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 tomake a set to be a RCIS. Finally, the proposed methods are illustrated by a reduced model of the lac operon in E. coli.


Introduction
In 1969, Kauffman [1] 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 [2] and immune response [3].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 [4].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][8][9][10][11][12][13][14][15], observability [16][17][18][19], disturbance decoupling problem [20] synchronization [21], function perturbations [22], optimal control [23][24][25][26], normalization problem [27], and others.The STP has also been widely applied in games [28,29] and asynchronous sequential machines [30,31].In detail, [28] 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 [29], 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, [30] and [31] 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 [32].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) [33], 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][35][36][37].In [33], 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 [38].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 [39], 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.

Preliminaries
In this section, we give some necessary properties of the STP and list some useful notations.
( (vi) Let 1 2  denote the row vector of length 2  with all entries being 1.
Definition ([4]).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 ([4]).Let  ∈ R  be a column and  be any matrix.en Definition ( [40]).Let  ∈ M × ,  ∈ M × .Then, the Khatri-Rao product is defined as The notation ∨ represents that when the two values denoted by  and  taking from D are the same, then ∨ = 0. Otherwise, the result equals 1.
A logical domain, denoted by D is defined as D = {0, 1}, and If identifying 0 ∼  2 2 and 1 ∼  1 2 , then Δ 2 ∼ D, and Δ 2  ∼ D  , where "∼" represents two different forms of the same object.If  ∈ D  , then we say  is in a scalar form.If  ∈ Δ 2  , 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 D  , while we use the notation  for its vector variable in Δ 2 .In short, The following lemma is important for the algebraic expressions of logical functions.
Definition ( [39]).Consider system (7) with disturbance inputs.For any given state  ∈ D  and when   () ≡ 1(   () ≡ 0), the discrete derivative of Boolean functions at state , denoted by   1 ()(   0 ()), 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   ∈ I(),  ∈ [1, ], and () = ≤  () ⋉  (  , ) . ( Theorem 9. Consider system ( ) with the fixed point  for any disturbance.If () has at most one in each column, then I  () 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 I  () beginning from any initial state   ∈ I(),  ∈ [1, ], in every step, which completes the proof.
Remark .It is learned from Theorem 9 that matrix () with dimension  ×  (not 2  × 2  ) can be constructed, based on which the RIS is analyzed.Therefore, compared with the results obtained by STP in [35], the computational complexity is reduced from O(2  ) to O( 2 ).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.
Remark .As mentioned above that set D  is equivalent to Δ 2  .Based on STP, the algebraic expression of system (20) can be obtained.Therefore, it will be better to write set S 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  = , which will be designed.On one hand, for system (26) with () = (), () =   2  , and On the other hand, we define the following sets: Obviously,   = { 1 ,  2 , . . .,   }.For each and then denote For any integer V ∈ [1, ], define For each (  V ), we construct a set, denoted by W(  V ), as Then, the result about the existence of output feedback controllers can be obtained.
Moreover, if ( ) holds, then output feedback matrices under which  is a ROCIS are constructed as with Proof.(Sufficiency): Suppose that (36) holds.We construct controller (37).Then, for Moreover, for any and then it can be learned from (34) that (20).
(Necessity): Suppose that  is a ROCIS of system (20) with control () = (), and then, for any In fact, if (36) does not hold, then there exists an integer In this case, it can be learned from (35)

Complexity
Example .Let us consider a reduced model of the lac operon in E. coli [33,41]: where  1 ,  2 , and  3 are state variables denoting the lac mRNA, the lactose in high concentrations, and the lactose in medium concentrations, respectively;  1 ,  2 , and  3 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 where  (45)

Conclusion
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.
and we analyze the following two problems:(i) Problem 1: For a given output feedback matrix  ∈ L 2  ×2  , analyze whether the set  is a ROCIS of system (20) under control system () = ().