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 1 of 13 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAA

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) 2 of 13

TALK OUTLINE Observers robust to measurement disturbances in ISS sense: formulation and Lyapunov condition Application to robust synchronization: Lorenz chaotic system 2 of 13

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 3 of 13

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 4 of 13

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 5 of 13

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 6 of 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) 7 of 13

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 8 of 13

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 9 of 13

TALK OUTLINE Observers robust to measurement disturbances in ISS sense: formulation and Lyapunov condition Application to robust synchronization: Lorenz chaotic system 10 of 13

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) 11 of 13

APPLICATION EXAMPLE Lorenz system 12 of 13

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) 12 of 13

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] 13 of 13