Simultaneous fault detection and control for uncertain discrete-time stochastic systems with limited communication

Accepted Manuscript

Simultaneous fault detection and control for uncertain discrete-time stochastic systems with limited communication Zhaoke Ning, Jinyong Yu, Tong Wang PII: DOI: Reference:

S0016-0032(17)30485-4 10.1016/j.jfranklin.2017.09.016 FI 3152

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Journal of the Franklin Institute

Received date: Revised date: Accepted date:

20 January 2017 13 July 2017 20 September 2017

Please cite this article as: Zhaoke Ning, Jinyong Yu, Tong Wang, Simultaneous fault detection and control for uncertain discrete-time stochastic systems with limited communication, Journal of the Franklin Institute (2017), doi: 10.1016/j.jfranklin.2017.09.016

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Simultaneous fault detection and control for uncertain discrete-time stochastic systems with limited communication Zhaoke Ning,Jinyong Yu and Tong Wang

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Research Institute of Intelligent Control and Systems, Harbin Institute of Technology, Harbin 150001, China

Abstract

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In the paper, the simultaneous fault detection and control (SFDC) problem is concerned for uncertain discrete-time stochastic systems with limited communication. An integrated module with a filter and a dynamic output feedback controller is designed to achieve the desired fault detection (FD) and control objectives, simultaneously. The event-triggered technology is employed to save the network communication resources, while two event detectors are utilized to decrease the amount of data transmitted from the sensor to the SFDC module and from the SFDC module to the plant. A novel method is proposed to ensure that the obtained closed-loop model is robustly stochastically stable and satisfies the desired detection and control performances. Sufficient conditions are derived to obtain the parameters of filter, controller and event detectors. Finally, the validity of the design strategy is verified by a simulation case. Keywords: Discrete-time system, limited communication, SFDC, event detector, stochastic system.

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1. Introduction

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Due to the advantages of flexibility, higher reliability, simplified installation and maintainability, networked control systems (NCSs) have received great attentions recently and have been widely applied in many practical applications. For example, industrial control systems, health monitoring, intelligent transportation systems and remote control systems. For NCSs, all components are distributed at different places and the signals amount them are transmitted via a communication network [1, 2, 3, 4, 5]. However, the resources of communication network are limited in many actual systems, which may lead to some challenging problems. Therefore, it is essential to design methods to save energy and network bandwidth. The typical data transmission strategy (also known as time-triggered one) samples and transmits the data periodically regardless whether it is really necessary or not. Many unnecessary data may be transmitted and the network resources may be excessively utilized in this strategy. In order to overcome these problems, a smart event-triggered strategy has been widely investigated recently [6, 7, 8]. The basic idea of this strategy is to design an event detector to transmit useful data which satisfy the predefined event-triggered condition. Many existing control and filter design approaches were proposed under the event-triggered strategy in literatures. For example, the event-based controller design problems were discussed in [9, 10, 11, 12]. The authors developed the filter design methods under the event-triggered strategy for different complex systems [13, 14, 15, 16]. On the other hand, in order to increase the safety and reliability of practical industry processes, the problems of fault detection and isolation, fault-tolerant control, fault detection and control have been widely developed over the past decades [17, 18, 19, 20, 21, 22]. For SFDC problems, the main aim is to design a fault detector and controller such that: (1) the generated closed-loop system is stochastically stable; (2) the residual output is sensitive to fault and insensitive to disturbance simultaneously; (3) the controlled ∗ Corresponding

author:Jinyong Yu. E-mail:[email protected].

Preprint submitted to Elsevier

September 27, 2017

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output is robust to both the fault and disturbance. Many results for SFDC problems have been obtained for different systems in past few years [23, 24, 25]. To mention a few, an integrated controller and filter structure was proposed for linear uncertain systems [26]. The design approach of state feedback controller/residual generator was reported for discrete-time switched delay systems [27]. In [28], the event-triggered control and fault diagnosis scheme was formulated as a mixed H∞ , l1 and H2 optimisation problem, which was solved by linear matrix inequalities (LMIs). It is observed that above results are reported under the periodic sampling scheme, which may result in many unnecessary data being transmitted and the poor utilization of the limited communication resources. Therefore, it is important to design strategy to deal with SFDC problem under the event-triggered strategy. This is the main motivation of this study. Due to the frequent existence of the stochastic phenomena, the stochastic systems are widely applied in many practical fields and many works have been developed in the literatures in recent years [29, 30, 31, 32]. To mention a few, The state feedback and observer-based fuzzy control problems were discussed in [33, 34]. The filter problems were investigated in [35, 36]. The adaptive fuzzy tracking control problem was studied in [37]. The authors designed a fuzzy filter to detect the sensor fault for the nonlinear stochastic systems [38]. However, few related SFDC results were discussed for uncertain stochastic systems taking the event-triggered strategy into account. This is the another motivation of this study. In the paper, the SFDC problem is investigated for uncertain discrete-time stochastic systems under an event-triggered strategy. The main contributions of this paper are summarized as: (1) For the limited network communication resources, the event-triggered strategy is employed for uncertain discrete-time stochastic systems, while two event detectors are provided to decrease the amount of sensor signals and control signals, respectively; (2) With the consideration of stochastic model and the event-triggered strategy, a novel closed loop residual model is formulated for the stochastic stability analysis and the H∞ performance analysis; (3) Two different Lyapunov functions are proposed to achieve the desired fault detection and control objectives. The rest of this paper is organized as follows. In Section 2, the preliminaries are presented to formulate the SFDC problem under the event-triggered strategy. The design approach of the FD filer and controller is presented in Section 3. In Section 4, the validity of the design strategy is proved by a simulation case. The conclusion is drawn in the last section. Notations The matrix transpose and inverse are defined by “T ” and “–1”, respectively. I and 0 represent the identity and zero matrix with appropriate dimensions. The probability space is represented by (Ω, F, P), where Ω and F represent the sample space and the σ-algebras of subsets, respectively. E{·} is the expectation operator with respect to the probability measure P. He(A) is the sum of A and AT . The l2 ([0, ∞)) norm is represented by k · k2 . The nation kf (k)ke2 is the l2 norm in (Ω, F, P) and satisfies: (N ) (N ) X X T kf (k)ke2 = E kf (k)k2 = E f (k)f (k) . 2. Preliminaries and problem formulation

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The overall block diagram of the event-triggered SFDC design scheme for uncertain stochastic systems investigated in the paper is described in Fig.1. The SFDC module is designed to generate the residual signals to detect faults and offer the control signals to stabilize the system. In order to reduce the communication pressure and save the limited network resources, two event detectors are utilized to optimize the transmission data of the measurement output and control output, respectively. Consider the following uncertain stochastic model with disturbance and fault:  u(k) + Bd d(k) + Bf f (k) + Lx(k)ω(k),   x(k + 1) =A(k)x(k) + B(k)ˆ y(k) =Cx(k) + Dd d(k) + Df f (k) + N x(k)ω(k), (1)   z(k) =Ex(k) + Gˆ u(k) + Fd d(k) + Ff f (k), where x(k) ∈ Rn is the plant state, u ˆ(k) ∈ Rnu is the optimized control signals that are transmitted to the plant from the event detector, y(k) ∈ Rny is the measurement output of the plant, z(k) ∈ Rnz is the 2

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f(k)

z(k) Plant

‫ݑ‬ො(݇)

u(k)

‫ݕ‬ො(݇)

SFDC Module r(k)

Weighting system

ҧ ݂(݇)

+ e(k)

-

Event Detecor

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Event Detecor

y(k)

Figure 1: Schematic of event-triggered fault detection and control

E{ω(k)} = 0,

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controlled output of the plant, The disturbance input d(k) ∈ Rnd and the fault f (k) ∈ Rnf are defined to belong to l2 [0, ∞) norm bounded. The stochastic process ω(k) is a zero-mean real constant and belongs to the probability space (Ω, F, P) which is relative to an increasing family (Fk )k∈N of σ-algebras Fk ⊂ F generated by (ω(k))k∈N . Here the ω(k) is supposed to satisfy: E{ω(k)2 } = η,

k = 0, 1, ...,

(2)

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where η > 0. The matrices Bd , Bf , Dd , Df , Fd , Ff , C, E, G, L and N are the known parameters with appropriate dimensions, and A(k) = A + ∆A(k),

B(k) = B + ∆B(k),

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where A and B are known constant matrices. ∆A(k) and ∆B(k) indicate the uncertainties of the plant, which are assumed to satisfy:     ∆A(k) ∆B(k) = M F (k) N1 N2 , (3)

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where M , N1 and N2 are real matrices and F (k) is time-varying parameter which satisfies: F (k)T F (k) ≤ I,

∀k.

(4)

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The main purpose of the paper is to design the SFDC module, which consists of a FD filter and a controller. The filter is utilized to construct a residual signal for fault detection objective while the controller is for some control objectives. The representation of the SFDC module is shown as follows:  ˆ(k + 1) = Ar x ˆ(k) + Br yˆ(k),  x r(k) = Cr x ˆ(k) + Dr yˆ(k), (5)   u(k) = Kr x ˆ(k),

where x ˆ(k) ∈ Rn is the state vector of SFDC module; yˆ(k) ∈ Rny denotes the real input of SFDC module; r(k) ∈ Rnr and u(k) ∈ Rnu represent the generated residual signal and the control signal, respectively. Ar , Br , Cr , Dr and Kr are the parameter matrices of SFDC module to be designed. Then, the network transmission problem discussed in the paper will be studied. 2.1. Event detector In the actual physical system, the network communication capability should be considered. In some conditions, the limited network resources may not satisfy a large quantity of signal transmission. Thus, how 3

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to reduce the amount of signal transmitted through the network is worth to be studied. The event-triggered strategy is one of the most useful methods to save the limited network resources. The principle of the scheme is to decide and transmit the useful signals through the network with a predefined event-triggered condition. In the event-triggered strategy, the measurement output of the system y(k) is transmitted to the event detector. If the error between the current measured data y(k) and the previous transmitted one yˆ(k) satisfies the predefined event-triggered condition, the current measured data will not be transmitted to the SFDC module from the event detector. The predefined event-triggered condition is:

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(ˆ y (k) − y(k))T (ˆ y (k) − y(k)) ≤ 1 y T (k)y(k),

(6)

where 1 is the threshold of the event detector. The measured data of the system which not satisfies the condition (6) will be transmitted to the SFDC module. For the control signals transmission side, a similar event-triggered condition is utilized to determine the transmitted control signals: (ˆ u(k) − u(k))T (ˆ u(k) − u(k)) ≤ 2 uT (k)u(k),

(7)

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where u(k) and u ˆ(k) represent the current and the previous transmitted control signal, respectively. 2 is the threshold of event detector for the controller. The control signal will be sent to the system from the event detector only when the condition (7) is not satisfied. The transmission time sequences of the event detectors for measured and control signals are assumed as k0y , k1y , k2y ,..., and k0u , k1u , k2u ,..., respectively. The initial time is k0y = k0u = 0. Thus, the transmitted measured and control signals can also be described by: yˆ(k) = y(kiy ),

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and

y k ∈ [kjy , kj+1 ),

u ˆ(k) = u(kiu ),

u k ∈ [kju , kj+1 ).

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For better representation, the errors of measured and control data are defined by: ey (k) = yˆ(k) − y(k),

eu (k) = u ˆ(k) − u(k).

(8)

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Therefore, we can rewrite (6) − (7) that:

eTy (k)ey (k) ≤ 1 y(k)y T (k),

k ∈ Z+ 0.

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eTu (k)eu (k) ≤ 2 uT (k)u(k),

k ∈ Z+ 0,

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Remark 1. In the framework of the event-triggered strategy for SFDC problem studied in this paper, two event detectors are applied to avoid unnecessary signal transmission. The thresholds of event detectors 1 and 2 decide the transmission rates of the measured and control signals, respectively. The bigger the thresholds are, the less measured and control signals will be transmitted. It also should be point out that two event detectors are independent of each other. In other words, the event-triggered conditions (6) and (7) have the asynchronous property.

Remark 2. The event-triggered strategy is a smart method to save the limited communication resources, which is widely applied in many practical applications. For example, industrial control systems, health monitoring, intelligent transportation systems and remote control systems. The strategy utilizes a triggered condition to transmit the necessary data through the network and discard the unnecessary data purposely, which can reduce computation and increase the lifetime of the overall system.

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2.2. Fault Weighting System In the framework of fault detection scheme, a weighting fault function should be applied to improve the detection performance. The form of weighting fault function is f¯(z) = W (z)f (z), where W (z) is a weighting matrix which reflects the frequency weighting of the fault signal. One of the state-space realization is shown as follows: ( xw (k + 1) = Aw xw (k) + Bw f (k), (9) f¯(k) = Cw xw (k) + Dw f (k),

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where xw (k) ∈ Rnw is the weighting state; f¯(t) ∈ Rl is the weighting fault signal; the parameter matrices Aw , Bw , Cw and Dw are known. 2.3. Residual Evaluation In order to evaluate the generated residual signal, a residual evaluation function is essential to be designed. Moreover, the fault can be detected by comparing with the predefined threshold Jth . The structure of residual evaluation function adopted in this paper can be described by: E

k2 1 X rT (k)r(k) N k=k1

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,

N = k2 − k1 + 1,

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Jr (k) =

(

(10)

where k1 represents the initial evaluation time instant and N represents the length of evaluation time instant. The predefined threshold is as the following structure: Jth =

sup

Jr (k).

(11)

d∈l2 [0,∞),f =0

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Then, we can get the alarms for fault occurrence with the following relationship: ( Jr > Jth =⇒ faults =⇒ alarm, Jr ≤ Jth =⇒ no

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Denote

e(k) =r(k) − f¯(k),

faults.

 T v(k) = dT (k) f T (k) ,

T  ˆT (k) , ξ(k) = xTw (k) xT (k) x

    ˜ ˜ ˜2 (k) = B ˜2 + ∆B ˜ 2 (k), C˜ = −Cw Dr C Cr , N ˜ = 0 Dr N 0 , A(k) =A˜ + ∆A(k), B         Aw 0 0 0 Bw 0 0 ˜ =  Bd ˜1 =  0  , B ˜2 = B  , A BKr  , B Bf  , B A˜ =  0 0 Br C Ar Br Dd Br Df Br 0       0 0 0 0 0 0 0   ˜ ˜ = 0 ˜ 2 (k) = ∆B  , F˜ = Fd Ff , L 0 , ∆B ∆A(k) = 0 ∆A ∆BKr  , L 0 0 0 0 Br N 0 0     ˜ ˜ D = Dr Dd Dr Df − Dw , E = 0 E GKr .

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where

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referring to (1), (5), (9) and two event-triggered conditions (6) − (7), the following model can be obtained:  ˜ ˜ ˜1 ey (k) + B ˜2 (k)eu (k) + Lξ(k)ω(k), ˜ ξ(k + 1) =A(k)ξ(k) + Bv(k) +B   ˜ ˜ ˜ ξ(k)ω(k), (12) e(k) =Cξ(k) + Dv(k) + Dr ey (k) + N   ˜ z(k) =Eξ(k) + F˜ v(k) + Geu (k),

Problem: The SFDC problem studied in this paper is to design a FD filter and a controller with an event-triggered strategy such that: 5

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• With the condition of the external input v(k) = 0, the closed-loop system (12) is robustly stochastically stable. • Under zero-initial condition, the error of the residual e(k) satisfies: ke(k)ke2 ≤ γ1 kv(k)ke2 ,

(13)

for all v(k) 6= 0.

kz(k)ke2 ≤ γ2 kv(k)ke2 , for all v(k) 6= 0.

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• Under zero-initial condition, the control output z(k) satisfies: (14)

For further discussion, the following lemma is helpful to derive the main results studied in this paper. Lemma 1. Suppose A, D, S and W > 0 are real matrices and F T (k)F (k) < I. Then, for any real scalar  satisfies W −1 − −1 DT D > 0 and:

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(A + DF (k)S)T W(A + DF (k)S) ≤ AT (W −1 − −1 DDT )−1 A + S T S.

3. Main results

In this section, a novel approach with the event-triggered strategy is proposed to solve the SFDC problem for uncertain discrete-time stochastic systems with limited communication. Three theorems are given to achieve the robust stochastic stability of the closed-loop system (12) with the desired fault detection and control objectives. In addition, an optimization algorithm is formulated to obtain the parameters of filter, controller and event detectors.

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3.1. Condition for fault detection objective In this subsection, a LMI is proposed to achieve the fault detection objective.

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Theorem 1. For given filter and controller parameter matrices Ar , Br , Cr , Dr , Kr and the thresholds of the event-triggered conditions in (6) − (7) 1 and 2 , the closed-loop system (12) is robustly stochastically stable and has the desired fault detection level γ1 for all v(k) 6= 0 and admissible uncertainties if there exist a symmetric matrix P > 0 and a scalar 3 > 0 such that the following LMI is satisfied:  ˜ ˜T ˜ ˜1 B ˜2 0 0 0 0 0 0 0 A˜ B B −P + −1 3 MM  −1 ˜ ∗ −η P 0 0 0 0 0 0 L 0 0 0   ˜ ∗ ∗ −I 0 0 0 0 0 C˜ D Dr 0   ˜ ∗ ∗ ∗ −η −1 I 0 0 0 0 N 0 0 0   −1  ∗ ∗ ∗ ∗ −1 I 0 0 0 C˜1 D0 0 0  ˜0  ∗ ∗ ∗ ∗ ∗ −(1 η)−1 I 0 0 N 0 0 0  −1 ˜  ∗ ∗ ∗ ∗ ∗ ∗ −2 I 0 Kr 0 0 0  −1  ˜1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ − I N 0 0 N 2 3  −1  ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −P 0 0 0   ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −γ12 I 0 0   ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −I 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −I

                    

< 0. (15)

where ˜0 = N ˜r = K





0

N

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0

   ˜1 = 0 N1 N2 Kr , C˜1 = 0 ,N    ˜ = 0 MT 0 T . Kr , M 0



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Proof 1. Firstly, we establish the robust stochastic stability of uncertain closed-loop system (12). For this objective, the system (12) with v(k) = 0 is considered as follows: ˜ ˜1 ey (k) + B ˜2 (k)eu (k) + Lξ(k)ω(k), ˜ ξ(k + 1) =A(k)ξ(k) +B

(16)

choose the following Lyapunov function for system (16): V (k, ξ(k)) = ξ(k)T P −1 ξ(k).

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Then, the difference of V (k, ξ(k)) can be obtained:

(17)

∆V (k, ξ(k)) =E {V (k + 1, ξ(k + 1) | ξ(k))} − V (k, ξ(k))   ˜ ˜1 ey (k) + B ˜2 (k)eu (k) T = A(k)ξ(k) +B   × P −1 A1 ξ(k) + A3 f (k) + A4 δ(k) ˜ T P −1 L ˜ − P −1 )ξ(k), + ξ(k)T (η L and set T

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 δ(k) = ξ(k)T

eTy

eTu

(18)

.

Considering the event-triggered strategy (6) − (7), we can rewrite (18) as:

∆V (k, ξ(k)) ≤ ∆V (k, ξ(k)) + 1 y T (k)y(k) − eTy ey + 2 uT (k)u(k) − eTu eu = δ(k)T Υ(k)δ(k),

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where

(19)

(20)

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   T   T T  T T  T ˜ ˜ A˜ (k) A˜ (k) −P −1 0 0 L L −1  ˜ T  −1  T ˜       ∗ −I 0 + Υ(k) = P +η 0 P 0  B1 B1 T T ˜ ˜ ∗ ∗ −I 0 0 B2 (k) B2 (k)    T          T T ˜T ˜0 ˜T ˜r C˜ T C˜ N N K K r 0  1   1          +1  0 +η 0 0 0 0 .  + 2 0 0 0 0 0 0 0

Based on Lemma 1 and (3) − (4), one can be deduced that:

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 T     T T  T    T A˜T (k) A˜ (k) ∆A˜T (k)  ∆A˜T (k)   A˜  A˜ ˜ T  P −1  B ˜ T  = B ˜T  +  ˜T  +   B  P −1 B  0 0 1 1 1   ˜1T   T T T T T ˜ ˜ ˜ ˜ ˜ ∆B2 (k) ∆B2 (k) B2 (k) B2 (k) B2 B2     T      ˜T     ˜T T ˜T T A N N   A˜T   1 1 T T T T −1 T T ˜ ˜ ˜ ˜         F (k)M P F (k)M = B1 + 0 B1 + 0       B   B ˜T ˜T N2T N2T 2 2  T   T T  T  T T ˜ ˜ A˜ A˜ N N 1 1 −1 T −1  ˜ T  T ˜ ˜ ˜      + 3 0 = B1 (P − 3 M M ) 0  . B1 ˜T ˜T N2T N2T B B 2 2 

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(21)

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0 0 0 0 −−1 2 I ∗ ∗ ∗ ∗

0 0 0 0 0 −−1 3 I ∗ ∗ ∗

A˜ ˜ L C˜1 ˜0 N ˜ Kr ˜1 N −P −1 ∗ ∗

˜1 B 0 0 0 0 0 0 −I ∗

˜2 B 0 0 0 0 N2 0 0 −I



      .      

(22)

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Then, by Schur complement, (20) can be deduced that:  ˜ ˜T −P + −1 0 0 0 3 MM −1  ∗ −η P 0 0  −1  ∗ ∗ − I 0 1   ∗ ∗ ∗ −(1 η)−1 I  Υ(k) =  ∗ ∗ ∗ ∗   ∗ ∗ ∗ ∗   ∗ ∗ ∗ ∗   ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

Thus, it is easily to deduce that Υ(k) < 0 from (15) and the system (12) is robustly stochastically stable. Then, we establish the condition (13) for system (12) with zero initial condition and define:   α(k) = ξ T (k) v T (k) eTy (k) eTu (k) , by multiplying both sides on (15) with α(k) and αT (k), we can obtain:

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∆V (k, ξ(k)) + e(k)T e(k) − γ12 v(k)T v(k) + 1 y T (k)y(k) − eTy ey + 2 uT (k)u(k) − eTu eu < 0. According to the event-triggered strategy (6) − (7), we can verify that:

∆V (k, ξ(k)) + e(k)T e(k) − γ12 v(k)T v(k) < 0.

(23)

In order to obtain the desired performance (13), summing both sides of (23) over [0, N ], one can get: (N ) X T 2 T (24) E (e(k) e(k) − γ1 v(k) v(k)) + V (N, ξ(N + 1)) − V (0, ξ(0)) < 0,

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where V (N, ξ(N + 1)) > 0 and V (0, ξ(0)) = 0 for the zero initial condition, we have: (N ) X T 2 T E (e(k) e(k) − γ1 v(k) v(k)) < 0,

(25)

k=0

therefore, (13) is satisfied and the proof is completed.

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Then, the follow theorem provides a solvability condition for fault detection objective.

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Theorem 2. Consider the uncertain stochastic model (1), for given constants η and 3 , if there exist symmetric matrices P˜1 > 0, P˜3 > 0, matrices P˜2 , X, Y , S1 , S2 , AR , BR , CR , Dr , KR and the event thresholds 1 , 2 such that:   E11 0 0 0 0 0 0 0 E13 E14 E15 E16 E17  ∗ −η −1 E22 0 0 0 0 0 0 0 E24 0 0 0    ˜  ∗ ∗ −I 0 0 0 0 0 0 E D D 0  34 r    ∗ ∗ ∗ −η −1 I 0 0 0 0 0 E44 0 0 0     ∗ ∗ ∗ ∗ −−1 0 0 0 0 E54 D0 0 0  1 I    ∗ ∗ ∗ ∗ ∗ −(1 η)−1 I 0 0 0 E64 0 0 0     ∗ ∗ ∗ ∗ ∗ ∗ −−1 0 0 E74 0 0 0  2 I   < 0, −1  ∗ ∗ ∗ ∗ ∗ ∗ ∗ −3 I 0 E84 0 0 N2     ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −3 I 0 0 0 0     ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ E94 0 0 0     ∗ 0 0  ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −γ12 I    ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −I 0  ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 −I (26) 8

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where 





   B1 0 E11 = E22 = − , E15 = , E16 = , Y B 1 + BR D 0 BR       A0 X + B0 KR A0 M0 B0 E14 = , E13 = , E17 = , AR Y A 0 + BR C 0 Y M0 Y B0     L0 X L0 E24 = , E34 = −CW X + CR Dr C0 − CW , S1 Y L 0 + B R N0       E44 = S2 Dr N0 , E54 = C0 X C0 , E64 = N0 X N0 ,       Aw 0 E74 = KR 0 , E84 = N10 X + N2 KR N10 , A0 = , 0 A       P˜1 − He(X) P˜2 − 2I 0 0 Bw E94 = , B = , B = , 0 1 B Bd Bf ∗ P˜3 − He(Y )           0 0 0 L0 = , C0 = 0 C , N0 = 0 N , D0 = Dd Df , M0 = , 0 L M         CW = Cw 0 , DW = 0 Dw , E0 = 0 E , N10 = 0 N1 . P˜2 P˜3

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P˜1 ∗

Furthermore, the desired parameter matrices of SFDC module (5) can be calculated by: Ar = V −1 (AR − Y A0 X − BR C0 X − Y B0 KR )U −1 , Br = V −1 BR , Cr = (CR − Dr C0 X)U −1 , Kr = KR U −1 ,

(27)

where V and U are nonsingular matrices and satisfy:

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V U = I − Y X.

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Proof 2. From the inequality (26), one notes that:    P˜1 P˜2 X + XT 0< < ˜ ∗ ∗ P3

2I Y +YT



(28)

,

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which means that X and Y are nonsingular. Then, it is easily to obtain that:    −1   −T  X + XT 2I X X −I = (Y X − I)X −1 + X −T (Y X − I)T > 0, −I ∗ Y +YT

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which implies that Y X − I is nonsingular. Thus, there exist nonsingular matrices V and U satisfying (28). Define     X (I − XY T )V −T I YT ,Φ = , S1 = Y L0 X + BR N0 X, Ξ= 0 VT U −U Y T V −T     (29) P1 P2 P˜1 P˜2 −T −1 P = =Ξ Ξ , S2 = Dr N0 X. ∗ P3 ∗ P˜3 Pre- and post-multiplying (26) by diag{Ξ−T , Ξ−T , I, I, I, I, I, I, I, Ξ−T , I, I, I} and diag{Ξ−1 , Ξ−1 , I, I, I,

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I, I, I, I, Ξ−1 , I, I, I}, respectively, and based on (29), one can obtain the following matrix inequality:  ˜ ˜ ˜ ˜1 −P 0 0 0 0 0 0 0 M AΦ B B  ∗ −1 ˜ −η P 0 0 0 0 0 0 0 LΦ 0 0   ∗ ˜ ˜ ∗ −I 0 0 0 0 0 0 CΦ D D  r  ˜Φ ∗ ∗ −η −1 I 0 0 0 0 0 N 0 0  ∗   ∗ ∗ ∗ ∗ −−1 0 0 0 0 C˜1 Φ D0 0 1 I  −1 ˜0 Φ  ∗ ∗ ∗ ∗ ∗ −( η) I 0 0 0 N 0 0 1  ˜rΦ  ∗ ∗ ∗ ∗ ∗ ∗ −−1 0 0 K 0 0 2 I  −1  ∗ ˜1 Φ ∗ ∗ ∗ ∗ ∗ ∗ − I 0 N 0 0 3   ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ − I 0 0 0 3  T  ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ P − Φ − Φ 0 0   ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −γ12 I 0   ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −I ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0

˜2 B 0 0 0 0 0 0 N2 0 0 0 0 −I

(30)

Then, we can note that (P − Φ)T P −1 (P − Φ) ≥ 0 implies −ΦT P −1 Φ ≤ P − Φ − ΦT . Similarly, perform congruence transformations to (30) by diag{I, I, I, I, I, I, I, I, I, Φ−1 , I, I, I}. we can obtain (15) from (30). This completes the proof. 3.2. Condition for control objective

Similar to the previous subsection, the control objective can be achieved by the follow theorem.

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Theorem 3. Consider the uncertain stochastic model (1), for given constants γ2 and 3 , the closed-loop system (12) is robustly stochastically stable and satisfies the desired control performance (14) if there exist symmetric matrices P˜1 > 0, P˜3 > 0, matrices P˜2 , X, Y , S1 , S2 , AR , BR , CR , Dr , KR and the event thresholds 1 , 2 such that the following matrix inequality holds:   E11 0 0 0 0 0 0 E13 E14 E15 E16 E17  ∗ −η −1 E22 0 0 0 0 0 0 E24 0 0 0    ¯34 ˜   ∗ ∗ −I 0 0 0 0 0 E F 0 G   −1  ∗  ∗ ∗ − I 0 0 0 0 E D 0 0 54 0 1   −1   ∗ ∗ ∗ ∗ −( η) I 0 0 0 E 0 0 0 1 64   −1  ∗  ∗ ∗ ∗ ∗ − I 0 0 E 0 0 0 74 2  < 0,  −1  ∗ ∗ ∗ ∗ ∗ ∗ −3 I 0 E84 0 0 N2     ∗ ∗ ∗ ∗ ∗ ∗ ∗ −3 I 0 0 0 0     ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ E94 0 0 0     ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −γ22 I 0 0     ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −I 0  ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 −I (31) where   ¯34 = E0 X + GKR E0 , E and the other matrix parameters are the same as that shown in Theorem 2

Proof 3. The proof is similar to the proof of Theorem 1 and 2 and is thus omitted.

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                      

< 0.

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3.3. Algorithm The following algorithm is proposed to obtain the parameters of fault detection filter, controller and event detectors. Step 1: According to Theorem 2-3, for given constants η and 3 , we can obtain the matrix parameters X, Y , AR , BR , CR , DR and KR by solving the following minimization problem with LMI tools. min

γ1 + γ2 + u1 + u2

s.t. (26), (31).

V U = I − Y X.

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where γ1 and γ2 represent the optimal fault detection performance level and control performance level, respectively; 1 = u−1 and 2 = u−1 represent the maximum communication thresholds of two event 1 2 detectors. Step 2: Based on the parameters X and Y obtained in Step 1, one can compute V and U by:

Step 3: From the Equation (27), one can obtain the desired fault detection filter and controller parameters.

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4. Illustrative example

In this section, a physical motion system [39] is considered to demonstrate the effectiveness of the proposed design strategy. The motion system is a single axis linear motor driven system and the dynamic equation is described by:        0 0 1 x1 x˙ 1 + Kv Kf u, (32) = B x2 x˙ 2 0 −M M

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the physical meanings of model parameters are given as:

x1 , actual position of the moving platform x2 , actual speed of the moving platform

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B , the viscous friction coefficient M , the mass of the moving platform Kv , the current amplification coefficient

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Kf , the proportion constant of the armature current

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According to the model identification method described in [39], we can get the above continuous-time model parameters. Under sampling period T = 0.05s, the discrete-time model parameters can be obtained as follows:     1 0.0335 0.0172 A= ,B = . 0 0.4209 0.6016 Due to the restriction of the identification technology, the model uncertainty always exists. In addition, there also exists the external disturbance, stochastic perturbations and possible faults. We assume other parameters in (1) as follows:       0.2 0.1 0.5 0.4 L= Bd = , Bf = , 0 −0.1 0.2 0.6     C = 1 0 , N = 0.1 0.3 , Dd = 0.6, Df = 0.8,   E = 0.5 0 , G = 0.2, Fd = 0.1, Ff = 0.1,     0.2 0.05 M= , N1 = , N2 = 0.1. 0.1 0.1 11

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The parameters of the fault weighting system are assumed to be: Aw = 0.5, Bw = 0.25, Cw = 1.0, Dw = 0.5.

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In the example, the stochastic process ω(k) is assumed to satisfy (2) with η = 0.01, 3 = 1.2. Using the algorithm 3.3, the parameters of fault detection filter and controller (5) can be calculated by:     0.5000 −0.0020 0.0010 0.0161 Ar =  0.0000 0.0434 0.0277  , Br =  0.9404  , 0.0000 −1.8924 0.3048 0.3485   Cr = 1.0000 0.0079 0.0031 , Dr = 0.0229,   Kr = 0.0000 −2.6471 −0.1949 .

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the performance indexes γ1 = 3.5957, γ2 = 3.6761 and the event detector parameters 1 = 0.0148, 2 = 0.0065. In the following, we choose the following external disturbance signal d(k) and the fault signal f (k) to exhibit the effectiveness of the obtained fault detection filter and controller: ( 0.5 ∗ sin(k), 30 ≤ k ≤ 170, d(k) = 0, else. ( 1, 50 ≤ k ≤ 150, f (k) = 0, else.

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The simulation results are shown in Fig.2-Fig.7. Among them, the measured output and controlled output are shown in Fig.2. It is observed that the controlled output z(k) is robust to both the external disturbance and the fault using the designed controller. The detection threshold and the residual evaluation curves under fault case and fault free case are shown in Fig.3. Based on (10) − (11), we can conclude the designed detection threshold Jth = 1.7705 × 10−3 and Jr (50) = 1.8848 × 10−3 > Jth . Thus, the fault will be detected when it occurs at k=50. The control signals and residual signals under the event-triggered strategy are exhibited in Fig.4 and Fig.5, respectively. Fig.6 and Fig.7 demonstrate release instants of sensor and control data, respectively. From Fig.6, only 96 sensor data packets are transmitted to the SFDC module under the event-triggered strategy and the sensor data transmission rate is 48%. It is also observed that 103 control data packets are transmitted to the plant and the control data transmission rate is 51.5% from the Fig.7. Above all, the designed fault detection filter and controller can ensure the robust stochastic stability of the closed-loop system with desired control objective (see Fig.2) and fault detection objective (see Fig.3). In addition, the amount of the sensor and control data transmitted through the network are decreased efficiently (see Fig.6-Fig.7). The network communication resources are saved significantly.

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z(k) y(k)

5 4 3 2 1 0 −1 0

50

100 Time in samples(k)

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Measured output and controlled output

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0.014

Fault case Fault free Threshold

0.012

0.008 0.006

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Residual evaluation function

0.01

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Figure 2: Measured output and controlled output

0.004

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0.002 0 0

50

100

Time in samples(k)

150

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Figure 3: Residual evaluation and threshold

u(k)

2 0

Control input signals

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4

−2 −4 −6 −8 −10 −12 −14 0

50

100 Time in samples(k)

150

Figure 4: Control input signals

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0.3

r(k)

0.25

Residual signal

0.2 0.15

0.05 0 −0.05 0

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100 Time in samples(k)

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30

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Figure 5: Residual output signals

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5

0 0

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100 Time in samples(k)

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Figure 6: Release time intervals of sensor data

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Release time intervals

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25 20 15 10 5 0 0

50

100 Time in samples(k)

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Figure 7: Release time intervals of control signals

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To make a comparison between the proposed design method with the existing time-triggered results for uncertain stochastic systems, we obtain the fault detection filter and controller based on the methods in [20, 40]. The comparison results are shown in Table I, from which we can conclude that: (1) the desired fault detection and control objectives can be achieved by transmitting 96 sensor data and 103 control data, while the transmission rate of sensor data and control data are 48% and 51.5%, respectively; (2) Compared with the existing results in [20, 40], the proposed strategy can achieve better resource utilization by sacrificing some fault detection and control performance. Moreover, the proposed strategy is helpful to reduce computation and increase the lifetime of the overall system; Table 1: The comparison between the proposed method and the existing one

Design methods Release number for sensor data Release number for control data Fault detection performance Control performance

The method in [20, 40] 200 200 3.4299 3.5696

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5. Conclusion

The proposed method 96 103 3.5957 3.6761

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In this paper, the SFDC problem has been investigated for uncertain discrete-time stochastic systems with limited communication. Firstly, an event-triggered strategy with two event detectors is introduced to decrease the amount of sensor and control data that are transmitted through the network. A more efficient utilisation of limited resources in real network system can be achieved. Secondly, an integrated fault detection filter and dynamic output feedback controller is designed to achieve the robust stochastic stability of the closed-loop system and guarantee the desired fault detection and control objectives. Then, under these considerations, a novel algorithm is presented and adopted to co-design the desired filter, controller and the event detectors parameters. Finally, a physical motion system example is provided to demonstrate the effectiveness of this design methodology.

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Acknowledgment

6. References References

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This work was partially supported by the National Natural Science Foundation of China (61673133).

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