Stability of switched systems: a Lie-algebraic condition (

We present a sucient condition for asymptotic stability of a switched linear system in terms of the Lie algebra generated by the individual matrices. Namely, if this Lie algebra is solvable, then the switched system is exponentially stable for arbitrary switching. In fact, we show that any family of linear systems satisfying this condition possesses a quadratic common Lyapunov function. We also discuss the implications of this result for switched nonlinear systems. c 1999 Elsevier Science B.V. All rights reserved.

• Publication year

  1999

• Venue

  Systems & Control Letters

• Publication date

  1999-07-01

• Fields of study

  Mathematics, Engineering

• Identifiers
  DOI 10.1016/S0167-6911(99)00012-2
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

• Common Lyapunov Function for Exponentially Stable Nonlinear Systems

  2001cited by this paper
• Lie-Algebraic Stability Criteria for Switched Systems

  2001cited by this paper
• On the stability and existence of common Lyapunov functions for stable linear switching systems

  1998cited by this paper
• Linear Matrix Inequalities In System And Control Theory

  1998cited by this paper
• A solution to the common Lyapunov function problem for continuous-time systems

  1997cited by this paper
• Control Using Logic-Based Switching

  1997cited by this paper
• A sufficient condition for the existence of a common Lyapunov function for two second order linear systems

  1997cited by this paper
• On a common quadratic Lyapunov function for widely distant systems

  1997cited by this paper
• On the existence of a Lyapunov function for a family of switching systems

  1996cited by this paper
• Stability of discrete linear inclusion

  1995cited by this paper
• Linear Matrix Inequalities in Systems and Control Theory

  1994cited by this paper
• A common Lyapunov function for stable LTI systems with commuting A-matrices

  1994cited by this paper
• Criteria of asymptotic stability of differential and difference inclusions encountered in control theory

  1989cited by this paper
• Absolute characteristic exponent of a class of linear nonstatinoary systems of differential equations

  1988cited by this paper
• Matrix analysis

  1985cited by this paper
• Notes on Lie algebras

  1969cited by this paper

• Analysis of Switched Linear Systems Using the Invariant Ellipsoid Technique

  2025cites this paper
• Stabilization and Uniformity of Controllability of Switched Linear Systems with Unstable Subsystems

  2025cites this paper
• Correlation Study of Discrete Time for Markovian Jump Nonlinear Systems

  2025cites this paper
• Estimation and output feedback stabilization for switched linear discrete-time systems with delayed-outputs

  2025cites this paper
• Asynchronous H∞ Control of Switched Systems and Its Applications: An Admissible Chain-Dependent Average Dwell Time Switching Strategy

  2025cites this paper
• Hybrid dynamical systems modeling of power systems

  2025cites this paper
• Command governor schemes for dynamical systems subject to rate-bounded disturbances

  2025cites this paper
• Performance estimation of continuous-time autonomous switched linear systems via square coordinate transformations

  2024cites this paper
• Semi-time-dependent stabilization results for asynchronously switched systems: An extended persistent dwell time switching strategy involving admissible transition information

  2024cites this paper
• Period-sampled switching event-triggered control of discrete-time impulsive switched systems under two asynchronous behaviors

  2024cites this paper
• Input-to-state stability of impulsive switched systems under mode-dependent event-triggered impulsive control

  2024cites this paper
• Exponential stability analysis of switched nonlinear systems: A linearization method

  2024cites this paper
• Coordinated predictive control of networked multiagent systems via distributed cloud computing with time-varying transmission delays

  2024cites this paper
• Switched Flow Matching: Eliminating Singularities via Switching ODEs

  2024cites this paper
• Stability Analysis of Asynchronous Impulsive Switched T-S Fuzzy Systems Based on the Admissible Edge-dependent Scheme

  2024cites this paper
• Uniformity of Controllability and Stabilization of Switched Linear Time-Varying Systems

  2024cites this paper
• Finite time stability condition for switched systems under restricted switching

  2024cites this paper
• Mean-square stability of discrete-time switched systems under modeled random switching

  2023cites this paper
• Observer Design for A Class of Switched Linear Systems via A Parametric Approach

  2023cites this paper
• Regret-Guaranteed Safe Switching with Minimum Cost: LQR Setting with Unknown Dynamics

  2023cites this paper
• An approach to the stability of reset switched systems using dynamics without reset

  2023cites this paper
• Bounded switched linear swarms

  2023cites this paper
• Uniform Global Stability of Switched Nonlinear Systems in the Koopman Operator Framework

  2023cites this paper
• Exponential Consensus of Multiple Agents Over Dynamic Network Topology: Controllability, Connectivity, and Compactness

  2023cites this paper
• Optimal Control Strategy for Switched Nonlinear Systems Based on Fire Hawk Optimizer Algorithm

  2023cites this paper
• Commutation relations and stability of switched systems: a personal history

  2023influential citation
• Exponential stabilization of asynchronously switched linear systems with constant time‐delay: Observer‐based event‐triggered control

  2023influential citation
• Robust predictive control of perturbed switched linear systems with stable and unstable subsystems in the presence of uncertainties

  2023cites this paper
• Reachable set estimation for switched linear systems with state-dependent switching and bumpless transfer based event-triggered control.

  2023cites this paper
• The minimum principle of hybrid optimal control theory

  2023cites this paper
• Switching controller synthesis for discrete-time switched linear systems with average dwell time

  2023cites this paper
• Optimal decision strategy for discrete-time Markovian jump linear systems

  2022cites this paper
• CMMSE: An approach to the stability of Reset Switched Systems using a dynamics without reset

  2022cites this paper
• Existence, stability and controllability results for a class of switched evolution system with impulses over arbitrary time domain

  2022cites this paper
• Stability analysis of switched ARX models and application to learning with guarantees

  2022cites this paper
• A new method for control of nonlinear state-dependent switched systems

  2022cites this paper
• A convex optimization approach to synthesizing state feedback data-driven controllers for switched linear systems

  2022cites this paper
• Stability of Logical Dynamic Systems With a Class of Constrained Switching

  2022cites this paper
• Construction of a Lyapunov function for a linear large-scale periodic system with possibly unstable subsystems

  2022cites this paper
• Stability of linear systems with bounded switching intervals

  2022cites this paper
• Incremental nonlinear stability analysis of stochastic systems perturbed by Lévy noise

  2021cites this paper
• Stability of bimodal planar linear switched systems

  2021cites this paper
• Model Predictive Control based on FSP-ADT Switching Sequence Constraint

  2021cites this paper
• New stability criteria of switched systems with unstable modes under a weighted ADT scheme

  2021cites this paper
• Stability Analysis of Conewise Affine Dynamical Systems Using Conewise Linear Lyapunov Functions

  2021cites this paper
• Passivity and Reduced-order Feedback Passification of Discrete-Time Switched Systems with Mode-Dependent Persistent Dwell Time

  2021cites this paper
• The existence and uniqueness of the solutions of the nonlinear on–off switched systems with switching at variable times

  2021cites this paper
• Automated Controller Calibration by Kalman Filtering

  2021cites this paper
• Stability and stabilization of impulsive switched system with inappropriate impulsive switching signals under asynchronous switching

  2021cites this paper
• Lie Algebraic Criteria for Stability of Switched Systems of Differential Algebraic Equations (DAEs)

  2021influential citation
• APiecewiseSmoothControl-LyapunovFunctionFramework for SwitchingStabilization

  2021cites this paper
• Stability of bimodal planar switched linear systems with both stable and unstable subsystems

  2021cites this paper
• Stability and stabilization of linear periodic systems: A commutator calculus approach

  2021cites this paper
• Synthesis and L2‐gain analysis for switched systems under event‐triggered switching

  2021cites this paper
• Simultaneous and Sequential Control Design for Discrete-Time Switched Linear Systems Using Semi-Definite Programming

  2021cites this paper
• Stabilizing backstepping controller design for arbitrarily switched complex nonlinear system

  2020cites this paper
• Influence of the Number of Thresholds on the Dynamics of Models with Switchings of the Biological Systems

  2020cites this paper
• Robust integral sliding mode control for uncertain switched systems under arbitrary switching rules

  2020cites this paper
• Max-min Lyapunov functions for switched systems and related differential inclusions

  2020cites this paper
• Exponential stability of switched positive systems with all modes being unstable

  2020cites this paper
• Stability Analysis of DC-DC Converters Applying Lie Algebra

  2020influential citation
• Event‐based switching control for networked switched systems under nonperiodic DoS jamming attacks

  2020cites this paper
• Lie-Algebraic Criterion for Stability of Switched Differential-Algebraic Equations

  2020cites this paper
• Nonlinear Analysis: Hybrid Systems

  2020cites this paper
• Optimal control of discrete-time switched linear systems

  2020cites this paper
• Finite-Time Stability and Stabilization of Discrete-Time Switching Markov Jump Linear System

  2020cites this paper
• Topological entropy of switched linear systems: general matrices and matrices with commutation relations

  2020cites this paper
• Load/Frequency Control in the presence of Renewable Energy Systems: a Reference-Offset Governor approach.

  2020cites this paper
• Robust stabilization of non-linear non-autonomous control systems with periodic linear approximation

  2020cites this paper
• A limit inferior Φ ‐dependent average dwell time approach for stability analysis of switched systems

  2020cites this paper
• Hybrid Nonovershooting Set-Point Pressure Regulation for a Wet Clutch

  2020cites this paper
• Input-to-state stability for impulsive switched systems with incommensurate impulsive switching signals

  2020cites this paper
• Switching controller synthesis for discrete-time switched linear systems with average dwell time

  2020cites this paper
• Existence of Initial Condition Independent Stabilising Switching Functions

  2020cites this paper
• Structured Feedback Synthesis for Stability and Performance of Switched Systems

  2020cites this paper
• System with Switchings as Models of Regulatory Modules in Genomic Cell Systems

  2020cites this paper
• Normal forms of matrix words for stability analysis of discrete-time switched linear systems

  2020cites this paper
• SPR based design conditions for quadratic stability of multi-mode switched linear systems

  2020cites this paper
• Stability Analysis of Switched Systems Under Φ-Dependent Average Dwell Time Approach

  2020cites this paper
• Stability Analysis of Linear Impulsive Systems using Lie-Algebraic Methods

  2020cites this paper
• A new approach to stability analysis of switched systems composed of two subsystems with average dwell time switching

  2020cites this paper
• Ensuring the Stability of Power Systems Against Dynamic Load Altering Attacks: A Robust Control Scheme Using Energy Storage Systems

  2020cites this paper
• Data-driven switching modeling for MPC using Regression Trees and Random Forests

  2020cites this paper
• Stability analysis for discrete-time switched systems with stable and unstable modes based on a weighted average dwell time approach

  2020cites this paper
• Exponential Stability of Impulsive Switched Nonlinear Systems with Time Delay: The Average Dwell Time Approach

  2019cites this paper
• Global exponential stability of 2D switched positive nonlinear systems described by the Roesser model

  2019cites this paper
• Comparison principle for linear differential equations with periodic impulsive action

  2019cites this paper
• Robust integral sliding mode control strategy under designed switching rules for uncertain switched linear systems via multiple Lyapunov functions

  2019cites this paper
• Formal Synthesis of Stabilizing Controllers for Periodically Controlled Linear Switched Systems

  2019cites this paper
• A stability criterion for asynchronously switched linear systems via sampled‐data control

  2019influential citation
• Neural networks-based adaptive finite-time control of switched nonlinear systems under time-varying actuator failures

  2019cites this paper
• Global mean square exponential stability and stabilization of uncertain switched delay systems with Lévy noise and flexible switching signals

  2019cites this paper
• Introduction

  2019cites this paper
• Dwell time estimation algorithm for switched affine linear systems

  2019cites this paper
• Switch Controller Design and H~ Performance Achievement

  2019cites this paper
• Persistent Bounded Disturbance Rejection for Switched Linear Systems

  2019cites this paper
• Normal forms for discrete-time switched linear systems

  2019cites this paper
• Stable, Concurrent Controller Composition for Multi-Objective Robotic Tasks

  2019cites this paper
• Nonlinear and Hybrid Feedbacks with Continuous-Time Linear Systems. (Rétroactions non linéaires et hybrides avec systèmes linéaires à temps continu)

  2019cites this paper
• Stability of abstract linear switched impulsive differential equations

  2019cites this paper