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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"><html xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd"> <head> <meta http-equiv="Content-Type" content="text/html; charset=utf-8"> <!--This HTML is auto-generated from an M-file.To make changes, update the M-file and republish this document. --> <title>Learning PID Tuning III: Performance Index Optimization</title> <meta name="generator" content="MATLAB 7.5"> <meta name="date" content="2008-02-08"> <meta name="m-file" content="optimalpidtuning"><style>body { background-color: white; margin:10px;}h1 { color: #990000; font-size: x-large;}h2 { color: #990000; font-size: medium;}/* Make the text shrink to fit narrow windows, but not stretch too far in wide windows. */ p,h1,h2,div.content div { max-width: 600px; /* Hack for IE6 */ width: auto !important; width: 600px;}pre.codeinput { background: #EEEEEE; padding: 10px;}@media print { pre.codeinput {word-wrap:break-word; width:100%;}} span.keyword {color: #0000FF}span.comment {color: #228B22}span.string {color: #A020F0}span.untermstring {color: #B20000}span.syscmd {color: #B28C00}pre.codeoutput { color: #666666; padding: 10px;}pre.error { color: red;}p.footer { text-align: right; font-size: xx-small; font-weight: lighter; font-style: italic; color: gray;} </style></head> <body> <div class="content"> <h1>Learning PID Tuning III: Performance Index Optimization</h1> <introduction> <p>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. </p> </introduction> <h2>Contents</h2> <div> <ul> <li><a href="#1">Performance Indices</a></li> <li><a href="#2">Initial Controller Parameters</a></li> <li><a href="#3">Example 1:</a></li> <li><a href="#4">Example 2:</a></li> </ul> </div> <h2>Performance Indices<a name="1"></a></h2> <p>The performance indices used here are: 1. The Integral of Squared Error (ISE)</p> <p><img vspace="5" hspace="5" src="optimalpidtuning_eq15604.png"> </p> <p>2. The Integral of Absolute Error (IAE)</p> <p><img vspace="5" hspace="5" src="optimalpidtuning_eq15709.png"> </p> <p>3. The Integral of Time Multiply Squared Error (ITSE)</p> <p><img vspace="5" hspace="5" src="optimalpidtuning_eq16912.png"> </p> <p>4. The Integral of Time multiply Absolute Error (ITAE)</p> <p><img vspace="5" hspace="5" src="optimalpidtuning_eq17017.png"> </p> <h2>Initial Controller Parameters<a name="2"></a></h2> <p>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. <a href="http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=18561&objectType=FILE">The stability margin based Ziegler-Nichols</a> is used for initial controller parameters and for performance comparison. </p> <h2>Example 1:<a name="3"></a></h2> <p>Consider a 4th-order system:</p> <p><img vspace="5" hspace="5" src="optimalpidtuning_eq14275.png"> </p> <p>Compare closed-loop performance of PID controllers designed with different performance indices.</p><pre class="codeinput">G=zpk([],[-3 -2 -1 0],1); <span class="comment">% The plant</span>C1=optimPID(G,3,1); <span class="comment">% PID-Control, ISE index</span>C2=optimPID(G,3,2); <span class="comment">% PID-Control, IAE index</span>C3=optimPID(G,3,3); <span class="comment">% PID-Control, ITSE index</span>C4=optimPID(G,3,4); <span class="comment">% PID-Control, ITAE index</span>K=znpidtuning(G,3); <span class="comment">% Ziegler-Nichols stability margin tuning</span>t=0:0.1:30;y1=step(feedback(C1*G,1),t); <span class="comment">%Closed-loop step response of C1</span>y2=step(feedback(C2*G,1),t); <span class="comment">%Closed-loop step response of C2</span>y3=step(feedback(C3*G,1),t); <span class="comment">%Closed-loop step response of C3</span>y4=step(feedback(C4*G,1),t); <span class="comment">%Closed-loop step response of C4</span><span class="comment">%Closed-loop step response of K</span>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,<span class="string">'--'</span>,<span class="string">'Linewidth'</span>,2)legend(<span class="string">'ISE'</span>,<span class="string">'IAE'</span>,<span class="string">'ITSE'</span>,<span class="string">'ITAE'</span>,<span class="string">'Z-N'</span>,<span class="string">'Location'</span>,<span class="string">'Best'</span>)grid<span class="comment">% The comparison shows that the ITSE index leads to the best PID</span><span class="comment">% controller.</span></pre><img vspace="5" hspace="5" src="optimalpidtuning_01.png"> <h2>Example 2:<a name="4"></a></h2> <p>A 4th-order system with a repeated pole.</p> <p><img vspace="5" hspace="5" src="optimalpidtuning_eq8507.png"> </p> <p>Compare closed-loop performance of PI controllers.</p><pre class="codeinput">G=tf(1,[1 4 6 4 1]); <span class="comment">% The plant</span>C1=optimPID(G,2,1); <span class="comment">% PID-Control, ISE index</span>C2=optimPID(G,2,2); <span class="comment">% PID-Control, IAE index</span>C3=optimPID(G,2,3); <span class="comment">% PID-Control, ITSE index</span>C4=optimPID(G,2,4); <span class="comment">% PID-Control, ITAE index</span>K=znpidtuning(G,2); <span class="comment">% Ziegler-Nichols stability margin tuning</span>t=0:0.1:40;y1=step(feedback(C1*G,1),t); <span class="comment">%Closed-loop step response of C1</span>y2=step(feedback(C2*G,1),t); <span class="comment">%Closed-loop step response of C2</span>y3=step(feedback(C3*G,1),t); <span class="comment">%Closed-loop step response of C3</span>y4=step(feedback(C4*G,1),t); <span class="comment">%Closed-loop step response of C4</span><span class="comment">%Closed-loop step response of K</span>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,<span class="string">'--'</span>,<span class="string">'Linewidth'</span>,2)legend(<span class="string">'ISE'</span>,<span class="string">'IAE'</span>,<span class="string">'ITSE'</span>,<span class="string">'ITAE'</span>,<span class="string">'Z-N'</span>,<span class="string">'Location'</span>,<span class="string">'Best'</span>)grid<span class="comment">% This time the ITAE index gives the best design.</span></pre><img vspace="5" hspace="5" src="optimalpidtuning_02.png"> <p class="footer"><br> Published with MATLAB® 7.5<br></p> </div> <!--##### SOURCE BEGIN #####%% 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,'REPLACE_WITH_DASH_DASH','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,'REPLACE_WITH_DASH_DASH','Linewidth',2)
legend('ISE','IAE','ITSE','ITAE','Z-N','Location','Best')
grid
% This time the ITAE index gives the best design.
##### SOURCE END #####--> </body></html>⌨️ 快捷键说明
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