Adaptive model predictive control for constrained, linear time varying systems

We consider a discrete-time, linear time varying (LTV), multiple input, multiple output (MIMO) system with nu inputs and ny outputs. The system is known to be asymptotically stable, but the exact dynamics and the way they change over time are not known. We denote the vector of control inputs at time step t ∈ Z by u(t) = [u1(t), . . . , unu(t)] T , where ui(t) ∈ R, i = 1, . . . , nu are the individual plant inputs and T stands for the matrix transpose operator. In addition, we denote the vector of plant outputs by y(t) = [y1(t), . . . , yny (t)] T , where yj(t) ∈ R, j = 1, . . . , ny are the individual plant outputs. At each time step, the dynamic relation between the inputs and the outputs can be described by a linear model of the following form: