OUTPUT CONSENSUS OF HETEROGENEOUS LINEAR MULTI-AGENT SYSTEMS BY EVENT-TRIGGERED CONTROL

Transcription

1 OUTPUT CONSENSUS OF HETEROGENEOUS LINEAR MULTI-AGENT SYSTEMS BY EVENT-TRIGGERED CONTROL Gang FENG Department of Mechanical and Biomedical Engineering City University of Hong Kong July 25, 2014 Department of Mechanical and Biomedical Engineering, City University of Hong Kong 1 / 35

2 OUTLINES Introduction Background Problem Statement Research Methodologies Main Challenges Internal Reference Model Event-Triggered Control for Homogeneous Systems Main Results A Sufficient and Necessary Condition Event-Triggered Control Design Feasibility Self-Triggered Control Design An Example System Model Simulations Conclusions Department of Mechanical and Biomedical Engineering, City University of Hong Kong 2 / 35

3 OUTLINES Introduction Background Problem Statement Research Methodologies Main Challenges Internal Reference Model Event-Triggered Control for Homogeneous Systems Main Results A Sufficient and Necessary Condition Event-Triggered Control Design Feasibility Self-Triggered Control Design An Example System Model Simulations Conclusions Department of Mechanical and Biomedical Engineering, City University of Hong Kong 2 / 35

4 OUTLINES Introduction Background Problem Statement Research Methodologies Main Challenges Internal Reference Model Event-Triggered Control for Homogeneous Systems Main Results A Sufficient and Necessary Condition Event-Triggered Control Design Feasibility Self-Triggered Control Design An Example System Model Simulations Conclusions Department of Mechanical and Biomedical Engineering, City University of Hong Kong 2 / 35

5 OUTLINES Introduction Background Problem Statement Research Methodologies Main Challenges Internal Reference Model Event-Triggered Control for Homogeneous Systems Main Results A Sufficient and Necessary Condition Event-Triggered Control Design Feasibility Self-Triggered Control Design An Example System Model Simulations Conclusions Department of Mechanical and Biomedical Engineering, City University of Hong Kong 2 / 35

6 OUTLINES Introduction Background Problem Statement Research Methodologies Main Challenges Internal Reference Model Event-Triggered Control for Homogeneous Systems Main Results A Sufficient and Necessary Condition Event-Triggered Control Design Feasibility Self-Triggered Control Design An Example System Model Simulations Conclusions Department of Mechanical and Biomedical Engineering, City University of Hong Kong 2 / 35

7 Introduction CONTENTS 1 Introduction 2 Research Methodologies Background Problem Statement 3 Main Results 4 An Example 5 Conclusions Department of Mechanical and Biomedical Engineering, City University of Hong Kong 3 / 35

8 Introduction Background Multi-Agent System (MAS) Modeling/describing some collective behaviors of some animals. Wide applications in engineering problems. Some engineering applications of the MAS. ( One fundamental problem: consensus problem. Department of Mechanical and Biomedical Engineering, City University of Hong Kong 4 / 35

9 Introduction Background Two Kinds of MASs Early researches focused on MAS with identical dynamics. called homogeneous MAS. ẋ i = Ax i + Bu i, i = 1,, N, (1.1) In many applications, the agents.dynamics are non-identical. called heterogeneous MAS. ẋ i = A i x i + B i u i y i = C i x i, i = 1,, N, (1.2) All agents communicate with each other through a communication graph G. Department of Mechanical and Biomedical Engineering, City University of Hong Kong 5 / 35

10 Introduction Background Consensus Problem Homogeneous MAS: state consensus problem. Definition: Design u i such that lim x i(t) x j (t) = 0, i, j = 1,, N, (1.3) t holds for any finite x i (0), i = 1,, N. Heterogeneous MAS: output consensus problem. Definition: Design u i such that lim y i(t) y j (t) = 0, i, j = 1,, N, (1.4) t holds for any finite x i (0), i = 1,, N. Remark 1: Output consensus problem includes state consensus problem as a special case. Department of Mechanical and Biomedical Engineering, City University of Hong Kong 6 / 35

11 Introduction Background Why Event-Triggered Strategy? Individual agents equipped with microprocessors and some actuation modules. - On-board energy and resources are limited. - Energy-saving control schemes are needed. To reduce the communication load. - Proposed in stabilization problem for a single system [Tabuada (2007)]. - Better performance than traditional periodical sampling [Astrom & Bernhardsson (1999, 2002)]. - Mainly applied to some MASs with simple agent dynamics. Department of Mechanical and Biomedical Engineering, City University of Hong Kong 7 / 35

12 Introduction Problem Statement Problem Statement Design an event-triggered control scheme, such that the output consensus problem of heterogeneous MAS (1.2) can be solved. - Control input u i can only access information from itself and its neighboring agents. - Event-triggered strategy should be integrated in the control scheme. Department of Mechanical and Biomedical Engineering, City University of Hong Kong 8 / 35

13 Methodologies CONTENTS 1 Introduction 2 Research Methodologies 3 Main Results Main Challenges Internal Reference Model Event-Triggered Control for Homogeneous Systems 4 An Example 5 Conclusions Department of Mechanical and Biomedical Engineering, City University of Hong Kong 9 / 35

14 Methodologies Main Challenges Main Challenges Heterogeneity Problem. - Different dynamics: A i A j, B i B j for i j. - Different state dimensions. Use of event-triggered strategy. - Convergence analysis. - Development of event-triggering condition based on local information. - Feasibility analysis. Department of Mechanical and Biomedical Engineering, City University of Hong Kong 10 / 35

15 Methodologies Internal Reference Model Internal Reference Model Introduction of internal reference models. Consensus is achieved The internal reference models of the heterogeneous MAS. Department of Mechanical and Biomedical Engineering, City University of Hong Kong 11 / 35

16 Methodologies Internal Reference Model Key Idea Designed to be of identical dynamics. η i = Sη i + ũ i. (2.1) Regarded as a homogeneous MAS. - Exchanging information: states of the internal reference models. - To reduce the communication load: event-triggered strategy. Objective: (η i η 0 ) 0 as t, i = 1,, N. η 0 can be regarded as a reference signal for each agent. Department of Mechanical and Biomedical Engineering, City University of Hong Kong 12 / 35

17 Methodologies Event-Triggered Control for Homogeneous Systems Event-Triggered Control for Homogeneous MAS For a homogeneous MAS - Define the combined measurement as - Define the measurement error as Control law for each agent: ẋ i = Ax i + Bũ i, i = 1,, N, (2.2) q i(t) = N j=1 aij (xj(t) xi(t)). (2.3) ẽ i(t) = q i(t i k) q i(t). (2.4) ũ i (t) = K q i (t i k), t [ t i k, t i k+1), (2.5) with {t0 i, ti 1, } will be determined by the triggering condition. Triggering condition h(ẽ i (t), q i (t)) = 0. (2.6) Department of Mechanical and Biomedical Engineering, City University of Hong Kong 13 / 35

18 Methodologies Event-Triggered Control for Homogeneous Systems A Previous Result Lemma 1 [Hu et al. (2014)]: Under the assumptions that (A, B) is stabilizable and the undirected communication graph G is connected, there always exists at least one solution P > 0 for the following inequality PA + A T P αpbb T P + βi n 0, (2.7) where 0 < α 2λ 2, β 2λ N, with λ 2 and λ N the Fiedler eigenvalue and the largest eigenvalue of the Laplacian matrix of G, respectively. Then, letting K = B T P, the state consensus of homogeneous multi-agent system (2.2) can be achieved by the control law (2.5) and the following triggering condition h(ẽ i (t), q i (t)) = ẽ i (t) γ i q i (t) = 0. (2.8) where γ i = σi a(2 aρ) ρ with σ i (0, 1), ρ = PBB T P, and a being a positive number satisfying a < 2 ρ. Department of Mechanical and Biomedical Engineering, City University of Hong Kong 14 / 35

19 Main Results CONTENTS 1 Introduction 2 Research Methodologies 3 Main Results A Sufficient and Necessary Condition Event-Triggered Control Design Feasibility Self-Triggered Control Design 4 An Example 5 Conclusions Department of Mechanical and Biomedical Engineering, City University of Hong Kong 15 / 35

20 Main Results A Sufficient and Necessary Condition Event-Triggered Control Scheme Output feedback controller. η i = Sη i + K N j=1 ξ i = A i ξ i + B i u i + H i (C i ξ i y i ) ( a ij ηj (tk) i η i (tk) i ), t [ tk, i tk+1 i ) u i = K 1i ξ i + K 2i η i, i = 1,, N. (3.1) Design parameters: S, K, H i, K 1i, K 2i ; Triggering time sequence: {t i 0, ti 1, }. Event-triggering condition. h(e i (t), q i (t)) = e i (t) γ i q i (t) = 0, (3.2) where q i (t) = j N i (η j (t) η i (t)), e i (t) = q i (t i k ) q i(t) and γ i can be calculated by utilizing Lemma 1. Questions: Under what condition, the problem can be solved by (3.1) and (3.2)? How to design those parameters? Department of Mechanical and Biomedical Engineering, City University of Hong Kong 16 / 35

21 Main Results A Sufficient and Necessary Condition Assumptions & Result Assumptions: - Each pair of (A i, B i) is stabilizable. - Each pair of (A i, C i) is detectable. - The undirected communication graph G is connected. Theorem 1: Consider the heterogeneous linear multi-agent system (1.2) under Assumptions 1-3. The output consensus problem can be solved by the proposed controller (3.1) with the triggering condition (3.2) if and only if there exists (S, R), such that the following equations have solutions (Π i, Γ i ) for i = 1,, N, where S, R, Π i and Γ i all have compatible dimensions, A i Π i + B i Γ i = Π i S. (3.3) C i Π i = R. (3.4) Department of Mechanical and Biomedical Engineering, City University of Hong Kong 17 / 35

22 Main Results A Sufficient and Necessary Condition Notes Remark 2: Internal reference model is used to generate a virtual reference signal η 0 for each agent. The dynamics and output of the virtual reference signal η 0 (t) = Sη 0 (t) y(t) = Rη 0 (t) with (y i (t) y(t)) 0 as t, i = 1,, N. Department of Mechanical and Biomedical Engineering, City University of Hong Kong 18 / 35

23 Main Results Event-Triggered Control Design Parameters Design Design procedure. - Step 1: Choose proper matrices S and R, such that (3.3) and (3.4) have solution pairs (Π i, Γ i), i = 1,, N. - Step 2: Choose proper Λ i, such that A i + B iλ i is Hurwitz. Let K 1i, K 2i be as follows, K 1i = Λ i, K 2i = Γ i Λ iπ i. (3.5) - Step 3: Choose a proper H i, such that A i + H ic i is Hurwitz. - Step 4: Let K = P, where P > 0 satisfies the following inequality PS + S T P αpp + βi m 0. (3.6) - Step 5: Let t i 0, t i 1,, i = 1,, N, to be determined by the proposed triggering condition (3.2). Department of Mechanical and Biomedical Engineering, City University of Hong Kong 19 / 35

24 Main Results Feasibility Exclusion of Singular Triggering Singular Triggering. - Definition: - Method: To prove that if t i k exists and q i ( t i k ) 0, the next triggering time t i k+1 exists and q i ( t i k+1 ) 0. - Requirement: The proof holds for any agent. Corollary 1: Consider the heterogeneous multi-agent system (1.2) and the control scheme (3.1). No agent will exhibit singular triggering behavior. Department of Mechanical and Biomedical Engineering, City University of Hong Kong 20 / 35

25 Main Results Feasibility Exclusion of Zeno behavior Zeno behavior. - Definition: - Method: To prove that the length of inter-event interval is strictly positive. - Requirement: The proof holds for any agent. Corollary 2: Consider the heterogeneous multi-agent system (1.2) and the control scheme (3.1). No agent will exhibit Zeno behavior. - Calculate the inter-event interval for ith agent tk+1 i tk i > 1 ( S s i S ln k where s i k and α i k are two positive constants. α i k ) (3.7) Department of Mechanical and Biomedical Engineering, City University of Hong Kong 21 / 35

26 Main Results Self-Triggered Control Design Drawback of Event-Triggered Strategy In the proposed event-triggering condition h(e i (t), q i (t)) = e i (t) γ i q i (t) = 0, continuous monitoring of e i (t) and q i (t) are required. How to avoid this constraint? Department of Mechanical and Biomedical Engineering, City University of Hong Kong 22 / 35

27 Main Results Self-Triggered Control Design Self-Triggered Strategy Main idea: Estimate the next triggering time based on the measurement at previous triggering time. We propose the following condition e i (t) γ i 2 + 2γ 2 i qi ( t i k ) = s i k which implies h (e i (t), q i (t)) 0. Calculate the time it will be needed for e i (t) to increase to s i k. Department of Mechanical and Biomedical Engineering, City University of Hong Kong 23 / 35

28 Main Results Self-Triggered Control Design Self-Triggering Condition Calculate the increasing rate of e i (t). d dt e i (t) S s i k + w i (t), ( ) where w i (t) = (d i P S) q i t i k P j N i q j (t j, ) k (t) with t j k (t) latest triggering time for agent j, j N i. Self-Triggering Rule If no neighboring agent is triggered ahead of agent i, then t i k+1 = being the s i k S s i k +wi(ti k ). Otherwise, if one neighboring agent j is triggered first at time t, then update w i (t) as w i (t ) and calculate the left time which will be needed for e i (t) to increase to s i k. The feasibility of self-triggering rule: omitted here. Department of Mechanical and Biomedical Engineering, City University of Hong Kong 24 / 35

29 Main Results Self-Triggered Control Design Result Theorem 2: Consider the heterogeneous linear multi-agent system (1.2) under Assumptions 1-3. The output consensus problem can be solved by the proposed controller (3.1) and the self-triggering rule if and only if there exists (S, R), such that (3.3) and (3.4) always have solutions (Π i, Γ i ) for i = 1,, N, where S, R, Π i and Γ i all have compatible dimensions. Department of Mechanical and Biomedical Engineering, City University of Hong Kong 25 / 35

30 An Example AN EXAMPLE 1 Introduction 2 Research Methodologies System Model Simulations 3 Main Results 4 An Example 5 Conclusions Department of Mechanical and Biomedical Engineering, City University of Hong Kong 26 / 35

31 An Example System Model System Model Consider the following heterogeneous MAS [Wieland et al. (2011)]. ẋ i = c i x i u i 0 d i a i b i ( ) y i = x i, i = 1, 2, 3, 4, where parameters {a i, b i, c i, d i } are set as {1, 1, 1, 0}, {10, 2, 1, 0}, {2, 1, 1, 10} and {2, 1, 1, 1}, respectively. The communication graph The communication graph G. Department of Mechanical and Biomedical Engineering, City University of Hong Kong 27 / 35

32 An Example System Model Control Design Event-triggered control scheme is designed according to Steps In Step 1: ( ) ( ) S =, R = In Step 2: Λ i = ( ), K 1i = Λ i = ( 1 1 ( 1 ), ) K 2i = Γ i Λ iπ i = d i b i In Step 3: We set H i = 10 10, i = 1,, ( ) In Step 4: K = P = In Step 5: Threshold in the triggering condition can be calculated γ i = Triggering time sequence can be calculated by event-triggering condition or self-triggering rule. Department of Mechanical and Biomedical Engineering, City University of Hong Kong 28 / 35

33 An Example Simulations Simulations Simulation results. Output response of all agents via event-triggered control scheme. Note: y i = col(y i1, y i2 ) and e y ij = y i1 y j1. Department of Mechanical and Biomedical Engineering, City University of Hong Kong 29 / 35

34 An Example Simulations Simulations Continued Simulation results. Output response of all agents via self-triggered control scheme. Note: y i = col(y i1, y i2 ) and e y ij = y i1 y j1. Department of Mechanical and Biomedical Engineering, City University of Hong Kong 30 / 35

35 An Example Simulations Simulations Continued Performances comparison between two proposed control schemes Control scheme T s (sec) Triggering numbers for agents Event-triggered Self-triggered where T s is defined as a minimum time, such that, y i (t) y j (t) when t T s, for any agents i, j. Less settling time while more triggering numbers are needed to reach output consensus by the self-triggered control scheme. Department of Mechanical and Biomedical Engineering, City University of Hong Kong 31 / 35

36 Conclusions CONTENTS 1 Introduction 2 Research Methodologies Conclusions References 3 Main Results 4 An Example 5 Conclusions Department of Mechanical and Biomedical Engineering, City University of Hong Kong 32 / 35

37 Conclusions Conclusions Conclusions Output consensus problem of heterogeneous linear MASs has been studied. A novel event-triggered control scheme has been proposed. Feasibility of the proposed control scheme has been discussed. A novel self-triggered control scheme has been proposed. Department of Mechanical and Biomedical Engineering, City University of Hong Kong 33 / 35

38 Conclusions References References [1] P. Tabuada, Event-triggered real-time scheduling of stabilizing control tasks, IEEE Transactions on Automatic Control, 52(9)(2007)1680õ1685. [2] K. J. Åström, B. M. Bernhardsson, Comparison of periodic and event based sampling for first-order stochastic systems, In Proceedings of the 14th IFAC World congress., 11(1999) [3] K. J. Åström, B. M. Bernhardsson, Comparison of riemann and lebesque sampling for first order stochastic systems, In Proceedings of the 41st IEEE Conference on Decision and Control, 2(2002) [4] R. Olfati-Saber, J. A. Fax, R. M. Murray, Consensus and cooperation in networked multiagent systems, Proceedings of the IEEE, 95(1)(2007)215õ233. [5] W. Ren, On consensus algorithms for double-integrator dynamics, IEEE Transactions on Automatic Control, 53(2008) [6] P. Wieland, R. Sepulchre, F. Allgöwer, An internal model principle is necessary and sufficient for linear output synchronization, Automatica, 47(2011) [7] Y. Fan, G. Feng, Y. Wang, C. Song, Distributed event-triggered control of multi-agent systems with combinational measurements, Automatica, 49(2013) [8] W. Hu, L. Liu, G. Feng, Consensus of linear multi-agent systems by distributed eventtriggered strategy, In Proceedings of the 19th World Congress of the International Federation of Automatic Control, to appear. Department of Mechanical and Biomedical Engineering, City University of Hong Kong 34 / 35

39 Thank You THANK YOU Thank You! Department of Mechanical and Biomedical Engineering, City University of Hong Kong 35 / 35