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 and a supply function that dictates the ISS gain of the system. In order to reduce conservatism, a set of parameters is assigned to both the Lyapunov and the supply function, respectively. A set-valued map, which generates admissible sets of parameters for each state and input, is defined such that the corresponding parameterized Lyapunov and supply functions enjoy the standard ISS inequality. It is demonstrated that the so-obtained parameterized ISS inequality offers nonconservative analysis conditions, even when Lyapunov 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 effectiveness of the proposed synthesis methodology is illustrated for a benchmark test case for model predictive control, that is, a continuous stirred tank reactor.