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CONTROL with LIMITED INFORMATION Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign

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REASONS for SWITCHING Nature of the control problem Sensor or actuator limitations Large modeling uncertainty Combinations of the above

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Plant Controller INFORMATION FLOW in CONTROL SYSTEMS

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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 Coarse sensing Theoretical interest

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[ 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

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Caveat: This doesnt work in general, need robustness from controller 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 Technical tools: Input-to-state stability (ISS) Lyapunov functions Small-gain theorems Hybrid systems

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QUANTIZATION EncoderDecoder QUANTIZER finite subset of is partitioned into quantization regions

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QUANTIZATION and ISS

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– assume glob. asymp. stable (GAS) QUANTIZATION and ISS

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no longer GAS

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QUANTIZATION and ISS quantization error Assume class

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Solutions that start in enter and remain there This is input-to-state stability (ISS) w.r.t. measurement errors In time domain: QUANTIZATION and ISS quantization error Assume

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

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DYNAMIC QUANTIZATION

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– zooming variable Hybrid quantized control: is discrete state

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

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

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After ultimate bound is achieved, recompute partition for smaller region DYNAMIC QUANTIZATION – zooming variable Hybrid quantized control: is discrete state ISS from to small-gain condition Proof: Can recover global asymptotic stability

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

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QUANTIZATION and DELAY where Can write hence

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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:

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FINAL RESULT Need: small gain true

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FINAL RESULT Need: small gain true

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

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

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

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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:

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NETWORKED CONTROL SYSTEMS [ Nešić–L ] NCS: Transmit only some variables according to time scheduling protocol Examples: round-robin, TOD (try-once-discard) QCS: Transmit quantized versions of all variables NQCS: Unified framework combining time scheduling and quantization Basic design/analysis steps: Design controller ignoring network effects Apply small-gain theorem to compute upper bound on maximal allowed transmission interval (MATI) Prove discrete protocol stability via Lyapunov function

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ACTIVE PROBING for INFORMATION dynamic (changes at sampling times) (time-varying) PLANT QUANTIZER CONTROLLER EncoderDecoder very small

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NONLINEAR SYSTEMS sampling times Example: Zoom out to get initial bound Between samplings

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NONLINEAR SYSTEMS is divided by 3 at the sampling time Let Example: Between samplings grows at most by the factor in one period The norm on a suitable compact region (dependent on )

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Pick small enough s.t. NONLINEAR SYSTEMS (continued) grows at most by the factor in one period is divided by 3 at each sampling time The norm If this is ISS w.r.t. as before, then

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ROBUSTNESS of the CONTROLLER ISS w.r.t. Same condition as before (restrictive, hard to check) checkable sufficient conditions ([ Hespanha-L-Teel ]) ISS w.r.t. Option 1. Option 2. Look at the evolution of

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LINEAR SYSTEMS

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[ Baillieul, Brockett-L, Hespanha, Nair-Evans, Petersen-Savkin,Tatikonda ] Between sampling times, where is Hurwitz 0 divided by 3 at each sampling time grows at most by in one period amount of static info provided by quantizer sampling frequency vs. open-loop instability global quantity:

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HYBRID SYSTEMS as FEEDBACK CONNECTIONS continuous discrete [ Nešić–L, 05, 06 ] Other decompositions possible Can also have external signals

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SMALL – GAIN THEOREM Small-gain theorem [ Jiang-Teel-Praly 94 ] gives GAS if: Input-to-state stability (ISS) from to : ISS from to : (small-gain condition)

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SUFFICIENT CONDITIONS for ISS [ Hespanha-L-Teel 08 ] # of discrete events on is ISS from to if: and ISS from to if ISS-Lyapunov function [ Sontag 89 ] :

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LYAPUNOV – BASED SMALL – GAIN THEOREM Hybrid system is GAS if: and # of discrete events on is

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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 ]

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quantization error Zoom in: where ISS from to with gain small-gain condition! ISS from to with some linear gain APPLICATION to DYNAMIC QUANTIZATION

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RESEARCH DIRECTIONS Modeling uncertainty (with L. Vu) Disturbances and coarse quantizers (with Y. Sharon) Avoiding state estimation (with S. LaValle and J. Yu) Quantized output feedback (with H. Shim) Performance-based design (with F. Bullo) Vision-based control (with Y. Ma and Y. Sharon)

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