A Guaranteed Cost LMI-Based Approach for Event-Triggered Average Consensus in Multi-Agent Networks

Transcription

1 A Guaranteed Cost LMI-Based Approach for Event-Triggered Average Consensus in Multi-Agent Networks Amir Amini, Arash Mohammadi, Amir Asif Electrical and Computer Engineering,, Montreal, Canada. Concordia Institute for Information System Engineering,, Canada. 5 th IEEE Global Conference on Signal and Information Processing. GlobalSIP / 14

2 Outline Problem Statement and Objectives Problem Statement and Objectives GlobalSIP / 14

3 Event-triggered Average Consensus 1 x i (t): The state of agent i 2 ˆx i (t): The last transmitted state of agent i up to time t Average Consensus Definition: lim t x i (t) 1 N N x j (0) = 0, 1 i N. (1) j=1 GlobalSIP / 14

4 Motivation and Objective Motivation Transmission saving for average consensus in multi-agent networks with bandwidth constrained environments. Adapting Guaranteed Cost approach to event-triggered average consensus. Objective Achieving event-triggered average consensus with restricted guaranteed operational cost. Compute optimal parameters to achieve average consensus with small number of transmission. GlobalSIP / 14

5 Event-triggered Average Consensus 1 Agent model: ẋ i (t) = u i (t), 1 i N; 2 Last transmitted state: ˆx i (t) = x i (t i k ), t [ti k, ti k+1 ). 3 Controller: u i (t) = j N i a ij ( ˆx i (t) ˆx j (t) ), 4 Error: e i (t) = ˆx i (t) x i (t) Closed-loop system: ( ) ẋ(t) = L x(t) + e(t), (2) L : Laplacian Matrix; x(t) = [ x 1 (t),..., x N (t) ] T, ˆx(t) = [ ˆx 1 (t),..., ˆx N (t) ] T, e(t) = [ e 1 (t),..., e N (t) ] T GlobalSIP / 14

6 Event-triggered Average Consensus Event-triggering function: Transmit new state if e i (t) exceeds the threshold φ ˆX i (t) ˆX i (t) : N i ˆx i (t) N i j=1 ˆx j(t), Positive scalar φ: The transmission threshold to be computed How to design the optimal value for transmission threshold φ? If φ = 0 Constant Transmission Inadequate small φ Waste of communication resources Inadequate large φ No consensus agreement GlobalSIP / 14

7 Cost function Problem Statement and Objectives The proposed cost function: J = 0 ( ) x T (t)rx(t)+u T (t)qu(t) dt (3) R and Q: given positive definite weighting matrices. Matrix R assigns desired penalty on deviation of the states x(t) from the target value. Matrix Q assigns desired penalty on control input u(t). If there exists a positive scalar J such that the cost J associated with the event-triggered average consensus process satisfies J J, then J is said to be a guaranteed cost. How to restrict (minimize) J? GlobalSIP / 14

8 Converting Consensus problem to Stability problem In order to use the Lyapunov stability theorem and incorporate the cost function J in parameter design, Consensus in transformed to an equivalent stability problem ( ) ẋ(t) = L x(t) + e(t) } {{ } Consensus problem ( ẋ r (t) = L ) x r (t) + e r (t) } {{ } Stability problem x r (t) = ˆLx(t), e r (t) = ˆLe(t), and L = ˆLLˆL. ˆL : The reduced order Laplacian matrix. The global Event-triggering condition: ( ) TM e T r (t)e r (t) e r (t)+x r (t) T φ 2 M( e r (t)+x r (t) ). (4) GlobalSIP / 14

9 Computing Optimal Transmission Threshold Lyapunov Candidate: V (t) = x T r (t)px r (t) Incorporating the Lyapunov Stability theorem and proposed cost with this inequality: V (t) + x T r (t)rx r (t) + u T (t)qu(t) < 0 (5) If (5) is satisfied V (t) < 0 The system is stable ( lim t x r (t) = 0) Reaching average consensus Integrating (5) results in V ( ) V (0) + 0 x T r Rx r (t) + u T (t)qu(t) dt < 0, which is equivalent to J < [V (0) = x T r (0)Px r (0)]. }{{} J GlobalSIP / 14

10 Computing Optimal Transmission Threshold Compute Transmission Threshold φ from: φ = τ 1 γ 1 (6) which is conditioned on the solvability of the following convex optimization problem min γ,τ,p S.t: To enlarge φ {}}{ τ + γ + To restrict J {}}{ trace(p 2 ) PL L T P+R PL (LˆL ) T M T τi (LˆL ) T M T Q 1 0 γi < 0 (7) GlobalSIP / 14

11 Experimental Results Laplacian Matrix: L = Given optimization values: R = ri, Q = qi, with r = 10, q = 0.1 Computed Parameters: φ = , J = 13327, J = J < J State Trajectories Control Input ( a ) Time (sec) ( b ) Time (sec) GlobalSIP / 14

12 Experimental Results How different choices for R and Q affect the average consensus process? TI : Total iteration to reach average consensus AT : Average number of state transmission instants Table: The effect of weighting matrices R, and Q on the event-triggered average consensus performance r q φ TI AT J J GlobalSIP / 14

13 Conclusion 1 The data transmission threshold φ is affected by a different selection of weighting matrices R and Q. 2 A larger φ causes a faster consensus convergence rate with fewer number of transmissions which is at the expense of more cost J. 3 For a fixed Q, increasing R results in obtaining a relatively larger value for φ. Future Work 1 Guaranteed Cost average consensus in networks with random link failures. 2 Guaranteed Cost average consensus in networks with time-varying communication delay. GlobalSIP / 14

14 Question? Problem Statement and Objectives Thank You GlobalSIP / 14