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84 Introduction to Control Theory

Suppose the robot is representing a system whose input is and whose
output is the joint velocity q. Let the sum of the kinetic energy and potential
energy of the robot be denote by the Hamiltonian H and recall [Ortega and
Spong 1988]

then

which proves that from to q, the rigid robot is a passive system.

A passive system is in effect one that does not create energy.

Positive-Real Systems
If the system under consideration is linear and time-invariant, then passivity
is equivalent to positivity and may be tested in the frequency domain
[Narendra and Taylor 1973]. In fact, let us describe positive-real systems
and discuss some of their properties. Consider the multi-input-multi-output
linear time-invariant system

where x is an n vector, u is an m vector, y is a p vector, A, B, C, and D are of
the appropriate dimensions. The corresponding transfer function matrix is

P(s)=C(sI-A) B+D
-1
We will assume that the system has an equal number of inputs and outputs,
i.e. p=m. To simplify our notation we will denote the Hermitian part of a
real, rational transfer matrix T(s) by where s*
is the complex conjugate of s. A number of definitions have been given for
SPR functions and matrices [Narendra and Taylor 1973]. It appears that the
most useful definition for control applications is the following.

DEFINITION 2.9–2 An m×m matrix T(s) of proper real rational functions
which is not identically zero is positive-real or PR if

Copyright © 2004 by Marcel Dekker, Inc.