On the Equivalence Between Dissipativity and Optimality of Discontinuous Nonlinear Regulators for Filippov Dynamical Systems

The ℒ1 controller synthesis for piecewise continuous nonlinear systems via set invariance principles

“…Theorem 3. Suppose that K (∋ 0) is a convex generalized controlled invariance domain of Σ and u(x) = 𝜓(x) is defined on x ∈ O ∩ D j as (30). Then, there exist control values u(x) = 𝜓(x) at x ∈ O ∩ M, by which ( 9) holds.…”

“…Theorem 4 (Existence of the  1 admissible controller). If K is a convex and compact generalized controlled invariance domain of Σ, then there exists an  1 admissible controller u(x) = 𝜓(x) defined on the neighborhood O of K determined as (30) and (39).…”

“…For a more rigorous argument on the treatment of nonlinear systems with non-unique solutions, there have been a number of studies on piecewise continuous nonlinear systems. [24][25][26][27][28][29][30][31] This is a class of discontinuous dynamical systems, in which the types of linear systems, 8,9 continuous nonlinear systems, 11 and switched nonlinear systems 32 are involved as described shown in Figure 1. Here, solutions to piecewise continuous nonlinear systems should be described based on the arguments on Filippov's differential equations (inclusions) 33 because they cannot be defined on the classical Caratheodory-based arguments.…”

“…For a more rigorous argument on the treatment of nonlinear systems with non‐unique solutions, there have been a number of studies on piecewise continuous nonlinear systems 24‐31 . This is a class of discontinuous dynamical systems, in which the types of linear systems, 8,9 continuous nonlinear systems, 11 and switched nonlinear systems 32 are involved as described shown in Figure 1.…”

“…Here, solutions to piecewise continuous nonlinear systems should be described based on the arguments on Filippov's differential equations (inclusions) 33 because they cannot be defined on the classical Caratheodory‐based arguments. In a theoretical aspect for dealing with piecewise continuous nonlinear systems, various studies on stability and $$$ {\mathcal{L}}_2 $$$‐induced norms have been introduced recently in References 27‐31, respectively, but no argument on the $$$ {\mathcal{L}}_1 $$$ controller synthesis is discussed in those studies. With respect to a practical effectiveness, furthermore, there have been also various studies on employing piecewise continuous nonlinear systems to describe real systems such as frictional effects, 24 neural networks with discontinuous activation functions, 25 sliding mode control systems 26 and so on.…”