NONLINEAR CONTROL with LIMITED INFORMATION Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign Plenary talk, 2 nd Indian Control Conference, Hyderabad, Jan 5, of 21

Plant Controller INFORMATION FLOW in CONTROL SYSTEMS 2 of 21

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 3 of 21

[ 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 21

Caveat: This doesn’t 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 5 of 21

QUANTIZATION EncoderDecoder QUANTIZER finite subset of is partitioned into quantization regions 6 of 21

QUANTIZATION and ISS 7 of 21

– assume glob. asymp. stable (GAS) QUANTIZATION and ISS 7 of 21

QUANTIZATION and ISS no longer GAS 7 of 21

QUANTIZATION and ISS quantization error Assume class 7 of 21

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 7 of 21

LINEAR SYSTEMS Quantized control law: 9 feedback gain & Lyapunov function Closed-loop: (automatically ISS w.r.t. ) 8 of 21

DYNAMIC QUANTIZATION 9 of 21

DYNAMIC QUANTIZATION – zooming variable Hybrid quantized control: is discrete state 9 of 21

DYNAMIC QUANTIZATION – zooming variable Hybrid quantized control: is discrete state 9 of 21

Zoom out to overcome saturation DYNAMIC QUANTIZATION – zooming variable Hybrid quantized control: is discrete state 9 of 21

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 [ L–Nešić, ’05, ’06, L–Nešić–Teel ’14 ] 9 of 21

QUANTIZATION and DELAY Architecture-independent approach Based on the work of Teel Delays possibly large QUANTIZER DELAY 10 of 21

QUANTIZATION and DELAY where Can write hence 11 of 21

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 21

FINAL RESULT Need: small gain true 13 of 21

FINAL RESULT Need: small gain true 13 of 21

FINAL RESULT solutions starting in enter and remain there Can use “zooming” to improve convergence Need: small gain true 13 of 21

EXTERNAL DISTURBANCES [ Nešić–L ] State quantization and completely unknown disturbance 14 of 21

EXTERNAL DISTURBANCES [ Nešić–L ] State quantization and completely unknown disturbance 14 of 21

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 21

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 15 of 21

ACTIVE PROBING for INFORMATION dynamic (changes at sampling times) (time-varying) PLANT QUANTIZER CONTROLLER EncoderDecoder very small 16 of 21

NONLINEAR SYSTEMS sampling times Example: Zoom out to get initial bound Between samplings 17 of 21

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 ) 17 of 21

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 18 of 21

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 19 of 21

LINEAR SYSTEMS 20 of 21

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: 20 of 21

OTHER RESEARCH DIRECTIONS Modeling uncertainty (with L. Vu) Disturbances and coarse quantizers (with Y. Sharon) Multi-agent coordination (with S. LaValle and J. Yu) Quantized output feedback and observers (with H. Shim) Vision-based control (with Y. Ma and Y. Sharon) Performance-based design (with F. Bullo) Quantized control of switched systems 21 of 21