Determining a terminal set for a constrained MPC

[Saeed]

Given a Constrained MPC problem with polytopic constraints as GX<=g and HX<=h, I have computed its associated terminal cost and terminal linear state feedback controller as X'PX and KX, respectively using YALMIP easily. Thanks to YALMIP.

Now, I want to find the terminal set for this CMPC problem. What should I do?

I mean, what optimisation problem should be solved with YALMIP.

It would be appreciated if someone could help me with this issue.

Regards,

Saeed.

[Johan Löfberg]

Use MPT instead. It has ready-made functions for this

[Saeed]

Dear Johan,

Many thanks for your quick and helpful response.

Regards,

Saeed

[MohiF]

Dear Johan,

Thanks so much for your supports.

Do you have any toolbox recommended to determine terminal invariant set for nonlinear system?

Could you please if there are any analytic approaches for determining invariant set for  nonlinear systems?

regards,

Mohi. 

[Johan Löfberg]

You can formulate various invariance and attraction stuff using sum-of-squares, and try to solve those. There are numerous papers on the topic

[MohiF]

Dear John,

Could you please recommend me a paper regarding ''Computing invarience using sum-of-squares''. I'm not familiar with the topic.

Thanks so much.

[Johan Löfberg]

I don't have any specific reference. Google is your friend

[MohiF]

Hhhmmmm

Thanks Dr. John

And when such an invariance is formulated, Solve it using Yalmip??

[Johan Löfberg]

Yes, YALMIP has a sum-of-squares framework

[MohiF]

Dear Johan,

Having considered your PhD thesis, I managed to construct an ellipsoidal positive invariant set through a Determinant Maximisation problem.

Thanks for your explicit explanation.

As you know, given a lyapunov function, its level surfaces are the boundaries of positively invariant sets. Before determining an ellipsoidal PIS, I had constructed a lyapunov function for the unconstrained system controlled by a nominal state feedback controller, to be used as a Terminal cost.

The encountered problems are:

1)  Level surfaces associated to lyapunov function must be used as PIS or the ellipsoidal set found through determinant maximisation?

2)  the feedback controllers which have found through different optimisation problems, may be different. Is there a problem?

 I hope you'll excuse me for discussing such a problem here. It may be helpful to other researchers.

Best Regards,

[Johan Löfberg]

You can derive the PIS anyway you want, and you can use any feedback you want. 

[Gaara]

Dear Johan,

I actually used MPT to compute the terminal set (X_f), but I need yalmip to compute the MPC. So how can I include the final set constraint (x(N) \in X_f) in Yalmip?

Thanks

[Johan Löfberg]

You would simply extract the data defining the set and add the constraint, x^TPx<=1 or Fx<=f or whatever the terminal set is. Nothing magic

[Gaara]

Yes sorry, I put the question in a wrong way! That is what I was trying to do but I did not know the syntax. I found it and it's obviously the trivial one: polytope.A and polytope.b.

Thanks