Cartesian Products of Z-Matrix Networks: Factorization and Interval Analysis

Transcription

1 Nabi-Abdolyouse) Factorization University (Distributed andofinterval Washington Space Analysis Systems1 Lab / 27(D Cartesian Products of Z-Matrix Networks: Factorization and Interval Analysis Airlie Chapman and Mehran Mesbahi (work also in part with Marzieh Nabi-Abdolyouse) Distributed Space Systems Lab (DSSL) University of Washington

2 Nabi-Abdolyouse) Factorization University (Distributed andofinterval Washington Space Analysis Systems2 Lab / 27(D The Network in the Dynamics General Dynamics ẋ(t) = f (G, x(t), u(t)) y(t) = g(g, x(t), u(t)) Network Graph Spectrum Random Graphs Automorphisms Graph Factorization System Dynamics Rate of convergence Random Matrices Homogeneity Decomposition 1 State Dynamics 2 Controllability

3 Nabi-Abdolyouse) Factorization University (Distributed andofinterval Washington Space Analysis Systems3 Lab / 27(D The Network in the Dynamics First Order, Linear Time Invariant model ẋ i (t) = w ii x i (t) + w ij x j (t)+u i (t) i j y i (t) = x i (t) Dynamics ẋ(t) = A(G)x(t) + B(S)u(t) y(t) = C(R)x(t) A(G) = w 11 w 12 w 1n. w 21 w wn 1,n w n1 w n,n 1 w nn A(G): Z-matrix, e.g. Laplacian (w ii = w ij ) and Advection matrices (w ii = w ji ) Input node set S = {v i, v j,...}, B(S) = [e i, e j,...] Output node set R= {v p, v q,...}, C(R) = [e p, e q,...] T

4 Nabi-Abdolyouse) Factorization University (Distributed andofinterval Washington Space Analysis Systems4 Lab / 27(D Graph Cartesian Product Cartesian product G H Vertex set: V (G H) = V (G) V (H) Edge set: (x 1, x 2 ) (y 1, y 2 ) is in G H if x 1 y 1 and x 2 = y 2 or x 1 = y 1 and x 2 y 2

5 Nabi-Abdolyouse) Factorization University (Distributed andofinterval Washington Space Analysis Systems5 Lab / 27(D Graph Cartesian Product Cartesian product G H Vertex set: V (G H) = V (G) V (H) Edge set: (x 1, x 2 ) (y 1, y 2 ) is in G H if x 1 y 1 and x 2 = y 2 or x 1 = y 1 and x 2 y 2

6 Nabi-Abdolyouse) Factorization University (Distributed andofinterval Washington Space Analysis Systems6 Lab / 27(D Graph Cartesian Product Cartesian product G H Vertex set: V (G H) = V (G) V (H) Edge set: (x 1, x 2 ) (y 1, y 2 ) is in G H if x 1 y 1 and x 2 = y 2 or x 1 = y 1 and x 2 y 2

7 Nabi-Abdolyouse) Factorization University (Distributed andofinterval Washington Space Analysis Systems7 Lab / 27(D Part 1: Factoring the State Dynamics

8 Nabi-Abdolyouse) Factorization University (Distributed andofinterval Washington Space Analysis Systems8 Lab / 27(D State Dynamics Factorization Lemma 1: Uncontrolled Dynamics Consider the dynamics ẋ(t) = A(G 1 G 2... G n )x(t). Then, when properly initialized, one has x(t) = x 1 (t) x 2 (t) x n (t) where ẋ i (t) = A(G i )x i (t).

9 Nabi-Abdolyouse) Factorization University (Distributed andofinterval Washington Space Analysis Systems9 Lab / 27(D State Dynamics Factorization

10 Nabi-Abdolyouse) Factorization University (Distributed and of Washington Interval Space Analysis Systems 10 Lab / 27(D Graph Factorization A graph can be factored as well as composed... Theorem (Sabidussi 1960) Every connected graph can be factored as a Cartesian product of prime graphs. Moreover, such a factorization is unique up to reordering of the factors. Primes: G = G 1 G 2 implies that either G 1 or G 2 is K 1 Number of prime factors is at most log G Algorithms Feigenbaum (1985) - O ( V 4.5) ( Winkler (1987) - O V 4) from isometrically embedding graphs by Graham and Winkler (1985) Feder (1992) - O ( V E ) Imrich and Peterin (2007) - O ( E ) C++ implementation by Hellmuth and Staude

11 Nabi-Abdolyouse) Factorization University (Distributed and of Washington Interval Space Analysis Systems 11 Lab / 27(D Controlled Factorization Lemma Lemma 2: Controlled Dynamics Consider the dynamics ẋ(t) = A(G 1 G 2... G n )x(t) + B(S 1 S 2 S n )u(t). y(t) = C(R 1 R 2 R n )x(t) Then, when properly initialized and u(t) = u 1 (t) u 2 (t) u n (t), one has y(t) = y u (t) + y f (t), y u (t) = y 1u (t) y 2u (t) y nu (t) y f (t) = t 0 ẏ 1f (τ) ẏ 2f (τ) ẏ nf (τ)dτ where ẋ i (t) = A(G i )x i (t) + B(S i )u i (t) and y i (t) = C(R i )x i (t).

12 Nabi-Abdolyouse) Factorization University (Distributed and of Washington Interval Space Analysis Systems 12 Lab / 27(D Controlled Factorization Lemma

13 Nabi-Abdolyouse) Factorization University (Distributed and of Washington Interval Space Analysis Systems 13 Lab / 27(D Controlled Factorization Lemma Lemma 2: Controlled Dynamics Consider the dynamics ẋ(t) = A( y(t) = C( G i )x(t) + B( S i )u(t) R i )x(t) Then, when properly initialized and u(t) = u i (t), one has t y(t) = y iu (t) + ẏ if (τ)dτ 0 where ẋ i (t) = A(G i )x i (t) + B(S i )u i (t) and y i (t) = C(R i )x i (t).

14 Nabi-Abdolyouse) Factorization University (Distributed and of Washington Interval Space Analysis Systems 14 Lab / 27(D Controlled Factorization - Idea of the Proof Firstly Also Using As well as A(G 1 G 2 ) = A(G 1 ) I + I A(G 2 ) = A(G 1 ) A(G 2 ) e A(G 1) A(G 2 ) = e A(G 1) e A(G 2) B(S 1 S 2 ) = B(S 1 ) B(S 2 ) e A(G 1 G 2 )t Bu(t) = (e A(G 1)t e A(G 2)t )(B(S 1 ) B(S 2 ))(u 1 (t) u 2 (t)) = e A(G 1)t B(S 1 )u 1 (t) e A(G 2)t B(S 2 )u 2 (t) The proof follows from these observations.

15 Nabi-Abdolyouse) Factorization University (Distributed and of Washington Interval Space Analysis Systems 15 Lab / 27(D Approximate Graph Products A(G) [ A(G 1 G 2 ), A(G 1 G 2 ) ]

16 Nabi-Abdolyouse) Factorization University (Distributed and of Washington Interval Space Analysis Systems 16 Lab / 27(D Approximate Factorization Lemma Lemma 3: Approximation Consider the dynamics ẋ(t) = A(G)x(t) + B(S)u(t), y(t) = C(R)x(t) A(G) [ A( G i ), A( G i ) ], S [ S i, S i ], R = [ R i, R i ] and positive u(t) [ u i (t), u i (t)]. Then, when properly initialized, one has t y iu (t) + 0 t ẏ if (τ)dτ y(t) y iu (t) + ẏ if (τ)dτ 0 where ẋ i (t) = A(G i )x i (t) + B(S i )u i (t), y i (t) = C(R i )x i (t) and similarily for y i (t).

17 Nabi-Abdolyouse) Factorization University (Distributed and of Washington Interval Space Analysis Systems 17 Lab / 27(D Approximate Graph Products Can a graph be approximately factored as well as approximately composed? Distance between graphs d(g, H) is the smallest integer k such that V (G ) V (H ) + E(G ) E(H ) k. A graph G is a k-approximate graph product if there is a product H such that d(g,h) k. Lemma (Hellmuth 2009) For xed k all Cartesian k-approximate graph products can be recognized in polynomial time.

18 Nabi-Abdolyouse) Factorization University (Distributed and of Washington Interval Space Analysis Systems 18 Lab / 27(D Approximate Factorization Lemma: An Example x x x x x x x x x x x x x x x x time time time time

19 Nabi-Abdolyouse) Factorization University (Distributed and of Washington Interval Space Analysis Systems 19 Lab / 27(D Part 2: Factoring Controllability (work with Marzieh Nabi-Abdolyouse)

20 Nabi-Abdolyouse) Factorization University (Distributed and of Washington Interval Space Analysis Systems 20 Lab / 27(D Controllability Dynamics are controllable if for any x(0), x f and t f there exists an input u(t) such that x(t f ) = x f. Signicant in networked robotic systems, human-swarm interaction, network security, quantum networks. Challenging to establish for large networks Known families of controllable graphs for selected inputs Paths Circulants Grids Distance regular graphs

21 Nabi-Abdolyouse) Factorization University (Distributed and of Washington Interval Space Analysis Systems 21 Lab / 27(D Controllability Factorization - Product Control Theorem 1: Product Controllability The dynamics ẋ(t) = A( y(t) = C( G i )x(t) + B( S i )u(t) R i )x(t) where A( G i ) has simple eigenvalues is controllable/observable if and only if ẋ i (t) = A(G i )x i (t) + B(S i )u i (t) y i (t) = C(R i )x i (t) is controllable/observable for all i.

22 Nabi-Abdolyouse) Factorization University (Distributed and of Washington Interval Space Analysis Systems 22 Lab / 27(D Controllability Factorization - Idea of the Proof Popov-Belevitch-Hautus (PBH) test (A, B) is uncontrollable if and only if there exists a left eigenvalue-eigenvector pair (λ, v) of A such that v T B = 0. Eigenvalue and eigenvector relationship: A(G 1 ) A(G 2 ) A(G 1 G 2 ) Eigenvalue λ i µ j λ i + µ j Eigenvector v i u j v i u j Also (v i u i ) T (B(S 1 ) B(S 2 )) = v T i B(S 1 ) u T i B(S 2 ) The proof follows from these observations.

23 Nabi-Abdolyouse) Factorization University (Distributed and of Washington Interval Space Analysis Systems 23 Lab / 27(D Controllability Factorization - Layered Control Theorem 2: Layered Controllability The dynamics ẋ(t) = A( y(t) = C( G i )x(t) + B( S i )u(t) R i )x(t) where S i = R i = V (G i ) for i = 2,...,n is controllable/observable if and only if is controllable/observable. ẋ 1 (t) = A(G 1 )x 1 (t) + B(S 1 )u 1 (t) y 1 (t) = C(R 1 )x 1 (t)

24 Nabi-Abdolyouse) Factorization University (Distributed and of Washington Interval Space Analysis Systems 24 Lab / 27(D Controllability Factorization - Layered Control

25 Nabi-Abdolyouse) Factorization University (Distributed and of Washington Interval Space Analysis Systems 25 Lab / 27(D Uncontrollability through Symmetry Proposition (Rahmani and Mesbahi 2006) (A(G), B(S)) is uncontrollable if there exists an automorphism of G which xes all inputs in the set S (i.e., S is not a determining set.) The determining number of a graph G, denoted Det(G), is the smallest integer r so that G has a determining set S of size r. Corollary (A(G), B(S)) is uncontrollable if S < Det(G).

26 Nabi-Abdolyouse) Factorization University (Distributed and of Washington Interval Space Analysis Systems 26 Lab / 27(D Breaking Symmetry Automorphism group for graph Cartesian products The automorphisms for a connected G is generated by the automorphisms of its prime factors. Proposition: Automorphism group for graph Cartesian products For controllable pairs (A(G 1 ), B(S 1 )) and (A(G 2 ), B(S 2 )) where S 1 = Det(G 1 ) and S 2 = 1. Then S = S 1 S 2 is the smallest input set such that A(G 1 G 2, B(S)) is controllable.

27 Nabi-Abdolyouse) Factorization University (Distributed and of Washington Interval Space Analysis Systems 27 Lab / 27(D Conclusion Explored decomposition of Z-matrix based network into smaller factor-networks Provided exact and approximate factorization of state dynamics Presented a factorization of controllability - a product and layered approach Linked the factors symmetry to smallest controllable input set Future work involves examining other graph products in network dynamics