MEETING THE NEED FOR ROBUSTIFIED NONLINEAR SYSTEM THEORY CONCEPTS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign Plenary lecture, ACC, Seattle, 6/13/08 1 of 36

TALK OVERVIEW ContextConcept Control with limited information Input-to-state stability Adaptive control Minimum-phase property Stability of switched systems Observability 2 of 36

TALK OVERVIEW ContextConcept Control with limited information Input-to-state stability Adaptive control Minimum-phase property Stability of switched systems Observability 2 of 36

Plant Controller INFORMATION FLOW in CONTROL SYSTEMS 3 of 36

INFORMATION FLOW in CONTROL SYSTEMS Limited communication capacity Need to minimize information transmission Event-driven actuators Coarse sensing 3 of 36 Theoretical interest

[ Brockett, Delchamps, Elia, Mitter, Nair, Savkin, Tatikonda, Wong,… ] Deterministic & stochastic models Tools from information theory Mostly for linear plant dynamics BACKGROUND Previous work: Unified framework for quantization time delays disturbances Our goals: Handle nonlinear dynamics 4 of 36

Caveat: This doesn’t work in general, need robustness from controller OUR APPROACH (Goal: treat nonlinear systems; handle quantization, delays, etc.) Model these effects as deterministic additive error signals, Design a control law ignoring these errors, “Certainty equivalence”: apply control combined with estimation to reduce to zero Technical tools: Input-to-state stability (ISS) Lyapunov functions Small-gain theorems Hybrid systems 5 of 36

QUANTIZATION EncoderDecoder QUANTIZER finite subset of is partitioned into quantization regions 6 of 36

QUANTIZATION and INPUT-to-STATE STABILITY 7 of 36

– assume glob. asymp. stable (GAS) QUANTIZATION and INPUT-to-STATE STABILITY 7 of 36

QUANTIZATION and INPUT-to-STATE STABILITY 7 of 36 no longer GAS

quantization error Assume class QUANTIZATION and INPUT-to-STATE STABILITY 7 of 36

Solutions that start in enter and remain there This is input-to-state stability (ISS) w.r.t. measurement errors In time domain: [ Sontag ’89 ] quantization error Assume class QUANTIZATION and INPUT-to-STATE STABILITY 7 of 36 class, e.g.

Solutions that start in enter and remain there This is input-to-state stability (ISS) w.r.t. measurement errors In time domain: [ Sontag ’89 ] quantization error Assume class QUANTIZATION and INPUT-to-STATE STABILITY 7 of 36 class, e.g.

LINEAR SYSTEMS Quantized control law: 9 feedback gain & Lyapunov function Closed-loop: (automatically ISS w.r.t. ) 8 of 36

DYNAMIC QUANTIZATION 9 of 36

DYNAMIC QUANTIZATION – zooming variable Hybrid quantized control: is discrete state 9 of 36

DYNAMIC QUANTIZATION – zooming variable Hybrid quantized control: is discrete state 9 of 36

Zoom out to overcome saturation DYNAMIC QUANTIZATION – zooming variable Hybrid quantized control: is discrete state 9 of 36

After the ultimate bound is achieved, recompute partition for smaller region DYNAMIC QUANTIZATION – zooming variable Hybrid quantized control: is discrete state Can recover global asymptotic stability 9 of 36

QUANTIZATION and DELAY Architecture-independent approach Based on the work of Teel Delays possibly large QUANTIZER DELAY 10 of 36

QUANTIZATION and DELAY where hence Can write 11 of 36

SMALL – GAIN ARGUMENT if then we recover ISS w.r.t. [ Teel ’98 ] Small gain: Assuming ISS w.r.t. actuator errors: In time domain: 12 of 36

FINAL RESULT Need: small gain true 13 of 36

FINAL RESULT Need: small gain true 13 of 36

FINAL RESULT solutions starting in enter and remain there Can use “zooming” to improve convergence Need: small gain true 13 of 36

EXTERNAL DISTURBANCES [ Nešić–L ] State quantization and completely unknown disturbance 14 of 36

EXTERNAL DISTURBANCES [ Nešić–L ] State quantization and completely unknown disturbance 14 of 36

Issue: disturbance forces the state outside quantizer range Must switch repeatedly between zooming-in and zooming-out Result: for linear plant, can achieve ISS w.r.t. disturbance (ISS gains are nonlinear although plant is linear; cf. [ Martins ]) EXTERNAL DISTURBANCES [ Nešić–L ] State quantization and completely unknown disturbance After zoom-in: 14 of 36

TALK OVERVIEW ContextConcept Control with limited information Input-to-state stability Adaptive control Minimum-phase property Stability of switched systems Observability 15 of 36

OBSERVABILITY and ASYMPTOTIC STABILITY Barbashin-Krasovskii-LaSalle theorem: (observability with respect to ) observable => GAS Example: is glob. asymp. stable (GAS) if s.t. is not identically zero along any nonzero solution (weak Lyapunov function) 16 of 36

SWITCHED SYSTEMS Want: stability of for classes of Many results based on common and multiple Lyapunov functions [ Branicky, DeCarlo, … ] In this talk: weak Lyapunov functions is a collection of systems is a switching signal is a finite index set 17 of 36

SWITCHED LINEAR SYSTEMS [ Hespanha ’04 ] Want to handle nonlinear switched systems and nonquadratic weak Lyapunov functions Theorem (common weak Lyapunov function): observable for each Need a suitable nonlinear observability notion Switched linear system is GAS if infinitely many switching intervals of length s.t.. 18 of 36

OBSERVABILITY: MOTIVATING REMARKS Several ways to define observability (equivalent for linear systems) Benchmarks: observer design or state norm estimation detectability vs. observability LaSalle’s stability theorem for switched systems No inputs here, but can extend to systems with inputs Joint work with Hespanha, Sontag, and Angeli 19 of 36

STATE NORM ESTIMATION This is a robustified version of 0-distinguishability where (observability Gramian) Observability definition #1: where 20 of 36

OBSERVABILITY DEFINITION #1: A CLOSER LOOK Initial-state observability: Large-time observability: Small-time observability: Final-state observability: 21 of 36

DETECTABILITY vs. OBSERVABILITY Observability def’n #2: OSS, and can decay arbitrarily fast Detectability is Hurwitz Observability can have arbitrary eigenvalues A natural detectability notion is output-to-state stability (OSS): [ Sontag-Wang ] where 22 of 36

and Theorem: This is equivalent to definition #1 ( small-time obs. )definition #1 Definition: For observability, should have arbitrarily rapid growth OSS admits equivalent Lyapunov characterization: OBSERVABILITY DEFINITION #2: A CLOSER LOOK 23 of 36

STABILITY of SWITCHED SYSTEMS Theorem (common weak Lyapunov function): s.t. Switched system is GAS if infinitely many switching intervals of length Each system is observable: 24 of 36

MULTIPLE WEAK LYAPUNOV FUNCTIONS Example: Popov’s criterion for feedback systems Instead of a single, can use a family with additional hypothesis on their joint evolution: linear system observable positive real See also invariance principles for switched systems in: [ Lygeros et al., Bacciotti–Mazzi, Mancilla-Aguilar, Goebel–Sanfelice–Teel ] Weak Lyapunov functions: 25 of 36

TALK OVERVIEW ContextConcept Control with limited information Input-to-state stability Adaptive control Minimum-phase property Stability of switched systems Observability 26 of 36

MINIMUM-PHASE SYSTEMS [ Byrnes–Isidori ] Nonlinear (affine in controls): zero dynamics are (G)AS Robustified version [ L–Sontag–Morse ]: output-input stability Linear (SISO): stable zeros, stable inverse implies minimum-phase for nonlinear systems (when applicable) reduces to minimum-phase for linear systems (SISO and MIMO) 27 of 36 where for some ;

UNDERSTANDING OUTPUT-INPUT STABILITY uniform over : OSS (detectability)w.r.t. extended output, 28 of 36 Output-input stability detectability + input-bounding property: Sufficient Lyapunov condition for this detectability property:

CHECKING OUTPUT-INPUT STABILITY For systems affine in controls, can use structure algorithm for left-inversion to check the input-bounding property equation for is ISS w.r.t. Example 1: can solve for and get a bound on can get a bound on OUTPUT-INPUT STABLE 29 of 36 Detectability: Input bounding:

CHECKING OUTPUT-INPUT STABILITY Example 2: 30 of 36 For systems affine in controls, can use structure algorithm for left-inversion to check the input-bounding property Does not have input-bounding property: can be large near while stays bounded zero dynamics, minimum-phase Output-input stability allows to distinguish between the two examples

FEEDBACK DESIGN Output stabilization state stabilization ( r – relative degree) Output-input stability guarantees closed-loop GAS In general, global normal form is not needed Example: global normal form GAS zero dynamics not enough for stabilization Apply to have with Hurwitz It is stronger than minimum-phase: ISS internal dynamics 31 of 36

CASCADE SYSTEMS For linear systems, recover usual detectability If: is detectable (OSS) is output-input stable then the cascade system is detectable (OSS) through extended output 32 of 36

ADAPTIVE CONTROL Linear systems Controller Plant Design model If: plant is minimum-phase system inside the box is output-stabilized controller and design model are detectable then the closed-loop system is detectable through (“tunable” [ Morse ’92 ]) 33 of 36

ADAPTIVE CONTROL If: plant is minimum-phase system inside the box is output-stabilized controller and design model are detectable then the closed-loop system is detectable through e (“tunable” [ Morse ’92 ]) Nonlinear systems Controller Plant Design model 34 of 36

ADAPTIVE CONTROL If: plant is output-input stable system inside the box is output-stabilized controller and design model are detectable then the closed-loop system is detectable through e (“tunable” [ Morse ’92 ]) Nonlinear systems Controller Plant Design model 34 of 36

ADAPTIVE CONTROL Nonlinear systems If: plant is output-input stable system in the box is output-stabilized controller and design model are detectable then the closed-loop system is detectable through e (“tunable” [ Morse ’92 ]) Controller Plant Design model 34 of 36

ADAPTIVE CONTROL Nonlinear systems If: plant is output-input stable system in the box is output-stabilized controller and design model are detectable then the closed-loop system is detectable through and its derivatives (“weakly tunable”) Controller Plant Design model 34 of 36

TALK SUMMARY ContextConcept Control with limited information Input-to-state stability Adaptive control Minimum-phase property Stability of switched systems Observability 35 of 36

ACKNOWLEDGMENTS Roger Brockett Steve Morse Eduardo Sontag Financial support from NSF and AFOSR Colleagues, students and staff at UIUC Everyone who listened to this talk Special thanks go to: 36 of 36