Page 752 - Mechanical Engineers' Handbook (Volume 2)

5 Stability 743

The computation of the state transition matrix (k, k ) for time-varying, discrete-time
0
systems is difficult, as it is for continuous-time systems. For small or moderate values of the
order n of the system, numerical evaluation of Eq. (52) is appropriate:

F(l) k k 0
k 1
(k, k ) l k 0 (52)
0
I k k 0
For larger values of n, analytical and numerical methods for determining (k, k ) in special
0
cases are available. 2

5 STABILITY

Since state-space formulation is applicable to a large class of dynamic systems, the question
of stability for systems represented in state space is quite a complex one. A more general
consideration of stability than that used for SISO, LTI systems would indicate that stability
of dynamic systems is not really a property of the systems but is more properly associated
4
with isolated equilibrium points of dynamic systems. A particular point x in state space is
e
an equilibrium point of a dynamic system if, in the absence of inputs, the system state x is
equal to x for time t t for continuous-time systems or for k k for discrete-time systems.
0
0
e
For linear systems described by the state-space equations given in Section 2, the only isolated
equilibrium point is at the origin in state space. For nonlinear systems, there may be a number
of isolated equilibrium points. Any isolated equilibrium point can be shifted to the origin in
4
state space by a simple change of state variables. The stability definitions to be given assume
therefore that the equilibrium point is at the origin in state space and that the system is
unforced. Only the more commonly used types of stability will be defined.
The origin is a stable equilibrium point if, for any given value 0, there exists a
number ( , t ) 0 such that, if the norm x(t ) , then the norm x(t) for all t
0
0
t . The norm of a vector x may be defined as the Euclidean norm:
0
x(t) x (t) (53)

n
2
i 1 i
The origin is asymptotically stable if, in addition to being stable, there exists a number
(t ) 0 such that whenever x(t ) (t ) the following condition is satisfied:
0
0
0
lim x(t) 0 (54)
t→
If and are not functions of t in the previous definitions, the origin is said to be uniformly
0
stable or uniformly asymptotically stable, respectively. If (t ) can be arbitrarily large, the
0
origin is said to be globally asymptotically stable. Extension of these stability definitions to
discrete-time systems is straightforward and merely requires that the sequence numbers k,
k be used instead of the time instants t, t , respectively, in the definitions already given.
0
0
Additional types of stability that depend on the inputs to the system have been defined by
Brogan and Kuo. 7
4
For LTI systems, the conditions for stability reduce to conditions on the eigenvalues of
the system matrix A or F and are summarized in Table 6. These eigenvalues are the roots
of the system characteristic equation as well, as shown in Section 7. They may be computed