Fault Detection for 2-D Continuous-Discrete State-Delayed Systems in Finite Frequency Domains

This paper investigates the fault detection problem for two dimensional (2-D) continuous discrete state-delay Roesser systems in finite frequency domains. Two performance indexes <inline-formula> <tex-math notation="LaTeX">$H_{\infty }$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$H_{-}$ </tex-math></inline-formula> are used to measure the fault sensitivity and the disturbance robustness in finite frequency. Based on this, the fault detection problem is converted into a filtering problem by designing a filter to generate a residual signal. By the generalized KYP lemma, convex design conditions are obtained, which are expressed in terms of linear matrix inequalities (LMIs). An example is provided to demonstrate the feasibility and effectiveness of the proposed method.


I. INTRODUCTION
During the past decades, fault detection has attracted more attention and a lot of detection approaches have been presented. Wherein, one of the main method for fault detection technology researches is the model-based fault detection method [1], [2]. The main goal of fault detection is to distinguish faults from disturbances. It is common to construct a residual signal by designing fault detection observer or filter to minimize the influence of disturbances and maximize the influence of faults simultaneously [3], [4]. Then, compare the residual signal with a predefined threshold, if the residual exceeds the threshold, an alarm is generated. Moreover, in practice, fault usually emerge in the low frequency domain [5], e.g., actuator failures in flight control systems [6]. The generalized Kalman-Yakubivich-Popov (KYP) lemma [7], which establishes an equivalence between the finite frequency condition and LMIs, allows researchers to better tailor specific frequency and solve the fault detection problems. For instance, the fault detection problem in finite frequency for one-dimension (1-D) systems was studied in [2], [8].
The associate editor coordinating the review of this manuscript and approving it for publication was Jun Shen .
On the other hand, in recent years, a large attention has been paid to two-dimension (2-D) systems, which can be continuous-continuous, discrete-discrete or continuousdiscrete settings. The Roesser state-space model [9] is one of the most representative one. Based on this model, a number of methodologies and techniques have been developed for analysis and synthesis of 2-D system [10]- [13]. Recently, based on the generalized KYP lemma for 2-D Roesser system model, the fault detection problem for 2-D systems has been reported in the literature [14]- [20]. By the generalized KYP lemma for 2-D discrete Roesser systems in [21], the fault detection observer and filter design is formulated as a multiobjective optimization problem in [14]- [16], respectively. In [17], the problem of fault detection observer design for 2-D continuous Roesser systems was studied. Similar problem for 2-D continuous-discrete Roesser systems was discussed in [20]. More recently, the fault detection observer design method have been extend to 2-D continuous nonlinear systems and 2-D T-S fuzzy systems [18], [19].
As we all know, time-delay phenomenon, which usually cause system instability, is widespread in the practical engineering. Therefore, the study of time-delay systems fault detection problem has important theoretical significance. However, the time-delay phenomenon was not VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ considered in [20]. Recentlly, a generalized KYP lemma for 2-D continuous-discrete state-delay Roesser systems was given in [22]. Based on this work, in the paper, we focus on the fault detection filtering for 2-D continuous-discrete statedelay Roesser systems. This fact motivates the present work. This paper discuss the fault detection problem for a class of 2-D continuous-discrete state-delay systems described by the Roesser model. Different from existing results [20], the state-delay is considered. Moreover, a fault detection filter is designed to satisfy a finite-frequency H − index and a finite-frequency H ∞ index simultaneously. The remainder of the paper is organized as follows. The problem statement and preliminaries are presented in Section 2. Section 3 presents the main results of the paper, where a finite-frequency fault detection method is proposed for 2-D continuous-discrete state-delay Roessor systems. Section 4 gives an example to illustrate the effectiveness of the proposed method. Section 5 concluded this paper.
Notation: Throughout this paper, we use R m×n , C m×n , H m×n , R, R + and Z + to represent the m × n real matrix set, complex matrix set, Hermite matrix set, real numbers set, positive real numbers set and positive integers set, respectively. The superscript T , * denote the real matrix transpose and the complex matrix transpose, respectively. L l=0 denotes the Cartesian product of sets A l . X i denotes the diagonal matrix with diagonal entries X 1 , . . . , X n , where X i could be numbers or matrices. The symbols σ max (·) and σ min (·) denote the spectral norm of a matrix.

II. PRELIMINARIES AND PROBLEM FORMULATION
In this paper, we consider the 2-D continuous-discrete state-delay systems with state-space equations where x h (t, k) ∈ R n h and x v (t, k) ∈ R n v are the horizontal state and vertical state, respectively. y(t, k), f (t, k) and d(t, k) are the external output, fault input and disturbance input vectors. The exogenous disturbance d(t, k) is assumed energy-bounded in the paper. τ l (l = h, v) are the constant state delays of the system which satisfying 0 < τ l ≤τ l andτ h ∈ R + , τ v ,τ v ∈ Z + .τ l are the upper bound of statedelays. In the following discussion, unless specifically noted otherwise, the subscript ''l" represents either the subscript h or v. Matrices A, A τ ∈ R n×n (n = n h + n v ), B f ∈ R n×n f , B d ∈ R n×n d , C, C τ ∈ R n y ×n , D f ∈ R n y ×n f and D d ∈ R n y ×n d are system matrices.
In the paper, we assume the frequency variables ω f l of the fault input f (t, k) and the frequency variables ω d l of disturbance input d(t, k) satisfy ω f l ∈ U f l and ω d l ∈ U d l , respectively. The frequency ranges U f l and U d l have the following low frequency range: In this paper, we are interested in designing a fault detection filter in the following form: is the output estimation. MatricesÂ,B andĈ are the filter matrices to be determined. Let Then, we obtain the filtering error dynamic system: [12], the transfer function from the fault f (t, k) to residual output r(t, k) of the error dynamic system (Ẽ) is given by Similarly, let f (t, k) = 0, the transfer function from the disturbance d(t, k) to residual output r(t, k) of the error dynamic system (Ẽ) can be written as To formulate the fault detection problem, the following two definitions are needed, which are similar to the definition of H ∞ index in [22] and H − index in [14]. where where Then, the fault detection filter design problem to be addressed in this paper can be expressed as follows.
Given a system (E) and γ d > 0, γ f > 0, the fault detection observer described by (Ê) is defined such that the error dynamic system (Ẽ) satisfies the following conditions: (i) the system (Ẽ) is asymptotically stable; The following lemmas will be used in the paper.

III. MAIN RESULTS
Before presenting the main results of this paper, we first present the following conclusions in this section. According to Theorem 1 and Theorem 2 in [22], for system (E), let f (t, k) = 0, we have the following Corollaries: Corollary 1: Consider the system (E) with f (t, s) = 0. Given finite frequency ranges U f d and scalarsτ h ∈ R + , τ v ∈ Z + , for any delays τ l satisfying 0 < τ l ≤τ l , if there exist matrixes S = S l , Z = Z l , P = P l , Q = Q l ∈ H n and Z 0, Q 0, such that where is a given symmetric matrix with appropriate dimension. 1 (2) and Then the following finite frequency condition holds: where G yd is the transfer function from d(t, k) to y(t, k). Corollary 2: The system (E) is asymptotically stable, if there exist matrices X = X l , S τ = S τ l , Z τ = Z τ l ∈ H n with X 0, S τ 0 and Z τ 0, such that where (2) and (7).
Theorem 1: Consider the error dynamic system (Ẽ d ). Given finite frequency ranges U d l , scalars α i > 0 (i = 1, . . . , 6), γ d > 0 andτ h ∈ R + ,τ v ∈ Z + , for any delays τ l satisfying 0 < τ l ≤τ l , the error dynamic system (Ẽ d ) satisfies specification (ii), if there exist matrices A, V, Y d i ∈ R n×n , W d 1 ∈ R n×n r , W d 2 ∈ R n×n f , B ∈ R n y ×n , C ∈ R n r ×n , Z d l 2 , S d l 2 , P d l 2 , Q d l 2 ∈ R n×n , and symmetric matrices Z d l 1 , VOLUME 8, 2020 where d = diag{I n r , −γ 2 d I n d } and Proof: For the matrix variables P d = P d l , 0 ≺ Q d = Q d l , S d = S d l and 0 ≺ Z d = Z d l with appropriate dimensions, it is easy to verify DenoteT := diag{T , T , T }, it is easy to prove that d = T T¯ dT . According to inequation (7), we have where Then, inequality (12) can be written as whereT 1 = T d diag{T T , I n r +n d } and Using Lemma 1, condition (13) is equivalent to the existence of a matrix Y such that Let Y ∈ R (2n+n r )×(6n+n r +n d ) be the following specific block form: whereT 2 = diag{T , I n r },T 3 = diag{T , I n r , T , T , I n d } and a i ∈ R + , i = 1, . . . , 6. Note that,T T 1T 1 = I 6n+n r +n d , pre-and postmultiplying (14) byT T 1 and T 1 and substituting (2) and (15) into (14) given (11). According to Corollary 1, specification (ii) holds if inequations (10) and (11) holds. The proof is completed.

D. DESIGN OF FAULT DETECTION FILTERS
Theorems 1-3 present a group of LMI conditions for the fault detection filter design. By combining Theorems 1-3, an algorithm is proposed to obtain the parameters of a desired filter.
Algorithm 1: Given adjustable parameters α i (i = 1, . . . , 6), weighting factors p, q ∈ R + , which satisfying p + q = 1, the upper bound of state delay τ l ∈ Z + , finite frequency ranges U f l and U d l , β and vector η, solve the following convex optimization problem Then, the filter parameter matrixes satisfyinĝ Remark 1: In Algorithm 1, a much better fault detection filter can be designed by choosing different adjustable parameters α i (i = 1, . . . , 6) for inequations in 1-3. However, up to now, there is no feasible method for optimizing these parameters [23]. Thus, we just choose same adjustable parameters to simply show the efficacy of α i in the paper.

E. RESIDUAL EVALUATION FUNCTION AND THRESHOLD
Motivated by [15] and [17], we choose the following residual evaluation function J r (t, k) and the threshold J th : where J r (t, k) is the residual evaluation function,t,k are the horizontal range and the vertical range of evaluation window, respectively. J th is the threshold.The occurrence of faults can be detected using the following logic rules:

IV. SIMULAIONS
Considering the 2-D continuous-discrete state-delay Roesser system in the form of E with  By (32), we can obtain the threshold J th = 0.1538. The simulation results are shown in Figs. 1-2. Fig. 1 depicts the residual evaluation function J r (t, k) and the threshold J th in three-dimensional space. Figs. 2 depicts the residual evaluation function J r (t, k) and the threshold J th in two-dimensional space. It can be seen from Fig. 2 that the fault can be detected at 10.8.

V. CONCLUSION
In the paper, by the generalized KYP lemma, the finite frequency H − and H ∞ indexes have been used to design the fault detection filters for 2-D continuous-discrete state-delay Roesser systems. Finite-frequency performance analysis conditions are firstly obtained. Convex filter design conditions are derived by constructing a hyperplane tangent combined with matrix inequality techniques. Then, an algorithm is proposed to construct a desired fault detection filter. The effectiveness of the proposed fault detection method is illustrated by an example. Furthermore, system parametric uncertainties are frequently encountered in many practical systems and often a primary source of instability and performance degradation of a control system. Thus, it is a worthiness subject to study system parametric uncertainties in the future.
LIWEI LI received the Ph.D. degree in control theory and control engineering from Northeastern University, Shenyang, China, in 2019. She is currently an Associate Professor with Nanjing Technology University, Nanjing, China. Her current research interests include Markov jump systems, decentralized control, and fault detection.
MOUQUAN SHEN received the Ph.D. degree in control theory and control engineering from Northeastern University, Shenyang, China, in 2011. He is currently a Professor with Nanjing Technology University, Nanjing, China. His current research interests include Markov jump systems, adaptive control, data-driven based control, robust control, and iterative learning control.