revise the following academic text written in latex, Adaptive MPC (AMPC) shares similar cost function as the LTV-MPC, with its prediction function adjusted at run time to compensate for the plant's nonlinear characterestics. Different from the LTV-MPC, the linearized matrices $A,B$ remains constant over the predictive horizon,  At each sampling instant, the matrix $A, B$ update the value based on the the state $x_0$ and input $u_{-1}$ of last sampling time as \ref{Gb}, while the state $x_{k+i}$ over the prediction horizon is predicted by \ref{Gc}.