A First Course in FUZZY and NEURAL CONTROL

44 CHAPTER 2. MATHEMATICAL MODELS IN CONTROL

for values ofsfor which this improper integral converges. Taking Laplace trans-
forms of the system of equations

x ̇(t)=Ax(t)+Bu(t) (2.35)

y(t)=Cx(t)+Eu(t)

withx(0) = 0,weobtain

sàx(s)=Axà(s)+Bàu(s) (2.36)

ày(s)=Càx(s)+Eàu(s)

Writingsàx(s)=sIàx(s)whereIis the identity matrix of the appropriate size,
we get
(sI−A)àx(s)=Bàu(s) (2.37)

and inverting the matrixsI−A,

ày(s)=C(sI−A)−^1 Bàu(s)+Euà(s) (2.38)

=

≥

C(sI−A)−^1 B+E

¥

uà(s)

Thus, thesystem transfer function

G(s)=C(sI−A)−^1 B+E (2.39)

satisfies
G(s)uà(s)=ày(s) (2.40)

and thus describes the ratio between the outputày(s)and the inputàu(s).The

ratio
C(s)
R(s)

=

ày(s)

àu(s)

is also known as theclosed-loop control ratio.
There are a number of other reasons whytransfer functions obtained from the
Laplace transform are useful. A system represented by a differential equation is
difficult to model as a block diagram, but the Laplace transform of the system
has a very convenient representation of this form. Another reason is that in
the solutionx(t)=eAtx(0) +

Rt

0 e

A(t−τ)Bu(τ)dτto the differential equation

model in Equation 2.35, the inputuappears inside the ìconvolution integral,î
whereas in the transfer model,àx(s)becomes a rational function multiplied by
the inputàu. Also, the transfer function in the frequency domain is analytic,
except at afinite number of poles, and analytic function theory is very useful
in analyzing control systems.
In the closed-loop feedback system shown in Figure 2.15,R(s)=àu(s)is the
reference input,C(s)=ày(s)is thesystem output,G(s)is the closed-loop
control ratio,E(s)is theerror,andH(s)is thefeedback transfer function.
From thisfigure we can derive the following relationships:

1. The output of the plant isC(s)=G(s)E(s).