filtering for multi-rate multi-sensor systems with randomly occurring sensor saturations under the p-persistent CSMA protocol

IET Control Theory & Applications Research Article

H

filtering for multi-rate multi-sensor systems with randomly occurring sensor saturations under the p-persistent CSMA protocol

ISSN 1751-8644

Received on 18th January 2019 Revised 13th November 2019 Accepted on 2nd January 2020 E-First on 14th May 2020 doi: 10.1049/iet-cta.2019.0085 www.ietdl.org

Yuxuan Shen

, Zidong Wang

, Bo Shen

, Fuad E. Alsaadi

1College of Information Science and Technology, Donghua University, Shanghai 200051, People's Republic of China

2The Engineering Research Center of Digitalized Textile and Fashion Technology, Ministry of Education, Shanghai 200051, People's Republic of China

3Department of Computer Science, Brunel University London, Uxbridge, Middlesex UB8 3PH, UK

4Department of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia E-mail: [email protected]

Abstract: In this study, the H_∞ filtering problem is studied for a class of discrete networked multi-rate multi-sensor systems with randomly occurring sensor saturations under the p-persistent carrier sense multiple access (CSMA) protocol. A set of mutually independent Bernoulli distributed white sequences is introduced to characterise the random occurrence of the sensor saturations. The p-persistent CSMA protocol is employed to decide which sensor is allowed to transmit its measurement to the filter at a certain time instant. By using the lifting technique, the multi-rate system is converted to a single-rate one for convenient analysis. The main purpose of the addressed problem is to design an H_∞ filter such that the filtering error dynamics is exponentially mean-square stable and the H_∞ performance requirement is satisfied simultaneously. Sufficient conditions are established on the existence of the desired H_∞ filters and the corresponding filter gains are then characterised by resorting to the feasibility of certain matrix inequalities. Finally, a numerical example is given to illustrate the effectiveness of the proposed filtering scheme.

1 Introduction

The past few decades have witnessed a fast development of the networked control systems (NCSs) due mainly to their extensive applications in many areas such as power grids [1], industrial control systems [2], traffic systems [3] and unmanned aerial vehicles [4]. As a result, the control and filtering problems of the NCSs have received considerable research interest and fruitful results have been available in the literature, see e.g. [5–15] and the references therein. For instance, in [5], the filtering problem has been investigated for the impulsive NCSs with random packet dropouts and randomly occurring non-linearities. In [7], the controller design problem has been studied for the NCSs with stochastic delays, where a delay compensation control approach has been proposed to deal with the network-induced delays. The problem of event-triggered fuzzy filtering has been studied in [9]

for the NCSs by developing a co-design method to guarantee the asymptotic stability and H_∞ performance requirement on the filtering error dynamics. In [11], the estimation problem has been discussed for a class of sampled linear systems where the measurements are subject to random time-delays and packet dropouts.

In the NCSs, the components are connected through the communication networks of limited communication capability due to physical constraints such as limited bandwidths [16–22].

Consequently, a great deal of research effort has been devoted to the so-called network-induced phenomena including network- induced time-delays [23, 24], missing measurements [25, 26], signal quantisations [27] and channel fadings [28, 29] that have posed significant challenges to the filtering problem for the NCSs.

As one of the most frequently encountered network-induced phenomena, the sensor saturation is caused by many reasons such as physical limits of the sensors and imperfect communication environments. Sensor saturation is inherently a kind of non-linear phenomena which, if not adequately handled, would deteriorate the system performance or even result in the instability. Recently, the filtering problem under sensor saturation has gained particular

research interest and many important results have been reported, see e.g. [30–36]. Nevertheless, in the literature concerning sensor saturation, it has been implicitly assumed that the occurrence of sensor saturations is deterministic, that is, the sensor always undergoes saturation. Such an assumption, however, does have its limitation when sensors are deployed in unattended environments such as power grids where sensors might frequently encounter some transient phenomena. Under this circumstance, the sensor saturations may occur in a probabilistic way and are randomly changeable in terms of their types and/or levels due to the random occurrence of network-induced phenomena such as random sensor failures, sensor ageing, or sudden environment changes [37]. This gives rise to the so-called randomly occurring sensor saturation (ROSS) that has been largely overlooked in the literature in spite of its significant engineering background. Therefore, it is of great importance to examine the influence of ROSS on the corresponding filter performance, which constitutes partial motivation of our current research.

It is worth mentioning that most existing NCS-related literature has been concerned with single-rate systems where the system plants and the measurement outputs are sampled at the same rate.

In practice, however, it is sometimes impossible to sample all the signals with a uniform rate due mainly to different types and physical features of the system plant and the sensors. Moreover, according to the engineering requirements, the sampling can be conducted at different rates. For example, a higher-rate sampling can help enhance the system performance at the cost of consuming more resource, while the sampling with a lower rate can save the implementation cost by sacrificing certain performance. Based on the above facts, using different sampling rates for different actuators/sensors has proven to be an effective yet economic way of achieving the performance specifications.

The multi-rate sampling strategy has been widely used in various practical applications, such as structural health monitoring [38], aluminium electrolysis cells [39] and power networks [40].

For example, in the structural health monitoring, increasing the sampling rate would improve signal-to-noise ratio of sampling

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quantisation, but the large data stream after the high sampling rate leads to a low signal processing speed. In the health monitoring system, real-time data processing is demanded, so it is necessary to reduce the processing speed of data stream after A/D conversion, and therefore the multi-rate sampling strategy is used. Due to the wide applications of the multi-rate sampling strategy, the corresponding filtering problem has gained much research attention in the past decade, see e.g. [41–45]. For example, a set of local event-triggered filters has been constructed in [41] for multi-rate systems with stochastic non-linearities and coloured measurement noises. In [44], the variance-constrained state estimation problem has been investigated for multi-rate systems with probabilistic sensor failures and quantisation effects.

Despite the fruitful results available for NCSs, the filter design problem for NCSs with communication protocols has received inadequate research attention. In real engineering practice, due to the limited capability of the shared communication network, data collisions occur frequently and not all the sensors are able to get the access to transmission simultaneously. As such, various communication protocols have been introduced to schedule the sensor accesses to certain shared networks according to some given priorities. Consequently, the control/filtering problems for NCSs with communication protocols have recently become quite hot research topics. For example, an observer-based controller has been designed in [46] for NCSs where the stochastic communication protocol is employed in both the sensor-to-controller network and the controller-to-actuator network. In [47], the exponential stability has been analysed for discrete-time linear systems with the Round- Robin protocol.

As one of the most widely used random access protocols, the p- persistent carrier sense multiple access (CSMA) protocol has been applied in communication networks such as Ethernet and 802.11a/b/g in many real-world applications [48]. For example, the p-persistent CSMA protocol is used to regulate the communication between the train and the train control centre in the communication-based train control systems [49]. The p-persistent CSMA protocol has a simple feature which is ‘listen before talk’, that is, the sensors always check if the network is busy before transmitting measurement. p-persistent CSMA protocol is different from those protocols (e.g. the Round-Robin protocol) which schedule sensors with a global policy. Under the p-persistent CSMA protocol, all the sensors are assigned with identical priority and compete with each other for the access of transmission at each time instant. It should be pointed out that the sensors in multi-rate NCS have different sampling periods, and therefore the number of the sensors participating the competition for the transmission access at each time instant is not a constant. Consequently, the probability that a certain sensor gets the access of transmission is time varying according to the p-persistent CSMA protocol. Such kind of dynamically changing probability brings substantial challenges to the filter design problem for multi-rate NCSs and it is the main motivation of this paper to overcome these challenges by investigating the filtering problem for multi-rate NCSs under the p- persistent CSMA protocol.

In this paper, the H_∞ filtering problem is studied for multi-rate NCSs with ROSSs under the p-persistent CSMA protocol. Our purpose is to design an H_∞ filter such that the filtering error dynamics is exponentially mean-square stable and the H_∞ performance requirement is guaranteed. By employing the Lyapunov stability theory, sufficient conditions are established to ensure the existence of the desired filter and the characterisation of the desired filter gains is then derived. Finally, a simulation example is presented to illustrate the effectiveness of the proposed filtering approach. The main contributions of this paper are highlighted as follows: (i) a combination of important factors contributing to the complexity of NCSs is investigated in a unified framework which comprises the multi-rate sampling, the ROSSs and the p-persistent CSMA protocol; (ii) a novel analytical method is developed to deal with the time-varying probability issue caused by the coupling of the multi-rate sampling and the p-persistent CSMA protocol; and (iii) the stochastic analysis approach is conducted to derive sufficient condition that ensures the

exponential mean-square stability as well as the H_∞ performance constraint.

Notations. The notation used here is fairly standard except where otherwise stated. ℝⁿ denotes the n-dimensional Euclidean space. I denotes the identity matrix of compatible dimension. A^T represents the transpose of A. ∥ x ∥ describes the Euclidean norm of a vector x. l2([0, ∞); ℝⁿ) is the space of square summable n- dimensional vector-valued functions. col{…} represents a column vector composed of elements and colnj=m{A(j)} = col{A(m), A(m + 1), …, A(n)}. diag{⋯} stands for a block-diagonal matrix, diagn{A} stands for diag{A, …, A}

n

and diagnj=m{A(j)} = diag{A(m), A(m + 1), …, A(n)}. E{x} stands for the expectation of stochastic variable x. ∑_{i, j} stands for

∑_{i = 1}^m ∑^q_{j = 0}^{i− 1}, ∑_{ij, ln} stands for ∑_{i = 1}^m ∑_{l = 1}^m ∑^q_{j = 0}^{i− 1}∑_{n = 0}^{ql− 1} and

∑_{ij ≠ ln} stands for ∑_{i = 1}^m ∑_{l = 1}^m ∑^q_{j = 0}^{i− 1}∑_{n = 0}^{ql− 1} (excluding the case for i = l and j = n).

2 Problem formulation and preliminaries Consider the following discrete-time system:

x(Tk+ 1) = Ax(Tk) + Bω(Tk)

z(Tk) = Mx(Tk) (1)

with m-sensor measurements under randomly occurring saturations yi(tki) = βi(tki)σ(Cix(tki)) + (1 − βi(tki))Cix(tki)

+Divi(tki) (i = 1, 2, …, m) (2) where x(Tk) ∈ ℝ^nx is the state vector, y_i(tki) ∈ ℝ^ny is the measurement output from sensor i, z(Tk) ∈ ℝ^nz is the signal to be estimated, ω(Tk) is the process noise belonging to l2([0, ∞), ℝ^nω), vi(tki) is the measurement noise for sensor i belonging to l2([0, ∞), ℝ^nv), and A, B, Ci, Di and M are known matrices with appropriate dimensions.

In this paper, to cater for the engineering practice, it is assumed that the state updating period of the system and the sampling periods of the sensors are allowed to be different. The state updating period for (1) is denoted by h ≜ Tk+ 1− Tk, and the sampling periods for (2) are integer multiples of the state updating period for (1), i.e. tki+ 1− tki= bih where bi (i = 1, 2, …, m) are positive integers. That is, the measurement from the ith sensor is generated at time instants t0i, t1i, …, tki, …, while the state of the system updates at time instants T0, T1, …, Tk, ….

The saturation function σ( ⋅ ):ℝ^ny→ ℝ^ny is defined as

σ(s) ≜ σ(s1) σ(s2) ⋯ σ(sny)^T (3) with

σ(sl) = sign(sl) min {sl, max, sl}, l = 1, 2, …, ny

where sign( ⋅ ) is the signum function and sl, max is the lth element of the saturation level.

The stochastic variables β_i(tki) (i = 1, 2, …, m) are mutually independent Bernoulli distributed white sequences taking values on 0 or 1 with

Prob{βi(tki) = 1} = μi

Prob{βi(tki) = 0} = 1 − μi

where μ_i∈ [0, 1] (i = 1, 2, …, m) are known constants.

1256 IET Control Theory Appl., 2020, Vol. 14 Iss. 10, pp. 1255-1265

In model (2), the stochastic variable βi(tki) is used to describe the random occurrence of the sensor saturation on the ith sensor.

β_i(tki) = 1 represents that the sensor saturation occurs on the ith sensor and β_i(tki) = 0 otherwise. Generally, the probability of the occurrence of the sensor saturation for a fixed sensor can be obtained via statistical experiments during a period of time.

Remark 1: In reality, owing to the fast-growing and the wide application of the digital technologies, there are more and more systems that can be described as discrete-time system. As such, the discrete-time system can closely reflect the engineering practice.

On the other hand, almost all practical systems are time varying and contain non-linearity which should be described as non-linear time-varying systems. Nevertheless, most of the non-linear time- varying systems can be approximately represented by the LTI systems with sufficient accuracy. Therefore, many practical systems can be described as the LTI systems. For example, the state-space model of the continuous stirred tank reactor (CSTR), the most generally employed bioreactor for biohydrogen production, can be described as a discrete LTI system [50]. The phenomenon of ROSS exists extensively in the practical engineering and the measurement model (2) is capable of closely reflecting such a phenomenon. Moreover, the measurement model (2) is transformed to the commonly used measurement model in [12] when sl, max approaches infinity. As such, the proposed measurement model in this paper is more general.

Remark 2: In real world, the multi-rate sampling strategy (sampling different signals with different rates) owns certain advantages from the engineering perspective and has been widely used in various practical applications [41]. To cater for the engineering practice, in this paper, it is assumed that the sampling rates of the sensors and the state evolving rate of the system are different. Note that, the developed filtering algorithm for multi-rate systems can be easily extended to traditional single-rate systems by simply letting bi= 1.

In this paper, the data transmissions from the sensors to the filter are realised by a communication network which, due to limited bandwidth, only allows one sensor to transmit its measurement at each time instant. The p-persistent CSMA protocol is used to schedule the m sensors with different measurement periods. Under the p-persistent CSMA protocol, when a sensor has measurement to transmit (i.e. at its sampling instant), it senses the communication network for idle or busy first. If the network is idle, the sensor begins to transmit the measurement with a probability p.

If the network is busy, the sensor does not transmit the measurement. When more than one sensor senses that the network is idle and begins to transmit simultaneously, a collision occurs and, in this situation, all the sensors abort their transmissions immediately, which means that no sensor gets the access of transmission.

Let y^¯i(tki) represent the received measurement of the filter from sensor i after transmitted through the communication network, then we have

y^¯i(tki) = αi(tki)βi(tki)σ(Cix(tki)) + αi(tki)Divi(tki)

+αi(tki)(1 − βi(tki))Cix(tki) (4) where the stochastic variable α_i(tki) is a Bernoulli distributed white sequence taking values on 0 or 1. α_i(tki) = 1 means that the transmission of measurement from sensor i at time instant tki is successful, and αi(tki) = 0 means that the transmission at time instant tki fails.

According to the above description of the p-persistent CSMA protocol and noting that the communication channel is idle at the beginning of each sampling instant, it is easily known that the probability of a sensor successfully transmitting its measurement at a given time instant is p(1 − p)^{N− 1} where N is the number of the sensors wanting to transmit data at this time instant [51]. Then, we know

Prob{αi(tki) = 1} = p(1 − p)^{N(tki ) − 1}

Prob{αi(tki) = 0} = 1 − p(1 − p)^{N(tki ) − 1} (5) where N(tki) ∈ {1, 2, …, m} represents the number of the sensors which try to transmit their measurements at time instant tki. It can be derived from (5) that

νi(tki) ≜ E{αi(tki)} = p^¯+ Δi(tki)

where p^¯= 1/2(p + p(1 − p)^{m− 1}) and Δi(tki) = p(1 − p)^{N(tki ) − 1}− p^¯. Then, we have

Δi(tki) ≤ Δ^¯, νi(tki) ≤ ν^¯ (6) where Δ^¯ = 1/2(p − p(1 − p)^{m− 1}) and ν^¯= p^¯+ Δ^¯.

According to the p-persistent CSMA protocol, we know that E{αi(tki)αj(tkj)} (i, j ∈ {1, 2, …, m}) satisfies

E{αi(tki)αj(tkj)} =

νi(tki)νj(tkj), if tki≠ tkj

0, if tki= t_k^j, i ≠ j ν_i(tki), if tki= t_k^j, i = j .

Noting (6), we have

E{αi(tki)αj(t_k^j)} − νi(tki)νj(t_k^j) ≤ ν^¯. (7)  

Remark 3: In the NCSs, the system components are connected through communication networks of limited communication capability. As such, data collisions may occur when more than one component transmits information through communication network simultaneously. Nevertheless, data collisions can be prevented through the utilisation of communication protocols that decide which component can transmit its data at a certain time instant. The p-persistent CSMA protocol is often used in Ethernet, which is a widely employed communication network in industry, to prevent the data from collisions [51]. Therefore, the networked system with p-persistent CSMA protocol studied in this paper caters for many industrial systems in a networked environment.

Remark 4: In this paper, the sensor itself in the multi-rate NCS (1) and (2) has individual sampling period and, subsequently, the number of the sensors competing for the access of measurement transmission is time varying (at different time instant) which, in turn, renders the time-varying probability that a certain sensor gets the access of transmission under the p-persistent CSMA protocol.

This kind of time-varying probability (i.e. the mathematical expectation due to the fact that the stochastic variable is Bernoulli distributed) creates a great deal of difficulty in the subsequent filter design. In (5)–(7), we have analytically given certain bounds on the time-varying probabilities, thereby facilitating the filter design in sequel.

Recall that system (1) evolves with a period h, while the measurements (4) are sampled with periods b_ih. Accordingly, the system under consideration is essentially a multi-rate system that largely complicates the system analysis/synthesis issues. An effective way to deal with the multi-rate system is to convert it to a certain single-rate system by using the well-known lifting technique.

By defining the least common multiple of bi(i = 1, 2, …, m) as L, we set q_i≜ L/bi and denote the sampling instant of the derived single-rate system as Tk, i.e. Tk+ 1− Tk= Lh. Then, it is obtained from (1) and (4) that

IET Control Theory Appl., 2020, Vol. 14 Iss. 10, pp. 1255-1265 1257

x^¯(Tk+ 1) = A^¯x^¯(Tk) + B^¯ω^¯(Tk) y^~_i(Tk) = α^¯i(Tk)β^¯i(Tk)σ^¯(C^¯ix^¯(Tk))

+α^¯i(Tk)(I − β^¯i(Tk))C^¯ix^¯(Tk) +α^¯i(Tk)D^¯iv^¯i(Tk)

z^¯(Tk) = M^¯x^¯(Tk)

(8)

where

x^¯(Tk) = colLj= 0− 1{x(Tk+ jh)}, z^¯(Tk) = colLj= 0− 1{z(Tk+ jh)}, v^¯i(Tk) = colqji − 1= 0{vi(Tk+ jbih)}, C^¯i= colqji − 1= 0{Ci,j}, y^~_i(Tk) = colqji − 1= 0{y^¯i(Tk+ jbih)}, D^¯i= diagqi{Di}, ω^¯(Tk) = col2jL=− 2L− 1{ω(Tk+ jh)}, M^¯ = diagL{M}, α^¯i(Tk) = diagqji − 1= 0{αi(Tk+ jbih)},

β^¯_i(Tk) = diagqji − 1= 0{βi(Tk+ jbih)}, σ^¯(C^¯ix^¯(Tk)) = colqji − 1= 0{σ(Cix(Tk+ jbih))}, A^{^}= col{A, A², …, A^L}, A^¯ = 0 0 ⋯ A^{^}, B^¯= col{BL, BL− 1, …, B1},

C_i,j= 0 ⋯ 0

j × bi

C_i 0 ⋯ 0 ,

B_j= A^L−jB A^{L−j− 1}B ⋯ B 0 ⋯ 0 . Next, by denoting

y^~(Tk) ≜ colmj= 1{y^~j(Tk)}, v^¯(Tk) ≜ colmj= 1{v^¯Tj(Tk)}, α^¯(Tk) ≜ diagmj= 1{α^¯j(Tk)}, β^¯(Tk) ≜ diagmj= 1{β^¯j(Tk)}, C^¯ ≜ colmj= 1{C^¯j}, D^¯ ≜ diagmj= 1{D^¯j},

σ^~(C^¯x^¯(Tk)) ≜ colmj= 1{σ^¯(C^¯jx^¯(Tk))},

the system (8) can be rewritten in the following compact form:

x^¯(Tk+ 1) = A^¯x^¯(Tk) + B^¯ω^¯(Tk) y^~(Tk) = α^¯(Tk)(I − β^¯(Tk))C^¯x^¯(Tk)

+α^¯(Tk)β^¯(Tk)σ^~(C^¯x^¯(Tk)) +α^¯(Tk)D^¯v^¯(Tk) z^¯(Tk) = M^¯x^¯(Tk) .

(9)

In this paper, the following filter is adopted:

x^{^}(Tk+ 1) = Kx^{^}(Tk) + Hy^~(Tk)

z^{^}(Tk) = M^¯x^{^}(Tk) (10) where H and K are the filter gains to be designed, x^{^}(Tk) is the estimate of x^¯(Tk) and z^{^}(Tk) is the estimate of z^¯(Tk).

Now, letting z^~(Tk) = z^¯(Tk) − z^{^}(Tk) and denoting

η(Tk) ≜ x^¯(Tk)

x^{^}(Tk), B^~≜ B^¯ 0

0 Hp^~D^¯ , ϖ(Tk) ≜ ω^¯(Tk) v^¯(Tk) ,

A^~≜ A^¯ 0

Hp^~(I − β^~)C^¯ K, E^~≜ 0

Hp^~β^~, F^¯i,j≜ 0 HF^→i,j, M^~ ≜ M^¯ −M^¯ , Λ ≜ I 0 , Λ^¯ ≜ 0 I ,

p^~≜ diagϱ{p^¯}, μ^¯i≜ diagqi{μi}, β^~≜ diagmj= 1{μ^¯j}, s_i,j≜

∑

a = 1

i q_a− (qi− j), ϱ ≜

∑

i = 1 m q_i,

F^→i,j≜ diag{0, ⋯, 0

s_{i, j}

, 1, 0, ⋯, 0},

an augmented system that governs the filtering error dynamics of the filter is obtained as

η(Tk+ 1) = A^~η(Tk) + B^~ϖ(Tk) + E^~σ^~(C^¯Λη(Tk))

+

∑

i, jΔi(Tk+ jbih)(1 − μi)F^¯i,jC^¯Λη(Tk)

+

∑

i, jΔi(Tk+ jbih)μiF^¯_i,jσ^~(C^¯Λη(Tk))

+

∑

i, jΔi(Tk+ jbih)F^¯i,jD^¯Λ^¯ϖ(Tk)

+

∑

i, j(αi(Tk+ jbih)(1 − βi(Tk+ jbih))

−νi(Tk+ jbih)(1 − μi))F^¯i,jC^¯Λη(Tk)

+

∑

i, j(αi(Tk+ jbih)βi(Tk+ jbih)

−νi(Tk+ jbih)μi)F^¯i,jσ^~(C^¯Λη(Tk))

+

∑

i, j(αi(Tk+ jbih)

−νi(Tk+ jbih))F^¯i,jD^¯Λ^¯ϖ(Tk) z^~(Tk) = M^~η(Tk) .

(11)

Definition 1: The augmented system (11) with ϖ(Tk) = 0 is said to be exponentially mean-square stable if there exist constants λ > 0 and 0 < ℏ < 1 such that

E{∥ η(Tk) ∥²} ≤ λℏ^TkE{∥ η(T0) ∥²} .

The main purpose of this paper is to design the filter in the form of (10) such that the following requirements are satisfied simultaneously:

1. the zero-solution of the augmented system (11) with ϖ(Tk) = 0 is exponentially mean-square stable; and

2. under zero initial condition, for a given disturbance attenuation level γ > 0 and all non-zero ϖ(Tk), the filtering error z^~(Tk) satisfies the following condition:

k = 0

∑

∞ E{∥ z^~(Tk) ∥²} < γ²

∑

k = 0

∞ ∥ ϖ(Tk) ∥². (12)

3 Main results

In this section, the exponentially mean-square stability and the H_∞ performance are first analysed for system (11) and the filter design problem is then studied.

Along the similar line in [32], we know from (3) that there exists a diagonal matrix Ω satisfying 0 < Ω < I and

σ^~(C^¯Λη(Tk)) − ΩC^¯Λη(Tk)^T

× σ^~(C^¯Λη(Tk)) − C^¯Λη(Tk) ≤ 0. (13) The following lemma will be used in obtaining our main results.

Lemma 1: (See the work of Guan et al [52]). For any real matrices Xi,j and Yl,n, constants gi,j and hl,n

(1 ≤ i, l ≤ m, 0 ≤ j ≤ qi− 1, 0 ≤ n ≤ ql− 1), and a symmetric positive definite matrix S, we have

2

∑

i = 1

m

∑

l = 1

m

∑

j = 0 qi− 1

n = 0

∑

ql− 1

gi,jhl,nXiT,jSYl,n

≤ ϱ

∑

i = 1

m

∑

j = 0 qi− 1

g_i²_,jX_i^T_,jSX_i,j+ ϱ

∑

i = 1

m

∑

j = 0 qi− 1

h_i²_,jY_i^T_,jSY_i,j.  

1258 IET Control Theory Appl., 2020, Vol. 14 Iss. 10, pp. 1255-1265

Theorem 1: Let the filter gains H, K and the matrix Ω be given.

The zero-solution of the augmented system (11) with ϖ(Tk) = 0 is exponentially mean-square stable if there exist a positive definite matrix P and a positive scalar ε satisfying

Γ^¯ = Γ^¯11 A^~TPE^~+ εΛ^TC^¯T(I + Ω)/2

∗ Γ^¯22

< 0 (14)

where

Γ^¯11=

∑

i, j 2ς^¯i,j+ 2ϱς^¯i²,j+ 2ϱ^¯ν^¯(1 − μi)²+ ϑ^¯i¹,j+ ϑ^¯i²,j

× Λ^TC^¯TF^¯iT,jPF^¯i,jC^¯Λ +

∑

i, j τ^¯i,j+ ς^¯i,jA^~TPA^~

−εΛ^TC^¯TΩC^¯Λ + A^~TPA^~− P, Γ^¯22=

∑

i, j 2τ^¯i,j+ 2ϱτ^¯i²,j+ 2ϱ^¯ν^¯μ_i²+ ϑ^¯i²,j+ ϑ^¯i³,jF^¯iT,jPF^¯i,j

+

∑

i, j τ^¯i,j+ ς^¯i,jE^~TPE^~+ E^~TPE^~− εI, ςi,j= Δi(Tk+ jbih)(1 − μi), ς^¯i,j= Δ^¯(1 − μi), τi,j= Δi(Tk+ jbih)μi, τ^¯i,j= Δ^¯μ_i, ϱ^¯= ϱ − 1, ϑi¹,j= νi(Tk+ jbih)(1 − μi) − νi²(Tk+ jbih)(1 − μi)², ϑ^¯i¹,j= ν^¯(1 − μi), ϑi²,j= νi²(Tk+ jbih)μi(μi− 1), ϑ^¯i²,j= ν^¯2μ_i(μi− 1), ϑ^¯i³,j= ν^¯μ_i,

ϑ_i³_,j= νi(Tk+ jbih)μi− νi²(Tk+ jbih)μi².  

Proof: Choosing the following Lyapunov function for system (11):

V(η(Tk)) = η^T(Tk)Pη(Tk), (15) the difference of the Lyapunov function can be written as

ΔV(η(Tk)) = E{V(η(Tk+ 1)) η(Tk)} − V(η(Tk)) . Calculating ΔV(η(Tk)) along the trajectory of system (11) with ϖ(Tk) = 0, we have

E{ΔV(η(Tk))} = E{η^T(Tk+ 1)Pη(Tk+ 1) − η^T(Tk)Pη(Tk)}

= E{η^T(Tk)(A^~TPA^~− P)η(Tk) +2η^T(Tk)A^~TPE^~σ^~(C^¯Λη(Tk)) +σ^~T(C^¯Λη(Tk))E^~TPE^~σ^~(C^¯Λη(Tk)) +2

∑

i, jς_i,jη^T(Tk)A^~TPF^¯_i,jC^¯Λη(Tk) +2

∑

i, jτi,jη^T(Tk)A^~TPF^¯i,jσ^~(C^¯Λη(Tk)) +2

∑

i, jς_i,jσ^~T(C^¯Λη(Tk))E^~TPF^¯_i,jC^¯Λη(Tk) +2

∑

i, jτi,jσ^~T(C^¯Λη(Tk))E^~TPF^¯i,jσ^~(C^¯Λη(Tk))

+

∑

ij, lnς_i,jς_l,nη^T(Tk)Λ^TC^¯TF^¯_i^T_,jPF^¯_l,nC^¯Λη(Tk)

+2

∑

ij, lnςi,jτl,nη^T(Tk)Λ^TC^¯TF^¯iT,jPF^¯l,nσ^~(C^¯Λη(Tk))

+

∑

ij, lnτi,jτl,nσ^~T(C^¯Λη(Tk))F^¯iT,jPF^¯l,nσ^~(C^¯Λη(Tk))

+

∑

ij ≠ lnδij,ln(1 − μi)(1 − μl)

× η^T(Tk)Λ^TC^¯TF^¯iT,jPF^¯l,nC^¯Λη(Tk) +2

∑

ij ≠ lnδ_ij,ln(1 − μi)μl

× η^T(Tk)Λ^TC^¯TF^¯iT,jPF^¯l,nσ^~(C^¯Λη(Tk))

+

∑

ij ≠ lnδ_ij,lnμ_iμ_lσ^~T(C^¯Λη(Tk))F^¯iT,jPF^¯_l,nσ^~(C^¯Λη(Tk)) +

∑

i, jϑi¹,jη^T(Tk)Λ^TC^¯TF^¯iT,jPF^¯i,jC^¯Λη(Tk) +2

∑

i, jϑi²,jη^T(Tk)Λ^TC^¯TF^¯iT,jPF^¯i,jσ^~(C^¯Λη(Tk)) +

∑

i, jϑi³,jσ^~T(C^¯Λη(Tk))F^¯iT,jPF^¯i,jσ^~(C^¯Λη(Tk)) where

δij,ln= E{αi(Tk+ jbih)αl(Tk+ nblh)}

−νi(Tk+ jbih)νl(Tk+ nblh) .

By using the elementary inequality 2a^Tb ≤ a^Ta + b^Tb (where a and b are vectors of appropriate dimensions) and Lemma 1, the following inequalities can be obtained:

2

∑

i, jςi,jη^T(Tk)A^~TPF^¯i,jC^¯Λη(Tk)

≤

∑

i, jς_i,jη^T(Tk)A^~TPA^~η(Tk)

+

∑

i, jςi,jη^T(Tk)Λ^TC^¯TF^¯iT,jPF^¯i,jC^¯Λη(Tk), 2

∑

i, jτ_i,jη^T(Tk)A^~TPF^¯_i,jσ^~(C^¯Λη(Tk))

≤

∑

i, jτi,jη^T(Tk)A^~TPA^~η(Tk)

+

∑

i, jτi,jσ^~T(C^¯Λη(Tk))F^¯iT,jPF^¯i,jσ^~(C^¯Λη(Tk)), 2

∑

i, jςi,jσ^~T(C^¯Λη(Tk))E^~TPF^¯i,jC^¯Λη(Tk)

≤

∑

i, jς_i,jσ^~T(C^¯Λη(Tk))E^~TPE^~σ^~(C^¯Λη(Tk))

+

∑

i, jς_i,jη^T(Tk)Λ^TC^¯TF^¯_i^T_,jPF^¯_i,jC^¯Λη(Tk), 2

∑

i, jτi,jσ^~T(C^¯Λη(Tk))E^~TPF^¯i,jσ^~(C^¯Λη(Tk))

≤

∑

i, jτ_i,jσ^~T(C^¯Λη(Tk))E^~TPE^~σ^~(C^¯Λη(Tk))

+

∑

i, jτi,jσ^~T(C^¯Λη(Tk))F^¯iT,jPF^¯i,jσ^~(C^¯Λη(Tk)),

ij ≠ ln

∑

≤

∑

ij ≠ lnδij,ln(1 − μi)²η^T(Tk)Λ^TC^¯TF^¯iT,jPF^¯i,jC^¯Λη(Tk),

IET Control Theory Appl., 2020, Vol. 14 Iss. 10, pp. 1255-1265 1259