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Input-State Incidence Matrix of Boolean Control Networks and Its Applications$ Input-State Incidence Matrix of Boolean Control Networks and Its ApplicationsI Yin Zhao, Hongsheng Qi, Daizhan Cheng The Key Laboratory of Systems & Control Institute of Systems Science, Academy of Mathematics & Systems Science, Chinese Academy of Sciences, Beijing 100190, P. R. China Abstract The input-state incidence matrix of control Boolean network is proposed. It is shown that this matrix contains complete information of the input-state mapping. Using it, an easily verifiable necessary and sufficient condition for the controllability of Boolean control network is obtained. The corresponding control which drives a point to a given reachable point is designed. Moreover, certain topological properties such as the fixed points and cycles of a Boolean control network are investigated. Then, as another application, a sufficient condition for the observability is presented. Finally, the results are extended to mix-valued logical control systems. Keywords: Boolean control network, input-state incidence matrix, reachability, observability, mix-valued logic 1. Introduction The Boolean network was firstly proposed by Kauffman for modeling complex and nonlinear biological systems [1,2,3]. Since then, it has been studies widely and applied to some other systems. The first interesting topic is the topological structure of a Boolean network. Many progresses have been done [4,5,6]. Another challenging and important topic is the control ISupported partly by NNSF 60736022, and 60221301 of China. Email address: [email protected],[email protected],[email protected] (Yin Zhao, Hongsheng Qi, Daizhan Cheng) Preprint submitted to Systems & Control Letters August 20, 2010 of Boolean networks. The main efforts have been put on the controllability of Boolean networks [7,8,9]. Recently, a new matrix product, called the semi-tensor product of ma- trices, has been proposed, and using it a logical mapping can be expressed into a matrix form [10]. Using them, a Boolean (control) network can be converted into a standard discrete-time dynamic (control) system [11, 12]. In this paper the same framework is utilized. We, therefore, give a brief review on this new technique. For statement ease, we introduce some nota- tions first. (i) Denote by Col(A) (Row(A)) the set of columns (rows) of a matrix A, and Coli(A) (Rowi(A)) the i-th column (row) of A. (ii) D := f0; 1g. i 1 2 n (iii) Let δn be the i-th column of the identity matrix In, and ∆n := fδn; δn; ··· ; δng. When n = 2 we simply use ∆ := ∆2. T (iv) 1k := [1 1 ··· 1] . | {z } k i1 i2 is (v) Assume a matrix M = [δn δn ··· δn ] 2 Mn×s , i.e., its columns, Col(M) ⊂ ∆n. Then M is called a logical matrix, and simply denoted as M = δn[i1 i2 ··· is]: The set of n × s logical matrices is denoted by Ln×s. (vi) A matrix B 2 Mn×s is called a Boolean matrix, if its entries bij 2 D, 8 i; j. The set of n × s Boolean matrices is denoted by Bn×s. (vii) Let A 2 Mn×mn, denoted by Blki(A) the i-th n × n square block of A, i = 1; 2 ··· ; m. A Boolean network with n nodes is described as 8 >x1(t + 1) = f1(x1(t); ··· ; xn(t)) < . (1) :> xn(t + 1) = fn(x1(t); ··· ; xn(t)); xi 2 D; n where fi : D !D, i = 1; ··· ; n are logical functions. Similarly, a Boolean control network with n network nodes, m input 2 nodes, and p outputs is described as 8 x1(t + 1) = f1(x1(t); ··· ; xn(t); u1(t); ··· ; um(t)) <> . (2) :> xn(t + 1) = fn(x1(t); ··· ; xn(t); u1(t); ··· ; um(t)); xi; uj 2 D; yj(t) = hj(x1(t); ··· ; xn(t)); j = 1; ··· ; p; n+m n where fi : D !D, i = 1; ··· ; n and hj : D !D, j = 1; ··· ; p are logical functions. Throughout this paper the matrix product is assumed to be semi-tensor product as A n B, and the symbol \n" is omitted in most places. We refer to [10] for details. 1 2 Identifying 1 ∼ δ2, 0 ∼ δ2, and using semi-tensor product of matrices, logic functions can be expressed by logical matrix, vectors, and their prod- ucts. For example, the truth table of some common logical functions such as \negation (:)", \conjunction (^)", \disjunction (_)", \conditional (!)", \biconditional ($)" and \exclusive or (_¯)" is given in Table1, and their structure matrices are given in Table2. Table 1: Truth Table p q :p p ^ q p _ q p ! q p $ q p_¯q 1 1 0 1 1 1 1 0 1 0 0 0 1 0 0 1 0 1 1 0 1 1 0 1 0 0 1 0 0 1 1 0 Table 2: Matrix Expressions of Logical Functions : Mn = δ2[2 1] ! Mi = δ2[1 2 1 1] _ Md = δ2[1 1 1 2] $ Me = δ2[1 2 2 1] ^ Mc = δ2[1 2 2 2] _¯ Mp = δ2[2 1 1 2] Then, in vector form we have :p = Mn n p and p ^ q = Mc n p n q, etc. n m p Set x = ni=1xi, u = ni=1ui, and y = ni=1yi. Using vector form, (1) and (2) can be expressed by the following (3) and (4) respectively, which are called the algebraic forms of (1) and (2) respectively. x(t + 1) = Lx(t); (3) 3 where L 2 L2n×2n is called structure matrix of the Boolean network. ( x(t + 1) = Lu(t)x(t) (4) y(t) = Hx(t); where L 2 L2n×2n+m , and H 2 L2p×2n . A matrix, A = (aij) 2 Mn×n is called the adjacency matrix (or incidence matrix) of the Boolean network (1), if ( 1; xj(t + 1) depends on xi(t) aij = 0; otherwise: If we consider the network graph, aij 6= 0, iff there is an edge from xi to xj. The difference between L of (3) and A is that L contains complete information of the network dynamics, precisely, (1) can be recovered from L easily, while A does not. The controllability and observability of Boolean control networks were investigated under this framework [13]. As a related topic, the controllable normal form and the realization of Boolean networks were discussed in [14]. The topological structure of Boolean network (without control) such as fixed points and cycles was investigated in [11]. We first give rigorous definitions for the controllability, observability, fixed points and cycles. Definition 1.1. Consider system (2). Denote its state space as X = Dn, and let X0 2 X . 1. X 2 X is said to be reachable from X0 at time s > 0, if we can find a se- quence of controls U(0) = fu1(0); ··· ; um(0)g, U(1) = fu1(1); ··· ; um(1)g, ··· , such that the trajectory of (2) with the initial value X0 and the con- trols fU(t)g, t = 0; 1; ··· will reach X at time t = s. The reachable set at time s is denoted by Rs(X0). The overall reachable set is denoted by 1 R(X0) = [s=1Rs(X0): 2. System (2) is said to be controllable at X0 if R(X0) = X . The system is said to be controllable if it is controllable at every X 2 X . Definition 1.2. Consider system (2). Denote by Y (t) = (y1(t); ··· ; yp(t)) 2 Dp. 4 0 0 1. X1 and X2 are said to be distinguishable, if there exists a control se- quence fU(0);U(1); ··· ;U(s)g, where s ≥ 0, such that 1 s+1 0 2 s+1 0 Y (s + 1) = y (U(s); ··· ;U(0);X1 ) 6= Y (s + 1) = y (U(s); ··· ;U(0);X2 ): (5) 0 0 2. The system is said to be observable, if any two initial points X1 ;X2 2 X are distinguishable. Definition 1.3. Consider system (2). Denote the input-state (product) space by p n S = f(U; X) j U = (u1; ··· ; um) 2 D ;X = (x1; ··· ; xn) 2 D g : Note that jSj = 2m+n. i i j j i 1. Let Si = (U ;X ) 2 S and Sj = (U ;X ) 2 S. Denote by U = i i i i i (u1; ··· ; um), X = (x1; ··· ; xn), etc. (Si;Sj), is said to be a directed edge, if Xi;U i;Xj satisfy (2). Precisely, j i i i i xk = fk(x1; ··· ; xn; u1; ··· ; um); k = 1; ··· ; n: The set of edges is denoted by E ⊂ S × S. 2. The pair (S; E) forms a directed graph, which is called the input-state transfer graph. 3.( S1;S2; ··· ;S`) is called a path, if (Si;Si+1) 2 E, i = 1; 2; ··· ; ` − 1. 4. A path (S1;S2; ··· ) is called a cycle, if Si+` = Si for all i, the smallest ` is called the length of the cycle. Particular, the cycle of length 1 is called a fixed point. Note that when the vector form of logical variables is used, the state space becomes X = ∆2n , similarly, the input-state space becomes S = ∆2m+n . According to the corresponding statement, it is easy to tell which form is used there. In this paper we propose a matrix, called the input-state incidence matrix. Using it, a neat result about to controllability of Boolean control networks is presented by verifying the controllability matrix. The corresponding con- troller is also designed. Based on the structure of reachable set, an easily verifiable sufficient condition for observability is also obtained. Then the 5 input-state incidence matrix is also used to reveal some topological struc- tures of Boolean control networks, including fixed points, cycles etc.
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