Compute Steady-State Gain

This example shows how to analyze a model predictive controller using cloffset. This function computes the closed-loop, steady-state gain for each output when a sustained, 1-unit disturbance is added to each output. It assumes that no constraints are active.

Define a state-space plant model.

A = [-0.0285 -0.0014; -0.0371 -0.1476];
B = [-0.0850 0.0238; 0.0802 0.4462];
C = [0 1; 1 0];
D = zeros(2,2);
CSTR = ss(A,B,C,D);

CSTR.InputGroup.MV = 1;
CSTR.InputGroup.UD = 2;

Create an MPC controller for the defined plant.

MPCobj = mpc(CSTR,1);
-->The "PredictionHorizon" property of "mpc" object is empty. Trying PredictionHorizon = 10.
-->The "ControlHorizon" property of the "mpc" object is empty. Assuming 2.
-->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.
   for output(s) y1 and zero weight for output(s) y2 

Specify tuning weights for the measured output signals.

MPCobj.W.OutputVariables = [1 0];

Compute the closed-loop, steady-state gain for this controller.

DCgain = cloffset(MPCobj)
-->Converting model to discrete time.
-->The "Model.Disturbance" property of "mpc" object is empty:
   Assuming unmeasured input disturbance #2 is integrated white noise.
-->Assuming output disturbance added to measured output channel #1 is integrated white noise.
   Assuming no disturbance added to measured output channel #2.
-->The "Model.Noise" property of the "mpc" object is empty. Assuming white noise on each measured output channel.
DCgain = 2×2

    0.0000    0.0000
    2.3272    1.0000

DCgain(i,j) represents the gain from the sustained, 1-unit disturbance on output j to measured output i.

The second column of DCgain shows that the controller does not react to a disturbance applied to the second output. This disturbance is ignored because the tuning weight for this channel is 0.

Since the tuning weight for the first output is nonzero, the controller reacts when a disturbance is applied to this output, removing the effect of the disturbance (DCgain(1,1) = 0). However, since the tuning weight for the second output is 0, this controller reaction introduces a gain for output 2 (DCgain(2,1) = 2.3272).

See Also

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