An explicit solution to constrained stabilization via polytopic tubes

This paper proposes a method to obtain stabilizing controllers for constrained linear systems with assigned sets of initial conditions. The controller synthesis method is based on invariant tubes and works for linear time-invariant systems and for linear systems with multiplicative uncertainties. Given a compact initial condition set, a sequence of sets and an associated sequence of control laws is computed such that the initial condition set is contained in the first set of the sequence and every state in any set of the sequence is controlled to the next set in the sequence while satisfying state and input constraints. Assumptions on the parameterizations of the sets and the control laws are given that guarantee recursive feasibility of the tube synthesis problem and convergence of the closed-loop trajectories. For a particular type of parameterization it is shown that these assumptions are satisfied. Numerical simulations are presented that illustrate the developed synthesis method.