1. ROBUST OBSERVERS and PECORA – CARROLL
SYNCHRONIZATION with LIMITED INFORMATION
Boris Andrievsky and Alexander L. Fradkov
(Institute for Problems of Mechanical Engineering of
Russian Academy of Sciences, St. Petersburg, Russia)
Daniel Liberzon
(University of Illinois, Urbana-Champaign, USA)
CDC, Melbourne, Dec 2017
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2. TALK OUTLINE Observers robust to measurement disturbances in
ISS sense: formulation and Lyapunov condition
[Shim–L, TAC, 2016]
Application to robust synchronization:
Lorenz chaotic system
(this paper)
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3. TALK OUTLINE Observers robust to measurement disturbances in
ISS sense: formulation and Lyapunov condition
Application to robust synchronization:
Lorenz chaotic system
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4. INPUT – to – STATE STABILITY (ISS)
[Sontag ’89]
INPUT – to – STATE STABILITY (ISS)
System is ISS if its solutions satisfy
class fcn
where
ISS existence of ISS Lyapunov function:
pos. def., rad. unbdd, function satisfying
Say that \rho is directly related to \gamma
Will later mention a variation on these Lyapunov conditions – asymptotic ratio
or equivalently
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5. ROBUST OBSERVER DESIGN PROBLEM
Sensors + + –
Plant
Plant:
Observer:
Full-order observer:
; reduced-order:
State estimation error:
Actually in the diagram the disturbance is additive but this is just a special case
Example supporting the last claim was constructed in Hyungbo’s paper with Andy (a bit complicated to give here)
Robustness issue: can have when
yet for arbitrarily small
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6. DISTURBANCE – to – ERROR STABILITY (DES)
Plant:
Observer:
Estimation error:
ISS-like robustness notion: call observer DES if
Known conditions for this [Sontag–Wang ’97, Angeli ’02] are very strong
Also, DES is coordinate dependent as global error convergence
is coordinate dependent:
Sontag-Wang: necessary condition (incremental OSS), Angeli: sufficient condition (incremental ISS after output injection)
I’m skipping the example of coordinate dependence, which can be realized by a suitable observer (see paper)
Path toward less restrictive, coordinate-invariant robustness
property: impose DES only as long as are bounded
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7. QUASI – DISTURBANCE – to – ERROR STABILITY (qDES)
Definition: observer is quasi-Disturbance-to-Error Stable
(qDES) if
such that
whenever
Emphasize dependence of beta and gamma on K
I chose to skip the fact that we need e_0 (e(0) might be 0 due to d(0)) and “a.e.” (e(t) can occasionally be large due to large d(t))
If we drop the squares from the example, then the ISS gain will depend on u but not on x
The qDES property is invariant to coordinate changes
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8. REDUCED – ORDER qDES OBSERVERS
Plant (after a coordinate change):
Observer:
Assume this is , then we have an
asymptotic observer: when
In this simplified presentation, output injection \ell is absorbed into coord change
This e is only one component of error, in the paper we call it \eps (slight abuse of notation)
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9. REDUCED – ORDER qDES OBSERVERS
Plant (after a coordinate change):
Observer:
assume this has norm
assumed to be
upper-bounded by
In the assumption on the 1st term of \dot V (2nd line) y plays the role of dummy variable, so d doesn’t really play a role
The bound on the last term is valid when |u|, |x| \le K
Then
whenever
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10. REDUCED – ORDER qDES OBSERVERS
Plant (after a coordinate change):
Observer:
Asymptotic ratio condition for qDES [L–Shim, TAC, 2015]:
If we have such that
The special case (known \alpha) applies to Lorenz example
then
when
Can estimate ISS gain but only if is known
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11. TALK OUTLINE Observers robust to measurement disturbances in
ISS sense: formulation and Lyapunov condition
Application to robust synchronization:
Lorenz chaotic system
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12. ROBUST PECORA – CARROLL SYNCHRONIZATION
Leader +
Follower
Robust synchronization: s.t.
whenever
Equivalently: follower is a reduced-order qDES observer for leader
Now no u
This is robust version of classical Pecora-Carroll scheme
Sticking to notation in qDES paper, not in synchronization papers (d is part of y)
Explain that x_1-dynamics in the observer are not needed, have static estimate; or if they are present in the follower’s model, it’s easy to make them track x_1 by (high-gain) feedback
\alpha_1,2 are bounds on V, not writing them (but can say it)
Sufficient condition from before: s.t.
and
(asymptotic ratio condition)
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13. APPLICATION EXAMPLE
Lorenz system
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14. APPLICATION EXAMPLE Can show is bounded using+
Observer
Lorenz system
Can show is bounded using
Can show qDES from to using
For arising from time sampling and quantization, we can
derive an explicit bound on synchronization error which is
inversely proportional to data rate (see paper for details)
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15. CONCLUSIONS Summary: Other applications:
Asymptotic ratio characterization of ISS
qDES observer concept and Lyapunov condition
Robust version of Pecora–Carroll synchronization scheme
Application example: Lorenz system with sampled and
quantized measurements
Other applications:
Quantized output feedback control [with H. Shim, TAC 2016]
Synchronization of electric power generators [with A.
Domínguez-Garcia, ongoing work]
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