[Paper Review] A Novel Unified Approach to Invariance in Control

Zoltán Horváth, Yunfei Song+1 more|arXiv (Cornell University)|May 20, 2014

Stability and Control of Uncertain Systems3 citations

TL;DR

This paper introduces a unified framework to determine necessary and sufficient conditions for invariance of convex sets—polyhedra, polyhedral cones, ellipsoids, and Lorenz cones—under linear continuous or discrete dynamical systems. By leveraging the S-lemma instead of traditional Lyapunov methods, the approach extends invariance analysis to general quadratic inequality sets, including nonconvex and unbounded ones, and derives continuous system conditions from discrete counterparts via Euler discretization.

ABSTRACT

In this paper, we propose a novel, unified, general approach to investigate sufficient and necessary conditions under which four types of convex sets, polyhedra, polyhedral cones, ellipsoids and Lorenz cones, are invariant sets for a linear continuous or discrete dynamical system. In proving invariance of ellipsoids and Lorenz cones for discrete systems, instead of the traditional Lyapunov method, our novel proofs are based on the S-lemma, which enables us to extend invariance conditions to any set represented by a quadratic inequality. Such sets include nonconvex and unbounded sets. Finally, according to the framework of our novel method, sufficient and necessary conditions for continuous systems are derived from the sufficient and necessary conditions for the corresponding discrete systems that are obtained by Euler methods.

Motivation & Objective

• To unify the analysis of invariance for four key convex sets—polyhedra, polyhedral cones, ellipsoids, and Lorenz cones—across linear continuous and discrete systems.
• To overcome limitations of traditional Lyapunov methods in analyzing invariance of ellipsoids and Lorenz cones by introducing the S-lemma as a foundational tool.
• To extend invariance conditions to any set defined by a quadratic inequality, including nonconvex and unbounded sets, thereby broadening applicability.
• To establish a systematic link between invariance conditions for continuous systems and their discrete approximations via Euler methods.

Proposed method

• Utilizes the S-lemma to derive invariance conditions for ellipsoids and Lorenz cones in discrete-time systems, replacing classical Lyapunov-based analysis.
• Applies the S-lemma to any set defined by a quadratic inequality, enabling invariance analysis for nonconvex and unbounded sets beyond standard convex cones.
• Derives invariance conditions for continuous systems by applying Euler discretization to the corresponding discrete-time systems, ensuring consistency between continuous and discrete frameworks.
• Establishes a general framework that unifies the treatment of invariance across different types of convex and quadratic sets under linear dynamics.
• Relies on convex analysis and quadratic forms to characterize invariance via algebraic conditions expressible as linear matrix inequalities (LMIs) or related constraints.
• Demonstrates that the same analytical structure applies to polyhedra and polyhedral cones, enabling a single unified treatment across all four set types.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient conditions for a linear discrete-time system to preserve ellipsoids and Lorenz cones as invariant sets?
• RQ2 How can the S-lemma be systematically applied to generalize invariance conditions beyond convex sets to any quadratic inequality-defined set?
• RQ3 What is the relationship between invariance conditions for continuous systems and their discrete approximations obtained via Euler methods?
• RQ4 Can a single unified framework be developed to analyze invariance for polyhedra, polyhedral cones, ellipsoids, and Lorenz cones under linear dynamics?
• RQ5 To what extent can the proposed method extend to nonconvex or unbounded sets defined by quadratic inequalities?

Key findings

• The S-lemma enables derivation of invariance conditions for ellipsoids and Lorenz cones in discrete systems without relying on Lyapunov functions.
• The method generalizes to any set defined by a quadratic inequality, including nonconvex and unbounded sets, significantly broadening the scope of invariance analysis.
• Invariance conditions for continuous-time systems are rigorously derived from their discrete counterparts obtained via Euler discretization.
• The framework unifies the treatment of four distinct types of convex sets—polyhedra, polyhedral cones, ellipsoids, and Lorenz cones—under linear dynamics.
• The approach provides a systematic, algebraic method to verify invariance using conditions expressible in terms of quadratic forms and matrix inequalities.
• The results demonstrate that the S-lemma is a more versatile tool than traditional Lyapunov methods for invariance analysis in quadratic set classes.