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Time-optimal state feedback stabilization of switched Boolean control networks
Author’s Accepted Manuscript Time-optimal State Feedback Stabilization of Switched Boolean Control Networks Yong Ding, Yuqian Guo, Yongfang Xie, Chunhua Yang, Weihua Gui www.elsevier.com/locate/neucom
PII: DOI: Reference:
S0925-2312(16)31560-0 http://dx.doi.org/10.1016/j.neucom.2016.12.044 NEUCOM17863
To appear in: Neurocomputing Received date: 29 September 2015 Revised date: 4 November 2016 Accepted date: 12 December 2016 Cite this article as: Yong Ding, Yuqian Guo, Yongfang Xie, Chunhua Yang and Weihua Gui, Time-optimal State Feedback Stabilization of Switched Boolean Control Networks, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2016.12.044 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Time-optimal State Feedback Stabilization of Switched Boolean Control NetworksI,II Yong Ding, Yuqian Guo, Yongfang Xie, Chunhua Yang, Weihua Gui School of Information Science and Engineering, Central South University, Hunan, China, 410083.
Abstract This paper investigates the time-optimal state-feedback stabilization of switched Boolean control networks (SBCNs). Based on the properties of the semi-tensor product (STP) of matrices, a technique is proposed to merge the switching signal and all inputs of subnetworks into a single input variable. This technique allows us to transfer a SBCN to an equivalent non-switching logical control network (LCN) and eases the analysis and design process significantly. So long as a state-feedback stabilizer is obtained for the resulting non-switching LCN, it can then be decomposed uniquely into the respective state feedbacks of the sub-networks and the state-dependent switching law. Based on this technique, the controllability and stabilisability of such SBCNs are solved and an algorithm for finding all time-optimal switching state feedbacks is proposed. An example is described to illustrate the main results and the design process proposed in this paper. Keywords: switched Boolean control network, semi-tensor product of matrices, stabilization, state-dependent switching law.
1. Introduction Boolean networks (BN) have attracted massive attention from various fields since Kauffman first introduced them for modelling genetic regulatory networks (GRNs) [1]. BNs are systems of logical evolution and can be applied to describe the behaviour of many living organisms at the system level. There are many other models for GRNs, including Bayesian networks and neural networks [2], but BNs are capable of capturing the main logical interconnections between genes while being convenient from the viewpoint of analysis and simulation. Owing to shortage of useful mathematical tools, many analysis and design problems related to BNs had been challenging for a long time. Recently, the semi-tensor product (STP) of matrices, a new matrix product, proposed by Cheng, was successfully applied to analyse BNs [3]. By using this tool, any logical expression can be expressed as a unified multi-linear form, and many fundamental and excellent results pertaining BNs have been published [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. Although the typical BN model is based on purely discrete dynamics, the dynamics of biological networks in practice often involve multiple instances of switching between different models [16]. For instance, different cellular fates, including cell proliferation, apoptosis and differentiation can be regarded as cellular states and regulation of the functional states of a cell can be described as switching from one mode to another upon exposure to an appropriate stimulus [17]. As another example, the cell cycle of eukaryotic cells consists of two basic I This work was jointly supported by the National Natural Science Foundation of China (61473315, 61074002, 61321003) and National Key Basic Research Development Program (973 Program) Sub-project (61325309) II Corresponding author: Y. Guo. Email: [email protected], Tel. +86 15074917920. Fax +86 731 88836876.
Preprint submitted to Neurocomputing
phases: S phase (DNA synthesis) and M phase (mitosis). These two phases are separated by phases G1 and G2. The M phase can be further divided into several phases, including prophase, metaphase, anaphase and telophase. Each phase of a cell cycle is triggered by a set of conditions or events. See [18] and the references therein for further information about the cell cycle. In addition, at the intercellular level, cell differentiation can be viewed as a switched system [16]. In the past few decades, many results related to switched systems have been published [19, 20, 21]. Recently, under the framework of STP of matrices, the stability and stabilisability of switched BNs (SBNs) have been investigated [22, 23, 24, 25]. Specifically, consistent stabilisability was studied in [22]. The concept of switching point reachability was introduced in [23], where point-wise stabilisability of SBNs and global stability of switched Boolean control networks (SBCNs) with arbitrary switching signals were solved. By constructing the switching-state incidence matrix, global stability at some limit cycle of SBNs was solved in [24]. In addition, the stability of SBCNs with impulsive effects was analysed in [25]. Optimal control problems for different situations have been solved in [27] and [28]. See [28, 29, 30, 31, 32] for controllability analyses and disturbance decoupling control design of SBCNs. It is worth pointing out that in some cases, external interventions can be applied to trigger switching of a system between different modes to achieve a certain purpose. For instance, in biology, (in)activation of genes can lead to certain cellular functional states or phenotypes [33], and interventions such as irradiation, chemicals and gene engineering methods can be used to (in)activate genes. This process can interfere with switching within the cell cycle or between cellular fates. For instance, during the G1 phase of cell cycle, if DNA damage occurs, the cell cycle can arrest in the transition from the G1 to the S phase, acDecember 15, 2016
2. Preliminaries
tivate DNA repairing mechanism, or initiating apoptosis when the damage is beyond repair. However, if gene p53 is deactivated, the cell cycle will continue to enter the S phase, enabling the damaged genetic materials to pass onto the next generation [33, 34, 35]. By contrast, differentiated cells are not cycling cells, which have different needs for dealing with DNA lesions. Thus, DNA damage response and DNA repair differ, depending on cellular fates [36]. In sum, different mechanisms are applied to different cellular fates. Therefore, in this work, we assume that the switching signal is designable. The main purpose of this work is to develop an algorithm to design time-optimal switching feedback stabilizers for SBCNs. The main contributions are as follows:
This section lists a few necessary preliminaries related to the STP of matrices, which will be used in this paper. Definition 1. [3] The STP of two matrices A ∈ Mm×n and B ∈ M p×q is (1) A n B = (A ⊗ I ns )(B ⊗ I ps ) where s = lcm(n, p) is the least common multiple of n and p, and ⊗ is the Kronecker product. Remark 1. The STP of matrices is a generalization of the conventional matrix product. Most fundamental properties of the conventional matrix product remain true. If n = p, then the STP degenerates to the conventional matrix product. Therefore, in this paper, the symbol n is omitted unless it is needed to be emphasized.
• A technique is developed to merge the switching signal and the control variable together and transfer a SBCN to an equivalent non-switching logical control network (LCN). The control variable of the resulting LCN is actually the STP of the control variable and the coded switching signal of the original SBCN. This technique eases the control design process significantly. For instance, so long as a state feedback for an LCN is obtained, it then can be decomposed uniquely into the sub-controllers and the switching signal of the original SBCN. In addition, the resulting switching state feedback of the original SBCN is time-optimal if and only if the state feedback of the LCN is time-optimal.
Some related concepts and notations are listed below. 1. D = {0, 1} represents the logical domain. 2. δin denotes the i-th column of the identity matrix In . ∆n := {δ1n , δ2n , · · · , δnn }, and for compactness, ∆ := ∆2 . ∆ is the vector form of the logical domain which can be identified with D under the equivalence 1 ∼ δ12 and 0 ∼ δ22 , where ∼ denotes logical equivalence. 3. δ0n := 0n×1 and 1n := [1, 1, · · · , 1]T . | {z } n
4. Bm×n represents the set of m × n Boolean matrices. A matrix A = (ai j ) is a Boolean matrix if ai j ∈ D. 5. The symbols ∧ and ∨ represent the logical operators AND and OR, respectively. 6. Assume that A1 , A2 , · · · , A s , A, B ∈ Bm×n , then Boolean addition is defined as A+B B := A∨B = (ai j ∨bi j ). Boolean s P summation is defined as (B) Ai := A1 ∨ A2 ∨ · · · ∨ A s .
• Based on the technique above, an algorithm for timeoptimal switching state feedbacks for SBCNs is proposed. The set of all the time-optimal switching state-feedbacks is characterized by a Boolean matrix. Precisely, a logical matrix is a logical sub-matrix of the obtained Boolean matrix if and only if it can be decomposed into a time-optimal switching state-feedback of the SBCN.
i=1
7. Coli (A) represents the i-th column of the matrix A. RowΣ (A) denotes the Boolean summation of all rows of A. 8. Ln×s represents the set of n × s logical matrices. A matrix B ∈ Bm×n is a logical matrix if Coli (B) ∈ ∆m ∀i = 1, 2, · · · , n. 9. δn [i1 · · · i s ] represents the Boolean matrix L ∈ Bn×s such i that Col j (L) = δnj , j = 1, 2, · · · , s. 10. Z≥0 denotes the set of non-negative integers. 11. Let A = (ai j ) ∈ Bm×n , B = (bi j ) ∈ Bn×p , their Boolean product is defined as A nB B = C = (ci j ) ∈ Bm×p , with n P ci j := (B) aik bk j . The Boolean product of X ∈ Bm×n
A preliminary result related to SBCN stabilization has been reported in [26], where a unified feedback gain was adopted for all subnetworks. In this paper, different feedback gains are adopted for different subnetworks. This provides great flexibility in control design, but the technique proposed in [26] does not directly apply any more. A new technique (summarized in Lemma 2) is developed to establish the design algorithm for this situation. The rest of this paper is organized as follows. In Section 2, a few necessary preliminaries related to the STP of matrices, including the definition of STP and some basic results, are introduced. In Section 3, the main objectives of this study are given. In Section 4, a few lemmas crucial for establishing the main results of this study are introduced and proved. Section 5 is the main part of this paper, and it describes stabilisability and time-optimal control design of SBCNs. In Section 6, an example is provided to illustrate the proposed method. Finally, our concluding remarks are presented in Section 7.
k=1
and Y ∈ B p×q is defined as X nB Y = X ⊗ I ns nB Y ⊗ I ps where s = lcm(n, p). The Boolean powers are defined as A(k) := A nB A nB · · · nB A. | {z } k
2
12. For logical matrices A1 ∈ L s×n and A2 ∈ Lm×n , A1 ∗ A2 ∈ L sm×n represents the Khatri-Rao product defined as Coli (A1 ∗ A2 ) := Coli (A1 ) n Coli (A2 ), i = 1, 2, · · · , n. 13. W[m,n] represents the swap matrix with index [m, n] defined as W[m,n] := [In ⊗ δ1m , In ⊗ δ2m , · · · , In ⊗ δm m ]. For any vector X ∈ ∆m and Y ∈ ∆n , Y X = W[m,n] XY holds. 14. Diag{A1 , · · · , Ak } denotes the block diagonal matrix whose (i, i)-block is Ai , i = 1, · · · , k, and all off-diagonal blocks are zero. 15. The power-reducing matrix Mr,N ∈ LN 2 ×N is defined as Mr,N := Diag{δ1N , δ2N , · · · , δNN }. For any x ∈ ∆N , x n x = Mr,N x holds. For simplicity, we denote Mr := Mr,2 . 16. For any K ∈ Bm×n , a logical matrix K ∈ Lm×n is called a logical sub-matrix of K if K ∧K = K. The set of all logical sub-matrices of K is denoted by S (K). Especially, for any Boolean vector x ∈ Bm×1 , we define S (x) := {z ∈ ∆m | z ∧ x = z}. For y ∈ B1×m , we define S T (y) = S (yT ).
process of transferring a BCN to its algebraic form is omitted here, but it can be found in [3]. This would not cause any problem because the method developed in and the results of this study are based directly on the algebraic form (3). Under the control sequence u = {u(t)}t∈Z≥0 and the switching signal σ, the solution to SBCN (3) with initial state x0 is denoted by x(t; x0 , u, σ). Definition 2. Considering SBCN (3), a given state Xe ∈ ∆2n is said to be stabilizable, if for any initial state x0 ∈ ∆2n , there exist a control sequence u and a switching signal σ such that x(t; x0 , u, σ) = Xe , ∀t ≥ T (x0 , u, σ),
where T (x0 , u, σ) is some non-negative integer depending on x0 , u and σ. Denote the smallest integer such that (4) holds by T m (x0 , u, σ) and the set of all stabilizing pairs (u, σ) for the initial state x0 by Uσ (x0 ) . Define
Proposition 1. [7] Any logical function f : ∆n2 → ∆2 can be expressed uniquely as f (x1 , x2 , · · · , xn ) = M f nni=1 xi
(4)
T m (x0 ) :=
∀xi ∈ ∆2
min
(u,σ)∈Uσ (x0 )
T m (x0 , u, σ).
The transient period of SBCN (3) is denoted by T m and defined as T m := max T m (x0 ). (5)
where the structure matrix M f ∈ L2×2n is determined uniquely by f .
x0 ∈∆2n
This paper aims to design switching state feedbacks of the form u(t) = Fσ(t) x(t) (6) σ(t) = φ(x(t))
3. Problem Setting A SBCN with n nodes, m control inputs and N subnetworks is described as X1 (t + 1) = f1σ(t) (X(t), U(t)) σ(t) X2 (t + 1) = f2 (X(t), U(t)) (2) .. . Xn (t + 1) = f σ(t) (X(t), U(t))
for SBCN (3), where Fi is the feedback gain for the ith subnetwork and φ : ∆2n → N. Suppose that Xe ∈ ∆2n is the state to be stabilized. The switching state feedback (6) is said to be stabilizing if for any initial state x0 ∈ ∆2n , it holds that
n
where X(t) = (X1 (t), X2 (t), . . . , Xn (t))T ∈ Dn , U(t) = (U1 (t), U2 (t), . . . , Um (t))T ∈ Dm are the states and the control inputs, respectively. Xi ∈ D, i = 1, 2, . . . , n, are logical variables. σ : Z≥0 → N := {1, 2, . . . , N} is the switching signal. fi j : Dm+n → D, i = 1, 2, . . . , n, j = 1, 2, . . . , N, are logical functions. We denote the vector forms of Xi and Ui by xi ∈ ∆2 and ui ∈ ∆2 , respectively. By the theory of STP [3], a natural oneto-one mapping from ∆n2 to ∆2n is given by
x(t; x0 , uFσ , σφ ) = Xe , ∀t ≥ T (x0 , uFσ , σφ ),
(7)
where T (x0 , uFσ , σφ ) is some non-negative integer depending on x0 , uFσ and σφ . uFσ and σφ are, respectively, the control sequence and the switching signal generated by (6). The switching state feedback (6) is said to be time-optimal if for any initial state x0 , (7) holds with T (x0 , uFσ , σφ ) = T m (x0 ).
(x1 , x2 , · · · , xn ) 7→ nni=1 xi (t) := x1 n x2 n · · · n xn . We denote the vector form of X(t) by x(t) := nni=1 xi (t) ∈ ∆2n , then x(t) and X(t) can be determined uniquely by each other. Similarly, we denote the vector form of U(t) by u(t) := m nm i=1 ui (t) ∈ ∆2 . By using Proposition 1 and the properties of STP of matrices, we can transfer SBCN (2) to the algebraic form x(t + 1) = Lσ(t) u(t)x(t), (3)
4. Some Lemmas Lemma 1. Any logical matrix F ∈ L2Nm ×2n can be decomposed uniquely as F = F1 ∗ F2 ∗ · · · ∗ F N , (8) where F1 , F2 , · · · , F N ∈ L2m ×2n . In addition, it holds that
where x(t) ∈ ∆2n and u(t) ∈ ∆2m . Lσ(t) ∈ L2n ×2n+m is the logical matrix determined uniquely by the original SBCN. The detailed
F x = (F1 x) n (F2 x) n · · · n (F N x) 3
∀x ∈ ∆2n .
1. The k-step controllability matrix Ck ∈ B2n ×2n is defined as ( 1, δi2n ∈ R(k) (δ2j n ) (Ck )i j = (11) 0, δi2n < R(k) (δ2j n ).
Proof. Suppose that F = δ2Nm [i1 i2 · · · i2n ]. i
By using Proposition 3 in [30] repeatedly, every column δ2jNm can be decomposed uniquely as i
s
s
2. The controllability matrix C ∈ B2n ×2n is defined as ( 1, δi2n ∈ R(δ2j n ) (C)i j = 0, δi2n < R(δ2j n ).
s
δ2jNm = δ2mj,1 n δ2mj,2 n · · · n δ2mj,N m
where 1 ≤ s j,t ≤ 2 . We define Ft := δ2m [s1,t s2,t · · · s2n ,t ],
Proposition 2. (Controllability Matrix) For SBCN (3), the following claims hold.
t = 1, 2, · · · , N.
1) The k-step controllability matrix is
Lemma 2. For any v1 , v2 , · · · , vN ∈ ∆2m and ω ∈ ∆N , there holds [v1 v2 · · · vN ]ω = Gωv1 v2 · · · vN , (9)
Ck = Q(k) ,
By definition of the Khatri-Rao product, (8) follows.
Q = (B)
N X (Li nB 12m ). i=1
G = [G1 G2 · · · G N ]
2) The controllability matrix is
(10)
with
C = (B) G j :=
⊗ I2m )W[2 jm ,2(N− j)m ] ,
j = 1, 2, · · · , N.
Ck .
(14)
Proof. Note that the switching signal σ can be re-expressed as σ(t) = µT ω(t),
(15)
µT := [1 2 · · · N]
(16)
ω : Z≥0 → ∆N .
(17)
where
With a mild abuse of terminology, we call ω the vector form of the switching signal. Define L := [L1 L2 . . . LN ], where Li is the structure matrix of the ith subnetwork. Then it holds that
RHS =(1T2(N−1)m ⊗ I2m )W[2 jm ,2(N− j)m ] v1 · · · v j−1 v j v j+1 · · · vN | {z } | {z } 2 jm
2n X k=1
Proof. The left hand side (LHS) of (9) actually defines a logical function from ∆N2Nm to ∆2m . According to Proposition 1, there must exist a unique logical matrix G such that (9) holds. In the following, we prove that the logical matrix G defined in (10) is the one. Set ω = δNj ∈ ∆N , the LHS of (9) equals v j , while the right hand side (RHS) is G j v1 v2 · · · vN . Hence,
2(N− j)m
=(1T2(N−1)m ⊗ I2m ) v j+1 · · · vN v1 · · · v j−1 v j | {z }
Lσ(t) = [L1 L2 . . . LN ]ω(t) = Lω(t).
2(N−1)m
=I2m v j = v j = LHS . Thus, (9) holds.
(13)
where
where G ∈ L2m ×N2Nm is defined as
(1T2(N−1)m
(12)
Thus, SBCN (3) can be expressed equivalently as x(t + 1) = Lω(t)u(t)x(t).
(18)
We define a new control variable uˆ as
5. Main Results
uˆ (t) := ω(t) n u(t).
5.1. Reachability and Stabilisability We briefly discuss the reachability of SBCNs here, which will be used to establish the design procedure in this paper.
Then, SBCN (3) can also be re-written as the LCN x(t + 1) = Lˆu(t)x(t).
Definition 3. Given SBCN (3) and let x0 , xd ∈ ∆2n . xd is said to be k-step reachable from x0 if there exist a control sequence u and a switching law σ such that x(k; x0 , u, σ) = xd . xd is said to be reachable from x0 if there exists a positive integer k such that xd is k-step reachable from x0 .
(19)
According to Proposition 3 in [30], for any uˆ ∈ ∆N2m , there exist two unique logical vectors ω ∈ ∆N and u ∈ ∆2m such that uˆ = ω n u, that is, any control sequence uˆ (t) for LCN (19) can be decomposed uniquely into the control sequence u(t) and the switching signal σ(t) = µT ω(t) for SBCN (3), such that these two LCNs produce the same solution for the same initial condition. Based on this observation, these two LCNs share the same controllability matrices. By following the same argument for the analyses of controllability of BCNs in [6, 3], the result follows.
The k-step reachable set and the reachable set from x are de(k) noted by R(k) (x) and R(x), respectively. We define R−1 (y) := (k) {x ∈ ∆2n | y ∈ R (x)}, and R−1 (y) := {x ∈ ∆2n | y ∈ R(x)}. Definition 4. (Controllability matrix) 4
Proof. By the definition of the new control variable ξ, SBCN (3) can be transformed to an equivalent form (24). According to Lemma 1, every state feedback (26) for LCN (24) can be decomposed uniquely into a switching state feedback for SBCN (3). Under such feedbacks, both networks generate the same solution for the same initial condition.
By using the controllability matrices obtained above, we have the following results. Theorem 1. The following claims hold. 1) SBCN (3) is stabilizable to Xe if and only if both the following statements hold: (a) XeT C1 Xe = 1; (b) XeT C = 1T2n . 2) If SBCN (3) is stabilizable to Xe , then
Remark 2. According to Theorem 2, to find a switching state feedback controller for SBCN (3), we can design a state feedback controller for LCN (24) first, and then decompose it to get the matrices H and Fi . In the following, we shall use this idea to find all time-optimal switching state feedback stabilizers for SBCN (3).
T m (x0 ) = min{k | XeT Ck x0 = 1} k≥0
and T m = min{k | XeT Ck = 1T2n }, k≥0
(20)
We are now ready to state the design process for time-optimal state feedback stabilizers of SBCN (3). According to the analysis above, we only need to design a time-optimal state-feedback for LCN (24) and then decompose it to obtain a time-optimal state feedback for SBCN (3). The time-optimal state-feedback design for LCN (24) is similar to the methods proposed in [8, 14]. A slightly different version of the design process is detailed in the following for integrity. Suppose that Xe is stabilizable. We then define a series of subsets as follows:
where C0 := I2n .
5.2. Switching State-feedback Design In the following, we discuss how to design switching state feedbacks of the form (6) for SBCN (3). By introducing µ and ω as in (16) and (17), respectively, the switching state feedback (6) can be expressed equivalently as u(t) = Fσ(t) x(t) (21) σ(t) = µT ω(t)
M0 M1
= {Xe } (1) = R−1 (Xe ) \ M0
(22)
M2
where Fi , i = 1, 2, · · · , N, and H are the logical matrices to be designed. We denote the feedback for the ith subnetwork by vi = Fi x. Then, we have
MTm
(2) = R−1 (Xe ) \ (M0 ∪ M1 ) .. . (T m ) = R−1 (Xe ) \ (M0 ∪ M1 ∪ · · · ∪ MTm −1 )
with ω(t) = Hx(t),
where T m is the transition period defined in (5) and by the definition, (k) R−1 (Xe ) = S T (XeT Ck ). (28)
u(t) = [v1 v2 · · · vN ]ω(t). N We define v(t) = ni=1 vi ∈ ∆2Nm . By lemma 2, it holds that
u(t) = Gω(t)v(t),
(i) (i+k) Since Xe is stabilizable, if x0 ∈ R−1 (Xe ), then x0 ∈ R−1 (Xe ) ∀k ∈ Z≥0 . Thus
(23)
where G is defined in (10). Substituting (23) into (18) gives
(1) (2) (T m ) M0 ⊆ R−1 (Xe ) ⊆ R−1 (Xe ) ⊆ · · · ⊆ R−1 (Xe ),
x(t + 1) =Lω(t)Gω(t)v(t)x(t) =LW[2m ,N]GW[N,N·2Nm ] Mr,N ω(t)v(t)x(t).
and for 1 ≤ i ≤ T m , Mi
We define a new control variable ξ(t) = ω(t) n v(t) ∈ ∆N·2Nm . Then, SBCN (3) is presented equivalently as the LCN ˜ x(t + 1) = Lξ(t)x(t)
(24)
L˜ = LW[2m ,N]GW[N,N·2Nm ] Mr,N .
(25)
(i) = R−1 (Xe ) \ (M0 ∪ · · · ∪ Mi−1 ) (i) (i−1) = R−1 (Xe ) \ R−1 (Xe ).
(29)
(i) This means that for any x0 ∈ Mi , x0 ∈ R−1 (Xe ) and x0 < ( j) R−1 (Xe ) ∀ j < i hold. Therefore, i is the smallest integer for Xe ∈ R(i) (x0 ). Thus
where
Mi = {x0 | T m (x0 ) = i},
Theorem 2. SBCN (3) is stabilizable to Xe by the switching state feedback (21)-(22), if and only if LCN (24) is stabilizable to Xe by state feedback
∀i = 0, 1, · · · , T m ,
and Mi ∩ M j = ∅,
i , j,
ξ(t) = K x(t)
(26)
M0 ∪ M1 ∪ · · · ∪ MTm = ∆2n .
K = H ∗ F1 ∗ F2 ∗ · · · ∗ F N .
(27)
In other words, the collection of subsets {Mi }0≤i≤Tm defines a partition of the state space ∆2n .
with 5
By the definition of K,
The key to obtain the time-optimal state feedback is to find all the control efforts for any given state that drives the system from Mq to Mq−1 or from M0 to itself. For any i = 1, 2, · · · , 2n , define S i ∈ LN2Nm+n ×N2Nm as S i = IN2Nm ⊗ δi2n .
S [Coli (K)] = S T (RowΣ Pi ) n j o Col j (Pi ) , δ02n , j = 1, · · · , N2Nm . = δN2 Nm ∈ ∆ N2Nm
(30)
Based on the above analysis, for any logical sub-matrix K of K,
By the definition of S i , we have Si
ξ(t) = K x(t)
= Diag{δi2n , · · · , δi2n } | {z }
is a time-optimal state feedback stabilizer for LCN (24). We summarize the design process for time-optimal state feedback stabilizers of SBCN (3) in the following.
N2Nm Nm
N2 i = [δ1N2Nm n δi2n δ2N2Nm n δi2n · · · δN2 Nm n δ2n ].
We divide the structure matrix L˜ into N2Nm blocks of equal size as L˜ = [L˜ 1 L˜ 2 · · · L˜ N2Nm ], where L˜ j ∈ L2n ×2n . Then, for any δi2n ∈ ∆2n ,
Algorithm 1. Step 1. Calculate the controllability matrices Ck and C as in (13) and (14), respectively. Step 2. Check the stabilisability by using Theorem 1. If Xe is stabilizable, then determine T m by using (20) and go to Step 3. Step 3. Construct the partition of the state space M0 , M1 , · · · , MTm by using (28) and (29). Calculate the indicator matrices Mq ∈ B2n ×2n as (31). Step 4. Calculate S i , i = 1, 2, · · · , 2n , as (30), and Pi ∈ B2n ×N2Nm for each δi2n ∈ ∆2n as (32). Step 5. Calculate the Boolean matrix K ∈ BN2Nm ×2n as (33). Step 6. Take any K ∈ S (K), and decompose it as
Nm ˜ N2Nm ˜ 2 Nm δi2n , · · · , Lδ ˜ i = [Lδ ˜ 1 Nm δi2n , Lδ δi2n ] LS N2 N2 N2 = [L˜ 1 δi2n L˜ 2 δi2n · · · L˜ N2Nm δi2n ].
˜ i ) = L˜ j δi n = Coli (L˜ j ), then, under the conObviously, Col j (LS 2 j trol effort ξ = δN2Nm , δi2n reaches Coli (L˜ j ) in one step. Besides, {Coli (L˜ j ) | j = 1, 2, · · · , N2Nm } = R(1) (δi2n ). For all 0 ≤ q ≤ T m , we denote the indicator matrix of Mq by Mq ∈ B2n ×2n and define it as j j δ2n , δ2n ∈ Mq Col j (Mq ) = (31) δ0n , δ j n < Mq . 2 2
K = H ∗ F1 ∗ F2 ∗ · · · ∗ F N by using Lemma 1, where H ∈ LN×2n and F1 , · · · , F N ∈ L2m ×2n . Then, H, F1 , F2 , · · · , F N are the matrices designed for the time-optimal switching state feedback (21)-(22).
Obviously, it holds that MqT = Mq . With this definition, if δi2n ∈ ˜ i) , Mq for some integer q with 1 ≤ q ≤ T m and Mq−1 Col j (LS δ02n , then the system can be driven from δi2n to Mq−1 . Similarly, ˜ i ) , δ0n , then the system can stay if δi2n ∈ M0 and M0 Col j (LS 2 within M0 . Based on this observation, for any δi2n ∈ ∆2n , we define Pi ∈ B2n ×N2Nm as ˜ i , δi n ∈ Mq , 1 ≤ q ≤ T m Mq−1 LS 2 Pi = (32) ˜ i, M0 LS δi2n ∈ M0 .
Remark 3. The Boolean matrix K defined in (33) characterizes all of the time-optimal switching state feedbacks for SBCN (3). Precisely, (21) is a time-optimal switching stabilizer for SBCN (3) if and only if K := H ∗ F1 ∗ F2 ∗ · · · ∗ F N is a logical sub-matrix of K.
When δi2n ∈ Mq , 1 ≤ q ≤ T m ,
6. Example
Col j (Pi ) = Mq−1 L˜ j δi2n .
Consider a SBCN
By the property of Mq , if Col j (Pi ) , δ02n , then Col j (Pi ) ∈ j (Mq−1 ∩ R(1) (δi2n )), that is, under the control effort ξ = δN2 Nm , 0 the trajectory reaches Mq−1 . If Col j (Pi ) = δ2n , then under the j control effort ξ = δN2 Nm , the trajectory can not reach Mq−1 . In a similar way, when δi2n ∈ M0 , Col j (Pi ) = M0 L˜ j δi2n . If Col j (Pi ) , δ02n , then Col j (Pi ) ∈ (M0 ∩ R(1) (δi2n )), that is, under j the control effort ξ = δN2 Nm , the state remains in M0 = {Xe }. The above analysis shows that if x(t) = δi2n and ξ(t) ∈ S T (RowΣ Pi ), then under this control effort, the trajectory of LCS (24) will go one step closer to Xe or stay at Xe . We define K ∈ BN2Nm ×2n as Coli (K) = (RowΣ Pi )T .
x(t + 1) = Lσ(t) u(t)x(t) with
(34)
L1 = δ8 [7 7 5 8 6 6 8 8 7 8 4 7 6 5 2 8 2 7 4 8 5 6 4 7 2 4 5 2 5 5 8 7] L2 = δ8 [7 1 5 4 3 1 8 2 2 1 6 5 4 1 8 7 8 8 2 2 3 2 1 7 2 1 4 4 1 7 2 2],
where x ∈ ∆8 , u ∈ ∆4 and σ : Z≥0 → N = {1, 2}. Both subnetworks of this SBCN are not stabilizable at any state. However, we shall show in the following that the SBCN is stabilizable to some states by switching state feedback. Precisely, by Proposition 2, we can calculate the k-step controllability matrices Ck
(33) 6
The state transfer graph of the closed-loop SBCN under the obtained switching state-feedback controller is shown in Figure 1. From Figure 1 and Table 1, one can see that any solution converges to δ68 in the least steps. Simple calculations show that for Xe = δ68 , there are in total 11010048 time-optimal switching state feedbacks.
Table 1: Transient periods T m (x0 ) (with Xe = δ68 )
x0
T m (x0 )
x0
T m (x0 )
4 3 1 2
δ58 δ68 δ78 δ88
1 0 3 4
δ18 δ28 δ38 δ48
7. Conclusion In this study, we investigated time-optimal stabilization of SBCNs by switching state feedbacks. A technique for merging the coded switching signal and the control variable of a SBCN was proposed. Using this technique, a SBCN can be transferred to an equivalent non-switching LCN with the merged variables as the new control input and the time-optimal control design method developed previously for non-switching BCNs applies. So long as a time-optimal state-feedback for the resulting nonswitching LCN is obtained, it can be decomposed to a timeoptimal switching state feedback for the original SBCN. By using this technique, in this paper, we calculated the controllability matrices of SBCNs and provided a necessary and sufficient condition for stabilisability of SBCNs. In addition, we proposed a time-optimal state-feedback design algorithm for SBCNs. The example presented herein illustrates the main results and the design process.
and the controllability matrix C. It is easy to check that for any x ∈ {δ48 , δ58 , δ68 , δ88 }, it holds that xT C1 x = 1 xT C = 1T . 8 Thus, SBCN (34) is stabilizable at any Xe ∈ {δ48 , δ58 , δ68 , δ88 } by Theorem 1. In the following, we take Xe = δ68 as an example to illustrate the algorithm for designing a time-optimal switching feedback. First of all, by Theorem 1, the transient periods T m (x0 ) for the respective initial states can be calculated, as listed in Table 1. From this table, we have T m = 4. Thus, by using (28) and (29), the state space can be partitioned into five subsets as M0 = {δ68 },
M1 = {δ38 , δ58 },
M3 = {δ28 , δ78 },
M2 = {δ48 },
M4 = {δ18 , δ88 }.
References
The indicator matrices Mq ∈ B8×8 , q = 0, 1, 2, 3, 4, for these subsets can be calculated by (31). For example, M3 = δ8 [0 2 0 0 0 0 7 0]. By following the remaining steps in Algorithm 1, S i ∈ L256×32 and Pi ∈ B8×32 , i = 1, · · · , 8, can be calculated. This process is omitted here to save space and finally, we get K ∈ B32×8 which is given by (35). Choose any logical matrix K such that K = K ∧ K. For instance, we take
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K = δ32 [3 14 22 30 1 4 11 25]. By Lemma 1, K can be decomposed uniquely as K =H ∗ F1 ∗ F2 =δ2 [1 1 2 2 1 1 1 2] ∗ δ4 [1 4 2 4 1 1 3 3] ∗ δ4 [3 2 2 2 1 4 3 1]. Thus, a time optimal state feedback for SBCN (34) is given by u(t) = Fσ(t) x(t) (36) σ(t) = µT Hx(t) with µT = [1 2] and H =δ2 [1 1 2 2 1 1 1 2] F1 =δ4 [1 4 2 4 1 1 3 3] F2 =δ4 [3 2 2 2 1 4 3 1]. 7
K =
11111111111111111101110111011101 00000000000011110000000000000000 00000000000000000100010001000100 00000000000000000100010001000100 11111111000000000000000000000000 11110000111100000000000000000000 00000000111100000000000000000000 00000000111111111111111111111111
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PII: DOI: Reference:
S0925-2312(16)31560-0 http://dx.doi.org/10.1016/j.neucom.2016.12.044 NEUCOM17863
To appear in: Neurocomputing Received date: 29 September 2015 Revised date: 4 November 2016 Accepted date: 12 December 2016 Cite this article as: Yong Ding, Yuqian Guo, Yongfang Xie, Chunhua Yang and Weihua Gui, Time-optimal State Feedback Stabilization of Switched Boolean Control Networks, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2016.12.044 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Time-optimal State Feedback Stabilization of Switched Boolean Control NetworksI,II Yong Ding, Yuqian Guo, Yongfang Xie, Chunhua Yang, Weihua Gui School of Information Science and Engineering, Central South University, Hunan, China, 410083.
Abstract This paper investigates the time-optimal state-feedback stabilization of switched Boolean control networks (SBCNs). Based on the properties of the semi-tensor product (STP) of matrices, a technique is proposed to merge the switching signal and all inputs of subnetworks into a single input variable. This technique allows us to transfer a SBCN to an equivalent non-switching logical control network (LCN) and eases the analysis and design process significantly. So long as a state-feedback stabilizer is obtained for the resulting non-switching LCN, it can then be decomposed uniquely into the respective state feedbacks of the sub-networks and the state-dependent switching law. Based on this technique, the controllability and stabilisability of such SBCNs are solved and an algorithm for finding all time-optimal switching state feedbacks is proposed. An example is described to illustrate the main results and the design process proposed in this paper. Keywords: switched Boolean control network, semi-tensor product of matrices, stabilization, state-dependent switching law.
1. Introduction Boolean networks (BN) have attracted massive attention from various fields since Kauffman first introduced them for modelling genetic regulatory networks (GRNs) [1]. BNs are systems of logical evolution and can be applied to describe the behaviour of many living organisms at the system level. There are many other models for GRNs, including Bayesian networks and neural networks [2], but BNs are capable of capturing the main logical interconnections between genes while being convenient from the viewpoint of analysis and simulation. Owing to shortage of useful mathematical tools, many analysis and design problems related to BNs had been challenging for a long time. Recently, the semi-tensor product (STP) of matrices, a new matrix product, proposed by Cheng, was successfully applied to analyse BNs [3]. By using this tool, any logical expression can be expressed as a unified multi-linear form, and many fundamental and excellent results pertaining BNs have been published [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. Although the typical BN model is based on purely discrete dynamics, the dynamics of biological networks in practice often involve multiple instances of switching between different models [16]. For instance, different cellular fates, including cell proliferation, apoptosis and differentiation can be regarded as cellular states and regulation of the functional states of a cell can be described as switching from one mode to another upon exposure to an appropriate stimulus [17]. As another example, the cell cycle of eukaryotic cells consists of two basic I This work was jointly supported by the National Natural Science Foundation of China (61473315, 61074002, 61321003) and National Key Basic Research Development Program (973 Program) Sub-project (61325309) II Corresponding author: Y. Guo. Email: [email protected], Tel. +86 15074917920. Fax +86 731 88836876.
Preprint submitted to Neurocomputing
phases: S phase (DNA synthesis) and M phase (mitosis). These two phases are separated by phases G1 and G2. The M phase can be further divided into several phases, including prophase, metaphase, anaphase and telophase. Each phase of a cell cycle is triggered by a set of conditions or events. See [18] and the references therein for further information about the cell cycle. In addition, at the intercellular level, cell differentiation can be viewed as a switched system [16]. In the past few decades, many results related to switched systems have been published [19, 20, 21]. Recently, under the framework of STP of matrices, the stability and stabilisability of switched BNs (SBNs) have been investigated [22, 23, 24, 25]. Specifically, consistent stabilisability was studied in [22]. The concept of switching point reachability was introduced in [23], where point-wise stabilisability of SBNs and global stability of switched Boolean control networks (SBCNs) with arbitrary switching signals were solved. By constructing the switching-state incidence matrix, global stability at some limit cycle of SBNs was solved in [24]. In addition, the stability of SBCNs with impulsive effects was analysed in [25]. Optimal control problems for different situations have been solved in [27] and [28]. See [28, 29, 30, 31, 32] for controllability analyses and disturbance decoupling control design of SBCNs. It is worth pointing out that in some cases, external interventions can be applied to trigger switching of a system between different modes to achieve a certain purpose. For instance, in biology, (in)activation of genes can lead to certain cellular functional states or phenotypes [33], and interventions such as irradiation, chemicals and gene engineering methods can be used to (in)activate genes. This process can interfere with switching within the cell cycle or between cellular fates. For instance, during the G1 phase of cell cycle, if DNA damage occurs, the cell cycle can arrest in the transition from the G1 to the S phase, acDecember 15, 2016
2. Preliminaries
tivate DNA repairing mechanism, or initiating apoptosis when the damage is beyond repair. However, if gene p53 is deactivated, the cell cycle will continue to enter the S phase, enabling the damaged genetic materials to pass onto the next generation [33, 34, 35]. By contrast, differentiated cells are not cycling cells, which have different needs for dealing with DNA lesions. Thus, DNA damage response and DNA repair differ, depending on cellular fates [36]. In sum, different mechanisms are applied to different cellular fates. Therefore, in this work, we assume that the switching signal is designable. The main purpose of this work is to develop an algorithm to design time-optimal switching feedback stabilizers for SBCNs. The main contributions are as follows:
This section lists a few necessary preliminaries related to the STP of matrices, which will be used in this paper. Definition 1. [3] The STP of two matrices A ∈ Mm×n and B ∈ M p×q is (1) A n B = (A ⊗ I ns )(B ⊗ I ps ) where s = lcm(n, p) is the least common multiple of n and p, and ⊗ is the Kronecker product. Remark 1. The STP of matrices is a generalization of the conventional matrix product. Most fundamental properties of the conventional matrix product remain true. If n = p, then the STP degenerates to the conventional matrix product. Therefore, in this paper, the symbol n is omitted unless it is needed to be emphasized.
• A technique is developed to merge the switching signal and the control variable together and transfer a SBCN to an equivalent non-switching logical control network (LCN). The control variable of the resulting LCN is actually the STP of the control variable and the coded switching signal of the original SBCN. This technique eases the control design process significantly. For instance, so long as a state feedback for an LCN is obtained, it then can be decomposed uniquely into the sub-controllers and the switching signal of the original SBCN. In addition, the resulting switching state feedback of the original SBCN is time-optimal if and only if the state feedback of the LCN is time-optimal.
Some related concepts and notations are listed below. 1. D = {0, 1} represents the logical domain. 2. δin denotes the i-th column of the identity matrix In . ∆n := {δ1n , δ2n , · · · , δnn }, and for compactness, ∆ := ∆2 . ∆ is the vector form of the logical domain which can be identified with D under the equivalence 1 ∼ δ12 and 0 ∼ δ22 , where ∼ denotes logical equivalence. 3. δ0n := 0n×1 and 1n := [1, 1, · · · , 1]T . | {z } n
4. Bm×n represents the set of m × n Boolean matrices. A matrix A = (ai j ) is a Boolean matrix if ai j ∈ D. 5. The symbols ∧ and ∨ represent the logical operators AND and OR, respectively. 6. Assume that A1 , A2 , · · · , A s , A, B ∈ Bm×n , then Boolean addition is defined as A+B B := A∨B = (ai j ∨bi j ). Boolean s P summation is defined as (B) Ai := A1 ∨ A2 ∨ · · · ∨ A s .
• Based on the technique above, an algorithm for timeoptimal switching state feedbacks for SBCNs is proposed. The set of all the time-optimal switching state-feedbacks is characterized by a Boolean matrix. Precisely, a logical matrix is a logical sub-matrix of the obtained Boolean matrix if and only if it can be decomposed into a time-optimal switching state-feedback of the SBCN.
i=1
7. Coli (A) represents the i-th column of the matrix A. RowΣ (A) denotes the Boolean summation of all rows of A. 8. Ln×s represents the set of n × s logical matrices. A matrix B ∈ Bm×n is a logical matrix if Coli (B) ∈ ∆m ∀i = 1, 2, · · · , n. 9. δn [i1 · · · i s ] represents the Boolean matrix L ∈ Bn×s such i that Col j (L) = δnj , j = 1, 2, · · · , s. 10. Z≥0 denotes the set of non-negative integers. 11. Let A = (ai j ) ∈ Bm×n , B = (bi j ) ∈ Bn×p , their Boolean product is defined as A nB B = C = (ci j ) ∈ Bm×p , with n P ci j := (B) aik bk j . The Boolean product of X ∈ Bm×n
A preliminary result related to SBCN stabilization has been reported in [26], where a unified feedback gain was adopted for all subnetworks. In this paper, different feedback gains are adopted for different subnetworks. This provides great flexibility in control design, but the technique proposed in [26] does not directly apply any more. A new technique (summarized in Lemma 2) is developed to establish the design algorithm for this situation. The rest of this paper is organized as follows. In Section 2, a few necessary preliminaries related to the STP of matrices, including the definition of STP and some basic results, are introduced. In Section 3, the main objectives of this study are given. In Section 4, a few lemmas crucial for establishing the main results of this study are introduced and proved. Section 5 is the main part of this paper, and it describes stabilisability and time-optimal control design of SBCNs. In Section 6, an example is provided to illustrate the proposed method. Finally, our concluding remarks are presented in Section 7.
k=1
and Y ∈ B p×q is defined as X nB Y = X ⊗ I ns nB Y ⊗ I ps where s = lcm(n, p). The Boolean powers are defined as A(k) := A nB A nB · · · nB A. | {z } k
2
12. For logical matrices A1 ∈ L s×n and A2 ∈ Lm×n , A1 ∗ A2 ∈ L sm×n represents the Khatri-Rao product defined as Coli (A1 ∗ A2 ) := Coli (A1 ) n Coli (A2 ), i = 1, 2, · · · , n. 13. W[m,n] represents the swap matrix with index [m, n] defined as W[m,n] := [In ⊗ δ1m , In ⊗ δ2m , · · · , In ⊗ δm m ]. For any vector X ∈ ∆m and Y ∈ ∆n , Y X = W[m,n] XY holds. 14. Diag{A1 , · · · , Ak } denotes the block diagonal matrix whose (i, i)-block is Ai , i = 1, · · · , k, and all off-diagonal blocks are zero. 15. The power-reducing matrix Mr,N ∈ LN 2 ×N is defined as Mr,N := Diag{δ1N , δ2N , · · · , δNN }. For any x ∈ ∆N , x n x = Mr,N x holds. For simplicity, we denote Mr := Mr,2 . 16. For any K ∈ Bm×n , a logical matrix K ∈ Lm×n is called a logical sub-matrix of K if K ∧K = K. The set of all logical sub-matrices of K is denoted by S (K). Especially, for any Boolean vector x ∈ Bm×1 , we define S (x) := {z ∈ ∆m | z ∧ x = z}. For y ∈ B1×m , we define S T (y) = S (yT ).
process of transferring a BCN to its algebraic form is omitted here, but it can be found in [3]. This would not cause any problem because the method developed in and the results of this study are based directly on the algebraic form (3). Under the control sequence u = {u(t)}t∈Z≥0 and the switching signal σ, the solution to SBCN (3) with initial state x0 is denoted by x(t; x0 , u, σ). Definition 2. Considering SBCN (3), a given state Xe ∈ ∆2n is said to be stabilizable, if for any initial state x0 ∈ ∆2n , there exist a control sequence u and a switching signal σ such that x(t; x0 , u, σ) = Xe , ∀t ≥ T (x0 , u, σ),
where T (x0 , u, σ) is some non-negative integer depending on x0 , u and σ. Denote the smallest integer such that (4) holds by T m (x0 , u, σ) and the set of all stabilizing pairs (u, σ) for the initial state x0 by Uσ (x0 ) . Define
Proposition 1. [7] Any logical function f : ∆n2 → ∆2 can be expressed uniquely as f (x1 , x2 , · · · , xn ) = M f nni=1 xi
(4)
T m (x0 ) :=
∀xi ∈ ∆2
min
(u,σ)∈Uσ (x0 )
T m (x0 , u, σ).
The transient period of SBCN (3) is denoted by T m and defined as T m := max T m (x0 ). (5)
where the structure matrix M f ∈ L2×2n is determined uniquely by f .
x0 ∈∆2n
This paper aims to design switching state feedbacks of the form u(t) = Fσ(t) x(t) (6) σ(t) = φ(x(t))
3. Problem Setting A SBCN with n nodes, m control inputs and N subnetworks is described as X1 (t + 1) = f1σ(t) (X(t), U(t)) σ(t) X2 (t + 1) = f2 (X(t), U(t)) (2) .. . Xn (t + 1) = f σ(t) (X(t), U(t))
for SBCN (3), where Fi is the feedback gain for the ith subnetwork and φ : ∆2n → N. Suppose that Xe ∈ ∆2n is the state to be stabilized. The switching state feedback (6) is said to be stabilizing if for any initial state x0 ∈ ∆2n , it holds that
n
where X(t) = (X1 (t), X2 (t), . . . , Xn (t))T ∈ Dn , U(t) = (U1 (t), U2 (t), . . . , Um (t))T ∈ Dm are the states and the control inputs, respectively. Xi ∈ D, i = 1, 2, . . . , n, are logical variables. σ : Z≥0 → N := {1, 2, . . . , N} is the switching signal. fi j : Dm+n → D, i = 1, 2, . . . , n, j = 1, 2, . . . , N, are logical functions. We denote the vector forms of Xi and Ui by xi ∈ ∆2 and ui ∈ ∆2 , respectively. By the theory of STP [3], a natural oneto-one mapping from ∆n2 to ∆2n is given by
x(t; x0 , uFσ , σφ ) = Xe , ∀t ≥ T (x0 , uFσ , σφ ),
(7)
where T (x0 , uFσ , σφ ) is some non-negative integer depending on x0 , uFσ and σφ . uFσ and σφ are, respectively, the control sequence and the switching signal generated by (6). The switching state feedback (6) is said to be time-optimal if for any initial state x0 , (7) holds with T (x0 , uFσ , σφ ) = T m (x0 ).
(x1 , x2 , · · · , xn ) 7→ nni=1 xi (t) := x1 n x2 n · · · n xn . We denote the vector form of X(t) by x(t) := nni=1 xi (t) ∈ ∆2n , then x(t) and X(t) can be determined uniquely by each other. Similarly, we denote the vector form of U(t) by u(t) := m nm i=1 ui (t) ∈ ∆2 . By using Proposition 1 and the properties of STP of matrices, we can transfer SBCN (2) to the algebraic form x(t + 1) = Lσ(t) u(t)x(t), (3)
4. Some Lemmas Lemma 1. Any logical matrix F ∈ L2Nm ×2n can be decomposed uniquely as F = F1 ∗ F2 ∗ · · · ∗ F N , (8) where F1 , F2 , · · · , F N ∈ L2m ×2n . In addition, it holds that
where x(t) ∈ ∆2n and u(t) ∈ ∆2m . Lσ(t) ∈ L2n ×2n+m is the logical matrix determined uniquely by the original SBCN. The detailed
F x = (F1 x) n (F2 x) n · · · n (F N x) 3
∀x ∈ ∆2n .
1. The k-step controllability matrix Ck ∈ B2n ×2n is defined as ( 1, δi2n ∈ R(k) (δ2j n ) (Ck )i j = (11) 0, δi2n < R(k) (δ2j n ).
Proof. Suppose that F = δ2Nm [i1 i2 · · · i2n ]. i
By using Proposition 3 in [30] repeatedly, every column δ2jNm can be decomposed uniquely as i
s
s
2. The controllability matrix C ∈ B2n ×2n is defined as ( 1, δi2n ∈ R(δ2j n ) (C)i j = 0, δi2n < R(δ2j n ).
s
δ2jNm = δ2mj,1 n δ2mj,2 n · · · n δ2mj,N m
where 1 ≤ s j,t ≤ 2 . We define Ft := δ2m [s1,t s2,t · · · s2n ,t ],
Proposition 2. (Controllability Matrix) For SBCN (3), the following claims hold.
t = 1, 2, · · · , N.
1) The k-step controllability matrix is
Lemma 2. For any v1 , v2 , · · · , vN ∈ ∆2m and ω ∈ ∆N , there holds [v1 v2 · · · vN ]ω = Gωv1 v2 · · · vN , (9)
Ck = Q(k) ,
By definition of the Khatri-Rao product, (8) follows.
Q = (B)
N X (Li nB 12m ). i=1
G = [G1 G2 · · · G N ]
2) The controllability matrix is
(10)
with
C = (B) G j :=
⊗ I2m )W[2 jm ,2(N− j)m ] ,
j = 1, 2, · · · , N.
Ck .
(14)
Proof. Note that the switching signal σ can be re-expressed as σ(t) = µT ω(t),
(15)
µT := [1 2 · · · N]
(16)
ω : Z≥0 → ∆N .
(17)
where
With a mild abuse of terminology, we call ω the vector form of the switching signal. Define L := [L1 L2 . . . LN ], where Li is the structure matrix of the ith subnetwork. Then it holds that
RHS =(1T2(N−1)m ⊗ I2m )W[2 jm ,2(N− j)m ] v1 · · · v j−1 v j v j+1 · · · vN | {z } | {z } 2 jm
2n X k=1
Proof. The left hand side (LHS) of (9) actually defines a logical function from ∆N2Nm to ∆2m . According to Proposition 1, there must exist a unique logical matrix G such that (9) holds. In the following, we prove that the logical matrix G defined in (10) is the one. Set ω = δNj ∈ ∆N , the LHS of (9) equals v j , while the right hand side (RHS) is G j v1 v2 · · · vN . Hence,
2(N− j)m
=(1T2(N−1)m ⊗ I2m ) v j+1 · · · vN v1 · · · v j−1 v j | {z }
Lσ(t) = [L1 L2 . . . LN ]ω(t) = Lω(t).
2(N−1)m
=I2m v j = v j = LHS . Thus, (9) holds.
(13)
where
where G ∈ L2m ×N2Nm is defined as
(1T2(N−1)m
(12)
Thus, SBCN (3) can be expressed equivalently as x(t + 1) = Lω(t)u(t)x(t).
(18)
We define a new control variable uˆ as
5. Main Results
uˆ (t) := ω(t) n u(t).
5.1. Reachability and Stabilisability We briefly discuss the reachability of SBCNs here, which will be used to establish the design procedure in this paper.
Then, SBCN (3) can also be re-written as the LCN x(t + 1) = Lˆu(t)x(t).
Definition 3. Given SBCN (3) and let x0 , xd ∈ ∆2n . xd is said to be k-step reachable from x0 if there exist a control sequence u and a switching law σ such that x(k; x0 , u, σ) = xd . xd is said to be reachable from x0 if there exists a positive integer k such that xd is k-step reachable from x0 .
(19)
According to Proposition 3 in [30], for any uˆ ∈ ∆N2m , there exist two unique logical vectors ω ∈ ∆N and u ∈ ∆2m such that uˆ = ω n u, that is, any control sequence uˆ (t) for LCN (19) can be decomposed uniquely into the control sequence u(t) and the switching signal σ(t) = µT ω(t) for SBCN (3), such that these two LCNs produce the same solution for the same initial condition. Based on this observation, these two LCNs share the same controllability matrices. By following the same argument for the analyses of controllability of BCNs in [6, 3], the result follows.
The k-step reachable set and the reachable set from x are de(k) noted by R(k) (x) and R(x), respectively. We define R−1 (y) := (k) {x ∈ ∆2n | y ∈ R (x)}, and R−1 (y) := {x ∈ ∆2n | y ∈ R(x)}. Definition 4. (Controllability matrix) 4
Proof. By the definition of the new control variable ξ, SBCN (3) can be transformed to an equivalent form (24). According to Lemma 1, every state feedback (26) for LCN (24) can be decomposed uniquely into a switching state feedback for SBCN (3). Under such feedbacks, both networks generate the same solution for the same initial condition.
By using the controllability matrices obtained above, we have the following results. Theorem 1. The following claims hold. 1) SBCN (3) is stabilizable to Xe if and only if both the following statements hold: (a) XeT C1 Xe = 1; (b) XeT C = 1T2n . 2) If SBCN (3) is stabilizable to Xe , then
Remark 2. According to Theorem 2, to find a switching state feedback controller for SBCN (3), we can design a state feedback controller for LCN (24) first, and then decompose it to get the matrices H and Fi . In the following, we shall use this idea to find all time-optimal switching state feedback stabilizers for SBCN (3).
T m (x0 ) = min{k | XeT Ck x0 = 1} k≥0
and T m = min{k | XeT Ck = 1T2n }, k≥0
(20)
We are now ready to state the design process for time-optimal state feedback stabilizers of SBCN (3). According to the analysis above, we only need to design a time-optimal state-feedback for LCN (24) and then decompose it to obtain a time-optimal state feedback for SBCN (3). The time-optimal state-feedback design for LCN (24) is similar to the methods proposed in [8, 14]. A slightly different version of the design process is detailed in the following for integrity. Suppose that Xe is stabilizable. We then define a series of subsets as follows:
where C0 := I2n .
5.2. Switching State-feedback Design In the following, we discuss how to design switching state feedbacks of the form (6) for SBCN (3). By introducing µ and ω as in (16) and (17), respectively, the switching state feedback (6) can be expressed equivalently as u(t) = Fσ(t) x(t) (21) σ(t) = µT ω(t)
M0 M1
= {Xe } (1) = R−1 (Xe ) \ M0
(22)
M2
where Fi , i = 1, 2, · · · , N, and H are the logical matrices to be designed. We denote the feedback for the ith subnetwork by vi = Fi x. Then, we have
MTm
(2) = R−1 (Xe ) \ (M0 ∪ M1 ) .. . (T m ) = R−1 (Xe ) \ (M0 ∪ M1 ∪ · · · ∪ MTm −1 )
with ω(t) = Hx(t),
where T m is the transition period defined in (5) and by the definition, (k) R−1 (Xe ) = S T (XeT Ck ). (28)
u(t) = [v1 v2 · · · vN ]ω(t). N We define v(t) = ni=1 vi ∈ ∆2Nm . By lemma 2, it holds that
u(t) = Gω(t)v(t),
(i) (i+k) Since Xe is stabilizable, if x0 ∈ R−1 (Xe ), then x0 ∈ R−1 (Xe ) ∀k ∈ Z≥0 . Thus
(23)
where G is defined in (10). Substituting (23) into (18) gives
(1) (2) (T m ) M0 ⊆ R−1 (Xe ) ⊆ R−1 (Xe ) ⊆ · · · ⊆ R−1 (Xe ),
x(t + 1) =Lω(t)Gω(t)v(t)x(t) =LW[2m ,N]GW[N,N·2Nm ] Mr,N ω(t)v(t)x(t).
and for 1 ≤ i ≤ T m , Mi
We define a new control variable ξ(t) = ω(t) n v(t) ∈ ∆N·2Nm . Then, SBCN (3) is presented equivalently as the LCN ˜ x(t + 1) = Lξ(t)x(t)
(24)
L˜ = LW[2m ,N]GW[N,N·2Nm ] Mr,N .
(25)
(i) = R−1 (Xe ) \ (M0 ∪ · · · ∪ Mi−1 ) (i) (i−1) = R−1 (Xe ) \ R−1 (Xe ).
(29)
(i) This means that for any x0 ∈ Mi , x0 ∈ R−1 (Xe ) and x0 < ( j) R−1 (Xe ) ∀ j < i hold. Therefore, i is the smallest integer for Xe ∈ R(i) (x0 ). Thus
where
Mi = {x0 | T m (x0 ) = i},
Theorem 2. SBCN (3) is stabilizable to Xe by the switching state feedback (21)-(22), if and only if LCN (24) is stabilizable to Xe by state feedback
∀i = 0, 1, · · · , T m ,
and Mi ∩ M j = ∅,
i , j,
ξ(t) = K x(t)
(26)
M0 ∪ M1 ∪ · · · ∪ MTm = ∆2n .
K = H ∗ F1 ∗ F2 ∗ · · · ∗ F N .
(27)
In other words, the collection of subsets {Mi }0≤i≤Tm defines a partition of the state space ∆2n .
with 5
By the definition of K,
The key to obtain the time-optimal state feedback is to find all the control efforts for any given state that drives the system from Mq to Mq−1 or from M0 to itself. For any i = 1, 2, · · · , 2n , define S i ∈ LN2Nm+n ×N2Nm as S i = IN2Nm ⊗ δi2n .
S [Coli (K)] = S T (RowΣ Pi ) n j o Col j (Pi ) , δ02n , j = 1, · · · , N2Nm . = δN2 Nm ∈ ∆ N2Nm
(30)
Based on the above analysis, for any logical sub-matrix K of K,
By the definition of S i , we have Si
ξ(t) = K x(t)
= Diag{δi2n , · · · , δi2n } | {z }
is a time-optimal state feedback stabilizer for LCN (24). We summarize the design process for time-optimal state feedback stabilizers of SBCN (3) in the following.
N2Nm Nm
N2 i = [δ1N2Nm n δi2n δ2N2Nm n δi2n · · · δN2 Nm n δ2n ].
We divide the structure matrix L˜ into N2Nm blocks of equal size as L˜ = [L˜ 1 L˜ 2 · · · L˜ N2Nm ], where L˜ j ∈ L2n ×2n . Then, for any δi2n ∈ ∆2n ,
Algorithm 1. Step 1. Calculate the controllability matrices Ck and C as in (13) and (14), respectively. Step 2. Check the stabilisability by using Theorem 1. If Xe is stabilizable, then determine T m by using (20) and go to Step 3. Step 3. Construct the partition of the state space M0 , M1 , · · · , MTm by using (28) and (29). Calculate the indicator matrices Mq ∈ B2n ×2n as (31). Step 4. Calculate S i , i = 1, 2, · · · , 2n , as (30), and Pi ∈ B2n ×N2Nm for each δi2n ∈ ∆2n as (32). Step 5. Calculate the Boolean matrix K ∈ BN2Nm ×2n as (33). Step 6. Take any K ∈ S (K), and decompose it as
Nm ˜ N2Nm ˜ 2 Nm δi2n , · · · , Lδ ˜ i = [Lδ ˜ 1 Nm δi2n , Lδ δi2n ] LS N2 N2 N2 = [L˜ 1 δi2n L˜ 2 δi2n · · · L˜ N2Nm δi2n ].
˜ i ) = L˜ j δi n = Coli (L˜ j ), then, under the conObviously, Col j (LS 2 j trol effort ξ = δN2Nm , δi2n reaches Coli (L˜ j ) in one step. Besides, {Coli (L˜ j ) | j = 1, 2, · · · , N2Nm } = R(1) (δi2n ). For all 0 ≤ q ≤ T m , we denote the indicator matrix of Mq by Mq ∈ B2n ×2n and define it as j j δ2n , δ2n ∈ Mq Col j (Mq ) = (31) δ0n , δ j n < Mq . 2 2
K = H ∗ F1 ∗ F2 ∗ · · · ∗ F N by using Lemma 1, where H ∈ LN×2n and F1 , · · · , F N ∈ L2m ×2n . Then, H, F1 , F2 , · · · , F N are the matrices designed for the time-optimal switching state feedback (21)-(22).
Obviously, it holds that MqT = Mq . With this definition, if δi2n ∈ ˜ i) , Mq for some integer q with 1 ≤ q ≤ T m and Mq−1 Col j (LS δ02n , then the system can be driven from δi2n to Mq−1 . Similarly, ˜ i ) , δ0n , then the system can stay if δi2n ∈ M0 and M0 Col j (LS 2 within M0 . Based on this observation, for any δi2n ∈ ∆2n , we define Pi ∈ B2n ×N2Nm as ˜ i , δi n ∈ Mq , 1 ≤ q ≤ T m Mq−1 LS 2 Pi = (32) ˜ i, M0 LS δi2n ∈ M0 .
Remark 3. The Boolean matrix K defined in (33) characterizes all of the time-optimal switching state feedbacks for SBCN (3). Precisely, (21) is a time-optimal switching stabilizer for SBCN (3) if and only if K := H ∗ F1 ∗ F2 ∗ · · · ∗ F N is a logical sub-matrix of K.
When δi2n ∈ Mq , 1 ≤ q ≤ T m ,
6. Example
Col j (Pi ) = Mq−1 L˜ j δi2n .
Consider a SBCN
By the property of Mq , if Col j (Pi ) , δ02n , then Col j (Pi ) ∈ j (Mq−1 ∩ R(1) (δi2n )), that is, under the control effort ξ = δN2 Nm , 0 the trajectory reaches Mq−1 . If Col j (Pi ) = δ2n , then under the j control effort ξ = δN2 Nm , the trajectory can not reach Mq−1 . In a similar way, when δi2n ∈ M0 , Col j (Pi ) = M0 L˜ j δi2n . If Col j (Pi ) , δ02n , then Col j (Pi ) ∈ (M0 ∩ R(1) (δi2n )), that is, under j the control effort ξ = δN2 Nm , the state remains in M0 = {Xe }. The above analysis shows that if x(t) = δi2n and ξ(t) ∈ S T (RowΣ Pi ), then under this control effort, the trajectory of LCS (24) will go one step closer to Xe or stay at Xe . We define K ∈ BN2Nm ×2n as Coli (K) = (RowΣ Pi )T .
x(t + 1) = Lσ(t) u(t)x(t) with
(34)
L1 = δ8 [7 7 5 8 6 6 8 8 7 8 4 7 6 5 2 8 2 7 4 8 5 6 4 7 2 4 5 2 5 5 8 7] L2 = δ8 [7 1 5 4 3 1 8 2 2 1 6 5 4 1 8 7 8 8 2 2 3 2 1 7 2 1 4 4 1 7 2 2],
where x ∈ ∆8 , u ∈ ∆4 and σ : Z≥0 → N = {1, 2}. Both subnetworks of this SBCN are not stabilizable at any state. However, we shall show in the following that the SBCN is stabilizable to some states by switching state feedback. Precisely, by Proposition 2, we can calculate the k-step controllability matrices Ck
(33) 6
The state transfer graph of the closed-loop SBCN under the obtained switching state-feedback controller is shown in Figure 1. From Figure 1 and Table 1, one can see that any solution converges to δ68 in the least steps. Simple calculations show that for Xe = δ68 , there are in total 11010048 time-optimal switching state feedbacks.
Table 1: Transient periods T m (x0 ) (with Xe = δ68 )
x0
T m (x0 )
x0
T m (x0 )
4 3 1 2
δ58 δ68 δ78 δ88
1 0 3 4
δ18 δ28 δ38 δ48
7. Conclusion In this study, we investigated time-optimal stabilization of SBCNs by switching state feedbacks. A technique for merging the coded switching signal and the control variable of a SBCN was proposed. Using this technique, a SBCN can be transferred to an equivalent non-switching LCN with the merged variables as the new control input and the time-optimal control design method developed previously for non-switching BCNs applies. So long as a time-optimal state-feedback for the resulting nonswitching LCN is obtained, it can be decomposed to a timeoptimal switching state feedback for the original SBCN. By using this technique, in this paper, we calculated the controllability matrices of SBCNs and provided a necessary and sufficient condition for stabilisability of SBCNs. In addition, we proposed a time-optimal state-feedback design algorithm for SBCNs. The example presented herein illustrates the main results and the design process.
and the controllability matrix C. It is easy to check that for any x ∈ {δ48 , δ58 , δ68 , δ88 }, it holds that xT C1 x = 1 xT C = 1T . 8 Thus, SBCN (34) is stabilizable at any Xe ∈ {δ48 , δ58 , δ68 , δ88 } by Theorem 1. In the following, we take Xe = δ68 as an example to illustrate the algorithm for designing a time-optimal switching feedback. First of all, by Theorem 1, the transient periods T m (x0 ) for the respective initial states can be calculated, as listed in Table 1. From this table, we have T m = 4. Thus, by using (28) and (29), the state space can be partitioned into five subsets as M0 = {δ68 },
M1 = {δ38 , δ58 },
M3 = {δ28 , δ78 },
M2 = {δ48 },
M4 = {δ18 , δ88 }.
References
The indicator matrices Mq ∈ B8×8 , q = 0, 1, 2, 3, 4, for these subsets can be calculated by (31). For example, M3 = δ8 [0 2 0 0 0 0 7 0]. By following the remaining steps in Algorithm 1, S i ∈ L256×32 and Pi ∈ B8×32 , i = 1, · · · , 8, can be calculated. This process is omitted here to save space and finally, we get K ∈ B32×8 which is given by (35). Choose any logical matrix K such that K = K ∧ K. For instance, we take
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K = δ32 [3 14 22 30 1 4 11 25]. By Lemma 1, K can be decomposed uniquely as K =H ∗ F1 ∗ F2 =δ2 [1 1 2 2 1 1 1 2] ∗ δ4 [1 4 2 4 1 1 3 3] ∗ δ4 [3 2 2 2 1 4 3 1]. Thus, a time optimal state feedback for SBCN (34) is given by u(t) = Fσ(t) x(t) (36) σ(t) = µT Hx(t) with µT = [1 2] and H =δ2 [1 1 2 2 1 1 1 2] F1 =δ4 [1 4 2 4 1 1 3 3] F2 =δ4 [3 2 2 2 1 4 3 1]. 7
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