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Output tracking control of Boolean control networks via state feedback: Constant reference signal case
Automatica 59 (2015) 54–59
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Brief paper
Output tracking control of Boolean control networks via state feedback: Constant reference signal case✩ Haitao Li a,b , Yuzhen Wang b,1 , Lihua Xie c a
School of Mathematical Science, Shandong Normal University, Jinan 250014, PR China
b
School of Control Science and Engineering, Shandong University, Jinan 250061, PR China
c
EXQUISITUS, Center for E-City, School of Electrical and Electronic Engineering, Nanyang Technological University, 639798, Singapore
article
info
Article history: Received 28 March 2014 Received in revised form 15 May 2015 Accepted 21 May 2015 Available online 17 June 2015 Keywords: Boolean control network Output tracking control State feedback Semi-tensor product of matrices
abstract This paper investigates the state feedback based output tracking control of Boolean control networks (BCNs) with a constant reference signal by using the semi-tensor product method. Based on the algebraic expression of BCNs and by constructing a series of reachable sets, a general procedure is proposed for the design of the state feedback laws for BCNs to track a constant reference signal. The study of an illustrative example shows that the obtained new results are effective in designing state feedback based output tracking controllers for BCNs to track a constant reference signal. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction Since Kauffman’s pioneering work (Kauffman, 1969) in the Boolean model of gene regulatory networks, the study of Boolean networks has attracted great attention of biologists, physicists and systems scientists. Consequently, many excellent results have been established for Boolean networks (Akutsu, Hayashida, Ching, & Ng, 2007; Ay, Xu, & Kahveci, 2009; Chaves, 2009; Drossel, Mihaljev, & Greil, 2005; Xiao, 2009; Xiao & Dougherty, 2007). In a Boolean network, each gene can take two possible values, 1 and 0, and its value (1 or 0) indicates its measured abundance (expressed or unexpressed; high or low). From a graphical perspective, genes in a Boolean network are nodes in this network and edges describe regulatory relationships between genes. As is well known, the ultimate goal of modeling gene regulatory networks as Boolean networks is to design effective therapeutic intervention strategies to influence the network dynamics to avoid
✩ The research was supported by the National Research Foundation of Singapore under grant NRF-CRP8-2011-03, the National Natural Science Foundation of China under grants 61374065 and 61320106011, and the Research Fund for the Taishan Scholar Project of Shandong Province. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Yoshito Ohta under the direction of Editor Richard Middleton. E-mail addresses: [email protected] (H. Li), [email protected] (Y. Wang), [email protected] (L. Xie). 1 Tel.: +86 531 88392515; fax: +86 531 88392205.
http://dx.doi.org/10.1016/j.automatica.2015.06.004 0005-1098/© 2015 Elsevier Ltd. All rights reserved.
undesirable cellular states. Hence, in order to manipulate Boolean networks, binary control inputs and outputs are added to the network dynamics, which yields Boolean control networks (BCNs). The control of BCNs is a fundamental issue in both systems biology and control theory. However, due to the lack of effective tools to deal with logical dynamics, the control of BCNs has been a challenging problem for a long time until the introduction of the semitensor product method (Cheng, Qi, & Li, 2011). The feature of this method is that one can convert the dynamics of a Boolean (control) network into a linear (bilinear) discrete-time system. Then, one can analyze Boolean (control) networks by using the classical control theory. Up to now, there have been many interesting works on the control of BCNs via this novel method, which include controllability and observability (Chen & Sun, 2014; Cheng, Li, & Qi, 2010; Cheng & Qi, 2009, 2010; Feng, Yao, & Cui, 2012; Fornasini & Valcher, 2013a; Laschov & Margaliot, 2012; Li & Sun, 2011a,b; Li & Wang, 2012; Zhang & Zhang, 2013; Zhao, Cheng, & Qi, 2010), disturbance decoupling (Cheng, 2011; Yang, Li, & Chu, 2013), optimal control (Fornasini & Valcher, 2014; Laschov & Margaliot, 2011; Zhao, Li, & Cheng, 2011), stability and stabilization (Cheng, Qi, Li, & Liu, 2011; Li & Wang, 2013; Li, Yang, & Chu, 2013), and other control problems (Cheng & Xu, 2013; Cheng & Zhao, 2011; Fornasini & Valcher, 2013b; Li & Chu, 2012; Wang, Zhang, & Liu, 2012; Xu & Hong, 2013; Zhang, 2012; Zhang & Feng, 2013; Zhao, Kim, & Filippone, 2013; Zou & Zhu, 2014). It is noted that in many practical gene regulatory networks, the state variables cannot be measured directly due to the limitation
H. Li et al. / Automatica 59 (2015) 54–59
of measurement conditions and the impact of immeasurable variables. In this case, one can use the measured outputs to track a desirable reference signal which corresponds to some desirable states. Thus, it is meaningful for us to design suitable controllers (therapeutic intervention) that steer the output of BCNs to an expected reference signal, called the output tracking control problem of BCNs in this paper. It should be pointed out that several types of therapeutic intervention have been proposed for Boolean networks until now. Among them, the theory of automatic control, such as the controllability and the optimal control, is an important type of therapeutic intervention, and thereby many suitable control strategies have been obtained based on the classical control theory (Akutsu et al., 2007; Choudhary, Datta, Bittner, & Dougherty, 2006). As one of the most important issues in the control theory, it is believed that the output tracking control can provide an effective way for the design of therapeutic intervention. However, to our best knowledge, there are no results available on the output tracking control of BCNs. In this paper, using the semi-tensor product method, we investigate how to design output tracking controllers for BCNs to track a constant reference signal. We propose a general procedure to design state feedback based output tracking controllers for BCNs. The key idea of this procedure is to stabilize the BCN to a set of states whose outputs are the given constant reference signal. Although this procedure is a generalization of the state feedback stabilization control design method proposed in Fornasini and Valcher (2013b) and Li et al. (2013), it has the following two differences/novelties: (1) The procedure proposed in this paper has wider applications (such as the output tracking control, and the stabilization to a stable region) because it can stabilize the BCN to a set of states. When the set of states only contains an element, the procedure degenerates to that of Fornasini and Valcher (2013b) and Li et al. (2013). (2) In our procedure, the set of states that is stabilized to is not fixed, while the equilibrium in Fornasini and Valcher (2013b) and Li et al. (2013) is fixed. How to determine a proper set of states whose outputs are the given constant reference signal is a very challenging problem in our procedure. In this paper, we solve this problem by using the input-state incidence matrix introduced in Zhao et al. (2010) (please see Theorem 2). The rest of this paper is organized as follows. Section 2 formulates the output tracking control problem studied in this work. Section 3 investigates the output tracking control of BCNs via state feedback and presents the main results of this paper. An illustrative example is given to support our new results in Section 4, which is followed by a brief conclusion in Section 5. Notation: The notation of this paper is fairly standard. D := {1, 0}. ∆n := {δnk : k = 1, . . . , n}, where δnk denotes the kth column of the identity matrix In . For compactness, ∆ := ∆2 . An n × t i i matrix M is called a logical matrix, if M = [δn1 δn2 · · · δnit ], and we express M briefly as M = δn [i1 i2 · · · it ], and denote the set of n × t logical matrices by Ln×t . Coli (A) denotes the ith column of the matrix A, and Rowi (A) stands for the ith row of the matrix A.
55
where X (t ) = (x1 (t ), x2 (t ), . . . , xn (t )) ∈ D n , U (t ) = (u1 (t ), . . . , um (t )) ∈ D m and Y (t ) = (y1 (t ), . . . , yp (t )) ∈ D p are the state, the control input and the output of the system (1), respectively, and fi : D m+n → D , i = 1, . . . , n and hj : D n → D , j = 1, . . . , p are logical functions. Given a control sequence {U (t ) : t ∈ N}, denote the state trajectory of the system (1) starting from an initial state X (0) ∈ D n by X (t ; X (0), U ), and the output trajectory of the system (1) starting from X (0) ∈ D n by Y (t ; X (0), U ). The output tracking control problem studied in this paper is to design a state feedback control in the form of
u1 (t ) = k1 (X (t )), .. . um (t ) = km (X (t )),
(2)
such that the output of the closed-loop system consisting of the system (1) and the control (2) tracks a given constant reference signal Yr = (yr1 , . . . , yrp ) ∈ D p , that is, there exists an integer τ > 0 such that Y (t ; X (0), U ) = Yr holds for ∀ X (0) ∈ D n and ∀ t ≥ τ , where ki : D n → D , i = 1, . . . , m are logical functions to be determined. In the following, we convert the system (1) and the state feedback control (2) into equivalent algebraic forms, respectively. To this end, we first recall the definition and some properties of the semi-tensor product of matrices. Definition 1 (Cheng et al., 2011). The semi-tensor product of two matrices A ∈ Rm×n and B ∈ Rp×q is A n B = (A ⊗ I αn )(B ⊗ I αp ),
(3)
where α = lcm(n, p) is the least common multiple of n and p, and ⊗ is the Kronecker product. It is noted that when n = p, the semi-tensor product of A and B becomes the conventional matrix product. Thus, the semi-tensor product is a generalization of the conventional matrix product. We can simply call it ‘‘product’’ and omit the symbol ‘‘n’’ if no confusion arises in the following. Proposition 1 (Cheng et al., 2011). matrices has the following properties:
The semi-tensor product of
(i) Let A ∈ Rm×n , B ∈ Rp×q and C ∈ Rr ×s . Then (A n B) n C = A n (B n C ). (ii) Let X ∈ Rt ×1 be a column vector and A ∈ Rm×n . Then X n A = (It ⊗ A) n X .
(4)
By identifying 1 ∼ δ21 and 0 ∼ δ22 , we have ∆ ∼ D , where ‘‘∼’’ denotes two different expressions of the same thing. In most places of this work, we use δ21 and δ22 to express logical variables and call them the vector form of logical variables. The following lemma is fundamental for the algebraic expression of logical functions.
and Disjunction, respectively.
Lemma 1 (Cheng et al., 2011). Let f (x1 , x2 , . . . , xs ) : D s → D be a logical function. Then, there exists a unique matrix Mf ∈ L2×2s , called the structural matrix of f , such that
2. Problem formulation
f (x1 , x2 , . . . , xs ) = Mf nsi=1 xi ,
0n := 0 · · · 0 . ‘‘¬’’, ‘‘∧’’ and ‘‘∨’’ denote Negation, Conjunction 0 n
where xi ∈ ∆ and
Consider the following Boolean control network:
x1 (t + 1) = f1 (X (t ), U (t )), x2 (t + 1) = f2 (X (t ), U (t )), .. . x n (t + 1) = fn (X (t ), U (t )); yj (t ) = hj (X (t )), j = 1, . . . , p,
(1)
nsi=1
(5)
xi = x1 n · · · n xs .
Using the vector form of logical variables and setting x(t ) = nni=1 p m p xi (t ) ∈ ∆2n , u(t ) = nm i=1 ui (t ) ∈ ∆2 and y(t ) = ni=1 yi (t ) ∈ ∆2 , by Lemma 1, one can convert (1) and (2) into
x(t + 1) = Lu(t )x(t ), y(t ) = Hx(t ),
(6)
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H. Li et al. / Automatica 59 (2015) 54–59
and u(t ) = Kx(t ),
(7)
respectively, where L ∈ L2n ×2m+n , H ∈ L2p ×2n and K ∈ L2m ×2n . p Moreover, the reference signal becomes yr = ni=1 yri = δ2αp , where r α is uniquely determined by yi , i = 1, . . . , p. Thus, the output tracking control problem becomes how to design the state feedback gain matrix K ∈ L2m ×2n . In the next section, we will study the designing of K .
Thus, x(kj ; x(0), u) ∈ S , ∀ 1 ≤ j ≤ 2n . Since S ⊆ R1 (S ), one can see that x(t ; x(0), u) ∈ S ,
∀ t ≥ τ , ∀ x(0) ∈ ∆2n ,
which implies that y(t ; x(0), u) = Hx(t ; x(0), u) = yr holds for ∀ t ≥ τ and ∀ x(0) ∈ ∆2n . Therefore, the output of the system (1) tracks yr by the state feedback control
3. Main results
u(t ) = δ2m [p1 p2 · · · p2n ]x(t ).
In this section, we establish a constructive procedure for the design of state feedback based output tracking controllers for BCNs. Consider the system (6) with L = δ2n [i1 i2 · · · i2m+n ] and the reference signal yr = δ2αp . We define a set, denoted by O (α), as
(Necessity) Suppose that the output of the system (1) tracks yr = δ2αp by a state feedback control, say, u(t ) = Kx(t ), K ∈ L2m ×2n . Then, the closed-loop system consisting of the system (6) and the control u(t ) = Kx(t ) becomes
O (α) = {r ∈ N : Colr (H ) = δ2αp , 1 ≤ r ≤ 2n }.
(8)
Note that the set {δ : r ∈ O (α)} contains all the states of the system (6) whose outputs form the vector yr . We presuppose O (α) ̸= ∅ in the following. Otherwise, if O (α) = ∅, the output tracking control problem is not solvable. For S ⊆ ∆2n , S ̸= ∅ and k = 1, 2, . . . , denote by Rk (S ) the set of all states controllable at S in k steps, that is, r 2n
Rk (S ) = x(0) ∈ ∆2n : there exists {u(t ) ∈ ∆2m :
t = 0, . . . , k − 1} such that x(k; x(0), u) ∈ S .
(9)
Then, we have the following result. Theorem 1. The output of the system (1) tracks the reference signal yr = δ2αp by a state feedback control, if and only if there exist a nonempty set S ⊆ {δ2r n : r ∈ O (α)} and an integer 1 ≤ τ ≤ 2n such that
S ⊆ R1 (S ), Rτ (S ) = ∆2n .
(10)
Proof (Sufficiency). Assuming that (10) holds, we prove that the output of the system (1) tracks yr by a constructed state feedback control. Set R◦k (S ) = Rk (S ) \ Rk−1 (S ),
k = 1, . . . , τ ,
(11)
where R0 (S ) := ∅. Then, it is easy to see that R◦k1 (S ) R◦k2 (S ) = τ
∅, ∀ k1 , k2 ∈ {1, . . . , τ }, k1 ̸= k2 , and k=1 Rk (S ) = ∆2n . Thus, for any integer 1 ≤ j ≤ 2n , there exists a unique integer 1 ≤ kj ≤ τ j such that δ2n ∈ R◦kj (S ). For the integer 1 ≤ j ≤ 2n with kj = 1, there exists an integer 1 ≤ pj ≤ 2m such that the integer l := (pj − 1)2n + j satisfies i 1 ≤ l ≤ 2m+n and δ2ln ∈ S. Similarly, for the integer 1 ≤ j ≤ 2n with 2 ≤ kj ≤ τ , there exists an integer 1 ≤ pj ≤ 2m such that the i integer l := (pj − 1)2n + j satisfies 1 ≤ l ≤ 2m+n and δ2ln ∈ Rkj −1 (S ). Now, we set K = δ2m [p1 p2 · · · p2n ] ∈ L2m ×2n . Then, under the control u(t ) = Kx(t ), along the trajectory of the system (6) starting j from any initial state x(0) = δ2n ∈ ∆2n , we have ◦
x(t + 1) = Lx(t ), y(t ) = Hx(t ),
(12) n
where L = LK Φn , and Φn = Diag{δ21n , δ22n , . . . , δ22n } is the powerreducing matrix satisfying x n x = Φn n x, ∀ x ∈ ∆2n . Denote the state trajectory of the system (12) starting from an initial state x(0) ∈ ∆2n by x(t ; x(0)), and the output trajectory of the system (12) starting from x(0) ∈ ∆2n by y(t ; x(0)). For the Boolean network (12), denote the set of states in the limit set (all the fixed points and cycles) by S. In addition, let Tt be the transient period2 of the system (12). Then, it is easy to see that (10) holds for S and τ = Tt ≤ 2n . Now, we prove that S ⊆ {δ2r n : r ∈ O (α)}. In fact, if S ̸⊆ {δ2r n : r ∈ O (α)}, then there exists δ2i n ∈ S with i ̸∈ O (α). Since δ2i n is a state located in some fixed point or cycle of the system (12), there exists a positive integer T such that δ2i n = x(nT ; δ2i n ) holds for all n ∈ N. Thus, y(nT ; δ2i n ) = Hx(nT ; δ2i n ) ̸= yr , ∀ n ∈ N, which is a contradiction to the fact that the output of the system (1) tracks yr = δ2αp by u(t ) = Kx(t ). Therefore, S ⊆ {δ2r n : r ∈ O (α)}. This completes the proof. From the proof of Theorem 1, we can design a state feedback based output tracking controller for the system (1) as follows. Corollary 1. Consider the system (6) with L = δ2n [i1 i2 · · · i2m+n ]. Suppose that there exist a non-empty set S ⊆ {δ2r n : r ∈ O (α)} and an integer 1 ≤ τ ≤ 2n such that (10) holds. For each integer 1 ≤ j ≤ 2n which corresponds to a unique integer 1 ≤ kj ≤ τ such j that δ2n ∈ R◦kj (S ), let 1 ≤ pj ≤ 2m be such that 1 ≤ l ≤ 2m+n and
i
δ2ln ∈ S , il 2n
δ
∈ Rkj −1 (S ),
for kj = 1, for 2 ≤ kj ≤ τ ,
(13)
where l = (pj − 1)2n + j. Then, the state feedback based output tracking control can be designed as u(t ) = Kx(t ) with K = δ2m [p1 p2 · · · p2n ].
(14)
Remark 1. Theorem 1 and Corollary 1 provide a constructive procedure to design state feedback based output tracking controllers for the system (1). This procedure contains the following steps: (1) Calculate O (α) according to (8).
i
x(1; x(0), u) = LKx(0)x(0) = δ2ln S, Rkj −1 (S ),
∈
if kj = 1, if 2 ≤ kj ≤ τ .
2 The transient period of the Boolean network (12) is the minimum number of transient steps that lead any initial state x(0) ∈ ∆2n to the limit set, Ω , which consists of all fixed points and cycles.
H. Li et al. / Automatica 59 (2015) 54–59
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(2) Find a non-empty set S ⊆ {δ2r n : r ∈ O (α)} and an integer 1 ≤ τ ≤ 2n such that (10) holds. (3) Calculate Rk (S ) and R◦k (S ), k = 1, . . . , τ according to (9) and (11), respectively. (4) Calculate pj , j = 1, 2, . . . , 2n such that (13) holds. (5) The state feedback gain matrix can be designed as K = δ2m [p1 p2 · · · p2n ]. One can easily see that the key to Theorem 1 and Corollary 1 is how to find a non-empty set S ⊆ {δ2r n : r ∈ O (α)} and an integer 1 ≤ τ ≤ 2n such that (10) holds. From the proof of Theorem 1, one should calculate a series of sets Rk (S ), k = 1, . . . , τ to verify (10). However, the calculations of Rk (S ), k = 1, . . . , τ are not computationally tractable because one needs to find a control sequence {u(0), . . . , u(k − 1)} for each x(0) ∈ Rk (S ). In the following, based on the input-state incidence matrix proposed by Zhao et al. (2010), we give a computationally tractable result on the verification of (10). To aid the readers, we first recall some useful results on the input-state incidence matrix of the system (1). For details, please refer to Zhao et al. (2010). Consider the algebraic form (6) with
Fig. 1. The network graph of the system (18).
is reachable from x(0) = δ2kn at the τ th step. Hence, ∆2n ⊆ Rτ (S ). This together with Rτ (S ) ⊆ ∆2n shows that Rτ (S ) = ∆2n .
(Sufficiency) If MSτ has zero column, say, Colk (MSτ ) = 0Tv . Then, j
any x(τ ) = δ2ln ∈ S , l = 1, . . . , v is not reachable from x(0) = δ2kn ∈ ∆2n at the τ th step, which is a contradiction to δ2kn ∈ Rτ (S ). Therefore, all the columns of MSτ are nonzero.
L = δ2n [i1 i2 · · · i2m+n ] ∈ L2n ×2m+n .
Based on Lemma 2, we have the following result.
Split L into 2m equal blocks as [L1 L2 · · · L2m ], where Li ∈ L2n ×2n . Set
Theorem 2. The set S = {δ21n , . . . , δ2vn } ⊆ {δ2r n : r ∈ O (α)} with 1 ≤ j1 < · · · < jv ≤ 2n satisfies (10), if and only if there exists an integer 1 ≤ τ ≤ 2n such that all the columns of both MS and MSτ are nonzero.
M =
2m
Li .
(15)
i=1 p
q
Then, x(s) = δ2n is reachable from x(0) = δ2n at the sth step, if and only if (M s )p,q > 0, where (M s )p,q denotes the (p, q)th element of the matrix M s . j j For a positive integer τ and a set S = {δ21n , . . . , δ2vn } with n 1 ≤ j1 < · · · < jv ≤ 2 , we have the following result. Lemma 2. (1) S ⊆ R1 (S ) if and only if all the columns of
Mj1 ,j1
··· .. .
Mj1 ,jv
Mjv ,j1
···
Mjv ,jv
. MS := ..
j
j
Remark 2. From Theorem 2, one only needs to calculate a set of matrices MSk , k = 1, . . . , τ to find S. Thus, compared with the calculations of Rk (S ), k = 1, . . . , τ , Theorem 2 is more computationally tractable. 4. An illustrative example In this section, we give an illustrative example to show how to use the results obtained in this paper to design state feedback based output tracking controllers for BCNs.
.. .
(16) Example 1. Consider the following Boolean control network (Li et al., 2013), which is a reduced model for the lac operon in the bacterium Escherichia coli:
are nonzero;
(2) Rτ (S ) = ∆2n if and only if all the columns of Rowj1 (M τ ) .. MSτ := .
x1 (t + 1) = ¬u1 (t ) ∧ (x2 (t ) ∨ x3 (t )), x2 (t + 1) = ¬u1 (t ) ∧ u2 (t ) ∧ x1 (t ), x3 (t + 1) = ¬u1 (t ) ∧ (u2 (t ) ∨ (u3 (t ) ∧ x1 (t ))),
(17)
τ
Rowjv (M )
are nonzero. Proof. Firstly, we prove Conclusion (1). (Necessity) Suppose that all the columns of MS are nonzero. For any k = 1, . . . , v , since Colk (MS ) ̸= 0Tv , there exists an integer j
1 ≤ l ≤ v such that Mjl ,jk > 0. Therefore, x(1) = δ2ln ∈ S is jk 2n
reachable from x(0) = δ ∈ S in one step. From the arbitrariness of k, we know that S ⊆ R1 (S ). (Sufficiency) Assume S ⊆ R1 (S ). If MS has zero column, say, j Colk (MS ) = 0Tv , then any x(1) = δ2ln ∈ S , l = 1, . . . , v is not j
reachable from x(0) = δ2kn ∈ S in one step. This is a contradiction j
to δ2kn ∈ R1 (S ). Thus, all the columns of MS are nonzero. Secondly, let us prove Conclusion (2). (Necessity) Assume that all the columns of MSτ are nonzero. For any k = 1, . . . , 2n , since Colk (MSτ ) ̸= 0Tv , there exists an integer 1 ≤ l ≤ v such that (M τ )jl ,k > 0, which implies that x(τ ) = δ2ln ∈ S j
(18)
where x1 , x2 and x3 are state variables which denote the lac mRNA, the lactose in high concentrations, and the lactose in medium concentrations, respectively; u1 , u2 and u3 are control inputs which represent the extracellular glucose, the high extracellular lactose, and the medium extracellular lactose, respectively. The network graph of the system (18) is shown in Fig. 1. In this example, we are interested in observing the genes x1 and x2 of the system (18). Then, the output equation is
y1 (t ) = x1 (t ), y2 (t ) = x2 (t ).
(19)
Our objective is to design state feedback controllers such that the output of the system (18) tracks the reference signal Yr = (1, 0). Using the vector form of logical variables and setting x(t ) = n3i=1 xi (t ), u(t ) = n3i=1 ui (t ) and y(t ) = n2i=1 yi (t ), by the semi-tensor product of matrices, we have the following algebraic form:
x(t + 1) = Lu(t )x(t ), y(t ) = Hx(t ),
(20)
58
H. Li et al. / Automatica 59 (2015) 54–59
where L = δ8 [8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8888888888888888 1115333711153337 3 3 3 7 4 4 4 8 4 4 4 8 4 4 4 8], and H = δ4 [1 1 2 2 3 3 4 4]. Moreover, the reference signal becomes yr = δ21 n δ22 = δ42 . A simple calculation gives O (2) = {3, 4}. Setting S = {δ83 } ⊆ 3 {δ8 , δ84 }, one can easily obtain that R1 (S ) = {δ81 , δ82 , δ83 , δ85 , δ86 , δ87 } and R2 (S ) = ∆8 . Thus, (10) holds for S = {δ83 } and τ = 2. A straightforward calculation shows that p1 = p2 = p3 = 7, p4 = 5 or 6 or 7, and pj = 5 or 6, ∀ j = 5, . . . , 8. Hence, we obtain 48 state feedback gain matrices as K = δ8 [7 7 7 p4 p5 p6 p7 p8 ], where p4 ∈ {5, 6, 7} and pj ∈ {5, 6}, j = 5, . . . , 8. For example, we set pj = 5, j = 4, . . . , 8. Then, K = δ8 [7 7 7 5 5 5 5 5], which corresponds to the following state feedback based output tracking controller: u1 (t ) = 0, u2 (t ) = ¬x1 (t ) ∨ ¬(x2 (t ) ∨ x3 (t )), u3 (t ) = 1.
(21)
5. Conclusion In this paper, we have investigated the output tracking control of BCNs with a constant reference signal by using the semi-tensor product method, and presented a number of new results. We have proposed a constructive procedure to design state feedback based output tracking control laws for BCNs. The key idea of this procedure is to stabilize the BCN to a set of states whose outputs are the given constant reference signal. Based on the input-state incidence matrix, we have presented a computationally tractable result to find a proper set of states whose outputs are the given constant reference signal. The study of an illustrative example has shown that the new results obtained in this paper are effective in designing state feedback based output tracking controllers for BCNs. Acknowledgments The authors would like to thank the Associate Editor and the reviewers for their constructive comments and suggestions which improved the quality of the paper. References Akutsu, T., Hayashida, M., Ching, W., & Ng, M. (2007). Control of Boolean networks: Hardness results and algorithms for tree structured networks. Journal of Theoretical Biology, 244(4), 670–679. Ay, F., Xu, F., & Kahveci, T. (2009). Scalable steady state analysis of Boolean biological regulatory networks. PLoS ONE, 4(12), e7992. Chaves, M. (2009). Methods for qualitative analysis of genetic networks. In Proc. 10th European control conference (pp. 671–676). Chen, H., & Sun, J. (2014). Output controllability and optimal output control of statedependent switched Boolean control networks. Automatica, 50(7), 1929–1934. Cheng, D. (2011). Disturbance decoupling of Boolean control networks. Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, 56(1), 2–10. Cheng, D., Li, Z., & Qi, H. (2010). Realization of Boolean control networks. Automatica, 46(1), 62–69. Cheng, D., & Qi, H. (2009). Controllability and observability of Boolean control networks. Automatica, 45(7), 1659–1667. Cheng, D., & Qi, H. (2010). A linear representation of dynamics of Boolean networks. Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, 55(10), 2251–2258. Cheng, D., Qi, H., & Li, Z. (2011). Analysis and control of Boolean networks: a semitensor product approach. London: Springer. Cheng, D., Qi, H., Li, Z., & Liu, J. B. (2011). Stability and stabilization of Boolean networks. International Journal of Robust and Nonlinear Control, 21(2), 134–156. Cheng, D., & Xu, X. (2013). Bi-decomposition of multi-valued logical functions and its applications. Automatica, 49(7), 1979–1985.
Cheng, D., & Zhao, Y. (2011). Identification of Boolean control networks. Automatica, 47(4), 702–710. Choudhary, A., Datta, A., Bittner, M. L., & Dougherty, E. R. (2006). Intervention in a family of Boolean networks. Bioinformatics, 22(2), 226–232. Drossel, B., Mihaljev, T., & Greil, F. (2005). Number and length of attractors in a critical Kauffman model with connectivity one. Physical Review Letters, 94(8), 088701-(1–4). Feng, J., Yao, J., & Cui, P. (2012). Singular Boolean networks: Semi-tensor product approach. Science China. Information Sciences, http://dx.doi.org/10.1007/s11432012-4666-8. Fornasini, E., & Valcher, M. E. (2013a). Observability, reconstructibility and state observers of Boolean control networks. Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, 58(6), 1390–1401. Fornasini, E., & Valcher, M. E. (2013b). On the periodic trajectories of Boolean control networks. Automatica, 49(5), 1506–1509. Fornasini, E., & Valcher, M. E. (2014). Optimal control of Boolean control networks. Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, 59(5), 1258–1270. Kauffman, S. (1969). Metabolic stability and epigenesis in randomly constructed genetic nets. Journal of Theoretical Biology, 22(3), 437. Laschov, D., & Margaliot, M. (2011). A maximum principle for single-input Boolean control networks. Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, 56(4), 913–917. Laschov, D., & Margaliot, M. (2012). Controllability of Boolean control networks via the Perron–Frobenius theory. Automatica, 48(6), 1218–1223. Li, R., & Chu, T. (2012). Complete synchronization of Boolean networks. IEEE Transactions on Neural Networks and Learning Systems, 23(5), 840–846. Li, F., & Sun, J. (2011a). Controllability of Boolean control networks with time delays in states. Automatica, 47(3), 603–607. Li, F., & Sun, J. (2011b). Controllability of probabilistic Boolean control networks. Automatica, 47(12), 2765–2771. Li, H., & Wang, Y. (2012). On reachability and controllability of switched Boolean control networks. Automatica, 48(11), 2917–2922. Li, H., & Wang, Y. (2013). Output feedback stabilization control design for Boolean control networks. Automatica, 49(12), 3641–3645. Li, R., Yang, M., & Chu, T. (2013). State feedback stabilization for Boolean control networks. Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, 58(7), 1853–1857. Wang, Y., Zhang, C., & Liu, Z. (2012). A matrix approach to graph maximum stable set and coloring problems with application to multi-agent systems. Automatica, 48(7), 1227–1236. Xiao, Y. (2009). A tutorial on analysis and simulation of Boolean gene regulatory network models. Current Genomics, 10, 511–525. Xiao, Y., & Dougherty, E. R. (2007). The impact of function perturbations in Boolean networks. Bioinformatics, 23(10), 1265–1273. Xu, X., & Hong, Y. (2013). Matrix approach to model matching of asynchronous sequential machines. Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, 58(11), 2974–2979. Yang, M., Li, R., & Chu, T. (2013). Controller design for disturbance decoupling of Boolean control networks. Automatica, 49(1), 273–277. Zhang, X. (2012). Semitensor product based adaptive control for attitude tracking of spacecraft with unknown external disturbances. Journal of Control Theory and Applications, 10(3), 292–296. Zhang, L., & Feng, J. (2013). Mix-valued logic-based formation control. International Journal of Control, 86(6), 1191–1199. Zhang, L., & Zhang, K. (2013). Controllability and observability of Boolean control networks with time-variant delays in states. IEEE Transactions on Neural Networks and Learning Systems, 24, 1478–1484. Zhao, Y., Cheng, D., & Qi, H. (2010). Input-state incidence matrix of Boolean control networks and its applications. Systems & Control Letters, 59(12), 767–774. Zhao, Y., Kim, J., & Filippone, M. (2013). Aggregation algorithm towards largescale Boolean network analysis. Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, 58(8), 1976–1985. Zhao, Y., Li, Z., & Cheng, D. (2011). Optimal control of logical control networks. Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, 56(8), 1766–1776. Zou, Y., & Zhu, J. (2014). System decomposition with respect to inputs for Boolean control networks. Automatica, 50(4), 1304–1309.
Haitao Li received the B.S. and M.S. degrees from the School of Mathematical Science, Shandong Normal University in 2007 and 2010, respectively, and the Ph.D. degree at the School of Control Science and Engineering, Shandong University in 2014. Since 2015, he has been with the School of Mathematical Science, Shandong Normal University, China, where he is currently an associate professor. From Jan. 2014 to Jan. 2015, he worked as a Research Fellow in Nanyang Technological University, Singapore. His research interests include logical dynamic systems, switched systems, etc. He received the ‘‘Best Student Paper Award’’ at the 10th World Congress on Intelligent Control and Automation, and the ‘‘Guan Zhao-Zhi Award’’ at the 31st Chinese Control Conference.
H. Li et al. / Automatica 59 (2015) 54–59 Yuzhen Wang graduated from Tai’an Teachers College in 1986, received his M.S. degree from Shandong University of Science & Technology in 1995 and his Ph.D. degree from the Institute of Systems Science, Chinese Academy of Sciences in 2001. Since 2003, he is a professor with the School of Control Science and Engineering, Shandong University, China, and now the Dean of the School of Control Science and Engineering, Shandong University. From 2001 to 2003, he worked as a Postdoctoral Fellow in Tsinghua University, Beijing, China. From Mar. 2004 to Jun. 2004, from Feb. 2006 to May 2006 and from Nov. 2008 to Jan. 2009, he visited City University of Hong Kong as a Research Fellow. From Sept. 2004 to May 2005, he worked as a visit Research Fellow at the National University of Singapore. His research interests include nonlinear control systems, Hamiltonian systems and Boolean networks. Prof. Wang received the Prize of Guan Zhao-Zhi in 2002, the Prize of Huawei from the Chinese Academy of Sciences in 2001, the Prize of Natural Science from Chinese Education Ministry in 2005, and the National Prize of Natural Science of China in 2008. Currently, he is an associate editor of IMA Journal of Math Control and Inform., and a Technical Committee member of IFAC (TC2.3).
59 Lihua Xie received the B.E. and M.E. degrees in electrical engineering from Nanjing University of Science and Technology in 1983 and 1986, respectively, and the Ph.D. degree in electrical engineering from the University of Newcastle, Australia, in 1992. Since 1992, he has been with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, where he is currently a professor and served as the Head of Division of Control and Instrumentation from 2011 to 2014. He held teaching appointments in the Department of Automatic Control, Nanjing University of Science and Technology
from 1986 to 1989. Dr. Xie’s research interests include robust control and estimation, sensor networks, and networked control systems. In these areas, he has published many journal papers and co-authored two patents and six books. He is an Editor-in-Chief of Unmanned Systems and has served as an editor of IET Book Series in Control and an Associate Editor of a number of journals including IEEE Transactions on Automatic Control, Automatica, IEEE Transactions on Control System Technology, and IEEE Transactions on Circuits and Systems-II. Dr. Xie is a Fellow of IEEE, a Fellow of IFAC and an IEEE Distinguished Lecturer.
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Brief paper
Output tracking control of Boolean control networks via state feedback: Constant reference signal case✩ Haitao Li a,b , Yuzhen Wang b,1 , Lihua Xie c a
School of Mathematical Science, Shandong Normal University, Jinan 250014, PR China
b
School of Control Science and Engineering, Shandong University, Jinan 250061, PR China
c
EXQUISITUS, Center for E-City, School of Electrical and Electronic Engineering, Nanyang Technological University, 639798, Singapore
article
info
Article history: Received 28 March 2014 Received in revised form 15 May 2015 Accepted 21 May 2015 Available online 17 June 2015 Keywords: Boolean control network Output tracking control State feedback Semi-tensor product of matrices
abstract This paper investigates the state feedback based output tracking control of Boolean control networks (BCNs) with a constant reference signal by using the semi-tensor product method. Based on the algebraic expression of BCNs and by constructing a series of reachable sets, a general procedure is proposed for the design of the state feedback laws for BCNs to track a constant reference signal. The study of an illustrative example shows that the obtained new results are effective in designing state feedback based output tracking controllers for BCNs to track a constant reference signal. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction Since Kauffman’s pioneering work (Kauffman, 1969) in the Boolean model of gene regulatory networks, the study of Boolean networks has attracted great attention of biologists, physicists and systems scientists. Consequently, many excellent results have been established for Boolean networks (Akutsu, Hayashida, Ching, & Ng, 2007; Ay, Xu, & Kahveci, 2009; Chaves, 2009; Drossel, Mihaljev, & Greil, 2005; Xiao, 2009; Xiao & Dougherty, 2007). In a Boolean network, each gene can take two possible values, 1 and 0, and its value (1 or 0) indicates its measured abundance (expressed or unexpressed; high or low). From a graphical perspective, genes in a Boolean network are nodes in this network and edges describe regulatory relationships between genes. As is well known, the ultimate goal of modeling gene regulatory networks as Boolean networks is to design effective therapeutic intervention strategies to influence the network dynamics to avoid
✩ The research was supported by the National Research Foundation of Singapore under grant NRF-CRP8-2011-03, the National Natural Science Foundation of China under grants 61374065 and 61320106011, and the Research Fund for the Taishan Scholar Project of Shandong Province. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Yoshito Ohta under the direction of Editor Richard Middleton. E-mail addresses: [email protected] (H. Li), [email protected] (Y. Wang), [email protected] (L. Xie). 1 Tel.: +86 531 88392515; fax: +86 531 88392205.
http://dx.doi.org/10.1016/j.automatica.2015.06.004 0005-1098/© 2015 Elsevier Ltd. All rights reserved.
undesirable cellular states. Hence, in order to manipulate Boolean networks, binary control inputs and outputs are added to the network dynamics, which yields Boolean control networks (BCNs). The control of BCNs is a fundamental issue in both systems biology and control theory. However, due to the lack of effective tools to deal with logical dynamics, the control of BCNs has been a challenging problem for a long time until the introduction of the semitensor product method (Cheng, Qi, & Li, 2011). The feature of this method is that one can convert the dynamics of a Boolean (control) network into a linear (bilinear) discrete-time system. Then, one can analyze Boolean (control) networks by using the classical control theory. Up to now, there have been many interesting works on the control of BCNs via this novel method, which include controllability and observability (Chen & Sun, 2014; Cheng, Li, & Qi, 2010; Cheng & Qi, 2009, 2010; Feng, Yao, & Cui, 2012; Fornasini & Valcher, 2013a; Laschov & Margaliot, 2012; Li & Sun, 2011a,b; Li & Wang, 2012; Zhang & Zhang, 2013; Zhao, Cheng, & Qi, 2010), disturbance decoupling (Cheng, 2011; Yang, Li, & Chu, 2013), optimal control (Fornasini & Valcher, 2014; Laschov & Margaliot, 2011; Zhao, Li, & Cheng, 2011), stability and stabilization (Cheng, Qi, Li, & Liu, 2011; Li & Wang, 2013; Li, Yang, & Chu, 2013), and other control problems (Cheng & Xu, 2013; Cheng & Zhao, 2011; Fornasini & Valcher, 2013b; Li & Chu, 2012; Wang, Zhang, & Liu, 2012; Xu & Hong, 2013; Zhang, 2012; Zhang & Feng, 2013; Zhao, Kim, & Filippone, 2013; Zou & Zhu, 2014). It is noted that in many practical gene regulatory networks, the state variables cannot be measured directly due to the limitation
H. Li et al. / Automatica 59 (2015) 54–59
of measurement conditions and the impact of immeasurable variables. In this case, one can use the measured outputs to track a desirable reference signal which corresponds to some desirable states. Thus, it is meaningful for us to design suitable controllers (therapeutic intervention) that steer the output of BCNs to an expected reference signal, called the output tracking control problem of BCNs in this paper. It should be pointed out that several types of therapeutic intervention have been proposed for Boolean networks until now. Among them, the theory of automatic control, such as the controllability and the optimal control, is an important type of therapeutic intervention, and thereby many suitable control strategies have been obtained based on the classical control theory (Akutsu et al., 2007; Choudhary, Datta, Bittner, & Dougherty, 2006). As one of the most important issues in the control theory, it is believed that the output tracking control can provide an effective way for the design of therapeutic intervention. However, to our best knowledge, there are no results available on the output tracking control of BCNs. In this paper, using the semi-tensor product method, we investigate how to design output tracking controllers for BCNs to track a constant reference signal. We propose a general procedure to design state feedback based output tracking controllers for BCNs. The key idea of this procedure is to stabilize the BCN to a set of states whose outputs are the given constant reference signal. Although this procedure is a generalization of the state feedback stabilization control design method proposed in Fornasini and Valcher (2013b) and Li et al. (2013), it has the following two differences/novelties: (1) The procedure proposed in this paper has wider applications (such as the output tracking control, and the stabilization to a stable region) because it can stabilize the BCN to a set of states. When the set of states only contains an element, the procedure degenerates to that of Fornasini and Valcher (2013b) and Li et al. (2013). (2) In our procedure, the set of states that is stabilized to is not fixed, while the equilibrium in Fornasini and Valcher (2013b) and Li et al. (2013) is fixed. How to determine a proper set of states whose outputs are the given constant reference signal is a very challenging problem in our procedure. In this paper, we solve this problem by using the input-state incidence matrix introduced in Zhao et al. (2010) (please see Theorem 2). The rest of this paper is organized as follows. Section 2 formulates the output tracking control problem studied in this work. Section 3 investigates the output tracking control of BCNs via state feedback and presents the main results of this paper. An illustrative example is given to support our new results in Section 4, which is followed by a brief conclusion in Section 5. Notation: The notation of this paper is fairly standard. D := {1, 0}. ∆n := {δnk : k = 1, . . . , n}, where δnk denotes the kth column of the identity matrix In . For compactness, ∆ := ∆2 . An n × t i i matrix M is called a logical matrix, if M = [δn1 δn2 · · · δnit ], and we express M briefly as M = δn [i1 i2 · · · it ], and denote the set of n × t logical matrices by Ln×t . Coli (A) denotes the ith column of the matrix A, and Rowi (A) stands for the ith row of the matrix A.
55
where X (t ) = (x1 (t ), x2 (t ), . . . , xn (t )) ∈ D n , U (t ) = (u1 (t ), . . . , um (t )) ∈ D m and Y (t ) = (y1 (t ), . . . , yp (t )) ∈ D p are the state, the control input and the output of the system (1), respectively, and fi : D m+n → D , i = 1, . . . , n and hj : D n → D , j = 1, . . . , p are logical functions. Given a control sequence {U (t ) : t ∈ N}, denote the state trajectory of the system (1) starting from an initial state X (0) ∈ D n by X (t ; X (0), U ), and the output trajectory of the system (1) starting from X (0) ∈ D n by Y (t ; X (0), U ). The output tracking control problem studied in this paper is to design a state feedback control in the form of
u1 (t ) = k1 (X (t )), .. . um (t ) = km (X (t )),
(2)
such that the output of the closed-loop system consisting of the system (1) and the control (2) tracks a given constant reference signal Yr = (yr1 , . . . , yrp ) ∈ D p , that is, there exists an integer τ > 0 such that Y (t ; X (0), U ) = Yr holds for ∀ X (0) ∈ D n and ∀ t ≥ τ , where ki : D n → D , i = 1, . . . , m are logical functions to be determined. In the following, we convert the system (1) and the state feedback control (2) into equivalent algebraic forms, respectively. To this end, we first recall the definition and some properties of the semi-tensor product of matrices. Definition 1 (Cheng et al., 2011). The semi-tensor product of two matrices A ∈ Rm×n and B ∈ Rp×q is A n B = (A ⊗ I αn )(B ⊗ I αp ),
(3)
where α = lcm(n, p) is the least common multiple of n and p, and ⊗ is the Kronecker product. It is noted that when n = p, the semi-tensor product of A and B becomes the conventional matrix product. Thus, the semi-tensor product is a generalization of the conventional matrix product. We can simply call it ‘‘product’’ and omit the symbol ‘‘n’’ if no confusion arises in the following. Proposition 1 (Cheng et al., 2011). matrices has the following properties:
The semi-tensor product of
(i) Let A ∈ Rm×n , B ∈ Rp×q and C ∈ Rr ×s . Then (A n B) n C = A n (B n C ). (ii) Let X ∈ Rt ×1 be a column vector and A ∈ Rm×n . Then X n A = (It ⊗ A) n X .
(4)
By identifying 1 ∼ δ21 and 0 ∼ δ22 , we have ∆ ∼ D , where ‘‘∼’’ denotes two different expressions of the same thing. In most places of this work, we use δ21 and δ22 to express logical variables and call them the vector form of logical variables. The following lemma is fundamental for the algebraic expression of logical functions.
and Disjunction, respectively.
Lemma 1 (Cheng et al., 2011). Let f (x1 , x2 , . . . , xs ) : D s → D be a logical function. Then, there exists a unique matrix Mf ∈ L2×2s , called the structural matrix of f , such that
2. Problem formulation
f (x1 , x2 , . . . , xs ) = Mf nsi=1 xi ,
0n := 0 · · · 0 . ‘‘¬’’, ‘‘∧’’ and ‘‘∨’’ denote Negation, Conjunction 0 n
where xi ∈ ∆ and
Consider the following Boolean control network:
x1 (t + 1) = f1 (X (t ), U (t )), x2 (t + 1) = f2 (X (t ), U (t )), .. . x n (t + 1) = fn (X (t ), U (t )); yj (t ) = hj (X (t )), j = 1, . . . , p,
(1)
nsi=1
(5)
xi = x1 n · · · n xs .
Using the vector form of logical variables and setting x(t ) = nni=1 p m p xi (t ) ∈ ∆2n , u(t ) = nm i=1 ui (t ) ∈ ∆2 and y(t ) = ni=1 yi (t ) ∈ ∆2 , by Lemma 1, one can convert (1) and (2) into
x(t + 1) = Lu(t )x(t ), y(t ) = Hx(t ),
(6)
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H. Li et al. / Automatica 59 (2015) 54–59
and u(t ) = Kx(t ),
(7)
respectively, where L ∈ L2n ×2m+n , H ∈ L2p ×2n and K ∈ L2m ×2n . p Moreover, the reference signal becomes yr = ni=1 yri = δ2αp , where r α is uniquely determined by yi , i = 1, . . . , p. Thus, the output tracking control problem becomes how to design the state feedback gain matrix K ∈ L2m ×2n . In the next section, we will study the designing of K .
Thus, x(kj ; x(0), u) ∈ S , ∀ 1 ≤ j ≤ 2n . Since S ⊆ R1 (S ), one can see that x(t ; x(0), u) ∈ S ,
∀ t ≥ τ , ∀ x(0) ∈ ∆2n ,
which implies that y(t ; x(0), u) = Hx(t ; x(0), u) = yr holds for ∀ t ≥ τ and ∀ x(0) ∈ ∆2n . Therefore, the output of the system (1) tracks yr by the state feedback control
3. Main results
u(t ) = δ2m [p1 p2 · · · p2n ]x(t ).
In this section, we establish a constructive procedure for the design of state feedback based output tracking controllers for BCNs. Consider the system (6) with L = δ2n [i1 i2 · · · i2m+n ] and the reference signal yr = δ2αp . We define a set, denoted by O (α), as
(Necessity) Suppose that the output of the system (1) tracks yr = δ2αp by a state feedback control, say, u(t ) = Kx(t ), K ∈ L2m ×2n . Then, the closed-loop system consisting of the system (6) and the control u(t ) = Kx(t ) becomes
O (α) = {r ∈ N : Colr (H ) = δ2αp , 1 ≤ r ≤ 2n }.
(8)
Note that the set {δ : r ∈ O (α)} contains all the states of the system (6) whose outputs form the vector yr . We presuppose O (α) ̸= ∅ in the following. Otherwise, if O (α) = ∅, the output tracking control problem is not solvable. For S ⊆ ∆2n , S ̸= ∅ and k = 1, 2, . . . , denote by Rk (S ) the set of all states controllable at S in k steps, that is, r 2n
Rk (S ) = x(0) ∈ ∆2n : there exists {u(t ) ∈ ∆2m :
t = 0, . . . , k − 1} such that x(k; x(0), u) ∈ S .
(9)
Then, we have the following result. Theorem 1. The output of the system (1) tracks the reference signal yr = δ2αp by a state feedback control, if and only if there exist a nonempty set S ⊆ {δ2r n : r ∈ O (α)} and an integer 1 ≤ τ ≤ 2n such that
S ⊆ R1 (S ), Rτ (S ) = ∆2n .
(10)
Proof (Sufficiency). Assuming that (10) holds, we prove that the output of the system (1) tracks yr by a constructed state feedback control. Set R◦k (S ) = Rk (S ) \ Rk−1 (S ),
k = 1, . . . , τ ,
(11)
where R0 (S ) := ∅. Then, it is easy to see that R◦k1 (S ) R◦k2 (S ) = τ
∅, ∀ k1 , k2 ∈ {1, . . . , τ }, k1 ̸= k2 , and k=1 Rk (S ) = ∆2n . Thus, for any integer 1 ≤ j ≤ 2n , there exists a unique integer 1 ≤ kj ≤ τ j such that δ2n ∈ R◦kj (S ). For the integer 1 ≤ j ≤ 2n with kj = 1, there exists an integer 1 ≤ pj ≤ 2m such that the integer l := (pj − 1)2n + j satisfies i 1 ≤ l ≤ 2m+n and δ2ln ∈ S. Similarly, for the integer 1 ≤ j ≤ 2n with 2 ≤ kj ≤ τ , there exists an integer 1 ≤ pj ≤ 2m such that the i integer l := (pj − 1)2n + j satisfies 1 ≤ l ≤ 2m+n and δ2ln ∈ Rkj −1 (S ). Now, we set K = δ2m [p1 p2 · · · p2n ] ∈ L2m ×2n . Then, under the control u(t ) = Kx(t ), along the trajectory of the system (6) starting j from any initial state x(0) = δ2n ∈ ∆2n , we have ◦
x(t + 1) = Lx(t ), y(t ) = Hx(t ),
(12) n
where L = LK Φn , and Φn = Diag{δ21n , δ22n , . . . , δ22n } is the powerreducing matrix satisfying x n x = Φn n x, ∀ x ∈ ∆2n . Denote the state trajectory of the system (12) starting from an initial state x(0) ∈ ∆2n by x(t ; x(0)), and the output trajectory of the system (12) starting from x(0) ∈ ∆2n by y(t ; x(0)). For the Boolean network (12), denote the set of states in the limit set (all the fixed points and cycles) by S. In addition, let Tt be the transient period2 of the system (12). Then, it is easy to see that (10) holds for S and τ = Tt ≤ 2n . Now, we prove that S ⊆ {δ2r n : r ∈ O (α)}. In fact, if S ̸⊆ {δ2r n : r ∈ O (α)}, then there exists δ2i n ∈ S with i ̸∈ O (α). Since δ2i n is a state located in some fixed point or cycle of the system (12), there exists a positive integer T such that δ2i n = x(nT ; δ2i n ) holds for all n ∈ N. Thus, y(nT ; δ2i n ) = Hx(nT ; δ2i n ) ̸= yr , ∀ n ∈ N, which is a contradiction to the fact that the output of the system (1) tracks yr = δ2αp by u(t ) = Kx(t ). Therefore, S ⊆ {δ2r n : r ∈ O (α)}. This completes the proof. From the proof of Theorem 1, we can design a state feedback based output tracking controller for the system (1) as follows. Corollary 1. Consider the system (6) with L = δ2n [i1 i2 · · · i2m+n ]. Suppose that there exist a non-empty set S ⊆ {δ2r n : r ∈ O (α)} and an integer 1 ≤ τ ≤ 2n such that (10) holds. For each integer 1 ≤ j ≤ 2n which corresponds to a unique integer 1 ≤ kj ≤ τ such j that δ2n ∈ R◦kj (S ), let 1 ≤ pj ≤ 2m be such that 1 ≤ l ≤ 2m+n and
i
δ2ln ∈ S , il 2n
δ
∈ Rkj −1 (S ),
for kj = 1, for 2 ≤ kj ≤ τ ,
(13)
where l = (pj − 1)2n + j. Then, the state feedback based output tracking control can be designed as u(t ) = Kx(t ) with K = δ2m [p1 p2 · · · p2n ].
(14)
Remark 1. Theorem 1 and Corollary 1 provide a constructive procedure to design state feedback based output tracking controllers for the system (1). This procedure contains the following steps: (1) Calculate O (α) according to (8).
i
x(1; x(0), u) = LKx(0)x(0) = δ2ln S, Rkj −1 (S ),
∈
if kj = 1, if 2 ≤ kj ≤ τ .
2 The transient period of the Boolean network (12) is the minimum number of transient steps that lead any initial state x(0) ∈ ∆2n to the limit set, Ω , which consists of all fixed points and cycles.
H. Li et al. / Automatica 59 (2015) 54–59
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(2) Find a non-empty set S ⊆ {δ2r n : r ∈ O (α)} and an integer 1 ≤ τ ≤ 2n such that (10) holds. (3) Calculate Rk (S ) and R◦k (S ), k = 1, . . . , τ according to (9) and (11), respectively. (4) Calculate pj , j = 1, 2, . . . , 2n such that (13) holds. (5) The state feedback gain matrix can be designed as K = δ2m [p1 p2 · · · p2n ]. One can easily see that the key to Theorem 1 and Corollary 1 is how to find a non-empty set S ⊆ {δ2r n : r ∈ O (α)} and an integer 1 ≤ τ ≤ 2n such that (10) holds. From the proof of Theorem 1, one should calculate a series of sets Rk (S ), k = 1, . . . , τ to verify (10). However, the calculations of Rk (S ), k = 1, . . . , τ are not computationally tractable because one needs to find a control sequence {u(0), . . . , u(k − 1)} for each x(0) ∈ Rk (S ). In the following, based on the input-state incidence matrix proposed by Zhao et al. (2010), we give a computationally tractable result on the verification of (10). To aid the readers, we first recall some useful results on the input-state incidence matrix of the system (1). For details, please refer to Zhao et al. (2010). Consider the algebraic form (6) with
Fig. 1. The network graph of the system (18).
is reachable from x(0) = δ2kn at the τ th step. Hence, ∆2n ⊆ Rτ (S ). This together with Rτ (S ) ⊆ ∆2n shows that Rτ (S ) = ∆2n .
(Sufficiency) If MSτ has zero column, say, Colk (MSτ ) = 0Tv . Then, j
any x(τ ) = δ2ln ∈ S , l = 1, . . . , v is not reachable from x(0) = δ2kn ∈ ∆2n at the τ th step, which is a contradiction to δ2kn ∈ Rτ (S ). Therefore, all the columns of MSτ are nonzero.
L = δ2n [i1 i2 · · · i2m+n ] ∈ L2n ×2m+n .
Based on Lemma 2, we have the following result.
Split L into 2m equal blocks as [L1 L2 · · · L2m ], where Li ∈ L2n ×2n . Set
Theorem 2. The set S = {δ21n , . . . , δ2vn } ⊆ {δ2r n : r ∈ O (α)} with 1 ≤ j1 < · · · < jv ≤ 2n satisfies (10), if and only if there exists an integer 1 ≤ τ ≤ 2n such that all the columns of both MS and MSτ are nonzero.
M =
2m
Li .
(15)
i=1 p
q
Then, x(s) = δ2n is reachable from x(0) = δ2n at the sth step, if and only if (M s )p,q > 0, where (M s )p,q denotes the (p, q)th element of the matrix M s . j j For a positive integer τ and a set S = {δ21n , . . . , δ2vn } with n 1 ≤ j1 < · · · < jv ≤ 2 , we have the following result. Lemma 2. (1) S ⊆ R1 (S ) if and only if all the columns of
Mj1 ,j1
··· .. .
Mj1 ,jv
Mjv ,j1
···
Mjv ,jv
. MS := ..
j
j
Remark 2. From Theorem 2, one only needs to calculate a set of matrices MSk , k = 1, . . . , τ to find S. Thus, compared with the calculations of Rk (S ), k = 1, . . . , τ , Theorem 2 is more computationally tractable. 4. An illustrative example In this section, we give an illustrative example to show how to use the results obtained in this paper to design state feedback based output tracking controllers for BCNs.
.. .
(16) Example 1. Consider the following Boolean control network (Li et al., 2013), which is a reduced model for the lac operon in the bacterium Escherichia coli:
are nonzero;
(2) Rτ (S ) = ∆2n if and only if all the columns of Rowj1 (M τ ) .. MSτ := .
x1 (t + 1) = ¬u1 (t ) ∧ (x2 (t ) ∨ x3 (t )), x2 (t + 1) = ¬u1 (t ) ∧ u2 (t ) ∧ x1 (t ), x3 (t + 1) = ¬u1 (t ) ∧ (u2 (t ) ∨ (u3 (t ) ∧ x1 (t ))),
(17)
τ
Rowjv (M )
are nonzero. Proof. Firstly, we prove Conclusion (1). (Necessity) Suppose that all the columns of MS are nonzero. For any k = 1, . . . , v , since Colk (MS ) ̸= 0Tv , there exists an integer j
1 ≤ l ≤ v such that Mjl ,jk > 0. Therefore, x(1) = δ2ln ∈ S is jk 2n
reachable from x(0) = δ ∈ S in one step. From the arbitrariness of k, we know that S ⊆ R1 (S ). (Sufficiency) Assume S ⊆ R1 (S ). If MS has zero column, say, j Colk (MS ) = 0Tv , then any x(1) = δ2ln ∈ S , l = 1, . . . , v is not j
reachable from x(0) = δ2kn ∈ S in one step. This is a contradiction j
to δ2kn ∈ R1 (S ). Thus, all the columns of MS are nonzero. Secondly, let us prove Conclusion (2). (Necessity) Assume that all the columns of MSτ are nonzero. For any k = 1, . . . , 2n , since Colk (MSτ ) ̸= 0Tv , there exists an integer 1 ≤ l ≤ v such that (M τ )jl ,k > 0, which implies that x(τ ) = δ2ln ∈ S j
(18)
where x1 , x2 and x3 are state variables which denote the lac mRNA, the lactose in high concentrations, and the lactose in medium concentrations, respectively; u1 , u2 and u3 are control inputs which represent the extracellular glucose, the high extracellular lactose, and the medium extracellular lactose, respectively. The network graph of the system (18) is shown in Fig. 1. In this example, we are interested in observing the genes x1 and x2 of the system (18). Then, the output equation is
y1 (t ) = x1 (t ), y2 (t ) = x2 (t ).
(19)
Our objective is to design state feedback controllers such that the output of the system (18) tracks the reference signal Yr = (1, 0). Using the vector form of logical variables and setting x(t ) = n3i=1 xi (t ), u(t ) = n3i=1 ui (t ) and y(t ) = n2i=1 yi (t ), by the semi-tensor product of matrices, we have the following algebraic form:
x(t + 1) = Lu(t )x(t ), y(t ) = Hx(t ),
(20)
58
H. Li et al. / Automatica 59 (2015) 54–59
where L = δ8 [8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8888888888888888 1115333711153337 3 3 3 7 4 4 4 8 4 4 4 8 4 4 4 8], and H = δ4 [1 1 2 2 3 3 4 4]. Moreover, the reference signal becomes yr = δ21 n δ22 = δ42 . A simple calculation gives O (2) = {3, 4}. Setting S = {δ83 } ⊆ 3 {δ8 , δ84 }, one can easily obtain that R1 (S ) = {δ81 , δ82 , δ83 , δ85 , δ86 , δ87 } and R2 (S ) = ∆8 . Thus, (10) holds for S = {δ83 } and τ = 2. A straightforward calculation shows that p1 = p2 = p3 = 7, p4 = 5 or 6 or 7, and pj = 5 or 6, ∀ j = 5, . . . , 8. Hence, we obtain 48 state feedback gain matrices as K = δ8 [7 7 7 p4 p5 p6 p7 p8 ], where p4 ∈ {5, 6, 7} and pj ∈ {5, 6}, j = 5, . . . , 8. For example, we set pj = 5, j = 4, . . . , 8. Then, K = δ8 [7 7 7 5 5 5 5 5], which corresponds to the following state feedback based output tracking controller: u1 (t ) = 0, u2 (t ) = ¬x1 (t ) ∨ ¬(x2 (t ) ∨ x3 (t )), u3 (t ) = 1.
(21)
5. Conclusion In this paper, we have investigated the output tracking control of BCNs with a constant reference signal by using the semi-tensor product method, and presented a number of new results. We have proposed a constructive procedure to design state feedback based output tracking control laws for BCNs. The key idea of this procedure is to stabilize the BCN to a set of states whose outputs are the given constant reference signal. Based on the input-state incidence matrix, we have presented a computationally tractable result to find a proper set of states whose outputs are the given constant reference signal. The study of an illustrative example has shown that the new results obtained in this paper are effective in designing state feedback based output tracking controllers for BCNs. Acknowledgments The authors would like to thank the Associate Editor and the reviewers for their constructive comments and suggestions which improved the quality of the paper. References Akutsu, T., Hayashida, M., Ching, W., & Ng, M. (2007). Control of Boolean networks: Hardness results and algorithms for tree structured networks. Journal of Theoretical Biology, 244(4), 670–679. Ay, F., Xu, F., & Kahveci, T. (2009). Scalable steady state analysis of Boolean biological regulatory networks. PLoS ONE, 4(12), e7992. Chaves, M. (2009). Methods for qualitative analysis of genetic networks. In Proc. 10th European control conference (pp. 671–676). Chen, H., & Sun, J. (2014). Output controllability and optimal output control of statedependent switched Boolean control networks. Automatica, 50(7), 1929–1934. Cheng, D. (2011). Disturbance decoupling of Boolean control networks. Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, 56(1), 2–10. Cheng, D., Li, Z., & Qi, H. (2010). Realization of Boolean control networks. Automatica, 46(1), 62–69. Cheng, D., & Qi, H. (2009). Controllability and observability of Boolean control networks. Automatica, 45(7), 1659–1667. Cheng, D., & Qi, H. (2010). A linear representation of dynamics of Boolean networks. Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, 55(10), 2251–2258. Cheng, D., Qi, H., & Li, Z. (2011). Analysis and control of Boolean networks: a semitensor product approach. London: Springer. Cheng, D., Qi, H., Li, Z., & Liu, J. B. (2011). Stability and stabilization of Boolean networks. International Journal of Robust and Nonlinear Control, 21(2), 134–156. Cheng, D., & Xu, X. (2013). Bi-decomposition of multi-valued logical functions and its applications. Automatica, 49(7), 1979–1985.
Cheng, D., & Zhao, Y. (2011). Identification of Boolean control networks. Automatica, 47(4), 702–710. Choudhary, A., Datta, A., Bittner, M. L., & Dougherty, E. R. (2006). Intervention in a family of Boolean networks. Bioinformatics, 22(2), 226–232. Drossel, B., Mihaljev, T., & Greil, F. (2005). Number and length of attractors in a critical Kauffman model with connectivity one. Physical Review Letters, 94(8), 088701-(1–4). Feng, J., Yao, J., & Cui, P. (2012). Singular Boolean networks: Semi-tensor product approach. Science China. Information Sciences, http://dx.doi.org/10.1007/s11432012-4666-8. Fornasini, E., & Valcher, M. E. (2013a). Observability, reconstructibility and state observers of Boolean control networks. Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, 58(6), 1390–1401. Fornasini, E., & Valcher, M. E. (2013b). On the periodic trajectories of Boolean control networks. Automatica, 49(5), 1506–1509. Fornasini, E., & Valcher, M. E. (2014). Optimal control of Boolean control networks. Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, 59(5), 1258–1270. Kauffman, S. (1969). Metabolic stability and epigenesis in randomly constructed genetic nets. Journal of Theoretical Biology, 22(3), 437. Laschov, D., & Margaliot, M. (2011). A maximum principle for single-input Boolean control networks. Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, 56(4), 913–917. Laschov, D., & Margaliot, M. (2012). Controllability of Boolean control networks via the Perron–Frobenius theory. Automatica, 48(6), 1218–1223. Li, R., & Chu, T. (2012). Complete synchronization of Boolean networks. IEEE Transactions on Neural Networks and Learning Systems, 23(5), 840–846. Li, F., & Sun, J. (2011a). Controllability of Boolean control networks with time delays in states. Automatica, 47(3), 603–607. Li, F., & Sun, J. (2011b). Controllability of probabilistic Boolean control networks. Automatica, 47(12), 2765–2771. Li, H., & Wang, Y. (2012). On reachability and controllability of switched Boolean control networks. Automatica, 48(11), 2917–2922. Li, H., & Wang, Y. (2013). Output feedback stabilization control design for Boolean control networks. Automatica, 49(12), 3641–3645. Li, R., Yang, M., & Chu, T. (2013). State feedback stabilization for Boolean control networks. Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, 58(7), 1853–1857. Wang, Y., Zhang, C., & Liu, Z. (2012). A matrix approach to graph maximum stable set and coloring problems with application to multi-agent systems. Automatica, 48(7), 1227–1236. Xiao, Y. (2009). A tutorial on analysis and simulation of Boolean gene regulatory network models. Current Genomics, 10, 511–525. Xiao, Y., & Dougherty, E. R. (2007). The impact of function perturbations in Boolean networks. Bioinformatics, 23(10), 1265–1273. Xu, X., & Hong, Y. (2013). Matrix approach to model matching of asynchronous sequential machines. Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, 58(11), 2974–2979. Yang, M., Li, R., & Chu, T. (2013). Controller design for disturbance decoupling of Boolean control networks. Automatica, 49(1), 273–277. Zhang, X. (2012). Semitensor product based adaptive control for attitude tracking of spacecraft with unknown external disturbances. Journal of Control Theory and Applications, 10(3), 292–296. Zhang, L., & Feng, J. (2013). Mix-valued logic-based formation control. International Journal of Control, 86(6), 1191–1199. Zhang, L., & Zhang, K. (2013). Controllability and observability of Boolean control networks with time-variant delays in states. IEEE Transactions on Neural Networks and Learning Systems, 24, 1478–1484. Zhao, Y., Cheng, D., & Qi, H. (2010). Input-state incidence matrix of Boolean control networks and its applications. Systems & Control Letters, 59(12), 767–774. Zhao, Y., Kim, J., & Filippone, M. (2013). Aggregation algorithm towards largescale Boolean network analysis. Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, 58(8), 1976–1985. Zhao, Y., Li, Z., & Cheng, D. (2011). Optimal control of logical control networks. Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, 56(8), 1766–1776. Zou, Y., & Zhu, J. (2014). System decomposition with respect to inputs for Boolean control networks. Automatica, 50(4), 1304–1309.
Haitao Li received the B.S. and M.S. degrees from the School of Mathematical Science, Shandong Normal University in 2007 and 2010, respectively, and the Ph.D. degree at the School of Control Science and Engineering, Shandong University in 2014. Since 2015, he has been with the School of Mathematical Science, Shandong Normal University, China, where he is currently an associate professor. From Jan. 2014 to Jan. 2015, he worked as a Research Fellow in Nanyang Technological University, Singapore. His research interests include logical dynamic systems, switched systems, etc. He received the ‘‘Best Student Paper Award’’ at the 10th World Congress on Intelligent Control and Automation, and the ‘‘Guan Zhao-Zhi Award’’ at the 31st Chinese Control Conference.
H. Li et al. / Automatica 59 (2015) 54–59 Yuzhen Wang graduated from Tai’an Teachers College in 1986, received his M.S. degree from Shandong University of Science & Technology in 1995 and his Ph.D. degree from the Institute of Systems Science, Chinese Academy of Sciences in 2001. Since 2003, he is a professor with the School of Control Science and Engineering, Shandong University, China, and now the Dean of the School of Control Science and Engineering, Shandong University. From 2001 to 2003, he worked as a Postdoctoral Fellow in Tsinghua University, Beijing, China. From Mar. 2004 to Jun. 2004, from Feb. 2006 to May 2006 and from Nov. 2008 to Jan. 2009, he visited City University of Hong Kong as a Research Fellow. From Sept. 2004 to May 2005, he worked as a visit Research Fellow at the National University of Singapore. His research interests include nonlinear control systems, Hamiltonian systems and Boolean networks. Prof. Wang received the Prize of Guan Zhao-Zhi in 2002, the Prize of Huawei from the Chinese Academy of Sciences in 2001, the Prize of Natural Science from Chinese Education Ministry in 2005, and the National Prize of Natural Science of China in 2008. Currently, he is an associate editor of IMA Journal of Math Control and Inform., and a Technical Committee member of IFAC (TC2.3).
59 Lihua Xie received the B.E. and M.E. degrees in electrical engineering from Nanjing University of Science and Technology in 1983 and 1986, respectively, and the Ph.D. degree in electrical engineering from the University of Newcastle, Australia, in 1992. Since 1992, he has been with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, where he is currently a professor and served as the Head of Division of Control and Instrumentation from 2011 to 2014. He held teaching appointments in the Department of Automatic Control, Nanjing University of Science and Technology
from 1986 to 1989. Dr. Xie’s research interests include robust control and estimation, sensor networks, and networked control systems. In these areas, he has published many journal papers and co-authored two patents and six books. He is an Editor-in-Chief of Unmanned Systems and has served as an editor of IET Book Series in Control and an Associate Editor of a number of journals including IEEE Transactions on Automatic Control, Automatica, IEEE Transactions on Control System Technology, and IEEE Transactions on Circuits and Systems-II. Dr. Xie is a Fellow of IEEE, a Fellow of IFAC and an IEEE Distinguished Lecturer.