Algebraic state space approach to logical dynamic systems and its applications

31 12 2014 12 DOI: 10.7641/CTA.2014.40777 Control Theory & Applications Vol. 31 No. 12 Dec. 2014, (, 100190) :. 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 100190, 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},, 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) : 2014 08 23; : 2014 11 02.. E-mail: dcheng@iss.ac.cn; Tel.: +86 10-62651445. : (61273013, 61333001, 61104065).

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.,.,

1634 31. 1 [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). 2. 3 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.

12 : 1635 3.2 (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 4.2.1 (Controllability).. [28]. [29]. [30 31].

1636 31 [32 34].,, [35] [36 37] [38 39] [40 41] [42],. 4.2.2 (Observability),.,, [28, 41]. [43]. [10],. [44]. [45]. 4.2.3 (Optimal control) [46 47]. [48]. [49 50]., [51] Riccati, [48]. 4.2.4 (Stability and stabilization) [52]. [53] [54]. [55]. [56] [57],. 4.2.5 (Decoupling) [58] ( ),. [59]. [60]. [61]., [62],. 4.2.6 (Identification and realization) [26]. [58]. [63]. [64],.,,.. 4.2.7 (Algorithms).,. [27],. [65 66]. 5 (Applications of algebraic state space approach) 5.1 (Biological systems and life sciences) : [27] T, ;, [56] [67], ;, [68]... 5.2 (Game theory) : [69 70] ; [48] ; [71] ; [72]. ( ),,.,.,,. 5.3 (Circuit design and fault diagnosis) : k, [73] ; [74].,,.,. 5.4 (Fuzzy control) : [75] ;

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