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80 Introduction to Control Theory

3. If u(t)∈L 2 , then .

EXAMPLE 2.7–11: Input/Output Stability of Rigid Robots
Consider the closed-loop robot of Example 2.7.9. Its input/output behavior
is described by a set of n decoupled differential equations

If . Then the transfer function between each
U i (s) and Q i (s) is

Note that all P i(s) are stable if k vi and k pi are both positive. Assume for the
purposes of illustration that k vi=3 and k pi=2, and that u i=sin (t). Note that u i
is bounded and let us find the output y i(t).

y i(t)=-0.2e +0.5e -0.32 cos(t+0.32)
-2t
-t
which is bounded above by 0.62 and below by -0.02. The derivative of y(t)
is also bounded. On the other hand, suppose the input is u i(t)=e then the
-3t
output is
-t
2t
y i(t)=0.5e -e- +0.5e -3t
Since lim t→∞ u i(t)=0, then so is lim t→∞ y i(t)=0.

2.8 Input/Output Stability

When dealing with nonlinear systems, stability in the sense of Lyapunov
does not necessarily imply that a bounded input will result in a bounded
output. This fact is shown in the next example.

Copyright © 2004 by Marcel Dekker, Inc.