Robust Output Controllability Analysis and Control Design for Incomplete Boolean Networks with Disturbance Inputs

Deng, Lei; Fu, Shihua; Li, Ying; Zhu, Peiyong

doi:https://doi.org/10.1155/2018/2940105

Mathematical Problems in Engineering

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Abstract Introduction Preliminaries Conclusions Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2018 | Article ID 2940105 | https://doi.org/10.1155/2018/2940105

Robust Output Controllability Analysis and Control Design for Incomplete Boolean Networks with Disturbance Inputs

Lei Deng,¹Shihua Fu,²Ying Li,²and Peiyong Zhu¹

Academic Editor: Sabri Arik

Received17 Mar 2018

Accepted01 Nov 2018

Published21 Nov 2018

Abstract

This paper addresses the problems of robust-output-controllability and robust optimal output control for incomplete Boolean control networks with disturbance inputs. First, by resorting to the semi-tensor product technique, the system is expressed as an algebraic form, based on which several necessary and sufficient conditions for the robust output controllability are presented. Second, the Mayer-type robust optimal output control issue is studied and an algorithm is established to find a control scheme which can minimize the cost functional regardless of the effect of disturbance inputs. Finally, a numerical example is given to demonstrate the effectiveness of the obtained new results.

1. Introduction

Boolean network (BN) was firstly introduced by Kauffman for modelling genetic regulatory networks (GRNs) [1]. Since then, it has been used to analyze and simulate cellular networks. A typical BN consists of nodes, and each can take one of the following two values: 1 or 0, representing the gene is expressed or not, respectively. Furthermore, the state evolution of each node can be determined by Boolean functions. Although BN is a simplified model, it becomes a powerful tool in analysing GRNs. And some significant results were presented [2, 3].

Recently, D. Cheng and coauthors have developed an algebraic state space representation (ASSR) of Boolean control networks (BCNs) by using the semitensor product (STP) of matrices [4]. Using the ASSR, many fundamental and landmark results about BCNs, which include but are not limited to controllability and observability [5–8], stability and stabilization [9–11], disturbance decoupling [12, 13], network synchronization [14, 15], output tracking [16], output regulation [17, 18], and optimal control [19], have been obtained in the last few years. The STP is an interesting research topic, and we can refer to [20–23] for some other applications of it.

As is well known, output-controllability and optimal output control issues are fundamental concepts in a control theory field. The output controllability describes the ability of an external input to move the output from any initial condition to any final condition in a finite time interval. The optimal output deals with the problem of finding an external input for a given system such that a certain optimality output criterion is achieved. A controllable system is not necessarily output controllable, and an output controllable system is not necessarily controllable. Hence, they are also two important structural properties in modern control theory which reflect the dominant ability of the control inputs over outputs [24, 25]. As a suitable model of GRNs, the study of output controllability (reachability) and optimal control is also meaningful for BCNs, and there are some literatures studying on both of them. Reference [26] discussed the output reachability of BCNs, and several necessary and sufficient conditions were obtained for the verification of output reachability. The output-controllability and optimal output control problems of a state-dependent switched BCN were studied in [27], in which the output controllability criteria were presented and an optimal output control design algorithm for the Mayer-type optimal output problem was established.

Robust control is a branch of control theory that deals with uncertainty [28]. It aims to achieve robust performance, such as controllability, in the presence of uncertain parameters or disturbances. It is known that, in practical networks such as GRNs, the disturbances are ubiquitous, which primarily stem from environmental changes, biological uncertainties, and experimental noises [29, 30]. These disturbance inputs may lead to instability of networks or make it difficult to understand the network dynamics. To be consistent with the dynamical behaviours of the practical networks, disturbances should be a characteristic which has to be taken into account when modelling the GRNs by BNs [31]. Thus, two interesting problems with biological significance are robust-output-controllability and robust optimal output control for BCNs with disturbance inputs, which deserve investigation for theoretical developments as well as for practical applications.

In order to solve two classical intellectual problems: the wolf-sheep-cabbage puzzle and the missionaries-cannibals problem, the authors of [32] presented a new logical control system (LCS), called the incomplete LCS, in which certain controls can only be applied to certain states. Moreover, in [33], authors modeled a networked evolutionary game with bankruptcy mechanism as an LCS whose certain control-states should be forbidden. These cases show that the investigation of incomplete BCNs is of practical significance. Unfortunately, to the best of authors’ knowledge, there is no result available for robust output controllability and robust optimal output control of incomplete BCNs with disturbance inputs, which motivates the study of this research.

This paper investigates the robust-output-controllability and robust optimal output control of incomplete BCNs with disturbance inputs. The main difficulties of extending the results obtained in [27, 34] contain the following two aspects: the dynamics of a BCN with disturbance inputs is much more complicated than a BCN and, in this paper, the disturbance inputs are arbitrary; the existence of control-state avoiding set makes the control law of this paper different from that of the existing results about BCNs since some control-states should be avoided when designing the control sequences. We present several necessary and sufficient conditions for the robust output controllability of incomplete BCNs with disturbance inputs. Moreover, a robust optimal output control design algorithm for the Mayer-type robust optimal output problem is also derived.

The remainder of this paper is organized as follows. Section 2 provides some preliminary results. In Section 3, we study the robust-output-controllability and robust optimal output control problems of incomplete BCNs with disturbance inputs, and present the main results of this paper. A numerical example is given in Section 4 to illustrate the validity of the obtained new results. Finally, conclusions are drawn in Section 5.

2. Preliminaries

Firstly, we list some notations which will be used throughout this paper.(i) and (ii) : the -th column of the identity matrix .(iii), and for compactness, . By identifying and , where “~” denotes two equivalent forms of the same object, the logical variable in then takes value from .(iv) is the set of columns (rows) of . is the -th column (row) of .(v): the set of real matrices. A matrix is called a logical matrix, if the columns of are elements of . Denote by the set of logical matrices.(vi) If , by definition it can be expressed as , and its shorthand is .(vii) A matrix is called a Boolean matrix, if its entries , for every . Denote by the set of Boolean matrices.(viii) A matrix is called an incomplete logical matrix, if where is an dimensional zero vector. Identifying with , an incomplete logical matrix can briefly expressed as , where certain could be 0. Denote by the set of incomplete logical matrices.(ix)For , is the Khatri-Rao product of A and B.

Some necessary definitions are given as follows.

Definition 1 (see [4]). For and , the STP, denoted by , is defined as follows: where l.c.m., denoting the least common multiple of and , and denotes the Kronecker product.

It is noted that the STP is a generalization of the ordinary matrix product. In this paper, we simply call it “product" and omit the symbol “" if no confusion arises.

Definition 2. The intersection of Boolean matrices is defined as

Definition 3. Let be a matrix; then we call if and only if , .

Finally, we give some conclusions which will be used in the remainder of the paper.

Lemma 4 (see [4]). Any logical function with logical arguments can be expressed in a multilinear form as where is unique, called the structure matrix of logical function .

Consider a disturbed BCN as follows:where denotes the state variable, denotes the control input, denotes the disturbance input, denotes the output variable, and , , and are Boolean functions. Denote the output trajectory of the system (6) by , where , , and represent the initial state, the control sequence, and the sequence of disturbance inputs, respectively.

In order to convert the system (6) into an algebraic form, we define , , and . Assume that the structure matrices for , are , , respectively, then the system (6) can be converted into the following algebraic form:where is called the state transition matrix, , and is the Khatri-Rao product.

3. Main Results

In this section, we study the robust-output-controllability and robust optimal output control of disturbed incomplete BCNs and present the main results of this paper.

3.1. An Algebraic Form for Disturbed Incomplete BCNs

First, we give the concept of disturbed incomplete BCNs.

Definition 5. Consider the disturbed BCN (7). Suppose certain controls are not applicable to certain states. Precisely, there exists a set of pairssuch that the control is not applicable to . is called the control-state avoiding set. Supposing that , then the disturbed BCN is called the disturbed incomplete BCN.

Consider the disturbed incomplete BCN (7) with control-state avoiding set and split into blocks as , where . For , a straightforward computation shows that , where . Thus, when and , we have where and . If , we let . In this case, the corresponding does not exist. We also denote it briefly as . Then, it is easy to obtain the following result.

Proposition 6. A disturbed incomplete BCN with control-state avoiding set as in (8) has its algebraic expression as in (7) except that where , and .

We denote the new blocks obtained from Proposition 6 by , that is,Let be the matrix obtained from an identity matrix substituting the columns with indexes by ; then can be calculated by .

Construct a new matrix as follows:and then it is easy to see that is the state transition matrix of a disturbed incomplete BCN with control-state avoiding set . In fact, can be further calculated as

Based on (7) and above analysis, the following result is obvious.

Proposition 7. Consider a disturbed incomplete BCN with control-state avoiding set . It can be expressed as an algebraic form:where is given in (12).

Proof. We only need to prove that the state trajectories of the disturbed incomplete logical system (14) are equivalent to that of the system (7) with control-state avoiding set .
Denote , where . It is obvious that and are equivalent. For any , we first prove the case of . A simple calculation shows thatWhen , we have Hence, in both cases, the state trajectories of the disturbed incomplete logical system (14) are equivalent to that of the system (7) with control-state avoiding set . Since and re equivalent, this completes the proof.

3.2. Robust Output Controllability Analysis of Disturbed Incomplete BCNs

Now, we give definitions of the robust output controllability for disturbed incomplete BCNs.

Definition 8. Consider the disturbed incomplete BCN (14).(i) is said to be -robust-output-reachable from the initial state , if one can find a control sequence such that for arbitrary disturbance inputs , the output at time satisfies , where and . The -robust-output-reachable set of the initial state is denoted by .(ii)The system (14) is said to be -robust-output-controllable at the initial state , if for any output state , one can find a control sequence such that, for arbitrary disturbance inputs , the output at time satisfies . The system (14) is said to be -robust-output-controllable, if the system is -robust-output-controllable at any .(iii)The system (14) is said to be robust-output-controllable at the initial state , if for any output state , one can find a positive integer and a control sequence such that, for arbitrary disturbance inputs , the output at time is . The system (14) is said to be robust-output-controllable, if the system is robust-output-controllable at any .

To obtain necessary and sufficient conditions for the robust output controllability of disturbed incomplete BCNs, we first consider the robust reachability of disturbed incomplete BCNs.

Consider the disturbed incomplete BCN (14), and denote by

Split into square blocks as

where . Then we have the following result.

Theorem 9. Consider the disturbed incomplete BCN (14). Let and be given. Then,
(i) is robust reachable from at the -th step, if and only ifwhere and denotes the th element of ;
(ii) is robust reachable from , if and only if

Proof.
(i) (Necessity). We prove it by induction on . Consider the case . It is assumed that can be robust reachable from at the first step; then one can find at least one control such that, for any disturbance input , the following equation will hold:Multiplying (21) from the left by yieldsFor each block , split it into equal blocks aswhere is given in (12). Then according to (22), we haveSince , one knows thatCombining (24) and (25), we have Thus, This proves the situation for .
For the induction step, we assume that, for any , if can be robust reachable from at the th step, then . In the following, we need to prove that (19) holds for . Assuming that can be robust reachable from at the th step, then there must exist a path from to at the th step and a path from to in one step. Applying the induction hypothesis, we have Therefore,The proof of necessity is completed.
(Sufficiency). We still prove it by induction on . From the proof of necessity, one can easily see that if , then (21) must hold, which implies that control can steer system (14) from to in one step under arbitrary disturbance inputs. This proves (19) for .
We assume that, for any , if , then can be robust reachable from at the th step. By (29), one can obviously see that if , then there exists at least a positive integer such that and . Based on the induction hypothesis, can be robust reachable from at the th step and can be robust reachable from in one step. Thus, can be robust reachable from at the th step.
The result (ii) can be easily obtained from (i).

From Theorem 9 and Definition 8, one can obtain the following robust output controllability results.

Theorem 10. Consider the disturbed incomplete BCN (14).
(i) is s-robust-output-reachable from the initial state , if and only if(ii) The system (14) is s-robust-output-controllable at the initial state , if and only if(iii) The system (14) is s-robust-output-controllable, if and only if(iv) The system (14) is robust-output-controllable at the initial state , if and only if(v) The system (14) is robust-output-controllable, if and only if

Proof.
(i) (Sufficiency). Assuming that (30) holds, and we need to prove that there exists a control sequence such that, for arbitrary disturbance inputs , the output satisfies . Sincethere must exist an integer such that and .
Based on Theorem 9, is robust reachable from at the th step. In other words, we can find a control sequence such that Correspondingly,which infers that is s-robust-output-reachable from .
(Necessity). Assuming that is s-robust-output-reachable from , then there exists a sequence of control signals under which regardless of the effect of disturbance inputs . Since letting , it is easy to check that and is robust reachable from at the th step. In other words, and . Hence, which means that (30) holds.
The results (ii) and (iii) can be derived from (i) and Definition 8 easily. Thus, we omit them.
Next, we prove result (iv).
(Sufficiency). Assuming that (33) holds, then for any , there exists an integer such that . Based on result (i), is s-robust-output-reachable from the initial state . For the arbitrariness of , we know system (14) is robust-output-controllable at the initial state .
(Necessity). Suppose that the system (14) is robust-output-controllable at the initial state . Then for any output , there exists an integer such that, for arbitrary disturbance inputs, the inequality holds, . Thus, which means that (33) holds.
The result (v) follows from (i) and Definition 8, and we omit it. The proof is completed.

Corollary 11. By Theorem 10, for a given initial state and an integer , one can successively obtain that

3.3. Robust Optimal Output Control of Disturbed Incomplete Boolean Control Networks

In this subsection, we investigate the Mayer-type robust optimal output control problem of disturbed incomplete Boolean control networks and present a new design procedure for the problem.

Consider a disturbed incomplete BCN with the algebraic state representation (14). The Mayer-type robust optimal output control problem can be described as follows: find a control sequence such that, for arbitrary disturbance inputs , the cost functionalis minimized under the given initial state , where is a given constant vector and is a fixed termination time.

Minimizing the cost functional is equal to find out the minimum value of under the following constraint:And, meanwhile, we need to design a control sequence which steers to the optimal terminal output regardless of the effect of disturbance inputs, where can be computed by (41).

It is noted that when , . Thus, to find out the minimum value of , we just need to compute the minimum value of .

According to the above discussion, we obtain the following algorithm to find a sequence of controls such that, for any disturbance inputs , the cost functional is minimized at the fixed termination time .

Algorithm 12.
Step 1. Compute the -robust-output-reachable set with the initial state by (41), denoted by ;
Step 2. Compute the optimal value =;
Step 3. Let . Find an integer such that and . Let ;
Step 4. Find two integers and such that and . Let and ;
Step 5. If , find such that and let , stop. Otherwise, replace by , and replace by . Go back to step 4.

Proposition 13. The control sequence generated by Algorithm 12 can minimize the cost functional (42) at the fixed termination time regardless of the effect of disturbance inputs.

Proof. Since can minimize the cost functional , we just need to show that the control sequence can steer the state to the objective output under arbitrary disturbance inputs.
Notice that ; by Theorem 10, we can find an integer such that and . Let . By (23), one knows that for arbitrary disturbance input , it follows the result To ensure that can be reached from in one step with the control for arbitrary disturbance input, the following results should be kept:Combining (25) and (45), we obtain .
In the same way, we can find integers and such that which implies that can be reached from in one step with the control for arbitrary disturbance.
Do the same procedure, we can finally obtain the control sequence which steers the state to the objective output regardless of the effect of disturbance inputs.

4. An Illustrative Example

In this section, we give an example to demonstrate the efficiency of the results obtained in this paper.

Consider an incomplete BCN with one disturbance input whose dynamic equations are given as follows:

In this example, the measured outputs are assumed as

Set and . Assume the control-state avoiding set . Then we can express the disturbed incomplete BCN (47) in its algebraic form aswhereand .

Skipping a tedious calculation, we can obtainwhere and can be calculated by (12) and (17). From Theorem 10, it is concluded that the system (49) is 3-robust-output-controllable. It is also robust-output-controllable.

Consider the Mayer-type robust optimal output control problem of the disturbed incomplete Boolean control network (49). The constant vector , and we aim to determine a control strategy such that, for arbitrary disturbance input, the cost functionalis minimized under the initial value and .

Now, we follow Algorithm 12 step by step as follows:

Step 1. Compute the 3-robust-output-reachable set with the initial value , and obtain that ;

Step 2. =min, and ;

Step 3. Let , and find such that and . So ;

Step 4. We can find such that , and . So and ;

Step 5. We can find such that , and . So and .

At last, we obtain an optimal control to minimize the cost functional regardless of the effect of disturbance inputs.

From another aspect, for the initial value , under the control , the state trajectories of the system (49) are . At this time, , . Hence, the control is an optimal control.

5. Conclusions

In this paper, we have investigated the robust-output-controllability and robust optimal output control problems of incomplete BCNs with disturbance inputs. We have proposed several necessary and sufficient conditions for the robust output controllability based on the algebraic representation of the system. Then, we have discussed the Mayer-type robust optimal output control issue and presented an algorithm to find a control scheme which can minimize the cost functional regardless of the effect of disturbance inputs. Finally, an illustrative example has been given to support the results. It should be pointed out that event-triggered control has attracted a great deal of attention from scholars in the last two decades. Hence, future works can study the following issues: output-controllability of higher-order incomplete BCNs via event-triggered control and event-triggered control for robust optimal output control of incomplete BCNs with disturbance inputs.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

All authors drafted the manuscript, and they read and approved the final version.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 11301247) and Science and Technology Project of Department of Education, Shandong Province (no. J15LI10).

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Copyright © 2018 Lei Deng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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