Forward Invariance of Sets for Hybrid Dynamical Systems (Part II)

Abstract

This article presents tools for the design of control laws inducing robust controlled forward invariance of a set for hybrid dynamical systems modeled as hybrid inclusions. A set has the robust controlled forward invariance property via a control law if every solution to the closed-loop system that starts from the set stays within the set for all future time, regardless of the value of the disturbances. Building on the first part of this article, which focuses on analysis (Chai and Sanfelice, 2019), in this article, sufficient conditions for generic sets to enjoy such a property are proposed. To construct invariance inducing state-feedback laws, the notion of robust control Lyapunov function for forward invariance is defined. The proposed synthesis results rely on set-valued maps that include all admissible control inputs that keep closed-loop solutions within the set of interest. Results guaranteeing the existence of such statefeedback laws are also presented. Moreover, conditions for the design of continuous state-feedback laws with minimum point-wise norm are provided. Major results are illustrated throughout this article in a constrained bouncing ball system and a robotic manipulator application.

Set-Invariance-based Interpretations for the L1 Performance of Nonlinear Systems

“…it immediately follows from (22) together with ( 25) that the inclusion of ( 24) is satisfied for all x ∈ (−1− , 1+ ). Thus, it is obvious from Theorem 2 that the continuous nonlinear system given by ( 21) and ( 22) satisfies the L 1 performance.…”

“…To put it another way, x(t; t 0 ) given by ( 26) becomes solutions of (21) for any t 0 > 0. Such a non-uniqueness of solutions as in ( 26) could be interpreted as arising from the fact that the vector field f (x) of (22) does not satisfy Lipschitz continuity at x = −1. This situation is quite crucial for a controller synthesis because the closed-loop system obtained through a feedback connection between a nominal system and a controller cannot satisfy Lipschitz property even if the vector field of the nominal system is Lipschitz continuous as discussed in Theorem 3.9 in [8] and Definition 2.3 in [10].…”

“…In contrast to this method, the arguments in this paper are not limited to Lipschitz continuous vector fields and can deal with piecewise continuous nonlinear differential dynamics. Furthermore, the ideas of Lyapunov functions and barrier certificates have been recently developed in [21][22][23][24] to establish set invariance results in an equivalent fashion to those in this paper, but they always require an involved task to constructing adequate scalar candidates for the Lyapunov functions and/or barrier certificates. In a comparison with these ideas, this paper can be interpreted as giving a more intuitive and systematic method for solving the L 1 analysis problem of nonlinear systems since it is not required in this paper to develop such scalar candidates.…”

Set‐invariance‐based interpretations for the L1 performance of nonlinear systems with non‐unique solutions

“…In contrast to the existing method assuming the Lipschitz continuity, the class of nonlinear differential equations taken in this article is not limited to that with Lipschitz continuous vector fields and contains those of non‐unique solutions. Furthermore, the ideas of Lyapunov functions and barrier certificates have been recently developed in References 20, 30, and 31 to establish set invariance results in an equivalent fashion to those in this article, but they always require an involved task to constructing adequate scalar candidates for the Lyapunov functions and/or barrier certificates. In a comparison with these ideas, this article can be interpreted as giving a more intuitive and systematic method for solving the $$$ {L}_1 $$$ analysis problem of nonlinear systems since it is not required in this article to develop such scalar candidates.…”

Regularity of Optimal Solutions and the Optimal Cost for Hybrid Dynamical Systems via Reachability Analysis

“…Numerous tools are available in the literature for the study of hybrid systems, in particular, for hybrid systems modeled as hybrid automata [10,11,12], impulsive systems [13,14], and hybrid inclusions [15,16]. The literature is rich in tools for the analysis of reachability [17,18,19], asymptotic stability [10,13,15], forward invariance [14,20,21], control design [16], and robustness [15,16]. On the other hand, optimality for hybrid systems is much less mature.…”