Optimal Communication Logics in Networked Control Systems Yonggang - PDF document

Optimal Communication Logics in Networked Control Systems Yonggang Xu João P. Hespanha Center for Control Engineering and Computation University of California Santa Barbara Typo in proceedings paper { eq. (22) and eq. above (11) } Please download corrected version from http://www.ece.ucsb.edu/~hespanha Distributed Control spatially distributed process to be controlled process process process (e.g., autonomous vehicles) sensing actuation controller controller controller node node node controller coordination supported by a communication network Goal : • minimize controller communication (stealth, bandwidth) • study the effect of non-ideal communication (delays, drops, blackouts) 1

Communication minimization controller controller controller node node node controller couplings supported by a communication network The “every bit-counts” paradigm… Goal : Design each controller to minimize the number of bits/second that need to be exchanged between nodes (quantization, compression, …) Domain: Media with little capacity and low-overhead protocols (bit at-a-time) E.g., underwater acoustic comm. between a small number of nodes. The “cost-per-message” paradigm… Goal : Design each controller to minimize the number of message exchanges paper focus between nodes (scheduling, estimation, …) Domain: Media shared by a large number of nodes with nontrivial media access control (MAC) protocol (packet at-a-time) E.g., 802.11 wireless comm. between a large number of nodes. Scenarios Rendezvous in minimum-time or using minimum-energy (in spite of disturbances) Group of autonomous agents cooperate in searching for a target (perhaps mobile—search & pursuit) 2

Prototype problem process process process controller controller controller node node node In this talk: � decoupled linear processes � coupled control objective (with stochastic disturbance d ) notation: x + ≡ x ( k +1) Prototype problem process process process controller controller controller node node node In this talk: � decoupled linear processes � coupled control objective (with stochastic disturbance d ) || C 1 x 1 – C 2 x 2 || 2 E.g., rendez-vous of two vehicles notation: x + ≡ x ( k +1) 3

Prototype problem process process process controller controller controller node node node In this talk: � decoupled linear processes Minimum-cost solution (centralized) (with stochastic disturbance d ) Completely decentralized solution Communication performance trade-off control cost optimal communication achievable cost/comm. pairs non-achievable cost/comm. pairs communication minimum-cost (centralized) minimum comm. needed for solvability (zero, when decentralized solution yields finite cost) 4

Centralized architecture i th local process i th local controller constant communication between local and external process(es) j th external process Closed-loop system for simplicity here we assume only two processes Estimator-based distributed architecture [Yook & Tilbury, i th local process Montestruque & Antsaklis, Xu & JH] i th local controller continuous open-loop estimator with discrete “updates” from the network i th local estimator for j th external process j th external process Closed-loop system additive perturbation w.r.t centralized equations 5

Communication logic network i th local estimator for j th external process j th external process fusion sched. logic logic How to fuse data? When to send data? Scheduling logic action With no noise & delay (for now…) x j ( k ) received from network at time k ⇓ Options… • periodically “best-estimate” T ∈ N a j ( k ) = { k divisible by T ?}, based on data received • feedback policy ⇓ a j ( k ) = F ( x j ( k ), …) • “optimal” … Optimal Scheduling Logic d is Gaussian i.i.d. with zero network mean and covariance Σ i th local estimator for j th external process j th external process fusion sched. logic logic Goals : minimize the estimation error ⇒ minimize cost-penalty w.r.t. centralized minimize the number of transitions ⇒ minimize communication bandwidth average L-2 norm average transmission rate relative weight of two criteria (will lead to Pareto-optimal solution) 6

Dynamic Programming solution for simplicity of notation 1. dropped j / i subscripts 2. A denotes A j + B j K jj 3. d is Gaussian with zero a ( k ) ∈ {0,1} mean and covariance Σ undiscounted average-cost problem Theorem dynamic programming (DP) operator 1. There exists J * ∈ R and bounded h * : R n → R such that 2. J * is the optimal cost and is achieved by the ( deterministic) static policy 3. h can be found by value iteration Dynamic Programming solution Proof outline: 1. e ( k ) is Markov and its transition distribution satisfies an Ergodic property (requires a mild restriction on the set of admissible policies omitted here) 2. T is a span-contraction [Hernandez-Lerma 96] 3. Result follows using standard arguments based on Banach’s Fixed-Point Theorem for semi-norms. Theorem dynamic programming (DP) operator 1. There exists J * ∈ R and bounded h * : R n → R such that 2. J * is the optimal cost and is achieved by the ( deterministic) static policy 3. h can be found by value iteration 7

Example (2-dim) local process 10 optimal scheduling e 2 λ = 100 5 a ( k ) a ( k ) = 1 = 0 0 -5 not ellipses! -10 -10 -5 0 5 10 e 1 Example (2-dim) local process 10 optimal scheduling e 2 λ = 100 5 large weight in comm. cost a ( k ) = 1 ⇓ a ( k large error threshold ) 0 = 0 ⇓ only communicate λ = 10 -5 when error is very large -10 -10 -5 0 5 10 e 1 8

Communication with latency network i th local estimator for j th external process j th external process fusion sched. logic logic delay of τ x j ( k ) is sched. logic fusion logic incorporated sends x j ( k ) recs. x j ( k ) into estimate k + τ k + τ+1 k τ = 0 in previous case Dynamic Programming solution a ( k ) ∈ {0,1} undiscounted average-cost problem Theorem 1. There exists J * ∈ R and bounded h * : R n → R such that 2. J * is the optimal cost and is achieved by the ( deterministic) static policy 3. h can be found by value iteration 9

Example (1-dim) 2 10 optimal scheduling estimation error variance optimal comm. vs. estim. error trade-off 1 10 estimation error, given all information enroute τ = 3 0 10 τ = τ = 1 τ = 0 2 with network latency same error variance communication rate -1 requires more bandwidth 10 .1 .25 .5 Conclusions process process process controller controller controller node node node communication network We constructed communications logics that minimize communication (measured in messages sending rate) We considered networks with (fixed) latency Study the effect of packet losses (especially important in wireless networks) Coupled control/communication-logic design Nonlinear processes Typo in proceedings paper { eq. (22) and eq. above (11) } Please download corrected version from http://www.ece.ucsb.edu/~hespanha 10