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2.7 Lyapunov Stability Theorems 77

EXAMPLE 2.7–9: Stability of PD Controllers for Rigid Robots

Consider the rigid robot example and the torque input of Example 2.3.2.
The resulting linear system is given by

T
T T
The equilibrium point is x e=[0 0 ] . It is then easy to find K p and K v to
stabilize the equilibrium point. In fact, let Q=I and consider the Lyapunov
equation of theorem 2.7.3
T
A P+PA=-I
which reduces to

where

The solution of these equations will provide a stabilizing controller for the
robot. In particular, the choices of K p and K v of Example 2.6.3(part 5) will
make the origin a GES equilibrium.

Convergence Rate
Although Lyapunov stability theory does not directly give an indication of
the transient behavior of the system, it may actually be used to estimate the
convergence rate for linear systems. In order to see this, consider the following
inequalities which are a result of Rayleigh-Ritz theorem 2.5.2,

T
x λ min(P)x x Px x λ max(P)x (2.7.2)
T
T

Copyright © 2004 by Marcel Dekker, Inc.