MPC using a linear model with a input express in a polynomials basis

Hello,

I want to use a MPC for my system which is linear : X(k+1)=A*X(k)+B*U(k) and for which i have already use a MPC unconstrained. I want to constraint the problem but this one is particular : U(k) is a polynomial basis : U(k) = [ 2*u1(k); 4*u1(k)^2 - 2; 8*u1(k)^3 - 12*u1(k); 16*u1(k)^4 - 48*u1(k)^2 + 12].

Obviously, the MPC choose the best control vector to minimize a cost function but this best control vector do not respect the polynomial basis.

I am a bit stuck here and don't know what to do, i tried to impose some constraint on the input but those are non linear ...

Does someone have any advice ?

Thanks,

Gilles