Mathematical Modelling and Control

: This article investigates the reference trajectory output tracking issue of Boolean control networks (BCNs) that have controls in the output. Firstly, to solve the problem, some necessary and su ffi cient conditions are obtained. The tracking problem is studied from the perspective of set and matrix calculation. Next, an algorithm for determining whether the output tracking issue is solvable is proposed. Furthermore, the controller design algorithm satisfying the solvability condition is given. Using our methods, we can track some trajectories that cannot be tracked in BCNs without controls in output. In addition, for better application in practice, the corresponding changes in the network transition matrix and output matrix under state, transition, and input constraints are considered. Finally, some examples are presented to illustrate the validity of our results.


Since Kauffman introduced a binary network called
Boolean network (BN) to study the evolution behavior in genetic regulatory networks (GRNs) [1], it has garnered considerable interest from biologists, systems scientists, and others.In a BN, variables are used to represent genes with values of 1 or 0, where 1 implies that the gene is active, and 0 implies that the gene is inactive.When there are external disturbances regulations, BNs are naturally extended to BCNs.In addition, probabilistic Boolean networks (PBNs) and switching Boolean networks (SBNs) are investigated due to the switching and random phenomena in GRNs [2,3].
In fact, BNs (BCNs) with logical function forms are nonlinear networks.Thus, traditional discrete-time linear system theory cannot be used to investigate research problems.Cheng et al. [4] invented the semi-tensor product (STP) of matrices to convert the logical dynamical system into a typical discrete-time linear system.Some methods and ideas in linear system theory can be applied to BNs (BCNs) by using the STP, which further helps scientists analyze and control GRNs for disease intervention [5] or work on feedback shift registers [6] and so on [7][8][9].Since then, various properties of BNs have been investigated, such as controllability, observability, stabilization and so on [10][11][12][13][14][15][16][17][18][19][20].Furthermore, as a helpful tool, the STP method also is used in hybrid dynamic systems and fuzzy systems [21][22][23][24][25][26].
In the practical GRNs, the state evolution is complicated and the measuring equipment has limitation [27].Therefore, it is an effective solution for scientists to measure the output and render the system output to track the desired signals to help study the system's dynamics.It is also of great significance in the application of robots and flight control [28][29][30].By reviewing the literature, we obtain that the current studies on output tracking issues of BCNs can be mainly divided into the two following categories according to the objectives of output tracking: track a constant reference signal and track a time-varying reference trajectory [27,[31][32][33][34][35].Methods for designing the state feedback control law were described in [27] to let the system's output track a constant reference signal stably after a finite time.In [31], the switching sequence is found by combining the method of set stabilization, so that the SBNs can track a given a given constant reference signal.[32] constructed an auxiliary system to help the outputs of a BCN track the outputs of a reference system.Since it may cost a lot to control over an infinite horizon, [33] considered tracking a given reference output trajectory.In summary, the output tracking problem is an inevitable and crucial issue for GRNs.
In addition, the output tracking problem has other practical significance.
For example, the motion of ships carrying heavy cargo needs to track a desired line reserved before, thereby minimizing financial burden [36].
Furthermore, considering that the output tracking research might help control the spread of future epidemics, we mainly care about tracking a given finite length of timevarying reference output trajectory.Moreover, the idea of adding controls to the output is motivated by the research problem investigated in [37].In the study of a functional system related to avalanche warning, context-alert of context model combined with terrain temperature, snow height, and accelerometer as the total input variables has a specific effect on the output of functional system [37].Hence, we add controls to the system output to describe these system models.It is worth highlighting that the authors in [33] studied the output tracking problem of BCNs.Nonetheless, the method in [33] fails to solve the problem in some cases.
To our knowledge, no study has been done considering the controls in the output to solve the output tracking problem.
The system considered in this paper has controls in both state transition and output models.Therefore, to a certain extent, the method discussed in this paper can realize the output tracking problem that cannot be realized in [33].In the following, we highlight the main contributions of this paper: • From the perspectives of set and matrix, for the BCN system with output having controls, we give some equivalent conditions to solve the reference trajectory output tracking issue.• Based on the obtained theorem, the authors construct an algorithm to determine whether the output tracking issue is solvable.
• If the problem can be solved, we develop an algorithm to obtain a feasible control sequence, allowing the system to track the reference output trajectory in a finite range.
The main structure of this paper is as follows.We introduce the preliminaries in Section 2. The main theorems for the solvability of output tracking issues are obtained in Section 3. Finally, some biological examples are proposed to verify our results in Section 4. A concise conclusion is shown in Section 5.

Preliminaries
First, for the convenience of subsequent description, some basic notations are given here.
• D := {0, 1}, real numbers are denoted by R and the set of positive integers is denoted by Z + .
• ∆ q := {δ i q |1 ≤ i ≤ q}, where δ i q represents an qdimensional column vector with the ith element being 1 and others are 0.
• The ith column and ith row of matrix Q are denoted by Col i (Q) and Row i (Q), respectively.
• We call a matrix Q ∈ M m×n logical matrix, if each column vector of it has only one element 1, and the other elements are 0. In this paper, denote the set of m × n-dimensional logical matrices by L m×n .Besides, B m×n denotes the set of m × n Boolean matrices, with all elements taking value from D.
• 1 k or 0 k is the k-dimensional column vector whose elements are all equal to 1 or 0, respectively.

Semi-tensor product
In this section, we introduce the definition of the STP and how to transform BNs from logical expressions to algebraic forms.Definition 2.1.[4] For two matrices P ∈ M a×b and Q ∈ M c×d , the semi-tensor product of them is defined by where l = lcm(b, c) is the least common multiple of b and c, and ⊗ is the Kronecker product.
When b = c, the STP of matrix is consistent with ordinary matrix multiplication.In this article, we will omit "⋉" without affecting the results to facilitate reading.Before using the STP to obtain the algebraic expression of BCNs, we show equivalence between Boolean variables and their vector forms.Here, 1 is equivalent to δ 1 2 and 0 is equivalent to δ 2 2 .
Lemma 2.1.[4] For a logical function g(x 1 , • • • , x n ) : D n → D, after changing all arguments x i to their vector form x i , i.e., x i ∈ ∆ 2 , i ∈ [1, n], it has an equivalent algebraic , where the structure matrix L g ∈ L 2×2 n can be uniquely determined by g.

Algebraic representation of BCNs
Consider the logical representation of a BCN is abbreviated as with its output system being where changing the logical form of variables to the vector form, we define where structure matrices are specified as Likewise, define y(t) = ⋉ p j=1 y j (t).Then the outputs are where . Therefore, we can determine the algebraic form of BCN (2.1) and its output (2.2) as where L, which can be obtained by referred to as the network transition matrix.Similarly, H is called an output matrix which can be expressed as where " * " is the Khatri-Rao product of matrices.all the control variables with a dummy matrix [4].Then, it will make the control sequences the same and larger dimensional than before for the convenience of subsequent studies.

Problem formulation
Based on the above STP work, we can analyze the algebraic form of BCNs to study the output tracking issue.For simplicity, denote a control sequence u t by Thus, the output of the system (2.5) under u t is shown as y(t, x(0), u t ).Then, there is Since we are investigating the problem of reference trajectory output tracking, there needs to be an assigned reference output trajectory y P : Mathematical Modelling and Control Volume 3, Issue 3, 256-266 Next, we will give the definition of reference trajectory output tracking and the solvable conditions for the output tracking issue will be discovered using the properties of BCNs.
We construct sets X(t), Λ(t) and Γ(t) alternatively as shown below to address the output tracking issue, where First, the state set obtained from state x(0) through one and Ω(t), i.e., It is essential to mention that for t ∈ [2, P] Remark 3.1.Ω(t) is the control-state pairs set whose elements can produce the desired output according to H. X(t) is the state set under control sequence which meets the condition of tracking, from the perspective of network transition matrix L of dynamical equation.To unify the dimensions of the two sets, we define the set Λ(t).

The solvability of the output tracking issue
Following the definition of sets X(t), Λ(t) and Γ(t), a theorem can be obtained to determine the solvability of the output tracking issue from the perspective of the set.
Theorem 3.2.Given an initial state x(0) = δ i 0 2 n and a positive integer P, the output tracking issue of BCN (2.5) is solvable if and only if γ(P) 0 2 m+n .
For γ(P) 0 2 m+n , we can take u(P)x(P) = δ j P 2 m+n ∈ Ξ γ(P) , and denote them by u(P) = δ k P 2 m , x(P) = δ i P 2 n , respectively.According to the construction of γ(t) in (3.4), it implies that y(P) = Hu(P)x(P) = y o (P).Then, we can choose } so that ∀t ∈ [1, P], y(t; x(0), u t ) = y o (t) holds, which means that there is one way that the output tracking goal of BCN (2.5) can be resolved.As for necessity, it is removed here since it is analogous to Theorem 3.1.

□ Theorem 3.1 gives the necessary and sufficient conditions
for the solvability of output tracking problem from the set viewpoint, which helps us to understand.In Theorem 3.2, the criterion is given by vector method, which is convenient for practical calculation and judgment.Then, Algorithm 1 is given below to illustrate the general steps for determining whether the output tracking issue can be solved.

Tracking control design
After we investigate whether the output tracking issue for a reference trajectory in a given finite time can be solved, we consider finding a control sequence satisfying the condition Algorithm 1 Determine the solvability of output tracking issue.
Input: x(0) = δ i 0 2 n , P, and the reference output trajectory y P with a known BCN (2.5) Input: x(0) = δ i 0 2 n , P, and the reference output trajectory y P with a known BCN (2.5) 1: Determine that whether the problem can be solved according to Algorithm 1.
Proof.From Algorithm 2, we can find that For any t ∈ [1, P], there is ) holds for all t ∈ [1, P], it implies that u ⋇ (t)x ⋇ (t) ∈ Ω(t).Therefore, it can be obtained that In daily life, constraints are everywhere.Taking the treatment of patients by doctors as an example, the state of patients or cells can be regarded as the state of the system, and the prescription drugs issued by doctors can be regarded as the control of the system.The purpose of doctors' treatment of patients is to hope that patients will enter a continuous healthy state, so some unhealthy and sick states should be avoided [38].In some cases, the type of drugs will be determined according to the state of the patient.
For example, when the patient is in the state of drinking alcohol, the doctor will not consider taking cephalosporin anti-inflammatory drugs, which is the transition constraint considered here [39].In addition, if some drugs were developed a long time ago, and with the increase of medical level, we find the use of these drugs may have undesirable side effects on the human body, then doctors will not consider prescribing these types of drugs.For example, furazolidone as an effective drug, is usually used to treat gastrointestinal diseases such as dysentery and enteritis caused by bacteria and protozoa.However, scientists found that it has potentially carcinogenic properties [40].
Hence, doctors won't use compound preparation containing furazolidone now, and it can be one of the examples for the input control constraints.Therefore, in controllers' design, we should also consider changes in state, transition and input constraints.
(1) State constraint When solving the output tracking issue of the system and designing the controllers, if a state x(t) = δ i 2 n should be avoided, then the ith row of matrix L should be At this point, we do not consider any transition from other In a BCN, a transition constraint forbids the control input u(t) for a state x(t).For example, if the transition from the state The input constraint means there are some controls that we cannot take.Suppose that u(t) = δ k 2 m has to be forbidden, then the k-th block of L and H will be null matrices, that is

Example
Example 4.1.Consider BCN in [33]: This is a simplified BCN model of lac operon in Escherichia coli.Applying the STP method, the following algebraic form can be obtained: where x(t At the beginning, we assume that x(0) = δ 2 8 , and give two reference output trajectories as: Table 1.Output trajectory 1.The following is to determine whether the output tracking issue can be solved for the given output trajectories 1 and 2.
We first consider whether the system can track the specified reference output trajectory 1.
Furthermore, we have It can be concluded that the output tracking issue of the system is solvable for reference output trajectory 1 since γ(4) 0 8 .Then, we can use the method in Algorithm 2 to find a feasible control sequence to realize output tracking for the output trajectory 1.The specific process is as follows.
It determines that the system can track reference output trajectory 2 since γ(4) 0 8 .Furthermore, the input sequence that meets the output tracking condition can be {u In [33], there does not have any control in the output system.The system can realize output tracking under output trajectory 1, but fails under output trajectory 2. As one of the innovations of this paper, we add control input to the output system.We can find in this example that it makes the system realize output tracking under two reference output trajectories.
Example 4.2.Consider a modified lactose operon model in Escherichia coli with 5 nodes [41]: Based on the STP method, we can transfer the model to the form as (2.5) with x Here, we are going to check under the condition that the initial state is x(0) = δ 20  32 , whether the output tracking issue can be solvable with the given output trajectory u ⋇ (0) = δ 3 4 .Therefore, using the control sequence {u ⋇ (0) = δ 3  4 , u ⋇ (1) = δ 1 4 , u ⋇ (2) = δ 4 4 , u ⋇ (3) = δ 1 4 }, the output tracking issue based on the given reference trajectory can be solved.

Conclusions
By using the STP, for the BCNs with controls in output, the reference trajectory output tracking issue is investigated.Some results to solve the output tracking issue are proposed and an algorithm is designed to judge the solvability.Moreover, the controllers' design algorithm is given by calculation when we have determined that the output tracking issue is solvable.To better apply the results to daily life, the corresponding changes in the network transition matrix and output matrix under some constraints simulated from reality are considered.In the end, the effectiveness of our results is shown by some examples.
An important direction for future research is to investigate output tracking problem by reinforcement learning method.
In addition, we can study the problems considered in this paper in BNs with time delay or switching signal.

Remark 2 . 1 .
Without the loss of generality, in the equations (2.1) and (2.2), the evolution of the state and the output, respectively, in the BCN model we give, is related to the same control sequence.If the state and output are affected by different control variables, we can still combine

Theorem 3 .
1 shows that ∀t ∈ [1, P], Γ(t) ∅ means that there exists u(t)x(t) satisfying both dynamical update condition and output tracking condition.It determines that the output tracking goal of BCN (2.5) can be achieved from the perspective of the set.Next, we consider giving another theorem to realize the goal through vector expression from the perspective of matrix calculation.
Besides, the network transition matrix and the output matrix are L = δ 8[8