Hybrid Systems with Continuous-time Inputs: Subtleties in Solution Concepts and Existence Results

Abstract

We study solution concepts and their properties for hybrid systems that can flow and jump, affected by continuous-time inputs. While the solution concepts, the existence of solutions and (forward) completeness properties are extensively discussed in the absence of (external) inputs, there are surprisingly few results when inputs are present, certainly in the case where the flow and jump sets depend on the inputs. Given the relevance of this class of hybrid systems for many applications such as hybrid or networked control for plants subject to disturbances or measurement noise, we discuss in this paper notions of solutions in the presence of inputs and show through various examples the subtleties that can occur. Moreover, we provide tools to guarantee the existence of solutions and results to establish completeness properties.

Reset-and-Hold Control of Systems With Time Delay

“…1) The HI framework for systems with inputs This section briefly describes the formalism of Hybrid Dynamical Systems adapted to systems with inputs [14] as will be used in this work. A hybrid system with inputs Σ for a state x ∈ R nx and input w ∈ W for some set W ⊆ R nw is given by…”

“…where f : R nx ⇒ R nx is the flow map, g : R nx ⇒ R nx is the jump map (f and g are both set-valued maps in general), and C, D ⊆ R nx × R nw are the flow and jump sets respectively. Note that as in [14], we do not regard w as a hybrid signal (as otherwise its domain must be known in advance, which is unrealistic in practice); instead, the space of admissible inputs w(t) will be taken as the set of piecewise continuous functions from R ≥0 to W . With this reduced set of inputs, the two concepts of solution considered in [14] coincide, and the results for existence of solutions and completeness therein developed easily follow.…”

“…From their definition in ( 14) and ( 10)- (11), it is clear that C and D are closed sets, and thus the hybrid basic condition #1 easily follows. Regarding condition #2, it directly follows from (14) that that F(φ) is convex and nonempty.…”

“…(Delay-dependent stability conditions) Consider the resetand-hold control system Σ ∆ given by (14). The set W is asymptotically stable for Σ ∆ if there exist matrices P > 0, Q > 0, X = X ⊤ , Y and Z > 0 such that the following conditions are satisfied:…”

Towards improving the estimation performance of a given nonlinear observer: a multi-observer approach

“…, where pt, jq, pt 1 , j 1 q P R ě0 ˆZě0 . We use the notion of solution for system (1) as given in [33,Definition 4]. Given a set U Ď R nu , L U is the set of all functions from R ě0 to U that are Lebesgue measurable and locally essentially bounded.…”