Controllability and observability of switched Boolean control networks

www.ietdl.org Published in IET Control Theory and Applications Received on 14th February 2012 Revised on 22nd July 2012 doi: 10.1049/iet-cta.2012.0362

ISSN 1751-8644

Controllability and observability of switched Boolean control networks L. Zhang J. Feng J.Yao School of Mathematics, Shandong University, Jinan 250100, People’s Republic of China E-mail: [email protected]

Abstract: The controllability and observability are investigated for a class of switched Boolean control networks (SBCNs). Using semi-tensor product of matrices, the dynamics of an SBCN can be transformed into an algebraic form. The modelinput-state (MIS) matrix of an SBCN is introduced and studied for the first time. This matrix contains complete information of the MIS mapping. A necessary and sufficient condition for the controllability of SBCN is obtained. The corresponding control and switching law, which drive a point to a given reachable point are designed. A sufficient condition for the observability of an SBCN is given. Moreover, under the assumption of controllability, one necessary and sufficient condition is derived for the observability.

1

Introduction

The Boolean network that was firstly introduced by Kauffman [1] becomes a hot topic in system biology, physics and system science. The authors of [2–4] have studied the topological structure of a Boolean network. Another important topic is the control of Boolean networks. Boolean control networks (BCNs) were proposed in [5] to describe gene-regulatory networks. The authors [6, 7] have studied the controllability of BCNs. Li and Sun [8] analyse the observability analysis of BCNs with impulsive effects. Necessary and sufficient conditions for the observability of the BCNs with impulsive effects are obtained. Using the similar method in [8], one can study the observability of the BCN with impulsive effects. In fact, the models of an SBN can be switched according to a defined probability, we call this kind of networks probabilistic Boolean networks (PBNs). Further works currently under submission by Zhao and Cheng have already given a strict proof for controllability of PBCNs. Recently, a new matrix product, namely, the semi-tensor product (STP) of matrices, was proposed in [9], which generalises the conventional matrix product to two arbitrary matrices. Using STP, a logical equation can be expressed as an algebraic one and the dynamics of a Boolean (control) network can be converted into a linear (bilinear) discretetime (control) system [10]. For finite value systems, Boolean network and its generalisations are proper models to describe them. On the other hand, switched systems are very important in control theory. Many natural and engineering systems appear with different models according to the environment changes. For instance, transmission systems, aircraft formation control, robot movement and traffic control are all

IET Control Theory Appl., 2012, Vol. 6, Iss. 16, pp. 2477–2484 doi: 10.1049/iet-cta.2012.0362

based on switched control network models. Recently, a new model switched Boolean network is proposed. Switched Boolean networks (SBCNs) have been used to biology systems [11]. In [11], an SBCN model is constructed for the acute myeloid leukaemia (AML) signaling network. AML is characterised by the rapid growth of abnormal white blood cells, which accumulate in the patient bone marrow and perturb the production of normal blood cells. It constructs a model of the AML signalling network by combining the signalling pathways involved in either myeloid differentiation or cell proliferation. However, there is not much theoretical results on SBCNs. Hwang and Lee [11] just uses a simple SBCN model to calculate the output: frequency of the activation. Therefore it is worthwhile to study SBCNs. This paper considers the controllability and observability of SBCNs. The key tools used in this paper are the STP of matrices and the model-input-state (MIS) matrix of SBCNs. The latter one is firstly introduced in this paper. This MIS matrix idea comes from the input-state incidence matrix of BCN in [12]. The form of the results in [12] is concise and the algorithm proposed can be easily programmed by computer. Motivated by [12], the MIS matrix is introduced to analyse the controllability and observability of SBCNs. This paper is organised as follows. Some necessary preliminaries including notations and the matrix expression of SBCNs are presented in Section 2. Section 3 introduces the MIS matrix and its some useful properties. Controllability, trajectory tracking and observability of SBCNs are investigated in Sections 4–6, respectively. An illustrative example is given in Section 7. Finally, Section 8 concludes the paper briefly.

2477 © The Institution of Engineering and Technology 2012

www.ietdl.org 2 2.1

where

Preliminaries Notations

cij = ai1 ×B b1j +B · · · +B ain ×B bnj :=

• Mm×n is the set of m × n real matrices. • Coli (M )(Rowi (M )) is the ith column (row) of matrix M . Col(M )(Row(M )) is the set of columns (rows) of matrix M . • D := {0, 1}. • δni is the ith column of the identity matrix In . • n := {δn1 , δn2 , . . . , δnn }. When n = 2 we simply use  := 2 . • Assume a matrix M = [δni1 , δni2 , . . . , δnis ] ∈ Mn×s . Then M is called a logical matrix, and denoted simply by M = δn [i1 i2 · · · is ] The set of n × s logical matrices is denoted by Ln×s . • B = [bij ] ∈ Mm×n is called a Boolean matrix if bij ∈ D, ∀i, j. The set of m × n Boolean matrices is denoted by Bm×n . • A ∈ Mm×sn . Equally split the columns of A into s blocks, then Blki (A) is the ith m × n block of A, i = 1, 2, . . . , s, that is A = [Blk1 (A) Blk2 (A) · · · Blks (A)]

• A = (aij ) ∈ Mm×n , B ∈ Mp×q . The Kronecker product of matrices is ⎡

a11 B ⎢ a21 B A ⊗ B := ⎢ ⎣ ... am1 B

a12 B a22 B .. . ···

··· ··· .. .

⎤ a1n B a2n B ⎥ .. ⎥ . ⎦

A(k) = A ×B · · · ×B A ∈ Bn×n



k

2.2

Matrix expression of SBCNs

A BCN with n-network nodes, m inputs, and p outputs can be described as (see (5)) where xi , ul ∈ D, i = 1, 2, . . . , n, l = 1, . . . , m, fi : Dn+m → D, i = 1, 2, . . . , n and hj : Dn → D, j = 1, 2, . . . , p are logical functions. Let X (t) = (x1 (t) · · · xn (t))T , U (t) = (u1 (t) · · · um (t))T , Y (t) = (y1 (t) · · · yp (t))T then (5) can be briefly expressed as  X (t + 1) = F(U (t), X (t)) Y (t) = H (X (t))

(1)

 1 0

fn (u1 (t), · · · , um (t), x1 (t), · · · , xn (t))

⎡

⎤ h1 (x1 (t), · · · , xn (t)) ⎢h2 (x1 (t), · · · , xn (t))⎥ ⎥ H (X (t)) = ⎢ .. ⎣ ⎦ . hp (x1 (t), · · · , xn (t))

(2)

x = y otherwise

where x, y ∈ D. • Define Boolean addition +B and Boolean product ×B as a +B b := a ∨ b,

a ×B b := a ∧ b,

a, b ∈ D

(3)

If A, B ∈ Bm×n , then A +B B := (aij +B bij ) ∈ Bm×n If A ∈ Bm×n and B ∈ Bn×p , then A ×B B = (cij ) ∈ Bm×p

(4)

Consider a switched BCN  X (t + 1) = Fσ (t) (U (t), X (t)) Y (t) = H (X (t))

(7)

(8)

(9)

where σ (t) : Z+ →  = {1, 2, . . . , N }

(10)

is the switching law. The switching models Fi , i = 1, 2, . . . , N are determined by ⎤ ⎡ i f1 (u1 (t), . . . , um (t), x1 (t), . . . , xn (t)) i ⎢f2 (u1 (t), . . . , um (t), x1 (t), . . . , xn (t))⎥ ⎥ Fi (U (t), X (t)) = ⎢ .. ⎦ ⎣ . fni (u1 (t), . . . , um (t), x1 (t), . . . , xn (t)) (11)

⎧ ⎪ x1 (t + 1) = f1 (u1 (t), u2 (t), . . . , um (t), x1 (t), x2 (t), . . . , xn (t)) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨x2 (t + 1) = f2 (u1 (t), u2 (t), . . . , um (t), x1 (t), x2 (t), . . . , xn (t)) .. . ⎪ ⎪ ⎪ ⎪ x (t + 1) = fn (u1 (t), u2 (t), . . . , um (t), x1 (t), x2 (t), . . . , xn (t)) ⎪ n ⎪ ⎩y (t) = h (x (t), x (t), . . . , x (t)), j = 1, . . . , p j j 1 2 n 2478 © The Institution of Engineering and Technology 2012

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⎡ ⎤ f1 (u1 (t), · · · , um (t), x1 (t), · · · , xn (t)) ⎢f2 (u1 (t), · · · , um (t), x1 (t), · · · , xn (t))⎥ ⎥ F(U (t), X (t)) = ⎢ .. ⎣ ⎦ .

where s = lcm(n, p) is the least common multiple of n and p. The matrix product in this paper is assumed to be STP. ¯ = x∨y

aik ×B bkj

If A ∈ Bn×n , we denote

• A ∈ Mm×n , B ∈ Mp×q . Then the STP of A and B is

•

B

k=1

where

· · · amn B

A  B := (A ⊗ Is/n )(B ⊗ Is/p )

n

(5)

IET Control Theory Appl., 2012, Vol. 6, Iss. 16, pp. 2477–2484 doi: 10.1049/iet-cta.2012.0362

www.ietdl.org Assume the switching law (10) is designable. Let i denote the model corresponding to Fi . In order to use matrix expression, we identify 1 ∼ δ21 and 0 ∼ δ22 , equivalently, D ∼ . Thus, we can equivalently consider the mapping F : Dn → Dk as a mapping F : n → k . Consequently, the BCN (5) can be expressed into its algebraic form [13] as  x(t + 1) = Lu(t)x(t) (12) y(t) = Hx(t) p

where x = ni=1 xi ∈ 2n , y = j=1 yj ∈ 2p , u = mj=1 uj ∈ 2m and L ∈ L2n ×2m+n is the structure matrix of system (5). Let Lλ be the structure matrix corresponding to the model λ . Then the algebraic form of SBCN (9) can be expressed as  x(t + 1) = Lσ (t) u(t)x(t) (13) y(t) = Hx(t) where {Lσ (t) , t = 1, 2, . . .} is a sequence of logical matrices. Lσ (t) is the structure matrix corresponding to the model (t). Here, we have an example of the algebraic form of an SBCN. Example 1: Let X (t) = (x1 (t) x2 (t))T , U (t) = u(t). Assume the dynamics of an SBCN is X (t + 1) = Fσ (t) (U (t), X (t))

(14)

where σ (t) : Z+ → {1, 2, 3, 4}. The switching models Fi , i = 1, 2, 3, 4 are     u ∧ x1 u ∧ x1 , F2 = F1 = u → x1 u → x2     u ∧ x2 u ∧ x2 F3 = , F4 = u → x2 u → x1 The corresponding structure matrices are L1 = δ4 [1 1 4 4 3 3 3 3],

L2 = δ4 [1 2 3 4 3 3 3 3]

L3 = δ4 [1 3 2 4 3 3 3 3],

L4 = δ4 [1 4 1 4 3 3 3 3]

3

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MIS matrices of SBCNs

We introduce the concept of MIS matrix to describe dynamic process of an SBCN in this section. It is a generalisation of incidence matrix of BCN in [12]. Denote U as the corresponding input set and χ as the state set of (13). First, we define the set of points in the product space as {Qi } = {λ } × U × χ, where i = 1, . . . , NQ , NQ = N × 2m × 2n and N , m, n are defined in Section 2. For instance, consider Example 1. We obtain NQ = 4 × 21 × 22 = 32 and the corresponding points Qi , i = 1, . . . , 32 are as follows: ⎧ {Q1 : (t) = 1 , u(t) = δ21 , x(t) = δ41 } := (1, 1, 1) ⎪ ⎪ ⎪ ⎪{Q : (t) =  , u(t) = δ 1 , x(t) = δ 2 } := (1, 1, 2) ⎪ 1 2 ⎪ 2 4 ⎪ ⎪ ⎪ .. ⎨ . ⎪{Qi : (t) = a , u(t) = δ2b , x(t) = δ4c } := (a, b, c) ⎪ ⎪ ⎪ .. ⎪ ⎪ ⎪ . ⎪ ⎪ ⎩ {Q32 : (t) = 4 , u(t) = δ22 , x(t) = δ44 } := (4, 2, 4) (16) According to (14) we draw the flow of ((t), u(t), (x1 (t), x2 (t))) on the product space {λ } × U × χ, which is called IET Control Theory Appl., 2012, Vol. 6, Iss. 16, pp. 2477–2484 doi: 10.1049/iet-cta.2012.0362

Fig. 1

Neighbourhood of Q1

the MIS dynamic graph. Since the graph is too large to be drawn here, we only give the neighbourhood of Q1 as in Fig. 1. Now we construct MIS matrix J ∈ BNQ ×NQ of (13) in the following way  Jij =

1 there exsits an edge from Qj to Qi 0 otherwise

(17)

If J is the MIS matrix of (14), we can write the first column of J from Fig. 1 as Col1 (J ) = [1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 ]T Continuing this process we can obtain the MIS matrix J . After observing each column of J , we find that ⎡ ⎤⎫ J0 ⎪ ⎬ ⎢ .. ⎥ J = ⎣ . ⎦ 8, ⎪ J ⎭

where J0 = [ L1 L2 L3 L4 ]

(18)

0

From the constructing procedure of J , it is easy to see that property (18) is also true for general case. That is, the MIS matrix of system (13) is ⎡ ⎤⎫ J0 ⎪ ⎬ ⎢ .. ⎥ m J = ⎣ . ⎦ 2 N, ⎪ J0 ⎭

where J0 = [ L1 L2 · · · LN ] ∈ L2n ×NQ (19)

J0 is called the basic block of J . From Proposition 2.5 in [12], we obtain the following result. Proposition 1: (J s+1 )0 = M s J0

(20)

where 2 N m

M=

Blki (J0 )

(21)

i=1

Proof: Replacing 2m with 2m N in Proposition 2.5 of [12], we can easily obtain the proposition.  In fact, we have the following proposition, which shows the physical meaning of MIS matrix. Proposition 2: Consider system (13). Denote the (i,j)th element of the sth power of its MIS matrix as (J s )ij = c. Then there are c paths from point Qj to Qi at sth step with proper controls and models. 2479 © The Institution of Engineering and Technology 2012

www.ietdl.org Proof: When s = 1 the conclusion is obvious. From the constructing procedure of the MIS matrix, we know that Jij means whether there exists a set of controls and a model such that Qi is reachable from Qj in one step by judging if Jij = 1 or not. Now assume (J s )ij is the number of paths from point Qj to Qi at sth step. Since a path from Qj to Qi at (s + 1)th step can be considered as a path from Qj to Qk at sth step and from Qk to Qi at one step. Thus c=

NQ

Theorem 1: Assume the MIS matrix of system (13) is J and J0 is the basic block of J . System (13) is reachable, iff

Jik (J )kj s

k=1

which is exactly (J s+1 )ij . The proof is completed.



From the proposition above the following result is obvious. Corollary 1: Consider the MIS matrix J of SBCN (13). Qi is reachable from Qj at sth step, iff Jijs > 0.

4

MB =

B

Blki (J0 ) (23)

i=1

Proof: From Corollary 1 and (21), we know that x(s) = δ2αn j is reachable from x(0) = δ2n at sth step, iff 2 N m

(Blki (J0(s) ))αj = (MB(s) )αj > 0

(24)

j

Then, x = δ2αn is reachable from x(0) = δ2n , iff 2(m+n) N −1

(MB(s) )αj > 0

(25)

Thus, x = δ2αn is reachable (from any initial states), iff 2(m+n) N −1 B

Rowα (MB(s) ) > 0

(26)

s=1

Hence we can conclude that system (13) is reachable iff Cs > 0.  Furthermore from Remark 1, we obtain the following theorem.

Definition 2: Consider the SBCN with form (9). 1. The SBCN is said to be controllable from Xi to Xj ∈ Dn , if there exist a switching law σ (t) : Z+ → {1, 2, . . .} and a control sequence {u(t), t = 0, 1, 2, . . . , s − 1}, such that the corresponding model-input pairs {(t), u(t), t = 0, 1, 2, . . . , s − 1}, where 0 < s < ∞, drive the trajectory from X (0) = Xi to X (s) = Xj ; 2. The SBCN is said to be controllable at Xi , if it is controllable from Xi to any Xj ∈ Dn ; 3. The SBCN is said to be controllable, if it is controllable at any Xi ∈ Dn . Remark 1: The SBCNs are finite-state systems, the reachability is obviously equivalent to the controllability. Remark 2: The X0 ∈ Dn in system (9) corresponds to X0 ∈ 2n in the equivalent system (13). Equally split the basic block of J (s) of (13) into 2m N blocks as

2480 © The Institution of Engineering and Technology 2012

MB > 0,

s=1

1. A state Xj ∈ Dn is said to be reachable from Xi , if there exist a switching law σ (t) : Z+ → {1, 2, . . .} and a control sequence {u(t), t = 0, 1, 2, . . . , s − 1}, such that the corresponding model-input pairs {(t), u(t), t = 0, 1, 2, . . . , s − 1}, where 0 < s < ∞, makes the trajectory of the controlled network reaching Xj from Xi at time s; 2. Xj ∈ Dn is said to be reachable, if it is reachable from any Xi ∈ Dn ; 3. The SBCN (9) is said to be reachable, if any Xj ∈ Dn is reachable.

where m, n, N are same as those defined in Section 2.

B

s=1

B

Definition 1: Consider the SBCN (9).

=

2 N m

(s)

i=1

In this section, we consider the reachability and controllability of SBCNs. We give the following two new definitions first.

[Blk1 (J0(s) ), Blk2 (J0(s) ), . . . , Blk2m N (J0(s) )]

Cs :=

2(m+n) N −1

B

Controllability of SBCNs

J0(s)

Using Boolean algebra, from Proposition 1, we know Blki (J0(s) ) = M (s−1) Blki (J0 ). Convert the two-index (i, j; N , 2m ) into a single-index (μ(i, j); 2m N ) [14], where μ(i, j) = 1, . . . , 2m N . By the constructing process, it is clear that Blkμ(i,j) (J0(s) ) corresponds to the μ(i, j)th model-input j pair ( = i , u = δ2m ). Moreover, each Colj [Blkμ(i,j) (J0(s) )] j corresponds to the state x = δ2n . Based on this observation, we can prove the following two theorems.

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Theorem 2: System (13) is controllable, iff Cs > 0

5 Trajectory tracking and control-switchinglaw design The purpose of this section is finding a control and a proper switching law, which drive x0 to xd . The trajectory from x0 to xd is not unique in general. In the following, we propose j an algorithm to find a shortest trajectory. Let x0 = δ2n and i xd = δ2n . We give the following algorithm. J0 is the basic block of MIS matrix J . Algorithm 1: Let the (i, j)th element of the controllability matrix [Cs ]i,j > 0.

• Step 1: Find the smallest s, such that in the block decomposed form J0(s) = [Blk1 (J0(s) ), . . . , Blk2m N (J0(s) )]

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IET Control Theory Appl., 2012, Vol. 6, Iss. 16, pp. 2477–2484 doi: 10.1049/iet-cta.2012.0362

www.ietdl.org 2. The system is said to be observable, if any two initial states X10 , X20 ∈ Dn are distinguishable.

Fig. 2

Consider MIS matrix J . Blkμ(i,j) (J0(s) ) corresponds to j the μ(i, j)th model-input pair ( = i , u = δ2m ). Moreover, j (s) each Colj [Blkμ(i,j) (J0 )] corresponds to the state x = δ2n . To exchange the running order of the indices between ((t), u(t)) and x(t), we use swap matrix to define

Algorithm 5.1 Step 1

there exists a block, Blkα (J0(s) ), which has its (i, j)th element [Blkα (J0(s) )]i,j > 0

(28)

Change single-index (α(α1 , α2 ); 2m N ) into double-index (α1 , α2 ; N , 2m ). Then, set (0) = α1 , u(0) = δ2αm2 and x(s) = δ2i n . If s = 1, stop. Else, go to next step (see Fig. 2). • Step 2: Find k, β, such that [Blkβ (J0 )]ik > 0; [Blkα (J0(s−1) )]kj > 0

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m

Change single-index (β(β1 , β2 ); 2 N ) into two-index (β1 , β2 ; N , 2m ). Then, set (s − 1) = β1 , u(s − 1) = δ2βm2 and x(s − 1) = δ2kn (see Fig. 3). • Step 3: If s − 1 = 1, stop. Else, replace s by s − 1 and replace i by k, and go back to Step 2. Proposition 3: As long as xd is reachable from x0 , the control sequence {u(0), . . . , u(s − 1)} and model sequence {(0), . . . , (s − 1)} generated by Algorithm 1 can drive the trajectory from x0 to xd . Proof: Since xd is reachable from x0 , there exists the smallest s such that [Blkα (J0(s) )]i,j > 0. That means if (0) = j α1 , u(0) = δ2αm2 and x(0) = δ2n , there exists at least one path i from x(0) to x(s) = δ2n . Hence, it is obvious that there must exist k such that x0 can reach δ2kn at (s − 1)th step with u(0) = δ2αm , and β(β1 , β2 ) such that u(s − 1) = δ2βm which makes Lδ2βm δ2kn = δ2i n . Equivalently, we can find k, β, such that [Blkβ (J0 )]ik > 0; [Blkα (J0(s−1) )]kj > 0 





In the same way, we can find β and k such that δ2kn can be   reached at (s − 2)th step and Lδ2βm δ2kn = δ2kn . Continuing this process, the switching law and the sequence of controls and states from x0 to xd can be obtained. 

6

Observability of SBCNs

In this section, we consider the observability of SBCNs. To this end, we first introduce the following definitions. Definition 3: Consider SBCN system (9). Let X10 , X20 ∈ Dn be two initial states and U (t) ∈ Dm be inputs. 1. X10 , X20 are said to be distinguishable, if there exist a switching law σ (t) : Z+ → 1, 2, . . ., an integer s  0, and a control sequence {U (t), t = 0, 1, 2, . . . , s}, such that the corresponding model-input sequence {((0), U (0)), . . . , ((s), U (s))},

s0

leads the two outputs at time s + 1 different. That is Y1 (s + 1) = Y2 (s + 1) IET Control Theory Appl., 2012, Vol. 6, Iss. 16, pp. 2477–2484 doi: 10.1049/iet-cta.2012.0362

J0(s) := J0(s) W[2n ,2m N ]

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j (J(s) ) ∈ B2n ×2m N , j = Denote the jth block of J0(s) as Blk 0 n 1, 2, . . . , 2 k (J(s) ) corresponds to x = δ kn , and Now each block Blk 0 2 k (J (s) )] corresponds to the modeleach block Colμ(i,j) [Blk 0 j input pair ( = i , u = δ2m ). Then we get the following lemma obviously. i (J(s) )] consists of all the state Lemma 1: The set Col[Blk 0 x(t + s) corresponding to x(t) = δ2i n by choosing 2m N kinds of possible model-input pairs. Using Lemma 1 we have the following sufficient condition for the jointly observability. It has the similar form as the SBN situation in [12]. Theorem 3: Consider system (13) and its MIS matrix J , J0 is the basic block of J . If m+n j (J(s) )] = 0 i (J(s) ) ∨ ¯ (H  Blk ∨2s=1 N −1 [(H  Blk 0 0

1  i < j  2n

(31)

then system (13) is observable. i (J(s) )] consists of Proof: According to Lemma 1, Col[Blk 0 all the possible state x(t + s) corresponding to x(t) = δ2i n . j (J(s) ) are i (J(s) ) and H  Blk Thus, the columns of H  Blk 0 0 all the possible outputs corresponding to two different states j δ2i n and δ2n , respectively. It is easy to see that (31) implies that at least at one step the outputs corresponding to δ2i n and j  δ2n are distinct. The proof is complete. In the following we give a necessary and sufficient condition for the observability of SBCNs under the assumption of controllability. The constructing technique comes from [15]. Split J0 into 2m N equal blocks as J0 = [B1 , B2 , . . . , B2m N ]

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where [B1 , B2 , · · · , B2m N ] := [Blk1 (J0 ), Blk2 (J0 ), . . . , Blk2m N (J0 )], Bi ∈ L2n ×2n , i = 1, . . . , 2m N . Define a sequence of sets of matrices i ∈ L2p ×2n , i = 0, 1, . . . , as ⎧ ⎪ 0 = {H } ⎪ ⎪ ⎪ m ⎪  ⎪ 1 = {HBi |i = 1, . . . , 2 N } ⎪ ⎨ .. . ⎪ ⎪ ⎪  = {HBi1 Bi2 · · · Bis |i1 , . . . , is = 1, . . . , 2m N } s ⎪ ⎪ ⎪ .. ⎪ ⎩ .

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Fig. 3

Algorithm 5.1 Step 2

Note that s ⊂ L2p ×2n is a finite set. Thus, there exists the smallest s∗ > 0 such that ∗

j ⊂

s 

k , ∀j > s∗

s ti+1 > tis + s,

k=1

We also use s as the matrix, consisting of its elements and arranging in a column. For instance ⎤ HB1 ⎢ HB2 ⎥ ⎥ 1 = ⎢ ⎣ ... ⎦ , HB2m N ⎡

In general, assume that we have a time sequence tis1 ···is , ik = 1, . . . , 2m N , k = 1, . . . , s. Convert the multi-index (i1 , i2 , . . . , is ; 2m N , . . . , 2m N ) into a single-index (ν(i1 i2 · · · is ); 2ms N s ). We assume that this time sequence satisfies

⎤ HB1 B1 ⎢ HB1 B2 ⎥ ⎥,··· 2 = ⎢ .. ⎦ ⎣ . HB2m N B2m N ⎡

i = 0, 1, . . . , 2sm N s − 1 with t0s := 0

Let ir = μ(l r , j r ), r = 1, . . . , s, we use s (tν(i ) = (l 1 ), 1 ···is )

j1

s u(tν(i ) = δ2m 1 ···is )

s (tν(i + 1) = (l 2 ), 1 ···is )

j1

s u(tν(i + 1) = δ2m 1 ···is )

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.. . js

s + s − 1) = (l s ), (tν(i 1 ···is )

s u(tν(i + s − 1) = δ2m 1 ···is )

Set the observability matrix for SBCNs as ⎤ 0 ⎢ 1 ⎥ ⎥ Os = ⎢ ⎣ ... ⎦ ⎡

Since system (13) is controllable there exists proper input such that x(tis1 ···is ) = x0 . Then we have (35)

⎡

⎤ y(t1s + s) ⎢ y(t2s + s) ⎥ ⎢ ⎥ = s x0 .. ⎣ ⎦ . s y(t2m N + s)

s∗ Then we obtain the following theorem. Theorem 4: Assume that system (13) is controllable. Then it is observable, iff

Note that here we assume that the sets of time sequences are apart from each other far enough. Precisely

rank(Os ) = 2n

t1s 1 · · · 1 > (t s−1 m )+s 2 m N 2m N

  · · · 2 N

(36)

s

Proof: (Sufficiency) Let the initial state be x0 . Define a time sequence {ti1 |i = 1, 2, . . . , 2m N } satisfying 1 ti+1 > ti1 + 1,

i = 0, 1, . . . , 2m N − 1 with t01 := 0 j

1 1 Using (tμ(i,j) ) = (i), u(tμ(i,j) ) = δ2m , μ(i, j) = 1, . . . , 2m N , i = 1, . . . , N , j = 1, . . . , 2m , it is easy to see that

⎡ ⎢ ⎢ ⎣

y(t11 y(t21

⎤

⎡

+ 1) + 1) ⎥ ⎢ ⎥=⎢ .. ⎦ ⎣ .

y(t21m N + 1)

HB1 x(t11 ) HB2 x(t21 )

⎤

⎥ ⎥ .. ⎦ . 1 HB2m N x(t2m N )

Since system (13) is controllable there exists proper input such that x(ti1 ) = x0 . Thus ⎤ y(t11 + 1) 1 ⎢ y(t2 + 1) ⎥ ⎥ = 1 x0 ⎢ .. ⎦ ⎣ . ⎡

y(t21m N + 1)

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s−1

Finally, we have ⎡

y(0) y(t11 + 1) y(t21 + 1) .. .

⎢ ⎢ ⎢ ⎢ ⎤ ⎢ ⎡ ⎢  0 x0 ⎢ 1 ⎢ 1 x0 ⎥ ⎢ y(t2m N + 1) ⎥ ⎢ O s x0 = ⎢ = .. ⎣ ... ⎦ ⎢ . ⎢ ⎢ y(t s∗ + s∗ ) s∗ x0 ⎢ ⎢ y(t1s∗ + s∗ ) ⎢ 2 ⎢ .. ⎣ .

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ := Y ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(37)

∗

y(t2ssm N s + s∗ )

Note that Y is a set of observed data. Then since rank(Os ) = 2n , x0 can be uniquely solved out as x0 = (OsT Os )−1 OsT Y (Necessity) By the definition of s∗ it is easy to see that Y contains all possible outputs. Now if rank(Os ) < 2n , one sees easily that in addition to x0 there exists at IET Control Theory Appl., 2012, Vol. 6, Iss. 16, pp. 2477–2484 doi: 10.1049/iet-cta.2012.0362

www.ietdl.org least another solution x0 of (37). Then x0 and x0 are not distinguishable. 

7

According to Theorem 2 system (38) is controllable. From (34), we know ⎡ 1 ⎢0 ⎢0 ⎢ ⎢1 ⎢ ⎢1 ⎢ ⎢0 ⎢ ⎢0 ⎢1 ⎢ ⎢1 ⎢ ⎢0 ⎢ ⎢1 ⎢ ⎢0 ⎢ ⎢1 ⎢0 ⎢ ⎢1 ⎢ ⎢0 2 = ⎢ ⎢1 ⎢ ⎢0 ⎢ ⎢1 ⎢0 ⎢ ⎢1 ⎢ ⎢0 ⎢ ⎢1 ⎢ ⎢0 ⎢ ⎢1 ⎢0 ⎢ ⎢1 ⎢ ⎢0 ⎢ ⎢1 ⎢ ⎢0 ⎣ 1 0

Illustrative example

Let X (t) = (x1 (t)x2 (t))T , U (t) = u(t). Assume the dynamics of an SBCN is 

X (t + 1) = Fσ (t) (U (t), X (t)) y(t) = x1 (t) ∨ x2 (t)

(38)

where σ (t) : Z+ → {1, 2}. The switching models Fi , i = 1, 2 are     u ∧ x1 u ∧ x2 F1 = , F2 = u → x1 u → x1 Then we obtain the structure matrices of F1 and F2 , respectively, as

L1 = δ4 [1 1 4 4 3 3 3 3],

L2 = δ4 [1 3 2 4 3 3 3 3]

The MIS matrix of the system is ⎡ ⎤ J0 ⎢J0 ⎥ J =⎣ ⎦ J0 J0

(39)

1 0 0 1 0 1 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0

0 1 0 1 1 0 0 1 1 0 1 0 1 0 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0

⎤ 0 1⎥ 0⎥ ⎥ 1⎥ ⎥ 0⎥ ⎥ 1⎥ ⎥ 0⎥ 1⎥ ⎥ 1⎥ ⎥ 0⎥ ⎥ 1⎥ ⎥ 0⎥ ⎥ 1⎥ 0⎥ ⎥ 1⎥ ⎥ 0⎥ ⎥ ∈ B32×4 0⎥ ⎥ 1⎥ ⎥ 1⎥ 0⎥ ⎥ 0⎥ ⎥ 1⎥ ⎥ 1⎥ ⎥ 0⎥ ⎥ 1⎥ 0⎥ ⎥ 1⎥ ⎥ 0⎥ ⎥ 1⎥ ⎥ 0⎥ ⎦ 1 0

(43)

and rank(2 ) = 4

where J0 = δ4 [1 1 4 4 3 3 3 3 1 3 2 4 3 3 3 3]. Then, we have ⎡ 1 ⎢0 MB = ⎣ 1 0

1 0 1 0

0 1 1 1

Since rank(2 )  rank(Os )  4 we obtain

⎤ 0 0⎥ 1⎦ 1

rank(Os ) = 4 (40)

Thus, system (38) is observable.

8 It follows that

MB(2)

⎡ 1 ⎢1 =⎣ 1 1

1 1 1 1

1 1 1 1

⎤ 0 1⎥ , 1⎦ 1

MB(3)

⎡ 1 ⎢1 =⎣ 1 1

1 1 1 1

1 1 1 1

⎤ 1 1⎥ 1⎦ 1

(41)

Finally, we obtain ⎡ 1 ⎢1 Cp = ⎣ 1 1

1 1 1 1

1 1 1 1

⎤ 1 1⎥ >0 1⎦ 1

IET Control Theory Appl., 2012, Vol. 6, Iss. 16, pp. 2477–2484 doi: 10.1049/iet-cta.2012.0362

(42)

Conclusions

In this paper, the controllability, trajectory tracking and observability of SBCNs have been investigated. A new concept, namely, the MIS matrix for SBCN, is introduced. In the light of MIS matrix, the complete information about possible trajectories has been put into a certain matrix. A necessary and sufficient condition for controllability and reachability has been derived. Then a necessary condition for observability is revealed for the first time. Moreover, under the assumption of controllability a necessary and sufficient condition for the observability is also obtained. The weakness of the approach proposed in this paper is the computational complexity. Since the MIS matrix is of dimension 2m+n N × 2m+n N and the observability matrix is also not small, especially when N is large. We can use STP toolbox based on MATLAB to calculate these matrices in PC. However, even in STP toolbox we cannot deal with the matrix with dimension bigger than 225 . 2483 © The Institution of Engineering and Technology 2012

www.ietdl.org 9

Acknowledgments

The work was partially supported by NNSF (60974137, 61174092 and 61174141) of China, Research Awards Yong and Middle-Aged Scienists of Shandong Province (BS2011SF009, BS2011DX019).

6 7 8 9

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IET Control Theory Appl., 2012, Vol. 6, Iss. 16, pp. 2477–2484 doi: 10.1049/iet-cta.2012.0362