Hello everyone.
Consider a constrained LTI system as: dx=Ax+Bu+ alpha*H*x, where 'alpha' is a varying parameter in [0 1] and H is a constant matrix, controlled by MPC.
In the case of linear constrained MPC, usually, stability is achieved by forcing the last element of the optimal state trajectory to lie in a terminal invarient set. In this set, A lyapunov function for the unconstrained system under the nominal control is known. If this Lyapunov function is used as the terminal cost in MPC cost function, stability of the closed-loop system under MPC strategy is guaranteed.
The issue is: How stability is guaranteed for the above-mentioned linear uncertain system?
I calculated MPC data (i.e. Q and R cost function matrices as well as terminal cost function) according to the state matrix 'A+H' by writing Lyapunov reduction equation and then constructing an LMI optimisation problem. So stability when 'alpha' is between (0 1) must be satisfied ! Am I right?
Note that I calculate the positive invariant set for the system: dx=Ax+Bu(+H*x)
and the equality constraint is: dx=Ax+Bu(+H*x) and the optimal input is applied to dx=Ax+Bu+ alpha*H*x. But the closed-loop system is unstable!!
What should be done to guarantee stability while 'alpha' is varying?
Kind Regards,
Instead of simple LMI if you use LMI with pole placement objective I think your problem will be solved.
You can try with robust MPC for uncertain systems.
Some references: homes.esat.kuleuven.be/~maapc/static/files/.../survey-robust-mpc.pdf
http://www.me.berkeley.edu/~yzhwang/MPC/robustMPC.html
www.mit.edu/people/jhow/papers/Richards_PhD.pdf
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