Solve a Finite Set MPC Problem in MATLAB

This example shows how to solve, in MATLAB®, an MPC problem in which some manipulated variables belong to a finite (discrete) set.

Create a Plant Model

Fix the random generator seed for reproducibility.

rng(0);

Create a discrete-time strictly proper plant with 4 states, two inputs and one output.

plant = drss(4,1,2);
plant.D = 0;

Set the sampling time to 0.1s, and increase the control authority of the first input, to better illustrate its control contribution.

plant.Ts = 0.1;
plant.B(:,1)=plant.B(:,1)*2;

Design the MPC Controller

Create an MPC controller with one second sampling time, 20 steps prediction horizon and 5 steps control horizon.

mpcobj = mpc(plant,0.1,20,5);

-->"Weights.ManipulatedVariables" is empty. Assuming default 0.00000.
-->"Weights.ManipulatedVariablesRate" is empty. Assuming default 0.10000.
-->"Weights.OutputVariables" is empty. Assuming default 1.00000.

Specify the first manipulated variable as belonging to a set of seven possible values (you could also specify the type as an integer using the instruction mpcobj.MV(1).Type = 'integer';)

mpcobj.MV(1).Type = [-1 -0.7 -0.3 0 0.2 0.5 1];

Use rate limits to enforce maximum increment and decrement values for the first manipulated variable.

mpcobj.MV(1).RateMin = -0.5;
mpcobj.MV(1).RateMax = 0.5;

Set limits on the second manipulated variable, whose default type (continuous) has not been changed.

mpcobj.MV(2).Min = -2;
mpcobj.MV(2).Max = 2;

Simulate the Closed Loop Using the sim Command and Plot Results

Set the number of simulation steps.

simsteps = 50;

Create an output reference signal equal to zero from steps 20 to 35 and equal to 0.6 before and after.

r = ones(simsteps,1)*0.6;
r(20:35) = 0;

Simulate the closed loop using the sim command. Return the plant input and output signals.

[YY,~,UU,~,~,~,status] = sim(mpcobj,simsteps,r);

-->Assuming output disturbance added to measured output #1 is integrated white noise.
-->"Model.Noise" is empty. Assuming white noise on each measured output.

Plot results.

figure(1)

subplot(211)    % plant output
plot([YY,r]);
grid
title("Tracking control");

subplot(223)    % first plant input
stairs(UU(:,1));
grid
title("MV(1) finite set ")

subplot(224)    % second plant input
stairs(UU(:,2));
grid
title("MV(2) continuous between [-2 2]")

As expected, the first manipulated variable is restricted to the values specified in the finite set (with jumps less than the specified limit), while the second one can vary continuously between -2 and 2. The plant output tracks the reference value after a few seconds.

Simulate the Closed Loop Using the mpcmove Command and Plot Results

Get handle to mpcobj state and initialize plant state.

xmpc = mpcstate(mpcobj);
x = xmpc.Plant;

Initialize arrays that store signals.

YY = []; RR = []; UU = []; XX = [];

Perform simulation using the mpcmove command to calculate the control actions.

for k = 1:simsteps
    XX = [XX;x']; % store plant state
    y = plant.C*x; % calculate plant output
    YY = [YY;y]; % store plant output
    RR = [RR;r(k)]; % store reference
    u = mpcmove(mpcobj,xmpc,y,r(k)); % calculate optimal mpc move
    UU = [UU;u']; % store plant input
    x = plant.A*x+plant.B*u; % update plant state
    % is the last line necessary since x=xmpc.Plant gets updated anyway?
end

Plot results.

figure(2)

subplot(211)    % plant output
plot([YY,r]);
grid
title("Tracking control");

subplot(223)    % first plant input
stairs(UU(:,1));
grid
title("MV(1) finite set")

subplot(224)    % second plant input
stairs(UU(:,2));
grid
title("MV(2) continuous between [-2 2]")

The simulation results are identical as the ones achieved using the sim command.