Consider the closed-loop control system shown in Fig. 2. Fig. 2. Discrete-time c

Question

Consider the closed-loop control system shown in Fig. 2. Fig. 2. Discrete-time control system. Suppose that the transfer function G(z) = 0.004683K(z + 0.9355)/(z - 1)(z - 0.8187) for a sampling period of T=0.1s. Design a phase lead digital controller GD (w) = 1 + tau w/1 + alpha tau w such that the maximum phase lead is 20 degree, the gain margin >5dB and the static velocity error constant Kv=5 sec-1. Find the gain K so that the static velocity error is satisfy Kv = Transform G(z) into G(w) by bilinear transformation and plot the bode diagram using Matlab. Design the phase lead compensator 4) Transform GD(w) into GD(z) by inverse bilinear transformation Assume that we have unit step input R(z) = 1/1 - z - 1 for Problem 5, 1, Determine the output in z plane C(z). Use Matlab to plot the unit step response of output C(k).

Explanation / Answer

All automatic control systems use -ve feedback for controlling a physical parameter like position, velocity, torque etc. The parameter which has to be controlled is sensed by a suitable transducers and fed back to the input, for comparison with the reference value. This subtraction of the sampled output signal with that of reference input is called as -ve feedback. The difference signal, called the "error" is then amplified to drive the system (referred to as actuation ) in such a manner that the output approaches the set reference value. In other words the system is designed to minimize the error signal. All practical loads have inertia and spring constants due to which there is a delay in actuation. Hence, even though a system may be designed for -ve feedback, due to inherent time lags, the feedback may turn into +ve feedback at certain frequencies. If the loop gain is more than unity at some frequency at which the feedback is +ve, the system will oscillate. Hence, in designing control systems great care has to be taken to avoid such situations.

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