INPUT-TO-STATE STABILITY of SWITCHED SYSTEMS

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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
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ISS under ADT SWITCHING
then switched system is ISS
VuChatterjeeL, Automatica, Apr 2007
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SKETCH of PROOF
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SKETCH of PROOF
Special cases
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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

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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
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PROBLEM FORMULATION

• Desirable fault detection (in power systems)

• Undesirable security (in multi-agent networked
  systems)

Related work SundaramHadjicostis,
MilleriouxDaafouz
Vidal et al., Babaali et al., De Santis et al.
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MOTIVATING EXAMPLE
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INVERTIBILITY of NON-SWITCHED SYSTEMS
Linear BrockettMesarovic, Silverman,
SainMassey, MorseWonham
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INVERTIBILITY of NON-SWITCHED SYSTEMS
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INVERTIBILITY of NON-SWITCHED SYSTEMS
Linear BrockettMesarovic, Silverman,
SainMassey, MorseWonham
Nonlinear Hirschorn, IsidoriMoog, Nijmeijer,
Respondek, Singh
SISO nonlinear system affine in control
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BACK to the EXAMPLE
We can check that each subsystem is invertible
For MIMO systems, can use nonlinear structure
algorithm
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SWITCH-SINGULAR PAIRS
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FUNCTIONAL REPRODUCIBILITY
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CHECKING for SWITCH-SINGULAR PAIRS
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
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MAIN RESULT
Idea of proof
The devil is in the details
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BACK to the EXAMPLE
Let us check for switched singular pairs
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OUTPUT GENERATION
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OUTPUT GENERATION
Recall our example again
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OUTPUT GENERATION
Recall our example again
switch-singular pair
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OUTPUT GENERATION
Recall our example again
wont match the given output
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OUTPUT GENERATION
Recall our example again
Case 2 switch at
No more switch-singular pairs
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OUTPUT GENERATION
Recall our example again
Case 2 switch at
No more switch-singular pairs
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OUTPUT GENERATION
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
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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