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

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

Plant Controller INFORMATION FLOW in CONTROL SYSTEMS

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

[ 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

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

QUANTIZATION EncoderDecoder QUANTIZER finite subset of is partitioned into quantization regions

QUANTIZATION and ISS

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

no longer GAS

QUANTIZATION and ISS quantization error Assume class

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

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

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 DYNAMIC QUANTIZATION – zooming variable Hybrid quantized control: is discrete state ISS from to small-gain condition Proof: Can recover global asymptotic stability

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

QUANTIZATION and DELAY where Can write hence

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:

FINAL RESULT Need: small gain true

FINAL RESULT Need: small gain true

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

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

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

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:

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

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

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

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 )

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

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

LINEAR SYSTEMS

[ 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:

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

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)

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 ] :

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 ]

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

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)