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

t 0

From the expression of q( t), one can verify that

t

d

t

t

−

g( τ ) k( τ ) dτ

q( t) y( t) − q( t) g( t)

k( s) y( s) ds =

e

t 0

k( s) y( s) ds

(3.3.38)

t

dt

0

t 0

Using (3.3.38) in (3.3.37) and integrating both sides of (3.3.37), we obtain

t

t

t

−

g( τ ) k( τ ) dτ

e

t 0

k( s) y( s) ds ≤

λ( s) q( s) ds

t 0

t 0

Therefore,

t

t

t

g( τ ) k( τ ) dτ

k( s) y( s) ds ≤ e t 0

λ( s) q( s) ds

t 0

t 0

t

t

s

g( τ ) k( τ ) dτ

−

g( τ ) k( τ ) dτ

= e t 0

λ( s) k( s) e

t 0

ds

t 0

t

t

=

λ( s) k( s) e

g( τ ) k( τ ) dτ

s

ds

(3.3.39)

t 0

102

CHAPTER 3. STABILITY

Using (3.3.39) in (3.3.35), the proof for the inequality (3.3.36) is complete.

Consider the special case where λ is a constant and g = 1. Define

t

q 1 = λ +

k( s) y( s) ds

t 0

From (3.3.35), we have

y( t) ≤ q 1( t)

Now

˙ q 1 = ky

Because k ≥ 0, we have

˙ q 1 ≤ kq 1

Let w = ˙ q 1 − kq 1 . Clearly, w ≤ 0 and

˙ q 1 = kq 1 + w

which implies

t

t

k( τ ) dτ

t

q

k( s) ds

1( t) = e t 0

q 1( t 0) +

e τ

w( τ ) dτ

(3.3.40)

t 0

Because k ≥ 0 , w ≤ 0 ∀t ≥ t 0 and q 1( t 0) = λ, it follows from (3.3.40) that t k( τ) dτ

y( t) ≤ q 1( t) ≤ λe t 0

and the proof is complete.

✷

The reader can refer to [32, 232] for alternative proofs of the B-G Lemma.

Other useful forms of the B-G lemma are given by Lemmas 3.3.8 and 3.3.9.

Lemma 3.3.8 (B-G Lemma II) Let λ( t) , k( t) be nonnegative piecewise

continuous function of time t and let λ( t) be differentiable. If the function

y( t) satisfies the inequality

t

y( t) ≤ λ( t) +

k( s) y( s) ds, ∀t ≥ t 0 ≥ 0

t 0

then

t k( s) ds

t

t

y( t) ≤ λ( t

˙

k( τ ) dτ

0) e t 0

+

λ( s) e s

ds, ∀t ≥ t 0 ≥ 0 .

t 0

3.3. INPUT/OUTPUT STABILITY

103

Proof Let

t

z( t) = λ( t) +

k( s) y( s) ds

t 0

it follows that z is differentiable and z ≥ y. We have

˙ z = ˙ λ + ky, z( t 0) = λ( t 0)

Let v = z − y, then

˙ z = ˙ λ + kz − kv

whose state transition matrix is

t

Φ( t, τ ) = exp

k( s) ds

τ

Therefore,

t

z( t) = Φ( t, t 0) z( t 0) +

Φ( t, τ )[ ˙ λ( τ ) − k( τ ) v( τ )] dτ

t 0

Because

t

Φ( t, τ ) k( τ ) v( τ ) dτ ≥ 0

t 0

resulting from Φ( t, τ ) , k( τ ) , v( τ ) being nonnegative, we have

t

z( t) ≤ Φ( t, t 0) z( t 0) +

Φ( t, τ ) ˙ λ( τ ) dτ

t 0

Using the expression for Φ( t, t 0) in the above inequality, we have

t

t

k( s) ds

t

y( t) ≤ z( t) ≤ λ( t

˙

k( τ ) dτ

0) e t 0

+

λ( s) e s

ds

t 0

and the proof is complete.

✷

Lemma 3.3.9 (B-G Lemma III) Let c 0 , c 1 , c 2 , α be nonnegative constants

and k( t) a nonnegative piecewise continuous function of time. If y( t) satisfies

the inequality

t

y( t) ≤ c 0 e−α( t−t 0) + c 1 + c 2

e−α( t−τ) k( τ ) y( τ ) dτ, ∀t ≥ t 0

t 0

then

t

c

k( s) ds

t

t

y( t) ≤ ( c

2

k( s) ds

0 + c 1) e−α( t−t 0) e

t 0

+ c 1 α

e−α( t−τ) ec 2 τ

dτ, ∀t ≥ t 0

t 0

104

CHAPTER 3. STABILITY

Proof The proof follows directly from Lemma 3.3.8 by rewriting the given inequal-

ity of y as

t

¯

y( t) ≤ λ( t) +

¯

k( τ )¯

y( τ ) dτ

t 0

where ¯

y( t) = eαty( t) , ¯

k( t) = c 2 k( t) , λ( t) = c 0 eαt 0 + c 1 eαt. Applying Lemma 3.3.8, we obtain

t

t

c

k( s) ds

t

eαty( t) ≤ ( c

2

k( s) ds

0 + c 1) eαt 0 e

t 0

+ c 1 α

eατ ec 2 τ

dτ

t 0

The result follows by multiplying each side of the above inequality by e−αt.

✷

The B-G Lemma allows us to obtain an explicit bound for y( t) from the

implicit bound of y( t) given by the integral inequality (3.3.35). Notice that

if y( t) ≥ 0 and λ( t) = 0 ∀t ≥ 0, (3.3.36) implies that y( t) ≡ 0 ∀t ≥ 0.

In many cases, the B-G Lemma may be used in place of the small gain

theorem to analyze a class of feedback systems in the form of Figure 3.1 as

illustrated by the following example.

Example 3.3.6 Consider the same system as in Example 3.3.5. We have

t

x( t) = eAtx(0) +

eA( t−τ) Bx( τ ) dτ

0

Hence,

t

|x( t) | ≤ α 1 e−α 0 t|x(0) | +

α 1 e−α 0( t−τ) B |x( τ) |dτ

0

i.e.,

t

|x( t) | ≤ α 1 e−α 0 t|x(0) | + α 1 e−α 0 t B

eα 0 τ |x( τ ) |dτ

0

Applying the B-G Lemma I with λ = α 1 e−α 0 t|x(0) |, g( t) = α 1 B e−α 0 t, k( t) = eα 0 t, we have

|x( t) | ≤ α 1 e−α 0 t|x(0) | + α 1 |x(0) |e−γt

where

γ = α 0 − α 1 B

Therefore, for |x( t) | to be bounded from above by a decaying exponential (which

implies that Ac = A + B is a stable matrix), B has to satisfy

α

B < 0

α 1

3.4. LYAPUNOV STABILITY

105

which is the same condition we obtained in Example 3.3.5 using the small gain

theorem. In this case, we assume that |x(0) | = 0, otherwise for x(0) = 0 we would

have λ( t) = 0 and |x( t) | = 0 ∀t ≥ 0 which tells us nothing about the stability of Ac. The reader may like to verify the same result using B-G Lemmas II and III.

3.4

Lyapunov Stability

3.4.1

Definition of Stability

We consider systems described by ordinary differential equations of the form

˙ x = f ( t, x) ,

x( t 0) = x 0

(3.4.1)

where x ∈ Rn, f : J ×B( r) → R, J = [ t 0 , ∞) and B( r)= {x∈Rn | |x| < r}.

We assume that f is of such nature that for every x 0 ∈ B( r) and every

t 0 ∈ R+, (3.4.1) possesses one and only one solution x( t; t 0 , x 0).

Definition 3.4.1 A state xe is said to be an equilibrium state of the

system described by (3.4.1) if

f ( t, xe) ≡ 0 for all t ≥ t 0

Definition 3.4.2 An equilibrium state xe is called an isolated equilib-

rium state if there exists a constant r > 0 such that B( xe, r) = {x |

|x − xe| < r} ⊂ Rn contains no equilibrium state of (3.4.1) other than

xe.

The equilibrium state x 1 e = 0 , x 2 e = 0 of

˙ x 1 = x 1 x 2 , ˙ x 2 = x 21

is not isolated because any point x 1 = 0 , x 2 = constant is an equilibrium

state. The differential equation

˙ x = ( x − 1)2 x

has two isolated equilibrium states xe = 1 and xe = 0.

106

CHAPTER 3. STABILITY

Definition 3.4.3 The equilibrium state xe is said to be stable (in the

sense of Lyapunov) if for arbitrary t 0 and

> 0 there exists a δ( , t 0)

such that |x 0 − xe| < δ implies |x( t; t 0 , x 0) − xe| < for all t ≥ t 0 .

Definition 3.4.4 The equilibrium state xe is said to be uniformly stable

(u.s.) if it is stable and if δ( , t 0) in Definition 3.4.3 does not depend on t 0 .

Definition 3.4.5 The equilibrium state xe is said to be asymptotically

stable (a.s.) if (i) it is stable, and (ii) there exists a δ( t 0) such that |x 0 −

xe| < δ( t 0) implies lim t→∞ |x( t; t 0 , x 0) − xe| = 0 .

Definition 3.4.6 The set of all x 0 ∈ Rn such that x( t; t 0 , x 0) → xe as

t → ∞ for some t 0 ≥ 0 is called the region of attraction of the equilibrium

state xe. If condition (ii) of Definition 3.4.5 is satisfied, then the equilibrium

state xe is said to be attractive .

Definition 3.4.7 The equilibrium state xe is said to be uniformly asymp-

totically stable (u.a.s.) if (i) it is uniformly stable, (ii) for every > 0 and

any t 0 ∈ R+ , there exist a δ 0 > 0 independent of t 0 and

and a T ( ) > 0

independent of t 0 such that |x( t; t 0 , x 0) − xe| <

for all t ≥ t 0 + T ( )

whenever |x 0 − xe| < δ 0 .

Definition 3.4.8 The equilibrium state xe is exponentially stable (e.s.)

if there exists an α > 0 , and for every

> 0 there exists a δ( ) > 0 such

that

|x( t; t 0 , x 0) − xe| ≤ e−α( t−t 0) for all t ≥ t 0

whenever |x 0 − xe| < δ( ) .

Definition 3.4.9 The equilibrium state xe is said to be unstable if it is

not stable.

When (3.4.1) possesses a unique solution for each x 0 ∈ Rn and t 0 ∈ R+,

we need the following definitions for the global characterization of solutions.

Definition 3.4.10 A solution x( t; t 0 , x 0) of (3.4.1) is bounded if there

exists a β > 0 such that |x( t; t 0 , x 0) | < β for all t ≥ t 0 , where β may depend on each solution.

3.4. LYAPUNOV STABILITY

107

Definition 3.4.11 The solutions of (3.4.1) are uniformly bounded (u.b.)

if for any α > 0 and t 0 ∈ R+ , there exists a β = β( α) independent of t 0

such that if |x 0 | < α, then |x( t; t 0 , x 0)

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