Using fminimax with a Simulink Model

This example shows how to tune the parameters of a Simulink® model. The model, optsim, is included when you run this example. The model includes a nonlinear process plant modeled as a Simulink block diagram.

Plant with Actuator Saturation

The plant is an under-damped third-order model with actuator limits. The actuator limits are a saturation limit and a slew rate limit. The actuator saturation limit cuts off input values greater than 2 units or less than –2 units. The slew rate limit of the actuator is 0.8 units/sec. The closed-loop response of the system to a step input is shown in Closed-Loop Response. You can see this response by opening the model (type optsim at the command line or click the model name), and selecting Run from the Simulation menu. The response plots to the scope.

Closed-Loop Response

The problem is to design a feedback control loop that tracks a unit step input to the system. The closed-loop plant is entered in terms of the blocks where the plant and actuator are located in a hierarchical Subsystem block. A Scope block displays output trajectories during the design process.

Closed-Loop Model

To optimize this system, minimize the maximum value of the output at any time $$t$$ between 0 and 100. (In contrast, in the example lsqnonlin with a Simulink Model, the solution involves minimizing the error between the output and the input signal.)

The code for this example is contained in the helper function runtrackmm at the end of this example. 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 $$g\le 0$$, the constraint in the function is g = -yout(20:100) + 0.95.

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.

Call runtrackmm.

[Kp,Ki,Kd] = runtrackmm

                  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     2.274e-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 value of the function tolerance and constraints 
are satisfied to within the 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 is just under 1. 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

Helper Function

The following code creates the runtrackmm helper function.

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