A First Course in FUZZY and NEURAL CONTROL

72 CHAPTER 2. MATHEMATICAL MODELS IN CONTROL

In standard international (SI) units, the armature constantKt,isequaltothe
motor constantKe. Therefore, if we letK=Kt=Ke,wecanwrite

T = Ki

e = Kθ ̇

LetJrepresent the inertia of the wheel andbthe damping constant. Also, letL
represent the inductance of the armature coil with resistanceR. With a voltage
V applied to the motor terminals as shown in Figure 2.39, we can write the
following coupled electromechanical equations based on Newtonís second law
and Kirchhoffís law:

J

d^2 θ

dt^2

+b

dθ

dt

= T

L

di

dt

+Ri+e = V

or

J

d^2 θ

dt^2

+b

dθ

dt

= Ki

L

di

dt

+Ri+K

dθ

dt

= V

Transfer function The coupled electromechanical equations form the basis
for obtaining the input-output relationship of the system. Using Laplace trans-
forms, and with zero initial conditions, the above equations can be expressed
as
Js^2 Θ(s)+bsΘ(s)=KI(s)

or
s(Js+b)Θ(s)=KI(s)

and
LsI(s)+RI(s)+KsΘ(s)=V(s)

or
(Ls+R)I(s)+KsΘ(s)=V(s)
Since our objective is to obtain a relationship between rotor angleΘ(s)and
the applied voltageV(s), we eliminate the currentI(s).Fromthefirst equation
we obtain

I(s)=

s(Js+b)

K

Θ(s)

Substituting this in the second equation and collecting terms, we get
μ
(Ls+R)

s(Js+b)

K

+Ks

∂

Θ(s)=V(s)

From this we obtain

Θ(s)

V(s)

=

1

h

(Ls+R)s(JsK+b)+Ks

i=

K

[LJs^3 +s^2 (RJ+bL)+s(bR+K^2 )]