Constrained control of linear systems: from interpolating control to model predictive control.

A fundamental problem in automatic control is the control of uncertain and/or time-varying plants with input and state or output constraints. Most elegantly, and theoretically most satisfying, the problem is solved by optimal control which, however, rarely gives a feedback solution, and oftentimes only a numerical solutions.

Therefore, in practice, the problem has seen many ad-hoc solutions, such as over-ride control, anti-windup. Another solution, that has become popular during the last decades is Model Predictive Control (MPC) where an optimal control problem is solved at each sampling instant, and the element of the control vector meant for the nearest sampling interval is applied. In spite of the increased computational power of control computers, MPC is at present mainly suitable for low-order, nominally linear systems. The robust version of MPC is conservative and computationally complicated, while the explicit version of MPC that gives a piecewise affine state feedback solution involves a very complicated division of the state space into polyhedral cells.

In this presentation a novel and computationally cheap solution is presented for uncertain and/or time-varying linear discrete-time systems with polytopic bounded control and state (or output) vectors, with bounded disturbances. The approach is based on the interpolation between a stabilizing, outer low-gain controller that respects the control and state constraints, and an inner, high-gain controller, designed by any method that has its robustly positively invariant set satisfying the constraints. A simple Lyapunov function is used for the proof of closed loop stability.

In contrast to MPC, the new interpolating controller is not necessarily employing an optimization criterion inspired by performance. In its explicit form, the cell partitioning is considerable simpler that the MPC counterpart. For the implicit version, the on-line computational demand can be restricted to the solution of at most two linear programming problems or one quadratic programming problem.

Several simulation examples are given, including uncertain linear systems with disturbances. Some examples are compared with MPC.