Page 754 - Mechanical Engineers' Handbook (Volume 2)

5 Stability 745

of as a generalized energy function. The requirement on the Lyapunov function is that it be
positive definite in some region about the origin in state space, the origin having been
assumed to be an isolated equilibrium point here. A function V(x), which is continuous and
has continuous partial derivatives, is said to be positive (negative) definite in some region
about the origin if it is zero at the origin and greater than (less than) zero everywhere else
in the specified region. If the function is greater than (less than) or equal to zero everywhere
in the specified region, it is said to be positive (negative) semidefinite. 4
Consider the unforced continuous-time system represented by the state equation

˙ x(t) f[x(t)] (57)
where
f(0) 0 (58)
If a positive-definite function V(x) can be determined in some region about the origin such
that its derivative with respect to time is negative semidefinite in , then the origin is a stable
equilibrium point. If dV/dt is negative definite, the origin is asymptotically stable. If the
region can be arbitrarily large and the conditions for asymptotic stability hold and if, in
addition, V(x) → as x → , the origin is a globally asymptotically stable equilibrium
point. Table 7 gives the corresponding stability conditions for nonlinear, time-invariant,
discrete-time systems. Extensions of the stability conditions for time-varying systems have
8
been described by Kalman and Bertram and DeRusso et al. 2
As an example of the application of the second method of Lyapunov, consider the
following nonlinear system:
˙ x x 2 (59)
1
3
˙ x ax bx x 1
0 2
0 2
2
where a , b 0 and both are not zero. The origin is an equilibrium point for this system
0 0
since, if both x and x are zero,
1 2
˙ x ˙x 0 (60)
2
1
Consider the following Lyapunov function:
2
V(x , x ) x x 2 2 (61)
1
2
1
It satisfies the conditions for positive definiteness in an arbitrarily large region about the
origin:
Table 7 Application of the Second Method of Lyapunov to Nonlinear, Time-Invariant, Discrete-Time
Systems
State equation x(k 1) f[x(k)]
f(0) 0
Lyapunov function Scalar function V[x(k)] positive definite in some region
about the origin
Condition for stability in V V[x(k 1)] V[x(k)] is negative semidefinite in
Condition for asymptotic stability in V V[x(k 1)] V[x(k)] is negative definite in
Condition for global asymptotic (i) can be arbitrarily large
stability (ii) V[x(k)] → as x(k) →