Question

1. A feedback control system is shown in the figure below. Suppose that our design objective is to find a controller Gc(S) of minimal complexity such that our closed-loop system can track a unit step input with a steady-state error of zero.

Answer 1

The controller is, Ge(s) = 0 The process transfer function is, G(s) - 17.00 Write the MATLAB code to plot the unit step response for proportional controller having a gain K. num1=10; den1=[1,10]; plant=tf(num1,den1); k=2; sys1=series(plant,k); %forward path transfer function% T=feedback(sys1,1); %Closed loop transfer function step(T) xlabel('Time(seconds)'),ylabel('y(t)'),grid title('step response for k=2')

Draw the step response. step response for k=2 0.7 0.76 0.6 y(t) O 0.05 0.1 0.15 0.2 Time(seconds) (seconds) 0.25 0.3

From the step response, it is observed that the final value of the output is, y()=0.67 Desired steady state value of the output is one. ºss = 1- y() Thus, the steady state error e is, =1-0.67 = 0.33 Thus, the steady state error to a step input, ess is 0.33.

(b) The controller is, Ge(s) = Ko + 5 =2+ 20 The process transfer function is, G(s) = 10. Write the MATLAB code to plot the unit step response for proportional controller having gains Ko and Kı. Ko=2; K1=20; num1=[Ko,K1]; den 1=[1,0]; Gc=tf(num1,den1); num2=[10]; den2=[1,10]; plant=tf(num2,den2); sys1=series(Gc,plant); %forward path transfer function% T=feedback(sys1,1); %Closed loop transfer function step(T) xlabel('Time(seconds)'),ylabel('y(t)'),grid title('step response for Pl controller')

Draw the step response. step response for PI controller 0.4......... 'o 0.1 0.3 0.2 Time(seconds) (seconds)

From the step response, it is observed that the final value of the output is, y()=1 Calculate the steady state error. ess = 1- y(0) =1-1 =0 Thus, the steady state error to a step input, essis O.

(c) From the plots of step response in part (a) and part (b), the steady state tracking error is 0.3 for proportional controller where as it is zero for PL controller. In terms of complexity, Pl controller tuning for desired response is difficult when compared to proportional controller.