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, 2015 1of 21
INFORMATION FLOW in CONTROL SYSTEMS Plant Controller 2of 21
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 Theoretical interest 3of 21
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 4of 21
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) Lyapunov functions Small-gain theorems Hybrid systems 5of 21
QUANTIZATION finite subset of Encoder Decoder QUANTIZER is partitioned into quantization regions 6of 21
QUANTIZATION and ISS 7of 21
QUANTIZATION and ISS assume glob. asymp. stable (GAS) 7of 21
QUANTIZATION and ISS no longer GAS 7of 21
QUANTIZATION and ISS Assume quantization error class 7 of 21
QUANTIZATION and ISS Assume quantization error Solutions that start in enter and remain there This is input-to-state stability (ISS) w.r.t. measurement errors In time domain: 7 of 21
LINEAR SYSTEMS feedback gain & Lyapunov function Quantized control law: 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
DYNAMIC QUANTIZATION zooming variable Hybrid quantized control: is discrete state Zoom out to overcome saturation 9 of 21
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 [L Nešić, 05, 06, L Nešić Teel 14] 9 of 21
QUANTIZATION and DELAY QUANTIZER DELAY Architecture-independent approach Delays possibly large Based on the work of Teel 10 of 21
QUANTIZATION and DELAY where Can write hence 11 of 21
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] 12 of 21
FINAL RESULT Need: small gain true 13 of 21
FINAL RESULT Need: small gain true 13 of 21
FINAL RESULT Need: small gain true solutions starting in enter and remain there Can use zooming to improve convergence 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
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]) 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 Prove discrete protocol stability via Lyapunov function Apply small-gain theorem to compute upper bound on maximal allowed transmission interval (MATI) 15 of 21
ACTIVE PROBING for INFORMATION PLANT QUANTIZER CONTROLLER dynamic (time-varying) dynamic (changes at sampling times) Encoder Decoder very small 16 of 21
Example: NONLINEAR SYSTEMS Zoom out to get initial bound sampling times Between samplings 17 of 21
NONLINEAR SYSTEMS Example: Between samplings Let The norm on a suitable compact region (dependent on ) grows at most by the factor in one period is divided by 3 at the sampling time 17 of 21
NONLINEAR SYSTEMS (continued) The norm grows at most by the factor in one period is divided by 3 at each sampling time Pick small enough s.t. If this is ISS w.r.t. as before, then 18 of 21
ROBUSTNESS of the CONTROLLER Option 1. ISS w.r.t. Same condition as before (restrictive, hard to check) Option 2. Look at the evolution of ISS w.r.t. checkable sufficient conditions ([Hespanha-L-Teel]) 19 of 21
LINEAR SYSTEMS 20 of 21
LINEAR SYSTEMS Between sampling times, grows at most by in one period divided by 3 at each sampling time global quantity: sampling frequency vs. open-loop instability amount of static info provided by quantizer where is Hurwitz 0 [Baillieul, Brockett-L, Hespanha, Nair-Evans, Petersen-Savkin,Tatikonda] 20 of 21
OTHER RESEARCH DIRECTIONS Quantized control of switched systems Quantized output feedback and observers (with H. Shim) Disturbances and coarse quantizers (with Y. Sharon) Modeling uncertainty (with L. Vu) Performance-based design (with F. Bullo) Multi-agent coordination (with S. LaValle and J. Yu) Vision-based control (with Y. Ma and Y. Sharon) 21 of 21