Model Predictive Control Toolbox 3.1.1

MPC Control of a Multi-Input Multi-Output Nonlinear System

Contents

This demonstration shows how to use MPC to control a multi-input multi-output nonlinear system. The system has 3 manipulated variables and 2 measured outputs.

Open-loop Model: Linearize Nonlinear System

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

The nonlinear model is described in the Simulink® diagram "nonlinear_mpcmodel.mdl"

sys='nonlinear_mpcmodel';

We linearize the nonlinear model at (0,0) using LINMOD

[A,B,C,D]=linmod(sys);

MPC Controller Setup

Let us define the plant model that will be used for MPC control. First, we get a discrete-time version of the linearized model:

Ts=.2;                          %Sampling time
model=c2d(ss(A,B,C,D),Ts);      %Convert to discrete time
clear A B C D

Assign names to I/O variables (note: the model has no physical meaning)

model.InputName={'Mass Flow';'Heat Flow';'Pressure'};
model.OutputName={'Temperature';'Level'};

Define I/O constraints and units

clear InputSpecs OutputSpecs
InputSpecs(1)=struct('Min',-3,'Max',3,'RateMin',-1000,'Ratemax',Inf,'Units',
'kg/s');
InputSpecs(2)=struct('Min',-2,'Max',2,'RateMin',-1000,'Ratemax',Inf,'Units',
'J/s');
InputSpecs(3)=struct('Min',-2,'Max',2,'RateMin',-1000,'Ratemax',Inf,'Units',
'Pa');
OutputSpecs(1)=struct('Min',-Inf,'Max',Inf,'Units','K');
OutputSpecs(2)=struct('Min',-Inf,'Max',Inf,'Units','m');

Define weights on manipulated and controlled variables

Weights=struct('ManipulatedVariables',[0 0 0],...
   'ManipulatedVariablesRate',[.1 .1 .1],...
   'OutputVariables',[1 1]);

Define prediction and control horizons, and set up the MPC object

p=5;
m=2;
MPCobj=mpc(model,Ts,p,m,Weights,InputSpecs,OutputSpecs);

MPC Simulation using Simulink®

Run simulation

Tfinal=8;
open_system('mpc_nonlinear')    % Open Simulink(R) Model
sim('mpc_nonlinear',Tfinal);    % Start Simulation

-->Integrated white noise added on measured output channel #1.
-->Integrated white noise added on measured output channel #2.
-->The "Model.Noise" property of the "mpc" object is empty. Assuming whit
e noise on each measured output channel.

MPC Simulation with Ramp Signals

Now we modify the MPC design in order to track ramp signals.

In order to track a ramp, a triple integrator is defined as an output disturbance model on both outputs

outdistmodel=tf({1 0;0 1},{[1 0 0 0],1;1,[1 0 0 0]});
setoutdist(MPCobj,'model',outdistmodel);

-->The "Model.Noise" property of the "mpc" object is empty. Assuming whit
e noise on each measured output channel.

Run simulation

Tfinal=12;
bdclose('mpc_nonlinear');
open_system('mpc_nonlinear_setoutdist') % Open Simulink(R) Model
sim('mpc_nonlinear_setoutdist',Tfinal); % Start Simulation

MPC Simulation without Constraints

Now we get a linear version of the MPC controller.

When the constraints are not active, the MPC controller behaves like a linear controller. We can then get the state-space form of the MPC controller

Get the linear equivalent in SS form of the MPC controller

LTIMPC=ss(MPCobj,'r');

The input to the linear controller LTIMPC is the vector [ym;r], where ym is the vector of measured outputs, and r is the vector of output references

Remove constraints from MPC controller

MPCobj.MV=[];

Run simulation

refs=[1;1];                         % output references
Tfinal=8;
bdclose('mpc_nonlinear_setoutdist');
open_system('mpc_nonlinear_ss')     % Open Simulink(R) Model
sim('mpc_nonlinear_ss',Tfinal);     % Start Simulation

-->The "Model.Noise" property of the "mpc" object is empty. Assuming whit
e noise on each measured output channel.

Compare Simulation Results

fprintf('Compare output trajectories: ||ympc-ylin||=%g\n',norm(ympc-ylin));
disp('The MPC controller and the linear controller produce the same closed-l
oop trajectories.');

Compare output trajectories: ||ympc-ylin||=1.81537e-014
The MPC controller and the linear controller produce the same closed-loop tr
ajectories.

bdclose('mpc_nonlinear_ss')