Customized Cost Function and Binary Manipulated Variable in (NL)MPC

Hi everyone!

I am trying to implement MPC to control the compressor of a model similar to the Residential Refrigerator Model (Residential Refrigerator - MATLAB & Simulink - MathWorks Deutschland). I used simulation data (inputs: Compressor on/off and Door open/closed, output: air temperature in the compartment) to derive a data-driven linear state-space model of the plant.

Additionally to temperature setpoint tracking, I want to use information about door openings to lower the energy consumption. Therefore, I thought about using a customized cost function, where an active compressor is penalized during a door opening.

Now to my problem:

To my knowledge, defining a customized cost function is only possible in NLMPC and not in linear MPC. But NLMPC is not really an option, because (to my knowledge) defining a binary MV (0 = compressor off, 1 = compressor on) is not supported in NLMPC. Is there a workaround to consider and lower the power consumption of the compressor without using a customized cost function? Or is there a way to use a customized cost function in linear MPC that I don't know of? Or is there a possibility to use binary MV in NLMPC?

I would be really grateful if someone could help me!

Answer 1

[Thomas Kuenzel]

Hi Sebastian, the following initial answer is not mine but from our colleague Gernot Schraberger, who is an expert on MPC:

1) I would use a constraint to limit the temperature or compressor power during door open phase. Constraints can be defined in matrix form E*u + F*y >=G (E,F,G are matrices, u and y vectors of the plant inputs and outputs). Without knowing in detail, how you want to express this penalty a formulation like -door - compressor >= -1.5 would work (door open=1, closed = 0, compressor on = 1, off =0).

2) Using a nonlinear MPC only for allowing a customized nonlinear cost function is not very efficient, especially the binary inputs cause problems here and NMPC runs typically much slower. I would not use that approach.

3) If for the first case with the constraints in linear MPC a nonlinear constraint will be needed, there are ways to formulate a piecewise linear constraint, but that has a certain complexity. If such an approach is needed, it would make sense to have a more detailed conversation.

Maybe this already helps. Please contact us if you are interested in more in-depth support!

[Sebastian]

Hi Thomas,

thank you very much for the response!

ad 2) I got the idea to use nonlinear MPC from the example: Control House Heating System Using Nonlinear Model Predictive Control With Neural State-Space Prediction Model - MATLAB & Simulink - MathWorks Deutschland. If I use a compressor with a continuous output instead of an on/off compressor, the problem would be pretty similar to the mentioned example. But when I try to identify a neural state space model (like in this example) with the same experimental data sets which I used to identify my linear state space model, the result is very unsatisfying.

In the case of a compressor with continuous output and if I would be able to identify a good nss model, would you still suggest sticking to linear MPC?

ad 1 + 3) This approach seems appealing to me, since I can work with linear MPC. I would appreciate further help from you regarding the formulation of piecewise linear constraints, since I am new to MPC and have no experience.

With kind regards,

Sebastian