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 1 of 23 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAA

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 2 of 23

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) 3 of 23

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] 4 of 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 4 of 23

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 ) 5 of 23

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) 6 of 23

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 6 of 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 7 of 23

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 8 of 23

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 9 of 23

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 10 of 23

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) 11 of 23

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 12 of 23

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 13 of 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 14 of 23

OBSERVER – BASED OUTPUT FEEDBACK REVISITED Plant Sensors Controller Observer 15 of 23

OBSERVER – BASED OUTPUT FEEDBACK REVISITED Sensors Controller Observer 15 of 23

OBSERVER – BASED OUTPUT FEEDBACK REVISITED Controller Observer 15 of 23

OBSERVER – BASED OUTPUT FEEDBACK REVISITED Controller Assuming full-order observer here 15 of 23

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 15 of 23

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” 16 of 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 17 of 23

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) 18 of 23

APPLICATION EXAMPLE #1 load Generator 1 Generator 2 19 of 23

APPLICATION EXAMPLE #1 Generator 1 Generator 2 control input (mechanical power) With integral control: desired frequency electrical load (slowly varying) 19 of 23

APPLICATION EXAMPLE #1 Generator 1 Generator 2 control input (mechanical power) With integral control: desired frequency electrical load (slowly varying) 19 of 23

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 19 of 23

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) 20 of 23

APPLICATION EXAMPLE #2 Lorenz system 21 of 23

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) 21 of 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 electric power generators Lorenz chaotic system 22 of 23

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 23 of 23