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closed-loop switching signals
Nonlinear Analysis: Hybrid Systems 22 (2016) 137–146
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Output tracking of switched Boolean networks under open-loop/closed-loop switching signals✩ Haitao Li a,∗ , Yuzhen Wang b a
School of Mathematical Science, Shandong Normal University, Jinan 250014, PR China
b
School of Control Science and Engineering, Shandong University, Jinan 250061, PR China
article
info
Article history: Received 29 August 2015 Accepted 4 April 2016 Keywords: Switched Boolean network Output tracking Switching signal design Semi-tensor product of matrices
abstract This paper addresses the output tracking problem of switched Boolean networks (SBNs) via the semi-tensor product method, and presents a number of new results. Firstly, the concept of switching-output-reachability is proposed for SBNs, based on which, a necessary and sufficient condition is presented for the output tracking of SBNs under arbitrary open-loop switching signal. Secondly, a constructive procedure is proposed for the design of closedloop switching signals for SBNs to track a constant reference signal. The study of an illustrative example shows that the obtained new results are very effective. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Boolean networks have attracted a great attention from biologists, physicists and systems scientists in the last half of a century [1–3]. Recently, a semi-tensor product method [4,5] has been established for the analysis and control of Boolean networks. The main contribution of this method is that one can convert the dynamics of a Boolean network into a linear discrete-time system, and then one can study Boolean networks by using the classical control theory. Using this novel method, many scientists have systematically investigated Boolean networks, and many excellent works have been obtained on the control of Boolean networks, which include controllability and observability [6–12], stability and stabilization [13,14], optimal control [15–17], and other control problems [18–26]. As was shown in [27–29], the dynamics of Boolean networks in practice is often governed by different switching models. For example, a Boolean control network can be regarded as a switched system by encoding the control inputs as a switching signal [8] (see Section 6). Moreover, an asynchronous Boolean network can be converted to a switched one by combining all the Boolean functions [30]. Thus, it is necessary for us to investigate switched Boolean networks (SBNs). In the last five years, some interesting results have been obtained for the controllability and stability of SBNs by using the semi-tensor product method. The controllability of SBNs was studied in [29], and a kind of switching-input-state incidence matrix was proposed for the controllability analysis. The output controllability and optimal output control of state-dependent SBNs were considered in [27], and some necessary and sufficient conditions were presented. The stability of SBNs under arbitrary switching signal was investigated in [28,30], and some necessary and sufficient conditions were established.
✩ The research was supported by the National Natural Science Foundation of China under grants 61374065 and 61503225, the Major International (Regional) Joint Research Project of the National Natural Science Foundation of China under grant 61320106011, the Research Fund for the Taishan Scholar Project of Shandong Province, and the Natural Science Foundation of Shandong Province under grant ZR2015FQ003. ∗ Corresponding author. E-mail addresses: [email protected] (H. Li), [email protected] (Y. Wang).
http://dx.doi.org/10.1016/j.nahs.2016.04.001 1751-570X/© 2016 Elsevier Ltd. All rights reserved.
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H. Li, Y. Wang / Nonlinear Analysis: Hybrid Systems 22 (2016) 137–146
It is noted that a basic issue in the control theory is to make the measured outputs of a plant track a desirable reference signal, which is called the output tracking problem. This problem is also very important for genetic regulatory networks. In many practical genetic regulatory systems, the state variables cannot be obtained directly due to the limitation of measurement conditions and the impact of immeasurable variables. In this case, one can use the measured outputs to track a desirable reference signal which corresponds to some desirable gene states. For example, in order to manipulate the large scale behavior of the lactose regulation system of the Escherichia coli bacteria, Julius et al. [31] proposed a novel feedback control architecture to make the fraction of induced cells in the population (the output of the system) attain a desired level (a given reference trajectory). As a suitable model of genetic regulatory networks, the output tracking problem of Boolean networks was studied in [15,22], respectively, and some necessary and sufficient conditions were presented. However, to our best knowledge, there are no results available on the output tracking of SBNs. In this paper, using the semi-tensor product method, we investigate the output tracking of SBNs under open-loop/closedloop switching signals, and present a number of new results. The main contributions of this paper are as follows. (i) The output tracking of SBNs under arbitrary open-loop switching signal is firstly studied in this work, and a necessary and sufficient condition is presented for this problem based on the switching-output-reachability. One can easily verify the condition with the help of MATLAB toolbox. (ii) Based on the construction of a series of matrices, a constructive procedure is proposed to design closed-loop switching signals for SBNs to track a constant reference signal. The procedure is computationally tractable. The rest of this paper is organized as follows. Section 2 recalls some necessary preliminaries on the semi-tensor product of matrices. Section 3 formulates the output tracking problem studied in this paper. Section 4 investigates the output tracking of SBNs under arbitrary open-loop switching signal, while Section 5 studies the output tracking of SBNs under closed-loop switching signals. An illustrative example is given to support our new results in Section 6, which is followed by a brief conclusion in Section 7. Notation: Z+ denote the sets of real numbers, natural numbers and positive integers, respectively. D := {1, 0}. R, N and 1n :=
1 · · · 1 . ∆n := {δnk : k = 1, . . . , n}, where δnk denotes the kth column of the identity matrix In . For 1 n
i
i
compactness, ∆ := ∆2 . An n × t matrix M is called a logical matrix, if M = [δn1 δn2 · · · δnit ], which is briefly expressed as M = δn [i1 i2 · · · it ]. Denote the set of n × t logical matrices by Ln×t . Coli (A) denotes the ith column of the matrix A, and Rowi (A) stands for the ith row of the matrix A. Blki (A) denotes the ith n × p block of an n × mp matrix A. 2. Preliminaries In this section, we recall some necessary preliminaries on the semi-tensor product of matrices. Definition 2.1 ([4]). The semi-tensor product of two matrices A ∈ Rm×n and B ∈ Rp×q is
A n B = A ⊗ I αn
B ⊗ I αp
,
(2.1)
where α = lcm(n, p) is the least common multiple of n and p, In denotes the n × n identity matrix, and ⊗ is the Kronecker product. When n = p, the semi-tensor product of A and B becomes the conventional matrix product. Thus, it is a generalization of the conventional matrix product. We omit the symbol ‘‘n’’ if no confusion arises in the following. The semi-tensor product of matrices has the following properties. Proposition 2.2 ([4]). (i) Let X ∈ Rt ×1 be a column vector and A ∈ Rm×n . Then X n A = (It ⊗ A) n X .
(2.2)
(ii) Let X ∈ Rm×1 and Y ∈ Rn×1 be two column vectors. Then Y n X = W[m,n] n X n Y ,
(2.3)
where W[m,n] = δmn [1 m + 1 · · · (n − 1)m + 1 2 m + 2 · · · (n − 1)m + 2
··· m m + m · · · (n − 1)m + m] ∈ Lmn×mn is the so-called swap matrix. Identify 1 ∼ δ21 and 0 ∼ δ22 , then ∆ ∼ D , where ‘‘∼’’ denotes two different forms of the same object. In most places of this work, we use δ21 and δ22 to express logical variables and call them the vector form of logical variables. The following lemma is fundamental for the matrix expression of logical functions.
H. Li, Y. Wang / Nonlinear Analysis: Hybrid Systems 22 (2016) 137–146
139
Lemma 2.3 ([4]). Let f (x1 , x2 , . . . , xs ) : D s → D be a logical function. Then, there exists a unique matrix Mf ∈ L2×2s , called the structural matrix of f , such that f (x1 , x2 , . . . , xs ) = Mf nsi=1 xi , where xi ∈ ∆ and
nsi=1
(2.4)
xi = x1 n · · · n xs .
For example, the structural matrices for Negation (¬), Conjunction (∧) and Disjunction (∨) are Mn = δ2 [2 1], Mc =
δ2 [1 2 2 2] and Md = δ2 [1 1 1 2], respectively. 3. Problem formulation
Consider the following switched Boolean network:
σ (t ) x1 (t + 1) = f1 (x1 (t ), x2 (t ), . . . , xn (t )), σ (t ) x2 (t + 1) = f2 (x1 (t ), x2 (t ), . . . , xn (t )), .. . σ (t ) xn (t + 1) = fn (x1 (t ), x2 (t ), . . . , xn (t )); yk (t ) = hk (x1 (t ), x2 (t ), . . . , xn (t )), k = 1, . . . , p,
(3.1)
where σ : N → W = {1, 2, · · · , w} is the switching signal, X (t ) = (x1 (t ), x2 (t ), . . . , xn (t )) ∈ D n and Y (t ) = (y1 (t ), . . . , yp (t )) ∈ D p are the state and the output of the system (3.1), respectively, and fij : D n → D , i = 1, . . . , n, j = 1, . . . , w and hk : D n → D , k = 1, . . . , p are logical functions. Given a switching signal σ : N → W , denote the state and output trajectories of the system (3.1) starting from an initial state X (0) ∈ D n by X (t ; X (0), σ ) and Y (t ; X (0), σ ), respectively. Definition 3.1. Given a reference signal Yr = (yr1 , . . . , yrp ) ∈ D p . The output of the system (3.1) is said to track Yr under arbitrary open-loop switching signal, if there exists an integer τ > 0 such that Y (t ; X (0), σ ) = Yr
(3.2)
holds for ∀ X (0) ∈ D , ∀ σ : N → W and ∀ t ≥ τ . n
Definition 3.2. Given a reference signal Yr = (yr1 , . . . , yrp ) ∈ D p . The output of the system (3.1) is said to track Yr under a closed-loop switching signal, if there exists a switching signal in the form of
σ (t ) = g (X (t )),
t ∈N
(3.3)
under which there exists an integer τ > 0 such that Y (t ; X (0), σ ) = Yr
(3.4)
holds for ∀ X (0) ∈ D and ∀ t ≥ τ , where g : D → W is a w -valued logical function [5]. n
n
In the following, we convert the system (3.1) and the closed-loop switching signal (3.3) into equivalent algebraic forms, respectively. p Using the vector form of logical variables and setting x(t ) = nni=1 xi (t ) ∈ ∆2n and y(t ) = ni=1 yi (t ) ∈ ∆2p , by Lemma 2.3, one can convert (3.1) into the following algebraic form:
x(t + 1) = Lσ (t ) x(t ), y(t ) = Hx(t ),
(3.5)
where Li ∈ L2n ×2n , i ∈ W and H ∈ L2p ×2n . Moreover, the reference signal becomes yr = ni=1 yri = δ2αp , where α is uniquely determined by yri , i = 1, . . . , p. i Similarly, identifying σ (t ) = i ∼ δw , i ∈ W and setting x(t ) = nni=1 xi (t ) ∈ ∆2n , (3.3) can be converted to the following form: p
σ (t ) = Gx(t ),
(3.6)
where G ∈ Lw×2n is the structural matrix of the w -valued logical function g. For details, please refer to [5]. In the following, we study the output tracking of the system (3.1) based on the algebraic forms (3.5) and (3.6). 4. Output tracking under arbitrary open-loop switching signal In this section, we investigate the output tracking of SBNs under arbitrary open-loop switching signal. To this end, we first propose the concept of switching-output-reachability of SBNs.
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Definition 4.1. Consider the system (3.1). Let X (0) ∈ D n . Then, Y = (y1 , . . . , yp ) ∈ D p is said to be s-switching-outputreachable from X (0), if one can find a switching signal σ : N → W such that Y (s; X (0), σ ) = Y . In order to give a necessary and sufficient condition for the switching-output-reachability of SBNs, we recall a useful result on the switching reachability of SBNs. For details, please refer to [30]. q
Lemma 4.2. Consider the system (3.5). Let xf = δ2n and x(0) = δ2r n be given. Then, xf is switching reachable from x(0) at time s ∈ Z+ , if and only if
(M s )q,r > 0,
(4.1)
where w
M =
Li ,
(4.2)
i=1
and (M s )q,r denotes the (q, r )-th element of M s . Based on Definition 4.1 and Lemma 4.2, we have the following result. Theorem 4.3. Consider the system (3.5). Let yf = δ2kp and x(0) = δ2r n be given. Then, yf is s-switching-output-reachable from x(0), if and only if
HM s
k,r
> 0.
(4.3)
Proof (Sufficiency). Suppose that (4.3) holds. Since
HM
s
k,r
=
2n
(H )k,q (M s )q,r > 0,
(4.4)
q =1
there exists an integer 1 ≤ q ≤ 2n such that (H )k,q = 1 and (M s )q,r > 0. q By Lemma 4.2, δ2n is switching reachable from δ2r n at time s, that is, one can find a switching signal σ : {0, . . . , s − 1} → W such that q
x(s; δ2r n , σ ) = δ2n . Thus, q
y(s; δ2r n , σ ) = H δ2n = Colq (H ) = δ2kp , which implies that yf is s-switching-output-reachable from x(0). (Necessity) Assume that δ2kp is s-switching-output-reachable from δ2r n . Then, there exists a switching signal σ : {0, . . . , s− 1} → W such that y(s; δ2r n , σ ) = δ2kp . Since y(s; δ2r n , σ ) = Hx(s; δ2r n , σ ), q q letting x(s; δ2r n , σ ) = δ2n , it is easy to obtain that Colq (H ) = δ2kp , and δ2n is switching reachable from δ2r n at time s, that is, (H )k,q = 1 and (M s )q,r > 0.
Therefore,
HM s
k ,r
=
2n (H )k,i (M s )i,r ≥ (H )k,q (M s )q,r > 0, i=1
which implies that (4.3) holds.
Remark 4.4. When δ2kp is s-switching-output-reachable from δ2r n , one can design a switching sequence to realize the switching-output-reachability by the following steps: (1) Find an integer 1 ≤ q ≤ 2n such that (H )k,q = 1 and (M s )q,r > 0. (2) Find an integer 1 ≤ α ≤ w such that
Blkα (M s−1 L)
where L = [L1
L2
q ,r
> 0,
···
Lw ]. Set σ (0) = α . If s = 1, stop. Otherwise, go to the next step.
H. Li, Y. Wang / Nonlinear Analysis: Hybrid Systems 22 (2016) 137–146
(3) Find two integers 1 ≤ j ≤ 2n and 1 ≤ β ≤ w such that Blkβ (L)
q ,j
Blkα (M s−2 L)
141
> 0 and
> 0.
j ,r
j
Set σ (s − 1) = β and x(s − 1) = δ2n . If s − 1 = 1, stop. Otherwise, replace s and q by s − 1 and j, respectively, and go to (2). In the following, based on the switching-output-reachability, we study the output tracking of the system (3.1) under arbitrary open-loop switching signal. We have the following result. Theorem 4.5. The output of the system (3.1) tracks the reference signal yr = δ2αp under arbitrary open-loop switching signal, if and only if there exists a positive integer s ≤ 2n such that
Rowα HM s
= ws 12n .
(4.5)
Proof (Sufficiency). Suppose that (4.5) holds. Firstly, we prove that 2p
HM s
i =1
= ws
i,j
(4.6)
holds for ∀ j = 1, 2, . . . , 2n and ∀ s ∈ Z+ . We prove (4.6) by induction. When s = 1, since Lq ∈ L2n ×2n , q = 1, . . . , w , we have
2n
k=1
(Lq )k,j = 1. Thus,
w w (M )k,j = (Lq )k,j = (Lq )k,j = w. 2n
2n
k =1
2n
k =1 q =1
q=1 k=1
Hence, 2p 2n 2p (H )i,k (M )k,j (HM )i,j = i=1 k=1
i =1
=w
2p (H )i,k = w, i =1
which implies that (4.6) holds for s = 1. Assume that (4.6) holds for s = l. Now, we consider the case of s = l + 1. In this case, it is easy to see that 2p
HM l+1
i =1
i,j
=
2p 2n
HM l
i=1 k=1
=
2p
i ,k
(M )k,j
2n HM (M )k,j l
i,k
i=1
k=1
2n
= wl
(M )k,j = w l+1 . k=1
Therefore, (4.6) holds for s = l + 1. By induction, (4.6) holds for any s ∈ Z+ . Secondly, we prove that if (4.5) holds, then
Rowα HM t
= wt 12n ,
∀ t ≥ s.
(4.7)
We also prove (4.7) by induction. It is obvious that (4.7) holds for t = s. Suppose that (4.7) holds for t = l ≥ s. Then, for the case of t = l + 1, we have
Rowα HM l+1
= Rowα HM l M = wl 12n M = w l+1 ,
which shows that (4.7) holds for t = l + 1. By induction, (4.7) holds for any t ≥ s. From (4.6) and (4.7), for any t ≥ s, one can obtain that
HM t
i ,j
= 0,
∀ i ̸= α, j = 1, . . . , 2n ,
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H. Li, Y. Wang / Nonlinear Analysis: Hybrid Systems 22 (2016) 137–146
and
HM t
α,j
= wt ,
∀ j = 1, . . . , 2n .
By Theorem 4.3, we know that under any switching signal σ (t ), the output trajectory of the system (3.5) starting from any j initial point x(0) = δ2n will stay at δ2αp after time s. Therefore, the output of the system (3.5) tracks the reference signal yr = δ2αp under arbitrary open-loop switching signal. (Necessity): Suppose that the output of the system (3.5) tracks the reference signal yr = δ2αp under arbitrary open-loop switching signal. We firstly show that under any switching signal σ (t ), the output trajectory of the system (3.5) starting j from any initial point x(0) = δ2n reaches yr = δ2αp in the time T (x(0), σ (t )) ≤ 2n , and then stays at yr forever. If not, then, one can find an initial point x(0), an integer T > 2n and a switching sequence π = {(0, i0 ), (1, i1 ), . . . , (T − 1, iT −1 )} such that under π , the shortest time that the output trajectory of the system (3.5) starting from x(0) reaches yr is T . Denote the state trajectory by {x(0), x(1), . . . , x(T − 1), x(T )}. Then, one can easily see that Hx(k) ̸= yr , ∀ 0 ≤ k ≤ T − 1. Since the number of different states for the system (3.5) is 2n , there must exist two integers 0 ≤ k1 < k2 ≤ T − 1 such that x(k1 ) = x(k2 ). Construct the following switching signal:
ik1 , t = γ (k2 − k1 ), ik1 +1 , t = γ (k2 − k1 ) + 1, σ (t ) = .. . ik2 −1 , t = (γ + 1)(k2 − k1 ) − 1,
(4.8)
where γ ∈ N. Then, under the switching signal σ (t ), the output trajectory of the system (3.5) starting from the initial point x(0) = x(k1 ) forms a cycle {Hx(k1 ), Hx(k1 + 1), . . . , Hx(k2 − 1)} with Hx(k) ̸= yr , ∀ k = k1 , k1 + 1, . . . , k2 − 1, which is a contradiction. Therefore, T (x(0), σ (t )) ≤ 2n holds for any σ (t ) and x(0). Now, we set s = maxx(0),σ (t ) {T (x(0), σ (t ))} ≤ 2n . By Theorem 4.3, one can obtain that HM s = 0, ∀ r ̸= α, j = 1, . . . , 2n r ,j
and
HM s
α,j
= ws ,
∀ j = 1, . . . , 2n ,
which implies that (4.5) holds.
5. Output tracking under closed-loop switching signals In this section, we study the output tracking of SBNs under closed-loop switching signals, and present a constructive procedure for the design of closed-loop switching signals. For the system (3.5) and the reference signal yr = δ2αp , define the following set:
O (α) = {r ∈ N : Colr (H ) = δ2αp , 1 ≤ r ≤ 2n }.
(5.1)
Note that the set {δ2r n : r ∈ O (α)} contains all the states of the system (3.5) whose outputs form the vector yr = δ2αp . We presuppose O (α) ̸= ∅ in the following. Otherwise, if O (α) = ∅, the output tracking problem of SBNs under closed-loop switching signals is not solvable. Given positive integers τ and j1 < · · · < jv ≤ 2n , define
Mj1 ,j1
.. .
M{j1 ,...,jv } :=
Mjv ,j1
··· .. .
Mj1 ,jv
···
Mjv ,jv
.. .
(5.2)
and
Rowj1 (M τ )
.. , . τ Rowjv (M )
M{τj1 ,...,jv } :=
(5.3)
where M is given in (4.2). We have the following result. Theorem 5.1. The output of the system (3.1) tracks the reference signal yr = δ2αp under a closed-loop switching signal, if and only if there exist positive integers τ ≤ 2n and ji ∈ O (α), i = 1, . . . , v with j1 < · · · < jv such that all the columns of both M{j1 ,...,jv } and M{τj1 ,...,jv } are nonzero.
H. Li, Y. Wang / Nonlinear Analysis: Hybrid Systems 22 (2016) 137–146 j
143
j
In order to prove Theorem 5.1, for the set Ω = {δ21n , . . . , δ2vn } and k ∈ Z+ , we define a sequence of sets as follows: j
j
Ωk = {x(0) ∈ ∆2n : there exist σ : {0, . . . , k − 1} → W and δ2in ∈ Ω such that x(k; x(0), σ ) = δ2in }.
(5.4)
Then, by Lemma 4.2 and a simple calculation, we have the following result. Lemma 5.2. (1) Ω ⊆ Ω1 if and only if all the columns of M{j1 ,...,jv } are nonzero; (2) Ωτ = ∆2n if and only if all the columns of M{τj1 ,...,jv } are nonzero. In the following, based on Lemma 5.2, we give the proof of Theorem 5.1. Proof of Theorem 5.1 (Sufficiency). Assuming that there exist positive integers τ ≤ 2n and ji ∈ O (α), i = 1, . . . , v with j1 < · · · < jv such that all the columns of both M{j1 ,...,jv } and M{τj1 ,...,jv } are nonzero, we design a closed-loop switching signal under which the output of the system (3.1) tracks yr . From Lemma 5.2, one can see that Ω ⊆ Ω1 and Ωτ = ∆2n . Set
Ωk◦ = Ωk \ Ωk−1 ,
k = 1, . . . , τ ,
where Ω0 := ∅. Obviously, Ωk1
(5.5)
τ
Ωk2 = ∅ holds for any k1 ̸= k2 , and k=1 Ωk = ∆2n . Therefore, for any integer 1 ≤ l ≤ 2n , there exists a unique integer 1 ≤ kl ≤ τ such that δ2l n ∈ Ωk◦l . For the system (3.5), let L = [L1 · · · Lw ] = δ2n [i1 i2 · · · iw2n ]. Then, for any x(t ) = δ2l n and σ (t ) = q ∈ W , it is ◦
◦
◦
easy to see that
i(q−1)2n +l
x(t + 1) = Lq δ2l n = δ2n
.
(5.6) i(p −1)2n +l
From (5.4) and (5.6), when kl = 1, there exists pl ∈ W such that δ2n l i(p −1)2n +l l 2n
∈ Ω ; when 2 ≤ kl ≤ τ , there exists
pl ∈ W such that δ ∈ Ωkl −1 . Set G = δw [p1 p2 · · · p ] ∈ L . Then, under the closed-loop switching signal σ (t ) = Gx(t ), for any initial state x(0) = δ2l n ∈ ∆2n , we have x(kl ; x(0), σ ) ∈ Ω , ∀ 1 ≤ l ≤ 2n . One can see from Ω ⊆ Ω1 that x(t ; x(0), σ ) ∈ Ω ,
2n
w×2n
∀ t ≥ τ , ∀ x(0) ∈ ∆2n ,
which implies that y(t ; x(0), σ ) = Hx(t ; x(0), σ ) = yr ,
∀ t ≥ τ , ∀ x(0) ∈ ∆2n .
Therefore, the output of the system (3.1) tracks yr under the designed closed-loop switching signal σ (t ) = δw [p1 p2 · · · p2n ]x(t ). (Necessity) Suppose that the output of the system (3.1) tracks yr = δ2αp by a closed-loop switching signal, say, σ (t ) = Gx(t ), G ∈ Lw×2n . Then, the closed-loop system consisting of the system (3.5) and the closed-loop switching signal becomes the following Boolean network:
x(t + 1) = [L1 · · · Lw ]GMr ,2n x(t ),
(5.7)
y(t ) = Hx(t ), n
where Mr ,2n = Diag{δ21n , δ22n , . . . , δ22n } ∈ L22n ×2n . Denote the set of states in all the attractors (fixed points and cycles) of j1 2n
jv 2n
(5.7) by Ω = {δ , . . . , δ }, j1 < · · · < jv , and the transient period of (5.7) by τ ≤ 2n . Then, a straightforward calculation shows that
Ω ⊆ Ω1 , Ωτ = ∆ 2 n , ji ∈ O (α), i = 1, . . . , v. By Lemma 5.2, all the columns of both M{j1 ,...,jv } and M{τj1 ,...,jv } are nonzero. This completes the proof.
(5.8)
Remark 5.3. The proof of Theorem 5.1 provides a constructive procedure to design closed-loop switching signals for the output tracking of SBNs, which contains the following steps:
• Step 1: Find positive integers τ ≤ 2n and ji ∈ O (α), i = 1, . . . , v with j1 < · · · < jv such that all the columns of both M{j1 ,...,jv } and M{τj1 ,...,jv } are nonzero. j
• Step 2: For Ω = {δ21n , . . . , δ2jvn }, calculate Ωk and Ωk◦ according to (5.4) and (5.5), respectively.
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H. Li, Y. Wang / Nonlinear Analysis: Hybrid Systems 22 (2016) 137–146
• Step 3: For any integer 1 ≤ l ≤ 2n which corresponds to a unique integer 1 ≤ kl ≤ τ such that δ2l n ∈ Ωk◦l , find an integer pl ∈ W such that i(p −1)2n +l Ω , if kl = 1 l δ2n ∈ (5.9) Ωkl −1 , if 2 ≤ kl ≤ τ . • Step 4: The closed-loop switching signal can be designed as σ (t ) = δw [p1
p2
···
p2n ]x(t ).
6. An illustrative example In this section, we apply the obtained results to the analysis of apoptosis networks. The network graph of the considered apoptosis network is shown in Fig. 1, where IAP (denoted by x1 ) stands for the concentration level (high or low) of the inhibitor of apoptosis proteins, C3a (denoted by x2 ) the concentration level of the active caspase 3, C8a (denoted by x3 ) the concentration level of the active caspase 8, and TNF (a stimulus, denoted by u) the concentration level of the tumor necrosis factor. The dynamics of the considered apoptosis network is as follows [2]: x1 (t + 1) = ¬x2 (t ) ∧ u(t ), x2 (t + 1) = ¬x1 (t ) ∧ x3 (t ), x3 (t + 1) = x2 (t ) ∨ u(t ).
(6.1)
By letting u = 1 and u = 0, respectively, the dynamics of the apoptosis network (6.1) becomes the following switched Boolean network [30]:
σ (t ) x1 (t + 1) = f1 (x1 (t ), x2 (t ), x3 (t )), σ (t ) x2 (t + 1) = f2 (x1 (t ), x2 (t ), x3 (t )), σ (t ) x3 (t + 1) = f3 (x1 (t ), x2 (t ), x3 (t )),
(6.2)
where σ : N → W = {1, 2} is the switching signal, f11 = ¬x2 , f21 = ¬x1 ∧ x3 and f31 = 1 correspond to u = 1, and f12 = 0, f22 = ¬x1 ∧ x3 and f32 = x2 correspond to u = 0. In the apoptosis network, {x1 = 1, x2 = 0} stands for the cell survival [2]. Thus, we are interested in observing x1 and x2 at each time instance, and have the following output equation:
y1 (t ) = x1 (t ), y2 (t ) = x2 (t ).
(6.3)
In the following, we study the following two problems. Problem 1. Can the output of the apoptosis network (6.2) track Yr = (1, 0) (the cell survival) under arbitrary stimulus (or switching signal)? Problem 2. Can the output of the apoptosis network (6.2) track Yr under a closed-loop stimulus (or switching signal)? It should be pointed out that the practical meaning of Problem 1 is to seek the possibility of steering the apoptosis network to the cell survival state under any amount of stimulus, while the practical meaning of Problem 2 is to design some proper amount of stimulus (depending on ‘‘IAP’’, ‘‘C3a’’ and ‘‘C8a’’) to drive the apoptosis network to the cell survival state. As was shown in [2], the apoptosis networks always present bistable characterization, which shows that both problems have no solutions. In the following, we will prove the conclusion by using Theorems 4.5 and 5.1. Using the vector form of logical variables and setting x(t ) = n3i=1 xi (t ), we have the following algebraic form for the system (6.2) with the output (6.3):
x(t + 1) = Lσ (t ) x(t ), y(t ) = Hx(t ),
(6.4)
where L1 = δ8 [7 7 3 3 5 7 1 3], L2 = δ8 [7 7 8 8 5 7 6 8], and H = δ4 [1 1 2 2 3 3 4 4]. Moreover, Yr ∼ yr = δ42 . A simple calculation gives 0 0 0 0 M = L1 + L2 = 0 0 2 0
0 0 0 0 0 0 2 0
0 0 1 0 0 0 0 1
0 0 1 0 0 0 0 1
0 0 0 0 2 0 0 0
0 0 0 0 0 0 2 0
1 0 0 0 0 1 0 0
0 0 1 0 . 0 0 0 1
H. Li, Y. Wang / Nonlinear Analysis: Hybrid Systems 22 (2016) 137–146
145
Fig. 1. The network graph of the apoptosis network.
For Problem 1, it is easy to obtain that Row2 (HM s ) ̸= 2s 18 holds for any integer 1 ≤ s ≤ 8. By Theorem 4.5, the output of the apoptosis network (6.2) cannot track Yr = (1, 0) under arbitrary stimulus. Now, we consider Problem 2. One can obtain that O (2) = {3, 4}. For j1 = 3 and j2 = 4, a straightforward computation shows that
M{j1 ,j2 } =
1 0
1 0
and
Col2 HM{τj1 ,j2 }
=
0 0
holds for any integer 1 ≤ s ≤ 8. Thus, the conditions of Theorem 5.1 cannot hold for j1 = 3 and j2 = 4. Similarly, one can prove that the conditions of Theorem 5.1 cannot hold for the choice of j = 3 or j = 4. Therefore, one cannot design a closed-loop stimulus such that the output of the apoptosis network (6.2) tracks Yr , which is consistent to the bistability of apoptosis networks [2]. 7. Conclusion In this paper, using the semi-tensor product method, we have studied the output tracking of SBNs under openloop/closed-loop switching signals. We have presented a necessary and sufficient condition for the output tracking of SBNs under arbitrary open-loop switching signal based on the switching-output-reachability of SBNs. We have proposed a constructive procedure to design closed-loop switching signals for SBNs to track a constant reference signal. Finally, we have applied the obtained new results to apoptosis networks, and shown their effectiveness. It should be pointed out that the method proposed in this paper can only deal with the output tracking problem of SBNs with a constant reference signal. If the tracking goal is a time-varying trajectory, one needs to develop new methods in future works. References [1] T. Akutsu, M. Hayashida, W. Ching, M. Ng, Control of Boolean networks: Hardness results and algorithms for tree structured networks, J. Theoret. Biol. 244 (4) (2007) 670–679. [2] M. Chaves, Methods for qualitative analysis of genetic networks, in: Proc. 10th European Control Conference, 2009, pp. 671–676. [3] S. Kauffman, Metabolic stability and epigenesis in randomly constructed genetic nets, J. Theoret. Biol. 22 (3) (1969) 437. [4] D. Cheng, H. Qi, Z. Li, Analysis and Control of Boolean Networks: A Semi-tensor Product Approach, Springer, London, 2011. [5] D. Cheng, H. Qi, Y. Zhao, An Introduction to Semi-tensor Product of Matrices and Its Applications, World Scientific, Singapore, 2012. [6] H. Chen, X. Li, J. Sun, Stabilization, controllability and optimal control of Boolean networks with impulsive effects and state constraints, IEEE Trans. Automat. Control 60 (3) (2015) 806–811. [7] D. Cheng, H. Qi, Controllability and observability of Boolean control networks, Automatica 45 (7) (2009) 1659–1667. [8] E. Fornasini, M.E. Valcher, Observability, reconstructibility and state observers of Boolean control networks, IEEE Trans. Automat. Control 58 (6) (2013) 1390–1401. [9] D. Laschov, M. Margaliot, Controllability of Boolean control networks via the Perron–Frobenius theory, Automatica 48 (6) (2012) 1218–1223. [10] F. Li, J. Sun, Controllability of Boolean control networks with time delays in states, Automatica 47 (3) (2011) 603–607. [11] Y. Zhao, D. Cheng, H. Qi, Input-state incidence matrix of Boolean control networks and its applications, Systems Control Lett. 59 (12) (2010) 767–774. [12] L. Zhang, K. Zhang, Controllability and observability of Boolean control networks with time-variant delays in states, IEEE Trans. Neural Netw. Learn. Syst. 24 (2013) 1478–1484. [13] D. Cheng, H. Qi, Z. Li, J.B. Liu, Stability and stabilization of Boolean networks, Internat. J. Robust Nonlinear Control 21 (2) (2011) 134–156. [14] R. Li, M. Yang, T. Chu, State feedback stabilization for Boolean control networks, IEEE Trans. Automat. Control 58 (7) (2013) 1853–1857. [15] E. Fornasini, M.E. Valcher, Feedback stabilization, regulation and optimal control of Boolean control networks, in: 2014 American Control Conference, 2014, pp. 1981–1986. [16] D. Laschov, M. Margaliot, A maximum principle for single-input Boolean control networks, IEEE Trans. Automat. Control 56 (4) (2011) 913–917. [17] Y. Zhao, Z. Li, D. Cheng, Optimal control of logical control networks, IEEE Trans. Automat. Control 56 (8) (2011) 1766–1776. [18] D. Cheng, H. Qi, A linear representation of dynamics of Boolean networks, IEEE Trans. Automat. Control 55 (10) (2010) 2251–2258.
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[19] J. Feng, J. Yao, P. Cui, Singular Boolean networks: Semi-tensor product approach, Sci. China Inf. Sci. (2012) http://dx.doi.org/10.1007/s11432-0124666-8. [20] C. Li, J. Sun, R. Sun, Stability analysis of a class of stochastic differential delay equations with nonlinear impulsive effects, J. Franklin Inst. 347 (2010) 1186–1198. [21] F. Li, J. Sun, Stability and stabilization of multivalued logical networks, Nonlinear Anal. RWA 12 (6) (2011) 3701–3712. [22] H. Li, Y. Wang, L. Xie, Output tracking control of Boolean control networks via state feedback: Constant reference signal case, Automatica 59 (2015) 54–59. [23] X. Xu, Y. Hong, Matrix approach to model matching of asynchronous sequential machines, IEEE Trans. Automat. Control 58 (11) (2013) 2974–2979. [24] M. Yang, R. Li, T. Chu, Controller design for disturbance decoupling of Boolean control networks, Automatica 49 (1) (2013) 273–277. [25] J. Suo, J. Sun, Asymptotic stability of differential systems with impulsive effects suffered by logic choice, Automatica 51 (1) (2015) 302–307. [26] J. Zhong, J. Lu, Y. Liu, J. Cao, Synchronization in an array of output-coupled Boolean networks with time delay, IEEE Trans. Neural Netw. Learn. Syst. 25 (2014) 2288–2294. [27] H. Chen, J. Sun, Output controllability and optimal output control of state-dependent switched Boolean control networks, Automatica 50 (7) (2014) 1929–1934. [28] F. Li, Global stability at a limit cycle of switched Boolean networks under arbitrary switching signals, Neurocomputing 133 (2014) 63–66. [29] H. Li, Y. Wang, On reachability and controllability of switched Boolean control networks, Automatica 48 (11) (2012) 2917–2922. [30] H. Li, Y. Wang, Z. Liu, Stability analysis for switched Boolean networks under arbitrary switching signals, IEEE Trans. Automat. Control 59 (7) (2014) 1978–1982. [31] A.A. Julius, A. Halasz, M.S. Sakar, H. Rubin, V. Kumar, G.J. Pappas, Stochastic modeling and control of biological systems: The lactose regulation system of Escherichia coli, IEEE Trans. Automat. Control 53 (2008) 51–65.
Contents lists available at ScienceDirect
Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs
Output tracking of switched Boolean networks under open-loop/closed-loop switching signals✩ Haitao Li a,∗ , Yuzhen Wang b a
School of Mathematical Science, Shandong Normal University, Jinan 250014, PR China
b
School of Control Science and Engineering, Shandong University, Jinan 250061, PR China
article
info
Article history: Received 29 August 2015 Accepted 4 April 2016 Keywords: Switched Boolean network Output tracking Switching signal design Semi-tensor product of matrices
abstract This paper addresses the output tracking problem of switched Boolean networks (SBNs) via the semi-tensor product method, and presents a number of new results. Firstly, the concept of switching-output-reachability is proposed for SBNs, based on which, a necessary and sufficient condition is presented for the output tracking of SBNs under arbitrary open-loop switching signal. Secondly, a constructive procedure is proposed for the design of closedloop switching signals for SBNs to track a constant reference signal. The study of an illustrative example shows that the obtained new results are very effective. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Boolean networks have attracted a great attention from biologists, physicists and systems scientists in the last half of a century [1–3]. Recently, a semi-tensor product method [4,5] has been established for the analysis and control of Boolean networks. The main contribution of this method is that one can convert the dynamics of a Boolean network into a linear discrete-time system, and then one can study Boolean networks by using the classical control theory. Using this novel method, many scientists have systematically investigated Boolean networks, and many excellent works have been obtained on the control of Boolean networks, which include controllability and observability [6–12], stability and stabilization [13,14], optimal control [15–17], and other control problems [18–26]. As was shown in [27–29], the dynamics of Boolean networks in practice is often governed by different switching models. For example, a Boolean control network can be regarded as a switched system by encoding the control inputs as a switching signal [8] (see Section 6). Moreover, an asynchronous Boolean network can be converted to a switched one by combining all the Boolean functions [30]. Thus, it is necessary for us to investigate switched Boolean networks (SBNs). In the last five years, some interesting results have been obtained for the controllability and stability of SBNs by using the semi-tensor product method. The controllability of SBNs was studied in [29], and a kind of switching-input-state incidence matrix was proposed for the controllability analysis. The output controllability and optimal output control of state-dependent SBNs were considered in [27], and some necessary and sufficient conditions were presented. The stability of SBNs under arbitrary switching signal was investigated in [28,30], and some necessary and sufficient conditions were established.
✩ The research was supported by the National Natural Science Foundation of China under grants 61374065 and 61503225, the Major International (Regional) Joint Research Project of the National Natural Science Foundation of China under grant 61320106011, the Research Fund for the Taishan Scholar Project of Shandong Province, and the Natural Science Foundation of Shandong Province under grant ZR2015FQ003. ∗ Corresponding author. E-mail addresses: [email protected] (H. Li), [email protected] (Y. Wang).
http://dx.doi.org/10.1016/j.nahs.2016.04.001 1751-570X/© 2016 Elsevier Ltd. All rights reserved.
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H. Li, Y. Wang / Nonlinear Analysis: Hybrid Systems 22 (2016) 137–146
It is noted that a basic issue in the control theory is to make the measured outputs of a plant track a desirable reference signal, which is called the output tracking problem. This problem is also very important for genetic regulatory networks. In many practical genetic regulatory systems, the state variables cannot be obtained directly due to the limitation of measurement conditions and the impact of immeasurable variables. In this case, one can use the measured outputs to track a desirable reference signal which corresponds to some desirable gene states. For example, in order to manipulate the large scale behavior of the lactose regulation system of the Escherichia coli bacteria, Julius et al. [31] proposed a novel feedback control architecture to make the fraction of induced cells in the population (the output of the system) attain a desired level (a given reference trajectory). As a suitable model of genetic regulatory networks, the output tracking problem of Boolean networks was studied in [15,22], respectively, and some necessary and sufficient conditions were presented. However, to our best knowledge, there are no results available on the output tracking of SBNs. In this paper, using the semi-tensor product method, we investigate the output tracking of SBNs under open-loop/closedloop switching signals, and present a number of new results. The main contributions of this paper are as follows. (i) The output tracking of SBNs under arbitrary open-loop switching signal is firstly studied in this work, and a necessary and sufficient condition is presented for this problem based on the switching-output-reachability. One can easily verify the condition with the help of MATLAB toolbox. (ii) Based on the construction of a series of matrices, a constructive procedure is proposed to design closed-loop switching signals for SBNs to track a constant reference signal. The procedure is computationally tractable. The rest of this paper is organized as follows. Section 2 recalls some necessary preliminaries on the semi-tensor product of matrices. Section 3 formulates the output tracking problem studied in this paper. Section 4 investigates the output tracking of SBNs under arbitrary open-loop switching signal, while Section 5 studies the output tracking of SBNs under closed-loop switching signals. An illustrative example is given to support our new results in Section 6, which is followed by a brief conclusion in Section 7. Notation: Z+ denote the sets of real numbers, natural numbers and positive integers, respectively. D := {1, 0}. R, N and 1n :=
1 · · · 1 . ∆n := {δnk : k = 1, . . . , n}, where δnk denotes the kth column of the identity matrix In . For 1 n
i
i
compactness, ∆ := ∆2 . An n × t matrix M is called a logical matrix, if M = [δn1 δn2 · · · δnit ], which is briefly expressed as M = δn [i1 i2 · · · it ]. Denote the set of n × t logical matrices by Ln×t . Coli (A) denotes the ith column of the matrix A, and Rowi (A) stands for the ith row of the matrix A. Blki (A) denotes the ith n × p block of an n × mp matrix A. 2. Preliminaries In this section, we recall some necessary preliminaries on the semi-tensor product of matrices. Definition 2.1 ([4]). The semi-tensor product of two matrices A ∈ Rm×n and B ∈ Rp×q is
A n B = A ⊗ I αn
B ⊗ I αp
,
(2.1)
where α = lcm(n, p) is the least common multiple of n and p, In denotes the n × n identity matrix, and ⊗ is the Kronecker product. When n = p, the semi-tensor product of A and B becomes the conventional matrix product. Thus, it is a generalization of the conventional matrix product. We omit the symbol ‘‘n’’ if no confusion arises in the following. The semi-tensor product of matrices has the following properties. Proposition 2.2 ([4]). (i) Let X ∈ Rt ×1 be a column vector and A ∈ Rm×n . Then X n A = (It ⊗ A) n X .
(2.2)
(ii) Let X ∈ Rm×1 and Y ∈ Rn×1 be two column vectors. Then Y n X = W[m,n] n X n Y ,
(2.3)
where W[m,n] = δmn [1 m + 1 · · · (n − 1)m + 1 2 m + 2 · · · (n − 1)m + 2
··· m m + m · · · (n − 1)m + m] ∈ Lmn×mn is the so-called swap matrix. Identify 1 ∼ δ21 and 0 ∼ δ22 , then ∆ ∼ D , where ‘‘∼’’ denotes two different forms of the same object. In most places of this work, we use δ21 and δ22 to express logical variables and call them the vector form of logical variables. The following lemma is fundamental for the matrix expression of logical functions.
H. Li, Y. Wang / Nonlinear Analysis: Hybrid Systems 22 (2016) 137–146
139
Lemma 2.3 ([4]). Let f (x1 , x2 , . . . , xs ) : D s → D be a logical function. Then, there exists a unique matrix Mf ∈ L2×2s , called the structural matrix of f , such that f (x1 , x2 , . . . , xs ) = Mf nsi=1 xi , where xi ∈ ∆ and
nsi=1
(2.4)
xi = x1 n · · · n xs .
For example, the structural matrices for Negation (¬), Conjunction (∧) and Disjunction (∨) are Mn = δ2 [2 1], Mc =
δ2 [1 2 2 2] and Md = δ2 [1 1 1 2], respectively. 3. Problem formulation
Consider the following switched Boolean network:
σ (t ) x1 (t + 1) = f1 (x1 (t ), x2 (t ), . . . , xn (t )), σ (t ) x2 (t + 1) = f2 (x1 (t ), x2 (t ), . . . , xn (t )), .. . σ (t ) xn (t + 1) = fn (x1 (t ), x2 (t ), . . . , xn (t )); yk (t ) = hk (x1 (t ), x2 (t ), . . . , xn (t )), k = 1, . . . , p,
(3.1)
where σ : N → W = {1, 2, · · · , w} is the switching signal, X (t ) = (x1 (t ), x2 (t ), . . . , xn (t )) ∈ D n and Y (t ) = (y1 (t ), . . . , yp (t )) ∈ D p are the state and the output of the system (3.1), respectively, and fij : D n → D , i = 1, . . . , n, j = 1, . . . , w and hk : D n → D , k = 1, . . . , p are logical functions. Given a switching signal σ : N → W , denote the state and output trajectories of the system (3.1) starting from an initial state X (0) ∈ D n by X (t ; X (0), σ ) and Y (t ; X (0), σ ), respectively. Definition 3.1. Given a reference signal Yr = (yr1 , . . . , yrp ) ∈ D p . The output of the system (3.1) is said to track Yr under arbitrary open-loop switching signal, if there exists an integer τ > 0 such that Y (t ; X (0), σ ) = Yr
(3.2)
holds for ∀ X (0) ∈ D , ∀ σ : N → W and ∀ t ≥ τ . n
Definition 3.2. Given a reference signal Yr = (yr1 , . . . , yrp ) ∈ D p . The output of the system (3.1) is said to track Yr under a closed-loop switching signal, if there exists a switching signal in the form of
σ (t ) = g (X (t )),
t ∈N
(3.3)
under which there exists an integer τ > 0 such that Y (t ; X (0), σ ) = Yr
(3.4)
holds for ∀ X (0) ∈ D and ∀ t ≥ τ , where g : D → W is a w -valued logical function [5]. n
n
In the following, we convert the system (3.1) and the closed-loop switching signal (3.3) into equivalent algebraic forms, respectively. p Using the vector form of logical variables and setting x(t ) = nni=1 xi (t ) ∈ ∆2n and y(t ) = ni=1 yi (t ) ∈ ∆2p , by Lemma 2.3, one can convert (3.1) into the following algebraic form:
x(t + 1) = Lσ (t ) x(t ), y(t ) = Hx(t ),
(3.5)
where Li ∈ L2n ×2n , i ∈ W and H ∈ L2p ×2n . Moreover, the reference signal becomes yr = ni=1 yri = δ2αp , where α is uniquely determined by yri , i = 1, . . . , p. i Similarly, identifying σ (t ) = i ∼ δw , i ∈ W and setting x(t ) = nni=1 xi (t ) ∈ ∆2n , (3.3) can be converted to the following form: p
σ (t ) = Gx(t ),
(3.6)
where G ∈ Lw×2n is the structural matrix of the w -valued logical function g. For details, please refer to [5]. In the following, we study the output tracking of the system (3.1) based on the algebraic forms (3.5) and (3.6). 4. Output tracking under arbitrary open-loop switching signal In this section, we investigate the output tracking of SBNs under arbitrary open-loop switching signal. To this end, we first propose the concept of switching-output-reachability of SBNs.
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Definition 4.1. Consider the system (3.1). Let X (0) ∈ D n . Then, Y = (y1 , . . . , yp ) ∈ D p is said to be s-switching-outputreachable from X (0), if one can find a switching signal σ : N → W such that Y (s; X (0), σ ) = Y . In order to give a necessary and sufficient condition for the switching-output-reachability of SBNs, we recall a useful result on the switching reachability of SBNs. For details, please refer to [30]. q
Lemma 4.2. Consider the system (3.5). Let xf = δ2n and x(0) = δ2r n be given. Then, xf is switching reachable from x(0) at time s ∈ Z+ , if and only if
(M s )q,r > 0,
(4.1)
where w
M =
Li ,
(4.2)
i=1
and (M s )q,r denotes the (q, r )-th element of M s . Based on Definition 4.1 and Lemma 4.2, we have the following result. Theorem 4.3. Consider the system (3.5). Let yf = δ2kp and x(0) = δ2r n be given. Then, yf is s-switching-output-reachable from x(0), if and only if
HM s
k,r
> 0.
(4.3)
Proof (Sufficiency). Suppose that (4.3) holds. Since
HM
s
k,r
=
2n
(H )k,q (M s )q,r > 0,
(4.4)
q =1
there exists an integer 1 ≤ q ≤ 2n such that (H )k,q = 1 and (M s )q,r > 0. q By Lemma 4.2, δ2n is switching reachable from δ2r n at time s, that is, one can find a switching signal σ : {0, . . . , s − 1} → W such that q
x(s; δ2r n , σ ) = δ2n . Thus, q
y(s; δ2r n , σ ) = H δ2n = Colq (H ) = δ2kp , which implies that yf is s-switching-output-reachable from x(0). (Necessity) Assume that δ2kp is s-switching-output-reachable from δ2r n . Then, there exists a switching signal σ : {0, . . . , s− 1} → W such that y(s; δ2r n , σ ) = δ2kp . Since y(s; δ2r n , σ ) = Hx(s; δ2r n , σ ), q q letting x(s; δ2r n , σ ) = δ2n , it is easy to obtain that Colq (H ) = δ2kp , and δ2n is switching reachable from δ2r n at time s, that is, (H )k,q = 1 and (M s )q,r > 0.
Therefore,
HM s
k ,r
=
2n (H )k,i (M s )i,r ≥ (H )k,q (M s )q,r > 0, i=1
which implies that (4.3) holds.
Remark 4.4. When δ2kp is s-switching-output-reachable from δ2r n , one can design a switching sequence to realize the switching-output-reachability by the following steps: (1) Find an integer 1 ≤ q ≤ 2n such that (H )k,q = 1 and (M s )q,r > 0. (2) Find an integer 1 ≤ α ≤ w such that
Blkα (M s−1 L)
where L = [L1
L2
q ,r
> 0,
···
Lw ]. Set σ (0) = α . If s = 1, stop. Otherwise, go to the next step.
H. Li, Y. Wang / Nonlinear Analysis: Hybrid Systems 22 (2016) 137–146
(3) Find two integers 1 ≤ j ≤ 2n and 1 ≤ β ≤ w such that Blkβ (L)
q ,j
Blkα (M s−2 L)
141
> 0 and
> 0.
j ,r
j
Set σ (s − 1) = β and x(s − 1) = δ2n . If s − 1 = 1, stop. Otherwise, replace s and q by s − 1 and j, respectively, and go to (2). In the following, based on the switching-output-reachability, we study the output tracking of the system (3.1) under arbitrary open-loop switching signal. We have the following result. Theorem 4.5. The output of the system (3.1) tracks the reference signal yr = δ2αp under arbitrary open-loop switching signal, if and only if there exists a positive integer s ≤ 2n such that
Rowα HM s
= ws 12n .
(4.5)
Proof (Sufficiency). Suppose that (4.5) holds. Firstly, we prove that 2p
HM s
i =1
= ws
i,j
(4.6)
holds for ∀ j = 1, 2, . . . , 2n and ∀ s ∈ Z+ . We prove (4.6) by induction. When s = 1, since Lq ∈ L2n ×2n , q = 1, . . . , w , we have
2n
k=1
(Lq )k,j = 1. Thus,
w w (M )k,j = (Lq )k,j = (Lq )k,j = w. 2n
2n
k =1
2n
k =1 q =1
q=1 k=1
Hence, 2p 2n 2p (H )i,k (M )k,j (HM )i,j = i=1 k=1
i =1
=w
2p (H )i,k = w, i =1
which implies that (4.6) holds for s = 1. Assume that (4.6) holds for s = l. Now, we consider the case of s = l + 1. In this case, it is easy to see that 2p
HM l+1
i =1
i,j
=
2p 2n
HM l
i=1 k=1
=
2p
i ,k
(M )k,j
2n HM (M )k,j l
i,k
i=1
k=1
2n
= wl
(M )k,j = w l+1 . k=1
Therefore, (4.6) holds for s = l + 1. By induction, (4.6) holds for any s ∈ Z+ . Secondly, we prove that if (4.5) holds, then
Rowα HM t
= wt 12n ,
∀ t ≥ s.
(4.7)
We also prove (4.7) by induction. It is obvious that (4.7) holds for t = s. Suppose that (4.7) holds for t = l ≥ s. Then, for the case of t = l + 1, we have
Rowα HM l+1
= Rowα HM l M = wl 12n M = w l+1 ,
which shows that (4.7) holds for t = l + 1. By induction, (4.7) holds for any t ≥ s. From (4.6) and (4.7), for any t ≥ s, one can obtain that
HM t
i ,j
= 0,
∀ i ̸= α, j = 1, . . . , 2n ,
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H. Li, Y. Wang / Nonlinear Analysis: Hybrid Systems 22 (2016) 137–146
and
HM t
α,j
= wt ,
∀ j = 1, . . . , 2n .
By Theorem 4.3, we know that under any switching signal σ (t ), the output trajectory of the system (3.5) starting from any j initial point x(0) = δ2n will stay at δ2αp after time s. Therefore, the output of the system (3.5) tracks the reference signal yr = δ2αp under arbitrary open-loop switching signal. (Necessity): Suppose that the output of the system (3.5) tracks the reference signal yr = δ2αp under arbitrary open-loop switching signal. We firstly show that under any switching signal σ (t ), the output trajectory of the system (3.5) starting j from any initial point x(0) = δ2n reaches yr = δ2αp in the time T (x(0), σ (t )) ≤ 2n , and then stays at yr forever. If not, then, one can find an initial point x(0), an integer T > 2n and a switching sequence π = {(0, i0 ), (1, i1 ), . . . , (T − 1, iT −1 )} such that under π , the shortest time that the output trajectory of the system (3.5) starting from x(0) reaches yr is T . Denote the state trajectory by {x(0), x(1), . . . , x(T − 1), x(T )}. Then, one can easily see that Hx(k) ̸= yr , ∀ 0 ≤ k ≤ T − 1. Since the number of different states for the system (3.5) is 2n , there must exist two integers 0 ≤ k1 < k2 ≤ T − 1 such that x(k1 ) = x(k2 ). Construct the following switching signal:
ik1 , t = γ (k2 − k1 ), ik1 +1 , t = γ (k2 − k1 ) + 1, σ (t ) = .. . ik2 −1 , t = (γ + 1)(k2 − k1 ) − 1,
(4.8)
where γ ∈ N. Then, under the switching signal σ (t ), the output trajectory of the system (3.5) starting from the initial point x(0) = x(k1 ) forms a cycle {Hx(k1 ), Hx(k1 + 1), . . . , Hx(k2 − 1)} with Hx(k) ̸= yr , ∀ k = k1 , k1 + 1, . . . , k2 − 1, which is a contradiction. Therefore, T (x(0), σ (t )) ≤ 2n holds for any σ (t ) and x(0). Now, we set s = maxx(0),σ (t ) {T (x(0), σ (t ))} ≤ 2n . By Theorem 4.3, one can obtain that HM s = 0, ∀ r ̸= α, j = 1, . . . , 2n r ,j
and
HM s
α,j
= ws ,
∀ j = 1, . . . , 2n ,
which implies that (4.5) holds.
5. Output tracking under closed-loop switching signals In this section, we study the output tracking of SBNs under closed-loop switching signals, and present a constructive procedure for the design of closed-loop switching signals. For the system (3.5) and the reference signal yr = δ2αp , define the following set:
O (α) = {r ∈ N : Colr (H ) = δ2αp , 1 ≤ r ≤ 2n }.
(5.1)
Note that the set {δ2r n : r ∈ O (α)} contains all the states of the system (3.5) whose outputs form the vector yr = δ2αp . We presuppose O (α) ̸= ∅ in the following. Otherwise, if O (α) = ∅, the output tracking problem of SBNs under closed-loop switching signals is not solvable. Given positive integers τ and j1 < · · · < jv ≤ 2n , define
Mj1 ,j1
.. .
M{j1 ,...,jv } :=
Mjv ,j1
··· .. .
Mj1 ,jv
···
Mjv ,jv
.. .
(5.2)
and
Rowj1 (M τ )
.. , . τ Rowjv (M )
M{τj1 ,...,jv } :=
(5.3)
where M is given in (4.2). We have the following result. Theorem 5.1. The output of the system (3.1) tracks the reference signal yr = δ2αp under a closed-loop switching signal, if and only if there exist positive integers τ ≤ 2n and ji ∈ O (α), i = 1, . . . , v with j1 < · · · < jv such that all the columns of both M{j1 ,...,jv } and M{τj1 ,...,jv } are nonzero.
H. Li, Y. Wang / Nonlinear Analysis: Hybrid Systems 22 (2016) 137–146 j
143
j
In order to prove Theorem 5.1, for the set Ω = {δ21n , . . . , δ2vn } and k ∈ Z+ , we define a sequence of sets as follows: j
j
Ωk = {x(0) ∈ ∆2n : there exist σ : {0, . . . , k − 1} → W and δ2in ∈ Ω such that x(k; x(0), σ ) = δ2in }.
(5.4)
Then, by Lemma 4.2 and a simple calculation, we have the following result. Lemma 5.2. (1) Ω ⊆ Ω1 if and only if all the columns of M{j1 ,...,jv } are nonzero; (2) Ωτ = ∆2n if and only if all the columns of M{τj1 ,...,jv } are nonzero. In the following, based on Lemma 5.2, we give the proof of Theorem 5.1. Proof of Theorem 5.1 (Sufficiency). Assuming that there exist positive integers τ ≤ 2n and ji ∈ O (α), i = 1, . . . , v with j1 < · · · < jv such that all the columns of both M{j1 ,...,jv } and M{τj1 ,...,jv } are nonzero, we design a closed-loop switching signal under which the output of the system (3.1) tracks yr . From Lemma 5.2, one can see that Ω ⊆ Ω1 and Ωτ = ∆2n . Set
Ωk◦ = Ωk \ Ωk−1 ,
k = 1, . . . , τ ,
where Ω0 := ∅. Obviously, Ωk1
(5.5)
τ
Ωk2 = ∅ holds for any k1 ̸= k2 , and k=1 Ωk = ∆2n . Therefore, for any integer 1 ≤ l ≤ 2n , there exists a unique integer 1 ≤ kl ≤ τ such that δ2l n ∈ Ωk◦l . For the system (3.5), let L = [L1 · · · Lw ] = δ2n [i1 i2 · · · iw2n ]. Then, for any x(t ) = δ2l n and σ (t ) = q ∈ W , it is ◦
◦
◦
easy to see that
i(q−1)2n +l
x(t + 1) = Lq δ2l n = δ2n
.
(5.6) i(p −1)2n +l
From (5.4) and (5.6), when kl = 1, there exists pl ∈ W such that δ2n l i(p −1)2n +l l 2n
∈ Ω ; when 2 ≤ kl ≤ τ , there exists
pl ∈ W such that δ ∈ Ωkl −1 . Set G = δw [p1 p2 · · · p ] ∈ L . Then, under the closed-loop switching signal σ (t ) = Gx(t ), for any initial state x(0) = δ2l n ∈ ∆2n , we have x(kl ; x(0), σ ) ∈ Ω , ∀ 1 ≤ l ≤ 2n . One can see from Ω ⊆ Ω1 that x(t ; x(0), σ ) ∈ Ω ,
2n
w×2n
∀ t ≥ τ , ∀ x(0) ∈ ∆2n ,
which implies that y(t ; x(0), σ ) = Hx(t ; x(0), σ ) = yr ,
∀ t ≥ τ , ∀ x(0) ∈ ∆2n .
Therefore, the output of the system (3.1) tracks yr under the designed closed-loop switching signal σ (t ) = δw [p1 p2 · · · p2n ]x(t ). (Necessity) Suppose that the output of the system (3.1) tracks yr = δ2αp by a closed-loop switching signal, say, σ (t ) = Gx(t ), G ∈ Lw×2n . Then, the closed-loop system consisting of the system (3.5) and the closed-loop switching signal becomes the following Boolean network:
x(t + 1) = [L1 · · · Lw ]GMr ,2n x(t ),
(5.7)
y(t ) = Hx(t ), n
where Mr ,2n = Diag{δ21n , δ22n , . . . , δ22n } ∈ L22n ×2n . Denote the set of states in all the attractors (fixed points and cycles) of j1 2n
jv 2n
(5.7) by Ω = {δ , . . . , δ }, j1 < · · · < jv , and the transient period of (5.7) by τ ≤ 2n . Then, a straightforward calculation shows that
Ω ⊆ Ω1 , Ωτ = ∆ 2 n , ji ∈ O (α), i = 1, . . . , v. By Lemma 5.2, all the columns of both M{j1 ,...,jv } and M{τj1 ,...,jv } are nonzero. This completes the proof.
(5.8)
Remark 5.3. The proof of Theorem 5.1 provides a constructive procedure to design closed-loop switching signals for the output tracking of SBNs, which contains the following steps:
• Step 1: Find positive integers τ ≤ 2n and ji ∈ O (α), i = 1, . . . , v with j1 < · · · < jv such that all the columns of both M{j1 ,...,jv } and M{τj1 ,...,jv } are nonzero. j
• Step 2: For Ω = {δ21n , . . . , δ2jvn }, calculate Ωk and Ωk◦ according to (5.4) and (5.5), respectively.
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• Step 3: For any integer 1 ≤ l ≤ 2n which corresponds to a unique integer 1 ≤ kl ≤ τ such that δ2l n ∈ Ωk◦l , find an integer pl ∈ W such that i(p −1)2n +l Ω , if kl = 1 l δ2n ∈ (5.9) Ωkl −1 , if 2 ≤ kl ≤ τ . • Step 4: The closed-loop switching signal can be designed as σ (t ) = δw [p1
p2
···
p2n ]x(t ).
6. An illustrative example In this section, we apply the obtained results to the analysis of apoptosis networks. The network graph of the considered apoptosis network is shown in Fig. 1, where IAP (denoted by x1 ) stands for the concentration level (high or low) of the inhibitor of apoptosis proteins, C3a (denoted by x2 ) the concentration level of the active caspase 3, C8a (denoted by x3 ) the concentration level of the active caspase 8, and TNF (a stimulus, denoted by u) the concentration level of the tumor necrosis factor. The dynamics of the considered apoptosis network is as follows [2]: x1 (t + 1) = ¬x2 (t ) ∧ u(t ), x2 (t + 1) = ¬x1 (t ) ∧ x3 (t ), x3 (t + 1) = x2 (t ) ∨ u(t ).
(6.1)
By letting u = 1 and u = 0, respectively, the dynamics of the apoptosis network (6.1) becomes the following switched Boolean network [30]:
σ (t ) x1 (t + 1) = f1 (x1 (t ), x2 (t ), x3 (t )), σ (t ) x2 (t + 1) = f2 (x1 (t ), x2 (t ), x3 (t )), σ (t ) x3 (t + 1) = f3 (x1 (t ), x2 (t ), x3 (t )),
(6.2)
where σ : N → W = {1, 2} is the switching signal, f11 = ¬x2 , f21 = ¬x1 ∧ x3 and f31 = 1 correspond to u = 1, and f12 = 0, f22 = ¬x1 ∧ x3 and f32 = x2 correspond to u = 0. In the apoptosis network, {x1 = 1, x2 = 0} stands for the cell survival [2]. Thus, we are interested in observing x1 and x2 at each time instance, and have the following output equation:
y1 (t ) = x1 (t ), y2 (t ) = x2 (t ).
(6.3)
In the following, we study the following two problems. Problem 1. Can the output of the apoptosis network (6.2) track Yr = (1, 0) (the cell survival) under arbitrary stimulus (or switching signal)? Problem 2. Can the output of the apoptosis network (6.2) track Yr under a closed-loop stimulus (or switching signal)? It should be pointed out that the practical meaning of Problem 1 is to seek the possibility of steering the apoptosis network to the cell survival state under any amount of stimulus, while the practical meaning of Problem 2 is to design some proper amount of stimulus (depending on ‘‘IAP’’, ‘‘C3a’’ and ‘‘C8a’’) to drive the apoptosis network to the cell survival state. As was shown in [2], the apoptosis networks always present bistable characterization, which shows that both problems have no solutions. In the following, we will prove the conclusion by using Theorems 4.5 and 5.1. Using the vector form of logical variables and setting x(t ) = n3i=1 xi (t ), we have the following algebraic form for the system (6.2) with the output (6.3):
x(t + 1) = Lσ (t ) x(t ), y(t ) = Hx(t ),
(6.4)
where L1 = δ8 [7 7 3 3 5 7 1 3], L2 = δ8 [7 7 8 8 5 7 6 8], and H = δ4 [1 1 2 2 3 3 4 4]. Moreover, Yr ∼ yr = δ42 . A simple calculation gives 0 0 0 0 M = L1 + L2 = 0 0 2 0
0 0 0 0 0 0 2 0
0 0 1 0 0 0 0 1
0 0 1 0 0 0 0 1
0 0 0 0 2 0 0 0
0 0 0 0 0 0 2 0
1 0 0 0 0 1 0 0
0 0 1 0 . 0 0 0 1
H. Li, Y. Wang / Nonlinear Analysis: Hybrid Systems 22 (2016) 137–146
145
Fig. 1. The network graph of the apoptosis network.
For Problem 1, it is easy to obtain that Row2 (HM s ) ̸= 2s 18 holds for any integer 1 ≤ s ≤ 8. By Theorem 4.5, the output of the apoptosis network (6.2) cannot track Yr = (1, 0) under arbitrary stimulus. Now, we consider Problem 2. One can obtain that O (2) = {3, 4}. For j1 = 3 and j2 = 4, a straightforward computation shows that
M{j1 ,j2 } =
1 0
1 0
and
Col2 HM{τj1 ,j2 }
=
0 0
holds for any integer 1 ≤ s ≤ 8. Thus, the conditions of Theorem 5.1 cannot hold for j1 = 3 and j2 = 4. Similarly, one can prove that the conditions of Theorem 5.1 cannot hold for the choice of j = 3 or j = 4. Therefore, one cannot design a closed-loop stimulus such that the output of the apoptosis network (6.2) tracks Yr , which is consistent to the bistability of apoptosis networks [2]. 7. Conclusion In this paper, using the semi-tensor product method, we have studied the output tracking of SBNs under openloop/closed-loop switching signals. We have presented a necessary and sufficient condition for the output tracking of SBNs under arbitrary open-loop switching signal based on the switching-output-reachability of SBNs. We have proposed a constructive procedure to design closed-loop switching signals for SBNs to track a constant reference signal. Finally, we have applied the obtained new results to apoptosis networks, and shown their effectiveness. It should be pointed out that the method proposed in this paper can only deal with the output tracking problem of SBNs with a constant reference signal. If the tracking goal is a time-varying trajectory, one needs to develop new methods in future works. References [1] T. Akutsu, M. Hayashida, W. Ching, M. Ng, Control of Boolean networks: Hardness results and algorithms for tree structured networks, J. Theoret. Biol. 244 (4) (2007) 670–679. [2] M. Chaves, Methods for qualitative analysis of genetic networks, in: Proc. 10th European Control Conference, 2009, pp. 671–676. [3] S. Kauffman, Metabolic stability and epigenesis in randomly constructed genetic nets, J. Theoret. Biol. 22 (3) (1969) 437. [4] D. Cheng, H. Qi, Z. Li, Analysis and Control of Boolean Networks: A Semi-tensor Product Approach, Springer, London, 2011. [5] D. Cheng, H. Qi, Y. Zhao, An Introduction to Semi-tensor Product of Matrices and Its Applications, World Scientific, Singapore, 2012. [6] H. Chen, X. Li, J. Sun, Stabilization, controllability and optimal control of Boolean networks with impulsive effects and state constraints, IEEE Trans. Automat. Control 60 (3) (2015) 806–811. [7] D. Cheng, H. Qi, Controllability and observability of Boolean control networks, Automatica 45 (7) (2009) 1659–1667. [8] E. Fornasini, M.E. Valcher, Observability, reconstructibility and state observers of Boolean control networks, IEEE Trans. Automat. Control 58 (6) (2013) 1390–1401. [9] D. Laschov, M. Margaliot, Controllability of Boolean control networks via the Perron–Frobenius theory, Automatica 48 (6) (2012) 1218–1223. [10] F. Li, J. Sun, Controllability of Boolean control networks with time delays in states, Automatica 47 (3) (2011) 603–607. [11] Y. Zhao, D. Cheng, H. Qi, Input-state incidence matrix of Boolean control networks and its applications, Systems Control Lett. 59 (12) (2010) 767–774. [12] L. Zhang, K. Zhang, Controllability and observability of Boolean control networks with time-variant delays in states, IEEE Trans. Neural Netw. Learn. Syst. 24 (2013) 1478–1484. [13] D. Cheng, H. Qi, Z. Li, J.B. Liu, Stability and stabilization of Boolean networks, Internat. J. Robust Nonlinear Control 21 (2) (2011) 134–156. [14] R. Li, M. Yang, T. Chu, State feedback stabilization for Boolean control networks, IEEE Trans. Automat. Control 58 (7) (2013) 1853–1857. [15] E. Fornasini, M.E. Valcher, Feedback stabilization, regulation and optimal control of Boolean control networks, in: 2014 American Control Conference, 2014, pp. 1981–1986. [16] D. Laschov, M. Margaliot, A maximum principle for single-input Boolean control networks, IEEE Trans. Automat. Control 56 (4) (2011) 913–917. [17] Y. Zhao, Z. Li, D. Cheng, Optimal control of logical control networks, IEEE Trans. Automat. Control 56 (8) (2011) 1766–1776. [18] D. Cheng, H. Qi, A linear representation of dynamics of Boolean networks, IEEE Trans. Automat. Control 55 (10) (2010) 2251–2258.
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