Stability PDF

1 Stability
1.1 Bounded Input-Bounded Output (BIBO) Stability

Definition: A system y = Hu is BIBO stable if for any bounded input u(t)
corresponds a bounded output y(t).
In general, the input u(t) and the output y(t) are bounded in the sense of a
signal norm! A scalar signal u(t) is bounded if
Mu < : ku(t)k = sup |u(t)| < Mu .

t0
Definition: A scalar function h(t) is absolutely integrable if Mh < such

that Z
|h( )| d < Mh .
0
Theorem: The SISO linear system with impulse response h(t) is BIBO stable if
and only if h(t) is absolutely integrable.
Theorem: The MIMO linear system with impulse response matrix

H(t) = (Hij (t)) is BIBO stable if and only if hij (t) is absolutely integrable for
all i, j.
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Corollary: The MIMO linear system with a rational and proper transfer matrix
H(s) = N (s)/d(s) is BIBO stable if and only if all poles of H(s), i.e., the roots
of d(s), are in the open left-half of the complex plane.
Proof: The impulse response hij (t) can be obtained from Hij (s) as

1 1 Nij (s)
hij (t) = L {Hij (s)} = L .
d(s)
Expanding Hij (s) in partial fractions we have
( m k ) m X ki
i
1
XX i
X
hij (t) = L j
= tj1 ei t
i=1 j=1
(s i ) i=1 j=1
where i denotes the ith root of d(s) with multiplicity ki . Therefore, hij (t) is
absolutely integrable if and only if the factors
tj1 ei t , j = 1, . . . , ki
are absolutely integrable. That is, iff R(i ) < 0.

Proof of Theorem (SISO): Sufficiency: For linear systems
Z
y(t) = h( )u(t ) d
0
and for SISO systems

Z

|y(t)| = h( )u(t ) d ,
Z 0
|h( )u(t )| d,
0
Z
|h( )||u(t )| d.
0
Therefore, since u(t) is bounded and h(t) is absolutely integrable

Z
|y(t)| |h( )||u(t )| d,
0 Z

Mu |h( )| d,
0
M u Mh
which implies y(t) is bounded.

Necessity: If h(t) is not absolutely integrable then there exists t such that
Z t
|h( )| d = .
0
Now for the particular input

(
+1, h(t) 0
u(t t) =
1, h(t) < 0
we have that
Z t Z t
y(t) = h( )u(t ) d = |h( )| d =
0 0
which implies y(t) is not bounded.

1.2 Internal Stability
The LTI system
x(t) = Ax(t) + Bu(t),
(1)
y(t) = Cx(t) + Du(t).
is BIBO stable iff H(s) = C(sI A)1 B + D has all poles on the open
left-half of the complex plane.
The LTI system (1) is internally stable iff all roots of d(s) = det(sI A) are on
the open left-half of the complex plane.
Internal stability = BIBO stability

Internal stability = BIBO stability +
controllability and observability

1.3 Lyapunov Stability
Consider the dynamic system
x = f (x).
Definition: x is an equilibrium point of (1.3) if f (x) = 0.
Definition: Let x = 0 be an equilibrium point of (1.3) and let Rn . It is
1) Stable if for each > 0 there is > 0 such that
kx(0)k < kx(t)k < , t 0.
2) Unstable if not stable.

3) Asymptotically stable if it is stable and > 0 can be chosen such that
kx(0)k < lim x(t) = 0.

t
Theorem: Let x = 0 be an equilibrium point of the dynamic system (1.3) and

x Rn . Let V : R be continuously differentiable and
V (0) = 0, V (x) > 0 x , x 6= 0,
and
V (x) 0, x .
Then x = 0 is stable. Moreover, if
V (x) < 0, x , x 6= 0
then x = 0 is asymptotically stable.
Proof: see H. K. Kalil, Nonlinear Systems, Chapter 3.

1.3.1 Lyapunov Stability for Linear Systems
Consider the LTI system
x = Ax. (2)
Theorem: The following statements about the linear system (2) are equivalent:
1) R(i (A)) < 0.
2) x = 0 is the unique equilibrium point of (2) and it is asymptotically stable.
Proof: Use the same tools to conclude that
x(t) = L (sI A)1 x(0)

so that
lim x(t) = 0
t
for any x(0) bounded iff R(i (A)) < 0. This implies A is nonsingular, which
implies that Ax = 0 x = 0 is the unique equilibrium point.

Theorem: The LTI system (2) is asymptotically stable if and only if, for any
matrix Q > 0, the Lyapunov equation
AT P + P A + Q = 0 (3)
has a unique solution P such that P > 0.
Proof: Sufficiency: Assume there exists P > 0 that solves (3). Then
V (x) = xT P x > 0, for all 0 =
6 x := Rn . Note that
d T
V (x) = x P x,
dt
= xT P x + xT P x,
= xT (AT P + P A)x,
= xT Qx < 0, 0 6= x Rn .
Therefore (2) is asymptotically stable.
Necessity: Assume (2) is asymptotically stable then R(i (A)) < 0. This can be
used to show that the Gramian
Z
T
P = eA t QeAt dt
0
converges to a finite value. Using stability we can also show that

T
lim eA t QeAt = 0.
t
Therefore, as we already know

Z Z
d T d T
AT P + P A = A T eA t QeAt dt + eA t QeAt dt A,
dt dt
Z 0 0
d AT t At
= e Qe dt,
0 dt
T
= lim (eA t QeAt ) Q = Q.
t
To prove that it is positive definite we assume that it is not, so that there exists
z 6= 0 such that (see a similar proof for the Controllability and Observability
Gramians)
Z
T

z P =z eA t QeAt dt = 0 eAt z 0, t 0 z = 0.
0

Finally, to prove that it is unique assume there exists P 6= P satisfying the
Lyapunov equation. Then
AT (P P ) + (P P )A = 0
and
h i h T i h T i
AT t T At T A t At A t At
e A (P P ) + (P P )A e = A e (P P )e + e (P P )e A,
d AT t
= e (P P )eAt = 0, t 0,
dt
which implies that
T
eA t (P P )eAt is constant t 0.
In particular, at t = 0 and t
T
(P P ) = lim eA t (P P )eAt = 0 P = P
t
REMARK #1: The proof works also if Q = C T C 0 and (A, C) is

observable.
REMARK #2: Internal stability Asymptotic stability.

1.4 Example

0 1
x = x
1 1
Lyapunov Equation (Q = I)
T
0 1 p1 p2 p1 p2 0 1 1 0
+ =
1 1 p2 p3 p2 p3 1 1 0 1
is equivalent to the linear system

0 2 0 p1 1
1 1 1 p2 = 0
0 2 2 p3 1
which has the unique solution

p1 3/2
p2 = 1/2 p1 p2 3/2 1/2
= >0
p2 p3 1/2 1
p3 1

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1 Stability

1.1 Bounded Input-Bounded Output (BIBO) Stability

Mu < : ku(t)k = sup |u(t)| < Mu .

Definition: A scalar function h(t) is absolutely integrable if Mh < such

Theorem: The MIMO linear system with impulse response matrix

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are absolutely integrable. That is, iff R(i ) < 0.

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and for SISO systems

Therefore, since u(t) is bounded and h(t) is absolutely integrable

which implies y(t) is bounded.

Now for the particular input

which implies y(t) is not bounded.

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Internal stability = BIBO stability

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kx(0)k < kx(t)k < , t 0.

2) Unstable if not stable.

kx(0)k < lim x(t) = 0.

Theorem: Let x = 0 be an equilibrium point of the dynamic system (1.3) and

V (0) = 0, V (x) > 0 x , x 6= 0,

then x = 0 is asymptotically stable.

Proof: see H. K. Kalil, Nonlinear Systems, Chapter 3.

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Proof: Use the same tools to conclude that

x(t) = L (sI A)1 x(0)

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converges to a finite value. Using stability we can also show that

Therefore, as we already know

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REMARK #1: The proof works also if Q = C T C 0 and (A, C) is

REMARK #2: Internal stability Asymptotic stability.

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