## Model Predictive Control Toolbox |

This example shows how to inspect the optimized sequence of manipulated variables computed by model predictive controller at each sample time.

The plant is a double integrator subject to input saturation.

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The basic setup of the MPC controller includes:

A double integrator as the prediction model

Prediction horizon of 20

Control horizon of 10

Input constraints

`-1 <= u(t) <= 1`

Define the MPC controller setup:

Ts = 0.1; % Sampling time p = 20; % Prediction horizon m = 10; % Control horizon mpcobj = mpc(tf(1,[1 0 0]),Ts,p,m); % MPC object mpcobj.MV = struct('Min',-1,'Max',1); % Input saturation constraints nu=1; % Number of manipulated variables

-->The "Weights.ManipulatedVariables" property of "mpc" object is empty. Assuming default 0.00000. -->The "Weights.ManipulatedVariablesRate" property of "mpc" object is empty. Assuming default 0.10000. -->The "Weights.OutputVariables" property of "mpc" object is empty. Assuming default 1.00000.

To run this example, Simulink® is required.

if ~mpcchecktoolboxinstalled('simulink') disp('Simulink(R) is required to run this example.') return end

Open the Simulink® model. You see an outport `mv.seq` in the MPC Controller block. Selecting **Enable control sequence outport** option in the MPC Controller Block Parameters dialog box creates this new outport. The outport exports the optimal control sequence computed by the controller at each sample time to MATLAB Workspace. The logged signal has size of [p, Nu, samples] where p is prediction horizon and Nu is the number of manipulated variables.

mdl = 'mpc_sequence'; open_system(mdl); % Open Simulink(R) Model sim(mdl); % Start simulation

-->Converting the "Model.Plant" property of "mpc" object to state-space. -->Converting model to discrete time. -->The "Model.Noise" property of the "mpc" object is empty. Assuming white noise on each measured output channel.

**Analyze Optimal Control Sequences**

Plot the optimal control sequence at specific time instants:

times = [0 0.2 1 2 2.1 2.2 3 3.5 5]; figure('Name','Optimal sequence history'); for t = 1:9 ct = times(t)*10+1; subplot(3,3,t); h = stairs(0:p-1,useq.signals.values(ct,:)); h.LineWidth = 1.5; hold on plot((0:p-1)+.5,useq.signals.values(ct,:),'*r'); xlabel('prediction step') ylabel('u') title(sprintf('Simulation time t=%5.1f',useq.time(ct))); grid axis([0 p -1.1 1.1]); hold off end

From the plot, you see that the MPC controller uses the first two seconds to bring the output very close to the new set point. The controller output is at the high limit (+1) for one second and then switched to the low limit (-1) for the next second, which is the best control strategy under the input constraint limits.

bdclose(mdl);