Algebraic state space approach to logical dynamic systems and its applications

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1 DOI: /CTA Control Theory & Applications Vol. 31 No. 12 Dec. 2014, (, ) :. 2 ( ) k ( ).,.,. R n Kalman,.,,.,. : ; ; ; ; : TP273 : A Algebraic state space approach to logical dynamic systems and its applications CHENG Dai-zhan, QI Hong-sheng (Key Laboratory of Systems and Control, Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing , China) Abstract: The logical dynamic system in this paper stands for the systems where the state variables can take only finite values. Particularly, when the number is 2 it is a classical logic (or Boolean logic); k-valued logic, and general finitely valued (general) logic. In recent years, using semi-tensor product of matrices the algebraic state space approach to logical dynamic systems has been developed and widely appreciated. It has been used to many engineering problems and to theoretical researches. Parallel to the Kalman state space approach to continuous state space dynamics where the differential equations or difference equations are used to describe the dynamic systems, the algebraic state space approach may provide a convenient platform for analyzing and control design of logical systems. The purpose of this paper is two fold: First, we give a brief survey on this new approach; then we introduce its current research topics and main results. Finally many applications and predict the potential of its further applications are presented. Key words: semi-tensor product; logical dynamic systems; algebraic state space equation; pure state and mixed state; control and game 1 (Introduction to logical dynamic D k, k 2 ( ) k, D k =k., D k = {1, 2,, k}., i D i,,. : k, D k = {0, 1 k 1,, k 2, 1};, D := k 1 D 2 = {T, F } D = {0, 1}, : T 1, F 0., D 3 = {1, 2, 3},, n 1 1) f : D k0 D ki ( ). y = f(x 1,, x n ), (1) : y D k0, x i D ki, i = 1,, n. 2) k 0 = k 1 = = k n = r, f r. r = 2,. n 3) f : R D ki ( ). (1). k 1 = k 2 = = k n = r, f r. r = 2,. 4) x 1 (t + 1) = f 1 (x 1 (t),, x n (t)), x 2 (t + 1) = f 2 (x 1 (t),, x n (t)),. x n (t + 1) = f n (x 1 (t),, x n (t)) (2) : ; : dcheng@iss.ac.cn; Tel.: : ( , , ).

2 12 : 1633, x i (t) D ki, f i, i = 1,, n. k 1 =k 2 = = k n = r, (2) r. r = 2,.,,, : x 1 (t + 1) = f 1 (x 1 (t),, x n (t), u 1 (t),, u m (t)), x 2 (t + 1) = f 2 (x 1 (t),, x n (t), u 1 (t),, u m (t)),. x n (t + 1) = f n (x 1 (t),, x n (t), u 1 (t),, u m (t)), y j (t) = h j (x 1 (t),, x n (t)), j = 1,, p, (3) : x i (t) D ki (i = 1,, n), u i (t) D ri (i = 1,, m), y i (t) D si (i = 1,, l), f i, h j.. R n ( ),,.,.,,. 1.,.,.. Jacob Monod, S. A. Kauffman 1969 [1 2]., [3 4].,,,,.,, [5 8].. [9].,., IEEE (CSS) M. E. Valcher [10] :,,.,.,.., D k, i δ i k, i = 1,, k, (4) δ i k I k i. k := Col(I k ), I k., (4), D k k, D k k.,.,,., A ( )µ 0 1.,. i S i = D k. r i 0, (r 1, r 2,, r k ) T (r i 0, i, k r i = 1). Υ k := {(r 1,, r k ) T r i 0, i; k r i = 1}. (1) (2). x i Υ ki, i = 1,, n,, ( ) ( )., (4). : : L M m n, δ i m, Col(L) m. m n L m n. L L m n, L, L = [δ i1 m δ i2 m δ in m ]. L = δ m [i 1 i 2 i n ]. : M M m n, Col(M) Υ m. m n Υ m n. 2 (Semi-tensor product of matrices) [11 13]. 2 A M m n, B M p q. n p t = lcm{n, p}. A B A B := (A I t/n )(B I t/p ), (5) Kronecker [14]. n = p,, 2.,.,

3 [12] n f : D ki D k0 1 1)., M f L k0 k, F (ag ± bh) = af G ± bf H, : n (af ± bg) H = (6) y = M f x i, (12) af H ± bg H, a, b R. M f f. 2) (F G) H = F (G H). (7),, Khatri-Rao.,. 3 W [m,n] M mn mn : W [m,n] = δ mn [ 1, m + 1,, (n 1)m + 1, 2, m + 2,, (n 1)m + 2,. m, 2m,, nm]. (8) W T [m,n] = W 1 [m,n] = W [n,m]. (9) 1) X R m, Y R n, W [m,n] X Y = Y X. (10) 2) X R m, Y R n, X Y W [m,n] = Y X. (11) 1) n = p, 2.,.,,. 2), : i), ii).,,.,.,.. 3 (Algebraic state space representation of logical dynamic 3.1 (Deterministic logical dynamic n f : D ki D k0, n f : ki k0. y =f(x 1, x 2,, x n ), k = n k i, 4 [14] M M m p, N M n p., M N Khatri-Rao M N := [Col 1 (M) Col 1 (N), Col 2 (M) Col 2 (N),. Col p (M) Col p (N)] M mn p. (13) (2)., x i (t + 1) = M i n M i f i. x j (t), i = 1,, n, (14) x(t) := n x j (t),. 2 [12] (2). L L k k, ( ) x(t + 1) = Lx(t), (15) L = M 1 M 2 M n, M i f i, i = 1,, n., u(t) := m u j (t), r = m r j ; y(t) := p y j (t), s = p s j, (3). 3 [13] (3). L L k kr H L s k, ( ) { x(t + 1) = Lu(t)x(t), (16) y(t) = Hx(t), : L = M 1 M 2 M n, M i f i, i = 1,, n; H = H 1 H 2 H p, H i h i, i = 1,, p.

4 12 : (Probabilistic logical dynamic (2). f i, f i =f j i, p j i, j =1,, n i ; i = 1,, n. (17), (2). [15]. f j i M j i. J i = {1, 2,, n i }, i = 1,, n; J = n J i. j J j = n j i, j i J i., j J x i (t + 1) = f ji i (x 1 (t),, x n (t)), i = 1,, n. f ji i M ji i,, j J (18) x(t + 1) = L j x(t), j J. (19) p j = n p ji i, j J. x(t + 1) = Lx(t), (20) L = j J P j L j Υ k k. 3.3 (Random logical dynamic [16]. (2). ( ) i N = {1, 2,, n}, i,, {xi (t + 1) = f i (x 1 (t),, x n (t)), (21) x j (t + 1) = x j (t), j i. (21) x(t + 1) = M i x(t), i = 1,, n. i, x(t + 1) = Mx(t), (22) M = 1 n n M i Υ k k.,,.. 5 1), x i Υ ki, i = 1,, n,, M Υ k k. x(t + 1) = Mx(t), (23) 2), x i Υ ki (i = 1,, n), u i Υ ri (i = 1,, m), y i Υ si (i = 1,, p),, { x(t + 1) = Lu(t)x(t), (24) y(t) = Hx(t), : L Υ k kr, H Υ s k. :, ( ) ;,, ( )., ; k ; k = 2. 4 (Advances in algebraic state space approach),.,,. 4.1 (Analysis of logical dynamic,,. [9, 17 19]. [20 21]. [22]. [23 25]. [26]. [27]. 4.2 (Control of logical dynamic (Controllability).. [28]. [29]. [30 31].

5 [32 34].,, [35] [36 37] [38 39] [40 41] [42], (Observability),.,, [28, 41]. [43]. [10],. [44]. [45] (Optimal control) [46 47]. [48]. [49 50]., [51] Riccati, [48] (Stability and stabilization) [52]. [53] [54]. [55]. [56] [57], (Decoupling) [58] ( ),. [59]. [60]. [61]., [62], (Identification and realization) [26]. [58]. [63]. [64],.,, (Algorithms).,. [27],. [65 66]. 5 (Applications of algebraic state space approach) 5.1 (Biological systems and life sciences) : [27] T, ;, [56] [67], ;, [68] (Game theory) : [69 70] ; [48] ; [71] ; [72]. ( ),,.,.,,. 5.3 (Circuit design and fault diagnosis) : k, [73] ; [74].,,.,. 5.4 (Fuzzy control) : [75] ;

6 12 : 1637 [76] ; [77] ; [78]., x, x Υ (Finite state automata and symbolic dynamics) [79] ; [80] ; [81] ; [82].,..,,.,. 5.6 (Graph theory and formation control) : [83] ; [84] ; [85].,., [29] (Coding theory and algorithm implementation) : [86] ; [48] ; [87].,,. 5.8 (Network inquiry and teleoperation) : [88] ; [89].. 6 (Conclusions),.,. (IFAC) Automatica /.,,.,.,,. /.,,,,! (References): [1] KAUFFMAN S A. Metabolic stability and epigenesis in randomly constructed genetic nets [J]. Journal of Theoretical Biology, 1969, 22(3): [2] WALDROP M M. Complexity The Emerging Science at the Edge of Order and Chaos [M]. New York: Simon & Schuster, [3] KAUFFMAN S A. The Origins of Order: Self-Organization and Selection in Evolution [M]. New York: Oxford University Press, [4] KAUFFMAN S A. At Home in the Universe [M], Oxford: Oxford University Press, [5] FARROW C, HEIDEL J, MALONEY H, et al. Scalar equations for synchronous Boolean networks with biological applications [J]. IEEE Transactions on Neural Networks, 2004, 15(2): [6] IRONS D J. Improving the efficiency of attractor cycle identification in Boolean network [J]. Physica D, 2006, 217(1): [7] HEIDEL J, MALONEY J, FARROW J, et al. Finding cycles in synchronous Boolean networks with applications to biochemical systems [J]. International Journal of Bifurcat Chaos, 2003, 13(3): [8] DROSSEL B, MIHALJEV T, GREIL F. Number and length of attractors in a critical Kauffman model with connectivity one [J/OL]. Physics Review Letters, 2005, 94: [9] CHENG D, QI H. A linear representation of dynamics of Boolean networks [J]. IEEE Transactions on Automatic Control, 2010, 55(10): [10] FORNASINI E, VALCHER M E. Observability, reconstructibility and state observers of Boolean control networks [J]. IEEE Transactions on Automatic Control, 2013, 58(6): [11],. [M]. 2. :, (CHENG Daizhan, QI Hongsheng. Semi-tensor Product of Matrices: Theory and Application [M]. 2nd ed. Beijing: Science Press, 2011.) [12] CHENG D, QI H, LI Z. Analysis and Control of Boolean Networks: a Semi-tensor Product Approach [M]. London: Springer, [13] CHENG D, QI H, ZHAO Y. An Introduction to Semi-tensor Product of Matrices and Its Applications [M]. Singapore: World Scientific, [14]. [M]. :, 2004.

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