1. Nonlinear Observers Robust to Measurement Errors and
their Applications in Control and Synchronization
Daniel Liberzon
Coordinated Science Laboratory and
Dept. of Electrical & Computer Eng.,
Univ. of Illinois at Urbana-Champaign
FoRCE online seminar, Mar 23, 2018
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2. INFORMATION FLOW in CONTROL SYSTEMS
Limited communication capacity
Coarse sensing
Security considerations
Errors of different kind – quantization (analog-to-digital conversion), time delays, etc.
Event-driven actuators
Theoretical interest
Limited information errors
need robust algorithms
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3. OBSERVER – BASED OUTPUT FEEDBACK CONTROL
Plant
Sensors
not much is known about this problem +
errors (e.g.,
quantization) +
error propagation
Controller
Observer
Moving closer to what we do here
Controller design robust to errors has been studied to a greater extent – will discuss later
Input-to-state stability (ISS) provides a framework for
quantifying robustness (graceful error propagation)
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4. TALK OUTLINE
Fresh look at input-to-state stability (ISS): asymptotic ratio
[L–Shim, An asymptotic ratio characterization of ISS, TAC 2015]
Observers robust to measurement disturbances:
formulation and Lyapunov condition
[Shim–L, Nonlinear observers robust to measurement disturbances
in an ISS sense, TAC 2016], see also [Shim–L–Kim, CDC 2009]
Hyungbo
Shim
Application to output feedback control design
Applications to robust synchronization
electric power generators [Ajala–Domínguez-Garcia–L, 2018]
Lorenz chaotic system [Andrievsky–Fradkov–L, CDC 2017, SCL 2018]
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5. TALK OUTLINE
Fresh look at input-to-state stability (ISS): asymptotic ratio
Observers robust to measurement disturbances:
formulation and Lyapunov condition
Application to output feedback control design
Applications to robust synchronization
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6. 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
or equivalently
Say that \rho is directly related to \gamma
GAS
ISS, e.g.:
( unbdd for )
(may have even if )
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7. ASYMPTOTIC – RATIO ISS LYAPUNOV FUNCTIONS
(1)
(2)
Definition: pos. def., rad. unbdd, function is an
asymptotic-ratio ISS Lyapunov function if
where , is continuous non-negative,
is non-decreasing for each , with , and
Explain the term “asymptotic ratio”
Actually \alpha_3 need not be unbounded, see revision to the paper
Can explain also the notation \alpha_3 (\alpha_1,2 are “sandwich” bounds for V)
Theorem: ISS asymptotic-ratio ISS Lyapunov function
Proof of follows from characterization of ISS via (2)
Proof of proceeds by constructing as in (1)
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8. ASYMPTOTIC – RATIO ISS LYAPUNOV FUNCTIONS
(1)
(2)
Definition: pos. def., rad. unbdd, function is an
asymptotic-ratio ISS Lyapunov function if
where , is continuous non-negative,
is non-decreasing for each , with , and
Explain trade-off with (1) and (2): may be easier to verify, but gives no direct info about ISS gain
Example (scalar): ,
No info about ISS gain
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9. TALK OUTLINE
Fresh look at input-to-state stability (ISS): asymptotic ratio
Observers robust to measurement disturbances:
formulation and Lyapunov condition
Application to output feedback control design
Applications to robust synchronization
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10. ROBUST OBSERVER DESIGN PROBLEM
Sensors + + –
Plant
Plant:
Observer:
Full-order observer:
; reduced-order:
State estimation error:
Animate and explain the diagram
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)
Allude to similarity with previous example about 0-GAS not implying ISS
Robustness issue: can have when
yet for arbitrarily small
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11. 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)
Coordinate-dependent behavior can be realized by a suitable observer (see example in paper)
Path toward less restrictive, coordinate-invariant robustness
property: impose DES only as long as are bounded
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12. QUASI – DISTURBANCE – to – ERROR STABILITY (qDES)
Definition: observer is quasi-Disturbance-to-Error Stable
(qDES) if
such that
whenever
Example:
Emphasize dependence of beta and gamma on K
Motivate: this is reasonable if we’re going to use a control that keeps x bounded
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
qDES but not DES
The qDES property is invariant to coordinate changes
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13. 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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14. 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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15. REDUCED – ORDER qDES OBSERVERS
Plant (after a coordinate change):
Observer:
Asymptotic ratio condition:
If we have such that
The special case applies to Lorenz example
then
when
Can estimate ISS gain but only if is known
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16. TALK OUTLINE
Fresh look at input-to-state stability (ISS): asymptotic ratio
Observers robust to measurement disturbances:
formulation and Lyapunov condition
Application to output feedback control design
Applications to robust synchronization
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17. OBSERVER – BASED OUTPUT FEEDBACK REVISITED
Plant
Sensors
Controller
Observer
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18. OBSERVER – BASED OUTPUT FEEDBACK REVISITED
Sensors
Controller
Observer
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19. OBSERVER – BASED OUTPUT FEEDBACK REVISITED
Controller
Observer
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20. OBSERVER – BASED OUTPUT FEEDBACK REVISITED
Controller
Assuming full-order observer here
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21. OBSERVER – BASED OUTPUT FEEDBACK REVISITED
Assume observer is qDES w.r.t. :
Assume controller is ISS w.r.t. :
Assuming full-order observer here
[Freeman, Fah, Jiang et al., Sanfelice–Teel, Ebenbauer et al.]
Cascade argument: closed-loop system is quasi-ISS
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22. APPLICATION to QUANTIZED OUTPUT FEEDBACK
quantizer
– quantization error
& upper bounds on s.t.
remain
Quantizer as disturbance generator
Contraction is guaranteed if quantization is fine enough
Can achieve asymptotic stability by dynamic “zooming”
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23. TALK OUTLINE
Fresh look at input-to-state stability (ISS): asymptotic ratio
Observers robust to measurement disturbances:
formulation and Lyapunov condition
Application to output feedback control design
Applications to robust synchronization
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24. ROBUST SYNCHRONIZATION and qDES OBSERVERS
Leader +
Follower
Robust synchronization: s.t.
whenever
Equivalently: follower is a reduced-order qDES observer for leader
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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25. APPLICATION EXAMPLE #1
load
Generator 1
Generator 2
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26. APPLICATION EXAMPLE #1 Generator 1 Generator 2
control input (mechanical power)
With integral control: desired frequency
electrical load (slowly varying)
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27. APPLICATION EXAMPLE #1 Generator 1 Generator 2
control input (mechanical power)
With integral control: desired frequency
electrical load (slowly varying)
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28. APPLICATION EXAMPLE #1 due to phase drift, will have at some time+
Generator 1
Generator 2
control input (mechanical power)
With integral control: desired frequency
electrical load (slowly varying)
Measurements:
PMU corrupted
by disturbance
Objective: connect 2nd generator when
gives DES (ISS) from to
Possible source of disturbance: spoofing attacks (Alejandro)
I will not give the control for 2nd generator
and will synchronize with error load variations
due to phase drift, will have at some time
can connect 2nd generator
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29. APPLICATION EXAMPLE #1 phase-dependent damping+
Generator 1
Generator 2
Extensions:
phase-dependent damping
analysis more challenging, but can still show
state boundedness and qDES from to
network case (microgrids)
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30. APPLICATION EXAMPLE #2
Lorenz system
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31. APPLICATION EXAMPLE #2 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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32. TALK OUTLINE
Fresh look at input-to-state stability (ISS): asymptotic ratio
Observers robust to measurement disturbances:
formulation and Lyapunov condition
Application to output feedback control design
Applications to robust synchronization
electric power generators
Lorenz chaotic system
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33. FUTURE WORK Nonlinear qDES observer design:
Identify system classes to which Lyapunov conditions apply
Develop more constructive procedures for observer design
Quantized output feedback control:
Relax ISS controller assumption
Study other coupled oscillator network models
Look for examples in other areas (e.g., vehicle formations)
Robust synchronization:
Papers and preprints available at liberzon.csl.illinois.edu
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