On the Scalability in Cooperative Control. Zhongkui Li. Peking University

Transcription

1 On the Scalability in Cooperative Control Zhongkui Li Peking University June 25, 2016 Zhongkui Li (PKU) Scalability June 25, / 28

2 Background Cooperative control is to have multiple autonomous agents work together to achieve collective group behaviors; e.g., consensus, formation, coverage control. Zhongkui Li (PKU) Scalability June 25, / 28

3 Distributed Control Agent Dynamics Inter-agent Interactions Cooperative Control Laws Agents: robots, UAVs, ships,... Zhongkui Li (PKU) Scalability June 25, / 28

4 Distributed Control Agent Dynamics Inter-agent Interactions Cooperative Control Laws Agents: robots, UAVs, ships,... Interactions: via relative sensing or communications; The information flow among agents can be characterized a graph Zhongkui Li (PKU) Scalability June 25, / 28

5 Distributed Control Agent Dynamics Inter-agent Interactions Cooperative Control Laws Agents: robots, UAVs, ships,... Interactions: via relative sensing or communications; The information flow among agents can be characterized a graph. Cooperative control laws: Enable the collective behaviors; are expected to be distributed, i.e., depending on only local information. Zhongkui Li (PKU) Scalability June 25, / 28

6 Outline Compared to a single system, cooperative MAS has many advantages, e.g., low operational costs, additional degree of redundancy, high robustness, strong adaptivity, flexible scalability. Zhongkui Li (PKU) Scalability June 25, / 28

7 Outline Compared to a single system, cooperative MAS has many advantages, e.g., low operational costs, additional degree of redundancy, high robustness, strong adaptivity, flexible scalability. In this lecture, we intend to discuss the scalability issue in the context of consensus control for general linear MASs. Zhongkui Li (PKU) Scalability June 25, / 28

8 Scalability Scalability: Cooperative control laws are independent of the scale of the network, i.e., they need not be redesigned even if new agents are added or existing agents are removed. Zhongkui Li (PKU) Scalability June 25, / 28

9 Scalability Scalability: Cooperative control laws are independent of the scale of the network, i.e., they need not be redesigned even if new agents are added or existing agents are removed. Scalability is vital for a multi-agent system being autonomous; relies on the design of fully distributed protocols using only local information. Zhongkui Li (PKU) Scalability June 25, / 28

10 Problem Statement Consider a network of N continuous-time linear agents, described by ẋ i = Ax i + Bu i, i =1,,N. (1) The communication topology is represented by a directed graph G Zhongkui Li (PKU) Scalability June 25, / 28

11 Problem Statement Consider a network of N continuous-time linear agents, described by ẋ i = Ax i + Bu i, i =1,,N. (1) The communication topology is represented by a directed graph G Only the local state information of neighboring agents is available. The objective is to design a consensus protocol depending on the local information such that lim t!1 kx i (t) x j (t)k =0. Zhongkui Li (PKU) Scalability June 25, / 28

12 Relative-state Consensus Protocol Based on relative states of neighboring agents, a static consensus protocol is given by u i = ck a ij (x i x j ), i =1,,N, (2) j=1 where c>0 is the coupling gain (weight), K is the feedback gain matrix, and [a ij ] denotes the adjacency matrix of G. Theorem [Li, Duan, Chen, & Huang, IEEE TCASI, 2010] Assuming that G contains a directed spanning tree, the static protocol (2) solves the consensus problem i A + c i BK, i =2,,N,areall Hurwitz, where i, i =2,,N, are the nonzero eigenvalues of the Laplacian matrix L. Zhongkui Li (PKU) Scalability June 25, / 28

13 Consensus Protocol Design Algorithm [Li, Duan, Chen, & Huang, IEEE TCASI, 2010] Assume that (A, B) is stabilizable. Then, the consensus protocol (2) can be constructed as: 1) Solve the LMI: AP + PA T 2BB T < 0 to get a P>0. Then, choose K = B T P 1 ; 1 2) Select the coupling gain c c th = min Re( i). i=2,,n Zhongkui Li (PKU) Scalability June 25, / 28

14 Consensus Protocol Design Algorithm [Li, Duan, Chen, & Huang, IEEE TCASI, 2010] Assume that (A, B) is stabilizable. Then, the consensus protocol (2) can be constructed as: 1) Solve the LMI: AP + PA T 2BB T < 0 to get a P>0. Then, choose K = B T P 1 ; 1 2) Select the coupling gain c c th = min Re( i). i=2,,n X A necessary and su cient condition for the feasibility of the LMI in step 1) is that (A, B) is stabilizable. X Algorithm 1 has a favorable decoupling feature: Step 1) is to design K, using only agent dynamics and a scalar c is used to deal with the communication graph. Zhongkui Li (PKU) Scalability June 25, / 28

15 Further Observations Algorithm [Li, Duan, Chen, & Huang, IEEE TCASI, 2010] Assume that (A, B) is stabilizable. Then, the consensus protocol (2) can be constructed as: 1) Solve the LMI: AP + PA T 2BB T < 0 to get a P>0. Then, choose K = B T P 1 ; 1 2) Select the coupling gain c c th = min Re( i). i=2,,n X The smallest real part of the nonzero eigenvalues is a global information, implying the consensus protocol cannot be designed and implemented in a fully distributed fashion. Zhongkui Li (PKU) Scalability June 25, / 28

16 Distributed Edge-Based Adaptive Protocols First consider the simple case with undirected graphs. Left: u i = ck P N j=1 a ij(x i x j ). Zhongkui Li (PKU) Scalability June 25, / 28

17 Distributed Edge-Based Adaptive Protocols First consider the simple case with undirected graphs. Left: u i = ck P N j=1 a ij(x i x j ). adaptively tune the coupling weights (di erently for the agents) Zhongkui Li (PKU) Scalability June 25, / 28

18 Distributed Edge-Based Adaptive Protocols First consider the simple case with undirected graphs. Left: u i = ck P N j=1 a ij(x i x j ). Right: Edge-based adaptive protocol. adaptively tune the coupling weights (di erently for the agents) Zhongkui Li (PKU) Scalability June 25, / 28

19 Distributed Edge-Based Adaptive Protocols The edge-based adaptive consensus protocol: u i = K c ij a ij (x i x j ), j=1 ċ ij = apple ij a ij (x i x j ) T (x i x j ), i =1,,N, where c ij (t) denotes the time-varying coupling weight for the edge (i, j) with c ij (0) = c ji (0), apple ij = apple ji > 0, K and are the feedback gain matrices. Zhongkui Li (PKU) Scalability June 25, / 28 (3)

20 Protocol Design Theorem [Li, Ren, Liu, & Fu, IEEE TAC, 2013] Assume that the communication graph G is undirected and connected. Then, the N agents in (1) reach consensus under the edge-based adaptive protocol (3) with K = B T P 1 and =P 1 BB T P 1,whereP>0is a solution to AP + PA T 2BB T < 0. Moreover, the coupling weights c ij converge to some finite steady-state values. Zhongkui Li (PKU) Scalability June 25, / 28

21 Protocol Design Theorem [Li, Ren, Liu, & Fu, IEEE TAC, 2013] Assume that the communication graph G is undirected and connected. Then, the N agents in (1) reach consensus under the edge-based adaptive protocol (3) with K = B T P 1 and =P 1 BB T P 1,whereP>0is a solution to AP + PA T 2BB T < 0. Moreover, the coupling weights c ij converge to some finite steady-state values. X The design of the adaptive protocol (3) depends only on the agent dynamics, thereby can be computed and implemented by each agent in a fully distributed fashion. Zhongkui Li (PKU) Scalability June 25, / 28

22 Distributed Node-Based Adaptive Protocols Left: Edge-based adaptive protocol. Right: Node-based adaptive protocol. Zhongkui Li (PKU) Scalability June 25, / 28

23 Distributed Node-Based Adaptive Protocols The node-based adaptive consensus protocol: u i = d i K d i = i [ a ij (x i x j ), j=1 a ij (x i x j )] T [ a ij (x i j=1 j=1 where d i (t) denotes the coupling weight for agent i. x j )], i =1,,N, (4) Zhongkui Li (PKU) Scalability June 25, / 28

24 Protocol Design Theorem [Li, Ren, Liu, & Xie, Automatica, 2013] Assume that the communication graph G is undirected and connected. Then, the N agents in (1) reach consensus under the node-based adaptive protocol (4) with K and designed as in Theorem 2. Moreover, the coupling weights d i converge to some finite steady-state values. Zhongkui Li (PKU) Scalability June 25, / 28

25 Distributed Adaptive Protocols for Directed Graphs The distributed adaptive protocols proposed earlier are applicable to only undirected graphs. How to design distributed adaptive protocols for the case with directed graphs is quite challenging. Zhongkui Li (PKU) Scalability June 25, / 28

26 Distributed Adaptive Protocols for Directed Graphs The distributed adaptive protocols proposed earlier are applicable to only undirected graphs. How to design distributed adaptive protocols for the case with directed graphs is quite challenging. Extending the node-based adaptive consensus protocol for undirected graphs, we propose a novel adaptive protocol for directed graphs: u i = d i i K d i =[ a ij (x i x j ), j=0 a ij (x i x j )] T [ a ij (x i x j )], j=0 j=0 (5) where d i (t) denotes the coupling weight with d i (0) 1 and i ( ) are monotonically nondecreasing functions satisfying i (s) 1 for s>0. Zhongkui Li (PKU) Scalability June 25, / 28

27 Distributed Adaptive Protocol Design Theorem [Li, Wen, Duan, & Ren, IEEE TAC, 2015] Suppose that G contains a directed spanning tree with the leader as the root node. Then, the consensus problem can be solved by the adaptive protocol (7) with K = B T P 1, =P 1 BB T P 1,and i =(1+ i T P 1 i ) 3,where i, P N j=0 a ij(x i x j ) and P>0is a solution to AP + PA T 2BB T < 0. Zhongkui Li (PKU) Scalability June 25, / 28

28 Distributed Adaptive Protocol Design Theorem [Li, Wen, Duan, & Ren, IEEE TAC, 2015] Suppose that G contains a directed spanning tree with the leader as the root node. Then, the consensus problem can be solved by the adaptive protocol (7) with K = B T P 1, =P 1 BB T P 1,and i =(1+ i T P 1 i ) 3,where i, P N j=0 a ij(x i x j ) and P>0is a solution to AP + PA T 2BB T < 0. X As the consensus error converges to zero, the functions i will converge to 1 and the adaptive protocol for directed graphs will reduce to the node-based adaptive protocol for undirected graphs. Zhongkui Li (PKU) Scalability June 25, / 28

29 Another Distributed Adaptive for Directed Graphs The proposed adaptive protocol for directed graphs with multiplicative terms: u i = d i i K i, i =(1+ i T P 1 i ) 3, d i = i T i, i, a ij (x i x j ), j=0 (6) The design relies on complicated integral Lyapunov function. Zhongkui Li (PKU) Scalability June 25, / 28

30 Another Distributed Adaptive for Directed Graphs The proposed adaptive protocol for directed graphs with multiplicative terms: u i = d i i K i, i =(1+ i T P 1 i ) 3, d i = i T i, i, a ij (x i x j ), j=0 (6) The design relies on complicated integral Lyapunov function. Another alternative adaptive protocol with additive terms [Lv, Li, Duan, & Feng, IJC, 2016]: u i =(d i + i )K i, i = T i P 1 i, d i = T i i. (7) Relies on modified quadratic Lyapunov function, much simplier. Zhongkui Li (PKU) Scalability June 25, / 28

31 Robustness Issue and Robustness Redesign Consider the robustness of the following node-based adaptive protocol: u i = d i K a ij (x i x j ), d i = i [ j=1 a ij (x i x j )] T [ a ij (x i x j )], j=1 with respect to external disturbances: where k! i kapple i. j=1 ẋ i = Ax i + Bu i +! i, i =1,,N, (8) Zhongkui Li (PKU) Scalability June 25, / 28

32 Robustness Issue and Robustness Redesign Consider the robustness of the following node-based adaptive protocol: u i = d i K a ij (x i x j ), d i = i [ j=1 a ij (x i x j )] T [ a ij (x i x j )], j=1 with respect to external disturbances: where k! i kapple i. j=1 ẋ i = Ax i + Bu i +! i, i =1,,N, (8) Note that d i are integrals of the nonnegative quadratic functions of x i x j,easytoseethatd i will grow unbounded, which is called the parameter drift phenomenon. Zhongkui Li (PKU) Scalability June 25, / 28

33 Robust Adaptive Consensus Protocols The focus is to redesign the distributed adaptive protocols to guarantee the boundedness of the consensus error and adaptive gains in the presence of bounded disturbances. Zhongkui Li (PKU) Scalability June 25, / 28

34 Robust Adaptive Consensus Protocols The focus is to redesign the distributed adaptive protocols to guarantee the boundedness of the consensus error and adaptive gains in the presence of bounded disturbances. Motivated by the modification technique, we propose a robust adaptive protocol as follows u i = d i N X j=1 d i = i [ ' i d i +( a ij F (x i x j ), a ij (x i x j ) T ) ( a ij (x i x j ))], j=1 j=1 (9) where ' i, i =1,,N, are small positive constants. Zhongkui Li (PKU) Scalability June 25, / 28

35 Robust Adaptive Protocol Design Theorem [Li & Duan, IET CTA, 2014] Assume that the communication graph G is undirected. The modified adaptive protocol (9) are designed as K = B T Q 1 and =Q 1 BB T Q 1,whereQ>0 is a solution to AQ + QA T + µq 2BB T < 0, (10) where µ>1. Then, both the consensus error and the adaptive gains d i are uniformly ultimately bounded. If ' i satisfies #, max{ i ' i }appleµ 1, then exponentially converges to D 1, { : k k 2 apple max (Q) (µ #) 2 [ max(l) min(q) i=1 2 i + 2 ' i ]}. i=1 Zhongkui Li (PKU) Scalability June 25, / 28

36 Robust Adaptive Protocol Design Theorem [Li & Duan, IET CTA, 2014] Assume that the communication graph G is undirected. The modified adaptive protocol (9) are designed as K = B T Q 1 and =Q 1 BB T Q 1,whereQ>0 is a solution to AQ + QA T + µq 2BB T < 0, (10) where µ>1. Then, both the consensus error and the adaptive gains d i are uniformly ultimately bounded. If ' i satisfies #, max{ i ' i }appleµ 1, then exponentially converges to D 1, { : k k 2 apple max (Q) (µ #) 2 [ max(l) min(q) i=1 2 i + 2 ' i ]}. i=1 X A su cient condition for LMI (10): (A, B) is controllable. Zhongkui Li (PKU) Scalability June 25, / 28

37 Distributed Tracking with a Leader of Nonzero Input Consider a group of N +1agents with general linear dynamics: ẋ i = Ax i + Bu i, i =0,,N. (11) The agent indexed by 0 is the leader and the rest are followers. One common assumption is that the leader s input u 0 is either equal to zero or available to all the followers. Not practical in many cases. Zhongkui Li (PKU) Scalability June 25, / 28

38 Distributed Tracking with a Leader of Nonzero Input Consider a group of N +1agents with general linear dynamics: ẋ i = Ax i + Bu i, i =0,,N. (11) The agent indexed by 0 is the leader and the rest are followers. One common assumption is that the leader s input u 0 is either equal to zero or available to all the followers. Not practical in many cases. The objective is to solve the distributed tracking for the case where u 0 is bounded (i.e., ku 0 k 2 apple ) and not accessible to any follower. Assumption 1: the communication graph G satisfies that the subgraph associated with the N followers is undirected and the leader has directed paths to all followers. Zhongkui Li (PKU) Scalability June 25, / 28

39 Static Tracking Protocol Based on the relative states of the neighboring agents, we propose the following distributed protocol: u i = c 1 K a ij (x i j=0 x j )+c 2 g(k a ij (x i x j )), (12) j=0 where c 1,c 2 > 0 are( the coupling weights, K is the feedback gain y kyk if kyk 6=0, matrix, and g(y) = 0 if kyk =0. Theorem [Li, Liu, Ren, & Xie, IEEE TAC, 2013] The distributed tracking problem is solved under the protocol (12) with c 1 1/ 2,c 2, K = B T P 1, where 2 is the smallest nonzero eigenvalue of L and P>0 is a solution to AP + PA T 2BB T < 0. Zhongkui Li (PKU) Scalability June 25, / 28

40 Scalability Global information: 2 of Laplacian matrix and the bound of the leader s control input u 0. Zhongkui Li (PKU) Scalability June 25, / 28

41 Scalability Global information: 2 of Laplacian matrix and the bound of the leader s control input u 0. To remove this limitation, we propose the following distributed adaptive protocol: u i = d i K d i = i [ a ij (x i j=1 x j )+d i g(k a ij (x i x j )), j=1 a ij (x i x j )] T [ a ij (x i x j )] j=1 + i kk j=1 a ij (x i x j )k 2. j=1 (13) Zhongkui Li (PKU) Scalability June 25, / 28

42 Scalability Theorem [Li, Liu, Ren, & Xie, IEEE TAC, 2013] Suppose that Assumptions 1 and 2 hold. The distributed tracking problem is solved by the above adaptive controller (13) with K = B T P 1 and =P 1 BB T P 1. Moreover, each coupling weight d i converges to some finite steady-state value. Zhongkui Li (PKU) Scalability June 25, / 28

43 Scalability Theorem [Li, Liu, Ren, & Xie, IEEE TAC, 2013] Suppose that Assumptions 1 and 2 hold. The distributed tracking problem is solved by the above adaptive controller (13) with K = B T P 1 and =P 1 BB T P 1. Moreover, each coupling weight d i converges to some finite steady-state value. X The discontinuity g( ) in the static and adaptive protocols (12) and (13) can result in chattering. Use the boundary layer technique to give a continuous approximation. Zhongkui Li (PKU) Scalability June 25, / 28

44 Scalability Theorem [Li, Liu, Ren, & Xie, IEEE TAC, 2013] Suppose that Assumptions 1 and 2 hold. The distributed tracking problem is solved by the above adaptive controller (13) with K = B T P 1 and =P 1 BB T P 1. Moreover, each coupling weight d i converges to some finite steady-state value. X The discontinuity g( ) in the static and adaptive protocols (12) and (13) can result in chattering. Use the boundary layer technique to give a continuous approximation. X Extensions to directed graphs are more challenging; see [Lv, Li, Duan, & Chen, Automatica, submitted]. Zhongkui Li (PKU) Scalability June 25, / 28

45 Conclusions In this lecture, we have presented our recent thoughts on the scalability issue in cooperative control. The scalability is ensured by designing fully distributed adaptive consensus protocols, which does not require any global information. Zhongkui Li (PKU) Scalability June 25, / 28

46 Conclusions In this lecture, we have presented our recent thoughts on the scalability issue in cooperative control. The scalability is ensured by designing fully distributed adaptive consensus protocols, which does not require any global information. Extensions to the case with only relative output information have been established; see [Li, Ren, Liu, & Xie, Automatica, 2013]. Scalability of other cooperative control problems can be similarly studied. Robustness issues with respect to matching uncertainties in the form of ẋ i = Ax i + B[u i + f i (x i,t)]; see [Li, Duan, & Lewis, Automatica, 2014]. Zhongkui Li (PKU) Scalability June 25, / 28

47 Nonlinear Control of Dynamic Networks More References Our recent book: Zhongkui Li, Zhisheng Duan, Cooperative Control of Multi-agent Systems: A Consensus Region Approach, CRC Press, Cooperative Control of Multi-Agent Systems A CONSENSUS REGION APPROACH Zhongkui Li Zhisheng Duan One recent survey: Zhongkui Li, Zhisheng Duan, Distributed consensus protocol design of general linear multi-agent systems: a consensus region approach, IET Control Theory and Applications, 8(18): , Zhongkui Li (PKU) Scalability June 25, / 28

48 Thanks! Zhongkui Li (PKU) Scalability June 25, / 28