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)