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28 Introduction to Control Theory

Y(s)=[C(sI-A) B+D] U(s)+C(sI-A) x(0) (2.2.8)
-1
-1
As mentioned previously, the transfer function is obtained as the relationship
between the input U(s) and the output Y(s) when x(0)=0, that is,

-1
Y(s)=[C(sI-A) B+D] U(s). (2.2.9)
The transfer function of this particular linear, time-invariant system is given
by

-1
P(s)=C(sI-A) B+D (2.2.10)

such that (see Fig.2.2.1)

Y(s)=P(s)U(s) (2.2.11)

EXAMPLE 2.2–3: Transfer Function of Double Integrator

Consider the system of Example 2.2.1. It is easy to see that the transfer
function is

Discrete-Time Systems
In the discrete-time case, a difference equation is used to described the system
as follows:

(2.2.12)

where a i, b i, i=0,…,n are scalar constants, y(k) is the output, and u(k) is the
input at time k. Note that the output at time k+n depends on the input at
time k+n but not on later inputs; otherwise, the system would be non-
causal.

Copyright © 2004 by Marcel Dekker, Inc.