## Description

1. Consider the discrete-time control system topology which is given in the Figure below. Let’s assume

that T = 0.5s.

+

− r(t) KP y1 ZOH ( t)

δ

T

y1

*(t)

δ

T

r*(t)

δ

T

+

KD

−

δ

T

y2

*(t)

δ y T 2 ( t)

(a) Let KD = 0 (i.e., no D action), then compute the pulese transfer functions Y1(z)

R(z)

and Y2(z)

R(z)

.

(b) For KD = 0, find a range of KP such that closed-loop system is stable (if possible).

(c) Let KD = 0 and KP = 2, then find a closed form expression for the step responses of the

closed-loop discrete system, i.e. y1[k] and y2[k] .

(d) For the same KP value, using MATLAB find and plot the step-responses of the closed loop discrete

system. Compare this result with the ones found in the previous part (e.g. you can plot both

results on the same figure).

(e) Now, you will analyze the general PD action. Let KP = 2, then compute the pulse transfer

functions Y1(z)

R(z)

and Y2(z)

R(z)

.

(f) For KP = 2, find a range for the KD values such that closed-loop system is stable.

(g) For KP = 2, choose a KD value that can can stabilize the system, then find a closed from

expression for the step responses of the closed-loop discrete system, i.e. y1[k] and y2[k].

(h) For the same (KP , KD) pair, using MATLAB find and plot the step-responses of the closed loop

discrete system. Compare these results with the ones found in the previous part, (g).

(i) Realize the block diagram topology in Simulink, and simulate the system with the same (KP , KD)

pair to find the unit-step responses of the closed-loop system. Compare this result with the ones

found in parts (g) and (h).

∗This document c M. Mert Ankarali

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