Quantized control for NCSs with communication constraints

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Quantized control for NCSs with communication constraints Jingjing Yan a, Yuanqing Xia b,∗, Chenglin Wen a a b

College of Electrical Engineering, Henan University of Technology, Zhengzhou 450052, China Department of Automatic Control, Beijing Institute of Technology, Beijing 100081, China

a r t i c l e

i n f o

Article history: Received 22 October 2016 Revised 18 March 2017 Accepted 20 June 2017 Available online xxx Communicated by Choon Ki Ahn Keywords: Asymptotic stability Time-varying systems Periodic protocol Data quantization

a b s t r a c t This paper studies the stabilization problem for networked control systems (NCSs) affected by data quantization, time-varying transmission intervals, time-varying transmission delays and communication constraints. Time-varying transmission intervals and delays, by limiting the upper and lower bounds of which, can be described by a two-dimensional convex region. Combined with the coupling role of data quantization, communication constraints mean that only one node can occupy the network and send its quantized values in each transmission. The order in which node transmits its quantized values is determined by a given periodic network protocol. By introducing a variable called proportionality coefficient of saturation value in well-known zoom strategy to deal with the complex coupling relationship between system states and quantized variables, some sufficient conditions are derived for reaching asymptotic stability of NCSs under properly designed quantizer parameters. A simulation example is given to illustrate the effectiveness of the theoretical analysis. © 2017 Elsevier B.V. All rights reserved.

1. Introduction The introduction of the network in the control systems has brought us a lot of convenience and advantages, such as decreased the complexity, increased the flexibility, improved the convenience of installation and maintenance, and reduced wiring and cost. However, networking the control systems also brings new challenges which can be divided into six categories in general: (i) data quantization; (ii) data packet loss; (iii) time-varying transmission intervals; (iv) time-varying transmission delays; (v) communication constraints, which impose that not all of the sensor and actuator signals can be transmitted simultaneously; (vi) networkinduced external disturbance. Combined with the performance constraints of the controlled object itself, such as actuator saturation, control algorithms should be pursued to handle with these communication imperfections and constraints simultaneously [1,2]. In recent years, much effort has been dedicated to the stability analysis of NCSs just affected by one or two of these phenomena [3–19]. For NCSs with (i)–(iii), a binary variable modeling stochastic sampling process and logarithmic quantization is introduced to describe the networked communication process in [20]. A randomly switched Takagi–Sugeno fuzzy system with multiple input-

∗

Corresponding author. E-mail addresses: [email protected] (J. Yan), [email protected], [email protected] (Y. Xia), [email protected] (C. Wen).

delay subsystems is proposed to model the nonlinear NCSs with (i), (ii) and (v) in [21]. The problem of achieving parameterized input to state stability with respect to (i), (ii) and (vi) for NCSs is studied in [22]. Delta operator approach is adopted in [23] to deal with the stabilization problem of NCSs with (i), (ii) and actuator saturation. Ne sˇ i c´ and Liberzon [24] formulate a unified controller design framework for NCSs with types (i), (iii) and (v). Cloosterman et al. [25] proposes a discrete-time model for NCSs that incorporates imperfections (ii)–(iv). Focusing on NCSs that are subject to (iii)–(v), Donkers et al. [26] presents a new modeling framework to derive the stability results. Under the influence of (iv)–(vi), the ultimate boundedness of the estimation error is guaranteed in [27]. Literature [28–31] research types (i), (ii) and (iv). All the above papers study the stability and stabilization issues of NCSs including three imperfections. For four imperfections, a novel NCSs model is described in [32], which includes multi-rate sampled-data, quantized signal, time-varying delay and packet dropout. In this paper, we will focus on the stability of NCSs with data quantization, timevarying transmission intervals, time-varying transmission delays and communication constraints, i.e., types (i), (iii), (iv) and (v). Donkers et al. [26] proposes skillfully a convex overapproximation method to analyze the stability of NCSs with imperfections (iii)–(v), but which is invalid to handle with data quantization simultaneously. To achieve the stability analysis of NCSs with (i)–(v), Loon et al. [33] promotes this convex overapproximation method, which ensures the stability under both uniform quantizer and logarithmic quantizer. Comparing the asymptotic stability under

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logarithmic quantizer with infinite quantization level and practical stability under uniform quantizer in [33], we will, by modifying the overapproximation procedure, pursue the asymptotic stability conditions by using a uniform quantizer with finite quantization level. As we know, time variability of quantizer parameters is the necessary condition ensuring asymptotic stability, under which the quantization errors can tend to zero. Zoom strategy proposed in [34] is a efficacious method to adjust the time-varying quantizer parameters. With the advantages of finite quantization level and adjustable variables, zoom strategy has been adopted by many articles to deal with the issues of quantization and data packet loss [19,22], or disturbance [22], or saturation [23]. But it has not been used to study the issues of (i), (iii), (iv) and (v) so far, which is one of the innovations of this paper. Due to the complex coupling relationship between system states and quantized variables, we introduce a variable called proportionality coefficient of saturation value to improve the zoom strategy, which ensures the unsaturation of the quantized variables, and guarantees that the system states load in the time-varying invariant regions at the same time. Above all, our contributions are three aspects with respect to earlier literature. First, a modified procedure is proposed to approximate the switched uncertain quantized system which is transferred from NCSs discussed here. Under this procedure, the decreasing rate of the related parameters is promoted, which allows that the overapproximation achieves tightness at a faster rate. Second, zoom strategy is adopted here to discuss the asymptotic stability of NCSs affected by imperfections (i), (iii), (iv) and (v), which has not been studied before. Third, a variable called proportionality coefficient of saturation value is introduced to improve the zoom strategy. Under which, the complex coupling relationship between system states and quantized variables can be dealt effectively, and thus the unsaturation of the quantized variables and the asymptotic stability of the closed-loop system can be guaranteed by adjusting the quantizer parameters suitably. The outline of this paper is as follows. In Section 2, we introduce the detailed model, network and problem descriptions. A method is given to write the NCSs model as a switched uncertain quantized system in Section 3. Next, we propose an improved procedure to overapproximate the NCSs according to a polytopic system in Section 4. In Section 5, we obtain the conditions to ensure the stability of the NCSs based on LMIs, and the detailed proof of which is given in Section 6. A numerical benchmark example is adopted to illustrate the effectiveness of the main results in Section 7 and conclusions are given in Section 8. The supplementary proofs of some lemmas and theorems are shown in Appendix. The notations used in this paper are very regular. Rn denotes the n-dimensional Euclidean space. R+ and N denote the set of positive real numbers and positive integers, respectively. We denote by · the standard Euclidean norm in Rn and the corresponding induced matrix norm in Rn×n . λmax (P) and λmin (P) denote the maximum eigenvalue and minimum eigenvalue of matrix P, respectively. The signal diag (A1 , . . . , An ) is used to denote a block-diagonal matrix with the diagonal elements A1 , . . . , An . AT ∈ Rm×n denotes the  transposed of matrix A ∈ Rn×m . For   brevity,  the symmetric matrix

[m5G;July 13, 2017;9:28]

A B BT C

is sometimes written as

A B ∗ C

. The

limits as s approaches t from above and below are denoted by lims↓t and lims↑t , respectively. We use co{A} to denote the convex hull of a set A. The signals · and · indicate the integer function upward and downward, respectively. 2. Model, network and problem descriptions 2.1. Model description The NCSs considered here are shown in Fig. 1, in which the plant is described by the following linear time-invariant (LTI)

Fig. 1. System configuration.

continuous-time equation:



x˙ p (t ) = A p x p (t ) + B p uˆ (t ) y(t ) = C p x p (t ),

(1)

where x p ∈ Rn p denotes the plant state, uˆ ∈ Rnu is as the most recently received control variable, y ∈ Rny denotes the output of the plant, t ∈ R+ denotes the time, and Ap , Bp , Cp are the given constant matrices with approximate dimensions. Due to the controller is typically implemented in a digital, and thus the discrete-time form, we design it as the following LTI discrete-time system:



xck+1 = Ac xck + Bc yˆk u(tk ) = Cc xck + Dc yˆ(tk ),

(2)

where xc ∈ Rnc denotes the controller state, yˆ ∈ Rny is as the most recently received output of the plant, u ∈ Rnu represents the controller output, and Ac , Bc , Cc , Dc are constant matrices. 2.2. Network description (i) Data quantization. The quantizer adopted here is the same as the one in [34], i.e.,

qμ (x(k )) = μ(k ) q

 x (k )  μ (k )

,

(3)

where μ(k ) ∈ R+ . Assume that the following conditions on qμ (·) are satisfied: I. If x(k) ≤ Mμ(k), then qμ (x(k )) − x(k ) ≤ μ(k ), II. If x(k) > Mμ(k), then qμ (x(k )) > Mμ(k ) − μ(k ), where M is the saturation value and the sensitivity. (ii) Time-varying transmission intervals and delays. For all k ∈ N, we assume that transmission interval hk and transmission delay τ k are all time-varying, and belong to the set  defined by

  : = (h, τ ) ∈ R2 |h ∈ [h, h¯ ], τ ∈ [τ , min{h, τ¯ } ), h¯ ≥ h > 0,  τ¯ ≥ τ ≥ 0 .

(iii) Communication constraints. At each transmission instant tk , k ∈ N, the plant outputs y(tk ) and controller outputs u(tk ) are sampled and quantized, and parts of the quantized value qμ (y(tk )) and qμ (u(tk )) are transmitted through the network. Assuming that the quantized values arrive at time instant rk which is called the arrival instant. The signal transmission situation illustrated above

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can refer to Fig. 1 in [26], in which y is replaced by qμ (y). It is worth mentioning that the controller states xck+1 are updated by using yˆk := limt ↓rk yˆ(t ). We assume that tk+1 > rk ≥ tk , k ∈ N. If transmission and reception occur, the outputs of the decoders, yˆ and uˆ, are updated according to the newly received values. Due to above analysis, for t ∈ (rk , rk+1 ], yˆ(t ) and uˆ (t ) are defined as:



yˆ(t ) = σk qμ (y(tk )) + (I − σk )yˆ(tk ) uˆ (t ) = σuk qμ (u(tk )) + (I − σuk )uˆ (tk ), y

3

:= μ(tk ), we get uˆk−1 = limt ↓rk−1 uˆ (t ) = uˆ (rk ) = uˆ (tk ) due to lims↑t uˆ (s ) = uˆ (t ) [26]. If we set





eA p hk = 0

Ahk









0 Bp C 0 0 ,B = ,C = p , Bc 0 0 Cc Ac

 ρ

y σk 0 eA p s ds 0 , ρ = 0 , σk = 0 I 0 I



I D= Dc

y

(4)

T

T

T

p y x¯k := [xk xck T ek euk T ] ∈ Rn

and

T Eku T ]

yT [Ek

with

0

σuk

(8)

n = n p + nc + nu + ny ,

Rny +nu ,

Ek := ∈ then the overall NCSs can be rewritten as the discrete-time switched uncertain quantized system represented by

where

iy = diag(γi,1 , . . . , γi,ny ), iu = diag(γi,1 , . . . , γi,nu ),



when σk = i. If the plant output yj and thus quantized plant output y qμ (yj ) is in node i, the elements of i , γ i, j , are equal to one for any i ∈ {1, . . . , N} and j ∈ {1, . . . , ny }. Otherwise, they are equal to zero. Similarly, if the controller output uj and thus quantized controller output qμ (uj ) is in node i, the elements of iu , γ i, j , are equal to one for any i ∈ {1, . . . , N} and j ∈ {1, . . . , nu }, and are zero elsewhere. To deal with communication constraints, the order in which node transmits its quantized values, that is the value of σ k , is determined by a given periodic network protocol satisfying

σk+N˜ = σk

(5)

˜ ∈ N. RR protocol is a special periodic protocol satisfying for some N σk+N = σk for the given N which denotes the number of channels to be selected by communicated signals. 2.3. Problem description The purpose of this paper is pursuing the asymptotic stability conditions for NCSs with imperfections data quantization, timevarying transmission intervals, time-varying transmission delays and communication constraints under uniform quantizer and periodic protocol. In the following part, we will first transfer the NCSs considered into switched uncertain quantized system, which is further approximated by a modified convex overapproximation method. With this setup, by introducing proportionality coefficient of saturation value in zoom strategy to deal with the complex coupling relationship between system states and quantized variables, the asymptotic stability conditions for the closed-loop system are obtained relying on the quantizer parameters designed suitably.

x¯k+1 =

Ahk + hk BDC C (I − Ahk − hk BDC )



+

hk BD − hk −τk Bσk x¯ I − D σk + C (hk −τk Bσk − hk BD ) k

−1

hk −τk Bσk E (D−1 − C hk −τk B )σk k

: = A˜ σk ,hk ,τk x¯k + B˜σk ,hk ,τk Ek .

(9)

The switching of above system is caused by the fact that only one node can access the network at each transmission instant because of communication constraints. The uncertainty is due to the timevarying of the transmission intervals and delays. Since the equivalence of system (9) and NCSs considered here, we will focus on the discussion of system (9) in the following part. It is worth mentioning that the switching time of system (9) relies on transmission instant tk , on which there are no constrains. Therefore, system (9) is indeed an arbitrary switching system, and the general dwell time and average dwell time methods are invalid to represent the system properties. If we set system (1) and controller (2) as open-loop discrete form:





xkp+1 = eA p hk xkp yk = C p xkp ,

xck+1 = Ac xck uk = Cc xck ,

(10)

it is obvious that B and D defined in (8) are zero matrix and identity matrix, respectively. This gives that the corresponding open-loop form of (9) is as:



x¯k+1 =



Ahk C ( I − Ahk )



0 0 x¯ + I − σk k σk



Ek : = A˜ oσk ,hk x¯k + B˜oσk ,hk Ek ,

(11)

which will be used in the zooming-out stage in the proof of Theorem 2.

3. Switched uncertain quantized system

4. A modified convex overapproximation method

In this part, NCSs described above, i.e., plant (1) combining with controller (2) under the influence of data quantization, time-varying transmission intervals, time-varying transmission delays and communication constraints is transformed into a discrete-time switched uncertain quantized system. To this end, the network-induced error is defined as:

Because of the nonlinear dependence of A˜ σk ,hk ,τk and B˜σk ,hk ,τk on hk and τ k , we will derive a polytopic system to overapproximate system (9) as [26] of the form





x¯k+1 =

L



α

l ¯ k Aσk ,l

¯ kC¯σk x¯k + B¯

l=1

eu (t ) = uˆ (t ) − u(t ) ey (t ) = yˆ(t ) − y(t )

(6)

+

L

 α

l ˆ k Bσk ,l

ˆ k Fˆσk Ek , + Dˆ

(12)

l=1

and the quantization error as



E (t ) = qμ (u(t )) − u(t ) E y (t ) = qμ (y(t )) − y(t ). u

By euk

:=

defining eu (t

k ),

y ek

(7)

p

xk := x p (tk ), :=

ey (t

k ),

Eku

uk :=

:= E u (t

k ),

u(tk ), y Ek

uˆk := limt ↓rk uˆ (t ),

:= E y (tk )

and

μk

where A¯ σ ,l ∈ Rn×n , B¯ ∈ Rn×3(n p +nc ) , C¯σ ∈ R3(n p +nc )×n , Bˆσ ,l ∈ Rn× ( ny +nu ) , Dˆ ∈ Rn×2(n p +nc ) and Fˆσ ∈ R2(n p +nc )×(ny +nu ) , for σ ∈ {1, . . . , N} and l ∈ {1, . . . , L} in which L denotes the num¯ ¯k ∈ ber of polytope vertices. Additive uncertain terms satisfy ˆ for any k ∈ N, where  ¯ and  ˆ defined below are ˆk ∈  and the sets of matrices with bounded norm in R3(n p +nc )×3(n p +nc )

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and R2(n p +nc )×2(n p +nc ) , respectively. Furthermore, unknown timevarying vector α k satisfies

αk = [αk1 , . . . , αkL ] ∈    L  := α ∈ RL  α l = 1, α l ≥ 0, l ∈ {1, . . . , L}





A˜ σ ,h,τ |(h, τ ) ∈  ⊆





B˜σ ,h,τ |(h, τ ) ∈  ⊆

L

  ¯ , (13a) α l A¯ σ ,l + B¯ ¯ C¯σ α ∈ , ¯ ∈ 

l=1 L



 ˆ . (13b) α l Bˆσ ,l + Dˆ ˆ Fˆσ α ∈ , ˆ ∈ 

Moreover, overapproximation (12) of (9) is said to be made arbitrarily tight if (13) holds and the following inclusion relations are achieved: L

    ¯ ⊆ co A˜ σ ,h,τ |(h, τ ) ∈  α l A¯ σ ,l + B¯ ¯ C¯σ α ∈ , ¯ ∈ 

l=1



L

 ¯¯  ¯¯  ≤ ε} + {

l=1



ˆˆ  ˆˆ  ≤ ε} + {

(14a)

(14b)

j

j

note the vertices of Sm . Furthermore, it is required that triangles M˜  set H = {S1 , . . . , SM˜ } satisfies Sj = ∅ i=1 Si = , Si = ∅ and Si ˜ }, i = j. for any i, j ∈ {1, . . . , M ˜ , τ˜ ) ∈ G , l ∈ {1, . . . , L}, Step (2). For all σ ∈ {1, . . . , N} and (h l l define

Bˆσ ,l := B˜σ ,h˜ ,τ˜ l l

¯ = diag(A p , 0 ). Then deStep (3). Construct the matrix ¯ as ¯ := T T −1 , where T is an invertible matrix compose ¯ with and = diag( 1 , . . . , K ) is the real Jordan form of ¯.

i ∈ Rni ×ni , i ∈ {1, . . . , K }, the ith real Jordan block of Step (4). For any i , i ∈ {1, . . . , K }, compute the worse case approximation error of H, that is,

δiEh = max

max

A δ˜i,m, α˜ ,

˜ }  3 α˜ j =1,α˜ j ≥0 m∈{1,...,M j=1

max

Eh δ˜i,m, α˜ ,

˜ }  3 α˜ j =1,α˜ j ≥0 m∈{1,...,M j=1

T −1 T BDC 0

C¯σ =



B¯ =

T −CT





0 T BD , −T −1 Bσ

−1

T −1 Bσ Fˆσ = T −1 Bσ

−1

T −CT

T −CT





× diag(δ1A I1 , . . . , δKA IK , δ1h I1 , . . . , δKh IK , δ1h−τ I1 , . . . , δKh−τ IK ), E

T Dˆ = 0

E

E

E



0 −CT

× diag(δ1h−τ I1 , . . . , δKh−τ IK , δ1h−τ I1 , . . . , δKh−τ IK ), E

E

E

E

(17)

σ ∈{1,...,N}

˜ L+1 , τ˜L+1 ) ∈  is added to G to Step (6). If ε > ε u , a pair (h ensure the tightness of overapproximation relating to ε u . In order ˜ L+1 , τ˜L+1 ), we pursue the triangle where to determine the pair (h the maximum approximation error is obtained by solving

(m∗ , α˜ ∗ ) ∈ arg

max 

˜ }, m∈{1,...,M

3 j=1

∗

α˜

α

j =1, ˜ j ≥0

δ˜ij∗ ,m,α˜

(18)

with

(i∗ , j∗ ) ∈ arg

max

i∈{1,...,K }, j∈{A,Eh ,Eh−τ }

δij .

Therefore, Sm∗ is the triangle to be pursued. Define

(d∗ , c∗ ) = arg max

d,c∈{1,2,3}

  (h˜ ldm∗ , τ˜ldm∗ ) − (h˜ lcm∗ , τ˜lcm∗ ) ,

b∗ = {1, 2, 3} \ {d∗ , c∗ }

˜ m , τ˜ m ), j ∈ {1, 2, 3} dewith l m ∈ {1, . . . , L}, j ∈ {1, 2, 3}. Here, (h l l j

max

Step (5). Construct



(16c)

j

σ ∈{1,...,N}

˜ l m , τ˜l m ), (h ˜ l m , τ˜l m ), (h ˜ l m , τ˜l m )} Sm = co{(h 1 1 2 2 3 3

˜ }  3 α˜ j =1,α˜ j ≥0 m∈{1,...,M j=1

l

j

  ε = max B¯ max {C¯σ }, Dˆ  max {Fˆσ } .

by

δiEh−τ = max

(16b)

in which Ii denotes the identity matrix of size ni , and calculate

for each σ ∈ {1, . . . , N} and arbitrarily small ε > 0. For a given threshold ε u > 0 such that (14) holds with any ε satisfying ε ≤ ε u , a procedure to obtain a tight overapproximation is formalized below. It is worth pointed out that the outline of the procedure is similar to that of [26]. We focus on the difference in the following, but also refer to some parts of the procedure of [26] in order to make the paper in a self-contained form. Procedure: ˜ , τ˜ ) ∈ Step (1). Select a desired ε u > 0. Choose distinct pairs (h l l , l ∈ {1, . . . , L}, satisfying coG =  with G = ∪Ll=1 {(h˜ l , τ˜l )}. Parti˜ triangles Sm , m ∈ {1, . . . , M ˜ } structured tion the region  into M

δiA = max

l



    ˆ ⊆ co B˜σ ,h,τ |(h, τ ) ∈  α l Bˆσ ,l + Dˆ ˆ Fˆσ α ∈ , ˆ ∈ 

A¯ σ ,l := A˜ σ ,h˜ ,τ˜ , l l

(16a)

   3 j   3j=1 α˜ j (h˜ lmj −τ˜lmj ) s  Eh−τ i  j=1 α˜ . δ˜i,m, = e ds α˜   ˜ m −τ˜ m h

and

l=1



  3 α˜ j h˜ 

i h˜ l m   i j=1 lmj A δ˜i,m, −  3j=1 α˜ j e j , α˜ = e    3 j   3j=1 α˜ j h˜ lmj s  Eh i ˜  δi,m,α˜ =  j=1 α˜ e ds , j

for any k ∈ N. Model (12) is an overapproximation of (9) if it holds that, for all σ ∈ {1, . . . , N},



in which α˜ = [α˜ 1 , α˜ 2 , α˜ 3 ]T and

˜m h l

l=1



[m5G;July 13, 2017;9:28]

Eh−τ δ˜i,m, α˜ ,

(15a)

and add the new pair

(h˜ L+1 , τ˜L+1 ) =













˜ m∗ , h ˜ m∗ + h ˜ m∗ − h ˜ m∗ /2, min τ˜ m∗ , τ˜ m∗ min h l ∗ l∗ l ∗ l∗ l ∗ l∗ c

d



 

c

d

d

+ τ˜l m∗∗ − τ˜l m∗∗ /2

(19)

c

d



c

˜ L+1 , τ˜L+1 ) is selected to the set G . Evidently, the new pair (h as the middle point of one edge of triangle Sm∗ . Redefine ˜ L+1 , τ˜L+1 )}, M ˜ := M ˜ + 1, G := G ∪ {(h







˜ L+1 , τ˜L+1 ), (h ˜ m∗ , τ˜ m∗ ), (h ˜ m∗ , τ˜ m∗ ) H := H \ Sm∗ ∪ co (h l ∗ l ∗ l∗ l∗



d

d

b

b

 

˜ L+1 , τ˜L+1 ), (h ˜ m∗ , τ˜ m∗ ), (h ˜ m∗ , τ˜ m∗ ) , ∪ co (h l∗ l∗ l∗ l∗ c

c

b

b

and return to Step (2). Step (7). If ε ≤ ε u , then the tightness of overapproximation defined by (13) and (14) is obtained, in which the sets ¯ ⊂ R3(n p +nc )×3(n p +nc ) and  ˆ ⊂ R2(n p +nc )×2(n p +nc ) are defined by 

(15b)

¯ = {diag( ¯ 1, . . . , ¯ 3K ) | ¯ i+ jL ∈ Rni ×ni ,  ¯ i+ jL   ≤ 1, i ∈ {1, . . . , K }, j ∈ {0, 1, 2}},

(15c)

ˆ = {diag( ˆ 1, . . . , ˆ 2K ) | ˆ i+ jL ∈ Rni ×ni ,  ˆ i+ jL  

(20)

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≤ 1, i ∈ {1, . . . , K }, j ∈ {0, 1}}.

(21)

Remark 1. Comparing the above procedure with the one in [26], steps (1), (3), (4) are the same as each other. However, we add some definitions or equations relating to quantization error in steps (2), (5), (7). Furthermore, in the process of simulation, we found that the pair added method proposed in [26] does not have E

E

an obvious effect to decrease the values of δiA , δi h , δi h−τ and ε , hence we select another means illustrated in step (6) to decrease them, the ultimate goal of which is to ensure ε ≤ ε u .

⎡

Pi ⎢∗ ⎣∗ ∗

5

⎤

A¯ Ti,l Pi+1 B¯ T Pi+1 Pi+1 ∗

0 Ri,l ∗ ∗

C¯iT Ri,l 0 ⎥ > 0, ⎦ 0 Ri,l

(22)

where PN˜ +1 := P1 , λmin (Pi ) ≥ λmin (P) and λmax (Pi ) ≤ λmax (P), for any ˜ } and l ∈ {1, . . . , L}. Then we can select suitable μk such i ∈ {1, . . . , N that the polytopic system (12) with periodic protocol (5) is asymptotic stability, and thus system (1) with (2), (4) and protocol (5) is also asymptotic stability.

Remark 2. The above procedure is one of the main contributions of this paper. In fact, although literature [26] gives an skillful convex overapproximation method to analyze the stability of NCSs with imperfections (iii)–(v). However, it is invalid to handle with the impact of quantization simultaneously. The above procedure extending the overapproximation method in [26] can deal with (i), (iii), (iv) and (v) simultaneously, and meanwhile, achieve tightness with a faster speed.

Remark 4. From Theorem 2 above and Theorem IV.5 in [26], we get, with or without data quantization, the LMIs to be satisfied are identical. However, due to the additional restrictions, i.e., λmin (Pi ) ≥ λmin (P) and λmax (Pi ) ≤ λmax (P), the asymptotic stability conditions with quantization is indeed relatively conservative compared with the system without quantization, which is reasonable. Moreover, we see that the research focus lies in the design of quantizer parameters, especially for μk , the detailed design method of these parameters will be given in the following proof.

The following Theorem gives the tightness of overapproximation (12) under the above procedure.

6. Proof of Theorem 2

Theorem 1. Consider system (9) with (hk , τk ) ∈ , k ∈ N. If system (12) is obtained by above proceduce for some given ε u > 0, then (13) is satisfied and thus (12) is an overapproximation of (9). Moreover, the overapproximation (12) is ε tight with ε defined by (17) and ε ≤ εu .

6.1. “zooming-out” stage.

Proof. The detailed proof is shown in Appendix A.

5. Asymptotic stability conditions for the closed-loop system

In this stage, system (1) with controller (2) is set as open-loop form (11) for integer k ∈ [1, . . . , kν − 1], and as closed-loop form (9) for integer k ∈ [kv , ∞), where kν is defined below. Let μ(0 ) = 0, and increase μ fast enough to dominate the rate of eA p hk  and Ac , such as μk = max{eA p hk k , Ac k }, there will be a k ∈ N satisfying yk /μk  ≤ M − 3 and uk /μk  ≤ M − 3 , which imply qμk (yk ) ≤ Mμk − 2 μk and qμk (uk ) ≤ Mμk − 2 μk . Hence, we can define



k0 = min k ≥ 1 : qμk (yk ) ≤ Mμk − 2 μk , qμk (uk )



Lemma 1. If quantization error Ek satisfies limk→∞ Ek = 0, then under periodic protocol (5), asymptotic stability of system (1) with (2) and (4), with (hk , τk ) ∈ , k ∈ N, is equivalent to the one of system (9). Proof. The proof can be seen in Appendix B. Remark 3. If quantization error tends to zero, then the asymptotic stability of system (1) with (2) and (4), with (hk , τk ) ∈ , k ∈ N, with periodic protocol (5), can be guaranteed by proving the asymptotic stability of (12), with periodic protocol (5), α k ∈ , and ¯, ˆ , k ∈ N, using above Lemma and the fact that (12) is ¯ ∈ ˆ ∈ a tight overapproximation of (9). Then in the following part, we will focus on the stability analysis of polytopic system (12). As the main result of this paper, the next Theorem tells us that if the quantizer variables M, and μ are selected suitably, the asymptotic stability of the polytopic system (12) can always be guaranteed. Theorem 2. Taking saturation value M of sensitivity to be large enough such that

   λmax (P ) φ1 + ω M ≥ max Dc  , , λmin (P ) a where P is a positive-definite matrix, ω is an arbitrary positive constant, a and φ 1 are defined detailed in the following part. If there ˜ }, and matrices Ri,l ∈ R exist positive-definite matrices Pi , i ∈ {1, . . . , N with R := {diag(r1 I1 , . . . , rK IK , rK+1 I1 , . . . , r2K IK , r2K+1 I1 , . . . , r3K IK ) ∈ R3(n p +nc )×3(n p +nc ) |ri > 0} and Ii an identity matrix of size ni , satisfying

≤ M μk − 2 μk . Obviously,

the

integer

k0

satisfies

2 μk 0 ,

 q μ k ( u k 0 )  ≤ M μk 0 − 2 μk 0 , 0 M μk 0 − μ k 0 ,  u k 0  ≤ M μ k 0 − μ k 0 .

 q μ k ( y k 0 )  ≤ M μk 0 − 0 thus yk0  ≤

and

Let tkν is the first time instant on which yk or uk is saturated after k0 . Based on the definition of μk , the time instant tkν is everpresent. It is obvious that ykν −1 and ukν −1 are unsaturated and Ekyν −1  ≤ μkν −1 , Ekuν −1  ≤ μkν −1 . In the following, we will testify that yk and uk can set to be unsaturated at time instant tkν by designing the value of μkν suitably. In fact, open-loop system (11) tells us

x¯kv  ≤ A˜ oσk ≤

A˜ oσk

v−1

,hk

v−1

,hk

v−1

x¯kv −1  + B˜oσkv −1 ,hkv −1 Ekv −1 

v−1

x¯kv −1  + B˜oσkv −1 ,hkv −1  2 μkv −1

√

:= A.

(23)

Moreover, if μkν is chosen large enough such that



  λmin (P ) Mμkν ≥ max 1, C p , Cc  + Dc  + Dc C p  A λmax (P ) (24)

for any given positive-definite matrix P, combined with (23) gives that



x¯kν  ≤

λmin (P ) Maμkν λmax (P )

(25) −1

with a = {max{1, C p , Cc  + Dc  + Dc C p }} called proportionality coefficient of saturation value, which means that x¯kν belongs to the ellipsoid

R1 := {x¯k : x¯Tk P x¯k ≤ λmin (P )(Ma )2 μ2kν }.

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Next, we will illustrate that R1 is an invariant region, that is, if we set μk := μkν for any k ≥ kν , then x¯k will not leave R1 . It follows from x¯kν ∈ R1 that x¯kν  ≤ Maμkν , indicating

ykv  ≤ Cp x¯kv  ≤ Mμkν , ukv  ≤ Cc xckv  + Dc yˆ(tkv ) ≤ (Cc  + Dc  + Dc C p  )A ≤ M μk ν .

(26)

ν

ν

√ Ekν  ≤ 2 μkν .

(27)

To obtain the values of yˆ and uˆ in closed-loop discrete form system, we define

yˆ(tkv −1 ) := qμkv −1 (ykv −1 ),

uˆ (tkv −1 ) := qμkv −1 (ukv −1 ).

(28)

Select positive-definite matrices Pi satisfying PN˜ +1 := P1 , λmin (Pi ) ˜ }. Especially, let ≥ λmin (P) and λmax (Pi ) ≤ λmax (P) for i ∈ {1, . . . , N Pi+ jN˜ = P when i = σkν for any j ∈ {N ∪ {0}}. For system (12), define Lyapunov function as V (x¯k ) = x¯Tk Pi x¯k , with i = σk , which satisfies

V (x¯kν ) =

x¯Tkν Pi x¯kν

≤ λmin (Pi )(Ma )

2

μ

2 kν

(29)

with i = σkν by taking the definition of R1 into consideration. Let i = σk , one sees that

V (x¯k+1 ) = x¯Tk+1 Pi+1 x¯k+1 − x¯Tk Pi x¯k L 

T l ¯ ¯ ¯ ¯ ≤ x¯k αk Ai,l + B kCi T Pi+1

×

l=1 L



α

l ¯ k Ai,l

¯ kC¯i + B¯

(30)

max

max



T

Pi+1

l=1

L

l=1



αkl

l=1

Pi ¯ C¯i ) Pi+1 (A¯ i,l + B¯

  αkl Bˆi,l + Dˆ ˆ k Fˆi  .

(A¯ i,l + B¯ ¯ C¯i )T Pi+1



Pi+1

> 0.

(31)

A necessary and sufficient condition of (31) is that

Pi ¯ C¯i ) Pi+1 (A¯ i,l + B¯

(A¯ i,l + B¯ ¯ C¯i ) Pi+1 T

Pi+1



> 0,

(32)

I ¯ C¯i A¯ i,l + B¯

T

−Pi 0

0 Pi+1





I <0 ¯ C¯i A¯ i,l + B¯

I A¯ i,l

T

0 B¯

−Pi 0

0 Pi+1



I A¯ i,l



$

B¯ > 0.

(34)

Qi,l,α , ¯ = Pi −

L



α

l ¯ k Ai,l

¯ C¯i + B¯



T

L

Pi+1

l=1



α

l ¯ k Ai,l

¯ C¯i + B¯

> 0,

l=1

which implies

 L   L   l 

l ¯   ¯ i,l + B¯ ¯i T Pi+1 ¯ ¯ ¯ ¯ A A α α + − P C B C i i k k i,l  l=1

l=1

=  − Qi,l,α , ¯  ≤ λmax (−Qi,l,α , ¯ ) = −λmin (Qi,l,α , ¯ ) < 0

(35)

¯ , i ∈ {1, . . . , N} and l ∈ {1, . . . , L}. ¯ ∈ for any α ∈ , Now, we obtain using (30) and (35) that, when i = σkν ,

V (x¯k+1 ) = x¯Tk+1 Pi+1 x¯k+1 − x¯Tk Pi x¯k

√ ≤ −βx¯k 2 + ξ1 x¯k  2 μkν + ξ2 2 2 μ2kν

= −β (x¯k  − φ1 μkν )(x¯k  − φ2 μkν )



with 2 ( ξ1 +

%



β := minα ∈, ¯ ∈¯ ,i∈{1,...,N},l∈{1,...,L} λmin (Qi,l,α , ¯ ) , ξ12 +βξ2 )

√

and φ2 :=

%

2 ( ξ1 −

ξ12 +βξ2 )

(36)

φ1 :=

β

based on (27). The last

λmax (P ) (φ + ω ) (Ma )−1 λmin (P ) 1

(37)



0 B¯

for some fixed ω > 0. Set μk = μkν for any k ≥ kν and take M of to be large enough such that  < 1, we can derive that R1 ⊃ B is an invariant region, that is, x¯k will not leave R1 for any k ≥ kν .

λmin (P )(Ma )2 − λmax (P )φ12 2 β 2 ω ( φ 1 − φ2 + ω )

and η = η ˜ , then η ∈ N by virtue of  < 1. Especially, if we set μk = μkν for any tk ∈ [tkν , tkν +η ), then x¯Tk Px¯k ≤ λmin (P )(Ma )2 μ2kν , and thus x¯k  ≤ Maμkν holds based on the fact that R1 is an invariant region. The unsaturation of yk and uk can be obtained for any tk ∈ [tkν , tkν +η ) similar to the analysis of (26). After applying √ Ek  ≤ 2 μk , we get

V (x¯k+1 ) = x¯Tk+1 Pi+1 x¯k+1 − x¯Tk Pi x¯k ≤ −β (x¯k  − φ1 μkν )(x¯k  − φ2 μkν )

(38)

˜ }. Under the basis of above for any tk ∈ [tkν , tkν +η ) and i ∈ {1, . . . , N inequality, we will certify that

(39)

holds with i = σkν . In order for (39) to hold, defining

(33)

¯ , i ∈ {1, . . . , N} and l ∈ {1, . . . , L}. Based on Lemma IV.1 ¯ ∈ for all in [26], a sufficient condition to ensure (33) is





x¯Tkν +η Pi+η x¯kν +η ≤ λmax (P )(φ1 + ω )2 2 μ2kν

¯ C¯i )T Pi+1 (A¯ i,l + B¯ ¯ C¯i ) > 0, and thus yielding Pi − (A¯ i,l + B¯







η˜ :=

  ¯ C¯i )T Pi+1 ( Ll=1 α l A¯ i,l + B¯ ¯ C¯i ) − Pi < 0 Consider that ( Ll=1 αkl A¯ i,l + B¯ k is equivalent to L

I <0 0

At the beginning of zooming-in stage, let us now introduce constants

l=1

αkl Bˆi,l + Dˆ ˆ k Fˆi



0 C¯i

6.2. “zooming-in” stage.

 L T L    l

l  ¯ ˆ ˆ ˆ ¯ ¯ × αk Bi,l + D ˆ k Fi Pi+1 αk Ai,l + B ¯ kCi  ,  L 



Using Schur complement, we can rewrite the condition (34) as (22). So far, it sees that if (22) is satisfied, we can define Qi,l,α , ¯ as

=

¯ , ˆ ,i∈{1,...,N},l∈{1,...,L} ˆ ∈ α ∈, ¯ ∈

× 

0 Ri,l

# A¯ T 0 − i,l Pi+1 A¯ i,l T ¯ Ri,l B

Pi − C¯iT Ri,l C¯i 0



− Pi x¯k

where

ˆ ,i∈{1,...,N},l∈{1,...,L} α ∈, ˆ ∈

−Ri,l 0

expression is negative outside the ball B = {x¯k : x¯k  ≤ φ1 μkν }. Define the scaling factor  by the formula



l=1

T

I 0

with Ri,l ∈ R, which is equal to

β

+ ξ1 x¯k Ek  + ξ2 Ek 2 ,

ξ2 : =

0 + ¯ Ci

√

l=1

ξ1 : =





Up to this point, the unsaturation of yk and uk at time instant tkν is verified, which means that quantization error satisfies Eky  ≤ μkν and Eku  ≤ μkν , yielding



[m5G;July 13, 2017;9:28]

˜ 2 := {x¯k : x¯T Pi x¯k ≤ λmax (P )(φ1 + ω )2 2 μ2 } R k kν ˜ 2 ⊃ B and thus R ˜ 2 is an invariant region for any with i = σk , then R tk ∈ [tkν , tkν +η ). Next, we will use proof by contradiction to testify (39). In fact, if (39) dose not hold, then it must be

x¯Tkν +η Pi+η x¯kν +η > λmax (P )(φ1 + ω )2 2 μ2kν ,

(40)

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and thus x¯k  > (φ1 + ω ) μkν

for any tk ∈ [tkν , tkν +η ) based ˜ 2 is an on λmax (Pi ) ≤ λmax (P ), ∀i ∈ {1, . . . , N} and the fact that R invariant region. Using (38) with i = σkν , we can also derive

V (x¯kν +η−1 ) = x¯Tkν +η Pi+η x¯kν +η − x¯Tkν +η−1 Pi+η−1 x¯kν +η−1

≤ −β (x¯kν +η−1  − φ1 μkν )(x¯kν +η−1  − φ2 μkν )

< −βω (φ1 − φ2 + ω ) 2 μ2kν . Similarly, for any j ∈ {1, . . . , η}, we obtain

V (x¯kν +η− j ) = x¯Tkν +η− j+1 Pi+η− j+1 x¯kν +η− j+1 − x¯Tkν +η− j Pi+η− j x¯kν +η− j < −βω (φ1 − φ2 + ω ) 2 μ2kν , which implies

x¯Tkν +η Pi+η x¯kν +η − x¯Tkν P x¯kν < −βω (φ1 − φ2 + ω ) 2 μ2kν η

Fig. 2. Selections of pairs and triangles.

< −βω (φ1 − φ2 + ω ) 2 μ2kν η˜ =

λmax (P )φ12 2 μ2kν − λmin (P )(Ma )2 μ2kν .

However, with i = σkν , the following result can be easily derived from (40) and the definition of R1 :

x¯Tkν +η Pi+η x¯kν +η − x¯Tkν P x¯kν >

λmax (P )(φ1 + ω )2 2 μ2kν

⎡

1.380 −0.581 ⎢ p x˙ (t ) = ⎣ 1.067 0.048

−0.208 −4.290 4.273 4.273

0 ⎢5.679 +⎣ 1.136 1.136

0 0 −3.146 0

⎡

− λmin (P )(Ma )2 μ2kν >

λmax (P )φ12 2 μ2kν − λmin (P )(Ma )2 μ2kν .

The above two inequalities are contradict with each other, thus we get

x¯Tkν +η Pi+η x¯kν +η ≤ λmax (P )(φ + ω )2 2 μ2kν = λmin (P )(Ma )2 (μkν )2



x¯kν +η ∈ R2 := {x¯k : x¯Tk Pi x¯k ≤ λmin (P )(Ma )2 (μkν )2 }. Let μk = μkν for any tk ∈ [tkν +η , tkν +2η ), similar analysis results in

x¯Tkν +2η Pi+2η x¯kν +2η ≤ λmin (P )(Ma )2 (2 μkν )2 with i = σkν . Similarly, if we set μk =  j−1 μkν for any tk ∈ [tkν +( j−1 )η , tkν + jη ), where j ∈ N, one has x¯Tk + jη Pi+ jη x¯kν + jη ≤

x¯kν + jη  < Ma j μkν := Maμkν + jη

y(t ) =

1 0

0 2

1 0

⎤

⎤

−5.676 0.675 ⎥ p x (t ) 5.893 ⎦ −2.104

⎥ˆ ⎦u(t ),



−1 p x (t ). 0

The discrete-time controller is designed as

with i = σkν , indicating

λmin (P )(Ma )2 ( j μkν )2 , implying



6.715 0 −6.654 1.343

ν

(41)

by λmin (Pi ) ≥ λmin (P ), ∀i ∈ {1, . . . , N}. At the same time, the unsaturation of yk and uk can be obtained for any [tkν +( j−1 )η , tkν + jη ) similar to the analysis of (26). Hence, if j → ∞, implies tk = tkν + jη → ∞, one has μkν + jη → 0 based on  < 1, and thus limtk →∞ x¯k = 0. Hereto, we have proved the asymptotic stability of polytopic system (12). Moreover, similar to the analysis of (26), we have that the quantization error Ek satisfies limk→∞ Ek = 0, which guarantees the asymptotic stability of system (1) with (2) and (4) by Lemma 3. Remark 5. Proportionality coefficient of saturation value, which is introduced to deal with the complex coupling relationship between system states and quantized variables, plays an important role in the whole derivation process. The introduction of this variable is one innovation of this paper. 7. Illustrative example This section provides simulation example to illustrate the theoretical analysis. A well-known benchmark example, see, e.g., [26] and [35–39], consisting of a model of a batch reactor, is given by

xck+1 = uk =



0.9048 0 −3 0







0 0 xc + 0.9048 k 0.0095



0 c 0 x + 1 k 5





0.0095 yˆk , 0

−2 yˆ(tk ). 0

Remark 6. The output matrix Cp defined here is different from the p one in [26] and [35–39] where y2 (t) is equal to x2 (t ) rather than p p 2x2 (t ). In fact, if x2 (t ) can be obtained, we can always get the double value of it by adding an amplifier at the suitable place. The controller defined above is also different from the one in the existing literature. All the changes here are to meet the requirements of Theorem 2 and increase the convergence rate of the closed-loop system states. The whole simulation process can be divided into six steps: Step 1: We assume 1 = diag{1, 0, 1, 0}, 2 = diag{0, 1, 0, 1}, τ = 0, τ¯ = 0.01, hnom = 0.01, h = hnom − 0.005, h¯ = hnom + 0.005 ˜ , τ˜ ), l ∈ {1, . . . , 6} and and a desired εu = 0.02 > 0. The pairs (h l l triangles Sm , m ∈ {1, . . . , 5} are shown in Fig. 2. Direct calculations give ε = 0.064 > εu based on (17), which means that a new pair (h˜ 7 , τ˜7 ) should be added. ˜ 7 , τ˜7 ) is selected as (hnom , τ ) according Step 2: The new pair (h to (19), and thus S2 is divided into S2 and S7 . In this case, we get ε = 0.018 which guarantees the tightness with εu = 0.02. p Step 3: If the initial states are selected as x0 = [12, 30, 42, 1]T , xc0 = [24.5, 10]T and transmission intervals from time instant 1 to 15 as h = [0.0102, 0.0077, 0.0053, 0.0146, 0.010 0 0.0146, 0.0 052, 0.0 074, 0.0 054, 0.0144, 0.0143, 0.0 051, 0.0 098, 0.0063, 0.0141], then under the open-loop form (10), k0 = 10 and kv = 12 can be gotten from Fig. 3. Step 4: Solving LMI (22) gives λmax (P1 ) = 5.7166, λmin (P1 ) = 0.1130, λmax (P2 ) = 5.8371 and λmin (P2 ) = 0.0459 which satisfy the conditions on Theorem 2. By setting P2 = P, it gives a = 0.0556.

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Fig. 5. State trajectories of controller. Fig. 3. Selections of k0 and kv .

Fig. 6. State trajectories of network-induced errors. Fig. 4. State trajectories of plant.

Step 5: Given ω = 103 and = 10−4 , direct calculations tell us β = 0.0064, ξ1 = 1.8456, ξ2 = 3.8815, φ1 = 820.6 and φ2 = −1.4844, thus we can select M satisfying





50 = M ≥ max

 D c  ,

 λmax (P ) φ1 + ω = 37. λmin (P ) a

Moreover, according to  = 0.7388 and A = 58.6395, μkν can be selected as

238 =

 μkν ≥ max 1, Cp , Cc  + Dc    λmax (P ) + Dc C p  A /M = 237.9681. λmin (P )

Step 6: State trajectories of system (9) are shown in Figs. 4–6, which illustrate the asymptotic stability of (9). The norm of system states is shown in Fig. 7 to explain the validity of (41). ¯ and Remark 7. Randomness of α˜ j , j ∈ {1, 2, 3}, α l , l ∈ {1, . . . , 7}, ˆ results in the non-uniqueness of above variables calculated. Remark 8. In the simulation, we found that (1) the pair added E

E

method proposed here can decrease the values of δiA , δi h , δi h−τ and ε more quickly than the one in [26], which keeps the tightness of overapproximation at a faster rate; (2) the sufficient conditions

Fig. 7. The norm of system states.

proposed in this paper are valid to ensure the asymptotic stability of NCSs affected by data quantization, time-varying transmission intervals, time-varying transmission delays and communication constraints; (3) the variable called proportionality coefficient of saturation value introduced can deal with the complex coupling relationship between system states and quantized variables. These

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Appendix A Proof Theorem 1: In view of the definitions of overapproximation and tightness, we only need to verify that inclusion relations (13) and (14) hold. Since that the selection method of new pair in step (6) does not influence the theorem verification, we can ensure (13a) and (14a) the same as Theorem III.2 in [26]. To verify (13b) and (14b), the following equation is considered:

B˜σ ,h,τ −

L

l=1



Fig. 8. State trajectories of plant without quantization.

are contributions of this paper and can be verified through the simulation method. Remark 9. If the benchmark model is selected as the one in [26], it indeed can not find suitable positive-definite matrix Pi and quantizer parameters satisfying (22), λmin (Pi ) ≥ λmin (P) and λmax (Pi ) ≤ λmax (P) simultaneously, which means that Theorem 2 is invalid in this case. However, when the system is not affected by quantization, the sufficient conditions ensuring asymptotic stability is only relating to (22) [26], it can be solved by LMI toolbox. Therefore, the asymptotic stability conditions with quantization obtained in this paper is indeed relatively conservative compared with the system without quantization, this is logical. Remark 10. For the benchmark model described in this section, sufficient conditions given in Theorem 2 can be guaranteed. However, when the system is irrelevant to quantization, the system states can be shown in Fig. 8, from which we see that the system is asymptotic stability and the convergence speed is extremely higher than the one of the system with quantization. Therefore, improving the convergence speed of the system with quantization is one of our future research directions. 8. Conclusion

α l Bˆσ ,l

 h−τ Bσ − Ll=1 α l h˜ l −τ˜l Bσ  = (D−1 − C h−τ B )σ − Ll=1 α l (D−1 − C h˜ l −τ˜l B )σ

 (h−τ − Ll=1 α l h˜ l −τ˜l )Bσ L = −C (h−τ − l=1 α l h˜ −τ˜ )Bσ l l



 h−τ − Ll=1 α l h˜ l −τ˜l 0 I 0 L = 0 −C 0 h−τ − l=1 α l h˜ l −τ˜l

Bσ × Bσ 

L  l h−τ s 0 T 0 ˜ l −τ˜l e ds l=1 α h =  L l h−τ s 0 −CT 0 ˜ l −τ˜l e ds l=1 α h

T −1 B × −1 σ T Bσ ˘ α ,h,τ Fˆσ . := D¯ Similar to the proof of Theorem III.2 in [26], by defining

 ˘m = 



˘ E1h−τ , . . . , ˘ EKh−τ , ˘ E1h−τ , . . . , ˘ EKh−τ diag

 Eh−τ  ˘  i

 ≤

3 j=1

max

α˜ j =1,α˜ j ≥0

Eh−τ δ˜i,m, α˜ , i ∈ {1, . . . , K }

and E E E E U˘ := diag(δ1h−τ I1 , . . . , δKh−τ IK , δ1h−τ I1 , . . . , δKh−τ IK ),

This paper focuses on the stability analysis of NCSs with data quantization, time-varying transmission intervals, time-varying transmission delays and communication constraints. Under periodic protocol and uniform quantization strategy, sufficient conditions are derived to ensure the asymptotic stability of NCSs. Due to the convergence speed is slow under our method, how to improve the convergence speed of the system affected by quantization is one of our future research directions. Moreover, since the data packet loss is also the network-induced communication imperfection which influence the closed-loop system performance, our future work will include the discussion of the stability of NCSs with data packet loss simultaneously.

˘ m ⊆ U˘  ˆ with  ˆ defined as in (21). Define Dˆ as ˘ α ,h,τ ∈  we get

Acknowledgments

Proof Lemma 1: If system (1) with (2) and (4) is asymptotic stability, it is obvious that the sampling system (9) is also asymptotic stability. Conversely, assume that system (9) is asymptotic stability, it only need to show that the intersample behavior of system (1) with (2) and (4) is bounded relating to a linear function of the system states at the transmission instants [26]. To this end, we introduce an additional variable t˜ := t − tk for any t ∈ (tk , tk+1 ], solving system (1) on t˜ ∈ (0, τk ] yields

This work was supported by National Natural Science Foundation of China (61403125, 61225015, 61333005, 61304258, 61503123), Fundamental Research Funds for the Henan Provincial Colleges and Universities in Henan University of Technology (2015RCJH15), and Natural Science Foundation of Henan Province Education Department (15A413012).

E E E E Dˆ = D¯ × diag(δ1h−τ I1 , . . . , δKh−τ IK , δ1h−τ I1 , . . . , δKh−τ IK ),

it is obtained that (13b) holds. Finally, tightness (14b) can be ensured due to the fact that produce terminates not until ε ≤ ε μ .

Appendix B

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J. Yan et al. / Neurocomputing 000 (2017) 1–11

x p (tk + t˜) = eA pt˜x p (tk ) + =e

x (tk ) +

A p t˜ p

 

t˜ 0 t˜ 0

eA p (t˜−s ) dsB p uˆ (tk ) eA p s dsB p (DcC p xkp + Cc xck + Dc eyk + euk ),

which deduces, for t˜ ∈ (τk , hk ],

x p (tk + t˜) = eA pt˜x p (tk ) + = eA pt˜x p (tk ) +  −

t˜−τk 0

 τk 0



0

t˜

eA p (t˜−s ) dsB p uˆ (tk ) +



t˜

τk

eA p (t˜−s ) dsB p lim uˆ (t ) t ↓rk

eA p s dsB p (DcC p xkp + Cc xck + Dc eyk + euk )

eA p s dsB p σuk (euk − Eku ).

Combined above two inequalities, the bound on x p (tk + t˜) with t˜ ∈ (0, hk ] can be achieved as

x p (tk + t˜) ≤ eA pt˜x p (tk ) + 



t˜ 0

eA p s dsB p (DcC p xkp 

+ Cc xck  + Dc eyk  + euk  )  t˜−τk + eA p s dsB p σuk (euk  + Eku  ). 0

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J. Yan et al. / Neurocomputing 000 (2017) 1–11 Yuanqing Xia received the M.S. degree in fundamental mathematics from Anhui University, Hefei, China, in 1998, and the Ph.D. degree in control theory and control engineering from the Beijing University of Aeronautics and Astronautics, Beijing, China, in 2001. He is currently a Professor with the School of Automation, Beijing Institute of Technology, Beijing. His current research interests include the fields of networked control systems, robust control, active disturbance rejection control, and flight control.

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Chenglin Wen received the B.S. and M.S. degrees in Mathematics from Henan University, Kaifeng, China, and Zhengzhou University, Zhengzhou, China, and the Ph.D. degree from Northwestern Polytechnical University, Xian, China, in 1986, 1996, and 1999, respectively. He is now a Chair Professor in the College of Electrical Engineering, Henan University of Technology, Zhengzhou, China. His current research interests include multi-sensor networked information fusion theory, multi-target tracking, fault diagnosis of complex systems and devices, reliability assessment and health control, recognition, and tracking of hypersonic vehicle. He is a Committee Member of Intelligent Automation Committee and Process Fault Diagnosis and Security Committee of Chinese Association of Automation.

Please cite this article as: J. Yan et al., Quantized control for NCSs with communication constraints, Neurocomputing (2017), http://dx.doi.org/10.1016/j.neucom.2017.06.040