Robust Output Tracking of Boolean Control Networks over Finite Time

Abstract

With an increase in tracking time, the operating cost of the controller will increase accordingly. Considering the biological applications of Boolean control networks (BCNs), it is necessary to study the control problem of BCNs over finite time. In this paper, we study the output tracking problem of a BCN with disturbance inputs in a given finite time. First, the logical form of BCNs is transformed into an algebraic form using the semi-tensor product (STP) method. Then, the robust output tracking problems of a reference output trajectory and the outputs of a reference system over finite time are transformed into the robust reachability problem of the BCNs. Based on the truth matrix technique, two necessary and sufficient conditions are provided for the trackability of the reference outputs over finite time. Moreover, two algorithms are proposed to design the controllers in the case of the traceable outputs. It should be pointed out that the truth matrix method we used here has some unique features, including its simple computation and concise expression. Finally, two illustrative examples are presented to demonstrate the results in this paper.

1. Introduction

The Boolean network, which was proposed by Kauffman in [1], is an effective tool for modeling gene regulatory networks. In the Boolean network, the values of each node are estimated as “1” and “0” to represent the active state and the inactive state of the genes, respectively. The state of each node at time $t+1$ is determined by the states of its adjacent nodes at time t, and the coupling relationships between the nodes can be expressed by Boolean functions. In addition to gene regulatory networks, Boolean networks have been applied to many other fields, such as artificial neural networks [2], cellular signaling pathways [3], and engineering [4], which are quite different from the continuous models [5,6]. In order to manipulate the dynamics of Boolean networks, exogenous control inputs are introduced, which generate the Boolean control networks (BCNs).

In previous studies, the investigation of BCNs has mainly been based on the mean-field approach, computer simulation, and experimentation. However, it is not easy to systematically investigate the evolutionary processes of the BCNs using these methods. To overcome this difficulty, the semi-tensor product (STP) method to investigate the BCNs was introduced by Prof. Cheng [7]. Based on the STP theory, the evolutionary dynamics of a BCN can be converted into an equivalent algebraic form, which makes it possible for the conventional analysis tools in control theory to be used to analyze and control BCNs. A lot of the results for BCNs are obtained using the STP theory, including but not limited to stability and stabilization [8], function perturbation [9], and disturbance coupling [10,11]. Moreover, the STP method can also be applied to finite automata [12], networked evolutionary games [13], feedback shift registers [14,15], and many other fields [16,17,18].

Influenced by immeasurable variables and limited by measurement methods, some state variables in many Boolean networks cannot be directly measured. As a result of this situation, the states of Boolean networks are usually studied by measuring their outputs. Therefore, some appropriate controllers are designed to make the measured outputs track an ideal signal trajectory. This is called the output trajectory tracking problem of BCNs. In addition, there is another kind of output tracking problem, which aims to track the outputs of a reference system. For example, when investigating the large-scale behaviors of a lactose regulation system with the Escherichia coli bacteria, Julius and co-authors presented a novel feedback control approach to make the proportion of induced cells, which is the output of the lactose regulation system, attain the optimal range [19].

These two kinds of output tracking problems have been studied by some scholars via the STP method. Using the augmented system method, the output tracking problem of (switched) BCNs with a reference system have also been discussed [20,21]. More research results on output tracking problems are shown in Figure 1.

Figure 1. Research status of output tracking problems.

It should be noted that most papers investigate output tracking problems over infinite time. However, considering the cost of the controllers, research on output tracking and the corresponding control design over infinite time is impractical. Thus, it is of great significance to study the output tracking problem of BCNs over finite time. The finite-time tracking control problem of BCNs have been discussed, and some interesting results have been given [22]. Those results were then extended to probabilistic BCNs, and some necessary and sufficient conditions were established in order to discuss the traceability of the reference output trajectory with probability one [23].

As stated above, there are some results on the output tracking problem over finite time, but these results did not take the impact of disturbances into account. As the disturbances are ubiquitous in the process of modeling BCNs, some unanticipated behaviors may occur in the system. Taking gene regulation as an example, it is a noisy process in essence, which is always affected by intracellular and extracellular disturbances. Intracellular disturbances may be gene mutation, duplication, or the deletion of fragments in gene recombination, while extracellular disturbances may be environmental stimulation. These disturbance inputs may obstruct the control strategies to maintain the cellular state of biological systems inside an ideal domain. Therefore, it is very necessary to design the control scheme in order to deal with the negative effects of the disturbances. However, as far as we know, few studies have been proposed to discuss these two kinds of robust output tracking problems of BCNs with disturbances over finite time.

Using the truth matrix method, the robust output tracking problems of BCNs over finite time are investigated in this paper. The main results are as follows. First, two necessary and sufficient conditions are provided for the trackability of a reference output trajectory and the outputs of a reference system with finite time. Then, two algorithms are proposed to design the state feedback controllers in the case of traceable outputs. Compared with the existing results, the main advantage of the truth matrix method used in this paper is that the computations involved are very easy and the control corresponding to each state can be determined from the non-zero columns of the truth matrices; on the other hand, the technique used in previous works cannot give the state feedback control directly.

The rest of this paper is organized as follows. Some necessary symbols and results are given in Section 2. The main results are shown in Section 3. Two illustrative examples are presented to confirm the results of this study in Section 4. Finally, a brief conclusion is provided in Section 5.

2. Preliminaries

In this section, some necessary notations and results are given in Table 1.

Table 1. Notations.

Notations	Definitions
$\mathbb{R}$	set of real numbers
${\mathbb{R}}_{v\times r}$	set of $v\times r$ real matrices
$\mathcal{D}$	$\{1,0\}$
${\mathcal{D}}^a$	${\displaystyle \underset{a}{\underbrace{\mathcal{D}\times \mathcal{D}\times \cdots \times \mathcal{D}}}}$
${\mathrm{Col}}_r\left(C\right)$	the r-th column of matrix C
$I_n$	n-order identity matrix
${\delta }_n^r$	${\mathrm{Col}}_r\left(I_n\right)$
${\Delta }_n$	$\{{\delta }_n^r\mid r=1,2,\cdots ,n\}$ ($\Delta :={\Delta }_2$)
${\delta }_n[v_1{v}_2\cdots v_r]$	$\left[{\delta }_n^{v_1}{\delta }_n^{v_2}\cdots {\delta }_n^{v_r}\right]$
${\mathcal{L}}_{v\times r}$	set of $v\times r$ logical matrices
$\mid ·\mid$	the cardinality of a set
⊗	the Kronecker product
*	the Khatri–Rao product
${\mathbf{0}}_r$ (${\mathbf{1}}_r$)	the column vector of length r with all entries equal to 0 (1)
$[a:b]$	set of integer numbers x satisfying $a\le x\le b$
$A^{\mathrm{T}}$	transposition of matrix A
$A_{i,j}$	the $(i,j)$-element of matrix A

Definition 1

([7]).Let λ be the least common multiple of b and p. The STP of $B\in {\mathbb{R}}_{a\times b}$ and $C\in {\mathbb{R}}_{p\times q}$ is defined as follows:

$$B⋉C=(B\otimes I_{\frac{\lambda }{b}})(C\otimes I_{\frac{\lambda }{p}}).$$

(1)

Remark 1

([7]).The STP is a generalization of the conventional matrix product; thus, “⋉” can be omitted in the following.

Lemma 1

([7]).Let $Z\in {\mathbb{R}}_{r\times 1}$, $C\in {\mathbb{R}}_{a\times b}$. Then,

$$ZC=(I_r\otimes C)Z.$$

(2)

To express a Boolean function in algebraic form, we denote “1” ∼ “${\delta }_2^1$” and “0" ∼ “${\delta }_2^2$”, where “∼” represents an equivalence relation. Then, we have the following lemma.

Lemma 2

([7]).Let $f:{\mathcal{D}}^r\mapsto \mathcal{D}$ be a Boolean function. Then, there exists a unique $M_f\in {\mathcal{L}}_{2\times 2^r}$, such that

$$\begin{array}{c}\hfill f(x_1,x_2,\cdots ,x_r)=M_f{⋉}_{i=1}^r{x}_i,\end{array}$$

(3)

where $x_i\in \Delta$, and $M_f$ is the structure matrix of f.

3. Main Results

3.1. Trackability of a Reference Output Trajectory

In this part, the trackability of a reference output trajectory with finite length is considered. First, the dynamics of the BCN with disturbance inputs is described as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{x}_i(t+1)=f_i(U\left(t\right),X\left(t\right),\Xi \left(t\right)),i\in [1:n],\hfill & \\ y_j\left(t\right)=h_j\left(X\left(t\right)\right),j\in [1:p],\hfill \end{array}\right.\end{array}$$

(4)

where $X\left(t\right)=(x_1\left(t\right),x_2\left(t\right),\cdots ,x_n\left(t\right))\in {\mathcal{D}}^n$, $U\left(t\right)=(u_1\left(t\right),u_2\left(t\right),\cdots ,u_m\left(t\right))\in {\mathcal{D}}^m$, $\Xi \left(t\right)=({\xi }_1\left(t\right),{\xi }_2\left(t\right),\cdots ,{\xi }_r\left(t\right))\in {\mathcal{D}}^r$, and $Y\left(t\right)=(y_1\left(t\right),y_2\left(t\right),\cdots ,y_p\left(t\right))\in {\mathcal{D}}^p$ are the states, inputs, disturbances, and outputs of the BCN (4), respectively. $f_i:{\mathcal{D}}^{m+n+r}\mapsto \mathcal{D}$ and $h_j:{\mathcal{D}}^p\mapsto \mathcal{D}$ are Boolean functions.

In this subsection, we consider the design of the state feedback controller, which is expressed as

$$u_i\left(t\right)=g_i\left(X\left(t\right)\right),i\in [1:m]$$

(5)

in order to track a reference output trajectory $Y^1,Y^2,\cdots ,Y^c$, where $Y^t=(y_1^t,y_2^t,\cdots ,y_p^t),t\in [1:c]$, and $Y^i\ne Y^j$ when $i\ne j$, and $g_i$ is a Boolean function. A definition of the trackability of the reference output trajectory is presented below.

Definition 2.

The reference output trajectory $Y^1,Y^2,\cdots ,Y^c$ is trackable for system (4) from a given initial state if there exists a state feedback control (5), such that $Y\left(t\right)=Y^t$ for any disturbance ξ and $t\in [1:c]$.

Using the STP method, (4) and (5) can be converted into algebraic forms, as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}x(t+1)=L\xi \left(t\right)u\left(t\right)x\left(t\right),\hfill & \\ y\left(t\right)=Hx\left(t\right),\hfill \end{array}\right.\end{array}$$

(6)

and

$$u\left(t\right)=Gx\left(t\right),$$

(7)

where $x\left(t\right)={⋉}_{i=1}^n{x}_i\left(t\right)\in {\Delta }_{2^n}$, $u\left(t\right)={⋉}_{i=1}^m{u}_i\left(t\right)\in {\Delta }_{2^m}$, $\xi \left(t\right)={⋉}_{i=1}^r{\xi }_i\left(t\right)\in {\Delta }_{2^r}$ and $y\left(t\right)={⋉}_{j=1}^p{y}_j\left(t\right)\in {\Delta }_{2^p}$, $L\in {\mathcal{L}}_{2^n\times 2^{m+n+r}}$, $H\in {\mathcal{L}}_{2^p\times 2^n}$ and $G\in {\mathcal{L}}_{2^m\times 2^n}$ are the state transition matrix, the output-state incidence matrix, and the state feedback gain matrix of original system (6).

Denote ${\mathcal{D}}^a\sim {\Delta }_{2^a}$, then the algebraic form of $Y^t$ is ${\delta }_{2^n}^{i_t},t\in [1:c]$. Denote

$${\Lambda }_t:=\{{\delta }_{2^n}^j\mid H{\delta }_{2^n}^j={\delta }_{2^p}^{i_t}\},$$

(8)

where $t\in [1:c]$. The output of states in ${\Lambda }_t$ are consistent with $Y^t$. Based on this, we can convert the robust output tracking problem into the robust reachability problem of system (6), where the definition of robust reachability is given below.

Definition 3

([24]).Consider system (6). $\left(\mathrm{i}\right)$ For a state ${\delta }_{2^n}^i$ and a nonempty set $S\subseteq {\Delta }_{2^n}$, S is one step robustly reachable from ${\delta }_{2^n}^i$ if there exists a control ${\delta }_{2^m}^j$, such that $L{\delta }_{2^r}^a{\delta }_{2^m}^j{\delta }_{2^n}^i\in S$ under any disturbance ${\delta }_{2^r}^a\in {\Delta }_{2^r}$.

$\left(\mathrm{ii}\right)$ For two nonempty sets $S_1$ and $S_2$, $S_1$ is one step robustly reachable from $S_2$ if for any state $x_0\in S_2$, $S_1$ is one step robustly reachable from $x_0$.

Next, we consider the trackability of $Y^1,Y^2,\cdots ,Y^c$ from a given initial state ${\delta }_{2^n}^{i^{*}}$. First, denote ${\Lambda }_{c-1}^{\prime }:={\Lambda }_{c-1}\backslash {\delta }_{2^n}^{i^{*}}$ and construct a truth matrix $T_{{\Lambda }_c\mid {\Lambda }_{c-1}^{\prime }}$, as follows:

$$\begin{array}{c}\hfill {\left(T_{{\Lambda }_c\mid {\Lambda }_{c-1}^{\prime }}\right)}_{j,i}=\left\{\begin{array}{c}1,\mathrm{if}L{\delta }_{2^r}^a{\delta }_{2^m}^j{\delta }_{2^n}^i\in {\Lambda }_c,{\delta }_{2^n}^i\in {\Lambda }_{c-1}^{\prime },\forall a\in [1:2^r],\hfill \\ 0,\mathrm{if}\mathrm{otherwise}.\hfill \end{array}\right.\end{array}$$

(9)

Let ${\tilde{\Lambda }}_{c-1}:=\{{\delta }_{2^n}^i\mid {\mathrm{Col}}_i\left(T_{{\Lambda }_c\mid {\Lambda }_{c-1}^{\prime }}\right)\ne {\mathbf{0}}_{2^m}\}.$ If ${\tilde{\Lambda }}_{c-1}=\varnothing$, then for any ${\delta }_{2^n}^i\in {\Lambda }_{c-1}^{\prime }$, there is no control ${\delta }_{2^m}^j$ such that $L{\delta }_{2^r}^a{\delta }_{2^m}^j{\delta }_{2^n}^i\in {\Lambda }_c$ under all disturbances ${\delta }_{2^r}^a\in {\Delta }_{2^r}$. This means that ${\Lambda }_c$ is not one step reachable from any state in ${\Lambda }_{c-1}^{\prime }$ under the influence of disturbances. Thus, in this case, $Y^1,Y^2,\cdots ,Y^c$ is untraceable by system (6). If ${\tilde{\Lambda }}_{c-1}\ne \varnothing$, then for any ${\delta }_{2^n}^i\in {\tilde{\Lambda }}_{c-1}$, there exists at least a control ${\delta }_{2^m}^j$ such that $L{\delta }_{2^r}^a{\delta }_{2^m}^j{\delta }_{2^n}^i\in {\Lambda }_c$ under any disturbance ${\delta }_{2^r}^a\in {\Delta }_{2^r}$, which means that ${\Lambda }_c$ is one step robustly reachable from ${\tilde{\Lambda }}_{c-1}$.

Similarly, $T_{{\tilde{\Lambda }}_{c-1}\mid {\Lambda }_{c-2}^{\prime }}$ can be constructed as follows:

$$\begin{array}{c}\hfill {\left(T_{{\tilde{\Lambda }}_{c-1}\mid {\Lambda }_{c-2}^{\prime }}\right)}_{j,i}=\left\{\begin{array}{c}1,\mathrm{if}L{\delta }_{2^r}^a{\delta }_{2^m}^j{\delta }_{2^n}^i\in {\tilde{\Lambda }}_{c-1},{\delta }_{2^n}^i\in {\Lambda }_{c-2}^{\prime },\forall a\in [1:2^r],\hfill \\ 0,\mathrm{if}\mathrm{otherwise},\hfill \end{array}\right.\end{array}$$

(10)

where ${\Lambda }_{c-2}^{\prime }:={\Lambda }_{c-2}\backslash {\delta }_{2^n}^{i^{*}}$. Let ${\tilde{\Lambda }}_{c-2}:=\{{\delta }_{2^n}^i\mid {\mathrm{Col}}_i\left(T_{{\tilde{\Lambda }}_{c-1}\mid {\Lambda }_{c-2}^{\prime }}\right)\ne {\mathbf{0}}_{2^m}\}.$ If ${\tilde{\Lambda }}_{c-2}=\varnothing$, then trajectory $Y^1,Y^2,\cdots ,Y^c$ is untraceable. If ${\tilde{\Lambda }}_{c-2}\ne \varnothing$, then ${\tilde{\Lambda }}_{c-1}$ is one step robustly reachable from ${\tilde{\Lambda }}_{c-2}$.

After iterative calculations, the results confirm that if ${\tilde{\Lambda }}_{c-1},{\tilde{\Lambda }}_{c-2},\cdots ,{\tilde{\Lambda }}_1$ are not empty sets, then the truth vector $T_{{\tilde{\Lambda }}_1\mid {\delta }_{2^n}^{i^{*}}}$ can be expressed as follows:

$$\begin{array}{c}\hfill {\left(T_{{\tilde{\Lambda }}_1\mid {\delta }_{2^n}^{i^{*}}}\right)}_j=\left\{\begin{array}{c}1,\mathrm{if}L{\delta }_{2^r}^a{\delta }_{2^m}^j{\delta }_{2^n}^{i^{*}}\in {\tilde{\Lambda }}_1,\forall a\in [1:2^r],\hfill \\ 0,\mathrm{if}\mathrm{otherwise}.\hfill \end{array}\right.\end{array}$$

(11)

Through the above discussion, a necessary and sufficient condition is thus given for the traceability of the reference output trajectory $Y^1,Y^2,\cdots ,Y^c$.

Theorem 1.

The reference output trajectory $Y^1,Y^2,\cdots ,Y^c$ is traceable by system (6) from the given initial state ${\delta }_{2^n}^{i^{*}}$ if and only if

$$T_{{\tilde{\Lambda }}_1\mid {\delta }_{2^n}^{i^{*}}}\ne {\mathbf{0}}_{2^m}.$$

(12)

Proof. 

(Sufficiency): If $T_{{\tilde{\Lambda }}_1\mid {\delta }_{2^n}^{i^{*}}}\ne {\mathbf{0}}_{2^m}$, then we have ${\tilde{\Lambda }}_1\ne \varnothing$, and there exists at least a control ${\delta }_{2^m}^j$ such that $L{\delta }_{2^r}^a{\delta }_{2^m}^j{\delta }_{2^n}^{i^{*}}\in {\tilde{\Lambda }}_1$ under any disturbance ${\delta }_{2^r}^a\in {\Delta }_{2^r}$, which means that ${\tilde{\Lambda }}_1$ is one step robustly reachable from ${\delta }_{2^n}^{i^{*}}$. As ${\tilde{\Lambda }}_1\ne \varnothing$, we can easily verify that ${\tilde{\Lambda }}_2,{\tilde{\Lambda }}_3,\cdots ,{\tilde{\Lambda }}_{c-1}$ are not empty sets, and that for any state ${\delta }_{2^n}^i\in {\tilde{\Lambda }}_t$, there exists at least a control ${\delta }_{2^m}^j$ such that $L{\delta }_{2^r}^a{\delta }_{2^m}^j{\delta }_{2^n}^i\in {\tilde{\Lambda }}_{t+1}$ under any disturbance ${\delta }_{2^r}^a\in {\Delta }_{2^r}$, which means that ${\tilde{\Lambda }}_{t+1}$ is one step robustly reachable from ${\tilde{\Lambda }}_t$, $t\in [1:c-1]$. Thus, ${\tilde{\Lambda }}_c$ is robustly reachable from ${\delta }_{2^n}^{i^{*}}$ by iterative calculations. As the output of the states in ${\Lambda }_t$ is consistent with $Y^t$, then $Y^1,Y^2,\cdots ,Y^c$ is traceable by system (6) from the initial state ${\delta }_{2^n}^{i^{*}}$.

(Necessity): If the reference output trajectory $Y^1,Y^2,\cdots ,Y^c$ is traceable from ${\delta }_{2^n}^{i^{*}}$, then we can confirm that ${\tilde{\Lambda }}_{c-1},{\tilde{\Lambda }}_{c-2},\cdots ,{\tilde{\Lambda }}_1\ne \varnothing$, and that ${\tilde{\Lambda }}_{t+1}$ is one step robustly reachable from ${\tilde{\Lambda }}_t$, where ${\tilde{\Lambda }}_c={\Lambda }_c,t\in [1:c-1]$. Moreover, there exists at least a control ${\delta }_{2^m}^j$ such that ${\tilde{\Lambda }}_1$ is one step robustly reachable from the state ${\delta }_{2^n}^{i^{*}}$. Thus, we can affirm that $T_{{\tilde{\Lambda }}_1\mid {\delta }_{2^n}^{i^{*}}}\ne {\mathbf{0}}_{2^m}$.    □

Remark 2.

The computational complexity of deriving the criteria for the traceability of the reference output trajectory is $O\left(({\sum }_{i=1}^{c-1}\mid {\Lambda }_i^{\prime }\mid +1)2^{m+n+r}\right)$. When studying the output tracking problem of a BCN in infinite time, the target set is generally unique. We first need to calculate a control invariant set in this target set and then design the controls such that all the initial states finally converge to this invariant set. However, if we consider the output tracking problem of a BCN in finite time, the target set is usually not unique. We need not compute the control invariant set of any set, but we should ensure that the given initial state can successively reach these target sets.

Then, we consider the design of the controller when the reference output trajectory $Y^1,Y^2,\cdots$, $Y^c$ is robustly traceable from ${\delta }_{2^n}^{i^{*}}$. Algorithm 1 is presented to search for all the controllers based on the truth matrix.

Algorithm 1 Design of controller for tracking a reference output trajectory

Input:${\Lambda }_1,{\Lambda }_2,\cdots ,{\Lambda }_c$ and ${\delta }_{2^n}^{i^{*}}$ Output:G Initialization: ${\tilde{\Lambda }}_c={\Lambda }_c$, $t=c-1$ for$t=c-1:1$do     Construct ${\Lambda }_t^{\prime }={\Lambda }_t\backslash {\delta }_{2^n}^{i^{*}}$ and truth matrices $$\begin{array}{c}\hfill {\left(T_{{\tilde{\Lambda }}_{t+1}\mid {\Lambda }_t^{\prime }}\right)}_{j,i}=\left\{\begin{array}{c}1,\mathrm{if}L{\delta }_{2^r}^a{\delta }_{2^m}^j{\delta }_{2^n}^i\in {\Lambda }_{t+1},{\delta }_{2^n}^i\in {\Lambda }_t^{\prime },\forall a\in [1:2^r],\hfill \\ 0,\mathrm{if}\mathrm{otherwise},\hfill \end{array}\right.\end{array}$$     where ${\tilde{\Lambda }}_t=\{{\delta }_{2^n}^i\mid {\mathrm{Col}}_i\left(T_{{\tilde{\Lambda }}_{t+1}\mid {\Lambda }_t^{\prime }}\right)\ne {\mathbf{0}}_{2^m}\}$ end for    Construct ${\left(T_{{\tilde{\Lambda }}_1\mid {\delta }_{2^n}^{i^{*}}}\right)}_j=\left\{\begin{array}{c}1,\mathrm{if}L{\delta }_{2^r}^a{\delta }_{2^m}^j{\delta }_{2^n}^{i^{*}}\in {\tilde{\Lambda }}_1,\forall a\in [1:2^r],\hfill \\ 0,\mathrm{if}\mathrm{otherwise}.\hfill \end{array}\right.$     Construct ${\mathrm{Col}}_i\left(G\right)\le \left\{\begin{array}{c}{T}_{{\Lambda }_1\mid {\delta }_{2^n}^{i^{*}}},i=i^{*},\hfill \\ {\mathrm{Col}}_i\left({\Sigma }_{t=1}^{c-1}{T}_{{\tilde{\Lambda }}_{t+1}\mid {\Lambda }_t^{\prime }}\right),{\mathrm{Col}}_i\left({\Sigma }_{t=1}^{c-1}{T}_{{\tilde{\Lambda }}_{t+1}\mid {\Lambda }_t^{\prime }}\right)\ne {\mathbf{0}}_{2^m},\hfill \\ {\mathbf{1}}_{2^m},\mathrm{if}\mathrm{otherwise}.\hfill \end{array}\right.$ returnG

Remark 3.

Assume that $Y^1,Y^2,\cdots ,Y^c$ are robustly traceable from ${\delta }_{2^n}^i$. If ${\delta }_{2^n}^i$ is robustly reachable from ${\delta }_{2^n}^j$, that is, ${\delta }_{2^n}^j$ belongs to the robust convergence region of ${\delta }_{2^n}^i$, then $Y^1,Y^2,\cdots ,Y^c$ are also robustly traceable from ${\delta }_{2^n}^j$.

3.2. Trackability of the Outputs of a Reference System

In this subsection, we consider continuously tracking the outputs of a reference system over finite time. At first, the dynamics of the reference BCN can be given as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\widehat{x}}_i(t+1)={\widehat{f}}_i\left(\widehat{X}\left(t\right)\right),i\in [1:n_1],\hfill & \\ {\widehat{y}}_j\left(t\right)={\widehat{h}}_j\left(\widehat{X}\left(t\right)\right),j\in [1:p],\hfill \end{array}\right.\end{array}$$

(13)

where $\widehat{X}\left(t\right)=({\widehat{x}}_1\left(t\right),{\widehat{x}}_2\left(t\right),\cdots ,{\widehat{x}}_{n_1}\left(t\right))\in {\mathcal{D}}^{n_1}$ and $\widehat{Y}\left(t\right)=({\widehat{y}}_1\left(t\right),{\widehat{y}}_2\left(t\right),\cdots ,{\widehat{y}}_p\left(t\right))\in {\mathcal{D}}^p$ are the states and the outputs of system (13), respectively, while ${\widehat{f}}_i:{\mathcal{D}}^{n_1}\mapsto \mathcal{D}$ and ${\widehat{h}}_j:{\mathcal{D}}^p\mapsto \mathcal{D}$ are Boolean functions.

Using the STP method, the algebraic form of system (13) can be expressed as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\widehat{x}(t+1)=\widehat{L}\widehat{x}\left(t\right),\hfill & \\ \widehat{y}\left(t\right)=\widehat{H}\widehat{x}\left(t\right),\hfill \end{array}\right.\end{array}$$

(14)

where $\widehat{x}\left(t\right)={⋉}_{i=1}^{n_1}{\widehat{x}}_i\left(t\right)\in {\Delta }_{2^{n_1}}$ and $\widehat{y}\left(t\right)={⋉}_{j=1}^p{\widehat{y}}_j\left(t\right)\in {\Delta }_{2^p}$, $\widehat{L}\in {\mathcal{L}}_{2^{n_1}\times 2^{n_1}}$ and $\widehat{H}\in {\mathcal{L}}_{2^p\times 2^{n_1}}$ are the state transition matrix and the output-state incidence matrix of reference system (14).

In this section, we want the outputs of system (6) to be consistent with that of reference system (14) for c consecutive times.

Hence, we have the following definition.

Definition 4.

The outputs of reference system (14) are traceable by system (6) for c consecutive times if there exist initial states $x_0$, ${\widehat{x}}_0$ and a state feedback control, such that $y\left(t\right)=\widehat{y}\left(t\right)$ for any disturbance ξ and any $t\in [1:c]$.

The augmented system approach is considered to solve this kind of robust output tracking problem. Denote $z\left(t\right):=x\left(t\right)\widehat{x}\left(t\right)$ and $w\left(t\right):=y\left(t\right)\widehat{y}\left(t\right)$. Then, an augmented system can be obtained with the following equation:

$$\begin{array}{cc}\hfill z(t+1)=& L\xi \left(t\right)u\left(t\right)x\left(t\right)\widehat{L}\widehat{x}\left(t\right)\hfill \\ \hfill =& L(I_{2^{n+m+r}}\otimes \widehat{L})\xi \left(t\right)u\left(t\right)x\left(t\right)\widehat{x}\left(t\right)\hfill \\ \hfill =& (L\otimes I_{2^{n_1}})(I_{2^{n+m+r}}\otimes \widehat{L})\xi \left(t\right)u\left(t\right)z\left(t\right)\hfill \\ \hfill =& (L\otimes \widehat{L})\xi \left(t\right)u\left(t\right)z\left(t\right).\hfill \end{array}$$

(15)

Moreover, the outputs of the augmented system can be expressed as follows:

$$\begin{array}{cc}\hfill w\left(t\right)=& Hx\left(t\right)\widehat{H}\widehat{x}\left(t\right)\hfill \\ \hfill =& H(I_{2^n}\otimes \widehat{H})x\left(t\right)\widehat{x}\left(t\right)\hfill \\ \hfill =& (H\otimes I_{2^p})(I_{2^n}\otimes \widehat{H})z\left(t\right)\hfill \\ \hfill =& (H\otimes \widehat{H})z\left(t\right).\hfill \end{array}$$

(16)

Then, the algebraic form of the augmented system can be rewritten as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}z(t+1)=K\xi \left(t\right)u\left(t\right)z\left(t\right),\hfill & \\ w\left(t\right)=Jz\left(t\right),\hfill \end{array}\right.\end{array}$$

(17)

where $K=L\otimes \widehat{L}\in {\mathcal{L}}_{2^{(n+n_1)}\times 2^{(n+n_1+m+r)}}$ and $J=H\otimes \widehat{H}\in {\mathcal{L}}_{2^{2p}\times 2^{(n+n_1)}}$ are the state transition matrix and the output-state incidence matrix of augmented system (17), respectively.

For augmented system (17), we consider the design of the state feedback controller, which has the following form:

$$u_i\left(t\right)=g_i^{\prime }(X\left(t\right),\widehat{X}\left(t\right)),i\in [1:m].$$

(18)

Then, (18) has the following algebraic form:

$$u\left(t\right)=Rz\left(t\right),$$

(19)

where $R=G_1^{\prime }*G_2^{\prime }*\cdots *G_m^{\prime }$, $G_i^{\prime }$ is the structure matrix of $g_i^{\prime }$.

If the outputs of reference system (14) can be traced by system (6) for c consecutive times, then there exist a state feedback control (19) and initial states $x_0$, ${\widehat{x}}_0$, such that

$$y\left(t\right)=\widehat{y}\left(t\right)$$

(20)

for any disturbance $\xi$ and any $t\in [1:c]$, which means that the state trajectory starting from $z_0$ of the augmented system (17) will stay c steps in set $\{x\widehat{x}\mid Hx=\widehat{H}\widehat{x}\}$, where $z_0=x_0{\widehat{x}}_0$.

Construct the following:

$$\Omega :=\{{\delta }_{2^{n+n_1}}^i\mid J{\delta }_{2^{n+n_1}}^i={\delta }_{2^p}^j⋉{\delta }_{2^p}^j,j\in [1:2^p]\}.$$

(21)

This type of robust output tracking problem can be converted into the robust reachability problem of augmented system (17).

Next, we consider the trackability of this kind of robust output tracking problem. Construct a truth matrix $T_{\Omega }$ as follows:

$$\begin{array}{c}\hfill {\left(T_{\Omega }\right)}_{j,i}=\left\{\begin{array}{c}1,\mathrm{if}K{\delta }_{2^r}^a{\delta }_{2^m}^j{\delta }_{2^{n+n_1}}^i\in \Omega ,{\delta }_{2^{n+n_1}}^i\in \Omega ,\forall a\in [1:2^r],\hfill \\ 0,\mathrm{if}\mathrm{otherwise}.\hfill \end{array}\right.\end{array}$$

(22)

Let ${\Omega }_1:=\{{\delta }_{2^{n+n_1}}^i\mid {\mathrm{Col}}_i\left(T_{\Omega }\right)\ne {\mathbf{0}}_{2^m}\};$ obviously, ${\Omega }_1\subseteq \Omega .$ If ${\Omega }_1=\varnothing$, then for any ${\delta }_{2^{n+n_1}}^i\in \Omega$, there is no control ${\delta }_{2^m}^j$ such that $K{\delta }_{2^r}^a{\delta }_{2^m}^j{\delta }_{2^{n+n_1}}^i\in \Omega$ under all disturbances ${\delta }_{2^r}^a\in {\Delta }_{2^r}$. Thus, the outputs of reference system (14) are untraceable by system (6). If ${\Omega }_1\ne \varnothing$, then for any ${\delta }_{2^{n+n_1}}^i\in {\Omega }_1$, there exists at least a control ${\delta }_{2^m}^j$ such that $K{\delta }_{2^r}^a{\delta }_{2^m}^j{\delta }_{2^{n+n_1}}^i\in \Omega$ under any disturbance ${\delta }_{2^r}^a\in {\Delta }_{2^r}$, which means that any state in ${\Omega }_1$ will still evolve in $\Omega$ within the subsequent time interval.

Similarly, we can construct $T_{{\Omega }_1}$ as follows:

$$\begin{array}{c}\hfill {\left(T_{{\Omega }_1}\right)}_{j,i}=\left\{\begin{array}{c}1,\mathrm{if}K{\delta }_{2^r}^a{\delta }_{2^m}^j{\delta }_{2^{n+n_1}}^i\in {\Omega }_1,{\delta }_{2^{n+n_1}}^i\in {\Omega }_1,\forall a\in [1:2^r],\hfill \\ 0,\mathrm{if}\mathrm{otherwise}.\hfill \end{array}\right.\end{array}$$

(23)

Let ${\Omega }_2:=\{{\delta }_{2^{n+n_1}}^i\mid {\mathrm{Col}}_i\left(T_{{\Omega }_1}\right)\ne {\mathbf{0}}_{2^m}\};$ obviously, ${\Omega }_2\subseteq {\Omega }_1\subseteq \Omega$. If ${\Omega }_2=\varnothing$, then the outputs of reference system (14) are untraceable by system (6). If ${\Omega }_2\ne \varnothing$, then any state in ${\Omega }_2$ will evolve in ${\Omega }_1$ in the subsequent time interval.

After iterative calculations, we verify that if ${\Omega }_1,{\Omega }_2,\cdots ,{\Omega }_{c-1}$ are not empty sets, then the truth matrix $T_{{\Omega }_{c-1}}$ can be expressed as follows:

$$\begin{array}{c}\hfill {\left(T_{{\Omega }_{c-1}}\right)}_{j,i}=\left\{\begin{array}{c}1,\mathrm{if}K{\delta }_{2^r}^a{\delta }_{2^m}^j{\delta }_{2^{n+n_1}}^i\in {\Omega }_{c-1},{\delta }_{2^{n+n_1}}^i\in {\Omega }_{c-1},\forall a\in [1:2^r],\hfill \\ 0,\mathrm{if}\mathrm{otherwise}.\hfill \end{array}\right.\end{array}$$

(24)

Denote ${\Omega }_c:=\{{\delta }_{2^{n+n_1}}^i\mid {\mathrm{Col}}_i\left(T_{{\Omega }_{c-1}}\right)\ne {\mathbf{0}}_{2^m}\}$; obviously, ${\Omega }_c\subseteq {\Omega }_{c-1}\subseteq \cdots \subseteq \Omega$.

After the above discussion, a necessary and sufficient condition is presented for the traceability of the outputs of reference system (14) for c consecutive times.

Theorem 2.

The outputs of reference system (14) are trackable by system (6) for c consecutive times if and only if

$${\Omega }_c\ne \varnothing .$$

(25)

Proof. 

(Sufficiency): If ${\Omega }_c\ne \varnothing$, then ${\Omega }_{c-1}\ne \varnothing$ and ${\Omega }_{c-1}$ is one step robustly reachable from ${\Omega }_c$. As ${\Omega }_{c-1}\ne \varnothing$, we can easily confirm that ${\Omega }_{c-2},{\Omega }_{c-3},\cdots ,{\Omega }_1\ne \varnothing$, and for any ${\delta }_{2^{n+n_1}}^i\in {\Omega }_{t+1}$, there exists at least a control ${\delta }_{2^m}^j$ such that $K{\delta }_{2^r}^a{\delta }_{2^m}^j{\delta }_{2^{n+n_1}}^i\in {\Omega }_t$ under any disturbance ${\delta }_{2^r}^a\in {\Delta }_{2^r}$, which means that ${\Omega }_t$ is one step robustly reachable from ${\Omega }_{t+1}$, $t\in [1:c-1]$. Thus, $\Omega$ is robustly reachable from ${\Omega }_c$. As ${\Omega }_c\subseteq {\Omega }_{c-1}\subseteq \cdots \subseteq \Omega$, we can confirm that the outputs of reference system (14) are traceable by system (6) for c consecutive times.

(Necessity): If the outputs of reference system (14) are trackable by system (6) for c consecutive times, then we can verify that there exists at least one initial state $z_0$, such that the state trajectory lasts c steps in $\Omega$. This means that ${\Omega }_1,{\Omega }_2,\cdots ,{\Omega }_{c-1}$ are not empty sets, and that the initial state ${\delta }_{2^n}^{i^{*}}$ will evolve into set ${\Omega }_{c-1}$. Thus, ${\Omega }_c\ne \varnothing .$    □

Remark 4.

The computational complexity of deriving the criteria for the traceability of the outputs of the reference system is $O\left(({\sum }_{i=0}^{c-1}\mid {\Omega }_i\mid )2^{m+n+n_1+r}\right)$, where ${\Omega }_0=\Omega .$

If the outputs of reference system (14) are continuously traceable for c consecutive times, then we give the method for designing the state feedback controllers. Algorithm 2 is presented to search for all the controllers based on the truth matrix method.

Algorithm 2 Design of controller for tracking the outputs of a reference system over finite time

Input:$\Omega$ Output:R Initialization: ${\Omega }_0=\Omega$, $t=0$ for$t=0:c-1$do     Construct $$\begin{array}{c}\hfill {\left(T_{{\Omega }_t}\right)}_{j,i}=\left\{\begin{array}{c}1,\mathrm{if}K{\delta }_{2^r}^a{\delta }_{2^m}^j{\delta }_{2^{n+n_1}}^i\in {\Omega }_t,{\delta }_{2^{n+n_1}}^i\in {\Omega }_t,\forall a\in [1:2^r],\hfill \\ 0,\mathrm{if}\mathrm{otherwise},\hfill \end{array}\right.\end{array}$$      and ${\Omega }_{t+1}:=\{{\delta }_{2^{n+n_1}}^i\mid {\mathrm{Col}}_i\left(T_{{\Omega }_t}\right)\ne {\mathbf{0}}_{2^m}\}$ end for for$t=0:c-2$do     Construct $$\begin{array}{c}\hfill {\left(T_{{\Omega }_t\mid ({\Omega }_{t+1}\backslash {\Omega }_{t+2})}\right)}_{j,i}=\left\{\begin{array}{c}1,\mathrm{if}K{\delta }_{2^r}^a{\delta }_{2^m}^j{\delta }_{2^{n+n_1}}^i\in {\Omega }_t,{\delta }_{2^{n+n_1}}^i\in ({\Omega }_{t+1}\backslash {\Omega }_{t+2}),\forall a\in [1:2^r],\hfill \\ 0,\mathrm{if}\mathrm{otherwise},\hfill \end{array}\right.\end{array}$$ end for    Construct ${\mathrm{Col}}_i\left(R\right)\le \left\{\begin{array}{c}{\mathbf{1}}_{2^m},{\mathrm{Col}}_i({\Sigma }_{t=0}^{c-2}{T}_{{\Omega }_t\mid ({\Omega }_{t+1}\backslash {\Omega }_{t+2})}+T_{{\Omega }_{c-1}})={\mathbf{0}}_{2^m},\hfill \\ {\mathrm{Col}}_i({\Sigma }_{t=0}^{c-2}{T}_{{\Omega }_t\mid ({\Omega }_{t+1}\backslash {\Omega }_{t+2})}+T_{{\Omega }_{c-1}}),\mathrm{if}\mathrm{otherwise}.\hfill \end{array}\right.$ returnR

4. Illustrative Examples

In this section, two illustrative examples are presented to verify the main results in this paper. Some basic logical operators are given: “¬” (negation), “∧” (conjunction), “∨” (disjunction), “→” (conditional), “↔” (biconditional), “↑” (not and).

Example 1.

Consider the following simplified model for the E. coli lactose operon network with disturbances:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{x}_1(t+1)=\neg u_1\left(t\right)\wedge (x_2\left(t\right)\vee x_3\left(t\right)),\hfill & \\ x_2(t+1)=(\neg u_1\left(t\right)\wedge u_2\left(t\right))\wedge (x_1\left(t\right)\vee \xi \left(t\right)),\hfill & \\ x_3(t+1)=\neg u_1\left(t\right)\wedge (u_2\left(t\right)\vee (u_3\left(t\right)\wedge x_1\left(t\right))),\hfill & \\ y_1\left(t\right)=x_1\left(t\right),\hfill & \\ y_2\left(t\right)=x_3\left(t\right),\hfill \end{array}\right.\end{array}$$

(26)

where $x_1,x_2$, and $x_3$ are state variables representing lactose messenger RNA, high-concentration lactose, and medium-concentration lactose, respectively; $u_1,u_2$, and $u_3$ are control inputs, indicating extracellular glucose, high exolactose, and medium exolactose, respectively; $y_1$ and $y_2$ are the outputs; ξ is the disturbance input. The network diagram of system (26) is shown in Figure 2:

Figure 2. Network diagram of system (26).

Then, the algebraic form of system (26) can be expressed as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}x(t+1)=L\xi \left(t\right)u\left(t\right)x\left(t\right),\hfill & \\ y\left(t\right)=Hx\left(t\right),\hfill \end{array}\right.\end{array}$$

(27)

where $x\left(t\right)={⋉}_{i=1}^3{x}_i\left(t\right)\in {\Delta }_8$, $u\left(t\right)={⋉}_{i=1}^3{u}_i\left(t\right)\in {\Delta }_8$, $\xi \left(t\right)\in \Delta$ and $y\left(t\right)={⋉}_{p=1}^2{y}_p\left(t\right)\in {\Delta }_4$, and

$$\begin{array}{cc}\hfill & L={\delta }_8[88888888\cdots 44484448],\hfill \\ \hfill & H={\delta }_4\left[12123434\right].\hfill \end{array}$$

Assume that the reference output trajectory is as shown below.

t	1	2	3	4
Y	${\delta }_4^1$	${\delta }_4^2$	${\delta }_4^3$	${\delta }_4^4$

We then detect whether the initial state ${\delta }_8^5$ can track this output trajectory. According to (8), we have ${\Lambda }_1=\{{\delta }_8^1,{\delta }_8^3\}$, ${\Lambda }_2=\{{\delta }_8^2,{\delta }_8^4\}$, ${\Lambda }_3=\{{\delta }_8^5,{\delta }_8^7\}$, and ${\Lambda }_4=\{{\delta }_8^6,{\delta }_8^8\}$.

By Algorithm 1, we can obtain the following:

$${\tilde{\Lambda }}_3=\left\{{\delta }_8^7\right\},{\tilde{\Lambda }}_2=\left\{{\delta }_8^4\right\},{\tilde{\Lambda }}_1=\{{\delta }_8^1,{\delta }_8^3\}$$

and

$$T_{{\tilde{\Lambda }}_4\mid ({\Lambda }_3\backslash {\delta }_8^5)}=\left[\begin{array}{cccccccc}0& 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right],T_{{\tilde{\Lambda }}_3\mid ({\Lambda }_2\backslash {\delta }_8^5)}=\left[\begin{array}{cccccccc}0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right],$$

$$T_{{\tilde{\Lambda }}_2\mid ({\Lambda }_1\backslash {\delta }_8^5)}=\left[\begin{array}{cccccccc}0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 1& 0& 1& 0& 0& 0& 0& 0\end{array}\right],T_{{\tilde{\Lambda }}_1\mid {\delta }_8^5}={\left[\begin{array}{cccccccc}0& 0& 0& 0& 1& 1& 0& 0\end{array}\right]}^{\mathrm{T}}.$$

As $T_{{\tilde{\Lambda }}_1\mid {\delta }_8^5}\ne {\mathbf{0}}_8$, we can verify that this reference output trajectory is traceable by system (26) from ${\delta }_8^5$ under any disturbance.

Next, according to Algorithm 1, one state feedback gain matrix G can be given as follows:

$$G={\delta }_8\left[81875328\right].$$

Thus, to track the reference output trajectory from ${\delta }_8^5$ by system (26), the state feedback controller (7) can be expressed as follows:

$$u\left(t\right)={\delta }_8\left[81875328\right]x\left(t\right).$$

(28)

To make the results more visual, we show the evolutionary trajectories of the initial state ${\delta }_8^5$ under controller (28) in Figure 3.

Figure 3. Evolutionary trajectories of initial state ${\delta }_8^5$ under controller (28).

If we study the output tracking problem of BCN (26) over infinite time, we should give a constant reference signal or a periodic signal and then confirm the target state set according to the output signals. Next, we need to calculate a control invariant set in this target set and make the given initial state ${\delta }_8^5$ converge to this invariant set. However, if we study the output tracking problem in finite time, we just need to ensure that state ${\delta }_8^5$ can successively reach sets ${\Lambda }_1$, ${\Lambda }_2$, ${\Lambda }_3$, and ${\Lambda }_4$.

Example 2.

Consider a BCN with the following disturbances:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{x}_1(t+1)=\xi \left(t\right)\wedge [(u_1\left(t\right)\to x_1\left(t\right))\uparrow (u_2\left(t\right)\vee x_2\left(t\right))],\hfill & \\ x_2(t+1)=(u_1\left(t\right)\wedge x_2\left(t\right))\leftrightarrow (u_2\left(t\right)\vee x_1\left(t\right)),\hfill & \\ y\left(t\right)=x_1\left(t\right)\wedge x_2\left(t\right),\hfill & \end{array}\right.\end{array}$$

(29)

where $x_1$ and $x_2$ are the state variables, $u_1$ and $u_2$ are control inputs, y is the output, and ξ is the disturbance input. The network diagram for system (29) is shown in Figure 4:

Figure 4. Network diagram of system (29).

BCN (29) can be converted into the following algebraic form:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}x(t+1)=L\xi \left(t\right)u\left(t\right)x\left(t\right),\hfill & \\ y\left(t\right)=Hx\left(t\right),\hfill \end{array}\right.\end{array}$$

(30)

where $x\left(t\right)={⋉}_{i=1}^2{x}_i\left(t\right)\in {\Delta }_4$, $u\left(t\right)={⋉}_{i=1}^2{u}_i\left(t\right)\in {\Delta }_4$, $\xi \left(t\right)\in \Delta$, $y\left(t\right)\in \Delta$, and

$$\begin{array}{cc}\hfill & L={\delta }_4\left[34343423242424133434342344444413\right],\hfill \\ \hfill & H={\delta }_2\left[1222\right].\hfill \end{array}$$

A reference BN is shown below:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\widehat{x}}_1(t+1)={\widehat{x}}_3\left(t\right),\hfill \\ {\widehat{x}}_2(t+1){=}^{\neg }({\widehat{x}}_1\left(t\right)\vee {\widehat{x}}_2\left(t\right))\vee ({\widehat{x}}_1\left(t\right)\wedge {\widehat{x}}_2\left(t\right)\wedge {\widehat{x}}_3\left(t\right)),\hfill \\ {\widehat{x}}_3(t+1){=}^{\neg }{\widehat{x}}_1\left(t\right)\wedge {\widehat{x}}_3\left(t\right),\hfill \\ \widehat{y}\left(t\right)=[{\widehat{x}}_1\left(t\right)\wedge ({\widehat{x}}_2\left(t\right)\leftrightarrow {\widehat{x}}_3\left(t\right))]\vee {[}^{\neg }{\widehat{x}}_1\left(t\right)\wedge ({\widehat{x}}_2\left(t\right)\to {\widehat{x}}_3\left(t\right))],\hfill \end{array}\right.\end{array}$$

(31)

where ${\widehat{x}}_1$, ${\widehat{x}}_2$, and ${\widehat{x}}_3$ are the state variables, and $\widehat{y}$ is the output. The network diagram for system (31) is shown in Figure 5.

Figure 5. Network diagram of system (31).

Let $\widehat{x}\left(t\right)={⋉}_{i=1}^2{\widehat{x}}_i\left(t\right)$, and we can obtain the algebraic form of system (31) as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\widehat{x}(t+1)=\widehat{L}\widehat{x}\left(t\right),\hfill \\ \widehat{y}\left(t\right)=\widehat{H}\widehat{x}\left(t\right),\hfill \end{array}\right.\end{array}$$

(32)

where $\widehat{L}={\delta }_8\left[28483816\right]$ and $\widehat{H}={\delta }_2\left[12211211\right].$

Denote $z\left(t\right):=x\left(t\right)\widehat{x}\left(t\right)$ and $w\left(t\right):=y\left(t\right)\widehat{y}\left(t\right)$, then an augmented system can be generated by combining system (30) and reference system (32) as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}z(t+1)=K\xi \left(t\right)u\left(t\right)z\left(t\right),\hfill & \\ w\left(t\right)=Jz\left(t\right),\hfill \end{array}\right.\end{array}$$

(33)

where $K={\delta }_{32}[1824202419241722\cdots 1824202419241722]$ and $J={\delta }_4[12211211$

$3443343334433433433433].$

We consider continuously tracing the outputs of reference system (32) with system (31) for 3 consecutive times.

Assume that the initial state of the augmented system is ${\delta }_{32}^{23}$. According to (21), we can calculate the pairs of states that keep the outputs of systems (31) and (32) consistent, and we can further confirm that

$$\Omega =\{{\delta }_{32}^1,{\delta }_{32}^4,{\delta }_{32}^5,{\delta }_{32}^7,{\delta }_{32}^8,{\delta }_{32}^{10},{\delta }_{32}^{11},{\delta }_{32}^{14},{\delta }_{32}^{18},{\delta }_{32}^{19},{\delta }_{32}^{22},{\delta }_{32}^{26},{\delta }_{32}^{27},{\delta }_{32}^{30}\}.$$

By Algorithm 2, we can obtain the following values: ${\Omega }_1=\{{\delta }_{32}^1,{\delta }_{32}^5,{\delta }_{32}^8,{\delta }_{32}^{18},{\delta }_{32}^{19}\}$, ${\Omega }_2=\{{\delta }_{32}^1,{\delta }_{32}^5,{\delta }_{32}^{18}\}\ne \varnothing .$ Thus, the outputs of reference system (32) are continuously tracked by system (31) for three consecutive times.

According to Algorithm 2, we can obtain a possible state feedback gain matrix R as follows:

$$R={\delta }_4\left[13244123411423213424214323211423\right].$$

Thus, in order to continuously track the outputs of reference system (32) with system (30) for three consecutive times, the state feedback controller (19) can be expressed as follows:

$$u\left(t\right)={\delta }_4\left[13244123411423213424214323211423\right]x\left(t\right)\widehat{x}\left(t\right).$$

(34)

The evolutionary trajectories of the initial state ${\delta }_{32}^{23}$ under controller (34) is shown in Figure 6.

Figure 6. Evolutionary trajectories of initial state ${\delta }_{32}^{23}$ under control (34).

When studying the output tracking problem in infinite time, there must be some state trajectories in set $\Omega$ that are evolving all the time, which requires set $\Omega$ to contain at least one control invariant set. However, when we study the output tracking problem in finite time, we just need to ensure that there exists a state trajectory that can evolve in set $\Omega$ for a finite number of steps.

5. Conclusions

Taking advantage of the STP method, classical control theory has been used to solve two kinds of robust output tracking problems of BCNs. First, two necessary and sufficient conditions were provided for the trackability of a reference output trajectory and the outputs of a reference system with finite length. Then, two algorithms were proposed to design the state feedback controllers in the case where the outputs could be tracked. The STP method has to perform a large amount of computation when dealing with networks with many nodes; thus, two illustrative examples using a few nodes were presented to demonstrate the theory results in this paper. The methodology and outline of this paper are shown in Figure 7:

Figure 7. The methodology and outline of this paper.

Our method can also be used to study the stability and control problems of many other finite-valued dynamic systems. In the future, we will study the influence of disturbance inputs on the consensus of multi-agent systems over finite fields, and we will propose some criteria for the finite-time consistency of these systems.