optimalpidtuning.m

optailpiduse of the model reference adaptive control method! Lypanuov stability theory used to desig

%% Learning PID Tuning III: Performance Index Optimization
% Most PID tuning rules are based on first-order plus time delay
% assumption of the plant hence cannot ensure the best control performance.
% Using mordern optimization techniques, it is possible to tune a PID
% controller based on the actual transfer function of the plant to optimize
% the closed-loop performance.

%% Performance Indices
% The performance indices used here are:
% 1. The Integral of Squared Error (ISE)
%
% $$I_1 = \int_0^\infty e^2(t) dt$$
%
% 2. The Integral of Absolute Error (IAE)
%
% $$I_2 = \int_0^\infty |e(t)| dt$$
%
% 3. The Integral of Time Multiply Squared Error (ITSE)
%
% $$I_3 = \int_0^\infty te^2(t) dt$$
%
% 4. The Integral of Time multiply Absolute Error (ITAE)
%
% $$I_4 = \int_0^\infty t|e(t)| dt$$
%

%% Initial Controller Parameters
% Like mose optimization problems, the control performance optimization is
% non-convext, hence a trap of local minimum is inevitable. To counteract
% this, the initial controller parameters is set to be those determined by
% one of existing tuning rules. In this way, the controller derived is at
% least better than that determined by the tuning method.
% <http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=18561&objectType=FILE The stability margin based Ziegler-Nichols> 
% is used for initial controller parameters and for performance comparison.

%% Example 1: 
% Consider a 4th-order system:
%
% $$G=\frac{1}{s(s+1)(s+2)(s+3)}$$
%
% Compare closed-loop performance of PID controllers designed with
% different performance indices. 
G=zpk([],[-3 -2 -1 0],1);   % The plant
C1=optimPID(G,3,1);   % PID-Control, ISE index
C2=optimPID(G,3,2);   % PID-Control, IAE index
C3=optimPID(G,3,3);   % PID-Control, ITSE index
C4=optimPID(G,3,4);   % PID-Control, ITAE index
K=znpidtuning(G,3);   % Ziegler-Nichols stability margin tuning
t=0:0.1:30;
y1=step(feedback(C1*G,1),t); %Closed-loop step response of C1
y2=step(feedback(C2*G,1),t); %Closed-loop step response of C2
y3=step(feedback(C3*G,1),t); %Closed-loop step response of C3
y4=step(feedback(C4*G,1),t); %Closed-loop step response of C4
%Closed-loop step response of K
y=step(feedback(G*(K.kc*(1+tf(1,[K.ti 0])+tf([K.td 0],1))),1),t);
plot(t,y1,t,y2,t,y3,t,y4,t,y,'--','Linewidth',2)
legend('ISE','IAE','ITSE','ITAE','Z-N','Location','Best')
grid

% The comparison shows that the ITSE index leads to the best PID
% controller.

%% Example 2:
% A 4th-order system with a repeated pole.
%
% $$G(s)=\frac{1}{(s+1)^4}$$
%
% Compare closed-loop performance of PI controllers.
G=tf(1,[1 4 6 4 1]);   % The plant
C1=optimPID(G,2,1);   % PID-Control, ISE index
C2=optimPID(G,2,2);   % PID-Control, IAE index
C3=optimPID(G,2,3);   % PID-Control, ITSE index
C4=optimPID(G,2,4);   % PID-Control, ITAE index
K=znpidtuning(G,2);   % Ziegler-Nichols stability margin tuning
t=0:0.1:40;
y1=step(feedback(C1*G,1),t); %Closed-loop step response of C1
y2=step(feedback(C2*G,1),t); %Closed-loop step response of C2
y3=step(feedback(C3*G,1),t); %Closed-loop step response of C3
y4=step(feedback(C4*G,1),t); %Closed-loop step response of C4
%Closed-loop step response of K
y=step(feedback(G*(K.kc*(1+tf(1,[K.ti 0]))),1),t);
plot(t,y1,t,y2,t,y3,t,y4,t,y,'--','Linewidth',2)
legend('ISE','IAE','ITSE','ITAE','Z-N','Location','Best')
grid

% This time the ITAE index gives the best design.