STABILITY OF SWITCHED SYSTEMS Daniel Liberzon Coordinated - PowerPoint PPT Presentation

STABILITY OF SWITCHED SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign U.S.A. DISC HS, June 2003

SWITCHED vs. HYBRID SYSTEMS Switched system: • is a family of systems • is a switching signal Switching can be: • State-dependent or time-dependent • Autonomous or controlled Details of discrete behavior are “abstracted away” Properties of the continuous state : stability

STABILITY ISSUE Asymptotic stability of each subsystem is necessary for stability

STABILITY ISSUE unstable Asymptotic stability of each subsystem is necessary but not sufficient for stability (This only happens in dimensions 2 or higher)

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 Reduces to standard GAS notion for non-switched systems

COMPARISON FUNCTIONS is of class if • for each fixed • as for each class function Example: GUES GUAS:

COMMON LYAPUNOV FUNCTION Lyapunov theorem: is GAS iff pos def rad unbdd function s.t. Similarly: is GUAS iff s.t. where is positive definite

COMMON LYAPUNOV FUNCTION (continued) Unless is compact and is continuous, is not enough Example: as if

CONVEX COMBINATIONS Define Corollary: is GAS Proof:

SWITCHED LINEAR SYSTEMS LAS for every σ GUES ∃ common Lyapunov function but not necessarily quadratic: (LMIs)

COMMUTING STABLE MATRICES => GUES = = P { 1 , 2 }, A A A A 1 2 2 1 … σ = σ = σ = σ = 1 2 1 2 t s t s t 1 2 1 2 A t A s A 2 t A 1 s = … x ( t ) x ( 0 ) 2 k 1 k 1 1 e e e e A t t A s s + + + + ( ... ) ( ... ) = → 2 k 1 1 k 1 e e x ( 0 ) 0 ∃ quadratic common Lyap fcn: A T + = − P P A I 1 1 1 1 A T + = − P P A P 2 2 2 2 1

LIE ALGEBRAS and STABILITY = ∈ g { A , p P } Lie algebra: p L A = − [ A , A ] A A A A Lie bracket: 1 2 1 2 2 1 + 1 k 1 k k k g = = ⊂ ∃ = g , g is nilpotent if s.t. g [ g , g ] g k g 0 U ( = + ( 1 ) k ) ( k 1 ) ( k ) ( k ) ( k ) = ∃ = ⊂ g is solvable if s.t. g g , k g 0 g [ g , g ] g

SOLVABLE LIE ALGEBRA => GUES g Lie’s Theorem: is solvable triangular form Example: exponentially fast exp fast 0 ∃ diagonal quadratic common Lyap fcn

MORE GENERAL LIE ALGEBRAS = ⊕ g r s Levi decomposition: radical (max solvable ideal) s is compact => GUES, quadratic common Lyap fcn • s is not compact => not enough info in Lie algebra •

NONLINEAR SYSTEMS • Commuting systems => GUAS • Linearization (Lyapunov’s indirect method) • Nothing is known beyond this

REMARKS on LIE-ALGEBRAIC CRITERIA • Checkable conditions • Independent of representation • In terms of the original data • Not robust to small perturbations

SYSTEMS with SPECIAL STRUCTURE • Triangular systems • Feedback systems • passivity conditions • small-gain conditions • 2-D systems

TRIANGULAR SYSTEMS Recall: for linear systems, triangular => GUAS For nonlinear systems, not true in general Example: For stability need to know Not necessarily true

INPUT-TO-STATE STABILITY (ISS) Linear systems: is AS is ISS: • bounded bounded • Nonlinear systems: but bdd bdd, is input-to-state stable (ISS) if For switched systems, triangular + ISS => GUAS

FEEDBACK SYSTEMS: ABSOLUTE STABILITY Hurwitz Circle criterion: quadratic common Lyapunov function is strictly positive real (SPR): For this reduces to SPR (passivity) Popov criterion not suitable: depends on

FEEDBACK SYSTEMS: SMALL-GAIN THEOREM Hurwitz Small-gain theorem: quadratic common Lyapunov function

TWO-DIMENSIONAL SYSTEMS Necessary and sufficient conditions for GUES known since 1970s worst-case switching ∃ quadratic common Lyap fcn <=> convex combinations of Hurwitz

WEAK LYAPUNOV FUNCTION Barbashin-Krasovskii-LaSalle theorem: is GAS if pos def rad unbdd function s.t. • (weak Lyapunov function) • is not identically zero along any nonzero solution (observability with respect to ) Example: => GAS observable

COMMON WEAK LYAPUNOV FUNCTION Theorem: is GAS if • • observable for each • s.t. there are infinitely many switching intervals of length Extends to nonlinear switched systems and nonquadratic common weak Lyapunov functions using a suitable nonlinear observability notion

TWO BASIC PROBLEMS • Stability for arbitrary switching • Stability for constrained switching

MULTIPLE LYAPUNOV FUNCTIONS = = − x f ( x ) , x f ( x ) & & GAS 1 2 − V 1 , V respective Lyapunov functions 2 V ( t ) σ ( t ) = x f ( x ) & σ is GAS t σ = σ = σ = σ = 1 2 1 2 Very useful for analysis of state-dependent switching

MULTIPLE LYAPUNOV FUNCTIONS decreasing sequence V ( t ) σ ( t ) = x f ( x ) & σ is GAS decreasing sequence t σ = σ = σ = σ = 1 2 1 2

DWELL TIME − ≥ τ t 1 t , , ... t t The switching times satisfy + 1 2 i i D = = − x f ( x ) , x f ( x ) & & GES 1 2 dwell time − V 1 , V respective Lyapunov functions 2

DWELL TIME − ≥ τ t 1 t , , ... t t The switching times satisfy + 1 2 i i D = = − x f ( x ) , x f ( x ) & & GES 1 2 ∂ V 2 2 1 ≤ ≤ ≤ − λ a | x | V ( x ) b | x | , f ( x ) V ( x ) 1 1 1 1 1 1 ∂ x ∂ V 2 2 ≤ ≤ 2 ≤ − λ a | x | V ( x ) b | x | , f ( x ) V ( x ) 2 2 2 ∂ 2 2 2 x < Need: V ( t ) V ( t ) 1 2 1 0 σ = σ = σ = 1 2 1 t t t t 0 1 2

DWELL TIME − ≥ τ t 1 t , , ... t t The switching times satisfy + 1 2 i i D = = − x f ( x ) , x f ( x ) & & GES 1 2 ∂ V 2 2 1 ≤ ≤ ≤ − λ a | x | V ( x ) b | x | , f ( x ) V ( x ) 1 1 1 1 1 1 ∂ x ∂ V 2 2 ≤ ≤ 2 ≤ − λ a | x | V ( x ) b | x | , f ( x ) V ( x ) 2 2 2 ∂ 2 2 2 x < Need: V ( t ) V ( t ) 1 2 1 0 < 1 must be b b τ − λ 1 ≤ ≤ 1 V 1 t ( 2 ) V ( 2 t ) e 2 D V ( 1 t ) a 2 a 2 2 2 b b τ b b τ − λ − λ + λ 1 2 ( ) ≤ 2 D e V ( 1 t ) ≤ 1 2 2 D e V ( 0 t ) 1 1 a a a a 1 2 2 1 1

AVERAGE DWELL TIME − T t ≤ + N ( T , t ) N τ σ 0 AD ( T t , ) # of switches on average dwell time − < τ = 0 − T t no switching: cannot switch if N AD 0 − < τ = 1 − T t dwell time: cannot switch twice if N 0 AD = x f ( x ) & σ

AVERAGE DWELL TIME − T t ≤ + N ( T , t ) N τ σ 0 AD ( T t , ) # of switches on average dwell time = x f ( x ) & σ = α ≤ ≤ α x f ( x ) (| x | ) V ( x ) (| x | ) & 1 p 2 σ ∂ V => p is GAS ≤ − λ f ( x ) V ( x ) p p ∂ x µ log τ if > ≤ µ ∈ V ( x ) V ( x ), p , q P AD λ p q

SWITCHED LINEAR SYSTEMS = x A x & σ τ σ • GUES over all with large enough AD • Finite induced norms for = + x A x B u & σ σ = y C x σ • The case when some subsystems are unstable

STATE-DEPENDENT SWITCHING Switched system unstable for some no common But switched system is stable for (many) other switch on the axes is a Lyapunov function

STATE-DEPENDENT SWITCHING Switched system unstable for some no common But switched system is stable for (many) other Switch on y -axis level sets of level sets of => GAS t σ = σ = σ = 1 2 1

MULTIPLE WEAK LYAPUNOV FUNCTIONS Theorem: is GAS if • (each is a weak Lyapunov function) • observable for each • s.t. there are infinitely many switching intervals of length • For every pair of switching times s.t. have

STABILIZATION by SWITCHING = = − x & A 1 , & x x A x both unstable 2 = α + − α α ∈ Assume: stable for some A A ( 1 ) A ( 0 , 1 ) 1 2 A T + PA < P 0