On parameterized dissipation inequalities and receding horizon robust control

This paper considers the standard input-to-state stability (ISS) inequality for discrete-time nonlinear systems, which involves a candidate Lyapunov function (LF) and a supply function that dictates the ISS gain of the system. To reduce conservatism, a set of parameters is assigned to both the LF and the supply function. A set-valued map, which generates admissible sets of parameters for each state and input, is defined such that the corresponding parameterized LF and supply function enjoy the standard ISS inequality. It is demonstrated that the so-obtained parameterized ISS inequality offers non-conservative analysis conditions, even when LFs and supply functions with a particular structure, such as quadratic forms, are considered. For bounded inputs, it is then shown how parameterized ISS inequalities can be used to synthesize a closed-loop system with an optimized envelope of trajectories. An implementation method based on receding horizon optimization is proposed, along with a recursive feasibility and complexity analysis. The advances provided by the proposed synthesis methodology are illustrated for a continuous stirred tank reactor.