Infinity Norms as Lyapunov Functions for Model Predictive Control of Constrained PWA Systems

Switching Stabilizability for Continuous-Time Uncertain Switched Linear Systems

“…For feedback stabilization, the switching signals are assumed to be given or restricted, while the continuous-variable control laws, in the form of state or output feedback, are designed to stabilize the switched systems under these given switching signals. Several classes of switching signals are considered in the literature, for example arbitrary switching [7], [11], slow switching [4], [13], restricted switching induced by partitions of the state space [6], [17], [29], [30] etc. The focus of this paper is on a typical switching stabilizability problem, where the switching signals are assumed to be free design variable, while no continuous-variable inputs are considered.…”

Lyapunov Functions, Stability and Input-to-State Stability Subtleties for Discrete-Time Discontinuous Systems

“…These results, together with the fact that existence of a possibly discontinuous USL function is equivalent to GAS [18], [20], issue a strong warning regarding existing nominally stabilizing state-feedback synthesis methods for discrete-time discontinuous systems, including both static feedback approaches [3]- [7] and MPC techniques [8]- [11]. This warning motivates the recent results on global input-to-state stabilization of discrete-time PWA systems [21] and input-to-state stabilizing (sub-optimal) MPC of discontinuous systems [22].…”

“…The tests developed in this section require that for each i 2 J , V i is a continuous function that satisfies (8), and as such must be defined on cl(0i ) and, furthermore, it is defined on: (i) P n for Proposition IV.2 and (ii) cl(0 i ) 8 B for some 2 (0;1) for Theorem IV.3. These are additional requirements with respect to USL functions, which in principle, only require that each Vi is defined on 0 i .…”

“…Notice that previous results on stability of discrete-time PWA systems [3]- [7] only indicated that continuous Lyapunov functions may be more difficult to find than discontinuous ones, while in fact a continuous Lyapunov function might not even exist. As such, a valid warning regarding nominally stabilizing state-feedback synthesis methods for discrete-time discontinuous systems, including both static feedback approaches [3]- [7] and MPC techniques [8]- [11] arises. These synthesis methods lead to a stable, possibly discontinuous closed-loop system and often rely on discontinuous Lyapunov functions.…”

Adaptive Cruise Controller Design: A Comparative Assessment for Pwa Systems

“…An equivalent model, the mixed logical dynamical (MLD) model (Bemporad and Morari, 1999), is used to compute the control law implicitly or explicitly (Borrelli, 2003). Several variants that consider robustness (Kerrigan and Mayne, 2002) or stability properties (Lazar et al, 2005) were also considered. Methods based on the construction of a piecewise Lyapunov function have been developed in (Hedlund and Rantzer, 1999).…”