The transfer function 20/(s+1) is your controller. As it is, it does not give you a 50-degree phase margin. You need to modify this controller to achieve the desired phase margin. There are couple ways of doing this:
- Either reduce the gain of the controller to achieve a larger phase margin, but at a lower frequency.
- Or add additional phase using controller terms such as lead-lags.
% Initial compensator
C = zpk([], -1, 20);
% The plant
G = zpk(-20, [-2 -5], 1);
% Phase and gain margin of the open-loop transfer function with initial
% compensator
L = C*G;
margin(L)
% Modify the compensator C to obtain the desired phase margin (this part is
% up to you, since this seems like a homework question).
C = zpk([], -2, 10) % This is my trivial change, will not give 50 degrees phase margin.
L = C*G;
margin(L)
% Bode plot of the compentator
bode(C)
% Closed-loop system
T = L / (1 + L);
bode(T)
step(T)
