CONTROL with LIMITED INFORMATION Daniel Liberzon Coordinated Science
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
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 • Theoretical interest
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 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])
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)
ACTIVE PROBING for INFORMATION PLANT QUANTIZER CONTROLLER dynamic (time-varying) dynamic (changes at sampling times) Encoder Decoder very small
NONLINEAR SYSTEMS Example: Zoom out to get initial bound sampling times Between samplings
NONLINEAR SYSTEMS Example: Between samplings Let The norm on a suitable compact region (dependent on ) • grows at most by the factor • is divided by 3 at the sampling time in one period
NONLINEAR SYSTEMS (continued) The norm • grows at most by the factor • is divided by 3 at each sampling time Pick small enough s. t. If this is ISS w. r. t. as before, then in one period
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])
LINEAR SYSTEMS
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 where amount of static info provided by quantizer is Hurwitz 0 [Baillieul, Brockett-L, Hespanha, Nair-Evans, Petersen-Savkin, Tatikonda]
HYBRID SYSTEMS as FEEDBACK CONNECTIONS continuous discrete • Other decompositions possible • Can also have external signals [L–Nešić, ’ 05, ’ 06, L–Nešić–Teel ’ 14]
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 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 La. Salle 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!
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. La. Valle and J. Yu) • Vision-based control (with Y. Ma and Y. Sharon)
- Slides: 42