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30 Introduction to Control Theory

The MIMO case is similar to the continuous-time case and is given by

(2.2.17)

where A is n×n, B is n×m, C is p×n, and D is p×m.
In many practical cases, such as in the control of robots, the system is a
continuous-time system, but the controller is implemented using digital
hardware. This will require the designer to translate between continuous-
and discrete-time systems. There are many different approaches to
“discretizing” a continuous-time system, some of which are discussed in
Chapter 3. The interested reader in this very important aspect of the control
problem is referred to [Åström and Wittenmark 1996], [Franklin et al.
1997].

EXAMPLE 2.2–4: Double Integrator in Discrete Time

Recall Example 2.2.1 which presented a model of the double integrator or
Newton’s system. One discrete-time version of the differential equation is
given by the following difference equation

where T is the sampling period in seconds. If we choose x 1(k)=y(k) and
x 2(k)=x 1(k+1), we obtain the state-space description

Transfer Function Representation
In a similar fashion to the continuous-time case, a linear, time-invariant,
discrete-time system given by (2.2.17) may be described in the Z-transform
domain, from input U(z) to output Y(z) by its transfer function P(z) such
that

Y(z)=P(z)U(z)

Copyright © 2004 by Marcel Dekker, Inc.