2.4. STABILITY 37
Figure 2.13. (a) Underdamped response (b) Undamped responsethe initial response is to oscillate beforeachieving steady state are referred to
asunderdamped systems(Figure 2.13 (a)). An underdamped response is
characterized by overshoot, and anundamped response(Figure 2.13 (b)) by
sustained oscillation.
A certain amount of oscillation is tolerable in the system response. For
example, if a change in the output of a plant is desired, the input to the plant
is changed in the form of a step change. Upon receiving this step change in
input, we expect the plant to respond quickly so that the desired output can
be obtained and maintained as rapidly as possible. We can let the plant output
have a fast rate of rise so that the desired output can be achieved quickly. In
doing so, we need to allow for a small overshoot and then control the response to
exhibit a frequency of oscillation that is adequately damped to bring the plant
response towards the desired value in the shortest possible time. A detailed
description of the response characteristics of the system is necessary both for
analysis and design.
2.4.2 Stability of linear systems..................
Consider the special case
f(x,t)=Ax(t) (2.31)whereAis a constantn◊nmatrix. IfAis nonsingular, that is, ifdetA 6 =0,
then the system described by Equation 2.31 has a unique equilibrium point,
namely 0. For this situation, we can simply talk about the stability of the linear
system. Its analysis is based upon the following theorem.
Theorem 2.1The linear systemx ̇=Axis asymptotically stable if and only if
all eigenvalues of the matrixAhavenegativerealparts.
Proof.The solution ofx ̇=Axis
x(t)=x 0 eAt