Research Article | Open Access
Robust Invariant Set Analysis of Boolean Networks
In this paper, the robust invariant set (RIS) of Boolean (control) networks with disturbances is investigated. First, for a given fixed point, consider a special set called immediate neighborhoods of the fixed point; then a discrete derivative of Boolean functions at the fixed point is used to analyze the robust invariance, based on which a sufficient condition is obtained. Second, for more general sets, the robust output control invariant set (ROCIS) of Boolean control networks (BCNs) is investigated by semitensor product (STP) of matrices. Then, under a given output feedback controller, we obtain a necessary and sufficient condition to check whether a given set is robust control invariant set (RCIS). Furthermore, output feedback controllers are designed to make a set to be a RCIS. Finally, the proposed methods are illustrated by a reduced model of the lac operon in E. coli.
In 1969, Kauffman  firstly used Boolean networks (BNs), which are a kind of logical networks to study genetic regulatory networks. Then, BNs have attracted extensive attention and have become abstract modeling schemes in other different fields such as neural networks  and immune response . In BNs, each gene expression has two states as “1” and “0” to represent “on” and “off,” respectively. Moreover, interactions between the states of each gene depend on Boolean functions, which are composed of logical operators such as disjunction and conjunction and so on. The state evolution for each gene is updated by a Boolean function about the states of current neighborhoods.
When control inputs are added into BNs, then BNs can be called BCNs. Similarly, control inputs take two values: “0” implies that the application of that intervention is ceased at that time point, and “1” means that some interventions are applied in BNs. A new matrix product called semitensor product (STP) of matrices was proposed to investigate BNs and BCNs . Based on this, a BN (or BCN) can be converted into the corresponding algebraic form by calculating its unique transition matrix. Therefore, many fundamental and interesting problems have been investigated for BNs and BCNs, such as the controllability [5, 6], stabilization [7–15], observability [16–19], disturbance decoupling problem  synchronization , function perturbations , optimal control [23–26], normalization problem , and others. The STP has also been widely applied in games [28, 29] and asynchronous sequential machines [30, 31]. In detail,  investigated the evolutionarily stable strategy of finite evolutionary networked games by STP and then designed event-triggered controllers such that systems could converge globally. In , the stochastic set stabilization of random evolutionary Boolean games was further investigated, and a constructive algorithm was proposed to calculate stochastic reachable sets. On the other hand,  and  investigated the reachability and skeleton matrix by STP, respectively.
Usually, some external disturbance inputs always exist in systems. For example, cancer can be defined as failures in the healthy mechanisms of biological systems, and it has been classified as a kind of genetic uncertainties consisting of mutations . Therefore, designing controllers is of great importance such that the set of desirable cellular states of BCNs with disturbance inputs is robust. In other words, if the trajectories of BCNs with some initial states reach a given set, which is called robust control invariant set (RCIS) , then those trajectories will never leave the set no matter what disturbances are. The RCIS has attracted many scholars’ attention and obtained many results [34–37]. In , Li et al. used STP to study the RCIS of BCNs and presented an effective procedure to design state feedback controllers. However, due to the limitation of measurement conditions and the impact of immeasurable variables, measured output information rather than state information is always used to analyze and control systems . Therefore, we will design output feedback controllers such that the trajectories of BCNs starting from some initial states in a given set will never leave the given set, which is called robust output control invariant set (ROCIS). On the other hand, Robert analyzed the local convergence of BNs by the discrete derivative of Boolean functions at a fixed point , which is a novel approach to investigate BNs. Motivated by this, we use the discrete derivative method to investigate the RIS, which makes the computational complexity reduced compared with STP. To the best of our knowledge, there is no result concerning the ROCIS of BCNs. The contributions of this paper are listed as follows:(i)The discrete derivative is used to analyze the robust invariance.(ii)A necessary and sufficient condition is derived to check whether a given set is ROCIS under a given output feedback controller.(iii)Output feedback controllers are designed to make a given set be a ROCIS.
The rest of this paper is organized as follows. Section 2 reviews some notations and preliminary results, which will be used in the latter. Section 3 presents the main results. Section 4 ends the paper with a brief conclusion.
In this section, we give some necessary properties of the STP and list some useful notations.(i)Let be the set for integers .(ii)Denote by matrix the set of real matrices.(iii)Set , where is the -th column of identity matrix .(iv)Let stand for the -th column of the matrix B.(v)Matrix is called a logical matrix, and simply denote it by .(vi)Let denote the row vector of length with all entries being .
Definition 1 (). For any given two matrices and , the STP of and is defined as where is the least common multiple of and and is the Kronecker product.
Proposition 2 (). Let be a column and be any matrix. Then
Definition 3 (). Let . Then, the Khatri-Rao product is defined as
The notation represents that when the two values denoted by and taking from are the same, then . Otherwise, the result equals .
A logical domain, denoted by is defined as , and . If identifying and , then , and , where “” represents two different forms of the same object. If , then we say is in a scalar form. If , we say is in a vector form. In the sequel, to distinguish the scalar form and the vector form of a variable, we use the notation for a variable in , while we use the notation for its vector variable in . In short, , but .
The following lemma is important for the algebraic expressions of logical functions.
Lemma 4 (). Consider a logical function . There exists a unique matrix , called the structure matrix of , such that where .
For example, the structure matrix of logical function is . And the structure matrix of logical function is .
Definition 5. Assume that there are -dimensional row vectors and , , , , respectively. The distance vector denoted by between and is defined; that is, .
Assume that , and then define a bijection , wherewith . Furthermore, we can also define , wherewith and ,
For example, assume that ; then, by (5), one has that .
3. Main Results
We consider the following system with disturbance inputs:where and . is a logical function.
For each , there exists a unique structure matrix , by Lemma 4. Let and , then system (7) can be converted into the following forms: where and . Multiplying all the equations in (8) together, we getwhere with is the Khatri-Rao product . Split into equal blocks denoted by , and then one has
Given a state denoted by , we denote the immediate neighborhoods of by ; that is, where represents the -th column of identify matrix . For example, assume that , and then, based on the above definition, one has that . Assume that , where , . Let .
Definition 6. Assume that two -dimensional Boolean row vectors and . We say that if, for any , .
For example, there are two vectors as and . We have , and , and then .
Definition 7. Consider system (7), and assume that is one of the fixed points for any disturbance. If, for any and disturbance input , , then is said to be a RIS.
Definition 8 (). Consider system (7) with disturbance inputs. For any given state and when , the discrete derivative of Boolean functions at state , denoted by , is an Boolean matrix with its elements given as follows: or
Based on Definition 8, we further define another Boolean matrix; that is, It can be learned from the construction of that for any , , and , we get
Theorem 9. Consider system (7) with the fixed point for any disturbance. If has at most one 1 in each column, then is a RIS.
Proof. If each column of can only be zero vector or a basis vector, then it can be learned from (15) that the state can be guaranteed in the set beginning from any initial state , , in every step, which completes the proof.
Example 10. Consider a BN with the fixed point being (1, 1) for any disturbance:One has and . Let and , respectively, and we can get and Therefore, Since and , then Theorem 9 holds. Therefore, is a RIS.
Remark 11. It is learned from Theorem 9 that matrix with dimension (not ) can be constructed, based on which the RIS is analyzed. Therefore, compared with the results obtained by STP in , the computational complexity is reduced from to . Unfortunately, the method of discrete derivative can only be used to analyze some special systems in the form of (7). In the sequel, we will further analyze more general sets and design output feedback controllers such that the set is robust for any disturbance inputs.
In the following, consider a BCN with nodes, outputs, disturbance inputs, and inputs aswhere and are logical functions, , .
Then, the definition of a ROCIS of BCNs is presented as follows.
Definition 12. Consider system (20). A nonempty set is said to be a ROCIS, if there exists an output feedback control aswhere are Boolean functions such that, for the closed-loop system consisting of (20) and (21), implies .
It can be learned from Lemma 4 that we can find unique structure matrices , , and for logical functions , , and in (20) and (21), respectively, , . Then the equivalent forms of (20) and (21) areandrespectively, where , , , and .
Assume that a given nonempty set with , and we analyze the following two problems:(i)Problem 1: For a given output feedback matrix , analyze whether the set is a ROCIS of system (20) under control system .(ii)Problem 2: Design output feedback controllers such that is a ROCIS of system (20).
Theorem 13. For a given set and an output feedback gain matrix , set is a ROCIS of system (20) under the output feedback control if and only if, for any , , and then .
Proof. It is easy to see that the sufficiency holds, and we only need to prove the necessity. Consider (28); since is a ROCIS of system (20) under the output feedback control , then, for any , , and any , one hasFrom the arbitrariness of and , the conclusion holds.
Remark 14. As mentioned above that set is equivalent to . Based on STP, the algebraic expression of system (20) can be obtained. Therefore, it will be better to write set in vector form in Theorem 13 compared with Definition 12.
In the following, we discuss Problem 2, and then controller (25) will be designed. Suppose that and , which will be designed. On one hand, for system (26) with , , and , one has that On the other hand, we define the following sets: Obviously, . For each , and then denote
For any integer , defineFor each , we construct a set, denoted by , asThen, the result about the existence of output feedback controllers can be obtained.
Proof. (Sufficiency): Suppose that (36) holds. We construct controller (37). Then, for , one has Moreover, for any , , and , , we have that , and then it can be learned from (34) that , for all . Therefore, is a ROCIS of system (20).
(Necessity): Suppose that is a ROCIS of system (20) with control , and then, for any , , one can obtain that , , which means that (36) holds.
In fact, if (36) does not hold, then there exists an integer such that . In this case, it can be learned from (35) that and . Assume that , i.e., , then we have . Since is a ROCIS of system (20) with control , then , which is a contradiction to . As a result, for any , .
Example 16. Let us consider a reduced model of the lac operon in E. coli [33, 41]:where , and are state variables denoting the lac mRNA, the lactose in high concentrations, and the lactose in medium concentrations, respectively; , and are inputs variables representing the extracellular lactose, the high extracellular lactose, and the medium extracellular lactose, respectively; is an external disturbance. The output equations are given by And the robust set .
Obviously, , and then . Let , , and , and then we have where and In this example, one can get , and , and then we have and . Therefore, , and . From (34), we have , which means that , and . By Theorem 15, is a ROCIS of system 40a and 40b with the disturbance , and the output feedback gain matrices are with , , and , . For example, let , , and , and then the corresponding output feedback control is
In this paper, the RIS of BNs (BCNs) with disturbances was investigated. A special set called immediate neighborhood of a given fixed point was considered; then the discrete derivative of Boolean functions at the fixed point was used to analyze the robust invariance; based on this a sufficient condition was obtained. Furthermore, the ROCIS of BCNs was considered by STP. A necessary and sufficient condition was obtained to check whether a given set is RCIS under a given output feedback controller. Finally, for a given set, output feedback controllers were designed such that the set is RCIS.
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
This work was partially supported by the National Natural Science Foundation of China under Grant nos. 11671361 and 61573008, the China Postdoctoral Science Foundation under Grant nos. 2015M580378 and 2016T90406, and the Zhejiang Provincial Natural Science Foundation of China under Grant No. LD19A010001, and the National Training Programs of Innovation and Entrepreneurship under Grants 201610345020 and 201710345009.
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