Page 753 - Mechanical Engineers' Handbook (Volume 2)

744 State-Space Methods for Dynamic Systems Analysis

Table 6 Stability Criteria for Linear, Time-Invariant Systems
Continuous-Time System Discrete-Time System
˙ x(t) Ax(t) Bu x(k 1) Fx(k) Gu(k)
Eigenvalues of A Eigenvalues of F
are s i ic j
ic are z i ic j
ic
Asymptotically
stable ic 0 for all roots z i 1 for all roots
Stable ic 0 for all repeated roots and ic 0 z i 1 for all repeated roots and
for all simple roots z i 1 for all simple roots
Unstable ic 0 for any simple root or ic 0 z i 1 for any simple root or
for any repeated root z i 1 for any repeated root

explicitly by numerical methods. Alternatively, stability criteria such as the Routh–Hurwitz
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5
criterion for continuous-time systems or the Jury test for discrete-time systems may be
applied. The conditions for asymptotic stability of such systems can also be shown to be
sufficient for other types of stability depending on the input, such as bounded-input, bounded-
output stability. 4
For continuous-time, LTV systems, the necessary and sufficient condition for the origin
to be a stable equilibrium point is that there exists a number N(t ) such that the norm of the
0
transition matrix satisfies the following condition:
(t, t ) N(t ) for t t 0 (55)
0
0
4
If, in addition, (t, t ) → 0as t → , the system is globally asymptotically stable. The
0
norm of the matrix may be defined as the spectral norm:
T
(t, t ) max (x x) (56)
T
0 x 1
The corresponding stability conditions for linear, discrete-time systems are obtained simply
by substituting the sequence numbers k and k for time instants t and t , respectively, in the
0
0
development. Time-varying systems that satisfy the property that the state converges expo-
nentially with time to the zero state are said to be exponentially stable. 12 For LTI systems,
of course, asymptotic stability is the same as exponential stability.
Stability considerations for nonlinear systems are more complex. For unforced second-
order nonlinear systems, the phase-plane method is useful for examining the stability of
equilibrium points of the system. The phase plane has the state variables as the coordinates.
The state-space equations are used to derive analytical expressions for the trajectories or to
draw the trajectories by graphical means. The phase portraits can then be examined to de-
termine the equilibrium points and their stability. Application of the phase-plane method is
described by DeRusso et al. for continuous-time systems and by Kuo for discrete-time
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systems.
Stability analysis of high-order nonlinear systems represented in state space can be done
using the second method of Lyapunov. This is a technique requiring considerable ingenuity
for effective use and provides sufficient conditions for stability rather than necessary and
sufficient conditions. 8
Lyapunov’s method for nonlinear, unforced, time-invariant systems requires the defini-
tion of a scalar function of state V(x) called the Lyapunov function. The latter may be thought