# Time-Varying MPC Control of a Time-Varying Plant

This example shows how the Model Predictive Control Toolbox™ can use time-varying prediction models to achieve better performance when controlling a time-varying plant.

The following MPC controllers are compared:

Linear MPC controller based on a time-invariant average model

Linear MPC controller based on a time-invariant model, which is updated at each time step

Linear MPC controller based on a time-varying prediction model.

### Time-Varying Linear Plant

In this example, the plant is a single-input-single-output 3rd order time-varying linear system with poles, zeros and gain that vary periodically with time.

The plant poles move between being stable and unstable at run time, which leads to a challenging control problem.

Generate an array of plant models at `t`

= `0`

, `0.1`

, `0.2`

, ..., 10 seconds.

Models = tf; ct = 1; for t = 0:0.1:10 Models(:,:,ct) = tf([5 5+2*cos(2.5*t)],[1 3 2 6+sin(5*t)]); ct = ct + 1; end

Convert the models to state-space format and discretize them with a sample time of 0.1 second.

Ts = 0.1; Models = ss(c2d(Models,Ts));

### MPC Controller Design

The control objective is to track a step change in the reference signal. First, design an MPC controller for the average plant model. The controller sample time is 0.1 second.

sys = ss(c2d(tf([5 5],[1 3 2 6]),Ts)); % prediction model p = 3; % prediction horizon m = 3; % control horizon mpcobj = mpc(sys,Ts,p,m);

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

Set hard constraints on the manipulated variable and specify tuning weights.

mpcobj.MV = struct('Min',-2,'Max',2); mpcobj.Weights = struct('MV',0,'MVRate',0.01,'Output',1);

Set the initial plant states to zero.

x0 = zeros(size(sys.B));

### Closed-Loop Simulation with Implicit MPC

Run a closed-loop simulation to examine whether the designed implicit MPC controller can achieve the control objective without updating the plant model used in prediction.

Set the simulation duration to 5 seconds.

Tstop = 5;

Use the `mpcmove`

command in a loop to simulate the closed-loop response.

yyMPC = []; uuMPC = []; x = x0; xmpc = mpcstate(mpcobj); fprintf('Simulating MPC controller based on average LTI model.\n'); for ct = 1:(Tstop/Ts+1) % Get the real plant. real_plant = Models(:,:,ct); % Update and store the plant output. y = real_plant.C*x; yyMPC = [yyMPC,y]; % Compute and store the MPC optimal move. u = mpcmove(mpcobj,xmpc,y,1); uuMPC = [uuMPC,u]; % Update the plant state. x = real_plant.A*x + real_plant.B*u; end

-->Assuming output disturbance added to measured output #1 is integrated white noise. -->"Model.Noise" is empty. Assuming white noise on each measured output. Simulating MPC controller based on average LTI model.

### Closed-Loop Simulation with Adaptive MPC

Run a second simulation to examine whether an adaptive MPC controller can achieve the control objective.

Use the `mpcmoveAdaptive`

command in a loop to simulate the closed-loop response. Update the plant model for each control interval, and use the updated model to compute the optimal control moves. The `mpcmoveAdaptive`

command uses the same prediction model across the prediction horizon.

yyAMPC = []; uuAMPC = []; x = x0; xmpc = mpcstate(mpcobj); nominal = mpcobj.Model.Nominal; fprintf(['Simulating MPC controller based on LTI model,' ... ' updated at each time step t.\n']); for ct = 1:(Tstop/Ts+1) % Get the real plant. real_plant = Models(:,:,ct); % Update and store the plant output. y = real_plant.C*x; yyAMPC = [yyAMPC, y]; % Compute and store the MPC optimal move. u = mpcmoveAdaptive(mpcobj,xmpc,real_plant,nominal,y,1); uuAMPC = [uuAMPC,u]; % Update the plant state. x = real_plant.A*x + real_plant.B*u; end

Simulating MPC controller based on LTI model, updated at each time step t.

### Closed-Loop Simulation with Time-Varying MPC

Run a third simulation to examine whether a time-varying MPC controller can achieve the control objective.

The controller updates the prediction model at each control interval and also uses time-varying models across the prediction horizon, which gives MPC controller the best knowledge of plant behavior in the future.

Use the `mpcmoveAdaptive`

command in a loop to simulate the closed-loop response. Specify an array of plant models rather than a single model. The controller uses each model in the array at a different prediction horizon step.

yyLTVMPC = []; uuLTVMPC = []; x = x0; xmpc = mpcstate(mpcobj); % Nominal conditions are constant over the prediction horizon. Nominals = repmat(nominal,3,1); fprintf(['Simulating MPC controller based on time-varying model, ' ... 'updated at each time step t.\n']); for ct = 1:(Tstop/Ts+1) % Get the real plant. real_plant = Models(:,:,ct); % Update and store the plant output. y = real_plant.C*x; yyLTVMPC = [yyLTVMPC, y]; % Compute and store the MPC optimal move. u = mpcmoveAdaptive(mpcobj,xmpc,Models(:,:,ct:ct+p),Nominals,y,1); uuLTVMPC = [uuLTVMPC,u]; % Update the plant state. x = real_plant.A*x + real_plant.B*u; end

Simulating MPC controller based on time-varying model, updated at each time step t.

### Performance Comparison of MPC Controllers

Compare the closed-loop responses.

t = 0:Ts:Tstop; figure subplot(2,1,1); plot(t,yyMPC,'-.',t,yyAMPC,'--',t,yyLTVMPC); grid legend('Implicit MPC','Adaptive MPC','Time-Varying MPC', ... Location="SouthEast") title('Plant Output'); subplot(2,1,2) plot(t,uuMPC,'-.',t,uuAMPC,'--',t,uuLTVMPC) grid title('Control Moves');

Only the time-varying MPC controller is able to bring the plant output close enough to the desired setpoint.

### Closed-Loop Simulation of Time-Varying MPC in Simulink

To simulate time-varying MPC control in Simulink®, pass the time-varying plant models to `model`

inport of the Adaptive MPC Controller block.

```
xmpc = mpcstate(mpcobj);
mdl = 'mpc_timevarying';
open_system(mdl);
```

Run the simulation.

```
sim(mdl,Tstop);
fprintf('Simulating MPC controller based on LTV model in Simulink.\n');
```

Simulating MPC controller based on LTV model in Simulink.

Plot the MATLAB® and Simulink time-varying simulation results.

figure subplot(2,1,1) plot(t,yyLTVMPC,t,ysim,'o'); grid legend('mpcmoveAdaptive','Simulink','Location','SouthEast') title('Plant Output'); subplot(2,1,2) plot(t,uuLTVMPC,t,usim,'o') grid title('Control Moves');

The closed-loop responses in MATLAB and Simulink are identical.

## See Also

### Functions

### Objects

`mpc`

|`mpcstate`

|`mpcmoveopt`