Pinning Control and Controllability of Complex Networks

Transcription

1 Pinning Control and Controllability of Complex Networks Guanrong (Ron) Chen City University of Hong Kong, China Joint work with Xiaofan Wang, Lin Wang Shanghai Jiao Tong University, China Xiang Li, Baoyu Hou Fudan University, China 35th Chinese Control Conference, Chengdu, July 27-29, 2016

2 Dedicated to the Memory of Rudolf E Kalman ( )

3 Motivational Examples

4 Example: C. elegans In its Neural Network: Neurons: 300~500 Synapses: 2500~7000

5 Excerpt The worm Caenorhabditis elegans has 297 nerve cells. The neurons switch one another on or off, and, making 2345 connections among themselves. They form a network that stretches through the nematode s millimeter-long body. How many neurons would you have to commandeer to control the network with complete precision? The answer is, on avergae: Adrian Cho, Science, 13 May 2011, vol. 332, p 777 Here, control = stimuli

6 Another Example very few individuals (approximately 5%) within honeybee swarms can guide the group to a new nest site. I.D. Couzin et al., Nature, 3 Feb 2005, vol. 433, p 513 These 5% of bees can be considered as controlling or controlled agents Leader-Followers network

7 Now mathematically o Given a network of identical dynamical systems (e.g., ODEs) o Given a specific control objective (e.g., synchronization) o Assume: a certain class of controllers (e.g., local linear statefeedback controllers) have been chosen to use

8 Questions: Objective: To achieve the control goal with good performance How many controllers to use? Where to put them? (which nodes to pin ) --- Pinning Control

9 Network Model Linearly coupled network: dx dt i f N ( xi ) c aijhx n j x R j1 i i 1,2,..., N - a general assumption is that f (.) is Lipschitz - coupling strength c > 0 and H - input matrix - coupling matrices (undirected): A [ ] a ij NN A: If node i connects to node j (j i), then aij = aji = 1; else, aij = aji = 0; and a where - degree of node i ii d i d i Note: For undirected networks, A is symmetrical; for directed networks, it is not so

10 What kind of controllers? How many? Where? dx dt i f N ( xi ) c aijhxj + i 1,2,..., N j1 u i ( ui xi ) dx dt i N f ( x ) c a Hx x i j1 ij j i i i 1,2,..., N i 1 0 if if to control not control Q: How many? Which i? 1 i

11 Pinning Control: Our Research Progress Wang X F, Chen G, Pinning control of scale-free dynamical networks, Physica A, 310: , Li X, Wang X F, Chen G, Pinning a complex dynamical network to its equilibrium, IEEE Trans. Circ. Syst. I, 51: , Sorrentino F, di Bernardo M, Garofalo F, Chen G, Controllability of complex networks via pinning, Phys. Rev. E, 75: , Yu W W, Chen G, Lu J H, Kurths J, Synchronization via pinning control on general complex networks, SIAM J. Contr. Optim., 51: , Chen G, Pinning control and synchronization on complex dynamical networks, Int. J. Contr., Auto. Syst., 12: , Xiang L, Chen F, Chen G. Pinning synchronization of networked multi-agent systems: Spectral analysis. Control Theory Tech., 13: 45-54, 2015.

12 Controllability Theory

13 In retrospect,

14 State Controllability Linear Time-Invariant (LTI) system xt ( ) Ax( t) Bu( t) x 1 (t) x(t 0 ) x(t) x R u R n A R B R p : state vector : control input nn n p : state matrix : control input matrix C. K. Chui and G. Chen, Linear Systems and Optimal Control, Springer, 1989 x 3 (t) x 2 (t) [Concept] State Controllable: The system orbit can be driven by an input from any initial state to the origin in finite time

15 State Controllability Theorems (i) Kalman Rank Criterion The controllability matrix Q has full row rank: Q BAB A B n 1 [ ] (ii) Popov-Belevitch-Hautus (PBH) Test The following relationship holds: xt ( ) Ax( t) Bu( t) v T A v T T, v B 0 : v : eigenvalue of A nonzero left eigenvactor with

16 What about networks? -- Some earlier attempts o Leader-follower multi-agent systems H.G. Tanner, CDC, 2004 o Pinning state-controllability of complex networks F. Sorrentino, M. di Bernardo, F. Garofalo, G. Chen, Phys. Rev. E, 2007 o Structural controllability of complex networks Y.Y. Liu, J.J. Slotine, A.L. Barabási, Nature, 2011

17 Structural Controllability A network of single-input/single-output (SISO) node systems, where the node systems can be of higher-dimensional

18 Structural Controllability In the controllability matrix Q: Q BAB A B n 1 [ ] All 0 are fixed There is a realization of independent nonzero parameters such that Q has full row-rank Example 1: a 0 Q 0 d Realization: All admissible parameters a 0, d 0 Example 2: Frobinius Canonical Form Q =

19 Examples: Structure matters 2 C = [B, A B, A B] b 1 0 a21 0 b 1 0 a21 0 b 1 0 a21 0 b 1 0 a21 a23a a a 0 a 0 0 a a a 0 a a a 32 21, 31, , rank C3 n ran k C 2 n 3 rank C3 n rank C? controllable uncontrollable controllable controllable? Partially controllable Structurally controllable Y.Y. Liu, J.J. Slotine, and A.L. Barabási, Nature, 2011

20 In retrospect: large-scale systems theory Structural Controllability (and Structural Observability) 1. C.T. Lin, Structural Controllability, IEEE Trans. Auto Contr., 19(3): , R.W. Shields, J.B. Pearson, Structural Controllability of Multiinput Linear Systems, IEEE Trans. Auto. Contr., 21(2): , K. Glover, L.M. Silverman, Characterization of Structural Controllability, IEEE Trans. Auto. Contr., 21(4): , C.T. Lin, System Structure and Minimal Structure Controllability, IEEE Trans. Auto. Contr., 22(5): , S. Hosoe, K. Matsumoto, On the Irreducibility Condition in the Structural Controllability Theorem, IEEE Trans. Auto. Contr., 24(6): , H. Mayeda, On Structural Controllability Theorem, IEEE Trans. Auto. Contr., 26(3): , A. Linnemann, A Further Simplification in the Proof of the Structural Controllability Theorem, IEEE Trans. Auto. Contr., 31(7): , J. Willems, Structural Controllability and Observability, Syst. Contr. Lett., 8(1): 5-12, 1986

21 Building Blocks Cactus is the minimum structure which contains no inaccessible nodes and no dilations

22 Structural Controllability Theorem The following two criteria are equivalent: 1. Algebraic: The LTI control system (A,B) is structurally controllable 2. Geometric: The digraph G (A,B) is spanned by a cactus C.T. Lin, IEEE Trans. Auto. Contr., 1977

23 Matching in Directed Networks Matching: a set of directed edges without common heads and tails Unmatched node: the tail node of a matching edge Perfect matching Maximum matching: Cannot be extended Perfect matching: All nodes are matched nodes Maximum but not perfect matching

24 Minimum Inputs Theorem Q: How many? A: The minimum number of inputs N D needed is: Case 1: If there is a perfect matching, then N D = 1 Case 2: If there is no perfect matching, then N D = number of unmatched nodes Q: Where to put them? A: Case 1: Anywhere Case 2: At unmatched nodes b3 b3 u3 Y. Y. Liu, J. J. Slotine, and A. L. Barabási, Nature, 2011

25 State Controllability A network of multi-input/multi-output (MIMO) node systems, where the node systems are of higher-dimensional

26 Some Earlier Progress (Note: All nodes are subject to control input) T. Zhou, Automatica (2015)

27 continued T. Zhou, Automatica (2015)

28 A Network of Multi-Input/Multi-Output LTI Systems Node system x yi Cx i Axi Bui i x n i R y m i R u p i R Networked system Networked system with external control N i i ij j i i j1 x Ax Hy, y Cx, i 1,2,, N N i i ij j i i j1 x Ax HCx Bu, i 1,2,, N Some notations Node system (A,B,C) Coupling matrix H 1: with external control 0: without external control i i Network structure L[ ] R N N External control inputs diag( 1,, N ) ij L. Wang, X.F. Wang, G.R. Chen and W.K.S. Tang, Automatica (2016)

29 Some counter-intuitive examples Network structure Node system Networked MIMO system H L 1 0 structurally controllable 1 0 A B 0 C 01 (A,B) is controllable state uncontrollable (A,C) is observable

30 Some counter-intuitive examples Network structure Node system Networked MIMO system H L 1 0 structurally controllable 1 0 A B 1 C 01 (A,B) is uncontrollable state controllable (A,C) is observable coupling matrix H is important Hou B Y, Li X, Chen G (2016)

31 A Network of Multi-Input/Multi-Output LTI Systems A necessary and sufficient condition x Ax HCx Bu k, y i i ij j ik j1 k1 N l mlj Dx j j1 L[ ] R N N [ ij ] R ij N s Ns x i u y k l n R, i 1, p R, k 1, q R, l 1, N s r State Controllable If and only if Matrix equations T T XB 0, L XHC X ( I A) has is Xa unique 0. solution X = 0 L. Wang, X.F. Wang, G.R. Chen and W.K.S. Tang, Automatica (2016)

32 General Topology with SISO Nodes N i i ij j i i j1 x Ax HCx Bu, i 1,2,, N x R n y R m u R p L[ ] R N N ij 1 i i diag(,, ) i N A network with SISO nodes is controllable if and only if (A,H) is controllable, (A,C) is observable, for any s( A) and ( s), L 0 if 0, 1 1 for any ( ), (, ),with ( ) s A rank I L N C si A H, C( si A) B. L. Wang, X.F. Wang and G. Chen (2016)

33 Some most recent progress

34 edge Temporally Switching Networks Adjacency matrix: a ji are constants, but appear and disappear in a temporal manner Division of time durations Network topology is temporally switching time B. Y. Hou, X. Li, G. Chen, IEEE Trans. Circ. Syst. Part I (2016) time

35 State Controllability of Temporally Switching Systems Temporally Switching Systems State Controllability: Necessary and Sufficient Condition State Controllable if and only if Controllability matrix C = e A m t t m 1 e A 2 t 2 t 1 C 1,, e A m t t m 1 C m 1, C m has full rank, where C i = A n 1 i B,, A i B, B

36 Structural Controllability of Temporally Switching Networks Temporally Switching Systems Structural Controllability: There exist a set of parameter values such that Necessary and Sufficient Condition Structural if and only if Controllability Controllability matrix C = e A m t t m 1 e A 2 t 2 t 1 C 1,, e A m t t m 1 C m 1, C m has full rank for some set of parameter values

37 Research Outlook General networks of linear time-varying (LTV) node-systems General networks of non-identical node-systems General temporal networks of LTI or LTV node-systems Some special types of networks of nonlinear node-systems There are more, of course

38 Thanks

39 References Müller F J, Schuppert A. Few inputs can reprogram biological networks. Nature, 478(7369): E4-E4, Banerjee S J, Roy S. Key to Network Controllability. arxiv , Nacher J C, Akutsu T. Dominating scale-free networks with variable scaling exponent: heterogeneous networks are not difficult to control. New Journal of Physics. 4(7): , Cowan N J, Chastain E J, Vilhena D A, et al. Nodal dynamics, not degree distributions, determine the structural controllability of complex networks. PloS One, 7(6): e38398, Nepusz T, Vicsek T. Controlling edge dynamics in complex networks. Nature Physics. 8(7): , Xiang L, Zhu J H, Chen F, Chen G. Controllability of weighted and directed networks with nonidentical node dynamics, Mathematical Problems in Engineering. ID , Mones E, Vicsek L, Vicsek T. Hierarchy measure for complex networks. PloS one, 7(3): e33799, Liu Y-Y, Slotine J-J, Barabási A-L. Control centrality and hierarchical structure in complex networks. PLoS ONE 7(9): e44459, 2012.

40 References Wang B, Gao L, Gao Y. Control range: a controllability-based index for node significance in directed networks. Journal of Statistical Mechanics, 2012(04): P04011, Wang W X, Ni X, Lai Y C, et al. Optimizing controllability of complex networks by minimum structural perturbations. Physical Review E, 85(2): , Yan G, Ren J, Lai Y C, et al. Controlling complex networks: How much energy is needed?. Physical Review Letters, 108(21): , Zhou T, On the controllability and observability of networked dynamic systems, Automatica, 52: 63-75, Yan G, Tsekenis G, Barzel B, Slotine J-J, Liu Y-Y, Barabási AL. Spectrum of controlling and observing complex networks, Nature Physics, doi: / nphys3422, Motter, A E, Networkcontrology, Chaos, 25: , 2015 Liu Y Y, Barabasi A-L, Control principles of complex networks, 2015 (arxiv: v1). Wang L, Chen G, Wang X F, Tang W K S. Controllability of networked MIMO systems, Automatica, 2016, 69: Hou B Y, Li X, Chen G. Structural controllability of temporally switching networks, IEEE Trans. Circ. Syst. I, 2016, in press.