2
2
1
0 ( τ ) dτ ≥
−
e− 2 γ 1 T
|Y 1( t) | 2
t
2
2 γ 1
α
T − T
T
+
1 T 0
− 1 − α
+ 1
|Y
2
T
2 T 0
2 | 2
(4.8.28)
0
T 0
Because the inequality (4.8.28) is satisfied for all T, T with T > T 1, let us first
choose T such that T > T 1 and
k 2
k 2
k
−
1 e− 2 γ
2
1 T
≥
2
2 γ 1
4
4.8. PARAMETER CONVERGENCE PROOFS
235
Now choose T to satisfy
α 1 T 0
T − T
T
− 1 − α
+ 1
≥ β
2
T
2 T 0
1
0
T 0
for a fixed β 1. We then have
t+ T
y 20( τ) dτ ≥ α |Y 1( t) | 2 + |Y 2( t) | 2
t
where α = min β 1 , k 2 . Hence, (4.8.22) is UCO.
✷
4
4.8.2
Proof of Corollary 4.3.1
Consider equations (4.3.30), (4.3.35), i.e.,
˙ e = Ace + Bc( −˜
θ φ − n 2 s)
˙˜ θ = Γ φ
(4.8.29)
= Cc e
that describe the stability properties of the adaptive law. In proving Theorem 4.3.1,
we have also shown that the time derivative ˙
V of
e P
˜
θ Γ − 1 ˜
θ
V =
ce +
2
2
where Γ = Γ > 0 and Pc = Pc > 0, satisfies
˙
V ≤ −ν 2
(4.8.30)
for some constant ν > 0. Defining
A
1
P
A( t) =
c − BcCc n 2 s
−Bcφ
,
C = [ C
c
0
Γ φC
c
0] , P =
c
0
2
0
Γ − 1
and x = [ e , ˜
θ ] , we rewrite (4.8.29) as
˙ x = A( t) x,
= C x
and express the above Lyapunov-like function V and its derivative ˙
V as
V
= x P x
˙
V
= 2 x P Ax + x ˙
P x
= x ( P A + A P + ˙
P ) x ≤ −ν x CC x = −ν 2
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CHAPTER 4. ON-LINE PARAMETER ESTIMATION
where ˙
P = 0. It therefore follows that P , as defined above, satisfies the inequality
A ( t) P + P A( t) + ν CC ≤ O
Using Theorem 3.4.8, we can establish that the equilibrium ee = 0 , ˜
θe = 0 (i.e.,
xe = 0) of (4.8.29) is u.a.s, equivalently e.s., provided ( C, A) is a UCO pair.
According to Lemma 4.8.1, ( C, A) and ( C, A + KC ) have the same UCO
property, where
B
K =
cn 2
s
−Γ φ
is bounded. We can therefore establish that (4.8.29) is UCO by showing that
( C, A + KC ) is a UCO pair. We write the system corresponding to ( C, A + KC )
as
˙
Y 1 = AcY 1 − Bcφ Y 2
˙
Y 2 = 0
(4.8.31)
y 0 = Cc Y 1
Because φ is PE and Cc ( sI − Ac) − 1 Bc is stable and minimum phase (which is
implied by Cc ( sI − Ac) − 1 Bc being SPR) and ˙ φ ∈ L∞, it follows from Lemma 4.8.3
(iii) that
τ
φf ( τ) =
Cc eAc( τ−σ) Bcφ( σ) dσ
t
is also PE; therefore, there exist constants α 1 , α 2 , T 0 > 0 such that
1
t+ T 0
α 2 I ≥
φ
T
f ( τ ) φf ( τ ) dτ ≥ α 1 I,
∀t ≥ 0
0
t
Hence, applying Lemma 4.8.4 to the system (4.8.31), we conclude that ( C, A +
KC ) is UCO which implies that ( C, A) is UCO. Therefore, we conclude that the
equilibrium ˜
θe = 0 , ee = 0 of (4.8.29) is e.s. in the large.
✷
4.8.3
Proof of Theorem 4.3.2 (iii)
The parameter error equation (4.3.53) may be written as
˙˜ θ = A( t)˜ θ
(4.8.32)
y 0 = C ( t)˜
θ
where A( t) = −Γ φφ , C ( t) = − φ , y
m 2
m
0 =
m. The system (4.8.32) is analyzed
using the Lyapunov-like function
˜
θ Γ − 1 ˜
θ
V =
2
4.8. PARAMETER CONVERGENCE PROOFS
237
that led to
˙
(˜
θ φ)2
V = −
= − 2 m 2
m 2
along the solution of (4.8.32). We need to establish that the equilibrium ˜
θe = 0 of
(4.8.32) is e.s. We achieve that