input-to-state stability of switched systems debasish chatterjee, linh vu, daniel liberzon...
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INPUT-TO-STATE STABILITY of
SWITCHED SYSTEMS
Debasish Chatterjee, Linh Vu, Daniel Liberzon
Coordinated Science Laboratory andDept. of Electrical & Computer Eng.,Univ. of Illinois at Urbana-Champaign
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
• Stability margin
• Integral ISS (with stability margin)
• 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
INVERTIBILITY of SWITCHED SYSTEMS
Aneel Tanwani, Linh Vu, Daniel Liberzon
Coordinated Science Laboratory andDept. of Electrical & Computer Eng.,Univ. of Illinois at Urbana-Champaign
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 switch-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
CONCLUSIONS
• Showed how results on stability under slow switching
extend in a natural way to external stability (ISS)
• Studied new invertibility problem: recovering both the
input and the switching signal
• Both problems have applications in control design
• General motivation/application: analysis and design
of complex interconnected systems