A First Course in FUZZY and NEURAL CONTROL

80 CHAPTER 2. MATHEMATICAL MODELS IN CONTROL

Then Equation (2.46) gives
x 1 (t)=−t+1 (2.49)

Therefore, the nominal trajectory for which Equations (2.46) and (2.47) are to
be linearized is described by

x 01 (t)=−t+1 (2.50)

x 02 (t)=1 (2.51)

Now evaluating the coefficients of Equation (2.45), we get

∂f 1 (t)

∂x 1 (t)=0

∂f 1 (t)

∂x 2 (t)=

2

x^32 (t)

∂f 1 (t)

∂u(t)=0

∂f 2 (t)

∂x 1 (t)=u(t)

∂f 2 (t)

∂x 2 (t)=0

∂f 2 (t)

∂u(t)=x^1 (t)

Using Equation (2.43), we get

4 x ̇ 1 (t)=

2

x^302 (t)

4 x 2 (t) (2.52)

4 x ̇ 2 (t)=u 0 (t) 4 x 1 (t)+x 01 (t) 4 u(t) (2.53)

Substituting Equations (2.50) and (2.51) into (2.52) and (2.53), we get

∑

4 x ̇ 1 (t)

4 x ̇ 2 (t)

∏

=

∑

02

00

∏∑

4 x 1 (t)

4 x 2 (t)

∏

+

∑

0

1 −t

∏

4 u(t) (2.54)

The set of Equations (2.54) represents linear state equations with time-varying
coefficients.

2.10Exercisesandprojects

1. Show that the following systems are either linear or nonlinear, and either
   time-varying or time-invariant.

(a)y(t)=v(t)dtd(v(t))

(b)

v(t)

−→ N 1

q(t)

−→ N 2

y(t)

−→whereN 1 andN 2 are linear systems connected

in cascade.

2. Prove or disprove the following statements.

(a) In a linear system, if the response tov(t)isy(t), then the response

toRe(v(t))isRe(y(t))whereRe(x)denotes the real part of the

complex numberx.

(b) In a linear system, if the response tov(t)isy(t), then the response

todtd(v(t))isdtd(y(t)).