Output Tracking of Boolean Control Networks With Impulsive Effects

In this article, the output tracking of Boolean control networks with impulsive effects (BCNs-IE) is discussed. Based on structure matrices of BCNs-IE and controllability matrices with respect to state subsets, stable complex attractors are studied. By constructing an auxiliary BCN-IE and using stable complex attractors, a necessary and sufficient condition for the output tracking problem is proposed. In addition, an algorithm to design state feedback controllers for the output tracking problem is provided. Moreover, an example is given to show the validity of the obtained results.


I. INTRODUCTION
To model genetic regulation networks, Kauffman [1] proposed Boolean networks (BNs) with variables taking values 0 or 1. Since then, BNs has been widely used in biological fields [2]- [5]. Consider the influence of external environment as a control variable of a BN, Boolean control networks (BCNs) [6] were introduced. As a creative mathematical tool for studying BNs and BCNs, the semi-tensor product (STP) of matrices was presented and applied to convert BNs and BCNs into linear (bilinear) discrete-time systems [7]. Thanks to the application of STP, many excellent works about BNs and BCNs are emerged, such as controllability and observability [8]- [10], stability and stabilization [11]- [13], optimal control problem [14]- [16] and other related problems [17]- [19].
As a significant issue in the control theory, the purpose of output tracking is to design suitable controllers that steer the output of a system to a given reference signal. In biological systems, the automated monitoring of cell populations in a high-throughput, high-content environment depends on accurate cell tracking of individual cell that display various behaviors. Reference [20] presented a cell tracking approach, which explicitly models cell behaviors in graph-theoretic frameworks. In addition, many modern live-cell imaging experiments are technologies that automatically track and analyze the motion of objects in time-lapse microscopy images [21].
The associate editor coordinating the review of this manuscript and approving it for publication was Jianquan Lu . Moreover, reference [22] designed controllers to drive a certain amount of Escherichia coli to an desired state. There is no doubt that [22] provides an example of the output tracking of genetic regulatory networks.
Based on the analysis above, it is meaningful to concern the output tracking problem of BCNs. By constructing matrices reflecting the (output) reachability, the output regulation problem of BCNs was solved in [23]- [25]. In order to track the output of a time-varying reference signal, reference [26] established a bilinear equation, which reflects the relationship between states and outputs. And reference [27] introduced an auxiliary system, calculated the set of control attractors and control invariant subsets of this system. The method proposed in [27] was also used to handle the output tracking of BCNs driven by a constant reference signal [28]. Furthermore, inspired by cycles and control invariant subsets, reference [29] defined the concept of stable complex attractor, which is similar to cycles, and contained in control invariant subsets. Therefore, it is possible to investigate the output tracking problem via stable complex attractors.
Impulsive systems serve as basic models to research such dynamical process, whose states undergo abrupt changes at certain instants. Due to the extensive applications in many fields, impulsive systems have received considerable attention. For example, reference [30] studied the controllability of complex-valued impulsive systems with time-varying delay in control input. Reference [31] concerned the Lyapunov stability problem for impulsive systems via event-triggered VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ impulsive control. And other types of stability problem for impulsive systems were considered in [32]- [35]. In biological networks, to describe the dynamic process of sudden changes of states, BCNs with impulsive effects (BCNs-IE) were discussed firstly in [36]. Many related problems about BCNs-IE are solved, such as control problems [37]- [39], stability and stabilization [40]- [42], output tracking [43] and so on. Although the output tracking problem of BCNs-IE has been addressed in [43], stable complex attractors were not considered to deal with this problem, and the corresponding controller was not designed. Therefore, this article re-discusses the output tracking problem of BCNs-IE on the basis of stable complex attractors. The main contributions of this article are shown as follows.
(1) Stable complex attractors of BCNs-IE are defined. Based on structure matrices of BCNs-IE and controllability matrices with respect to state subsets, an algorithm to find all stable complex attractors of BCNs-IE is given.
(2) According to the original BCN-IE and the time-varying reference signal, an auxiliary BCN-IE is constructed. Motivated by the results proposed in [27] and [29], a necessary and sufficient condition for the output tracking problem of BCNs-IE is presented.
(3) Combined the necessary and sufficient condition for the output tracking problem with properties of stable complex attractors, an approach to design state feedback controllers for this problem is provided.
The rest of this article is organised as follows. Section II reviews some necessary notations and the algebraic form of BCNs-IE. In Section III, stable complex attractors are studied. A necessary and sufficient condition for the output tracking problem of BCNs-IE is proposed. Besides, a method is presented for the controller design of this problem. Section IV provides an example to show the effectiveness of our main results. Some concluding remarks are given in Section V.

II. PRELIMINARIES
In this section, we introduce some necessary preliminaries about STP and BCNs-IE, which will be used throughout this article.
• 0 is a zero matrix.
• Given two matrices P ∈ M m×n and Q ∈ M p×q . The STP of P and Q is defined as where ⊗ is the Kronecker product, t is the least common multiple of n and p. Without confusion, the symbol is omitted.
To expressed logical functions in algebraic forms, two lemmas are presented.

B. MATRIX EXPRESSION OF BCNs-IE
A BCN-IE is described as where {t k |t k < t k+1 , k ∈ Z + } ⊂ Z + is an impulsive time sequence. f i , g i and h j are Boolean functions. x i ∈ D, u j ∈ D and y k ∈ D are state variables, input variables and output variables, respectively.
From Formula (2), it is not hard to find that a BCN-IE can be considered as a switched BCN with a specific switching signal. If BCN-IE (2) is controllable, then there exists a proper control sequence such that initial state δ i 2 n reaches each state in time 2 n+1 . This result has been given in [46].
Lemma 3 [46]: Consider BCN-IE (2) as a switched BCN with a specific switching signal. Then the system is controllable iff each state of the system reaches any state in time 2 n+1 with proper controls.
Remark 1: By analyzing BCN-IE (2), the state changes abruptly at a prescribed time. It shows that the impulsive effect this article studied is triggered by time. But reference [42] concerned the other impulsive effect, which is triggered by system states. It is the first difference between these two impulsive effects. In addition, as shown in Lemma 3, BCN-IE (2) can be viewed as a switched BCN with a specific switching signal. However, states of state-triggered impulsive BNs may jump successively at a single time instant. Reference [42] points out that it makes state-triggered impulsive BNs different from switched systems. Let x(t; x 0 , u(t)) (y(t; x 0 , u(t))) represent the state (output) at time t with initial state x 0 under control input u(t).

III. MAIN RESULTS
This section discusses stable complex attractors of BCNs-IE, considers the output tracking of BCNs-IE, and designs state feedback controllers for the output tracking problem.

A. STABLE COMPLEX ATTRACTORS
To begin with, the definition of stable complex attractor is proposed.
Definition 1: A set ⊂ 2 n is called a complex attractor of BCN-IE (2), if for ∀x i , x j ∈ , there exist an integer T ij and a control sequence A set ⊂ 2 n is said to be stable, if ∀x i ∈ and ∀x k ∈ , x k is not reachable from x i .
As shown in Section I, stable complex attractors are similar to cycles and contained in control invariant subsets. Some explication are given in the following.
If is a cycle, then for ∀x i , x j ∈ , there exists an integer Because the concept of cycle is defined in BNs, one knows that x(t; x i ) ∈ holds all the time for ∀x i ∈ . From the definition of stable complex attractor, one sees the similarity between cycles and stable complex attractors.
A subset is called a control invariant subset, if there exist a control sequence {u(t)|t ∈ N} such that x(t; x i , u(t)) ∈ holds for ∀x i ∈ , t ≥ 1. It is easy to find that stable complex attractors are control invariant subsets. But a control invariant subset may not be a stable complex attractor. The reasons are as follows. State x i ∈ may reach state x j ∈ . And state x i ∈ may not be reachable from x k ∈ . Thus, stable complex attractors are contained in control invariant subsets.
Based on matrices M and L 2 , the following proposition is provided.
Row i (M | ω ) = 2 m 1 T l and L 2 | ω ∈ L l×l . Proof 1: By Definition 1, is stable in BCN-IE (2), if and only if for ∀x i ∈ and ∀x k ∈ , x k is not reachable from x i . It is equivalent to Row ω (Col ω (M )) = 0 and Row ω (Col ω (L 2 )) = 0. That is L 2 | ω ∈ L l×l . Since M = Sgn(M ), we also get Now we construct such a matrix C , which reflects the controllability of states in . Suppose t q ≤ 2 n+1 < t q+1 , where q ∈ Z + is a constant. A sequence of matrices r ⊂ B l×l , r = 1, 2, . . . , q + 1 is calculated Then the controllability matrix with respect to is With no doubt, (C ) kj = 1 implies that system (2) reaches state δ i k 2 n from state δ i j 2 n . If = 2 n , then C 2 n is the controllability matrix of BCN-IE (2).
According to the controllability matrix with respect to , the following result is obtained.
Proof 2: According to Definition 1, it is easy to see that this result holds.
By Propositions 1 and 2, an approach to find all stable complex attractors of BCNs-IE is given.
Step 1: Compute the controllability matrix C 2 n . If C 2 n = 1 2 n ×2 n , then 2 n is the unique stable complex attractor of BCN-IE (2). Otherwise, go to the next step.
Step 2: On the basis of Proposition 1, find ω i satisfying To provide a necessary and sufficient condition for the output tracking problem, a useful result about stable complex attractors is presented.
And there is at least one stable complex attractor contained in {δ i 0 2 n , δ i 1 2 n , . . . , δ i k 2 n }. It is a contradiction. Hence, each state of BCN-IE (2) can reach one of stable complex attractors.
Definition 2: The output of BCN-IE (2) is said to track the output of time-varying reference signal (5), if there exist an integer T and a control sequence {u(t)|t ∈ N} such that y(t; x(0), u(t)) =ŷ(t;x(0)) holds for ∀x(0) ∈ 2 n ,x(0) ∈ 2n and ∀t ≥ T . Now we construct an auxiliary BCN-IE to study the solvability of the output tracking problem. Consider BCN-IE (2) and time-varying reference signal (5).  (5), if and only if there exist an integer T and a control sequence {u(t)|t ∈ N} such thatȳ(t;x(0), u(t)) ∈ holds for ∀x(0) ∈ 2 n+n , ∀t ≥ T . Thus, it equals to the conclusion that there exist an integer T and a control sequence {u(t)|t ∈ N} such that x(t;x(0), u(t)) ∈ holds for ∀x(0) ∈ 2 n+n , ∀t ≥ T . It means that contains one complex attractor at least. Therefore, the output tracking problem of BCN-IE (2) can be discussed by using stable complex attractors of BCN-IE (6). Proof 4: Based on the analysis above, we only need to prove that i ∩ = ∅, i = 1, 2, . . . , ν, if and only if there exist an integer T and a control sequence {u(t)|t ∈ N} such thatx(t;x(0), u(t)) ∈ holds for ∀x(0) ∈ 2 n+n , ∀t ≥ T .
(Sufficiency.) Assume there exists a stable complex attractor i such that i ∩ = ∅. Then one gets x(t;x(0), u(t)) ∈ for each initial statex(0) ∈ 2 n+n satisfyingx(t;x(0), u(t)) ∈ i . It is a contradiction. Hence, for each stable complex attractor i , i ∩ = ∅. Remark 2: Consider a constant reference signal y r = δ α 2 p , 1 ≤ α ≤ 2 p . It is easy to see that = {δ j 2 n |Col j (H ) = y r }. According to the analysis above, the output of BCN-IE (2) tracks the constant reference signal y r , if and only if there are an integer T and a control sequence {u(t)|t ∈ N} such that x(t; x(0), u(t)) ∈ holds for ∀x(0) ∈ 2 n , ∀t ≥ T . From the proof of Theorem 1, it equals to i ∩ = ∅, i = 1, 2, . . . , ν, where 1 , . . . , ν are all stable complex attractors of BCN-IE (2). Hence, Theorem 1 can be used to solve the output tracking problem driven by a constant reference signal.
Remark 3: This article studies the output tracking problem of BCNs-IE, while this problem of BCNs has been discussed in [23]- [28]. A time-varying reference output trajectory is concerned in this article. But the tracking object [23], [25], [28] considered is a constant reference signal, which is a special case of time-varying reference signal. References [23]- [28] do not used stable complex attractors to solve the output tracking problem. And the methods they proposed have been introduced in Section I.
Although [43] and this article investigate the output tracking problem of BCNs-IE, there are some differences between these two papers. From Theorems 3.7 and 3.8 of [43], the corresponding results hold under the assumption that is an L 1 -invariant set and anL 2 -invariant set. However, Theorem 1 of this article holds without any preconditions. The control sequence required in Theorem 3.7 (3.8) is a state (an output) feedback control sequence, while Theorem 1 does not have this requirement. In addition, [43] does not use stable complex attractors to address the output tracking problem. And the corresponding controller is not designed in [43].

C. DESIGN OF STATE FEEDBACK CONTROLLERS
In order to design the state feedback controller for the output tracking problem of BCNs-IE, we prove the following theorem firstly.
According to Theorem 1, The output of BCN-IE (2) tracks the output of reference signal (5), if and only if i ∩ = ∅, i = 1, 2, . . . , ν. From this result and {ϒ i |i = 1, . . . , ν}, the state feedback matrix K can be gotten by the following steps. Denote i j ). Repeat this process until Do this process for each state subset ϒ i , i = 1, 2, . . . , ν, the state feedback matrix K is obtained.
Based on the results above, an approach to design the state feedback controller is proposed.
Algorithm 2: Consider BCN-IE (2) and reference signal (5). Design the state feedback control u(t) = Kx(t)x(t) such that the output of BCN-IE (2) tracks the output of reference signal (5).
Step 3: Compute i ∩ , i = 1, 2, . . . , ν. If there exists i such that i ∩ = ∅, then the output of BCN-IE (2) does not track the output of reference signal (5). Otherwise, go to the next step.
Step 5: For each state δ α 2 n+n ∈ ϒ i , i = 1, . . . , ν, the state feedback matrix K is obtained by the following formula