We use Lyapunov-like functions and similar arguments as in the example
above to analyze the stability of a wide class of adaptive schemes considered
throughout this book.
3.4. LYAPUNOV STABILITY
119
3.4.4
Lyapunov’s Indirect Method
Under certain conditions, conclusions can be drawn about the stability of the
equilibrium of a nonlinear system by studying the behavior of a certain linear
system obtained by linearizing (3.4.1) around its equilibrium state. This
method is known as the first method of Lyapunov or as Lyapunov’s indirect
method and is given as follows [32, 232]: Let xe = 0 be an equilibrium state
of (3.4.1) and assume that f ( t, x) is continuously differentiable with respect
to x for each t ≥ 0. Then in the neighborhood of xe = 0, f has a Taylor
series expansion that can be written as
˙ x = f ( t, x) = A( t) x + f 1( t, x)
(3.4.15)
where A( t) = ∇f |x=0 is referred to as the Jacobian matrix of f evaluated
at x = 0 and f 1( t, x) represents the remaining terms in the series expansion.
Theorem 3.4.5 Assume that A( t) is uniformly bounded and that
|f
lim sup 1( t, x) | = 0
|x|→ 0 t≥ 0
|x|
Let ze = 0 be the equilibrium of
˙ z( t) = A( t) z( t)
The following statements are true for the equilibrium xe = 0 of (3.4.15):
(i)
If ze = 0 is u.a.s. then xe = 0 is u.a.s.
(ii) If ze = 0 is unstable then xe = 0 is unstable
(iii) If ze = 0 is u.s. or stable, no conclusions can be drawn about the
stability of xe = 0 .
For a proof of Theorem 3.4.5 see [232].
Example 3.4.9 Consider the second-order differential equation
m¨
x = − 2 µ( x 2 − 1) ˙ x − kx
where m, µ, and k are positive constants, which is known as the Van der Pol os-
cillator. It describes the motion of a mass-spring-damper with damping coefficient
120
CHAPTER 3. STABILITY
2 µ( x 2 − 1) and spring constant k, where x is the position of the mass. If we define
the states x 1 = x, x 2 = ˙ x, we obtain the equation
˙ x 1 = x 2
k
2 µ
˙ x 2 = − x
( x 2
m 1 − m
1 − 1) x 2
which has an equilibrium at x 1 e = 0 , x 2 e = 0. The linearization of this system
around (0 , 0) gives us
˙ z 1
0
1
z
=
1
˙ z
2 µ
2
− k
z
m
m
2
Because m, µ > 0 at least one of the eigenvalues of the matrix A is positive and
therefore the equilibrium (0 , 0) is unstable.
3.4.5
Stability of Linear Systems
Equation (3.4.15) indicates that certain classes of nonlinear systems may be
approximated by linear ones in the neighborhood of an equilibrium point or,
as often called in practice, operating point. For this reason we are interested
in studying the stability of linear systems of the form
˙ x( t) = A( t) x( t)
(3.4.16)
where the elements of A( t) are piecewise continuous for all t ≥ t 0 ≥ 0, as a
special class of the nonlinear system (3.4.1) or as an approximation of the
linearized system (3.4.15). The solution of (3.4.16) is given by [95]
x( t; t 0 , x 0) = Φ( t, t 0) x 0
for all t ≥ t 0, where Φ( t, t 0) is the state transition matrix and satisfies the
matrix differential equation
∂ Φ( t,t
∂t
0) = A( t)Φ( t, t 0) ,
∀t ≥ t 0
Φ( t 0 , t 0) = I
Some additional useful properties of Φ( t, t 0) are
(i)
Φ( t, t 0) = Φ( t, τ)Φ( τ, t 0) ∀t ≥ τ ≥ t 0 (semigroup property)
(ii) Φ( t, t 0) − 1 = Φ( t 0 , t)
(iii)
∂ Φ( t, t
∂t
0) = −Φ( t, t 0) A( t 0)
0
3.4. LYAPUNOV STABILITY
121
Necessary and sufficient conditions for the stability of the equilibrium state
xe = 0 of (3.4.16) are given by the following theorems.
Theorem 3.4.6 Let Φ( t, τ ) denote the induced matrix norm of Φ( t, τ ) at
each time t ≥ τ . The equilibrium state xe = 0 of (3.4.16) is
(i)
stable if and only if the solutions of (3.4.16) are bounded or equivalently
c( t 0) = sup Φ( t, t 0) < ∞
t≥t 0
(ii) u.s. if and only if
c 0 = sup c( t 0) = sup sup Φ( t, t 0)
< ∞
t 0 ≥ 0
t 0 ≥ 0
t≥t 0
(iii) a.s. if and only if
lim Φ( t, t 0) = 0
t→∞
for any t 0 ∈ R+
(iv) u.a.s. if and only if there exist positive constants α and β such that
Φ( t, t 0) ≤ αe−β( t−t 0) ,
∀t ≥ t 0 ≥ 0
(v) e.s. if and only if it is u.a.s.
(vi) a.s., u.a.s., e.s. in the large if and only if it is a.s., u.a.s., e.s., respec-
tively.
Theorem 3.4.7 [1] Assume that the elements of A( t) are u.b. for all t ∈
R+ . The equilibrium state xe = 0 of the linear system (3.4.16) is u.a.s. if
and only if, given any positive definite matrix Q( t) , which is continuous in
t and satisfies
0 < c 1 I ≤ Q( t) ≤ c 2 I < ∞
for all t ≥ t 0 , the scalar function defined by
∞
V ( t, x) = x
Φ ( τ, t) Q( τ )Φ( τ, t) dτ x
(3.4.17)
t
exists (i.e., the integral defined by (3.4.17) is finite for finite values of x and
t) and is a Lyapunov function of (3.4.16) with
˙
V ( t, x) = −x Q( t) x
122
CHAPTER 3. STABILITY
It follows using the properties of Φ( t, t 0) that P ( t)= ∞Φ ( τ, t) Q( τ)Φ( τ, t) dτ
t
satisfies the equation
˙
P ( t) = −Q( t) − A ( t) P ( t) − P ( t) A( t) (3.4.18)
i.e., the Lyapunov function (3.4.17) can be rewritten as V ( t, x) = x P ( t) x,
where P ( t) = P ( t) satisfies (3.4.18).
Theorem 3.4.8 A necessary and sufficient condition for the u.a.s of the
equilibrium xe = 0 of (3.4.16) is that there exists a symmetric matrix P ( t)
such that
γ 1 I ≤ P ( t) ≤ γ 2 I
˙
P ( t) + A ( t) P ( t) + P ( t) A( t) + νC( t) C ( t) ≤ O
are satisfied ∀t ≥ 0 and some constant ν > 0 , where γ 1 > 0 , γ 2 > 0 are constants and C( t) is such that ( C( t) , A( t)) is a UCO pair (see Definition 3.3.3).
When A( t) = A is a constant matrix, the conditions for stability of the
equilibrium xe = 0 of
˙ x = Ax
(3.4.19)
are given by the following theorem.
Theorem 3.4.9 The equilibrium state xe = 0 of (3.4.19) is stable if and
only if
(i)
All the eigenvalues of A have nonpositive real parts.
(ii) For each eigenvalue λi with Re {λi} = 0 , λi is a simple zero of the
minimal polynomial of A (i.e., of the monic polynomial ψ( λ) of least
degree such that ψ( A) = O).
Theorem 3.4.10 A necessary and sufficient condition for xe = 0 to be a.s.
in the large is that any one of the following conditions is satisfied 1 :
(i)
All the eigenvalues of A have negative real parts
1Note that (iii) includes (ii). Because (ii) is used very often in this book, we list it
separately for easy reference.
3.4. LYAPUNOV STABILITY
123
(ii) For every positive definite matrix Q, the following Lyapunov matrix
equation
A P + P A = −Q
has a unique solution P that is also positive definite.
(iii) For any given matrix C with ( C, A) observable, the equation
A P + P A = −C C
has a unique solution P that is positive definite.
It is easy to verify that for the LTI system given by (3.4.19), if xe = 0 is
stable, it is also u.s. If xe = 0 is a.s., it is also u.a.s. and e.s. in the large.
In the rest of the book we will abuse the notation and call the matrix
A in (3.4.19) stable when the equilibrium xe = 0 is a.s., i.e., when all the
eigenvalues of A have negative real parts and marginally stable when xe = 0
is stable, i.e., A satisfies (i) and (ii) of Theorem 3.4.9.
Let us consider again the linear time-varying system (3.4.16) and suppose
that for each fixed t all the eigenvalues of the matrix A( t) have negative real
parts. In view of Theorem 3.4.10, one may ask whether this condition for
A( t) can ensure some form of stability for the equilibrium xe = 0 of (3.4.16).
The answer is unfortunately no in general, as demonstrated by the following
example given in [232].
Example 3.4.10 Let
− 1 + 1 . 5 cos2 t
1 − 1 . 5 sin t cos t
A( t) =
− 1 − 1 . 5 sin t cos t
− 1 + 1 . 5 sin2 t
The eigenvalues of A( t) for each fixed t,
√
λ( A( t)) = −. 25 ± j. 5 1 . 75
have negative real parts and are also independent of t. Despite this the equilibrium
xe = 0 of (3.4.16) is unstable because
e. 5 t cos t
e−t sin t
Φ( t, 0) =
−e. 5 t sin t e−t cos t
is unbounded w.r.t. time t.
124
CHAPTER 3. STABILITY
Despite Example 3.4.10, Theorem 3.4.10 may be used to obtain some
sufficient conditions for a class of A( t), which guarantee that xe = 0 of
(3.4.16) is u.a.s. as indicated by the following theorem.
Theorem 3.4.11 Let the elements of A( t) in (3.4.16) be differentiable 2 and
bounded functions of time and assume that
(A1) Re{λi( A( t)) } ≤ −σs ∀t ≥ 0 and for i = 1 , 2 , . . . , n where σs > 0 is some constant.
(i) If
˙
A ∈ L 2 , then the equilibrium state xe = 0 of (3.4.16) is u.a.s. in
the large.
(ii) If any one of the following conditions:
1
(a) t+ T
˙
A( τ ) dτ ≤ µT + α
t
0 , i.e., (
˙
A ) 2 ∈ S( µ)
(b) t+ T
˙
A( τ ) 2 dτ ≤ µ 2 T + α
t
0 , i.e.,
˙
A ∈ S( µ 2)
(c)
˙
A( t) ≤ µ
is satisfied for some α 0 , µ ∈ R+ and ∀t ≥ 0 , T ≥ 0 , then there exists a
µ∗ > 0 such that if µ ∈ [0 , µ∗) , the equilibrium state xe of (3.4.16) is
u.a.s. in the large.
Proof Using (A1), it follows from Theorem 3.4.10 that the Lyapunov equation
A ( t) P ( t) + P ( t) A( t) = −I
(3.4.20)
has a unique bounded solution P ( t) for each fixed t. We consider the following
Lyapunov function:
V ( t, x) = x P ( t) x
Then along the solution of (3.4.16) we have
˙
V = −|x( t) | 2 + x ( t) ˙
P ( t) x( t)
(3.4.21)
From (3.4.20), ˙
P satisfies
A ( t) ˙
P ( t) + ˙
P ( t) A( t) = −Q( t) ,
∀t ≥ 0
(3.4.22)
where Q( t) = ˙
A ( t) P ( t) + P ( t) ˙
A( t). Because of (A1), it can be verified [95] that
∞
˙
P ( t) =
eA ( t) τ Q( t) eA( t) τ dτ
0
2The condition of differentiability can be relaxed to Lipschitz continuity.
3.4. LYAPUNOV STABILITY
125
satisfies (3.4.22) for each t ≥ 0, therefore,
∞
˙
P ( t) ≤ Q( t)
eA ( t) τ
eA( t) τ dτ
0
Because (A1) implies that eA( t) τ ≤ α 1 e−α 0 τ for some α 1 , α 0 > 0 it follows that
˙
P ( t) ≤ c Q( t)
for some c ≥ 0. Then,
Q( t) ≤ 2 P ( t)
˙
A( t)
together with P ∈ L∞ imply that
˙
P ( t) ≤ β ˙
A( t) ,
∀t ≥ 0
(3.4.23)
for some constant β ≥ 0. Using (3.4.23) in (3.4.21) and noting that P satisfies
0 < β 1 ≤ λmin( P ) ≤ λmax( P ) ≤ β 2 for some β 1 , β 2 > 0, we have that
˙
V ( t) ≤ −|x( t) | 2 + β ˙
A( t) |x( t) | 2 ≤ −β− 1
˙
2 V ( t) + ββ− 1
1
A( t) V ( t)
therefore,
t
−
( β− 1 −ββ− 1
˙
A( τ ) ) dτ
V ( t) ≤ e
t
2
1
0
V ( t 0)
(3.4.24)
Let us prove (ii) first. Using condition (a) in (3.4.24) we have
V ( t) ≤ e−( β− 1 −ββ− 1 µ)( t−t
α
2
1
0) eββ?