Robust Controllability and Observability of Boolean Control Networks under Different Disturbances

Wang, Yuanhua; Zhang, Xiao; Hao, Yaqi

doi:https://doi.org/10.1155/2019/1813594

Mathematical Problems in Engineering

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2019 | Article ID 1813594 | https://doi.org/10.1155/2019/1813594

Robust Controllability and Observability of Boolean Control Networks under Different Disturbances

Yuanhua Wang,¹Xiao Zhang,²and Yaqi Hao²

Academic Editor: Carlos-Renato Vázquez

Received09 Oct 2018

Revised03 Dec 2018

Accepted20 Dec 2018

Published02 Jan 2019

Abstract

This paper investigates robust controllability and observability of Boolean control networks under disturbances. Firstly, under unobservable disturbances, some sufficient conditions are obtained for robust controllability of BCNs. Then an algorithm is proposed to construct the least control sequences which drive the trajectory from a state to a given reachable state. If the disturbances are observable, by defining the order-preserving system, an efficient sufficient condition is obtained for robust controllability of BCNs. Finally, the robust observability problem is converted into an equivalent robust controllability via set controllability and is solved by using the results obtained for set controllability. Some numerical examples are presented to illustrate the obtained results.

1. Introduction

Boolean network (BN) was first proposed by Kauffman to model gene regularity networks [1]. It has been widely used in some other fields, such as biology and system science [2, 3]. To manipulate BN, external control inputs and outputs have been added to the logical dynamics, which yields Boolean control networks (BCNs) [4].

Recently, a new matrix product, called the semi-tensor product (STP) of matrices, has been proposed [5]. Using this method, an algebraic state space representation (ASSR) has been developed for dealing with some classical control problems of logical dynamic systems [6, 7], including controllability and observability [8–11], stability and stabilization [12–15], optimal control [16, 17], and disturbance decoupling [18, 19]. One may refer to [20–27] for some other applications.

It is known that the external disturbances are ubiquitous and may lead the network dynamics to some unexpected behaviour. Therefore, when modeling a gene regularity network, disturbances should be considered [28, 29]. In recent years, the disturbance decoupling problem of BCNs has been investigated in [18, 19]. Moreover, the global robust stability and stabilization of BCNs have been studied in [30]. However, to our best knowledge, there is few literatures devoted to the robust controllability and observability of BCNs, though they have obvious importance for BCNs with disturbances. Especially for robust observability problems, they are much harder to be understood and verified, compared with robust controllability problems. A recent work [31] proposes a new method to investigate observability of BCNs by converting it to a problem of set controllability. Under a fundamental framework of [31], we investigate robust controllability and observability problem in our earlier conference paper [32]. Regarding [32], the novelties of this paper consist of the following: (i) we discuss two kinds of control inputs, respectively, for robust controllability under unobservable disturbances: one is a free Boolean sequence, and the other inputs are logical variables satisfying certain input nteworks. For these two cases, we, respectively, give some sufficient conditions for robust controllability. But in [32], we only consider the case that control inputs are free; (ii) we give an actual example to illustrate our theoretical results; please see Example 22. But there is no such example in [32].

The main contributions of this paper include the following: (i) For unobservable disturbances, some sufficient conditions for robust controllability and observability of BCNs are obtained, respectively; (ii) based on the definition of order-preserving system, some easily verifiable sufficient conditions for robust controllability and observability under observable disturbances are presented, respectively; (iii) a general algorithm is proposed to design the least control sequences, which performs the required robust controllability or observability of BCNs.

The rest of this paper is organized as follows: Section 2 presents some preliminaries on the STP of matrices and set controllability of BCNs. Section 3 studies the robust controllability of BCNs under unobservable disturbances and gives an algorithm for designing the least control sequences. Section 4 discusses the robust controllability of BCNs under observable disturbances. The robust observability of BCNs is considered in Section 5. Section 6 is a brief conclusion.

Notation. The notations of this paper are fairly standard. denotes the set of real matrices. () is the set of columns (rows) of and () is the -th column (row) of . . , where is the -th column of the identity matrix . . denotes a matrix with all entries equal to . A matrix is called a logical matrix if . Denote by the set of logical matrixes. If , it is briefly denoted as . Denote by the set of Boolean matrices. Let ; then is the Boolean addition (with respect to and ). A matrix means all the entries are positive; that is, , .

2. Preliminaries

2.1. Semi-Tensor Product of Matrices

This subsection is a brief survey on STP of matrices [5].

Definition 1. Let , , and be the least common multiple of and . The STP of and is defined aswhere is the Kronecker product.

When , the STP becomes the conventional matrix product. Therefore, the STP is a generalization of the conventional matrix product. We can omit the symbol without confusion.

Proposition 2. (1) Let be a column and be a matrix. Then(2) Let and be two columns. Then , where is called the swap matrix, which is defined by

By identifying and , we have . Using vector expression, a logical function can be expressed as its algebraic form.

Theorem 3. Let be a Boolean function. Then there exists a unique matrix , called the structure matrix of , such that in the vector form we have

Theorem 4. Let and , where , , are logical variables and , are logical matrices. Thenwhere is the Khatri-Lao product.

2.2. Set Controllability of BCNs

A BCN is described aswhere , , are state variables, , , are controls, and , , and , , are Boolean functions.

Definition 5 (see [5]). System (6) is (1)controllable from to , if there are and a sequence of control , such that driven by these controls the trajectory can go from to ;(2)controllable at , if it is controllable from to any ;(3)controllable, if it is controllable at any .

Using Theorems 3 and 4, (6) can be converted into its algebraic form aswhere , , , and , .

Defineand setwhich is called the controllability matrix. Then we have the following result.

Theorem 6 (see [33]). Consider system (6) (by free control sequence). Assume its controllability matrix is ; then system (6) is (1)controllable from states to , if and only if ;(2)controllable at , if and only if ;(3)controllable, if and only if .

Denote by the set of states for BCNs. Assuming , the index column vector of , denoted by , is defined as

Define the set of initial sets and the set of destination sets , respectively, as follows:

Using initial sets and destination sets, the set controllability is defined as follows.

Definition 7. Consider system (6) with a set of initial sets and a set of destination sets . System (6) is (1)set controllable from to , if there exist and , such that is controllable from ;(2)set controllable at , if, for any , the system is controllable from to ;(3)set controllable, if it is set controllable at any .

Using and defined in (11), we define the initial index matrix and the destination index matrix , respectively, as

Then we define a matrix, called the set controllability matrix, as

Note that hereafter all the matrix products are assumed to be Boolean product (). Hence the symbol is omitted.

Definition 8 (see [31]). Consider system (6) with and as defined in (11). Moreover, the corresponding set controllability matrix is , which is defined by (13). Then system (6) is (1)set controllable from to , if and only if ;(2)controllable at , if and only if ;(3)set controllable, if and only if

3. Robust Controllability of BCNs under Unobservable Disturbances

Consider the following disturbed BCN:withwhere are states, controls, disturbances, and outputs, respectively, , , , . Using vector expression with , , , and , the algebraic forms of (14) and (15) are expressed aswithwhere , .

First, the robust controllability of disturbed BCNs is defined as follows.

Definition 9. System (14) is (1)robust controllable from to , if there exist and a control sequence , such that, under any , , the state can be driven from to ;(2)robust controllable at , if it is robust controllable from to any ;(3)robust controllable, if it is robust controllable at any .

In this paper, we consider two kinds of control inputs.(i)The control is a free Boolean sequence.(ii)The control inputs are logical variables satisfying certain input networks, such as

3.1. Robust Controllability with a Free Control Sequence

For the algebraic form (16) of system (14), defineandwhich is called the robust controllability matrix; then we have the following result.

Theorem 10. Consider system (14) under unobservable disturbances. Assume ; then system (14) is(1)robust controllable from to , if ;(2)robust controllable at , if ;(3)robust controllable, if .

Proof. It is easy to see that the system is controllable from to ; then there exists a finite control sequence such thatBecause of the construction of matrix , particularly , at any step no matter which disturbance happens, we can always find the corresponding to realize the corresponding one step transfer in (21). Hence, system (14) is controllable from to . (The detailed argument is similar to the proof of Theorem 6.)

Since robust controllability for any disturbance is a strong requirement, next we consider robust controllability under a constrained set of disturbance inputs. First, a constrained set for disturbance inputs is denoted by , where is the cardinality of set . Define the robust controllability matrix under the constrained disturbances set aswhereThen we have the following result.

Corollary 11. Consider system (14) under the constrained disturbances set . Assume ; then system (14) is(1)robust controllable from to , if ;(2)robust controllable at , if ;(3)robust controllable, if .

Remark 12. The main advantage of Theorem 10 is when the iterative steps increase, the corresponding matrices, which are defined in (19) and (20), do not increase their dimensions. Hence it is easily verifiable and computable. The main disadvantage is that this result is not necessary; refer to the following example.

Example 13. Consider the following BCN with unobservable disturbance:where , , and Assuming and , then the state trajectory can be presented in Figure 1. It is easy to see that, under unobservable disturbance, system (24) is robust controllable from to . However, by immediate calculation, we haveIt follows thatObviously, . Hence we conclude that Theorem 10 is only sufficient but not necessary.

3.2. Robust Controllability with Input Networks

Using the STP method, the input networks (18) can be expressed in its algebraic form aswhere .

Putting (16) and (27) together, we haveSet ; then (28) can be converted intowhere .

Similar to the argument in [34], we construct the set of initial sets and the set of destination sets aswhere

Then it is easy to see that each corresponds to a unique state . Hence system (29) is set controllable from to , which is equivalent to the controllability from state to of system (14).

Theorem 14. System (14) with inputs (18) is robust controllable, if and only if the extended system (29) is robust set controllable with respect to and , which are defined in (30).

Define , and Using (12), we have the robust set controllable matrix asSimilar to Theorem 10, we have an easily verifiable sufficient condition, which is shown as follows.

Theorem 15. Consider system (14) with inputs (18). Assuming , then system (14) is (1)robust controllable from to , if ;(2)robust controllable at , if ;(3)robust controllable, if .

3.3. Control Design

For system (14) with any disturbance, assume can be driven to by a proper free control sequence. Since the trajectory from to is in general not unique, in the following sequel, we try to find the shortest one. We propose the following “back stepping" algorithm.

Algorithm 16. Set and . Assume the -th element of robust controllability matrix, .(i)Step 1. Define (at most) -step controllability matrix as Find smallest , such that .(ii)Step 2. If , then . Find such that , . Set ; stop. Else, find one , such that and .(iii)Step 3. Find such that (note that is not unique, but there is at least one such block). Set .(iv)Step 4. Set and replace by . Go back to Step 2.

The following proposition is an immediate consequence of Algorithm 16.

Proposition 17. The control sequence generated by Algorithm 16 is a least sequence of controls, which can drive the trajectory from to . Moreover, the corresponding trajectory is , which is also produced by the algorithm.

It is easily verified that Algorithm 16 can also be applied to the case with the constrained disturbances set.

Example 18. Consider the following BCN with unobservable disturbance:Setting , we can get its algebraic form aswhere , It is easy to calculate thatIt follows thatAccording to Theorem 10, since , we can see that system (32) is robust controllable from to . What is the least control sequence for driving to ?
Using Algorithm 16, we can find the smallest , such that . It is easy to find that and . Choose , such that , . Set
According to Step 4 in Algorithm 16, set , and replace by . Then we have . Find (or ), such that , . Set
Hence the least control sequence is , which can drive to , and the trajectory is .
For system (32) with constrained disturbance set , using (22) and (23), we haveandAccording to Corollary 11, we conclude that system (32) with is robust controllable at .

4. Robust Controllability of BCNs under Observable Disturbances

Assume the disturbance at time ; that is, the disturbance is dependent on time , where ; then the dynamic model at time iswhere .

Now we consider the overall robust controllability of system (14) by a free control sequence. It is easily extended to the case with networked inputs (18).

We give a necessary condition as follows, which is obvious, because the constant disturbance is legal.

Lemma 19. Assuming system (14) is robust controllable, then each subsystem (38) is robust controllable.

Then we consider the robust controllability problem under the following assumption:

(A1) Each subsystem (38) with is controllable, .

For each pair of states and , we define the distance between them under model .

It is noted that, under assumption (A1), the distance is well defined for any and any . If we only consider the reachability to , then assumption (A1) can be replaced by the following assumption.

(A2) Each subsystem with is reachable to the fixed , .

Next we define the order-preserving of system (14).

Definition 20. System (14) is (1)order-preserving at , if for any , and any , we have(2)order-preserving, if it is order-preserving at any .

Theorem 21. Consider system (14) with observable disturbances. (1)Assume (A2) is satisfied and system (14) is order-preserving at the fixed . Then it is robust reachable to ; i.e., it is robust controllable from any to .(2)Assume (A1) is satisfied and system (14) is order-preserving at any . Then it is robust controllable.

Proof. We only prove for the fixed state case. The proof for general case is exactly the same. If we know the model at time is , then we can design the control to drive to , whereSince the number of all possible models is , after finite times, at least one model should be reached as many times as we wish. Because of the order-preserving, it is easy to see that at each time the distance is decreasing at least one. Hence the distance will eventually reach zero. That is, system (14) will reach .

Example 22 (see [35]). Consider a lac operon regulatory network modelwhere , and are state variables, which denote the mRNA, the LacZ polypeptide, and the internal lactose, respectively, and are control inputs, which represent the external lactose and the external glucose, respectively. The variable equal to (or ) means the corresponding molecular has a high (or low) concentration. In order to investigate the robust controllability of the model, we introduce an artificial disturbance in , which becomes .
Using the vector form of logical variables and setting , , by using the STP, we have the following algebraic form:whereAssume the disturbance is observable at time ; then we havewhereAccording to (8) and (9), we calculate thatwhere , .
It is obvious that both subsystems are not controllable. Hence system (43) does not satisfy Assumption (A1). From the above matrices and , it is easy to see that both subsystems are all reachable to . Using (39), for and , respectively, we can figure out that , , and , . Obviously, this system is order-preserving at . According to Theorem 21, we conclude that system (43) is robust reachable to ; i.e., it is robust controllable from any to .

5. Robust Observability of BCNs

First, we give the definition of robust observability for disturbed BCNs.

Definition 23. System (14)-(15) under unobservable disturbances is robust observable, if for any two initial states , and under any disturbance , there exists a control sequence , such that the corresponding output sequences are distinct; that is,

As suggested in [31], we consider two kinds of state pairs. A pair is -indistinguishable if . Otherwise, is called -distinguishable.

Next, we split the product state space into a partition of three components as

Using algebraic form (16), we construct a dual system asSet ; we haveThen (51) can be expressed as a new systemwhere .

Now the observability problem of system (14)-(15) can be converted into a set controllability problem of the extended system (52). Construct the set of initial sets and the set of destination sets as follows:andNote that (53) means each state pair in is an element of , while (54) means has only one element, which is . Then we have the following result.

Theorem 24. Consider system (14)-(15) under unobservable disturbances with and as defined in (53) and (54), respectively. The system is robust observable, if system (52) is robust set controllable. Precisely speaking, the conditions in Theorem 10 are satisfied.

Example 25. Recall Example 18. Assuming that the output iswe have its algebraic form aswhere .
It is easy to figure out thatandSet ; we have a new system aswhere
We calculatewhere
Using and , we haveIt follows thatAccording to Theorem 24, since , we conclude that system (32) with output (55) is robust observable at .

Similarly, we can obtain an easily verifiable sufficient condition for the robust observability of BCNs under observable disturbances.

Theorem 26. System (14)-(15) with observable disturbances is robust observable, if system (52) is robust set ontrollable from to . Precisely speaking, the conditions in Theorem 21 are satisfied.

6. Conclusion

Robust controllability and observability of disturbed BCNs have been investigated in this paper. Some sufficient conditions for robust controllability under unobservable and observable disturbances have been obtained, respectively. By constructing an extended system for a given BCN, an observability problem can be converted to a set controllability problem of its extended system. Then some corresponding conditions for robust observability are obtained for two cases, respectively. A universal algorithm is presented to construct the least control sequences to realize the required robust controllability.

The results in this paper can easily be extended to mix-valued logical systems.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

An earlier version of this paper was presented as a conference paper at the 37th Chinese Control Conference, 25-27, July, Wuhan, 2018. We would like to thank Prof. Daizhan Cheng for his valuable discussion. This work was supported by National Natural Science Foundation (NNSF) of China under Grants 61773371 and 71701115.

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Copyright © 2019 Yuanhua Wang 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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