Output regulation of Boolean control networks with stochastic disturbances
2017; Institution of Engineering and Technology; Volume: 11; Issue: 13 Linguagem: Inglês
10.1049/iet-cta.2016.1675
ISSN1751-8652
Autores Tópico(s)Bacterial Genetics and Biotechnology
ResumoIET Control Theory & ApplicationsVolume 11, Issue 13 p. 2097-2103 Special Issue: Recent Developments in Logical Networks and its ApplicationsFree Access Output regulation of Boolean control networks with stochastic disturbances Hongwei Chen, Hongwei Chen School of Mathematics, Southeast University, Nanjing, 210096 People's Republic of ChinaSearch for more papers by this authorJinling Liang, Corresponding Author Jinling Liang jinlliang@seu.edu.cn School of Mathematics, Southeast University, Nanjing, 210096 People's Republic of China Department of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah, 21589 Saudi ArabiaSearch for more papers by this author Hongwei Chen, Hongwei Chen School of Mathematics, Southeast University, Nanjing, 210096 People's Republic of ChinaSearch for more papers by this authorJinling Liang, Corresponding Author Jinling Liang jinlliang@seu.edu.cn School of Mathematics, Southeast University, Nanjing, 210096 People's Republic of China Department of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah, 21589 Saudi ArabiaSearch for more papers by this author First published: 01 September 2017 https://doi.org/10.1049/iet-cta.2016.1675Citations: 12AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract This study addresses the output regulation problem of Boolean control networks (BCNs) in the presence of exogenous disturbances, which are characterised by two-valued mutually independent random logical variables. By using the semi-tensor product technique, the BCNs with stochastic perturbations are represented in their compact algebraic forms, and then a relating augmented system is constructed, which facilitates the analysis of the output regulation problem for the BCNs. As a consequence, necessary and sufficiency criteria are obtained to ensure the existence of the state feedback controllers which efficaciously solve the considered problem. Moreover, executable constructive procedure is also proposed for the controller design. Two simulation examples are exploited in order to illustrate the effectiveness of the proposed criteria as well as the controller design scheme. 1 Introduction Genetic regulatory networks (GRNs) are the mechanisms that genes encode proteins and some of which in turn regulate the gene expressions by their own productions [1]. Current developments in DNA microarray technology have made it possible to measure the mRNA-levels for thousands of genes simultaneously, which give raise to a large volume of experimental data [2]. However, it is still a challenging task to effectively utilise these data and explain the interactions between genes and proteins. Theoretical analysis on GRNs with mathematical models has become one method of choice for investigating this challenging problem over the last decades, which yields important insights into gene function, interaction and evolution. Generally speaking, there are mainly two formal types of genetic network models including the continuous model [3] and the discrete one (such as the Boolean model [4]). In the continuous case, the whole network is described by a set of differential (or difference) equations that faithfully simulate the dynamics of small GRNs. However, it should be pointed out that when simulating certain aspects of regulatory networks, the continuous model sometimes captures more dynamical details than required. For example, when solely focusing on the sequence of biochemical activation patterns in a cell, regardless of their exact biochemical timing, it has been shown in [5] that the much simpler discrete dynamical models might be more sufficient. Experimental results have further confirmed that highly simplified network models based on Boolean states (i.e. ON and OFF) with discrete dynamics are capable of predicting the dynamical activation patterns of GRNs in living cells [6]. A specific example is the temporal sequence of cell cycle activation patterns in yeasts Schizosaccharomyces pombe that can be dutifully reproduced by Boolean model [6]. Therefore, dynamics analysis of GRNs characterised by Boolean networks (BNs) has quickly become an attractive area of research and received great attention over the past decades. When investigating the BN model, one of the main objectives is to describe the manner in which cells execute and control the normal functions, and then explore how the abnormal functions resulted from breakdown in regulation. Hence, researches on control-related problems of BNs may reveal new insights into the translational genome and enable us to develop therapies based on the disruption or mitigation of aberrant gene functions. In order to have more appropriate models, binary control inputs and outputs are introduced into the BNs aiming at tackling the control problem of GRNs, which yields the Boolean control networks (BCNs) [7]. In the last decades, concept of semi-tensor product (STP) of matrices has been employed by Cheng et al. [8] to develop an algebraic representation of BCNs, which greatly facilitates the dynamics analysis of BNs. In recent years, a large amount of results have been reported in the literature on such topics that include, but are not limited to, controllability and observability [9–16], stability and stabilisation [17, 18], optimal control [19–21], network synchronisation [22–25] and so on. The output regulation problem concerned is to design a control law for a fixed plant such that its output tracks the reference inputs generated by an exosystem [26]. It has already been shown in [27] that this problem is also relevant to the study of GRNs. For instance, in order to manipulate the large-scale dynamics of the lactose regulation system of the Escherichia coli bacteria, a feedback control architecture has been proposed to make the fraction of induced cells in the population (i.e. the output of the system) attain a desired level which is described by a given reference trajectory [27]. Besides, in the study of feedback shift register (FSR) that can be regarded as a special class of BNs, a salient problem is to construct proper feedback functions such that the FSR can output an anticipated periodic sequence [28]. Recently, by resorting to the STP technique, regulation of the output trajectory to a given constant reference signal has been investigated in [29], and a general novel procedure for the design of state feedback laws has been proposed by constructing a series of reachable sets. This approach has been further adopted in [30] to deal with the output tracking control problem of the probabilistic BNs. In [31], based on an augmented system, the output regulation has been discussed for the BCNs, where the reference signals are produced by some external BN. On the other hand, stochastic fluctuations are inevitable in real-world gene expression data, which primarily stem from the probabilistic chemical reactions and the random variation of some key signalling proteins [32]. As is well known, the stochastic perturbations might cause instability of networks or make it difficult to know the network dynamics. To be consistent with the dynamical behaviours of the real-world systems, stochastic disturbances should be recognised as a characteristic that has to be taken into account when modelling the GRNs by BNs [33–36]. Therefore, an interesting problem with biological significance is to design a feedback control law for BCNs such that the network output can achieve the desired performance in the presence of a class of stochastic disturbances, which constitutes the main focus of the present research. This paper is concerned with the solvability and controller design for the output regulation of BCNs with stochastic disturbances, where the variables describing the stochastic perturbations are assumed to be mutually independent two-valued random logical variables. We aim to design a state feedback controller such that, for any initial states and stochastic disturbances, the output trajectories of the perturbed BCN will always track the reference signals produced by some other BN. By resorting to the STP technique, necessary and sufficient conditions are firstly derived for the solvability of the output regulation problem, which can be easily checked with the help of MATLAB toolbox (a MATLAB toolbox has been established by Cheng et al. in http://lsc.amss.ac.cn/dcheng/stp/STP.zip for some related computation). Then, based on the augmented system, an effective method is developed for the explicit controller design of the problem. Two illustrative examples provided show the effectiveness of the obtained theoretical results when tackling the output regulation of BCNs. The remaining part of this paper is organised as follows. A brief summary of some preliminaries on the problem considered is provided in Section 2. Section 3 formulates the output regulation problem. Solvability and controller design of the output regulation problem are investigated in detail in Section 4. Two numerical examples are given in Section 5 to illustrate the validity of the obtained results. Finally, conclusions are drawn in Section 6. 2 Preliminaries The following notations will be used throughout this paper: , and . and are the sets of non-negative integers and positive integers, respectively. Given with , denote the set by . means the set of all real matrices. , where is the i th column of identity matrix for . A matrix is called a logical matrix if , which is also expressed as for simplicity. The set of all logical matrices is denoted by . is used to represent the i th column of matrix A, and the superscript ' ' means the transpose. An Boolean matrix is an matrix with entries . The set of all Boolean matrices is denoted by . For two matrices X, , the inequality means that for all and . First, the definition and some basic properties of STP are introduced that are useful in our later discussion. Definition 1 [8].The STP of two matrices and (in particular, and ) is defined as where , and l.c.m. denotes the least common multiple. Lemma 1 [8].The STP of matrices has the following properties: If , then . Let the logical matrix and , then . Second, by identifying and , where ' ' means two different/equivalent forms of the same object, the logical variable in then takes value from . Then, an instrumental representation of the logical functions, needed in the sequel to derive the equivalent algebraic form of BNs, is given in the following lemma. Lemma 2 [8].Suppose that maps into , , and . Then there exists a unique matrix such that , where and . Finally, some definitions for the random logical algebra are presented based on the vector form of the logical variables. Definition 2.Let with . A function is called a random logical matrix if it takes value () in the set with associated probability satisfying . For the particular case , i.e. , we call it an m -valued random logical variable. 3 Problem formulation A BCN with stochastic disturbances can be described as follows: (1a)where time ; and are, respectively, the state variable and the control input at time t; is the output variable to be regulated; represents the exogenous mutually independent disturbance signals with its i th component being a two-valued random logical variable that satisfies where and ; and are logic functions. In this paper, it is further assumed that the exogenous disturbances are mutually independent. The dynamics of the associated reference BN is given by (1b)where time ; and denote the -dimensional state variable and the p -dimensional output variable, taking values in and , respectively; and are logic functions. The output regulation problem explored in this paper is to design a proper state feedback controller in the form of (2)where is a logic function to be designed from to , such that for any initial states and , and any exogenous disturbance signals under the probability constraint, the corresponding output trajectories of (1a) and (1b) exhibit identical dynamics in the sense of probability after finite time steps. To formally define the solvability of the output regulation problem, we introduce the following notations. Denote the output trajectory of (1a) by , where is the initial state, and are, respectively, the driven control signal and the disturbance sequence. Similarly, represent the output trajectory of (1b) by starting from the initial state . Definition 3.Consider system (1). For any disturbance sequence , the output regulation problem is said to be solvable if a state feedback controller in the form of (2) can be found under which there exists a positive integer such that holds for any and . To facilitate the analysis of the above problem, we convert system (1) into its algebraic form by resorting to the STP technique. Based on the vector forms of logical variables, and setting , and , from Lemma 2, one can obtain the following algebraic representation of BCN (1a): (3a)where is a R -valued random logical variable with ; and in which , and . Similarly, the reference BN (1b) and the state feedback controller (2) can be converted into (3b)and (4)respectively, where with , , and is called the state feedback matrix. It is worth mentioning that designing (2) is equivalent to solving (4) [8]. Thus, the output regulation problem reduces equivalently into finding the proper state feedback matrix G. In other words, instead of designing the state feedback controller (2) directly, we will first construct its algebraic form (4) and then convert it into its relating logical form. 4 Main results In Section 4.1, an augmented system generated by (3a) and (3b) is introduced to facilitate the analysis of the output regulation problem. Based on it, a necessary and sufficient condition for the existence of a state feedback controller that solves the addressed problem is established in Section 4.2. If the output regulation problem is solvable, one may ask how to design the state feedback matrix G. This issue is tackled in Section 4.3. 4.1 Augmented system Motivated by the fact that the state feedback controller (4) relies on both and , a new variable is introduced, which is a bijective map from to . With defined and (3), an augmented system can be obtained as (5)where with , and . By Definition 3, it is known that if the output regulation is realised by the state feedback controller (4), then the following equalities: hold with probability 1 for any , which imply that the output trajectory of the augmented system (5) starting from any initial state will reach set before the th time step, and then stay there forever with probability 1. Accordingly, it can be concluded that the state (namely, the anticipated state) whose output is included in plays an important role when analysing the output regulation problem for the BCNs. Let denote the set of all anticipated states, it follows from the output equation of (5) that (6)Based on the above discussions, it is known that the solvability of the output regulation problem reduces to the set stability of the augmented system (5), and the output regulation control problem turns into stabilising the augmented system (5) to an non-empty set with probability 1. Note that if , then the output regulation problem is solvable by any state feedback controller (4), which makes the problem trivial. Hence, in the following, we assume that . 4.2 Solvability of the output regulation problem In this subsection, solvability of the output regulation of BCNs is investigated based on the augmented system (5) generated by (3a) and (3b). To determine conditions under which the output regulation problem is solvable, a few preliminary definitions and results are necessary. Definition 4.Let with entries , define , where is the ceiling function [ is the smallest integer not less than x]. Definition 5.Consider system (5) with given state feedback controller (4), i.e. the state feedback matrix G is given in priori. For a given initial state set with , the reachable set at the k th time step starting from is defined as (7)where in which , and for . For the particular case , we call the reachable set at the k th time step with global initial state set . The following result presents some important properties for the reachable set , whose proof is similar to that given in [24] and hence omitted here. Lemma 3.Consider system (5) with given state feedback controller (4). (1) Let be the reachable set starting from the initial state set at the k th time step, then where . (2) If for some , , then holds for all . (3) If is the smallest positive integer such that , then , where c is the cardinality of . Now, we can give a necessary and sufficient condition for the solvability of the output regulation problem. Theorem 1.Consider system (5) with given state feedback controller (4). The output regulation problem is solvable if and only if there exist a logical matrix and a positive integer such that (8) To get the explicit algebraic expression of , some definitions for the logical sub-matrix are introduced as follows. Definition 6 [17].For any , a logical matrix is called a logical sub-matrix of if . Denote by the set of all logical sub-matrices of , i.e. Especially, for any non-zero , . Lemma 4.Consider the augmented system (5) with given state feedback controller (4). Let be the reachable set starting from the initial state set at the k th time step, then one has (9)where . Proof.From (4) and (5), we know Denote by the overall expected value of z and assume that . Then the above equation yields (10)which implies that This together with the definition of logical sub-matrix infer that It thus follows from (7) that (11)The proof is completed. □ By using the algebraic representation of , Theorem 1 can be restated in the following different while more explicit form. Corollary 1.Consider system (1) with algebraic representation (3) and augmented system (5). Let . Then the output regulation problem is solvable if and only if there exist a logical matrix and a positive integer such that (12) Remark 1.For a given state feedback controller in the form of (4), Corollary 1 infers that one only needs to examine finite time steps to ascertain whether or not it is a feasible solution for the output regulation problem of system (1). However, the condition in Corollary 1 is not computationally meaningful for the design of the state feedback matrix G since matrix inequality (12) is hard to be solved, and this motivates the following general algorithm for the design of the state feedback matrix G. 4.3 State feedback controller design In this subsection, an effective algorithm is proposed for designing the state feedback controller of the output regulation problem. To this end, we first define a sequence of sets recursively (13) Then we have the following lemma. Lemma 5.If , then holds for any integer . Proof.The result is to be drawn by induction on k. Suppose that , then there must exist one such that holds for arbitrary disturbance signal . This together with the condition yields which means , and thus .Now, let integer and assume that by induction. Suppose that . Then, for arbitrary disturbance signal , there exists such that Similarly, by applying the induction hypothesis, it is easy to see that which infers , and hence one has . The proof is completed. □ Theorem 2.Let (5) be the augmented system generated by (3a) and (3b). Then the output regulation problem is solvable if and only if there exist non-empty set and a positive integer such that (14)where c is the cardinality of . Proof (Sufficiency).Suppose that (14) holds, and we need to prove that, under arbitrary disturbance sequence , the output regulation problem is solvable [equivalently, the augmented system (5) is globally stabilised to with probability one] by constructing an appropriate state feedback controller. In what follows, we first construct the state feedback matrix . Observe that if (14) holds, by Lemma 5, we have which shows that can be represented as the union of disjoint sets as follows: Thus, for any integer , there exists a unique integer such that , where . If , take , then one can find a vector such that holds for arbitrary disturbance signal . Otherwise, one can also find a vector such that Let be the state feedback matrix in (4). In the following, we prove that, under arbitrary disturbance sequence , the augmented system (5) with state feedback controller will globally be stabilised to set . Noting that , it is enough to show that for every , , i.e. (15)whenever and , where and . We draw the conclusion by induction on k. Consider firstly the case . Let . It follows from the definition of that which implies that . Then, fix , one can successively obtain that and hence which means that (15) holds for .Now, let and assume that (15) holds for . For the induction step, consider the case . If , there is nothing to prove. Thus, we suppose that . Then one gets It thus follows that . Applying the induction hypothesis, we have By induction, one concludes that (15) holds for any positive integer k.(Necessity): Assume that for any disturbance sequence , the output regulation problem is solvable by a state feedback controller, say . By (10), the closed-loop system turns into (16)Let be the limit state set of the above augmented system. Then , and from (2) and (3) of Lemma 3, one gets that there exists a such that (17)where c is the cardinal number of set . By Definition 5, it is easy to verify that (18)holds for any , which yields Combining (17) and (18), one can obtain that Now, we need to prove that . If such an assertion is not true, then there exists a positive integer such that and . Notice that is the limit state set of the augmented system, take , there exists a positive integer T such that , i.e. for , where and . Note that where . Then, from the definition of , one obtains for , which contradicts with the definition of solvability of the output regulation problem. Thus, it is obtained that holds. The proof is completed. □ Remark 2.In the proof of Theorem 2 (the sufficiency part), a specific strategy has been provided to design the state feedback matrix G that solves the output regulation problem, which can be summarised in the following steps: Step 1: Calculate for . Step 2: For each , find the unique integer such that . Step 3: If , find a vector such that Else, find a vector such that . Step 4: The state feedback matrix is designed as . 5 Numerical examples In this section, two examples are presented to demonstrate the applicability of the obtained results. Example 1.Consider the BCN model of an apoptosis network presented in [37] where , and are, respectively, the concentration level (high or low) of the inhibitor of apoptosis proteins (IAP), active caspase 3 (C3a) and active caspase 8 (C8a); the concentration level of the tumour necrosis factor (TNF, a stimulus) is denoted by u, which is regarded as the control input.To investigate the output regulation problem of this model, an artificial stochastic disturbance is introduced in as where is a two-valued random logical variable satisfying (19)with . In the apoptosis network, a steady state with and (respectively, and ) means that the cell is dead (respectively, survival), which infers that and are the key nodes in this Boolean model. Thus, take and as the output variables, i.e. (20)Construct the associated reference BN with two nodes and two outputs as (21)where . Similarly, one can also obtain the corresponding augmented system as shown (5). Some tedious calculation shows that the output regulation problem for this system is unsolvable by designing a state feedback controller in the form of (4). Noting that the reference BN converges globally to the fixed point (the cell's survival state), this fact is in accordance with the bistability of the apoptosis network from the biological point of view. Example 2.Consider the following system consisting of two BNs, each with two nodes and two outputs: (22a) (22b) Setting , , and , the algebraic form of this network can be obtained as in (3), where , , and . By using the transformation , one derives the relating compact algebraic form (23)where and Then, the anticipated states set can be derived as . It is easy to check that Hence, the output regulation problem for BNs (22a) and (22b) is solvable according to Theorem 2. By implementing the algorithm proposed in Remark 2, one can obtain state feedback matrices as follows: where for ; otherwise, . 6 Conclusion In this paper, the output regulation problem of BCNs has been investigated by utilising the algebraic representation of the logical dynamics. Based on the augmented system, a necessary and sufficient condition has been presented for the solvability of the output regulation problem, which can be easily verified with the help of MATLAB toolbox. In addition, an effective method has also been proposed for the explicit controller design. Two illustrative examples have been given to demonstrate the applicability and usefulness of the obtained criteria. 7 Acknowledgments This work was supported in part by the National Natural Science Foundation of China under grant no. 61673110, the Six Talent Peaks Project for the High Level Personnel from the Jiangsu Province of China under grant no. 2015-DZXX-003, the Fundamental Research Funds for the Central Universities (no. 2242015K42009), the Scientific Research Foundations of Graduate School of Southeast University (grant no. YBJJ1560) and the Innovation Program of Jiangsu Province (grant no. KYZZ15_0050). 8 References 1Goodwin B.C.: ' Temporal organization in cells: a dynamic theory of cellular control processes' ( Academic Press, London, UK, 1963) 2Li S.P. 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