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CONTROL with LIMITED INFORMATION ; SWITCHING ADAPTIVE CONTROL Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign IAAC Workshop, Herzliya, Israel, June 1, 2009 SWITCHING CONTROL Plant: Classical continuous feedback paradigm: u P u y y P C u But logical decisions are often necessary: y P C1 C2 logic REASONS for SWITCHING • Nature of the control problem • Sensor or actuator limitations • Large modeling uncertainty • Combinations of the above REASONS for SWITCHING • Nature of the control problem • Sensor or actuator limitations • Large modeling uncertainty • Combinations of the above INFORMATION FLOW in CONTROL SYSTEMS Plant Controller INFORMATION FLOW in CONTROL SYSTEMS • Coarse sensing • Limited communication capacity • many control loops share network cable or wireless medium • microsystems with many sensors/actuators on one chip • Need to minimize information transmission (security) • Event-driven actuators BACKGROUND Previous work: [Brockett, Delchamps, Elia, Mitter, Nair, Savkin, Tatikonda, Wong,…] • Deterministic & stochastic models • Tools from information theory • Mostly for linear plant dynamics Our goals: • Handle nonlinear dynamics • Unified framework for • quantization • time delays • disturbances OUR APPROACH (Goal: treat nonlinear systems; handle quantization, delays, etc.) • Model these effects via deterministic error signals, • Design a control law ignoring these errors, • “Certainty equivalence”: apply control, combined with estimation to reduce to zero Caveat: This doesn’t work in general, need robustness from controller Technical tools: • Input-to-state stability (ISS) • Small-gain theorems • Lyapunov functions • Hybrid systems QUANTIZATION Encoder Decoder QUANTIZER is partitioned into quantization regions finite subset of QUANTIZATION and ISS QUANTIZATION and ISS – assume glob. asymp. stable (GAS) QUANTIZATION and ISS no longer GAS QUANTIZATION and ISS quantization error Assume class QUANTIZATION and ISS quantization error Assume Solutions that start in enter and remain there This is input-to-state stability (ISS) w.r.t. measurement errors In time domain: LINEAR SYSTEMS 9 feedback gain & Lyapunov function Quantized control law: Closed-loop: (automatically ISS w.r.t. ) DYNAMIC QUANTIZATION DYNAMIC QUANTIZATION – zooming variable Hybrid quantized control: is discrete state DYNAMIC QUANTIZATION – zooming variable Hybrid quantized control: is discrete state DYNAMIC QUANTIZATION – zooming variable Hybrid quantized control: is discrete state Zoom out to overcome saturation DYNAMIC QUANTIZATION – zooming variable Hybrid quantized control: is discrete state After ultimate bound is achieved, recompute partition for smaller region Can recover global asymptotic stability Proof: ISS from to ISS from to small-gain condition QUANTIZATION and DELAY QUANTIZER Architecture-independent approach Delays possibly large Based on the work of Teel DELAY QUANTIZATION and DELAY where Can write hence SMALL – GAIN ARGUMENT Assuming ISS w.r.t. actuator errors: In time domain: Small gain: if then we recover ISS w.r.t. [Teel ’98] FINAL RESULT Need: small gain true FINAL RESULT Need: small gain true FINAL RESULT Need: small gain true enter solutions starting in and remain there Can use “zooming” to improve convergence EXTERNAL DISTURBANCES [Nešić–L] State quantization and completely unknown disturbance EXTERNAL DISTURBANCES [Nešić–L] State quantization and completely unknown disturbance EXTERNAL DISTURBANCES [Nešić–L] State quantization and completely unknown disturbance After zoom-in: 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]) HYBRID SYSTEMS as FEEDBACK CONNECTIONS continuous discrete • Other decompositions possible • Can also have external signals [Nešić–L, ’05, ’06] SMALL – GAIN THEOREM Small-gain theorem [Jiang-Teel-Praly ’94] gives GAS if: • Input-to-state stability (ISS) from • ISS from • to to : : (small-gain condition) SUFFICIENT CONDITIONS for ISS • ISS from to • ISS from to if ISS-Lyapunov function [Sontag ’89]: if: and # of discrete events on [Hespanha-L-Teel ’08] is LYAPUNOV – BASED SMALL – GAIN THEOREM Hybrid system is GAS if: • • and # of discrete events on • is SKETCH of PROOF is nonstrictly decreasing along trajectories Trajectories along which is constant? None! GAS follows by LaSalle principle for hybrid systems [Lygeros et al. ’03, Sanfelice-Goebel-Teel ’07] APPLICATION to DYNAMIC QUANTIZATION ISS from to with some linear gain quantization error Zoom in: where ISS from to with gain small-gain condition! RESEARCH DIRECTIONS • Quantized output feedback • Performance-based design • Disturbances and coarse quantizers (with Y. Sharon) • Modeling uncertainty (with L. Vu) • Avoiding state estimation (with S. LaValle and J. Yu) • Vision-based control (with Y. Ma and Y. Sharon) http://decision.csl.uiuc.edu/~liberzon REASONS for SWITCHING • Nature of the control problem • Sensor or actuator limitations • Large modeling uncertainty • Combinations of the above MODELING UNCERTAINTY unmodeled dynamics parametric uncertainty 0 Also, noise and disturbance Adaptive control (continuous tuning) vs. supervisory control (switching) EXAMPLE Scalar system: , otherwise unknown (purely parametric uncertainty) stable not implementable Controller family: Could also take controller index set SUPERVISORY CONTROL ARCHITECTURE Supervisor candidate controllers Controller Controller ... Controller u1 u2 u Plant um ... – switching signal, takes values in – switching controller y TYPES of SUPERVISION • Prescheduled (prerouted) • Performance-based (direct) • Estimator-based (indirect) TYPES of SUPERVISION • Prescheduled (prerouted) • Performance-based (direct) • Estimator-based (indirect) OUTLINE • Basic components of supervisor • Design objectives and general analysis • Achieving the design objectives (highlights) OUTLINE • Basic components of supervisor • Design objectives and general analysis • Achieving the design objectives (highlights) SUPERVISOR y u y1 e1 Multi- y2 e2 ... Estimator y ep p ... ... Want to be small Then small indicates estimation errors: likely EXAMPLE Multi-estimator: => exp fast EXAMPLE disturbance Multi-estimator: => exp fast STATE SHARING Bad! Not implementable if The system produces the same signals is infinite SUPERVISOR y u y1 e1 1 e Monitoring y 2 Multi2 2 ... Signals Estimator y ep Generator p p ... ... ... Examples: EXAMPLE Multi-estimator: – can use state sharing SUPERVISOR y u y1 e1 1 e Monitoring y 2 Switching Multi2 2 ... Signals Estimator y Logic ep Generator p p ... ... ... Basic idea: Justification? Plant small => , controllers: small => plant likely in (“certainty equivalence”) gives stable closed-loop system => SUPERVISOR y u y1 e1 1 e Monitoring y 2 Switching Multi2 2 ... Signals Estimator y Logic ep Generator p p ... ... ... Basic idea: , controllers: Justification? Plant small => small => plant likely in only know converse! Need: small => gives stable closed-loop system => gives stable closed-loop system This is detectability w.r.t. DETECTABILITY Linear case: plant in closed loop with view as output Want this system to be detectable “output injection” matrix is Hurwitz asympt. stable SUPERVISOR y u y1 e1 1 Multi- y2 e2 Monitoring 2 Switching . Signals Estimator ..y Logic ep Generator p p ... ... ... We know: is small Switching logic (roughly): This (hopefully) guarantees that Need: is small small => stable closed-loop switched system This is switched detectability DETECTABILITY under SWITCHING Switched system: plant in closed loop with view as output Want this system to be detectable: Assumed detectable for each frozen value of Output injection: need this to be asympt. stable Thus needs to be “non-destabilizing” : • switching stops in finite time • slow switching (on the average) SUMMARY of BASIC PROPERTIES Multi-estimator: 1. At least one estimation error ( • when • is bounded for bounded ) is small & Candidate controllers: 2. For each , closed-loop system is detectable w.r.t. Switching logic: 3. is bounded in terms of the smallest 4. Switched closed-loop system is detectable w.r.t. provided this is true for every frozen value of conflicting : for 3, want to switch to for 4, want to switch slowly or stop SUMMARY of BASIC PROPERTIES Multi-estimator: 1. At least one estimation error ( • when • is bounded for bounded ) is small & Candidate controllers: 2. For each , closed-loop system is detectable w.r.t. Switching logic: 3. is bounded in terms of the smallest 4. Switched closed-loop system is detectable w.r.t. provided this is true for every frozen value of Analysis: 1 + 3 => is small 2 + 4 => detectability w.r.t. => state is small OUTLINE • Basic components of supervisor • Design objectives and general analysis • Achieving the design objectives (highlights) CANDIDATE CONTROLLERS fixed Controller y Plant u Multiestimator y yq eq CANDIDATE CONTROLLERS fixed y Plant P Controller C u Multiestimator E yq y Linear: overall system is detectable w.r.t. i. system inside the box is stable ii. plant is detectable Need to show: => i C, E => => => P, C, eq if E => ii P CANDIDATE CONTROLLERS fixed y Plant P Controller u C Multiestimator E yq y eq Linear: overall system is detectable w.r.t. i. system inside the box is stable ii. plant is detectable if Nonlinear: same result holds if stability and detectability are interpreted in the ISS / OSS sense: external signal CANDIDATE CONTROLLERS fixed y Plant P Controller C u Multiestimator E yq y eq Linear: overall system is detectable w.r.t. i. system inside the box is stable ii. plant is detectable if Nonlinear: same result holds if stability and detectability are interpreted in the integral-ISS / OSS sense: SWITCHING LOGIC: DWELL-TIME – dwell time – monitoring signals Initialize Wait time units Find ? no Detectability is preserved if Obtaining a bound on yes is large enough in terms of is harder Not suitable for nonlinear systems (finite escape) SWITCHING LOGIC: HYSTERESIS – hysteresis constant – monitoring signals Initialize Find no or (scale-independent) ? yes SWITCHING LOGIC: HYSTERESIS Initialize Find no finite, This applies to Linear, ? yes bounded => switching stops in finite time exp fast, bounded => average dwell time