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tf

Convert unconstrained MPC controller to linear transfer function form

Description

Use the Model Predictive Control Toolbox™ tf function to convert an unconstrained MPC controller to transfer function form (see mpc for background). The returned controller is equivalent to the original MPC controller MPCobj when no constraints are active. You can then use Control System Toolbox™ software for sensitivity analysis and other diagnostic calculations.

To create or convert a generic LTI dynamical system to zero/pole/gain form instead, see tf and Dynamic System Models.

example

ktf = tf(MPCobj) returns the linear discrete-time dynamic controller ktf, in transfer function form. ktf is equivalent to the MPC controller MPCobj when no constraint is active.

Examples

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Create a plant, a corresponding MPC object, and convert it to transfer function form.

mpcverbosity off;          % turn off mpc messaging
plant=tf(1,[1 1],0.2);     % create plant (0.2 seconds sampling time)
mpcobj=mpc(plant,0.2);     % create mpc object (0.2 second sampling time)

ktf=tf(mpcobj)             % convert mpc to transfer function

ktf =
 
  From input "MO1" to output "MV1":
    0.452 z^3 - 0.6781 z^2 - 1.506e-16 z
  -----------------------------------------
  z^3 - 1.001 z^2 + 0.0002642 z + 0.0006399
 
Sample time: 0.2 seconds
Discrete-time transfer function.

Plot the response to a step in the unmeasured disturbance input using both the sim command and the controller transfer function form.

% create a plant and the corresponding mpc object
mpcverbosity off;                  % turn off mpc messaging
plant=tf({1,1},{[1 1],[1 1]},0.2); % create plant (0.2 seconds sampling time)
plant=setmpcsignals(plant,'UD',2); % second input is a disturbance entering at mv
mpcobj=mpc(plant,0.2);             % create mpc object (0.2 second sampling time)

% set input and output disturbance models (remove integrators)
setindist(mpcobj,'model',tf(1))    % set input disturbance model to 1
setoutdist(mpcobj,'model',tf(1))   % set output disturbance model to 1

% closed loop output sensitivity
cloffset(mpcobj)
ans =
    1.4472

% convert the controller and calculate closed loop transfer function
ktf=tf(mpcobj);                    % convert mpc to transfer function
Muy=feedback(plant(:,1),ktf,1);    % closed loop transfer function from mv to y

% closed loop output sensitivity using transfer function
1+dcgain(Muy*ktf)
ans =
    1.4472

% plot closed loop response to a step on the measured input
step(Muy,10);                                     % simulate using step

% create option object to inject disturbance in simulation
SimOptions = mpcsimopt;                           % create object
SimOptions.UnmeasuredDisturbance = ones(50,1);    % specify unmeasured input disturbance

% simulate closed loop for 50 steps with sim and step and plot the response
[y,t,u,xp]=sim(mpcobj,50,0,[],SimOptions);           % simulate using sim

% overlap the sim results on the plot
hold on
stairs(t,y,'r')
hold off

Input Arguments

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Model predictive controller, specified as an MPC controller object. To create an MPC controller, use mpc.

Output Arguments

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Transfer function form of the MPC controller MPCobj when no constraint is active. This is also equivalent to tf(ss(MPCobj))

Introduced before R2006a