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Another approach to optimizing the control parameters in the Simulink^{®} model shown in Plant with Actuator Saturation is to use the fminimax function. In this case, rather than minimizing the error between the output and the input signal, you minimize the maximum value of the output at any time t between 0 and 100.
The code for this example, shown below, is contained in the function runtrackmm, in which the objective function is simply the output yout returned by the sim command. But minimizing the maximum output at all time steps might force the output to be far below unity for some time steps. To keep the output above 0.95 after the first 20 seconds, the constraint function trackmmcon contains the constraint yout >= 0.95 from t=20 to t=100. Because constraints must be in the form g ≤ 0, the constraint in the function is g = -yout(20:100)+.95.
Both trackmmobj and trackmmcon use the result yout from sim, calculated from the current PID values. To avoid calling the simulation twice, runtrackmm has nested functions so that the value of yout is shared between the objective and constraint functions. The simulation is called only when the current point changes.
The following is the code for runtrackmm:
function [Kp, Ki, Kd] = runtrackmm optsim % initialize Simulink(R) pid0 = [0.63 0.0504 1.9688]; % a1, a2, yout are shared with TRACKMMOBJ and TRACKMMCON a1 = 3; a2 = 43; % Initialize plant variables in model yout = []; % Give yout an initial value pold = []; % tracks last pid options = optimoptions('fminimax','Display','iter',... 'TolX',0.001,'TolFun',0.001); pid = fminimax(@trackmmobj,pid0,[],[],[],[],[],[],... @trackmmcon,options); Kp = pid(1); Ki = pid(2); Kd = pid(3); function F = trackmmobj(pid) % Track the output of optsim to a signal of 1. % Variables a1 and a2 are shared with RUNTRACKMM. % Variable yout is shared with RUNTRACKMM and % RUNTRACKMMCON. updateIfNeeded(pid) F = yout; end function [c,ceq] = trackmmcon(pid) % Track the output of optsim to a signal of 1. % Variable yout is shared with RUNTRACKMM and % TRACKMMOBJ updateIfNeeded(pid) c = -yout(20:100)+.95; ceq=[]; end function updateIfNeeded(pid) if ~isequal(pid,pold) % compute only if needed Kp = pid(1); Ki = pid(2); Kd = pid(3); myobj = sim('optsim','SrcWorkspace','Current'); yout = myobj.get('yout'); pold = pid; end end end
Copy the code for runtrackmm to a file named runtrackmm.m, placed in a folder on your MATLAB^{®} path.
When you run the code, it returns the following results:
[Kp,Ki,Kd] = runtrackmm Done initializing optsim. Objective Max Line search Directional Iter F-count value constraint steplength derivative Procedure 0 5 0 1.11982 1 11 1.184 0.07978 1 0.482 2 17 1.012 0.04285 1 -0.236 3 23 0.9995 0.007058 1 -0.0186 Hessian modified twice 4 29 0.9997 9.705e-07 1 0.00716 Hessian modified Local minimum possible. Constraints satisfied. fminimax stopped because the size of the current search direction is less than twice the selected value of the step size tolerance and constraints are satisfied to within the default value of the constraint tolerance. Kp = 0.5910 Ki = 0.0606 Kd = 5.5383
The last value in the Objective value column of the output shows that the maximum value for all the time steps is 0.9997. The closed loop response with this result is shown in the figure Closed-Loop Response Using fminimax.
This solution differs from the solution obtained in lsqnonlin with a Simulink Model because you are solving different problem formulations.