stability under constrained switching ; switched systems with inputs and outputs
DESCRIPTION
STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS. Daniel Liberzon. Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign. IAAC Workshop, Herzliya, Israel, June 1, 2009. TWO BASIC PROBLEMS. - PowerPoint PPT PresentationTRANSCRIPT
STABILITY under CONSTRAINED SWITCHING ;
SWITCHED SYSTEMS with INPUTS and OUTPUTS
Daniel Liberzon
Coordinated Science Laboratory andDept. of Electrical & Computer Eng.,Univ. of Illinois at Urbana-Champaign
IAAC Workshop, Herzliya, Israel, June 1, 2009
TWO BASIC PROBLEMS
• Stability for arbitrary switching
• Stability for constrained switching
MULTIPLE LYAPUNOV FUNCTIONS
Useful for analysis of state-dependent switching
– GAS
– respective Lyapunov functions
is GAS
MULTIPLE LYAPUNOV FUNCTIONS
decreasing sequence
decreasing sequence
[DeCarlo, Branicky]
GAS
DWELL TIME
The switching times satisfy
dwell time– GES
– respective Lyapunov functions
DWELL TIME
– GES
Need:
The switching times satisfy
DWELL TIME
– GES
Need:
The switching times satisfy
DWELL TIME
– GES
Need:
must be 1
The switching times satisfy
AVERAGE DWELL TIME
# of switches on average dwell time
– dwell time: cannot switch twice if
AVERAGE DWELL TIME
Theorem: [Hespanha ‘99] Switched system is GAS if
Lyapunov functions s.t. • .
•
•
Useful for analysis of hysteresis-based switching logics
# of switches on average dwell time
MULTIPLE WEAK LYAPUNOV FUNCTIONS
Theorem: is GAS if
• .
•
•
•
– milder than ADT
Extends to nonlinear switched systems as before
observable for each
s.t. there are infinitely many
switching intervals of length
For every pair of switching times
s.t.
have
APPLICATION: FEEDBACK SYSTEMS (Popov criterion)
Corollary: switched system is GAS if
• s.t. infinitely many switching intervals of length
• For every pair of switching times at
which we have
linear system observable
positive real
See also invariance principles for switched systems in: [Lygeros et al., Bacciotti–Mazzi, Mancilla-Aguilar, Goebel–Sanfelice–Teel]
Weak Lyapunov functions:
STATE-DEPENDENT SWITCHING
But switched system is stable for (many) other
Switched system
unstable for some
no common
switch on the axes
is a Lyapunov function
STATE-DEPENDENT SWITCHING
But switched system is stable for (many) other
level sets of level sets of
Switched system
unstable for some
no common
Switch on y-axis
GAS
STABILIZATION by SWITCHING
– both unstable
Assume: stable for some
STABILIZATION by SWITCHING
[Wicks et al. ’98]
– both unstable
Assume: stable for some
So for each
either or
UNSTABLE CONVEX COMBINATIONS
Can also use multiple Lyapunov functions
Linear matrix inequalities
SWITCHED SYSTEMS with INPUTS and OUTPUTS
• Background
• Input-to-state stability (ISS)
• Main results
• ISS under ADT switching
• Invertibility of switched systems
Outline:
INPUT-TO-STATE STABILITY (ISS)
classNonlinear gain functions:
Equivalent Lyapunov characterization [Sontag–Wang]:
without loss of generality,can replace by
ISS [Sontag ’89]:
classclass , e.g.
(means: pos. def., rad. unbdd.)
ISS under ADT SWITCHING
eachsubsystem
is ISS
[Vu–Chatterjee–L, Automatica, Apr 2007]
If has average dwell time
• .
•
•
class functions and constants
such that :
Suppose functions
then switched system is ISS
SKETCH of PROOF
1
1 Let be switching times on
Consider
Recall ADT definition:
2
3
SKETCH of PROOF
12
3
2
1
3
Special cases:
• GAS when
• ISS under arbitrary switching if (common )
• ISS without switching (single )
– ISS
VARIANTS
• Output-to-state stability (OSS) [M. Müller]
• Stochastic versions of ISS for randomly switched systems [D. Chatterjee]
• Some subsystems not ISS [Müller, Chatterjee]
finds application in switching adaptive control
• Integral ISS:
[Vu–L, Automatica, Apr 2008; Tanwani–L, CDC 2008]
SWITCHED SYSTEMS with INPUTS and OUTPUTS
• Background
• Input-to-state stability (ISS)
• Main results
• ISS under ADT switching
• Invertibility of switched systems
Outline:
PROBLEM FORMULATION
Invertibility problem: recover uniquely from for given
• Desirable: fault detection (in power systems)
Related work: [Sundaram–Hadjicostis, Millerioux–Daafouz]; [Vidal et al., Babaali et al., De Santis et al.]
• Undesirable: security (in multi-agent networked systems)
MOTIVATING EXAMPLE
because
Guess:
INVERTIBILITY of NON-SWITCHED SYSTEMS
Linear: [Brockett–Mesarovic, Silverman, Sain–Massey, Morse–Wonham]
INVERTIBILITY of NON-SWITCHED SYSTEMS
Linear: [Brockett–Mesarovic, Silverman, Sain–Massey, Morse–Wonham]
Nonlinear: [Hirschorn, Isidori–Moog, Nijmeijer, Respondek, Singh]
INVERTIBILITY of NON-SWITCHED SYSTEMS
Linear: [Brockett–Mesarovic, Silverman, Sain–Massey, Morse–Wonham]
Nonlinear: [Hirschorn, Isidori–Moog, Nijmeijer, Respondek, Singh]
SISO nonlinear system affine in control:
Suppose it has relative degree at :
Then we can solve for :
Inverse system
BACK to the EXAMPLE
We can check that each subsystem is invertible
For MIMO systems, can use nonlinear structure algorithm
– similar
SWITCH-SINGULAR PAIRS
Consider two subsystems and
is a switch-singular pair if such that
|||
FUNCTIONAL REPRODUCIBILITY
SISO system affine in control with relative degree at :
For given and , that produces this output
if and only if
CHECKING for SWITCH-SINGULAR PAIRS
is a switch-singular pair for SISO subsystems
with relative degrees if and only if
MIMO systems – via nonlinear structure algorithm
Existence of switch-singular pairs is difficult to check in general
For linear systems, this can be characterized by a
matrix rank condition
MAIN RESULT
Theorem:
Switched system is invertible at over output set
if and only if each subsystem is invertible at and
there are no switched-singular pairs
Idea of proof:
The devil is in the details
no switch-singular pairs can recover
subsystems are invertible can recover
BACK to the EXAMPLE
Let us check for switched singular pairs:
Stop here because relative degree
For every , and with
form a switch-singular pair
Switched system is not invertible on the diagonal
OUTPUT GENERATION
Recall our example again:
Given and , find (if exist) s. t.
may be unique for some but not all
OUTPUT GENERATION
Recall our example again:
switch-singular pair
Given and , find (if exist) s. t.
may be unique for some but not all
Solution from :
OUTPUT GENERATION
Recall our example again:
switch-singular pair
Given and , find (if exist) s. t.
may be unique for some but not all
Solution from :
OUTPUT GENERATION
Recall our example again:
Case 1: no switch at
Then up to
At , must switch to 2
But then
won’t match the given output
Given and , find (if exist) s. t.
may be unique for some but not all
OUTPUT GENERATION
Recall our example again:
Case 2: switch at
Given and , find (if exist) s. t.
may be unique for some but not all
No more switch-singular pairs
OUTPUT GENERATION
Recall our example again:
Given and , find (if exist) s. t.
may be unique for some but not all
Case 2: switch at
No more switch-singular pairs
OUTPUT GENERATION
Recall our example again:
Given and , find (if exist) s. t.
We also obtain from
We see how one switch can helprecover an earlier “hidden” switch
may be unique for some but not all
Case 2: switch at
No more switch-singular pairs