STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS
-
Upload
brian-hartman -
Category
Documents
-
view
30 -
download
5
description
Transcript of STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS
![Page 1: STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS](https://reader038.fdocuments.us/reader038/viewer/2022110210/56812af3550346895d8eda43/html5/thumbnails/1.jpg)
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
![Page 2: STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS](https://reader038.fdocuments.us/reader038/viewer/2022110210/56812af3550346895d8eda43/html5/thumbnails/2.jpg)
TWO BASIC PROBLEMS
• Stability for arbitrary switching
• Stability for constrained switching
![Page 3: STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS](https://reader038.fdocuments.us/reader038/viewer/2022110210/56812af3550346895d8eda43/html5/thumbnails/3.jpg)
MULTIPLE LYAPUNOV FUNCTIONS
Useful for analysis of state-dependent switching
– GAS
– respective Lyapunov functions
is GAS
![Page 4: STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS](https://reader038.fdocuments.us/reader038/viewer/2022110210/56812af3550346895d8eda43/html5/thumbnails/4.jpg)
MULTIPLE LYAPUNOV FUNCTIONS
decreasing sequence
decreasing sequence
[DeCarlo, Branicky]
GAS
![Page 5: STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS](https://reader038.fdocuments.us/reader038/viewer/2022110210/56812af3550346895d8eda43/html5/thumbnails/5.jpg)
DWELL TIME
The switching times satisfy
dwell time– GES
– respective Lyapunov functions
![Page 6: STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS](https://reader038.fdocuments.us/reader038/viewer/2022110210/56812af3550346895d8eda43/html5/thumbnails/6.jpg)
DWELL TIME
– GES
Need:
The switching times satisfy
![Page 7: STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS](https://reader038.fdocuments.us/reader038/viewer/2022110210/56812af3550346895d8eda43/html5/thumbnails/7.jpg)
DWELL TIME
– GES
Need:
The switching times satisfy
![Page 8: STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS](https://reader038.fdocuments.us/reader038/viewer/2022110210/56812af3550346895d8eda43/html5/thumbnails/8.jpg)
DWELL TIME
– GES
Need:
must be 1
The switching times satisfy
![Page 9: STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS](https://reader038.fdocuments.us/reader038/viewer/2022110210/56812af3550346895d8eda43/html5/thumbnails/9.jpg)
AVERAGE DWELL TIME
# of switches on average dwell time
– dwell time: cannot switch twice if
![Page 10: STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS](https://reader038.fdocuments.us/reader038/viewer/2022110210/56812af3550346895d8eda43/html5/thumbnails/10.jpg)
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
![Page 11: STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS](https://reader038.fdocuments.us/reader038/viewer/2022110210/56812af3550346895d8eda43/html5/thumbnails/11.jpg)
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
![Page 12: STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS](https://reader038.fdocuments.us/reader038/viewer/2022110210/56812af3550346895d8eda43/html5/thumbnails/12.jpg)
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:
![Page 13: STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS](https://reader038.fdocuments.us/reader038/viewer/2022110210/56812af3550346895d8eda43/html5/thumbnails/13.jpg)
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
![Page 14: STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS](https://reader038.fdocuments.us/reader038/viewer/2022110210/56812af3550346895d8eda43/html5/thumbnails/14.jpg)
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
![Page 15: STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS](https://reader038.fdocuments.us/reader038/viewer/2022110210/56812af3550346895d8eda43/html5/thumbnails/15.jpg)
STABILIZATION by SWITCHING
– both unstable
Assume: stable for some
![Page 16: STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS](https://reader038.fdocuments.us/reader038/viewer/2022110210/56812af3550346895d8eda43/html5/thumbnails/16.jpg)
STABILIZATION by SWITCHING
[Wicks et al. ’98]
– both unstable
Assume: stable for some
So for each
either or
![Page 17: STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS](https://reader038.fdocuments.us/reader038/viewer/2022110210/56812af3550346895d8eda43/html5/thumbnails/17.jpg)
UNSTABLE CONVEX COMBINATIONS
Can also use multiple Lyapunov functions
Linear matrix inequalities
![Page 18: STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS](https://reader038.fdocuments.us/reader038/viewer/2022110210/56812af3550346895d8eda43/html5/thumbnails/18.jpg)
SWITCHED SYSTEMS with INPUTS and OUTPUTS
• Background
• Input-to-state stability (ISS)
• Main results
• ISS under ADT switching
• Invertibility of switched systems
Outline:
![Page 19: STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS](https://reader038.fdocuments.us/reader038/viewer/2022110210/56812af3550346895d8eda43/html5/thumbnails/19.jpg)
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.)
![Page 20: STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS](https://reader038.fdocuments.us/reader038/viewer/2022110210/56812af3550346895d8eda43/html5/thumbnails/20.jpg)
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
![Page 21: STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS](https://reader038.fdocuments.us/reader038/viewer/2022110210/56812af3550346895d8eda43/html5/thumbnails/21.jpg)
SKETCH of PROOF
1
1 Let be switching times on
Consider
Recall ADT definition:
2
3
![Page 22: STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS](https://reader038.fdocuments.us/reader038/viewer/2022110210/56812af3550346895d8eda43/html5/thumbnails/22.jpg)
SKETCH of PROOF
12
3
2
1
3
Special cases:
• GAS when
• ISS under arbitrary switching if (common )
• ISS without switching (single )
– ISS
![Page 23: STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS](https://reader038.fdocuments.us/reader038/viewer/2022110210/56812af3550346895d8eda43/html5/thumbnails/23.jpg)
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:
![Page 24: STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS](https://reader038.fdocuments.us/reader038/viewer/2022110210/56812af3550346895d8eda43/html5/thumbnails/24.jpg)
[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:
![Page 25: STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS](https://reader038.fdocuments.us/reader038/viewer/2022110210/56812af3550346895d8eda43/html5/thumbnails/25.jpg)
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)
![Page 26: STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS](https://reader038.fdocuments.us/reader038/viewer/2022110210/56812af3550346895d8eda43/html5/thumbnails/26.jpg)
MOTIVATING EXAMPLE
because
Guess:
![Page 27: STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS](https://reader038.fdocuments.us/reader038/viewer/2022110210/56812af3550346895d8eda43/html5/thumbnails/27.jpg)
INVERTIBILITY of NON-SWITCHED SYSTEMS
Linear: [Brockett–Mesarovic, Silverman, Sain–Massey, Morse–Wonham]
![Page 28: STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS](https://reader038.fdocuments.us/reader038/viewer/2022110210/56812af3550346895d8eda43/html5/thumbnails/28.jpg)
INVERTIBILITY of NON-SWITCHED SYSTEMS
Linear: [Brockett–Mesarovic, Silverman, Sain–Massey, Morse–Wonham]
Nonlinear: [Hirschorn, Isidori–Moog, Nijmeijer, Respondek, Singh]
![Page 29: STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS](https://reader038.fdocuments.us/reader038/viewer/2022110210/56812af3550346895d8eda43/html5/thumbnails/29.jpg)
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
![Page 30: STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS](https://reader038.fdocuments.us/reader038/viewer/2022110210/56812af3550346895d8eda43/html5/thumbnails/30.jpg)
BACK to the EXAMPLE
We can check that each subsystem is invertible
For MIMO systems, can use nonlinear structure algorithm
– similar
![Page 31: STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS](https://reader038.fdocuments.us/reader038/viewer/2022110210/56812af3550346895d8eda43/html5/thumbnails/31.jpg)
SWITCH-SINGULAR PAIRS
Consider two subsystems and
is a switch-singular pair if such that
|||
![Page 32: STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS](https://reader038.fdocuments.us/reader038/viewer/2022110210/56812af3550346895d8eda43/html5/thumbnails/32.jpg)
FUNCTIONAL REPRODUCIBILITY
SISO system affine in control with relative degree at :
For given and , that produces this output
if and only if
![Page 33: STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS](https://reader038.fdocuments.us/reader038/viewer/2022110210/56812af3550346895d8eda43/html5/thumbnails/33.jpg)
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
![Page 34: STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS](https://reader038.fdocuments.us/reader038/viewer/2022110210/56812af3550346895d8eda43/html5/thumbnails/34.jpg)
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
![Page 35: STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS](https://reader038.fdocuments.us/reader038/viewer/2022110210/56812af3550346895d8eda43/html5/thumbnails/35.jpg)
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
![Page 36: STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS](https://reader038.fdocuments.us/reader038/viewer/2022110210/56812af3550346895d8eda43/html5/thumbnails/36.jpg)
OUTPUT GENERATION
Recall our example again:
Given and , find (if exist) s. t.
may be unique for some but not all
![Page 37: STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS](https://reader038.fdocuments.us/reader038/viewer/2022110210/56812af3550346895d8eda43/html5/thumbnails/37.jpg)
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 :
![Page 38: STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS](https://reader038.fdocuments.us/reader038/viewer/2022110210/56812af3550346895d8eda43/html5/thumbnails/38.jpg)
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 :
![Page 39: STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS](https://reader038.fdocuments.us/reader038/viewer/2022110210/56812af3550346895d8eda43/html5/thumbnails/39.jpg)
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
![Page 40: STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS](https://reader038.fdocuments.us/reader038/viewer/2022110210/56812af3550346895d8eda43/html5/thumbnails/40.jpg)
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
![Page 41: STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS](https://reader038.fdocuments.us/reader038/viewer/2022110210/56812af3550346895d8eda43/html5/thumbnails/41.jpg)
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
![Page 42: STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS](https://reader038.fdocuments.us/reader038/viewer/2022110210/56812af3550346895d8eda43/html5/thumbnails/42.jpg)
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