OUTPUT

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OUTPUT INPUT STABILITY
Daniel Liberzon
Coordinated Science Laboratory and Dept. of
Electrical Computer Eng., Univ. of Illinois at
Urbana-Champaign
MTNS 02
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MOTIVATION
s
ISS
stability (no outputs)
linear stable eigenvalues
detectability (no inputs)
linear stable unobserv. modes
minimum phase
? ? ?
linear stable zeros stable inverse
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MOTIVATION Adaptive Control
Plant
Design
Controller
model

• the system in the box is output-stabilized
• the plant is minimum-phase

If
Then the closed-loop system is detectable through
e (tunable Morse 92)
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DEFINITION
Call the system
output-input stable if integer N and
functions s.t.
where
Example
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UNDERSTANDING OUTPUT-INPUT STABILITY
Output-input stability
Uniform detectability w.r.t. extended output
Input-bounding property
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SISO SYSTEMS
For systems analytic in controls, can replace the
input-bounding property by
where is the first derivative containing u
For affine systems this reduces to relative
degree ( )
doesnt have this property
For affine systems in global normal form,
output-input stability ISS internal
dynamics
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MIMO SYSTEMS
Existence of relative degree no longer necessary
For linear systems reduces to usual minimum phase
notion
Input-bounding property via Hirschorns
algorithm
Example
Extensions Singhs algorithm, non-affine systems
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INPUT / OUTPUT OPERATORS
A system is output-input stable if and only if
its I/O mapping (for zero i.c.) is output-input
stable under suitable minimality assumptions
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APPLICATION FEEDBACK DESIGN
Example
( r relative degree)
Output-input stability guarantees closed-loop GAS
No global normal form is needed
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CASCADE SYSTEMS
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ADAPTIVE CONTROL
Plant
Controller

• the plant is output-input stable (Nr)
• the system in the box is input-to-output
• stable (IOS) from to

If
Then the closed-loop system is detectable through
(weakly tunable)
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SUMMARY
New notion of output-input stability

• applies to general smooth nonlinear control
  systems
• reduces to minimum phase for linear (MIMO)
  systems
• robust variant of Byrnes-Isidori minimum phase
  notion
• relates to ISS, detectability,
  left-invertibility
• extends to input/output operators

Applications

• Feedback stabilization
• Cascade connections
• Adaptive control
• More ?