Robust Adaptive Control by Petros A. Ioannou, Jing Sun - HTML preview

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