Transcript Lecture 1 - Daniel Liberzon
Slide 1
INTRODUCTION to SWITCHED SYSTEMS ;
STABILITY under ARBITRARY SWITCHING
Daniel Liberzon
Coordinated Science Laboratory and
Dept. of Electrical & Computer Eng.,
Univ. of Illinois at Urbana-Champaign
IAAC Workshop, Herzliya, Israel, June 1, 2009
Slide 2
SWITCHED and HYBRID SYSTEMS
Hybrid systems combine continuous and discrete dynamics
Which practical systems are hybrid?
cell division
thermostat
electric circuit
stick shift
walking
Which practical systems are not hybrid?
More tractable models of continuous phenomena
Slide 3
MODELS of HYBRID SYSTEMS
…
[Van der Schaft–Schumacher ’00]
continuous
discrete
[Nešić–L ‘05]
[Proceedings of HSCC]
Slide 4
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”
Discrete dynamics
classes of switching signals
Properties of the continuous state
: stability and beyond
Slide 5
STABILITY ISSUE
unstable
Asymptotic stability of each subsystem is
not sufficient for stability
Slide 6
TWO BASIC PROBLEMS
• Stability for arbitrary switching
• Stability for constrained switching
Slide 7
TWO BASIC PROBLEMS
• Stability for arbitrary switching
• Stability for constrained switching
Slide 8
GLOBAL UNIFORM ASYMPTOTIC STABILITY
GUAS is: Lyapunov stability
plus asymptotic convergence
GUES:
Slide 9
COMMON LYAPUNOV FUNCTION
is GUAS if (and only if)
where
is positive definite
quadratic
is GUES
s.t.
Slide 10
OUTLINE
Stability criteria to be discussed:
• Commutation relations (Lie algebras)
• Feedback systems (absolute stability)
• Observability and LaSalle-like theorems
Common Lyapunov functions will play a central role
Slide 11
COMMUTING STABLE MATRICES => GUES
(commuting Hurwitz matrices)
For
subsystems – similarly
Slide 12
COMMUTING STABLE MATRICES => GUES
Alternative proof:
quadratic common Lyapunov function
[Narendra–Balakrishnan ’94]
..
.
is a common Lyapunov function
Slide 13
NILPOTENT LIE ALGEBRA => GUES
Lie algebra:
Lie bracket:
Nilpotent means sufficiently high-order Lie brackets are 0
For example:
(2nd-order nilpotent)
Recall: in commuting case
In 2nd-order nilpotent case
Hence still GUES [Gurvits ’95]
Slide 14
SOLVABLE LIE ALGEBRA => GUES
Larger class containing all nilpotent Lie algebras
Suff. high-order brackets with certain structure are 0
Lie’s Theorem:
is solvable
triangular form
Example:
exponentially fast
exp fast
quadratic common Lyap fcn
diagonal
[Kutepov ’82, L–Hespanha–Morse ’99]
Slide 15
SUMMARY: LINEAR CASE
Lie algebra
w.r.t.
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)
Quadratic common Lyapunov function exists in all these cases
No further extension based on Lie algebra only [Agrachev–L ’01]
Slide 16
SWITCHED NONLINEAR SYSTEMS
Lie bracket of nonlinear vector fields:
Reduces to earlier notion for linear vector fields
(modulo the sign)
Slide 17
SWITCHED NONLINEAR SYSTEMS
• Commuting systems
GUAS
Can prove by trajectory analysis [Mancilla-Aguilar ’00]
or common Lyapunov function [Shim et al. ’98, Vu–L ’05]
• Linearization (Lyapunov’s indirect method)
• Global results beyond commuting case – ?
[Unsolved Problems in Math. Systems & Control Theory ’04]
Slide 18
SPECIAL CASE
globally asymptotically stable
Want to show:
Will show: differential inclusion
is GAS
is GUAS
Slide 19
OPTIMAL CONTROL APPROACH
Associated control system:
where
(original switched system
)
Worst-case control law [Pyatnitskiy, Rapoport, Boscain, Margaliot]:
fix
and small enough
Slide 20
MAXIMUM PRINCIPLE
(along optimal trajectory)
Optimal control:
is linear in
(unless
at most 1 switch
GAS
)
Slide 21
GENERAL CASE
Want:
polynomial of degree
(proof – by induction on
bang-bang with
switches
GAS
[Margaliot–L ’06, Sharon–Margaliot ’07]
)
Slide 22
REMARKS on LIE-ALGEBRAIC CRITERIA
• Checkable conditions
• In terms of the original data
• Independent of representation
• Not robust to small perturbations
In any neighborhood of any pair of
matrices
there exists a pair of matrices generating the entire
Lie algebra
[Agrachev–L ’01]
How to measure closeness to a “nice” Lie algebra?
Slide 23
FEEDBACK SYSTEMS: ABSOLUTE STABILITY
controllable
Circle criterion:
quadratic common Lyapunov function
is strictly positive real (SPR):
For
this reduces to
Popov criterion not suitable:
depends on
SPR (passivity)
Slide 24
FEEDBACK SYSTEMS: SMALL-GAIN THEOREM
controllable
Small-gain theorem:
quadratic common Lyapunov function
Slide 25
OBSERVABILITY and ASYMPTOTIC STABILITY
Barbashin-Krasovskii-LaSalle theorem:
is GAS if
s.t.
•
(weak Lyapunov function)
•
is not identically zero along any nonzero solution
(observability with respect to
)
Example:
observable
=> GAS
Slide 26
SWITCHED LINEAR SYSTEMS
[Hespanha ’04]
Theorem (common weak Lyapunov function):
Switched linear system is GAS if
•
s.t.
•
observable for each
•
.
infinitely many switching intervals of length
To handle nonlinear switched systems and
non-quadratic weak Lyapunov functions,
need a suitable nonlinear observability notion
Slide 27
SWITCHED NONLINEAR SYSTEMS
Theorem (common weak Lyapunov function):
Switched system is GAS if
•
•
s.t.
infinitely many switching intervals of length
• Each system
is norm-observable:
[Hespanha–L–Sontag–Angeli ’05]
INTRODUCTION to SWITCHED SYSTEMS ;
STABILITY under ARBITRARY SWITCHING
Daniel Liberzon
Coordinated Science Laboratory and
Dept. of Electrical & Computer Eng.,
Univ. of Illinois at Urbana-Champaign
IAAC Workshop, Herzliya, Israel, June 1, 2009
Slide 2
SWITCHED and HYBRID SYSTEMS
Hybrid systems combine continuous and discrete dynamics
Which practical systems are hybrid?
cell division
thermostat
electric circuit
stick shift
walking
Which practical systems are not hybrid?
More tractable models of continuous phenomena
Slide 3
MODELS of HYBRID SYSTEMS
…
[Van der Schaft–Schumacher ’00]
continuous
discrete
[Nešić–L ‘05]
[Proceedings of HSCC]
Slide 4
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”
Discrete dynamics
classes of switching signals
Properties of the continuous state
: stability and beyond
Slide 5
STABILITY ISSUE
unstable
Asymptotic stability of each subsystem is
not sufficient for stability
Slide 6
TWO BASIC PROBLEMS
• Stability for arbitrary switching
• Stability for constrained switching
Slide 7
TWO BASIC PROBLEMS
• Stability for arbitrary switching
• Stability for constrained switching
Slide 8
GLOBAL UNIFORM ASYMPTOTIC STABILITY
GUAS is: Lyapunov stability
plus asymptotic convergence
GUES:
Slide 9
COMMON LYAPUNOV FUNCTION
is GUAS if (and only if)
where
is positive definite
quadratic
is GUES
s.t.
Slide 10
OUTLINE
Stability criteria to be discussed:
• Commutation relations (Lie algebras)
• Feedback systems (absolute stability)
• Observability and LaSalle-like theorems
Common Lyapunov functions will play a central role
Slide 11
COMMUTING STABLE MATRICES => GUES
(commuting Hurwitz matrices)
For
subsystems – similarly
Slide 12
COMMUTING STABLE MATRICES => GUES
Alternative proof:
quadratic common Lyapunov function
[Narendra–Balakrishnan ’94]
..
.
is a common Lyapunov function
Slide 13
NILPOTENT LIE ALGEBRA => GUES
Lie algebra:
Lie bracket:
Nilpotent means sufficiently high-order Lie brackets are 0
For example:
(2nd-order nilpotent)
Recall: in commuting case
In 2nd-order nilpotent case
Hence still GUES [Gurvits ’95]
Slide 14
SOLVABLE LIE ALGEBRA => GUES
Larger class containing all nilpotent Lie algebras
Suff. high-order brackets with certain structure are 0
Lie’s Theorem:
is solvable
triangular form
Example:
exponentially fast
exp fast
quadratic common Lyap fcn
diagonal
[Kutepov ’82, L–Hespanha–Morse ’99]
Slide 15
SUMMARY: LINEAR CASE
Lie algebra
w.r.t.
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)
Quadratic common Lyapunov function exists in all these cases
No further extension based on Lie algebra only [Agrachev–L ’01]
Slide 16
SWITCHED NONLINEAR SYSTEMS
Lie bracket of nonlinear vector fields:
Reduces to earlier notion for linear vector fields
(modulo the sign)
Slide 17
SWITCHED NONLINEAR SYSTEMS
• Commuting systems
GUAS
Can prove by trajectory analysis [Mancilla-Aguilar ’00]
or common Lyapunov function [Shim et al. ’98, Vu–L ’05]
• Linearization (Lyapunov’s indirect method)
• Global results beyond commuting case – ?
[Unsolved Problems in Math. Systems & Control Theory ’04]
Slide 18
SPECIAL CASE
globally asymptotically stable
Want to show:
Will show: differential inclusion
is GAS
is GUAS
Slide 19
OPTIMAL CONTROL APPROACH
Associated control system:
where
(original switched system
)
Worst-case control law [Pyatnitskiy, Rapoport, Boscain, Margaliot]:
fix
and small enough
Slide 20
MAXIMUM PRINCIPLE
(along optimal trajectory)
Optimal control:
is linear in
(unless
at most 1 switch
GAS
)
Slide 21
GENERAL CASE
Want:
polynomial of degree
(proof – by induction on
bang-bang with
switches
GAS
[Margaliot–L ’06, Sharon–Margaliot ’07]
)
Slide 22
REMARKS on LIE-ALGEBRAIC CRITERIA
• Checkable conditions
• In terms of the original data
• Independent of representation
• Not robust to small perturbations
In any neighborhood of any pair of
matrices
there exists a pair of matrices generating the entire
Lie algebra
[Agrachev–L ’01]
How to measure closeness to a “nice” Lie algebra?
Slide 23
FEEDBACK SYSTEMS: ABSOLUTE STABILITY
controllable
Circle criterion:
quadratic common Lyapunov function
is strictly positive real (SPR):
For
this reduces to
Popov criterion not suitable:
depends on
SPR (passivity)
Slide 24
FEEDBACK SYSTEMS: SMALL-GAIN THEOREM
controllable
Small-gain theorem:
quadratic common Lyapunov function
Slide 25
OBSERVABILITY and ASYMPTOTIC STABILITY
Barbashin-Krasovskii-LaSalle theorem:
is GAS if
s.t.
•
(weak Lyapunov function)
•
is not identically zero along any nonzero solution
(observability with respect to
)
Example:
observable
=> GAS
Slide 26
SWITCHED LINEAR SYSTEMS
[Hespanha ’04]
Theorem (common weak Lyapunov function):
Switched linear system is GAS if
•
s.t.
•
observable for each
•
.
infinitely many switching intervals of length
To handle nonlinear switched systems and
non-quadratic weak Lyapunov functions,
need a suitable nonlinear observability notion
Slide 27
SWITCHED NONLINEAR SYSTEMS
Theorem (common weak Lyapunov function):
Switched system is GAS if
•
•
s.t.
infinitely many switching intervals of length
• Each system
is norm-observable:
[Hespanha–L–Sontag–Angeli ’05]