fminimaxwith 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
0 and 100.
The code for this example, shown below, is contained in the
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
20 seconds, the constraint function
yout >= 0.95 from
Because constraints must be in the form
g ≤ 0,
the constraint in the function is
g = -yout(20:100)+.95.
calculated from the current PID values. To avoid calling the simulation
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
The following is the code for
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 opt = simset('solver','ode5','SrcWorkspace','Current'); options = optimset('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); [~,~,yout] = sim('optsim',[0 100],opt); pold = pid; end end end
Copy the code for
runtrackmm to a file named
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 0.9996 0.00397 1 -0.0195 Hessian modified twice 4 29 0.9996 3.464e-05 1 0.000687 Hessian modified 5 35 0.9996 2.273e-09 1 -0.0175 Hessian modified twice Local minimum possible. Constraints satisfied. fminimax stopped because the predicted change in the objective function is less than the selected value of the function tolerance and constraints are satisfied to within the default value of the constraint tolerance. Kp = 0.5894 Ki = 0.0605 Kd = 5.5295
The last value in the
Objective value column
of the output shows that the maximum value for all the time steps
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