Using fminimax with a Simulink Model
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
endCopy 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 =
5.5383The 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.
Closed-Loop Response Using fminimax
