Decomposition with Respect to Outputs for Boolean Control Networks

Decompositionwith respect to outputs ⋆ for Boolean control networks. Yunlei Zou, Jiandong Zhu School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, PR China Abstract This paper investigates the problem of decomposition with respect to outputs for Boolean control networks (BCNs). First, with the linear expression of BCNs and the matrix semi-tensor product, some algebraic equivalent conditions for the decomposition are obtained. Second, a necessary and sufficient graphical condition for the decomposition with respect to outputs is given. Third, an effective method is proposed to reduce the computational burden in the realization of the decomposition. Finally, some examples are addressed to validate the effectiveness of the proposed method. Key words: Boolean control networks, Decomposition with respect to outputs, Semi-tensor product, Observability. 1 Introduction Li & Wang, 2012). Boolean networks (BNs) are a kind of discrete dynamical System decomposition is an important issue in the tra- systems described by logical variables and logical func- ditional linear control system theory, which makes clear tions. They are first proposed by Kauffman (1969) to the observable and controllable components of a system. describe and analyze cell regulation (Albert & Othmer, An interesting topic is ascertaining whether there exists 2003; Faure, Naldi, Chaouiya, & Thieffry, 2006). BNs a similar system decomposition for BCNs. As a mat- with additional inputs and outputs are usually called ter of fact, this topic has been discussed by Cheng, Li Boolean control networks (BCNs), and they have been and Qi (2010), where the controllable normal form, the paid great attention by biologists and control theory sci- observable normal form and the Kalman decomposition entists (Akutsu, Hayashida, Ching, & Ng, 2007; Cheng, form of BCNs are presented with the state-space anal- 2009;). In recent years, a semi-tensor product method of ysis method (Cheng & Qi, 2010b). The state space of BNs and BCNs was developed by Cheng and his collab- an n-dimensional BCN is defined as the set of all logical orators (Cheng, 2009; Cheng & Qi, 2009; Cheng & Qi, functions with respect to the n logical variables. Since the cardinal of the state space of an n-dimensional BCN 2010a). For a complete control theory framework of the n is 22 , generally speaking, the computational burden of arXiv:1407.2104v1 [math.OC] 14 Jun 2014 BCNs, we refer to the book written by Cheng, Qi, and Li (2011). Based on the linear algebraic expression of state space method is very heavy for large n although BCNs, many classical problems in control theory can be it displays great advantage in theoretical analysis. Fur- generalized to BCNs such as controllability, observabil- thermore, some additional regularity assumptions are ity, stabilization, disturbance decoupling and optimal imposed on the system model for achieving those decom- control (Cheng & Qi, 2009; Laschov & Margaliot, 2012; positions. Laschov, Margaliot, & Even, 2013; Fornasini & Valcher, 2013a; Fornasini & Valcher, 2013b; Cheng, 2011; Zhao, In our recent paper Zou & Zhu (2014), we have proposed Li & Cheng, 2011). Moreover, some of these results have the decomposition with respect to inputs for BCNs, which be further extended to different kinds of BCNs (Li, Yang, is just the controllable normal form proposed by Cheng, & Chu, 2014; Li & Sun, 2012; Feng, Yao, & Cui, 2013; Li and Qi (2010) under the regularity assumption on the largest uncontrollable subspace. But we did not use any ⋆ The research is supported in part by National Natural Sci- concepts of state spaces. A main contribution of Zou & ence Foundation (NNSF) of China under Grants 11271194, Zhu (2014) lies in that the system decomposition can be 61273115, 61074115 and 10701042. realized without the regularity condition on the largest Email addresses: [email protected] (Yunlei Zou), uncontrollable subspace. Moreover, a graphical condi- [email protected] (Jiandong Zhu). tion and a constructive algorithm for the decomposition Preprint submitted to Automatica 25 August 2021 with respect to inputs are proposed. Then, a natural z1(t +1)= fˆ1(z1(t), ··· ,zs(t),u1(t), ··· ,um(t)), idea is reconsidering the observable normal form and the . Kalman decomposition form for BCNs. zs(t +1)= fˆs(z1(t), ··· ,zs(t),u1(t), ··· ,um(t)), In this paper, we consider the decomposition with re- zs+1(t +1)= fˆs+1(z1(t), ··· ,zn(t),u1(t), ··· ,um(t)), spect to outputs in the framework of the linear algebraic . representation of BCNs. First, some necessary and suf- . (3) ficient algebraic conditions for the decomposition with z (t +1)= fˆ (z (t), ··· ,z (t),u (t), ··· ,u (t)), respect to outputs are proposed. Next, an equivalent n n 1 n 1 m ˆ graphical condition for the decomposition with respect y1(t)= h1(z1(t), ··· ,zs(t)), to outputs is derived. As the graphical condition is sat- . isfied, a constructive coordinate transformation to real- . ˆ ize the decomposition is obtained. Then, a method is yp(t)= hp(z1(t), ··· ,zs(t)). proposed to reduce the computational burden in realiz- ing the decomposition with respect to outputs. Finally, BCN (3) is called a decomposition with respect to outputs some examples are analyzed with our method. of order n − s. A decomposition with respect to outputs of the maximum order is called the maximum decompo- We organize this paper as follows. In section 2, some pre- sition with respect to outputs. We say that BCN (1) is liminaries and problem statement are provided. In sec- undecomposable with respect to outputs if the order of the tion 3, some algebraic conditions for the decomposition maximum decomposition with respect to outputs is 0. with respect to outputs are presented. In section 4, a graphical condition is obtained. In Section 5, a compu- Denote the real number field by R and the set of all the tation method is given. Finally, Section 6 gives a brief m × n real matrices by Rm×n. Let Col(A) be the set of conclusion. all the columns of matrix A and denote the ith column of i i A by Coli(A). Set ∆k = {δk|i =1, 2, ··· , k}, where δk = 2 Preliminaries and Problem Statement Coli(Ik) with Ik the k×k identity matrix. For simplicity, 1 2 we denote ∆ := ∆2 = {δ2,δ2}. A matrix L ∈ Rm×n is called a logical matrix if Col(L) ⊂ △m. Obviously, the Let D = {True = 1, False = 0}. Consider a BCN de- T T scribed by the logical equations logical matrix L satisfies 1mL = 1n , where 1n denote the n−dimensional column vector whose entries are all equal x1(t +1)= f1(x1(t), ··· , xn(t),u1(t), ··· ,um(t)), to1.Thesetofallthe m×r logical matrices is denoted by . Lm×r. For simplicity, we denote the logical matrix L = . i1 i2 ir [δm,δm, ··· ,δm] by δm[i1,i2, ··· ,ir]. Set A ∈ Rm×n, xn(t +1)= fn(x1(t), ··· , xn(t),u1(t), ··· ,um(t)), B ∈ Rp×q, and α = lcm(n,p) be the least common y1(t)= h1(x1(t), ··· , xn(t)), (1) multiple of n and p. The left semi-tensor product of A . and B is defined as (Cheng, Qi, & Li, 2011): . yp(t)= hp(x1(t), ··· , xn(t)), ⋉ α α A B = (A ⊗ I n )(B ⊗ I p ), where the state variables xi, the output variables yi and n+m where ⊗ is the Kronecker product. Obviously, the left the controls ui take values in D, fi : D → D and n semi-tensor product is a generalization of the traditional hi : D → D are logical functions. matrix product. So A ⋉ B can be written as AB. To ex- press the logical equations with linear algebraic method, Consider the logical mapping G : Dn → Dn defined by 1 elements in D are identified with vectors True ∼ δ2 and 2 False ∼ δ2 (Cheng, Qi, & Li, 2011). z1 = g1(x1, x2, ··· , xn), z2 = g2(x1, x2, ··· , xn), Proposition 1. (Cheng & Qi, 2010a) Let xi and ui ⋉n ⋉p . take values in ∆ and denote x = i=1xi, y = i=1yi, . (2) ⋉m u = i=1ui. Then the BCN (1) can be expressed in the zn = gn(x1, x2, ··· , xn). algebraic form x(t +1)= Lu(t)x(t), y(t)= Hx(t), (4) If G : Dn → Dn is a bijection, it is called a logical coordinate transformation. where L ∈ L2n×2n+m and H ∈ L2p×2n . We say that BCN (1) is decomposable with respect to Let z = T x be the algebraic form of logical coordinate outputs of order n − s, if there exists a logical coordinate transformation (2), where T is a permutation matrix. [1] ⋉s [2] ⋉n transformation (2), such that (1) becomes Set z = i=1zi and z = i=s+1zi. Then the decom- 2 position form (3) can be rewritten in the algebraic form n−s ≥ n−s˜, i.e. s ≤ s˜. On the other hand, from the defi- nition of largest unobservable subspace, it follows that [1] [1] z (t +1) = G1u(t)z (t), zs+1, ··· ,zn ∈ Oc = F(˜zs˜+1, z˜s˜+2, ··· , zñ). (7) [2] z (t +1) = G2u(t)z(t), (5) Thus, (7) implies that n − s ≤ n − s˜, namely s ≥ s˜. y(t) = Mz[1](t), Therefore, we have proved that s =s ˜, which implies that (6) is a maximum decomposition with respect to outputs of system (4). Conversely, bys ˜ = s, (7) and where G ∈L s× s+m , G ∈L n−s× n+m and M ∈L p× s . 1 2 2 2 2 2 2 2 Theorem 13 of Cheng, Li, & Qi (2010), we obtain that {zs+1, ··· ,zn} is a regular basis of Oc, that is, (5) is an For the BCN (1) with algebraic form (4), the problem observable normal form. ✷ of decomposition with respect to outputs is to find a coordinate transformation matrix T such that (4) has the Remark 1.

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