Robust H∞ filtering for Markov jump systems with mode-dependent quantized output and partly unknown transition probabilities

Accepted Manuscript

Robust H Filtering for Markov Jump Systems with Mode-dependent Quantized Output and Partly Unknown Transition Probabilities Deyin Yao, Renquan Lu, Yong Xu, Lijie Wang PII: DOI: Reference:

S0165-1684(17)30066-X 10.1016/j.sigpro.2017.02.010 SIGPRO 6405

To appear in:

Signal Processing

Received date: Revised date: Accepted date:

17 October 2016 22 December 2016 15 February 2017

Please cite this article as: Deyin Yao, Renquan Lu, Yong Xu, Lijie Wang, Robust H Filtering for Markov Jump Systems with Mode-dependent Quantized Output and Partly Unknown Transition Probabilities, Signal Processing (2017), doi: 10.1016/j.sigpro.2017.02.010

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

1

H IGHLIGHTS •

This paper considers the H∞ filtering for MJSs subject to mode-dependent quantized output and unknown transition probabilities, which are more practical in the realistic systems. Hence, the obtained results of our paper are more robust. Compared with those existing studies for MJSs, our study considers more complex conditions: (i) the output quantization signal yq (t) will make system

•

CR IP T

•

extremely complex to achieve the error dynamics. However, in our paper, the quantization error of the output is transformed into a bounded nonlinearity via a model transformation; (ii) the detail knowledge of transition probabilities is tough to obtain. Hence, more general stochastic stability

AC

CE

PT

ED

M

AN US

criteria can be derived through the research of our article.

February 23, 2017

DRAFT

ACCEPTED MANUSCRIPT

2

Robust H∞ Filtering for Markov Jump Systems with Mode-dependent Quantized Output and

CR IP T

Partly Unknown Transition Probabilities Deyin Yao, Renquan Lu, Yong Xu and Lijie Wang

AN US

Abstract

This paper addresses the problem of robust H∞ filtering design for uncertain Markov jump systems (MJSs) with mode-dependent quantized output and partly unknown transition probabilities. The data transmission and exchange are completed over a digital communication channel such that the system outputs need to be quantized before transmission. Attention is mainly concentrated on tackling the parameter uncertainty, unknown transition probabilities and output quantization. The parameter uncer-

M

tainty considered in this paper is assumed to be norm-bounded, and the exact information of transition probabilities matrix is partly unavailable. The quantization error of the output is transformed into a

ED

bounded nonlinearity via a model transformation. A mode-dependent filter is designed to ensure that the filtering error system is stochastically stable and has an H∞ noise attenuation performance index. Finally, simulation results are provided to illustrate the validity of the proposed results.

PT

Keywords: Markov jump systems (MJSs); Parameter uncertainty; H∞ filtering design; Quantization.

CE

This work is supported by the Funds for China National Funds for Distinguished Young Scientists (61425009), Guangdong Province Higher Vocational Colleges & Schools Pearl River Scholar approved in 2015, the Zhejiang Provincial Natural Science Foundation of China under Grant (R1100716), the China National 863 Technology Projects under Grant (2015BAF32B03-05),

AC

and the National Natural Science Foundation of China under Grants (61503106, 61374005, U1611262). D. Yao, R. Lu, and Y. Xu are with the School of Automation, Guangdong University of Technology, Guangzhou 510006,

Guangdong, China. Email: [email protected], [email protected] (Corresponding author) and [email protected] L. Wang is with the School of Mathematics and Physics, Bohai University, Jinzhou 121013, Liaoning, China. Email:

[email protected]

DRAFT

ACCEPTED MANUSCRIPT

3

I. I NTRODUCTION Markov jump systems (MJSs), recognized as a special type of Markov switching systems [1]–[7], have been extensively investigated during the past decades. Many remarkable theoretical results on the stability analysis, stabilization, and filtering problems for MJSs have been presented [8]–[17]. To mention a few, the stabilization problem for a type of stochastic MJSs was established in [9]. The problem of

CR IP T

Kalman filtering for a class of uncertain linear continuous-time MJSs was discussed in [15]. The system mode switching [18] is dominated via a Markov process, and in the jumping process, the mode switching is related to transition probabilities that determine the system behavior. The transition probabilities of MJSs are assumed to be known in previous work [19], [20]. However, the exact information of transition probabilities is difficult to achieve, especially in the network environment. Consequently, many studies on

AN US

MJSs with partly unknown transition probabilities were reported [21]–[24]. For instance, the H∞ filtering problem for a class of discrete-time Markov jump linear systems (MJLSs) with partly unknown transition probabilities was established in [23]. The authors investigated the problems for a class of continuous-time and discrete-time MJLSs subject to partly unknown transition probabilities in [22]. Networked control systems (NCSs) [25]–[34] have attracted considerable attention due to the merits of NCSs such as high reliability, low cost and simple installation. In NCSs, for filter design and feedback

M

control, output measurements are transmitted via a digital communication channel such that the system outputs must be quantized before transmission [35]–[42]. Recently, many theoretical results have been

ED

studied for NCSs with quantization [27], [28], [38], [43]–[49]. To mention a few, the problem of reset quantized state control for continuous-time linear system was studied in [28]. The problem of output

PT

feedback control for NCSs with limited communication capacity was established in [45]. Hence, the quantized system outputs will also appear in MJSs, which will give us some challenges to study the

CE

problem of filtering design for MJSs with quantized outputs. The system state is often unavailable in the control systems due to the restriction of the measured approach. Such an issue leads to many influential achievements on estimation and filtering design for

AC

a series of MJSs published in [23], [50]–[54]. The mode-independent and mode-dependent filter design approaches have been established, which can be found in [19], [23], [55]. However, the robust H∞

filtering design problem of MJSs subject to parameter uncertainty, mode-dependent quantized output and partly unknown transition probabilities is still not studied, which motivates us for this study. In this paper, the H∞ filtering design problem is considered for uncertain MJSs with partly unknown transition probabilities and mode-dependent quantized output. The elements of transition probabilities

DRAFT

ACCEPTED MANUSCRIPT

4

are partially available and the parameter uncertainty satisfies the norm-bounded form. A mode-dependent full-order filter is constructed to ensure that the filtering error system is stochastically stable and has an H∞ performance index. Finally, two examples are presented to exploit the effectiveness and applicability

of the proposed theoretical results. The remainder of this paper is arranged as follows. The description and some preliminaries of uncertain

CR IP T

MJSs are provided in Section II. Section III and Section IV develop the output measurement quantizer and the full-order filter design, respectively. The existence conditions of the synthesized filter are provided in Section V. Simulation results are given to testify the validity of the proposed approach in Section VI. Notations: The superscript “T ” represents the matrix transposition, and Rn shows the n-dimensional Euclidean space. The matrix X > 0 denotes that X is real symmetric and positive definite. λmin (P ) and λmax (P ) are the minimum and maximum eigenvalues of symmetric matrix P , respectively. Ip is

AN US

the identity matrix with dimension p. (Ω, F, P ) represents the probability space. Ω, F and P denote the

sample space, σ -algebra of subsets of the sample space and probability measure on F , respectively. The

notation diag{·} represents a diagonal matrix. E{·} and Pr (·) denote the mathematical expectation and transition probability, respectively. L2 [0, ∞) denotes the space of square integral vector function. The

M

symbol “∗” indicates a symmetric term.

II. S YSTEM DESCRIPTION AND PRELIMINARIES

ED

Let {rt , t ≥ 0} be a finite-state Markov jumping process taking discrete values in state space

S = {1, 2, . . . , N } . The evolution of Markov process rt is determined by transition probabilities Π =

CE

PT

(πij )N ×N (i, j = 1, 2, . . . , N ) denoted as follows:   π υ + o (υ) , i 6= j, ij Pr (rt+υ = j | rt = i) = (1)  1 + π υ + o (υ) , i = j, ii P where πij > 0 when i 6= j ; υ > 0 and limυ→0 o(υ)/υ = 0, πii = − N j=1, j6=i πij for each mode i.

AC

Consider the following MJSs defined on probability space (Ω, F, P ):    x˙ (t) = (A (rt ) + 4A (rt , t)) x (t) + (B (rt ) + 4B (rt , t)) w (t) ,      y (t) = C (r ) x (t) + D (r ) w (t) , t

t

  yq (t) = qrt (C (rt ) x (t) + D (rt ) w (t)) ,      z (t) = L (r ) x (t) , t

(2)

where x (t) ∈ Rn , y (t) ∈ Rp , yq (t) ∈ Rp and w (t) ∈ Rm are respectively the state vector, the measurement, the mode-dependent quantized output and the disturbance belonging to L2 [0, ∞). z (t) ∈ Rr

DRAFT

ACCEPTED MANUSCRIPT

5

denotes the signal to be estimated. A (rt ) ∈ Rn×n , B (rt ) ∈ Rn×m , C (rt ) ∈ Rp×n , D (rt ) ∈ Rp×m and

L (rt ) ∈ Rr×n are constant system matrices of associated dimensions. 4A (rt , t) and 4B (rt , t) are

parameter uncertainties. For rt = i ∈ S , system (2) can be expressed as    x˙ (t) = (Ai + 4Ai (t)) x (t) + (Bi + 4Bi (t)) w (t) ,      y (t) = C x (t) + D w (t) , i

  yq (t) = qi (Ci x (t) + Di w (t)) ,      z (t) = L x (t) , i

(3)

CR IP T

i

where A (rt ), 4A (rt , t), B (rt ), 4B (rt , t), C (rt ), D (rt ) and L (rt ) are respectively represented by Ai ,

4Ai (t), Bi , 4Bi (t), Ci , Di and Li .

Assume the system admissible uncertainties 4Ai (t) and 4Bi (t) satisfy

4Bi (t) = MBi FBi (t) N2i ,

AN US

4Ai (t) = MAi FAi (t) N1i ,

(4)

∀i ∈ S,

where MAi , MBi , N1i and N2i are constant matrices with compatible dimensions, and unknown timevarying matrix functions FAi (t) and FBi (t) satisfy T FAi (t) FAi (t) ≤ I,

T FBi (t) FBi (t) ≤ I,

(5)

∀i ∈ S.

M

Furthermore, we assume that some elements in matrix Π are unknown. For instance, the form of the

ED

transition probability matrix Π with three operation modes may be written as:   π11 ? ?     Π =  ? π22 ?  ,   π31 π32 π33

i Sκi , {j : πij is known} , Suκ , {j : πij is unknown} .

CE

P

PT

i with where “?” denotes the unknown elements. For ∀i ∈ S, define S = Sκi ∪ Suκ

Define πκi ,

j∈Sκi

πij in this paper.

The following definitions and lemmas are necessary for the main results of this paper.

AC

Definition 1: [56] Suppose that the stochastic positive functional candidate of system (3) is selected as

V (x (t) , i), which is twice differentiable on x (t). Then, define the infinitesimal operator LV (x (t) , i)

as

LV (x (t) , i) = lim + 4t→0

1 [E{V (x (t + 4t) , rt+4t ) | x (t) , rt = i} − V (x (t) , i)] . 4t

(6)

Definition 2: [57] For any r (t) = i ∈ S, and w (t) = 0, if there exist α ≥ 0 and β > 0 such that E{kx (t, x0 , r0 )k2 } ≤ α exp (−βt) E{kx0 k2 },

(7) DRAFT

ACCEPTED MANUSCRIPT

6

where x (t, x0 , r0 ) denotes the trivial solution of system (3), then the system (3) is exponentially stable in mean square sense. Lemma 1: [58] Assume that C2 (Rn ; R+ ) represents the family of non-negative functions on Rn , which is continuously twice differentiable, if there exist non-negative function V (x (t) , rt ) ∈ C2 (Rn ; R+ ) and positive scalars c1 , c2 and c3 for any r (t) = i ∈ S such that

LV (x (t) , i) ≤ −c3 kx (t)k2 ,

then the system (3) is exponentially stable in mean square sense.

CR IP T

c1 kx (t)k2 ≤ V (x (t) , i) ≤ c2 kx (t)k2 ,

(8)

Lemma 2: [57] For any real vectors x, y and Q > 0 with compatible dimensions, the following inequality holds: xT y + y T x ≤ xT Qx + y T Q−1 y.

(9)

AN US

Lemma 3: [14] Let γ > 0, for w (t) = 0, system (3) is exponentially stable in mean square, and for all nonzero w (t) ∈ L2 [0, ∞), the following inequality holds: Z ∞ Z ∞ 2 2 kz (t)k dt ≤ γ kw (t)k2 dt, 0

(10)

0

then system (3) is said to be stochastically stable with γ –disturbance attenuation performance.

M

III. O UTPUT M EASUREMENT Q UANTIZER

By extending the static logarithmic quantizer in [59], we can obtain the mode-dependent logarithmic

(j)

ED

quantizer, which is described as follows: h iT (1) (2) (p) qi (·) = qi (·) , qi (·) , · · · , qi (·) ,

i ∈ S,

(11)

j = 1, 2, · · · , p.

(12)

PT

where qi (·) are symmetric, i.e.,

CE

(j)

(j)

qi (yj (t)) = −qi (−yj (t)) , (j)

AC

For each i ∈ S, the set of quantized levels of qi (·) is denoted as   n  l o (i,j) (i,j) (i,j) (i,j) (i,j) νj = ±ηl |ηl = ρ(i,j) · η0 , l = ±1, ±2, · · · ∪ ±η0 ∪ {0} , 0 < ρ(i,j) < 1, η0 > 0,

(13)

(j)

where ρ(i,j) represents the quantizer density of the subquantizer qi (·) . For the mode-dependent loga(j)

rithmic quantizer, the corresponding qi (·) is defined as follows:   (i,j) i (i,j) (i,j) η η   ηl , if yj (t) ∈ 1+δl (i,j) , 1−δl (i,j)   (j) qi (yj (t)) = 0, if yj (t) = 0     −q (j) (−y (t)) , if y (t) < 0 i

j

(14)

j

DRAFT

ACCEPTED MANUSCRIPT

7

where δ (i,j) =

1−ρ(i,j) 1+ρ(i,j)

represent the quantizer parameters.

Remark 1: Note that the mode-dependent quantizer can be viewed as an extension of the static quantizer in [27], [28], [60], [61]. When S = {1} , the mode-dependent quantizer (14) becomes the traditional static

quantizer in [27], [28], [60], [61]. The mode-dependent quantizer in (14) is more useful for MJSs since the parameters ρ(i,j) are related to the structure of logarithmic quantizer. Then, the influence of the mode

CR IP T

jumping in the control plant (3) will be regulated by each subquantizer qi (yj (t)). In the following part, a model variation is provided to describe the output quantization qi (y (t)) as memoryless bounded nonlinearities. We define the matrices as: h i ∆i , diag δ (i,1) , δ (i,2) , · · · , δ (i,p) , It is not difficult to find 0 < ∆i < Ip .

i ∈ S.

AN US

We can denote the logarithmic quantizer in (14) as the following scalar sector condition:     (i,j) 1 − δ (i,j) yj2 (t) ≤ qj (yj (t)) · yj (t) ≤ 1 + δ (i,j) yj2 (t) , t ≥ 0, j = 1, 2, · · · , p.

The inequality (16) is equivalent to

(15)

(16)

[qi (y (t)) − (Ip − ∆i ) y (t)]T [qi (y (t)) − (Ip + ∆i ) y (t)] ≤ 0.

M

To design a filter for (3), we decompose the quantization function qi (·) into the following form: qi (y (t)) = (Ip − ∆i ) y (t) + qis (y (t)) ,

(17)

ED

where qis (y (t)) : Rp → Rp . It should be pointed out that the transformed nonlinearity qis (y (t)) belongs

PT

to the following set φs : n o φs , qis (y (t)) : Rp → Rp : qis (0) = 0, (qis (y (t)))T [qis (y (t)) − 2∆i y (t)] ≤ 0, y (t) ∈ Rp .

(18)

For simplicity, we define some matrices parameters as

CE

W1i , Ip − ∆i ,

W2i , Ip + ∆i ,

Wi , W2i − W1i = 2∆i ,

i ∈ S.

AC

Thus, (18) can be rewritten as n o φs , qis (y (t)) : Rp → Rp : qis (0) = 0, (qis (y (t)))T [qis (y (t)) − Wi y (t)] ≤ 0, y (t) ∈ Rp . Substituting (17) into (3) yields    x˙ (t) = (Ai + 4Ai (t)) x (t) + (Bi + 4Bi (t)) w (t) ,      y (t) = C x (t) + D w (t) , i

i

  yq (t) = W1i Ci x (t) + qis (y (t)) + W1i Di w (t) ,      z (t) = L x (t) , i

(19)

(20)

(21)

DRAFT

ACCEPTED MANUSCRIPT

8

Remark 2: As we know, the filtering error dynamics can be easily deduced according to the measured output y (t) in the conventional design of filtering. However, in this paper, it makes system (3) extremely complex to achieve the error dynamics in the existence of output quantization signal yq (t) . We decompose the quantization function qi (y (t)) into the form (17) to tackle the above difficulty.

CR IP T

IV. ROBUST Q UANTIZED F ILTER D ESIGN For system (3), the robust quantized filter of full order n is synthesized as   x ˆi yq (t) , ˆ˙ (t) = Aˆi x ˆ (t) + B  zˆ (t) = Cˆ x ˆ (t) , i

(22)

ˆi and Cˆi are filter parameter matrices to be designed for each where x ˆ (t) ∈ Rn and zˆ (t) ∈ Rr ; Aˆi , B

AN US

i ∈ S.

Define the estimation error ex (t) , x (t)− x ˆ (t) . Combining (21) with (22), the filtering error dynamics

are obtained as e˙ (t) =



˜i q s (y (t)) + B ¯i + 4B ¯i (t) w (t) , A¯i + 4A¯i (t) e (t) + B i

M

¯ i e (t) , ez (t) = L

where



MAi



MBi

MAi



 FAi (t)

PT

4A¯i (t) = 

MBi

CE

¯i (t) =  4B

T e (t) = xT (t) , eTx (t) ,

ED

ez (t) = z (t) − zˆ (t) ,





h

N1i 0

 FBi (t) N2i ,

i

,

¯i = L

h



A¯i = 

Ai

(23)

0



, ˆi W1i Ci Aˆi Ai − Aˆi − B     0 Bi ˜i =  ¯i =  , B , B ˆi ˆi W1i Di −B Bi − B

Li − Cˆi Cˆi

i

.

The aim of this paper is to establish a mode-dependent filter such that 1) the resulting filtering error

AC

system (23) is exponentially stable in mean square for w (t) = 0, and 2) under the zero initial condition, for any qi (·) ∈ [W1i , W2i ], kez (t)k < γ kw (t)k holds for all nonzero w (t) ∈ L2 [0, ∞). V. M AIN R ESULTS

The existence conditions of stochastic stability for the system (23) are given in the following theorem.

DRAFT

ACCEPTED MANUSCRIPT

9

Theorem 1: Given a scalar γ > 0, for each i ∈ S, if there exist matrices Xi > 0 ∈ Rn×n , Yi > 0 ∈

Xi Ai + ATi Xi + Xj ≥ 0, Qi + QTi + Yj ≥ 0,

i ∀j ∈ Suκ ,

Xi Ai + ATi Xi + Xj ≤ 0,

where



Ψ11i

∗

Ψ22i

∗

∗

∗

∗ 

j 6= i,

CiT WiT

Xi Bi

−Ri

Yi Bi − Ri W1i Di

−2I

Wi Di

M

TN −γ 2 I + (εi3 + εi4 ) N2i 2i   T XM Xi MBi Li − Cˆi 0 0  i Ai    ˆT  0 0 C Y M Y M i Ai i Bi  i , =      0 0 0 0 0   0 0 0 0 0 X
T = (1 + πκi ) Xi Ai + ATi Xi + (εi1 + εi2 ) N1i N1i + πij Xj + πii Xi ,

CE


Ψ22i = (1 + πκi ) Qi + QTi + 

   =    

AC ∆1i

j 6= i,

∗

(26) (27) (28) (29)



   ,   

ED

Ψ2i

T RT Ψ11i ATi Yi − QTi − CiT W1i i

PT

Ψ1i

   =    

j = i,

j = i,

i ∀j ∈ Suκ ,

i ∀j ∈ Suκ ,

(25)

AN US

Qi + QTi + Yj ≤ 0,

i ∀j ∈ Suκ ,

(24)

CR IP T

Rn×n , Qi , Ri and scalars εil (l = 1, 2, 3, 4) such that   Ψ1i Ψ2i  < 0, ∀i ∈ Sκi ,  ∗ Ψ3i   ∆1i ∆2i i  < 0, ∀i ∈ Suκ  , ∗ ∆3i

X

j∈Sκi , j6=i

πij Yj + πii Yi ,

j∈Sκi , j6=i

T RT ∆11i ATi Yi − QTi − CiT W1i i

CiT WiT

Xi Bi

∗

∆22i

−Ri

Yi Bi − Ri W1i Di

∗

∗

−2I

Wi Di

∗

∗

∗

TN −γ 2 I + (εi3 + εi4 ) N2i 2i



   ,   

DRAFT

ACCEPTED MANUSCRIPT

10

   =    

∆2i



Xi MAi Xi MBi 0

0

0

0

0

0



X

∆11i = 1 +

j∈Sκi , j6=i

∆22i = 1 +

j∈Sκi , j6=i



X

Li − Cˆi CˆiT

T

0 0



0



0

  Yi MAi Yi MBi  ,   0 0  0 0


T πij  Xi Ai + ATi Xi + (εi1 + εi2 ) N1i N1i + 


πij  Qi + QTi +

X

πij Yj ,

j∈Sκi , j6=i

Ψ3i = ∆3i = diag{−εi1 I, −εi3 I, −I, −εi2 I, −εi4 I},

Aˆi = Yi−1 Qi ,

πij Xj ,

j∈Sκi , j6=i

AN US

the filter parameter matrices are designed as

X

CR IP T



ˆi = Y −1 Ri , B i

i ∈ S.

Then, the system (23) is stochastically stable with γ –disturbance attenuation performance. Proof: Select the following stochastic Lyapunov-Krasovskii functional:

Xi

0

0

Yi

. When w (t) = 0, according to Definition 1, we have

LV1 (x, ex , i)

= 2e (t) P¯i e˙ (t) + eT (t) T

N X

πij P¯j e (t)

j=1

PT



= xT (t) Xi Ai + ATi Xi +

CE



(30)

M

where P¯i = 

V1 (x, ex , i) = eT (t) P¯i e (t) ,



ED



N X j=1





πij Xj  x (t) + eTx (t) Yi Aˆi + AˆTi Yi +



N X j=1

ˆi W1i Ci x (t) − 2eT (t) Yi B ˆi q s (y (t)) +2eTx (t) Yi Ai − Yi Aˆi − Yi B x i



πij Yj  ex (t)

AC

+2xT (t) Xi MAi FAi (t) N1i x (t) + 2eTx (t) Yi MAi FAi (t) N1i x (t)     N N X X ≤ xT (t) Xi Ai + ATi Xi + πij Xj  x (t) + eTx (t) Yi Aˆi + AˆTi Yi + πij Yj  ex (t)



j=1



j=1

ˆi W1i Ci x (t) − 2eT (t) Yi B ˆi q s (y (t)) +2eTx (t) Yi Ai − Yi Aˆi − Yi B x i

+2xT (t) Xi MAi FAi (t) N1i x (t) + 2eTx (t) Yi MAi FAi (t) N1i x (t) −2 (qis (y (t)))T (qis (y (t)) − 2∆i y (t)) .

(31)

DRAFT

ACCEPTED MANUSCRIPT

11

By Lemma 2, we have T T T T 2xT (t) Xi MAi FAi (t) N1i x (t) ≤ ε−1 i1 x (t) Xi MAi MAi Xi x (t) + εi1 x (t) N1i N1i x (t) ,

(32)

T T T T 2eTx (t) Yi MAi FAi (t) N1i x (t) ≤ ε−1 i2 ex (t) Yi MAi MAi Yi ex (t) + εi2 x (t) N1i N1i x (t) .

(33)

Substituting (32)-(33) into (31), we have

CR IP T

LV1 (x, ex , i) ≤ η T (t) Φi η (t) ,

where h



  Φi =  

,



TB ˆ T Yi C T W T Σ11i ATi Yi − AˆTi Yi − CiT W1i i i i

∗

Σ22i

∗

∗

Σ11i = Xi Ai +

Σ22i

iT

xT (t) eTx (t) (qis (y (t)))T

ATi Xi

+

 ˆi  , −Yi B  −2I

AN US

η (t) =

T ε−1 i1 Xi MAi MAi Xi

T = AˆTi Yi + Yi Aˆi + ε−1 i2 Yi MAi MAi Yi +

+ (εi1 + N X

(34)

T εi2 ) N1i N1i

+

N X

πij Xj ,

(35)

j=1

πij Yj .

(36)

j=1

PN

M

For i ∈ Sκi , the term Xi Ai + ATi Xi +

j=1 πij Xj

ED

Θi = Xi Ai + ATi Xi +

PT

= (1 +

AC

CE

+

j∈Sκi

X

N X

πij Xj

j=1


X πij ) Xi Ai + ATi Xi + πij Xj


j∈S

πκi )

X

j∈Sκi

πij Xi Ai + ATi Xi +

i uκ

= (1 + +

X

in (35) can be rewritten as follows:

Xi Ai +

ATi Xi

πij Xi Ai +




+

ATi Xi

i j∈Suκ

X

πij Xj

i uκ

j∈S

X

j∈S

πij Xj

i κ


+ Xj .

i and if i ∈ Then, by (24) and (28), we can achieve Θi < 0. For ∀j ∈ Suκ / Sκi , we can obtain   X X
Θi = 1 + πij  Xi Ai + ATi Xi + πij Xj j∈Sκi , j6=i




+πii Xi Ai + ATi Xi + Xi +

X

i , j6=i j∈Suκ

j∈Sκi , j6=i


πij Xi Ai + ATi Xi + Xj .

Hence, it is obvious that Θi < 0 can be attained by (25), (26) and (28).

DRAFT

ACCEPTED MANUSCRIPT

12

P Similarly, for i ∈ Sκi , the term AˆTi Yi + Yi Aˆi + N j=1 πij Yj in (36) can be simplified as:     X X
Ωi = 1 + πκi AˆTi Yi + Yi Aˆi + πij Yj + πij AˆTi Yi + Yi Aˆi + Yj . j∈Sκi

i j∈Suκ

i and if i ∈ S i , it is straightforward that Ω < 0 can be acquired by (24) and (29). Besides, Then, ∀j ∈ Suκ i κ

j∈Sκi , j6=i



  πij  AˆTi Yi + Yi Aˆi +

  +πii AˆTi Yi + Yi Aˆi + Yi +

Hence, we can get Ωi < 0 via (25), (27) and (29).

X

i , j6=i j∈Suκ

X

πij Yj

CR IP T

i and if i ∈ ∀j ∈ Suκ / Sκi , we have  X Ωi = 1 +

j∈Sκi , j6=i

  πij AˆTi Yi + Yi Aˆi + Yj .

AN US

Thus, if Φi < 0 holds, LV1 (x, ex , i) < 0 for ∀η (t) 6= 0, which implies

LV1 (x, ex , i) ≤ − min λmin {−Φi } kη (t)k2 . i∈S

According to Definition 2 and Lemma 1, we can obtain that system (3) with w (t) = 0 is exponentially stable in mean square sense.

Next, when w (t) 6= 0, the infinitesimal generator of stochastic Lyapunov functional V1 (ˆ x, e, i) is

M

deduced as follows:

ED

LV1 (x, ex , i) = η T (t) Φi η (t) + 2 (qis (y (t)))T Wi Di w (t)
¯i + 4B ¯i (t) w (t) . +2eT (t) P¯i B

According to EV1 (x, ex , i) = E

R∞ 0

(37)

LV1 (x, ex , i) dt > 0, for w (t) ∈ L2 [0, ∞), under zero-initial

AC

CE

PT

condition, the system H∞ performance can be transformed into the following form: Z ∞  T 2 T E ez (t) ez (t) − γ w (t) w (t) dt 0 Z ∞ 
T 2 T ≤ E ez (t) ez (t) − γ w (t) w (t) + LV1 (x, ex , i) dt 0 Z ∞ = E ξ T (t) Ξi ξ (t) dt,

(38)

0

DRAFT

ACCEPTED MANUSCRIPT

13

where ξ (t) =

h

iT , xT (t) eTx (t) (qis (y (t)))T wT (t)  Ξ Ξ12i CiT WiT Xi Bi  11i  ˆi ˆi W1i Di  ∗ Ξ22i −Yi B Yi Bi − Yi B Ξi =    ∗ ∗ −2I Wi Di  TN ∗ ∗ ∗ −γ 2 I + (εi3 + εi4 ) N2i 2i



   ,   

CR IP T

T T Ξ11i = Xi Ai + ATi Xi + ε−1 i1 Xi MAi MAi Xi + (εi1 + εi2 ) N1i N1i N  T   X −1 T ˆ ˆ + Li − Ci Li − Ci + εi3 Xi MBi MBi Xi + πij Xj , j=1

 T T ˆT Ξ12i = ATi Yi − AˆTi Yi − CiT W1i Bi Yi + Li − Cˆi Cˆi ,

AN US

Ξ22i

−1 T T ˆT ˆ = AˆTi Yi + Yi Aˆi + ε−1 i2 Yi MAi MAi Yi + εi4 Yi MBi MBi Yi + Ci Ci +

By linear matrix inequalities (LMIs) (24)-(29), we have Ξi < 0, which means ! kez (t)k < γ. E sup 06=w(t)∈L2 kw (t)k

N X

πij Yj .

j=1

mance. The proof is completed.

M

Hence, by Lemma 3, the system (23) is stochastically stable with γ –disturbance attenuation perfor

Remark 3: Note that (24), (26) and (27) in Theorem 1 will not be checked simultaneously owing to

ED

i = ∅. Sκi ∩ Suκ

VI. S IMULATION R ESULTS

CE

PT

Remark 4: Suppose β = γ 2 , and minimize β subject to (24)–(29), then we can achieve the optimal √ H∞ noise attenuation performance index β (γ = β).

AC

In this section, two examples are used to show the applicability of the proposed results.

DRAFT

ACCEPTED MANUSCRIPT

14

AC

CE

PT

ED

M

AN US

CR IP T

Example 1: The system matrices of system (2) with three subsystems are respectively given as follows:     −15 0.1 0.03 −5 1.1 1         A1 =  −0.2 −10 0.1  , A2 =  −1.9 −5 1 ,     −0.4 1.0 −17 −3.7 1.1 −10       −15 0.21 0.11 0.2 0.2 0.18 0.21             A3 =  −0.29 −10 0.1  , B1 =  0.5 0.2  , B2 =  0.49 0.21  ,       −0.47 1.5 −18 0.4 0.3 0.37 0.24       0.28 0.2   0.2 0.8 −0.3 0.19 0.6 −0.32    , C2 =  , B3 =  0.59 0.21  , C1 =    1.0 0.3 0.5 1.1 0.17 0.45 0.37 0.34       0.69 0 −0.2 0.4 0.29 0.8 −0.42 ,  , D2 =   , D1 =  C3 =  0 −0.28 0.3 −0.6 1.1 0.27 0.55       −0.2 0.47 0.5 0.2 0.2 0.46 0.23 0.25  , L1 =   , L2 =  , D3 =  −0.28 0.7 0.3 0.3 0.2 0.24 0.27 0.21         0.2 0.23 0.2       0.45 0.2 0.25       , MA1 =  L3 =   0.2  , MA2 =  0.19  , MA3 =  0.1  ,       0.3 0.3 0.4 0.1 0.08 0.08       0.1 0.13 0.1       h i       MB1 =  0.4  , MB2 =  0.46  , MB3 =  0.4  , N11 = 0.2 0.2 0.1 ,       0.2 0.25 0.2 h i h i h i N21 = 0.2 0.1 , N12 = 0.21 0.18 0.11 , N22 = 0.21 0.09 ,   h i h i 0.6 0 , N13 = 0.1 0.18 0.15 , N23 = 0.1 0.1 , W11 =  0 0.5       1.4 0 0.58 0 1.42 0  , W12 =   , W22 =  , W21 =  0 1.5 0 0.47 0 1.53       0.8 0 1.2 0 0.8 0  , W23 =   , W1 =  , W13 =  0 0.7 0 1.3 0 1

DRAFT

ACCEPTED MANUSCRIPT

15



W2 = 

0.84

0

0

1.06



,



W3 = 

0.4

0

0

0.6



,

CR IP T

FA1 (t) = FA2 (t) = FA3 (t) = 0.5 sin (t) , FB1 (t) = FB2 (t) = FB3 (t) = 0.2 cos (t) .   −0.8 ? ?     The transition matrix Π is given as Π =  ? −0.5 ? , where “?” denotes the unavail  1.5 1.5 −3  

able element. The external disturbance w (t) is selected as w (t) = 

tial conditions are respectively chosen as x (0) =

h

0 0 0

iT

10 (10+sin2 (t))·exp(2t) 10 (10+sin2 (t))·exp(2t)

and x ˆ (0) =

h

0 0 0

. The ini-

iT

. For the

logarithmic quantizer (14), the quantizer densities are chosen as ρ(1,1) = 0.4286, ρ(1,2) = 0.3333, (1,1)

given as η0

(1,2)

= η0

(2,1)

= η0

(2,2)

= η0

AN US

ρ(2,1) = 0.4085, ρ(2,2) = 0.3072, ρ(3,1) = 0.4125, ρ(3,2) = 0.3120. The initial quantizer points are (3,1)

= η0

(3,2)

= η0

= 0.0001. Then we can calculate that

δ (1,1) = 0.4000, δ (1,2) = 0.5000, δ (2,1) = 0.4200, δ (2,2) = 0.5300, δ (3,1) = 0.4200, δ (3,2) = 0.5200.

Utilize the Matlab LMI toolbox to address the LMIs (24)-(29), the optimal H∞ performance index is

AC

CE

PT

ED

M

γ = 0.5376, and the filter parameter matrices are calculated as follows:     −14.6271 0.2744 1.9903 0.2012 0.0008         ˆ Aˆ1 =  0.7180 , B =   −9.6393 3.4959 0.3455 0.2970  , 1     1.2082 1.5351 −14.1556 0.2507 0.0338     −5.3998 0.8508 1.6668 0.3332 −0.1698         ˆ ˆ A2 =  −0.3386 −6.9034 2.1192  , B2 =  0.7839 −0.1760  ,     −1.6607 −0.2117 −9.4602 0.5420 −0.3893     −19.0694 3.0489 −1.8534 0.1351 0.3349        ˆ3 =  Aˆ3 =  1.1580  0.1888 0.5430  , −5.7046 −13.1834  , B     2.0485 4.9669 −27.1456 0.2258 0.5242     −0.0316 0.3668 −0.1697 0.2920 0.5836 −0.4009  , Cˆ2 =  , Cˆ1 =  −0.2463 0.4718 −0.1845 0.1111 0.5348 −0.2740   0.4500 0.2000 0.2500 . Cˆ3 =  0.3000 0.3000 0.4000

The simulation results are provided in Figures 1–9. Figure 1 shows the switching signal r (t) . The

responses of zˆ1 (t) and z1 (t) and the responses of zˆ2 (t) and z2 (t) are respectively depicted in Figures

DRAFT

ACCEPTED MANUSCRIPT

16

2–3. The trajectories of y (t) and quantized measurement qi (y (t)) are depicted in Figures 4–5. The trajectories of qis (y1 (t)) and qis (y2 (t)) are respectively illustrated in Figures 6–7. In Figures 8–9, the error responses of ez1 (t) and ez2 (t) are shown, respectively.

4

CR IP T

r(t)

3

AN US

2

1

0

0

10 Time (Sec.)

15

20

M

Fig. 1. Trajectory of r (t).

5

PT

ED

Example 2: Consider the following linearized model from an F-404 aircraft engine system [16],   −1.46 0 2.428     A(t) =  0.1643 + 0.5η (t) −0.4 + η (t) −0.3788  ,   0.3107 0 −2.23

CE

with η (t) being an uncertain model parameter. Let η (t) subject to a Markov process r (t) with N = 3. The uncertainty η (t) takes values η (t) = −1 when r (t) = 1; η (t) = −2 when r (t) = 2; and η (t) = −3

AC

when r (t) = 3 . Under this setting, we  −1.4600 0   A1 =  −0.3357 −1.4000  0.3107 0  −1.4600 0   A3 =  −1.3357 −3.4000  0.3107 0

have

2.4280



  −0.3788  ,  −2.2300  2.4280   −0.3788  .  −2.2300



−1.4600

0

2.4280

  A2 =  −0.8357 −2.4000 −0.3788  0.3107 0 −2.2300



  , 

DRAFT

ACCEPTED MANUSCRIPT

17

0.03

0.025

zˆ1 z1

0.02

CR IP T

0.015

0.01

0.005

−0.005

0

5

10 Time (Sec.)

15

20

M

Fig. 2. Responses of zˆ1 (t) and z1 (t).

AN US

0

ED

0.03

0.025

zˆ2 z2

PT

0.02

CE

0.015

0.01

AC

0.005

0

−0.005

0

5

10 Time (Sec.)

15

20

Fig. 3. Responses of zˆ2 (t) and z2 (t).

DRAFT

ACCEPTED MANUSCRIPT

18

0.25

y1 (t) qi (y1 (t))

0.2

CR IP T

0.15

0.1

0

0

5

10 Time (Sec.)

15

20

M

Fig. 4. The trajectories of y1 (t) and qi (y1 (t)).

AN US

0.05

ED

0.05

0

PT

−0.05

y2 (t) qi (y2 (t))

CE

−0.1

−0.15

AC

−0.2

−0.25

−0.3

0

5

10 Time (Sec.)

15

20

Fig. 5. The trajectories of y2 (t) and qi (y2 (t)).

DRAFT

ACCEPTED MANUSCRIPT

19

0.16 0.14

qis (y1 (t))

0.12

CR IP T

0.1 0.08 0.06 0.04

0

0

5

10 Time (Sec.)

15

20

M

Fig. 6. The trajectory of qis (y1 (t)).

AN US

0.02

ED

0.02

0 −0.02

PT

qis (y2 (t))

−0.04

CE

−0.06 −0.08

AC

−0.1

−0.12 −0.14 −0.16

0

5

10 Time (Sec.)

15

20

Fig. 7. The trajectory of qis (y2 (t)).

DRAFT

ACCEPTED MANUSCRIPT

20

0.03

0.025

ez1 (t) 0.02

CR IP T

0.015

0.01

0.005

−0.005

0

5

10 Time (Sec.)

15

20

M

Fig. 8. The error response of ez1 (t).

AN US

0

ED

0.03

0.025

ez2 (t)

PT

0.02

CE

0.015

0.01

AC

0.005

0

−0.005

0

5

10 Time (Sec.)

15

20

Fig. 9. The error response of ez2 (t).

DRAFT

ACCEPTED MANUSCRIPT

21

AC

CE

PT

ED

M

AN US

CR IP T

The other matrices for system (2) are given by       0.2 0.2 0.8 0.1 0.8 0.2             B1 =  0.5 0.2  , B2 =  0.49 1.0  , B3 =  0.59 1.0  ,       0.4 0.3 0.7 0.4 0.37 0.34     0.2 0.8 −0.3 0.1 0.1 −0.32  , C2 =  , C1 =  1.0 0.3 0.5 1.0 0.27 0.45       0.29 0.8 −0.42 0.2 0.4 0.21 0.37  , D1 =   , D2 =  , C3 =  1.1 0.27 0.55 0.3 0.6 0.28 0.58       0.2 0.47 0.1 0.2 0.2 −0.26 0.13 0.15  , L1 =   , L2 =  , D3 =  0.28 0.7 0.1. 0.1 0.2 0.14 0.17 −0.21       0.5 0.23     −0.15 0.2 0.15     , MA1 =  L3 =   −0.5  , MA2 =  0.19  ,     0.1 0.1 −0.14 0.5 0.08         0.2 0.1 0.13 0.1                 MA3 =  0.1  , MB1 =  0.4  , MB2 =  0.46  , MB3 =  0.4  ,         0.08 0.2 0.25 0.2 h i h i h i N11 = , N = , N = 0.2 0.2 0.1 0.2 0.1 0.21 0.18 0.11 , 21 12 h i h i h i N22 = 0.21 0.09 , N13 = 0.1 0.18 0.15 , N23 = 0.1 0.1 ,       0.6 0 1.4 0 0.58 0  , W21 =   , W12 =  , W11 =  0 0.5 0 1.5 0 0.47       1.42 0 0.8 0 1.2 0  , W13 =   , W23 =  , W22 =  0 1.53 0 0.7 0 1.3       0.8 0 0.84 0 0.4 0  , W2 =   , W3 =  , W1 =  0 1.0 0 1.06 0 0.6 FA1 (t) = FA2 (t) = FA3 (t) = 0.5 sin (t) , 

  The transition matrix Π is assumed to be Π =  

FB1 (t) = FB2 (t) = FB3 (t) = 0.2 cos (t) .  −1 ? ?   0.5 −3 2.5 , where “?” denotes unknown element.  0.5 1.5 −2 DRAFT

ACCEPTED MANUSCRIPT

22



The external disturbance w (t) is assumed to be w (t) = 

respectively selected as x (0) =

h

0 0 0

iT

and x ˆ (0) =



1 (1+sin (t))·exp(2t) 1 (1+sin2 (t))·exp(2t) h iT 2

0 0 0

. The initial conditions are

. For the logarithmic quantizer

(14), the quantizer densities are set as ρ(1,1) = 0.4286, ρ(1,2) = 0.3333, ρ(2,1) = 0.4085, ρ(2,2) = 0.3072, (1,1)

ρ(3,1) = 0.4125, ρ(3,2) = 0.3120. The initial quantizer points are given as η0 =

(3,1) η0

=

(3,2) η0

= 0.0001. Then it can be calculated that

δ (1,1)

= 0.4000,

δ (1,2)

(2,1)

= η0

=

= 0.5000,

CR IP T

(2,2) η0

(1,2)

= η0

δ (2,1) = 0.4200, δ (2,2) = 0.5300, δ (3,1) = 0.4200, δ (3,2) = 0.5200. To solve the conditions (24)-(29) via

Matlab LMI toolbox, we have 

−4.8451 −2.4640

1.8326





1.5366 2.0944



PT

ED

M

AN US

       ˆ1 =  Aˆ1 =  −6.5035 −6.1002 −2.7414  , B  3.5765 4.3067  ,     −2.6950 −2.2420 −3.5749 1.7781 2.2669     −132.6391 −37.5768 −63.6000 14.3162 100.7903         ˆ ˆ A2 =  98.6762 25.9766 50.5635  , B2 =  −8.9782 −76.3135  ,     33.0106 9.4373 14.1669 −4.3720 −24.5819     −13.2803 −10.3228 10.8584 1.9384 3.2884        ˆ3 =  Aˆ3 =  −29.9677 −31.0496 20.7450  , B  5.9630 7.2761  ,     −13.2991 −12.8006 7.1356 2.7165 3.6655     −0.5429 −0.3015 0.9956 −0.2600 0.1300 0.1500  , Cˆ2 =  , Cˆ1 =  −0.3593 −0.2538 0.7667 0.1400 0.1700 −0.2100   −0.1500 0.2000 0.1500 . Cˆ3 =  0.1000 0.1000 −0.1400

CE

The minimum H∞ performance index is calculated as γ = 2.0099.

Figure 10 is one of the possible switching signals r (t). The responses of zˆ1 (t) and z1 (t) are depicted

AC

in Figure 11. Figure 12 shows the responses of zˆ2 (t) and z2 (t). In Figures 13–14, the trajectories of y (t) and qi (y (t)) are presented, respectively. The trajectories of qis (y1 (t)) and qis (y2 (t)) are shown in

Figures 15–16. The responses of ez1 (t) and ez2 (t) are shown in Figures 17–18. Remark 5: It can be seen that the designed H∞ filter can effectively attenuate the parameter uncertainty

and external disturbance from Figures 1-18. It can be observed from Figures 2–3 and Figures 11–12 that the estimated signal of the original system can effectively track the designed filter state. In Figures 8–9 and Figures 17–18, the estimated signals eventually converge to zero, which implies that the augmented DRAFT

ACCEPTED MANUSCRIPT

23

4

r(t)

CR IP T

3

2

0

0

5

10 Time (Sec.)

15

20

M

Fig. 10. Trajectory of r (t).

AN US

1

ED

0.08 0.06

zˆ1 z1

0.04

PT

0.02

0

CE

−0.02 −0.04

AC

−0.06 −0.08 −0.1 −0.12

0

5

10 Time (Sec.)

15

20

Fig. 11. Responses of zˆ1 (t) and z1 (t).

DRAFT

ACCEPTED MANUSCRIPT

24

0.06

0.04

zˆ2 z2

0.02

CR IP T

0

−0.02

−0.04

−0.08

0

5

10 Time (Sec.)

15

20

M

Fig. 12. Responses of zˆ2 (t) and z2 (t).

AN US

−0.06

ED

0.6

y1 (t) qi (y1 (t))

0.5

PT

0.4

AC

CE

0.3

0.2

0.1

0

−0.1

0

5

10 Time (Sec.)

15

20

Fig. 13. The trajectories of y1 (t) and qi (y1 (t)).

DRAFT

ACCEPTED MANUSCRIPT

25

0.9 0.8

y2 (t) qi (y2 (t))

0.7

CR IP T

0.6 0.5 0.4 0.3 0.2

0

0

5

10 Time (Sec.)

15

20

M

Fig. 14. The trajectories of y2 (t) and qi (y2 (t)).

AN US

0.1

ED

0.35

0.3

qis (y1 (t))

PT

0.25

0.2

CE

0.15

0.1

AC

0.05 0

−0.05

0

5

10 Time (Sec.)

15

20

Fig. 15. The trajectory of qis (y1 (t)).

DRAFT

ACCEPTED MANUSCRIPT

26

0.45 0.4

qis (y2 (t))

0.35 0.3

CR IP T

0.25 0.2 0.15 0.1 0.05

−0.05

0

5

10 Time (Sec.)

15

20

M

Fig. 16. The trajectory of qis (y2 (t)).

AN US

0

ED

0.16 0.14

ez1 (t)

PT

0.12

0.1

CE

0.08 0.06

AC

0.04 0.02 0

−0.02

0

5

10 Time (Sec.)

15

20

Fig. 17. The error response of ez1 (t).

DRAFT

ACCEPTED MANUSCRIPT

27

0.12

0.1

ez2 (t) 0.08

0.04

0.02

0

0

5

10 Time (Sec.)

15

20

AN US

−0.02

CR IP T

0.06

Fig. 18. The error response of ez2 (t).

system (23) is stochastically stable with γ –disturbance attenuation performance.

M

VII. C ONCLUSION

The H∞ filtering problem has been investigated for MJSs with parameter uncertainty, partly unknown

ED

transition probabilities and mode-dependent quantized output measurement. Compared with the MJSs with completely known transition probabilities, the MJSs considered in this paper are more general. By

PT

Lyapunov-like theory, the sufficient conditions of desired full-order filter are derived to guarantee the stochastic stability of the resulting filtering error system with H∞ performance index. The validity of the

CE

proposed theoretical results has been illustrated via two examples. Future work will pay attention to use fuzzy control approach [62]–[65] or adaptive control approach [66]–[70] to handle MJSs with nonlinear

AC

plant and mode-dependent quantization. R EFERENCES

[1] L. Liu, Q. Zhou, H. Liang, and L. Wang, “Stability and stabilization of nonlinear switched systems under average dwell time,” Applied Mathematics & Computation, vol. 298, pp. 77–94, 2017.

[2] L. Wu, R. Yang, P. Shi, and X. Su, “Stability analysis and stabilization of 2-D switched systems under arbitrary and restricted switchings,” Automatica, vol. 59, no. 1, pp. 206–215, 2015. [3] L. Wu, W. X. Zheng, and H. Gao, “Dissipativity-based sliding mode control of switched stochastic systems,” IEEE Transactions on Automatic Control, vol. 58, no. 3, pp. 785–791, 2013. DRAFT

ACCEPTED MANUSCRIPT

28

[4] P. Shi, X. Su, and F. Li, “Dissipativity-based filtering for fuzzy switched systems with stochastic perturbation,” IEEE Transactions on Automatic Control, vol. 61, no. 6, pp. 1694–1699, 2016. [5] X. Su, P. Shi, L. Wu, and Y. D. Song, “Fault detection filtering for nonlinear switched stochastic systems,” IEEE Transactions on Automatic Control, vol. 61, no. 5, pp. 1310–1315, 2016. [6] H. Li, L. Bai, Q. Zhou, R. Lu, and L. Wang, “Adaptive fuzzy control of nonstrict-feedback stochastic nonlinear systems with input saturation,” IEEE Transactions on Systems, Man and Cybernetics: Systems, DOI: 10.1109/TSMC.2016.2635678. [7] C.-M. Chen, D.-J. Guan, Y.-Z. Huang, and Y.-H. Ou, “Anomaly network intrusion detection using hidden Markov model,”

CR IP T

International Journal of Innovative Computing, Information and Control, vol. 12, no. 2, pp. 569–580, 2016.

[8] L. Wu, X. Su, and P. Shi, “Sliding mode control with bounded l2 gain performance of Markovian jump singular time-delay systems,” Automatica, vol. 48, no. 8, pp. 1929–1933, 2012.

[9] H. Li, H. Gao, P. Shi, and X. Zhao, “Fault-tolerant control of Markovian jump stochastic systems via the augmented sliding mode observer approach,” Automatica, vol. 50, no. 7, pp. 1825–1834, 2014.

[10] L. Wu, X. Su, and P. Shi, “Output feedback control of Markovian jump repeated scalar nonlinear systems,” IEEE Transactions on Automatic Control, vol. 59, no. 1, pp. 199–204, 2014.

AN US

[11] Z.-G. Wu, P. Shi, Z. Shu, H. Su, and R. Lu, “Passivity-based asynchronous control for Markov jump systems,” IEEE Transactions on Automatic Control, DOI: 10.1109/TAC.2016.2593742.

[12] M. Liu, D. W. C. Ho, and P. Shi, “Adaptive fault-tolerant compensation control for Markovian jump systems with mismatched external disturbance,” Automatica, vol. 58, no. C, pp. 5–14, 2015.

[13] H. Li, P. Shi, D. Yao, and L. Wu, “Observer-based adaptive sliding mode control for nonlinear Markovian jump systems,” Automatica, vol. 64, pp. 133–142, 2016. vol. 43, no. 10, pp. 1784–1790, 2007.

M

[14] Y. Niu, D. W. Ho, and X. Wang, “Sliding mode control for Itˆo stochastic systems with Markovian switching,” Automatica, [15] P. Shi, E.-K. Boukas, and R. K. Agarwal, “Kalman filtering for continuous-time uncertain systems with Markovian jumping

ED

parameters,” IEEE Transactions on Automatic Control, vol. 44, no. 8, pp. 1592–1597, 1999. [16] Z. Wang, Y. Liu, and X. Liu, “Exponential stabilization of a class of stochastic system with Markovian jump parameters and mode-dependent mixed time-delays,” IEEE Transactions on Automatic Control, vol. 55, no. 7, pp. 1656–1662, 2010.

PT

[17] H. Li, P. Shi, and D. Yao, “Adaptive sliding mode control of Markov jump nonlinear systems with actuator faults,” IEEE Transactions on Automatic Control, DOI: 10.1109/TAC.2016.2588885. [18] X. Mao, “Exponential stability of stochastic delay interval systems with Markovian switching,” IEEE Transactions on

CE

Automatic Control, vol. 47, no. 10, pp. 1604–1612, 2002. [19] C. E. De Souza, A. Trofino, and K. A. Barbosa, “Mode-independent filters for Markovian jump linear systems,” IEEE Transactions on Automatic Control, vol. 51, no. 51, pp. 1837–1841, 2006.

AC

[20] J. Xiong and J. Lam, “Stabilization of discrete-time Markovian jump linear systems via time-delayed controllers,” Automatica, vol. 42, no. 5, pp. 747–753, 2006.

[21] M. Karan, P. Shi, and C. Y. Kaya, “Transition probability bounds for the stochastic stability robustness of continuous-and discrete-time Markovian jump linear systems,” Automatica, vol. 42, no. 12, pp. 2159–2168, 2006.

[22] L. Zhang and E.-K. Boukas, “Stability and stabilization of Markovian jump linear systems with partly unknown transition probabilities,” Automatica, vol. 45, no. 2, pp. 463–468, 2009. [23] L. Zhang and E. K. Boukas, “Mode-dependent H∞ filtering for discrete-time Markovian jump linear systems with partly unknown transition probabilities,” Automatica, no. 6, pp. 1462–1467, 2009.

DRAFT

ACCEPTED MANUSCRIPT

29

[24] Y. Zhang, Y. He, M. Wu, and J. Zhang, “Stabilization for Markovian jump systems with partial information on transition probability based on free-connection weighting matrices,” Automatica, vol. 47, no. 1, pp. 79–84, 2011. [25] Y.-J. Liu, J. Li, S. Tong, and C. L. Chen, “Neural network control-based adaptive learning design for nonlinear systems with full-state constraints,” IEEE Transactions on Neural Networks & Learning Systems, vol. 27, no. 7, pp. 1–10, 2016. [26] K. Mathiyalagan, J. H. Park, R. Sakthivel, and S. M. Anthoni, “Robust mixed H∞ and passive filtering for networked Markov jump systems with impulses,” Signal Processing, vol. 101, pp. 162–173, 2014. [27] R. Lu, Y. Xu, A. Xue, and J. Zheng, “Networked control with state reset and quantized measurements: observer-based

CR IP T

case,” IEEE Transactions on Industrial Electronics, vol. 60, no. 11, pp. 5206–5213, 2013.

[28] R. Lu, F. Wu, and A. Xue, “Networked control with reset quantized state based on bernoulli processing,” IEEE Transactions on Industrial Electronics, vol. 61, no. 9, pp. 4838–4846, 2014.

[29] R. Lu, Y. Xu, and R. Zhang, “A new design of model predictive tracking control for networked control system under random packet loss and uncertainties,” IEEE Transactions on Industrial Electronics, vol. 63, no. 11, pp. 6999–7007, 2016. [30] H. Lin, H. Su, Z. Shu, and Z.-G. Wu, “Optimal estimation in UDP-like networked control systems with intermittent inputs: 2015.

AN US

Stability analysis and suboptimal filter design,” IEEE Transactions on Automatic Control, vol. 61, no. 7, pp. 1794–1809, [31] Y. Wu, X. Meng, L. Xie, R. Lu, H. Su, and Z.-G. Wu, “An input-based triggering approach to leader-following problems,” Automatica, vol. 75, pp. 221–228, 2017.

[32] Y. Wu, H. Su, P. Shi, and R. Lu, “Output synchronization of nonidentical linear multiagent systems,” IEEE Transactions on Cybernetics, vol. 47, no. 1, pp. 130–141, 2017.

[33] Y. Xu, R. Lu, P. Shi, H. Li, and S. Xie, “Finite-time distributed state estimation over sensor networks with round-robin

M

protocol and fading channels,” IEEE Transactions on Cybernetics, DOI: 10.1109/TCYB.2016.2635122. [34] H. Yan, F. Qian, H. Zhang, F. Yang, and G. Guo, “H∞ fault detection for networked mechanical spring-mass systems with incomplete information,” IEEE Transactions on Industrial Electronics, vol. 63, no. 9, pp. 5622–5631, 2016.

ED

[35] S. Dong, Z.-G. Wu, P. Shi, H. Su, and R. Lu, “Reliable control of fuzzy systems with quantization and switched actuator failures,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, DOI: 10.1109/TSMC.2016.2636222. [36] S. Dong, H. Su, P. Shi, R. Lu, and Z.-G. Wu, “Filtering for discrete-time switched fuzzy systems with quantization,” IEEE

PT

Transactions on Fuzzy Systems, DOI: 10.1109/TFUZZ.2016.2612699. [37] M. Liu, D. W. C. Ho, and Y. Niu, “Stabilization of Markovian jump linear system over networks with random communication delay,” Automatica, vol. 45, no. 2, pp. 416–421, 2009.

CE

[38] Y. Niu, T. Jia, X. Wang, and F. Yang, “Output-feedback control design for NCSs subject to quantization and dropout,” Information Sciences, vol. 179, no. 21, pp. 3804–3813, 2009. [39] R. Sombutkaew, O. Chitsobhuk, D. Prapruttam, and T. Ruangchaijatuporn, “Adaptive quantization via fuzzy classified

AC

priority mapping for liver ultrasound compression,” International Journal of Innovative Computing, Information and Control, vol. 12, no. 2, pp. 635–649, 2016.

[40] X. Tang and B. Ding, “Model predictive control of linear systems over networks with data quantizations and packet losses,” Automatica, vol. 49, no. 5, pp. 1333–1339, 2013.

[41] Y. Wu, H. Su, P. Shi, Z. Shu, and Z.-G. Wu, “Consensus of multiagent systems using aperiodic sampled-data control,” IEEE Transactions on Cybernetics, vol. 46, no. 9, pp. 2132–2143, 2016. [42] Y. Xu, R. Lu, H. Peng, and J. Chen, “Passive filter design for periodic stochastic systems with quantized measurements and randomly occurring nonlinearities,” Journal of the Franklin Institute, vol. 353, no. 1, pp. 144–159, 2015.

DRAFT

ACCEPTED MANUSCRIPT

30

[43] P. Shi, M. Liu, and L. Zhang, “Fault-tolerant sliding-mode-observer synthesis of Markovian jump systems using quantized measurements,” IEEE Transactions on Industrial Electronics, vol. 62, no. 9, pp. 5910–5918, 2015. [44] R. Lu, P. Shi, H. Su, Z.-G. Wu, and J. Lu, “Synchronization of general chaotic neural networks with nonuniform sampling and packet missing: a switched system approach,” IEEE Transactions on Neural Networks & Learning Systems, DOI:10.1109/TNNLS.2016.2636163. [45] E. Tian, D. Yue, and C. Peng, “Quantized output feedback control for networked control systems,” Information Sciences, vol. 178, no. 12, pp. 2734–2749, 2008.

CR IP T

[46] G. Wang, H. Bo, and Q. Zhang, “H∞ filtering for time-delayed singular Markovian jump systems with time-varying switching: A quantized method,” Signal Processing, vol. 109, pp. 14–24, 2015.

[47] Y. Xu, R. Lu, H. Peng, K. Xie, and A. Xue, “Asynchronous dissipative state estimation for stochastic complex networks with quantized jumping coupling and uncertain measurements,” IEEE Transactions on Neural Networks & Learning Systems, DOI:10.1109/TNNLS.2015.2503772.

[48] H. Zhang, Q. Hong, H. Yan, F. Yang, and G. Guo, “Event-based distributed H∞ filtering networks of 2DOF quarter-car suspension systems,” IEEE Transactions on Industrial Informatics, DOI: 10.1109/TII.2016.2569566.

AN US

[49] H. Yan, F. Qian, F. Yang, and H. Shi, “H∞ filtering for nonlinear networked systems with randomly occurring distributed delays, missing measurements and sensor saturation,” Information Sciences, vol. 370-371, pp. 772–782, 2015. [50] Z. Wang, J. Lam, and X. Liu, “Robust filtering for discrete-time Markovian jump delay systems,” IEEE Signal Processing Letters, vol. 11, no. 8, pp. 659–662, 2004.

[51] Z.-G. Wu, P. Shi, H. Su, and J. Chu, “Asynchronous l2 − l∞ filtering for discrete-time stochastic Markov jump systems with randomly occurred sensor nonlinearities,” Automatica, vol. 50, no. 1, pp. 180–186, 2014.

M

[52] Y. Xu, R. Lu, P. Shi, J. Tao, and S. Xie, “Robust estimation for neural networks with randomly occurring distributed delays and Markovian jump coupling,” IEEE Transactions on Neural Networks & Learning Systems, DOI10.1109/TNNLS.2016.2636325.

ED

[53] N. Liu, X. Lyu, Y. Zhu, and J. Fei, “Active disturbance rejection control for current compensation of active power filter,” International Journal of Innovative Computing, Information and Control, vol. 12, no. 2, pp. 407–418, 2016. [54] J. Tao, R. Lu, P. Shi, H. Su, and Z.-G. Wu, “Dissipativity-based reliable control for fuzzy Markov jump systems with

PT

actuator faults,” IEEE Transactions on Cybernetics, DOI: 10.1109/TCYB.2016.2584087. [55] S. Xu, J. Lam, and X. Mao, “Delay-dependent H∞ control and filtering for uncertain Markovian jump systems with time-varying delays,” IEEE Transactions on Circuits & Systems I Regular Papers, vol. 54, no. 9, pp. 2070–2077, 2007.

CE

[56] S. P. Meyn and R. L. Tweedie, Markov chains and stochastic stability.

Springer Science & Business Media, 2012.

[57] Y. Niu and D. W. Ho, “Robust observer design for Itˆo stochastic time-delay systems via sliding mode control,” Systems & Control Letters, vol. 55, no. 10, pp. 781–793, 2006.

AC

[58] X. Mao and C. Yuan, Stochastic differential equations with Markovian switching.

World Scientific, 2006.

[59] N. Elia and S. K. Mitter, “Stabilization of linear systems with limited information,” IEEE Transactions on Automatic Control, vol. 46, no. 9, pp. 1384–1400, 2001.

[60] M. Fu and C. E. D. Souza, “State estimation for linear discrete-time systems using quantized measurements,” Automatica, vol. 45, no. 12, pp. 2937–2945, 2009.

[61] H. Gao and T. Chen, “H∞ estimation for uncertain systems with limited communication capacity,” IEEE Transactions on Automatic Control, vol. 52, no. 11, pp. 2070–2084, 2006.

DRAFT

ACCEPTED MANUSCRIPT

31

[62] J. Wang, Y. Gao, J. Qiu, and C. K. Ahn, “Sliding mode control for nonlinear systems by T-S fuzzy model and delta operator approaches,” IET Control Theory and Applications, DOI: 10.1049/iet-cta.2016.0231. [63] H. Li, J. Wang, L. Wu, H. K. Lam, and Y. Gao, “Optimal guaranteed cost sliding mode control of interval type-2 fuzzy time-delay systems,” IEEE Transactions on Fuzzy Systems, DOI: 10.1109/TFUZZ.2017.2648855. [64] Q. Zhou, H. Li, C. Wu, L. Wang, and C. K. Ahn, “Adaptive fuzzy control of nonstrict-feedback nonlinear systems with unmodeled dynamics and input saturation using small-gain approach,” IEEE Transactions on Systems, Man and Cybernetics: Systems, DOI: 10.1109/TSMC.2016.2586108. input saturation,” Fuzzy Sets and Systems, DOI: 10.1016/j.fss.2016.11.002.

CR IP T

[65] Q. Zhou, C. Wu, and P. Shi, “Observer-based adaptive fuzzy tracking control of nonlinear systems with time delay and [66] Y.-J. Liu and S. Tong, “Barrier lyapunov functions for Nussbaum gain adaptive control of full state constrained nonlinear systems,” Automatica, vol. 76, pp. 143–152, 2017.

[67] G. Wen, C. L. P. Chen, Y. J. Liu, and Z. Liu, “Neural network-based adaptive leader-following consensus control for a class of nonlinear multiagent state-delay systems,” IEEE Transactions on Cybernetics, DOI: 10.1109/TCYB.2016.2608499. [68] Y.-J. Liu, S. Tong, C. L. P. Chen, and D. J. Li, “Neural controller design-based adaptive control for nonlinear MIMO

AN US

systems with unknown hysteresis inputs,” IEEE Transactions on Cybernetics, vol. 46, no. 1, pp. 9–19, 2015.

[69] Y.-J. Liu and S. Tong, “Optimal control-based adaptive NN design for a class of nonlinear discrete-time block-triangular systems,” IEEE Transactions on Cybernetics, vol. 46, no. 11, pp. 2670–2680, 2016.

[70] Y. Wu, R. Lu, P. Shi, H. Su, and Z.-G. Wu, “Adaptive output synchronization of heterogeneous network with an uncertain

AC

CE

PT

ED

M

leader,” Automatica, vol. 76, pp. 183–192, 2017.

DRAFT