Average-Consensus of Multi-Agent Systems with Direct Topology Based on Event-Triggered Control

Transcription

1 Outline Background Preliminaries Consensus Numerical simulations Conclusions Average-Consensus of Multi-Agent Systems with Direct Topology Based on Event-Triggered Control Department of Automation Nankai University Tianjin , China October 10, 2010

2 Outline Background Preliminaries Consensus Numerical simulations Conclusions OUTLINE 1 Outline 2 Background 3 Preliminaries 4 Event-triggered average-consensus The Control Law Consensus protocols Remarks 5 Numerical simulations Simulation settings Simulation results 6 Conclusions

3 Outline Background Preliminaries Consensus Numerical simulations Conclusions BACKGROUND I A critical problem in distributed coordinated control of multiple agents is the consensus problem and numerous contributions have been given. The fact for some real systems: Each agent of a MAS (multi-agent system) may has limited resources to gather information and actuate the individual agent controller updates. Example: Physically distributed sensor/actuator networks responsible for collecting and processing information, and to react to this information through actuation according to some ruling. 1 The scheduling algorithms based on event-triggered is one of effective method that can increase the life span of each agent in such systems. 1 P. Tabuada. Event-Triggered Real-Time Scheduling of Stabilizing Control Tasks[J]. IEEE Trans. Autom. Control, 2007, 52(9):

4 Outline Background Preliminaries Consensus Numerical simulations Conclusions BACKGROUND II In the current work, the framework proposed by Dimarogonas et al. in 2 is used to design the average-consensus protocol for MAS with direct and weighted topology. 2 D. V. Dimarogonas and K. H. Johansson. Event-triggered Control for Multi-Agent Systems[C]// Proceedings of IEEE CDC/CCC2009.China, 2009:

5 Outline Background Preliminaries Consensus Numerical simulations Conclusions Graph Theory I A weighted diagraph with N vertices: = (,, A). = {1, 2,..., N} A = (a ij ) R N N, a ij 0 the set of vertices the set of edges the adjacency matrix i = {j (j, i), j = i} is the neighbor set for agent i. L = D A is called the Laplacian of, with D = diag{d 1, d 2,..., d N } and d i = j i a ij. If there is a path between any two vertices of the digraph, then is called connected. The Laplacian of a connected graph has a single zero eigenvalue and the corresponding eigenvector is the vector of ones, 1 = [1, 1,..., 1] T R N. We denote by 0 = λ 1 (L) < λ 2 (L)... λ N (L) the eigenvalues of L for a connected and undirect graph.

6 Outline Background Preliminaries Consensus Numerical simulations Conclusions Consensus Problem I The dynamics for each agent of a MAS with weighted graph, x i = u i, i = {1, 2,..., N}, (1) where x i R is the state of agent i, u i denotes the control inputs, which is called a protocol or algorithm. Protocol u i is said to asymptotically solve the consensus problem, if and only if the states of agents satisfy for all i, j. Furthermore, if lim x i(t) x j (t) = 0, (2) t lim t x i(t) = 1 N x j (0), (3) protocol u i is said to asymptotically solve the average-consensus problem. j

7 Outline Background Preliminaries Consensus Numerical simulations Conclusions Notations I n the n n unitary matrix u u T u, u R m M max 1 i N λi {M T M}, M R m m M s 1 2 (M + M T )

8 Outline Background Preliminaries Consensus Numerical simulations Conclusions Lemmas Lemma 1 If diagraph is strongly connected, the solutions of Lx = 0 are x = a1, a R. Lemma 2 Let be a digraph with Laplacian L. Then L s = 1 2 (L + LT ) is a valid Laplacian matrix for an undirected graph if and only if is balanced.

9 Outline Background Preliminaries Consensus Numerical simulations Conclusions Lemmas Lemma 3 For a connected graph that is undirected with Laplacian matrix L, the following property holds Lemma 2 and 3 can be found in 3. x T Lx min x =0 x = λ 2(L). (4) 2 1 T x=0 3 R. Olfati-Saber and R. M. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Transactions on Automatic Control, 2004, 49(9): 1520C1533.

10 Outline Background Preliminaries Consensus Numerical simulations The Control Conclusions Law Consensus protocols Remarks Event-Triggered Average-Consensus I Our method: For each agent i, and t 0, we introduce a (state) measurement error e i (t), which will be given in the sequel. The discrete event time instants : when a condition f(e(t)) = 0 holds, where e(t) = [e T 1 (t),..., et N (t)]t. f(e(t)) is called the trigger function. The sequence of event-triggered executions is denoted by: t 0, t 1,..., where each t i is defined by f(e(t i )) = 0 for i = 0, 1,.... The control law for agent i u i (t) = j i a ij (x i (t i ) x j (t i )), t [t i, t i+1 ). (5)

11 Outline Background Preliminaries Consensus Numerical simulations The Control Conclusions Law Consensus protocols Remarks Event-Triggered Average-Consensus II Between control updates, the input u i, i holds constant and equals to the last control update. The collective form of (5) is where u(t) = Lx(t i ), t [t i, t i+1 ). (6) u(t) = [u 1 (t), u 2 (t),..., u N (t)] T, x(t) = [x 1 (t), x 2 (t),..., x N (t)] T. The state measurement error is defined as for t [t i, t i+1 ). e(t) = x(t i ) x(t), i = 0, 1,..., (7)

12 Outline Background Preliminaries Consensus Numerical simulations The Control Conclusions Law Consensus protocols Remarks Event-Triggered Average-Consensus III The closed loop system x(t) = L(x(t) + e(t)). (8) We will show next that the following trigger function can ensure the average-consensus for the MAS with interacting topology being weighted and balanced f(e) Le σλ 2 (L s ) Lx L, (9) where σ (0, 1). According to this trigger function, each node need not know initially the unique consensus point, which is better for the protocol designer.

13 Outline Background Preliminaries Consensus Numerical simulations The Control Conclusions Law Consensus protocols Remarks Average-consensus protocols I Theorem 1 Consider the system (1) with the protocol (6). Assume diagraph is strongly connected and balanced and the trigger function is (9), then the inter-event times {t i+1 t i } are lower bounded by τ = σλ 2 (L s ) L ( L + σλ 2 (L s )) (10) for any initial condition in R n. Proof: We can get the followings e = x, x = L(x + e) Lx + Le

14 Outline Background Preliminaries Consensus Numerical simulations The Control Conclusions Law Consensus protocols Remarks Average-consensus protocols II Now we denote z = Le Lx, then d dt z = (Le)T L x Le Lx Le (Lx)T L x Lx 3... Le L x Le Lx L (1 + z) 2. Le Lx L x + Lx 3 So that z satisfies the bound z(t) φ(t, ω 0 ), where φ(t, φ 0 ) is the solution of φ = L (1 + φ) 2, φ(t, φ 0 ) = φ 0. (11)

15 Outline Background Preliminaries Consensus Numerical simulations The Control Conclusions Law Consensus protocols Remarks Average-consensus protocols III According to (9) and the event trigger condition f(e) = 0, the inter-event times are bounded from below by τ that satisfies φ(τ, 0) = τ L 1 τ L We can also get φ(τ, 0) = σλ 2(L s ). So that τ satisfies L σλ 2 (L s ) L = τ L 1 τ L, which implies (10). Now, we have the following criteria for average-consensus.

16 Outline Background Preliminaries Consensus Numerical simulations The Control Conclusions Law Consensus protocols Remarks Average-consensus protocols IV Theorem 2 Assume diagraph of the network (1) is strongly connected and balanced. Protocol (6) with the trigger function (9) globally asymptotically solves the average-consensus problem for any σ (0, 1). Proof: Let y = Lx.we have 1 T y = 1 T Lx = 0 and L s is a valid Laplacian matrix for an undirected graph according to Lemma 2. By Lemma 3, we then obtain y T L s y λ 2 (L s ) y 2. The candidate ISS Lyapunov function is chosen as V (y) = 1 2 yt y. V = y y = y T L s y + y T LLe λ 2 (L s ) y 2 + y L Le (12) = (σ 1)λ 2 (L s ) y 2. (13)

17 Outline Background Preliminaries Consensus Numerical simulations The Control Conclusions Law Consensus protocols Remarks Average-consensus protocols V On the other hand, the inter-event times {t i+1 t i } are lower bounded by a strictly positive time τ by Theorem 1. The positive property of τ guarantees that the overall switched system does not exhibit Zeno behavior. Using the extension of LaSalle s Invariance Principle for hybrid systems, we can conclude that lim t y(t) = 0. We then obtain lim t x(t) = a1 for some a R according to Lemma 1. Now, denote by x(t) = 1 n i x i(t) the average of the agents states. Since diagraph is balanced, we get x(t) = 1 x i (t) = 1 n n 1T L(x(t) + e(t)) = 0. (14) i So that x(t) = 1 n i x i(0) x. Thus a = 1 n i x i(0) x, which means that the average-consensus has been reached.

18 Outline Background Preliminaries Consensus Numerical simulations The Control Conclusions Law Consensus protocols Remarks Remarks Remark 1 Since larger τ implies lower frequency of control updating, thus the larger τ, the better for the system. On the other hand, the smaller V means the faster convergence speed, which is better for the network. We know from (10) that τ is increasing with respect to σ. Then a larger σ leads to lower frequency of control updating for each agent while a smaller σ leads to faster system convergence according to (13).

19 Outline Background Preliminaries Consensus Numerical simulations The Control Conclusions Law Consensus protocols Remarks Remarks I Remark 2 The event-triggered average-consensus should be discussed for MAS with switching topology. The topology of a switching network : s(t), where s(t) : R J = {1, 2,..., N}. Γ = { 1, 2,..., N } : the set of topologies of the network. Suppose the sequence of time instants for the network to change its interaction topology is T 0, T 1,..., then the interaction topology for the MAS is s(ti ) in the time interval [T i, T i+1 ), i = 0, 1,.... The trigger function (9) is changed to λ 2 (L s s(t i ) ) L s(ti x ) f(e(t)) Le σ, (15) L s(ti )

20 Outline Background Preliminaries Consensus Numerical simulations The Control Conclusions Law Consensus protocols Remarks Remarks II where 0 < σ < 1, and the event is triggered when f(e(t)) = 0 or t = T i for each time interval [T i, T i+1 ). Assume all the diagraphs in Γ are strongly connected and balanced. We can obtain that for any arbitrary switching signal s( ), the proposed protocol with this trigger function in each topology switching interval [T i, T i+1 ) globally asymptotically solves the average-consensus problem.

21 Outline Background Preliminaries Consensus Numerical simulations Simulation Conclusions settings Simulation settings I Simulation results We consider a multi-agent system with four nodes and a fixed topology that is weighted and balanced. The adjacency matrix of the system is = , (16) which is an asymmetric matrix. The trigger function (9) is used and the initial value of each agent is randomly selected within the interval [ 10, 10] in our simulations. Each simulation lasts 40 seconds.

22 Outline Background Preliminaries Consensus Numerical simulations Simulation Conclusions settings Simulation results I Simulation results Figure 1: The dynamic of each node when σ = 0.9. We can see that this system has gain consensus.

23 Outline Background Preliminaries Consensus Numerical simulations Simulation Conclusions settings Simulation results II Simulation results Figure 2: The dynamic of Le when σ = 0.9. The time instants where Le(t) = 0 are the time instants for the system to actuate control updating. We can see that only a few actuation are needed for the system to reach average-consensus while a great number of control updating may be done in the continuous feedback scheme.

24 Outline Background Preliminaries Consensus Numerical simulations Conclusions Conclusions Average-consensus protocols of multi-agent systems with direct topology under event-triggered actuation updating rules were proposed and analyzed in this paper. Our method: The discrete time instants when each agent updates its control depends on a trigger function. An average-consensus protocol is proposed for networks with fixed interaction topology, the stability and influencing factors of which are also analyzed. The design of trigger functions for networks with variable topology is also discussed.

25 Outline Background Preliminaries Consensus Numerical simulations Conclusions Thank you!