Page 228 - Modern Control Systems

202 Chapter 3 State Variable Models

when we select the third state variable as JC 3 = dB/dt. We now require a differential
equation describing the motor rotation. When L = 0, we have the field current
i = v 2/R and the motor torque T m = K mi. Therefore,

r Km

and the motor torque provides the torque to drive the belts plus the disturbance or
undesired load torque, so that

Tin = T + T d.
The torque T drives the shaft to the pulley, so that

2
d e dti_
T = J— I + b— + r(T l- T 2).
dr dt
Therefore,

2
_ d 6
dx 3
dt ~ dt ' 2
Hence,

dx 2 T m TA 2kr
• x 3 - * i ,
dt
where

K m dy
-
T m = r-«2, and v 2 = -k\k 2— = -k xk 2x 2.
K dt
Thus, we obtain

dxT, —K mkik 2 b 2kr
Xl
~di = ~~W~ " 7* 3 (3.106)

Equations (3.104)-(3.106) are the three first-order differential equations required to
describe this system. The matrix differential equation is

0 -1 r
0
2k (3.107)
0 0 0
x = m x +
-}_
-2kr -K mk\k 2 -b
J
JR