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M(s) = P/I = Kt/Js2 = 500/s2 [rad/A]
Amp
Ka = 4 [Amp/V]
DAC
Kd = 0.0003 [V/count]
Encoder
Kf = 4N/2π = 318 [count/rad]
ZOH
2000/(s+2000)
Digital Filter
KP = 12.5, KD = 245, T = 0.001
Therefore,
D(z) = 1030 (z-0.95)/Z
Accordingly, the coefficients of the continuous filter are:
P = 50
D = 0.98
The filter equation may be written in the continuous equivalent form:
G(s) = 50 + 0.98s = .098 (s+51)
The system elements are shown in Fig. 10.7.
V
Σ
FILTER
ZOH
DAC
AMP
MOTOR
50+0.980s
2000
S+2000
0.0003
4
500
S2
ENCODER
318
Figure 10.7 - Mathematical model of the control system
The open loop transfer function, A(s), is the product of all the elements in the loop.
A = 390,000 (s+51)/[s2(s+2000)]
To analyze the system stability, determine the crossover frequency, ωc at which A(j ωc) equals one.
This can be done by the Bode plot of A(j ωc), as shown in Fig. 10.8.
Chapter 10 Theory of Operation • 166
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