Abstract
This paper realizes global stabilization for probabilistic Boolean control networks (PBCNs) with event-triggered state feedback control (ETSFC). Via the semitensor product (STP) of matrices, PBCNs with ETSFC are converted into discrete-time algebraic systems, based on which a necessary and sufficient condition is derived for global stabilization of PBCNs. Furthermore, an algorithm is presented to design a class of feasible event-triggered state feedback controllers for global stabilization. Finally, an illustrative example shows the effectiveness of the obtained result.
1. Introduction
As a central focus of genomic research, the way cellular systems fail in diseases has attracted much attention [1, 2]. Various mathematical and computational models have been constructed to describe the behavior of gene regulatory networks, such as Bayesian networks [3], differential equation [4], and Boolean networks (BNs) [5]. By right of its simple representation, BNs have attracted much attention, where the state of each gene is described by two levels, either inactive (0 or OFF) or active (1 or ON). The evolution of each node is related to the states of some other nodes, including itself sometimes, and determined by a series of logical functions at each discrete-time point. BNs with control inputs are named by Boolean control networks (BCNs), which are essentially switched systems with switching among different BNs [6].
Recently, a new matrix product, named the semitensor product (STP), is presented by Cheng et al. [7]. It converts the logical form of BCNs into the algebraic state space representation (ASSR). Compared with other methods, such as algebra geometry and symbolic dynamics, STP is more convenient and scientific in the study of BCNs. In this area, many interesting researches have been made, such as controllability [8–10], stabilization [11–16], optimal control [17], disturbance decoupling problems [18, 19], and function perturbation [20]. Besides, it has been proved that STP is effective in the study of logical systems [21, 22], game theory [23–25], fault detection [26], nonlinear shift register [27, 28], and so on.
However, the determinacy of a Boolean function limits the application of BCNs, due to the existence of the randomness and measuring noise in the real word. Therefore, it is necessary to extend the BCN to the PBCN, which can be regarded as an undetermined system switching between different constituent BCNs in terms of the probabilistic structure. Similar to BCNs, many extended results have been derived for PBCNs, such as controllability [29–31] and stabilization [31–33].
The stability and stabilization problems are two fundamental issues in the control of PBCNs. Reference [32] presented a necessary and sufficient condition to judge whether a PBCN can achieve global stabilization with probability one in the finite time by state feedback controllers, and the controller design is also realized. As mentioned in [31], a reachability matrix was defined to study the asymptotic stability and stabilization of PBCNs. But it is worth mentioning that the state feedback controllers here charge too much cost and need a relatively long transfer period sometimes. Synchronously, many interesting controllers were favorable for their unique properties [18, 34, 35]; motivated by which, we aim to design the ETSFC to overcome the cost of control and transfer period. The event-triggered control not only has a wide application in BCNs [18, 36] but also has smart grids [37], multiagent systems [38–43], and so on. For example, [18] first presented two kinds of event-trigger controllers to study the disturbance decoupling problems of BCNs, and some necessary and sufficient conditions were also obtained. In [36], authors used a class of event-triggered controls to realize global convergence for finite evolution and minimized the event set triggering the state at a special case. But the stabilization of PBCNs with ETSFC is still open but meaningful in the realistic world.
The main constructions of this paper mainly focus on the following two points. (i)A series of reachability sets are defined, via which, a necessary and sufficient condition for global stabilization of PBCNs controlled by ETSFC is designed.(ii)An algorithm is presented to design a class of event-triggered state feedback controllers to realize global stabilization of PBCNs.
The remaining part of this paper is constructed as follows. In Section 2, some notations are given and the STP of matrices is introduced. In Section 3, we devote to investigate the necessary and sufficient condition for stabilization of PBCNs with ETSFC and design controllers to realize global stabilization. Besides, an illustrative example is also given. The conclusion is provided in Section 4.
Notation 1. and . Denote the th column of matrix by and the set of all columns of matrix by . , where is an identity matrix. is the least common multiple of and . Denote the set of logical matrices by , where logical matrix satisfies . Denote the column vector of length with all entries equaling and by and , respectively. is called a -dimensional probabilistic vector if and , and the set of -dimensional probabilistic vectors is denoted by . A matrix is called a probabilistic matrix if holds for any . The set of all probabilistic matrices is denoted by . Define operator “” of two probabilistic column vectors E and F as , where if and only if , else . The Khatri-Rao product of n matrices is defined as .
2. Preliminaries
2.1. STP of Matrices
Definition 1 (see [7]). Define the STP of matrix and by and is the tensor (or Kronecker) product.
Remark 1. If , . Thus, STP is a generalization of the conventional matrix product. If no confusion arises, the symbol “” can be omitted.
Comparing with the general matrix product, the following pseudocommutative properties of STP are presented.
Proposition 1 (see [7]). Multiply a column matrix and any matrix , then
Proposition 2 (see [7]). Multiply two column matrices and , then where .
Let and , then . Any logical functions with variables can be expressed as the equivalent algebraic form by the following lemma.
Lemma 1. For , there exists a unique matrix , called the structure matrix of function , such that where and .
2.2. Algebraic Representation of PBCNs by ETSFC
A probabilistic Boolean model with nodes is where are logic states. At each time step, are chosen from a given finite logical function set with probability , where holds for any . Ultimately, there are a total of models. If we use the following matrix to denote the index set of models: then the th model can be defined by the th row of matrix , denoted by , where and . Moreover, the probability of the th model selected is computed to
Using Lemma 1, let ; we convert the logical form (5) into an algebraic form as where are chosen from with probability . are the structure matrices of Boolean functions , , .
Denote as the mathematical expectation of state at time ; the evolution of state expectation is where and .
Further, we multiply equations in (9), which leads to the following equation: where and is the Khatri-Rao product.
Similarly, PBCNs with inputs can be equivalently described as where and are states and inputs, respectively, and functions are logical functions. By Lemma 1, let and ; the dynamic of PBCNs (11) can be converted into the following algebraic discrete-time system: where are the structure matrix of logical function . Taking the expected value on both sides of (12), we have where , .
Multiplying (13) together, the algebraic state space representation is as follows: where is the control input and .
The principle of ETSFC is that the controller only works if equals to some designated states, consisting of set , and, in this case, the system switches into (14). Otherwise, the system remains (10) and the controller does not work. The state feedback controller is where are logical functions. Assume is the structure matrix of logical function , (15) can be converted into where the state feedback gain matrix .
Define control sign depending on . If , the system corresponds to dynamic (10), and makes the system correspond to PBCN (14). Therefore, the dynamic of PBCNs controlled by ETSFC is presented as where , is the ETSFC which only works when , that is, where set consists of the states triggering the state feedback controller (16).
3. Main Results
In this section, the event-triggered stabilization of PBCNs is investigated.
Definition 2. PBCN (14) is said to be globally stabilized to state with probability one, if for any initial state , there is a positive integer and a control sequence such that , .
Then, for a nonempty set , let , and can be expressed by induction as follows:
Furthermore, define as
Remark 2. can also be represented as follows:
Assuming and the basins of by , for , a series of sets are defined as follows: where , , , , and .
For the above sets, we have some useful properties below.
Proposition 3 (see [32]). One has (i), then , ,(ii)if , for some , then , .
Theorem 1. System (17) is globally stabilized to a designated state with probability one, if and only if, the following conditions are satisfied: (i) or there exists an integer satisfying .(ii)There exists an integer such that .
Proof 1. Sufficiency: condition (i) shows that holds by the designed controller. According to condition (ii) and the constructing procedure of and , for any initial state , it claims that or or . Therefore, there are three cases.
Case 1:If , there exists a positive integer satisfying , which implies that .Case 2:If , we can find a control such that .Case 3:If , there exists such that . The case in has been discussed before. Alternatively, the state in will enter .In a word, every state will arrive at with probability one.
Necessity: system (17) is globally stabilizable to with probability one; should be fixed.
Assume that condition (ii) does not hold; any state in cannot be steered into via the intermittent control, which is in contradiction with Definition 2.
In the following, we aim to prove that the smallest integer satisfying condition (ii) is no more than . It is enough to show that
where . We will show it by induction. When , if , that is, , thus . By Proposition 3, it holds that , which is a contradiction with Definition 2.
Suppose that , for some . It can be immediately obtained that
If , then
which means that . Proposition 3 implies that , which is a contradiction to the minimality of integer .
Moreover, it also holds that , that is to say, .
Assume that conditions (i) and (ii) in Theorem 1 hold, then the event-triggered state feedback controllers can be designed by the following algorithm.
| ||||||
Proof 2. We only need to prove that system (17) is globally stabilized to under the controller designed above. For the desired state , if , along the trajectory of (10), , for any , only if , that is, ; else, find a such that . Thus, system (17) can stay at the desired state with probability one.
For any initial state , we split it into three cases: or or .
Case 1:For , according to the proof of Theorem 1, it will reach .Case 2:For , there exists a such that . Because , it will trigger the controller (16), under which, .Case 3:For , there exists an integer such that for , we claim that it will enter into the set and the state in will enter into , which has been discussed in Case 2.By mathematical induction, system (17) will stabilize to .
Remark 3. If , the controller (16) will be triggered forever and we can only cut down the cost on transient routes.
Remark 4. When , PBCN (14) deduces a conventional BCN. The obtained results also can be applied to the event-triggered stabilization of BCNs. Thus, our result can be regarded as a generalization of that in [36] to some extent.
Example 1. Consider the following probabilistic Boolean network: where with probabilistic , , respectively, and with probabilistic , , respectively. Then we can obtain Assume the control system is given as Similarly, we obtain as follows: In a word, the dynamics of this system can be described as Next, we will design the ETSFC to stabilize system (31) to . By Algorithm 1, , , can be calculated as follows. Step 1:Since , then and there exists such that Step 2:By the dynamic , we can calculate , , and , so .Step 3:Let , then . Next, we consider dynamic . For , there exists such that . For , there exists such that . For , there exists such that . For , there exists such that .So and . Thus, the PBCN (31) is globally stabilized to via ETSFC by Theorem 1. In terms of Algorithm 1, set can be expressed by and one of the feasible state feedback matrices K can be .
4. Conclusion
In this technique, the global stabilization of PBCNs has been investigated by ETSFC. Under the framework of STP, the algebraic representation of PBCNs has been obtained from the logical form. A necessary and sufficient condition has been derived based on reachability sets. Finally, we have designed a class of feasible event-triggered state feedback controls to realize global stabilization. A numerical example has shown the effectiveness of our results.
Data Availability
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.
Acknowledgments
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 National Training Programs of Innovation and Entrepreneurship under Grants 201610345020 and 201710345009.