A LIE-ALGEBRAIC CONDITION for STABILITY of SWITCHED NONLINEAR SYSTEMS CDC ’04 Michael Margaliot Tel Aviv University, Israel Daniel Liberzon Univ. of Illinois.

Slides:

Advertisements

Similar presentations

COMMON WEAK LYAPUNOV FUNCTIONS and OBSERVABILITY Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois.

Advertisements

1 STABILITY OF SWITCHED SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign.

1 of 16 SMALL - GAIN THEOREMS of LASALLE TYPE for HYBRID SYSTEMS Daniel Liberzon (Urbana-Champaign) Dragan Nešić (Melbourne) Andy Teel (Santa Barbara)

NONLINEAR HYBRID CONTROL with LIMITED INFORMATION Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois.

CONTROL with LIMITED INFORMATION Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign.

OUTPUT – INPUT STABILITY Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign.

INPUT-TO-STATE STABILITY of SWITCHED SYSTEMS Debasish Chatterjee, Linh Vu, Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer.

TOWARDS ROBUST LIE-ALGEBRAIC STABILITY CONDITIONS for SWITCHED LINEAR SYSTEMS 49 th CDC, Atlanta, GA, Dec 2010 Daniel Liberzon Univ. of Illinois, Urbana-Champaign,

ISS of Switched Systems and Application to Adaptive Control

INTRODUCTION to SWITCHED SYSTEMS ; STABILITY under ARBITRARY SWITCHING

THE ROLE OF LIE BRACKETS IN STABILITY OF LINEAR AND NONLINEAR SWITCHED SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical &

IFAC AIRTC, Budapest, October 2000 On the Dynamic Instability of a Class of Switching System Robert Noel Shorten Department of Computer Science National.

NONLINEAR OBSERVABILITY NOTIONS and STABILITY of SWITCHED SYSTEMS CDC ’02 João Hespanha Univ. of California at Santa Barbara Daniel Liberzon Univ. of Illinois.

TOWARDS ROBUST LIE-ALGEBRAIC STABILITY CONDITIONS for SWITCHED LINEAR SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical &

1 Nonlinear Control Design for LDIs via Convex Hull Quadratic Lyapunov Functions Tingshu Hu University of Massachusetts, Lowell.

GRADIENT ALGORITHMS for COMMON LYAPUNOV FUNCTIONS Daniel Liberzon Univ. of Illinois at Urbana-Champaign, U.S.A. Roberto Tempo IEIIT-CNR, Politecnico di.

1 of 9 ON ALMOST LYAPUNOV FUNCTIONS Daniel Liberzon University of Illinois, Urbana-Champaign, U.S.A. TexPoint fonts used in EMF. Read the TexPoint manual.

1 Formal Models for Stability Analysis : Verifying Average Dwell Time * Sayan Mitra MIT,CSAIL Research Qualifying Exam 20 th December.

Lecture #13 Stability under slow switching & state-dependent switching João P. Hespanha University of California at Santa Barbara Hybrid Control and Switched.

STABILITY under CONSTRAINED SWITCHING Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign.

1 Stability of Hybrid Automata with Average Dwell Time: An Invariant Approach Daniel Liberzon Coordinated Science Laboratory University of Illinois at.

COMMUTATION RELATIONS and STABILITY of SWITCHED SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of.

1 Stability Analysis of Switched Systems: A Variational Approach Michael Margaliot School of EE-Systems Tel Aviv University Joint work with Daniel Liberzon.

1 A Lyapunov Approach to Frequency Analysis Tingshu Hu, Andy Teel UC Santa Barbara Zongli Lin University of Virginia.

1 Absolute Stability with a Generalized Sector Condition Tingshu Hu.

Modern Control Systems1 Lecture 07 Analysis (III) -- Stability 7.1 Bounded-Input Bounded-Output (BIBO) Stability 7.2 Asymptotic Stability 7.3 Lyapunov.

1 Stability Analysis of Continuous- Time Switched Systems: A Variational Approach Michael Margaliot School of EE-Systems Tel Aviv University, Israel Joint.

CONTROL with LIMITED INFORMATION ; SWITCHING ADAPTIVE CONTROL Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ.

1 of 12 COMMUTATORS, ROBUSTNESS, and STABILITY of SWITCHED LINEAR SYSTEMS SIAM Conference on Control & its Applications, Paris, July 2015 Daniel Liberzon.

QUANTIZED CONTROL and GEOMETRIC OPTIMIZATION Francesco Bullo and Daniel Liberzon Coordinated Science Laboratory Univ. of Illinois at Urbana-Champaign U.S.A.

CONTROL of NONLINEAR SYSTEMS with LIMITED INFORMATION Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of.

1 of 17 NORM - CONTROLLABILITY, or How a Nonlinear System Responds to Large Inputs Daniel Liberzon Univ. of Illinois at Urbana-Champaign, U.S.A. Workshop.

CONTROL of NONLINEAR SYSTEMS under COMMUNICATION CONSTRAINTS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ.

MEETING THE NEED FOR ROBUSTIFIED NONLINEAR SYSTEM THEORY CONCEPTS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng.,

Stability Analysis of Linear Switched Systems: An Optimal Control Approach 1 Michael Margaliot School of Elec. Eng. Tel Aviv University, Israel Joint work.

1 Stability Analysis of Linear Switched Systems: An Optimal Control Approach Michael Margaliot School of Elec. Eng. Tel Aviv University, Israel Joint work.

CONTROL with LIMITED INFORMATION Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign.

TUTORIAL on LOGIC-BASED CONTROL Part I: SWITCHED CONTROL SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng.,

QUANTIZED OUTPUT FEEDBACK CONTROL of NONLINEAR SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of.

OUTPUT – INPUT STABILITY and FEEDBACK STABILIZATION Daniel Liberzon CDC ’03 Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ.

QUANTIZATION and DELAY EFFECTS in NONLINEAR CONTROL SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ.

Lecture #11 Stability of switched system: Arbitrary switching João P. Hespanha University of California at Santa Barbara Hybrid Control and Switched Systems.

Daniel Liberzon Coordinated Science Laboratory and

Feedback Linearization Presented by : Shubham Bhat (ECES-817)

Mathe III Lecture 7 Mathe III Lecture 7. 2 Second Order Differential Equations The simplest possible equation of this type is:

AUTOMATIC CONTROL THEORY II Slovak University of Technology Faculty of Material Science and Technology in Trnava.

Domain of Attraction Remarks on the domain of attraction

COMMUTATION RELATIONS and STABILITY of SWITCHED SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of.

OUTPUT-INPUT STABILITY: A NEW VARIANT OF THE MINIMUM-PHASE PROPERTY FOR NONLINEAR SYSTEMS D. Liberzon Univ. of Illinois at Urbana-Champaign, USA A. S.

Feedback Stabilization of Nonlinear Singularly Perturbed Systems MENG Bo JING Yuanwei SHEN Chao College of Information Science and Engineering, Northeastern.

Lecture #12 Controller realizations for stable switching João P. Hespanha University of California at Santa Barbara Hybrid Control and Switched Systems.

STABILIZATION by QUANTIZED FEEDBACK : HYBRID CONTROL APPROACH Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ.

SMALL-GAIN APPROACH to STABILITY ANALYSIS of HYBRID SYSTEMS CDC ’05 Dragan Nešić University of Melbourne, Australia Daniel Liberzon Univ. of Illinois at.

NONLINEAR CONTROL with LIMITED INFORMATION Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at.

STABILITY of SWITCHED SYSTEMS – family of asymptotically stable systems – piecewise constant switching signal Want GUAS w.r.t. want GUES For switched linear.

Lecture #7 Stability and convergence of ODEs João P. Hespanha University of California at Santa Barbara Hybrid Control and Switched Systems NO CLASSES.

Stability Analysis of Positive Linear Switched Systems: A Variational Approach 1 Michael Margaliot School of Elec. Eng. -Systems Tel Aviv University, Israel.

TOWARDS a UNIFIED FRAMEWORK for NONLINEAR CONTROL with LIMITED INFORMATION Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer.

Ch 9.6: Liapunov’s Second Method In Section 9.3 we showed how the stability of a critical point of an almost linear system can usually be determined from.

Stability and instability in nonlinear dynamical systems

§7-4 Lyapunov Direct Method

Input-to-State Stability for Switched Systems

Autonomous Cyber-Physical Systems: Dynamical Systems

SWITCHED SYSTEMS Switched system: is a family of systems

SWITCHED SYSTEMS Switched system: is a family of systems

Sliding Mode : An Introduction

Lecture #10 Switched systems

Michael Margaliot School of Elec. Eng. -Systems

On Topological Entropy and Stability of Switched Linear Systems

Presentation transcript:

A LIE-ALGEBRAIC CONDITION for STABILITY of SWITCHED NONLINEAR SYSTEMS CDC ’04 Michael Margaliot Tel Aviv University, Israel Daniel Liberzon Univ. of Illinois at Urbana-Champaign, USA

SWITCHED vs. HYBRID SYSTEMS : stability Switching can be: State-dependent or time-dependent Autonomous or controlled Hybrid systems give rise to classes of switching signals Properties of the continuous state Switched system: is a family of systems is a switching signal Further abstraction/relaxation: diff. inclusion, measurable switching

STABILITY ISSUE unstable Asymptotic stability of each subsystem is not sufficient for stability

TWO BASIC PROBLEMS Stability for arbitrary switching Stability for constrained switching

TWO BASIC PROBLEMS Stability for arbitrary switching Stability for constrained switching

GLOBAL UNIFORM ASYMPTOTIC STABILITY GUAS is Lyapunov stability plus asymptotic convergence GUES:

SWITCHED LINEAR SYSTEMS Extension based only on L.A. is not possible [Agrachev & L ’01] Lie algebra w.r.t. Quadratic common Lyapunov function exists in all these cases Assuming GES of all modes, GUES is guaranteed for: commuting subsystems: nilpotent Lie algebras (suff. high-order Lie brackets are 0) e.g. solvable Lie algebras (triangular up to coord. transf.) solvable + compact (purely imaginary eigenvalues)

SWITCHED NONLINEAR SYSTEMS Global results beyond commuting case – ??? Commuting systems Linearization (Lyapunov’s indirect method) => GUAS [Mancilla-Aguilar, Shim et al., Vu & L] [Unsolved Problems in Math. Systems and Control Theory]

SPECIAL CASE globally asymptotically stable Want to show: is GUAS Will show: differential inclusion is GAS

OPTIMAL CONTROL APPROACH Associated control system: where (original switched system ) Worst-case control law [Pyatnitskiy, Rapoport, Boscain, Margaliot] : fix and small enough

MAXIMUM PRINCIPLE is linear in at most 1 switch (unless ) GAS Optimal control: (along optimal trajectory)

SINGULARITY Need: nonzero on ideal generated by (strong extremality) At most 2 switches GAS Know: nonzero on strongly extremal (time-optimal control for auxiliary system in ) constant control (e.g., ) Sussmann ’79:

GENERAL CASE GAS Want: polynomial of degree (proof – by induction on ) bang-bang with switches

THEOREM Suppose: GAS, backward complete, analytic s.t. and Then differential inclusion is GAS (and switched system is GUAS)