A First Course in FUZZY and NEURAL CONTROL

30 CHAPTER 2. MATHEMATICAL MODELS IN CONTROL

The system is calledtime invariantif the response tou(t−τ)isy(t−τ),
that is,
g(x(t−τ),u(t−τ)) =y(t−τ)

for anyfixedτ. Linear time invariant sets of state equations are the easiest to
manage analytically and numerically. Furthermore, the technique of lineariza-
tion of nonlinear systems is an important one, relying on the fact that if the
perturbationz(t)from some desired statex(t)is small, then a set of linear
equations inzcan be formed by neglecting all but thefirst terms in a Taylor
series expansion inz.
Alinear differential equationis one that can be written in the form

bn(x)y(n)+bn− 1 (x)y(n−1)+∑∑∑+b 1 (x)y^0 +b 0 (x)y=R(x) (2.23)

whereb 1 ,...,bn,andRare arbitrary functions ofx.WritingD for the dif-
ferentiation operatorDy=dy/dx, and letting a power ofDdenote repeated
differentiation, that is, using the notation

Dny=

dny

dxn

the left side of Equation 2.23 can be rewritten in the form

bn(x)Dny + bn− 1 (x)Dn−^1 y+∑∑∑+b 1 (x)Dy+b 0 (x)y

=

£

bn(x)Dn+bn− 1 (x)Dn−^1 +∑∑∑+b 1 (x)D+b 0 (x)

§

y

= p(D)y

Thus, the linear differential equation is of the form

p(D)y=R(x)

wherep(D)is a polynomial inDwith coefficientsbi(x). For such an equation,
the general solution has the form

y(x)=yh(x)+yp(x)

whereyh(x)is the homogeneous solution ñ that is,p(D)yh=0,andyp(x)is
a particular solution.
A linear system is modeled by linear differential equations. For a linear
system, the mathematical form of thestate modelis as follows:

x ̇(t)=Ax(t)+Bu(t) State equations

y(t)=Cx(t)+Du(t) Output equations

wherex(t)is then◊ 1 state vector;Ais ann◊nmatrix andBis ann◊k
matrix;u(t)is thek◊ 1 input vector;Cis anm◊nmatrix, andDis an
m◊kmatrix. Thus, for a linear system we have the power of linear algebra
and matrix theory at our disposal.