Soften hard constraints in MPC formulated as a QP-problem

I have problems introducing a slack variable to soften some constraints I have on the states. I use QUADPROG to solve the problem which is written in the form

min 0.5x'Hx
S.t Aeq*x = beq
    lb <= x <=ub

The vector x contains 3 states z1, z2 and z3 as well as the the calculated control input u. For instance, for prediction horizon 1 the vector will have 4 rows, for prediction horizon 2 it will grow to 8 etc. Now I have constraints on the input varible which I put in the vectors lb and ub. Everything works fine until I introduce constraints on the states. These are hard inequality constraints. As expected the problem becomes infeasible so I would need to relax them. Therein lies the issue, I do not know how to modify the setup in order to incorporate a slack variable.

I understand that the state vector has to be expanded by adding an extra variable s => 0 preferably in the end. Consequently I have to increase the size of H by 1, so the last diagonal element will be the weight on the slack varible which of course should be heavily penalized. Now if I would like to soften the constraint on the first state z1 I need to modify the first entry in ub and lb by adding the slack variable s.

So far I have only modified H as mentioned above but I do not know how to continue with the vectors ub and lb, which I assume also have to be modified in some way.

Thank you in advance for any advice!

Answer 1

[Anton Semechko]

You could try using 'fmincon' instead of 'quadprog', which would allow you to incorporate non-linear constraints on your state variables in terms of barrier functions.

Answer 2

[Ahmed Hatem]

Anton,

thanks for your answer. Is it really necessary to use fmincon and nonlinear constraints? It seems like it will just complicate matters. I have experience with nonlinear models and solvers and usually the solver is slow and if it is able to find a solution it is often not global optimal. Furthermore, I want to keep the time to solve the problem down since I aim to implement this controller. This is the reason why I want to use linear models only.

[Anton Semechko]

You make a good point. Using 'fmincon' was a bad suggestion on my part. There is pretty good tutorial on YouTube explaining how to convert the inequality constraints into equality constraints with slack variables. I am not sure if that will help, as it is a something that may already be done internally when calling 'quadprog'.

[Ahmed Hatem]

Well I already have a set of equality constraints defined by the matrix Aeq and the vector beq. I cannot really see how to combine those with the ones that arise once I convert the inequalities into equalities. I will basically end up with two different matrices Aeq and vectors beq. I hope there is a way I can solve this problem using the vectors ub and lb which I pass to QUADPROG.