## Description

1. Consider the discrete time control system block given in the Figure below.

+

− r(t) KP ZOH y ( t)

δ

T

y*(t)

δ

T

r*(t)

δ

T

+

KD

−

δ

T

Then answer/solve the following questions regarding this system representation.

(a) Let T = 0.5 s and KD = 0. Draw the root-locus of the closed-loop system with respect to P gain

KP ,

i. by hand following the procedures in the lecture notes (or textbook),

ii. as well as in MATLAB using the rlocus command.

Compare your hand solution and MATLAB output.

(b) Let T = 0.5 s and KD = 2. Draw the root-locus of the closed-loop system with respect to P gain

KP ,

i. by hand following the procedures in the lecture notes (or textbook),

ii. as well as in MATLAB using the rlocus command. In MATLAB version label important

points and associated gains.

Compare your hand solution and MATLAB output. In MATLAB version label important points

and associated gains.

(c) Let T = 0.5 s and KP = 2. Draw the root-locus of the closed-loop system with respect to D gain

KD. In this part you don’t need to draw by hand (but you can do it if you want to be sure).

However, root locus plot via MATLAB is mandatory.

(d) Referring to these root locus plots, select a (KP , KD) such that the closed-loop system is stable

and output does not show oscillatory behaviour.

∗This document c M. Mert Ankarali

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(e) Verify that your selected (KP , KD) pair results in a stable closed-loop system and output of the

system does not yield any oscillations. In order to verify this, perform a closed-loop step-response

simulation in Simulink or MATLAB.

(f) Based on your selected (KP , KD) gain estimate the settling time of the closed-loop hybrid (CTplant controlled with a DT-controller) system. Hint: You can use the mapping z = e

T s &

s = ln (z) /T to make a connection between z-domain poles and continious-time performance

specifications. Now estimate the settling time using the simulation that you performed for the

previous part. Compre both estimates.

2. Consider the modified version of the discrete time control system block which is given in the Figure

below. In this problem take T = 0.5 s.

+

− r(t) KP ZOH

δ

T

+

KD

−

δ

T

y(t) +

w[k]

+

n[k]

+ +

v[k]

+

+

In this block diagram representation y(t) (or y[k]) is the output, r(t) (or r[k]) is the reference signal,

where as w[k], v[k], n[k] are disturbances/noises that enters the system from different locations. Let’s

assume that we know the range of (KP , KD) values such that closed loop system is stable. For a given

(KP , KD) pair in this range answer/solve following questions.

(a) r(t) is a unit-step input and all other inputs to are equal to 0. Compute the steady-state error,

ess in terms of KP and KD.

(b) w[k] is a unit-step input and all other inputs are equal to 0. Compute the steady-state response,

yss in terms of KP and KD.

(c) v[k] is a unit-step input and all other inputs are equal to 0. Compute the steady-state response,

yss in terms of KP and KD.

(d) n[k] is a unit-step input and all other inputs are equal to 0. Compute the steady-state response,

yss in terms of KP and KD.

(e) Comment on the effects of KP and KD on the steady-state error and disturbance rejection performance.

(f) Now let (KP , KD) pair be equal to the one that you selected in Problem 1(d). Then “simulate”

(MATLAB or Simulink) all four cases (Parts (a)-(d)) separately, and compare your results found

in parts (a)-(b) and simulation results (for the selected (KP , KD) pair).

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