EEL3657.asv

%Setup a pid controller

%CL RESPONSE     RISE TIME       OVERSHOOT  SETTLING TIME  S-S ERROR
%Kp              Decrease        Increase   Small Change   Decrease
%Ki              Decrease        Increase   Increase       Eliminate
%Kd              Small Change    Decrease   Decrease       No Change

Kp = 1;
Kd = 1;
Ki = 1;
s=tf('s');
GH = (0.2*s +3.2)/((s+1)*(s+.8));

%%
% solve PID control
z=-log(.10)/sqrt(pi^2+log(.10)^2);
wn=4/(8*z);
N=[0 0 wn^2];
D=[1 2*wn*z wn^2];
p= roots(D);

S1 = p(1);
syms kd kp x
GHs = (0.2*S1 +3.2)/((S1+1)*(S1+.8));
%1+P*GHs=0, P = Kp + Kd*s +Ki/s
%Kp + Kd*S1 = -1/GHs-Ki/(S1)
% Ki = .1 small
% use imaginary part to solve Kd
Kd = sym2poly(solve(imag(S1)*kd==imag(-1/GHs-Ki/(S1)),kd));
%use real part to solve Kp
Kp = sym2poly(solve(kp+real(S1)*Kd==real(-1/GHs-Ki/(S1)),kp));
P = Kp+Kd*s+Ki/s;


%%
% find Kv
syms b;
Px = Kp + Ki/x + Kd*x;           %pid 
Gh = (0.2*x +3.2)/((x+1)*(x+.8));  % plant
c = (x+1/Tg)/(x+1/(b*Tg));   % lag
kv = limit((x*c*Px*Gh),0);
% set Beta to that value
B = double(solve(kv==5,b));

%setup a lag compensator
Kc = 1;  % choose Kc = 1
Tg = 0.05;
%B = 1.25; % 1.25;


Gc = (s+1/Tg)/(s+1/(B*Tg)); % = 1 initially (lag)



Px = Kp + Ki/x + Kd*x;           %pid
Cg = (x+1/Tg)/(x+1/(B*Tg));     

c = (x+1/Tg)/(x+1/(b*Tg));   % lag
limit((x*c*Px*Gh),0)
Kv = limit((x * Cg * Cl * Px * Gh),x,0);
fprintf('The value of Kv is %s\n',char(Kv));

% show root locus
figure() 
rlocus(P*Gc*GH)
sgrid
figure()
sys = feedback(P * Gc * GH,1);

% show unit step plot
step(sys)
t = 0:.01:100;
figure()
% show unit ramp plot
lsim(sys,t,t)
title('Response to Unit Ramp Input')
sse = abs(1-dcgain(sys));
fprintf('The sse is %f\n',sse);

minreal(tf(P * Gc * GH,1))