Transcript INPUT-TO-STATE STABILITY of SWITCHED SYSTEMS Debasish Chatterjee, Linh Vu, Daniel Liberzon Coordinated Science Laboratory and Dept.

INPUT-TO-STATE STABILITY of
SWITCHED SYSTEMS
Debasish Chatterjee, Linh Vu, Daniel Liberzon
Coordinated Science Laboratory and
Dept. of Electrical & Computer Eng.,
Univ. of Illinois at Urbana-Champaign
ISS under ADT SWITCHING
Suppose
class
functions
functions
and constants
such that
each
subsystem
is ISS
•
.
•
•
If
has average dwell time
then switched system is ISS
[Vu–Chatterjee–L, Automatica, Apr 2007]
:
SKETCH of PROOF
1
Let
be switching times on
Consider
2
3
Recall ADT definition:
1
SKETCH of PROOF
1
2
2
1
3
3
– ISS 
Special cases:
• GAS when
• ISS under arbitrary switching if
• ISS without switching (single
(common
)
)
VARIANTS
• Stability margin
• Integral ISS (with stability margin)
finds application in switching adaptive control
• Output-to-state stability (OSS) [M. Müller]
• Stochastic versions of ISS for randomly switched
systems [D. Chatterjee]
• Some subsystems not ISS [Müller, Chatterjee]
INVERTIBILITY of SWITCHED SYSTEMS
Aneel Tanwani, Linh Vu, Daniel Liberzon
Coordinated Science Laboratory and
Dept. 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)
• Undesirable: security (in multi-agent networked systems)
Related work: [Sundaram–Hadjicostis, Millerioux–Daafouz];
[Vidal et al., Babaali et al., De Santis et al.]
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
Then we can solve for
Inverse system
:
at
:
BACK to the EXAMPLE
We can check that each subsystem is invertible
– similar
For MIMO systems, can use nonlinear structure algorithm
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
For given
and
if and only if
,
at
:
that produces this output
CHECKING for SWITCH-SINGULAR PAIRS
is a switch-singular pair for SISO subsystems
with relative degrees
if and only if
For linear systems, this can be characterized by a
matrix rank condition
MIMO systems – via nonlinear structure algorithm
Existence of switch-singular pairs is difficult to check in general
MAIN RESULT
Theorem:
Switched system is invertible at
over output set
if and only if each subsystem is invertible at
there are no switch-singular pairs
Idea of proof:
no switch-singular pairs
subsystems are invertible
The devil is in the details
can recover
can recover
and
BACK to the EXAMPLE
Let us check for switched singular pairs:
Stop here because relative degree
For every
,
and
form a switch-singular pair
with
Switched system is not invertible on the diagonal
OUTPUT GENERATION
Given
and
, find
(if exist) s. t.
may be unique for some but not all
Recall our example again:
OUTPUT GENERATION
Given
and
, find
(if exist) s. t.
may be unique for some but not all
Recall our example again:
Solution from
:
switch-singular pair
OUTPUT GENERATION
Given
and
, find
(if exist) s. t.
may be unique for some but not all
Recall our example again:
Solution from
:
switch-singular pair
OUTPUT GENERATION
Given
and
, find
(if exist) s. t.
may be unique for some but not all
Recall our example again:
Case 1: no switch at
Then
At
up to
,
must switch to 2
But then
won’t match the given output
OUTPUT GENERATION
Given
and
, find
(if exist) s. t.
may be unique for some but not all
Recall our example again:
Case 2: switch at
No more switch-singular pairs
OUTPUT GENERATION
Given
and
, find
(if exist) s. t.
may be unique for some but not all
Recall our example again:
Case 2: switch at
No more switch-singular pairs
OUTPUT GENERATION
Given
and
, find
(if exist) s. t.
may be unique for some but not all
Recall our example again:
Case 2: switch at
No more switch-singular pairs
We see how one switch can help
recover an earlier “hidden” switch
We also obtain
from
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