This example shows how to obtain an LTI representation of an unconstrained MPC controller using
ss. You can use this to analyze the frequency response and performance of the controller.
Define a plant model. For this example, use the CSTR model described in Design Controller Using MPC Designer.
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; CSTR.OutputGroup.MO = 1; CSTR.OutputGroup.UO = 2;
Create an MPC controller for the defined plant using the same sample time, prediction horizon, and tuning weights described in Design MPC Controller at the Command Line.
MPCobj = mpc(CSTR,1,15);
-->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
MPCobj.W.ManipulatedVariablesRate = 0.3; MPCobj.W.OutputVariables = [1 0];
Extract the LTI state-space representation of the controller.
MPCss = ss(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 no disturbance added to measured output channel #1. -->The "Model.Noise" property of the "mpc" object is empty. Assuming white noise on each measured output channel.
Convert the original
CSTR model to discrete form using the same sample time as the MPC controller.
CSTRd = c2d(CSTR,MPCss.Ts);
Create an LTI model of the closed-loop system using
feedback. Use the manipulated variable and measured output for feedback, indicating a positive feedback loop. Using negative feedback would lead to an unstable closed-loop system, because the MPC controller is designed to use positive feedback.
CLsys = feedback(CSTRd,MPCss,1,1,1);
You can then analyze the resulting feedback system. For example, verify that all closed-loop poles are within the unit circle.
poles = eig(CLsys)
poles = 6×1 complex 0.5513 + 0.2700i 0.5513 - 0.2700i 0.6131 + 0.1110i 0.6131 - 0.1110i 0.9738 + 0.0000i 0.9359 + 0.0000i
You can also view the system frequency response.