Minimization of cost function on MPC

Hello,

I want to optimize a cost function of MPC which is output reference tracking as shown below:

min J(∆u) = ||xref(k + i) − x(k + i)||2 Q+ ||∆u(k + i)||2 R + ||xref(k + N) − x(k + N)||2 S, where Q,R and S are weight matrices.

Can anyone tell me how can I optimize this with constraint -1<∆u<1. Is it better use solver or can I optimize it without using solver?

If using solver, how can I convert into standard solver form?

If without using solver, which function to be used to minimize this cost function?

Any help is appreciated. Thanks in advance

[Sam Chak]

@Chandrakanth Pavanaskar, Not sure how to answer this without writing the entire code for the MPC...

Are you familiar with the minimization of the Linear Quadratic problem? I think you can start from there and then gradually evolve the problem to whatever you desire as in the MPC thing.

[Chandrakanth Pavanaskar]

@Sam Chak I just wanted to know how can I convert a reference tracking MPC problem into the standard vector form of quadprog solver?

[Chandrakanth Pavanaskar]

@Sam Chak How can I convert

J (N1 ,N2,Nu ) = transpose (y −w ) . R. ( y- w ) + ∆u .Q . ∆u

This equation into quadprog solver standard form?

[Sam Chak]

Hi @Chandrakanth Pavanaskar,

The cost function has some similarities like the one you would use in LQR. Since you have experience in LQR, you can write the code for the Cost function, and then modifying it to suit your application.

By the way, have your checked the examples in quadprog()?

[Chandrakanth Pavanaskar]

Hello @Sam Chak

I have a question related to Quadprog. Does Quadprog update the states for next iteration on its own or do we have to update state at every iteration once quadprog gives afeasible solution?

Basically, for x(k+2) is x(k+1) taken as input by quadprog automatically or do we have to assign x(k+1) as x(k) for next iteration while solving MPC.

Thank you!

[Sam Chak]

Hi @Chandrakanth Pavanaskar

I think it depend on the problem and how you want to call quadprog() to perform the minimization and update the desired parameter in each iteration. If it is a MPC problem, then I prefer to use mpc() command:

mpcobj = mpc(plant, ts, P, M, W, MV, OV, DV)

https://www.mathworks.com/help/mpc/ref/mpc.html

[Chandrakanth Pavanaskar]

Hi @Sam Chak,

I have not used mpc command or toolbox in my optimization proble. I created matrices which predicts future states and optimize the control variable. I just wanted to know how can I give initial state as 0 and update the state for each iteration as my control variable and state changes every iteration and without using mpc commands.

Thank you

Answer 1

[Sam Chak]

Hi @Chandrakanth Pavanaskar

You have a system model sys and I presume that you have written your own code for the ODE_solver to run the sys.

Under normal circumstances, you should be able to specify the initial states y0 in the ODE_solver.

The solver should produce the numerical solution array y that corresponds to a value returned in time vector t.

With the data, you can run your own optim_solver to produce some control action u.

Then, update the initial states y0 with final value of the states y(end) from the last iteration.

Finally, repeat the loop/steps by running the ODE_solver > the optim_solver until the specified number of iteration is completed.