[Paper Review] Forward Invariance of Sets for Hybrid Dynamical Systems (Part I)

Jun Chai, Ricardo G. Sanfelice|arXiv (Cornell University)|Aug 15, 2018

Extremum Seeking Control Systems4 citations

TL;DR

This paper proposes robust forward invariance conditions for hybrid dynamical systems modeled as hybrid inclusions with state and disturbance constraints. It establishes sufficient conditions—based on tangent cone and flow/jump map properties—for sets to remain invariant under all admissible disturbances, including Zeno and finite-time terminating solutions, with applications to Lyapunov-like sublevel sets and a DC/AC inverter system.

ABSTRACT

In this paper, tools to study forward invariance properties with robustness to dis- turbances, referred to as robust forward invariance, are proposed for hybrid dynamical systems modeled as hybrid inclusions. Hybrid inclusions are given in terms of dif- ferential and difference inclusions with state and disturbance constraints, for whose definition only four objects are required. The proposed robust forward invariance notions allow for the diverse type of solutions to such systems (with and without dis- turbances), including solutions that have persistent flows and jumps, that are Zeno, and that stop to exist after finite amount of (hybrid) time. Sufficient conditions for sets to enjoy such properties are presented. These conditions are given in terms of the objects defining the hybrid inclusions and the set to be rendered robust forward invariant. In addition, as special cases, these conditions are exploited to state results on nominal forward invariance for hybrid systems without disturbances. Furthermore, results that provide conditions to render the sublevel sets of Lyapunov-like functions forward invariant are established. Analysis of a controlled inverter system is presented as an application of our results. Academic examples are given throughout the paper to illustrate the main ideas.

Motivation & Objective

• To formalize robust forward invariance for hybrid inclusions with disturbances, ensuring solutions remain within a set under all admissible disturbances.
• To address the challenge of invariance in hybrid systems with complex solution behaviors, including Zeno, persistent flows, and finite-time termination.
• To derive sufficient conditions for robust and nominal forward invariance using the system's data, including flow and jump maps.
• To specialize results to sublevel sets of Lyapunov-like functions, enabling invariance verification via convexity or strict decrease at boundaries.
• To demonstrate applicability through analysis of a controlled single-phase DC/AC inverter system with numerical validation.

Proposed method

• Formalizes robust forward invariance using hybrid inclusions defined by four core components: flow and jump sets, flow and jump maps, and disturbance constraints.
• Introduces four distinct invariance notions, with two stronger ones requiring all maximal solution pairs to remain in the set under disturbances.
• Derives sufficient conditions based on tangent cone inclusion: flow vectors must point inward at boundary points of the set, and jump maps must preserve invariance.
• Applies Assumption 4.14 to ensure local Lipschitz continuity of the flow map on the boundary of the invariant set, preventing escape.
• Uses Lemma 4.12 and auxiliary results on set-valued maps to verify invariance conditions, particularly for Lyapunov-like sublevel sets.
• Employs a pointwise minimum norm selection scheme for controller synthesis in the follow-up work, with numerical validation via MATLAB HyEQ toolbox.

Experimental results

Research questions

• RQ1 How can robust forward invariance be formally defined for hybrid systems with disturbances, including Zeno and finite-time solutions?
• RQ2 What sufficient conditions ensure that a set remains invariant under all admissible disturbances in hybrid inclusions?
• RQ3 How can sublevel sets of Lyapunov-like functions be rendered forward invariant under nominal and robust conditions?
• RQ4 What conditions guarantee the existence and completeness of maximal solution pairs within an invariant set?
• RQ5 How can the proposed invariance conditions be applied to real-world hybrid systems, such as power electronic inverters?

Key findings

• Sufficient conditions for robust forward invariance are established using tangent cone inclusion of the flow map and jump map image within the set’s tangent cone at boundary points.
• Nominal forward invariance is recovered as a special case when disturbances are excluded, with conditions reducing to standard invariance criteria.
• For Lyapunov-like sublevel sets, invariance holds if the flow map points inward or if the function strictly decreases at the boundary, especially under convexity.
• The existence of nontrivial, complete solution pairs from every point in the invariant set is guaranteed when zero disturbances are admissible during flows.
• Numerical simulations of a DC/AC inverter system confirm that the closed-loop system remains within the target set for all initial conditions and disturbance inputs.
• The controller design ensures the output voltage behaves sinusoidal-like with the desired frequency, validated via FFT analysis in the supplementary material.