A Robust Event-Triggered Consensus Strategy for Linear Multi-Agent Systems with Uncertain Network Topology

A Robust Event-Triggered Consensus Strategy for Linear Multi-Agent Systems with Uncertain Network Topology Amir Amini, Amir Asif, Arash Mohammadi Electrical and Computer Engineering,, Montreal, Canada. Concordia Institute for Information System Engineering,, Canada. 2018 International Conference on Acoustics, Speech, and Signal Processing ICASSP 2018 1 / 16

Multi-agent Systems A multi-agent system consists of multiple agents that interact to achieve a cooperative objective. An agent can represent a moving vehicle, a sensor node, an electric bus, etc. Cooperative Objectives: Formation, Consensus, Containment, Rendezvous,... ICASSP 2018 2 / 16

Consensus Problem Statement and Objectives Consensus Control: Agent 1 Agent 4 Agent 2 Agent 3 The agreement value Consensus: To reach an agreement upon a common value. ICASSP 2018 3 / 16

Agent Dynamics Problem Statement and Objectives General Linear Agent Model : ẋ i (t) = Ax i (t) + Bu i (t), 1 i N (1) x i (t) R n : The state of agent i at time instant t; A R n n : System matrix (known and constant); B R n m : Input matrix (known and constant); u i (t) R m n : A proposed distributed control input; N : Number of agents in the network. ICASSP 2018 4 / 16

Consensus Problem Statement and Objectives Consensus Definition: For any initial condition x i (0), the consensus problem for (1) is said to be solved iff : Global sense : lim x i (t) x j (t) = 0, (1 i, j N), t Average sense : lim x i (t) 1 N t N j=1 x j(0) = 0, (1 i N), Average consensus is usually considered for first-order agents defined by ẋ i (t) = u i (t), with x i (0) as initial local observation. Key components in reaching consensus: Distributed control input u i (t), Information exchange between the neighbouring agents. ICASSP 2018 5 / 16

Event-triggered Consensus Controller (t) u i xˆ j ( t), j N i ˆ Agent Transmit x i ( t) (t) x i Event-trigger Node i 1 x i(t): The state of agent i 2 ˆx i(t): The last transmitted state of agent i up to time t The received information is subject to uncertainty due to existence of communication unreliabilities robustness is required ICASSP 2018 6 / 16

Motivation and Objective Motivation: Transmission saving for consensus in multi-agent systems with bandwidth constrained environments and unreliable channel. Objective: Achieve event-triggered consensus with a desired exponential rate of convergence (as opposed to asymptotic rate); Compute optimal consensus parameters to achieve consensus in presence of network uncertainties. ICASSP 2018 7 / 16

Main Features Problem Statement and Objectives Event-based disagreement vector : q i (t) = j N i ā ij (e A(t ti k i ) xi (t i k i ) e A(t tj k j ) xj (t j k j )) where ā ij is the uncertain (but norm-bounded) weight for channel link between agent i and j. Measurement error : e i (t) = e A(t ti k i ) xi (t i k i ) x i (t). Event-triggering function : given an event time t i k i, the next event for agent i is triggered at t = t i k i +1, where t i k i +1 = inf {t > ti k i e i (t) φ q i (t) 0 }, (2) φ > 0 : Transmission threshold to be designed. ICASSP 2018 8 / 16

Design unknown parameters The proposed control law : u i (t) = K i q i (t), (3) K i : Control gain to be designed. Question: How to design optimal 1 values for transmission threshold φ and control gain K i that guarantee an exponential rate of consensus in norm-bounded uncertain network channel? 1 maximize φ to minimize events, and minimize K i to minimize control force ICASSP 2018 9 / 16

Preliminary steps prior to optimization Consider the augmented closed-loop system; Convert the consensus problem into an equivalent stability problem Lyapunov stability method Obtain sufficient conditions and inequalities for uncertain connectivity links. ICASSP 2018 10 / 16

Compute optimal consensus parameters Solve the following convex optimization problem with desired convergence rate ζ min f = Θ i,µ,ɛ,τ j,p,ω 1,ω 2,ω 3,ω 4 [ ] [ Π1 Π S.t: Π = 2 ω1 τ < 0, 1 Π 3 1 [ ] [ ω3 I I ω4 I Θ > 0, T ] < 0, P To maximize φ To minimize K {}}{{}}{ i ω 1 + ω 2 + ω 3 + ω 4 (4) I ] < 0, [ ] ω2 µ < 0, 1 Θ i (1 i N), µ, ɛ, τ j (1 j 3), P, ω c (1 c 4) are decision variables; Block Matrix Π contains information about agent models, network connectivity, exponential convergence criterion, uncertainty upper bound, control gain K i, and transmission threshold φ. ICASSP 2018 11 / 16

Compute optimal consensus parameters Once the optimization problem (4) is solved, compute consensus parameters φ = τ 1 1 µ 1, and K i = B i P 1 Θ i, (1 i N) (5) Consensus parameters are bounded for the minimized objective function f = ω 1 + ω 2 + ω 3 + ω 4 φ (ω 1 ω 2 ) 1 4, K T i K i ω 3 ω4b 2 i B T i, (1 i N). (6) ICASSP 2018 12 / 16

Experimental Results A network of six second-order heterogeneous agents ṙ i (t) = v i (t), m i v i (t) = u i (t), (1 i 6), (7) r i (t) R: Position, v i (t) R: Velocity, m i : Inertia Consensus in this problem is to, distributively, reach a common position and velocity Laplacian Matrix (two unreliable links) L = 2.5 0 0 0.5 1 1 0 2 0 0 1 1 0 1 3 1 0 1 0 1 0 2 0 1 1 1 0 1 3 0 1 1 0.5 0 0 2.5 L = 2 0 0 0 1 1 0 2 0 0 1 1 0 1 3 1 0 1 0 1 0 2 0 1 1 1 0 1 3 0 1 1 0 0 0 2 Solve the optimization problem (4) to compute K i and φ (8) ICASSP 2018 13 / 16

Experimental Results Position & Velocity 10 5 0 (a) 0 1 2 3 4 xi(t) vi(t) Agents index number (b) 6 5 4 3 2 1 0 1 2 3 4 How different values for decay rate ζ affect the consensus process? Table 1: Consensus performance for varying ζ. Number of transmissions decay Consensus Objective per agent rate ζ time (sec) function f 1 2 3 4 5 6 0.2 262 295 333 318 369 321 10.57 401.19 0.3 133 154 175 180 164 142 4.81 406.84 0.4 68 58 95 194 50 68 3.51 411.27 Position & Velocity 10 5 0 0 1 ing to Lemma 1 given in [23], condition (20) is satisfied if 50 ICASSP 2018 14 / 16 ity

Problem Statement and Objectives 1 For a desired rate of convergence, robust event-triggered consensus is reached for norm-bounded uncertain networks; 2 Using convex optimization, the transmission threshold φ is maximized (to trigger minimum number of events) and control gain K i is minimized (to minimize the control force); 3 As convergence rate ζ is increased, the consensus time constantly gets reduced until the optimization problem becomes infeasible ICASSP 2018 15 / 16

Question? Problem Statement and Objectives Thank You ICASSP 2018 16 / 16