Question

-2IH yes, what is K ? 2. (25pts)Given a continuous control system in the following Sgure, the plant G)ad s( +2) the controller C(a) the controlleCla) 41.7 +441) s + 18.4 41.7(s +4.41) s 18.4 s(x +2) a). Sketch the Bode magnitude plots of the plant G(), the controller C(o), and the controlled plant G(o)C() What kind of controller C(a) is ? For a sampling period T, obtained the discretized controller in Z - transform by applying the Generalized Bilinear Transformation and derive the Euler and Tustin discretization as special cases. b).

Answer 1

a) The Matlab code is given by

clear all
clc
s=tf('s');
G=4/(s*(s+2))
C=((417*s)/10 + 183897/1000)/(s + 92/5)
figure(1)
bode(G)
figure(2)
bode(C)
figure(3)
bode(G*C)

The command window will show

G =

      4
---------
s^2 + 2 s

Continuous-time transfer function.

C =

417 s + 1839
------------
   10 s + 184

Continuous-time transfer function.

Bode plot for plant G(s) is

Bode Diagram 20 0-20 -60 -90 )-135 -180 10 10 10 10 Frequency (rad/s)

Bode plot for controller C(s)

Bode Diagram 35 30 25 20- 40 O 30 20 10 10 10 10 10 10 Frequency (rad/s)

Bode plot for compensated plant C(s)*G(s)

Bode Diagram 50 0 -50 -100 -90 F 8-1 35 -180 10° 2 3 10 10 10 10 Frequency (rad/s)

b) The Matlab code is given by

clc
clear all
close all
syms s T z
c=41.7*(s+4.41)/(s+18.4);
subs(41.7*(s+4.41)/(s+18.4),s,(2*(z - 1))/(T*(z + 1))) %Bilinear
subs(41.7*(s+4.41)/(s+18.4),s,(z-1)/T) %Eular
subs(41.7*(s+4.41)/(s+18.4),s,((2/T)*(z-1))/(z+1))%Tustin

Command window will show the following answers

Bilenear C(s)=

((417*(2*z - 2))/(10*T*(z + 1)) + 183897/1000)/((2*z - 2)/(T*(z + 1)) + 92/5)

Eular C(s)=

((417*(z - 1))/(10*T) + 183897/1000)/((z - 1)/T + 92/5)

Tustin C(s)=

((417*(z - 1))/(5*T*(z + 1)) + 183897/1000)/((2*(z - 1))/(T*(z + 1)) + 92/5)

Again the same is done

2 굳-! as)-41구(s+4.41) 5 5 + 18 14 be nud to replace s wn uoith in meet la bん rt 군 41 (2+1) /83,877

imi larly tb Eulan て Ce) 4172+ 183. 847 て S- 2 (Z ) 183.397 5T (2+1) ( 군