Fault Detection for Quantized Networked Control Systems

Abstract

The fault detection problem in the finite frequency domain for networked control systems with signal quantization is considered. With the logarithmic quantizer consideration, a quantized fault detection observer is designed by employing a performance index which is used to increase the fault sensitivity in finite frequency domain. The quantized measurement signals are dealt with by utilizing the sector bound method, in which the quantization error is treated as sector-bounded uncertainty. By using the Kalman-Yakubovich-Popov (GKYP) Lemma, an iterative LMI-based optimization algorithm is developed for designing the quantized fault detection observer. And a numerical example is given to illustrate the effectiveness of the proposed method.

1. Introduction

Recently, due to many advantages of networked control systems, such as lower cost, easier installation and maintenance, and higher reliability, NCSs have been found successfully industrial applications in automobiles, manufacturing plants, aircraft, HVAC, systems and unmanned vehicles. However, the insertion of the communication channels results in discrepancies between the data information to be transmitted and their associated remotely transmitted images, hence raising new interesting and challenging problems such as quantization, packet losses, and time delays. As is well known to all, quantization always exists in bandwidth limited networked systems and the performance of NCSs will be inevitably subject to the effect of quantization error. Hence, the quantization problem of NCSs has long been studied and many important results have been reported in [1–10] and the references therein. Two most pertinent references to this paper are the work [2] and the following work [3]. In [2], the problem of quadratic stabilization of discrete-time single-input single-output (SISO) linear time-invariant systems using quantized feedback is studied. In [3], the work of [2] is generalized to general multi-input multi-output (MIMO) systems and to control problems requiring performances. This is done using the so-called sector bound method, which is based on using a simple sector bound to model the quantization error. This method has been employed by quite a few researchers and many results have been given correspondingly [4–7] and so on.

On the other hand, fault detection (FD) is a very significant problem and has attracted a lot of attention in the past two decades. A fault is defined as an unpermitted deviation of at least one characteristic property of a variable from an acceptable behaviour. Such a fault disturbs the normal operation of an automatic system, thus causing an unacceptable deterioration of the performance of the system [11, 12]. To detect the fault, an observer is usually designed which generates the residual signal, and, by satisfying certain performances, the observer parameters are then determined. Up to now, the studies on the FD are mainly categorized into two classes depending on the fault frequency domain. There are many results that study the FD problem in the full frequency domain, such as [13, 14]. Recently, there are many studies considering the fault in finite frequency domain which much more accords with practice. Because in practice, faults are usually in the finite frequency domain; for example, for an incipient fault signal, the fault information is contained within a low frequency band as the fault development is slow as stated in [15]. Another important finite frequency fault is the actuator stuck fault whose frequency is zero. The stuck fault occurs when an aircraft control surface (such as the rudder or an aileron) is stuck at some fixed value as stated in [16]. And the stuck fault is also considered for the F-16 aircraft in [17]. So, the finite frequency domain method to FD has been paid more attention to many new results occur successively [18–20]. In networked control systems, FD problem also exists and is unavoidably. So far, there have been some studies on the FD problem of the networked control systems [21, 22]. But to the best of the authors’ knowledge, there has been no work considering the quantized FD problem in finite frequency domain for networked control systems.

Motivated by the above-mentioned reason, in this paper, the quantized FD observer design problem for networked control systems with logarithmic quantizers is studied. The quantization errors are modeled as sector-bounded uncertainties. By employing the GKYP method, a quantized FD observer design method is proposed with an iterative LMI-based optimization algorithm. Finally, a numerical example is given to show the effectiveness of the proposed method.

The organization of this paper is as follows. Section 2 presents the problem under consideration and some preliminaries. Section 3 gives design methods of quantized FD observer design strategies. In Section 4, an example is presented to illustrate the effectiveness of the proposed methods. Finally, Section 5 gives some concluding remarks.

Notation. For a matrix 𝐸, 𝐸^𝑇, 𝐸^⟂, and 𝐸^∗ denote its transpose, orthogonal complement, and complex conjugate transpose, respectively. And 𝐴^†=𝐴^𝑇(𝐴𝐴^𝑇)⁻¹ denotes its Moore-Penrose inverse. I denotes the identity matrix with an appropriate dimension. For a symmetric matrix, 𝐴>(≥)0 and 𝐴<(≤)0 denote positive (semi-) definiteness and negative (semi-) definiteness. The Hermitian part of a square matrix 𝑀 is denoted by 𝐻𝑒(𝑀)∶=𝑀+𝑀^∗. The symbol 𝐻_𝑛 stands for the set of 𝑛×𝑛 Hermitian matrices. The symbol “∗” within a matrix represents the symmetric entries. 𝜎_max(𝐺) and 𝜎_min(𝐺) denote maximum and minimum singular value of the transfer matrix 𝐺, respectively.

2. Problem Statement and Preliminaries

2.1. Problem Statement

Consider an LTI discrete-time system as 𝑥(𝑘+1)=𝐴𝑥(𝑘)+𝐵₁𝑓(𝑘)+𝐵₂𝑢_𝑐(𝑘),𝑦(𝑘)=𝐶𝑥(𝑘)+𝐷₁𝑓(𝑘), (1) where 𝑥(𝑘)∈𝑅^𝑛 is the state, 𝑢_𝑐(𝑘)∈𝑅^𝑢 is the control input, and 𝑓(𝑘)∈𝑅^𝑓 is the fault input vector, respectively. 𝐴, 𝐵₁, 𝐵₂, 𝐶, and 𝐷₁ are known constant matrices of appropriate dimensions. Without loss of generality, assume that (𝐴,𝐶) is observable and 𝐵₁ is of full column rank.

To formulate the quantized FD problem, consider the quantized FD observer with the following form.

Consider a dynamic observer-based control strategy for (1) with observer given by 𝑥(𝑘+1)=𝐴𝑥(𝑘)+𝐵₂𝑢_𝑐(𝑘)+𝐿𝑦_𝑐(𝑘)−𝑦_𝑐(𝑘)𝑦_𝑐(𝑘)𝑥(𝑘),𝑢=𝐶_𝑐(𝑘)𝑥(𝑘),𝑟(𝑘)=𝐾=𝑦_𝑐(𝑘)−𝑦_𝑐(𝑘), (2) where 𝑦_𝑐(𝑘)∈𝑅^𝑦 is the observer output, 𝑥(𝑘)∈𝑅^𝑛 is the state estimation of system (1), and 𝑟(𝑘)∈𝑅^𝑟 is the residual signal. 𝐿 is the observer gain to be designed. Due to the insertion of the communication channel, the measurement signals will be quantized before they are transmitted to the filter through the network. The quantizer is denoted as 𝑞[⋅]=[𝑞₁[⋅],𝑞₂[⋅],…,𝑞_𝑦[⋅]]^𝑇, which is assumed to be symmetric; that is, 𝑞_𝑗[−𝜀]=−𝑞_𝑗[𝜀], 𝑗=1,2,…,𝑦. In this paper, the quantizer is selected as a logarithmic one, and, for each 𝑞_𝑗[⋅], the quantization levels are given by 𝑉_𝑗=±𝜈_𝑖^(𝑗)∶𝜈_𝑖^(𝑗)=𝜌^𝑖_𝑗𝜈₀^(𝑗)∪{0},𝜈,𝑖=0,±1,±2,…_0(𝑗)>0,0<𝜌_𝑗<1. (3) As in [3, 4], the associated quantizer is defined as follows: 𝑞_𝑗[𝜀]=𝜌^𝑖_𝑗𝜈^𝑗₀1,if1+𝛿_𝑗𝜌^𝑖_𝑗𝜈^𝑗₀1<𝜀<1−𝛿_𝑗𝜌^𝑖_𝑗𝜈^𝑗₀,𝜀>00,if𝜀=0,−𝑞_𝑗[]𝛿−𝜀,if𝜀<0,_𝑗=1−𝜌_𝑗1+𝜌_𝑗. (4)(5) Then, based on the quantizer (4), the measurement signal at the filter end is with the form as 𝑦_𝑐(𝑘)𝑦(𝑘)=𝑞=𝑞¹𝑦₁(𝑘)⋮𝑞_𝑦𝑦_𝑦(𝑘)=(𝐼+Δ(𝑘))𝑦(𝑘), (6) where Δ(𝑘)=diagΔ₁(𝑘),Δ₂(𝑘),…,Δ_𝑦(𝑘,Δ_𝑖)(𝑘)∈−𝛿_𝑖,𝛿_𝑖. (7) Combining FD observer (2) with system (1) and the quantized measurement (6), the following quantized error dynamic system is obtained: 𝑥(𝑘+1)=𝐴+𝐵₂𝐾𝑥(𝑘)+𝐵₁𝑓(𝑘)−𝐵₂(𝑘),𝑒()=()𝑒(𝑘)(𝑘)(𝑘)+𝐵𝐾𝑒𝑘+1𝐴−𝐿𝐶−𝐿Δ𝐶𝑥₁−𝐿Δ(𝑘)𝐷₁𝑓(𝑘),𝑟(𝑘)=𝐶𝑒(𝑘)+Δ(𝑘)𝐶𝑥(𝑘)+(𝐼+Δ(𝑘))𝐷₁𝑓(𝑘), (8) where 𝑒(𝑘)=𝑥(𝑘)−𝑥(𝑘).

Facilitating the presentation, (8) can be rewritten as 𝑥_𝑒(𝑘+1)=𝐴𝑥_𝑒(𝑘)+𝐵₁𝑓(𝑘)𝑟(𝑘)=𝐶𝑥_𝑒(𝑘)+𝐷₁𝑓(𝑘), (9) where 𝑥_𝑒(𝑘)=^𝑒(𝑘)_𝑥(𝑘) and (𝑘)𝐶𝐴=𝐴−𝐿𝐶−𝐿Δ−𝐵₂𝐾𝐴+𝐵₂𝐾,𝐵₁=𝐵₁−𝐿(𝐼+Δ(𝑘))𝐷₁𝐵₁,𝐶=𝐶Δ(𝑘)𝐶,𝐷₁=(𝐼+Δ(𝑘))𝐷₁. (10) Note that the dynamics of the residual signal depends on the fault 𝑓(𝑘), to detect the fault effects; quantized observer (2) is designed in this work such that the following conditions are satisfied: (i):(𝑘)𝐶𝐴=𝐴−𝐿𝐶−𝐿Δ−𝐵₂𝐾𝐴+𝐵₂𝐾()isstable,ii:inf𝜎_min𝐺_𝑟𝑓𝑒^𝑗𝜃>𝛾,∀𝜃∈𝜐₁,𝜐₂, (11) where 𝐺_𝑟𝑓𝑒^𝑗𝜃∶=𝐶𝑒^𝑗𝜃𝐼−𝐴⁻¹𝐵₁−𝐿𝐷₁+𝐷₁. (12)

Remark 1. Condition (ii) is a finite frequency performance index used to increase the fault sensitivity. Note that 𝜐₁, 𝜐₂ are given scalars which reflects the frequency range of faults.

The problem addressed in this paper is as follows.

Quantized FD Control Problem. Considering the effects of the quantization, design a quantized FD observer such that the error system (8) is with high fault sensitivity in finite frequency domain.

2.2. Preliminaries

The following lemma presented will be used in this paper.

Lemma 2 (see [23]). Consider a transfer function matrix 𝐺(𝑒^𝑗𝜃)∶=𝐶(𝑒^𝑗𝜃𝐼−𝐴)⁻¹𝐵+𝐷; let a symmetric matrix Π and scalars 𝜈₁, 𝜈₂ be given; the following statements are equivalent.
(i) The finite frequency inequality 𝐺𝑒^𝑗𝜃𝐼^∗Π𝐺𝑒^𝑗𝜃𝐼<0,∀𝜃∈𝜈₁,𝜈₂. (13)
(ii) There exist matrices 𝑃,𝑄∈𝐇_𝑛 of appropriate dimensions, satisfying 𝑄>0, and 𝐴𝐵𝐼0^∗Ξ+𝐴𝐵𝐼0𝐶𝐷0𝐼^∗Π𝐶𝐷0𝐼<0, (14) where Ξ=−𝑃𝑒^{𝑗(𝜈1+𝜈2)/2}𝑄𝑒^{−𝑗(𝜈1+𝜈2)/2}𝑄𝑃−2cos𝜈₂−𝜈₁/2𝑄. (15)

Lemma 3 (Finsler’s Lemma). Let 𝑥∈𝑅^𝑛, 𝑄∈𝑅^𝑛×𝑛, and 𝑈∈𝑅^𝑛×𝑚. Let 𝑈^⟂ be any matrix such that 𝑈^⟂𝑈=0. The following statements are equivalent:(i)𝑥^∗𝑄𝑥<0, for all 𝑈^∗𝑥=0,𝑥≠0,(ii)𝑈^⟂𝑄𝑈^⟂∗=0,(iii)∃𝜇∈𝑅∶𝑄−𝜇𝑈𝑈^∗<0,(iv)∃𝑦∈𝑅^𝑚×𝑛∶𝑄+𝑈𝑌+𝑌^∗𝑈^∗<0.

Lemma 4 (Projection Lemma). Let Γ, Λ, Θ and be given. There exists a matrix 𝐹 satisfying Γ𝐹Λ+(Γ𝐹Λ)^𝑇+Θ<0 if and only if the following two conditions hold: Γ^⟂Θ<0,Λ^𝑇⟂ΘΛ^𝑇⟂𝑇<0. (16)

Lemma 5 (see [24]). For any real matrices 𝑌, 𝑀, 𝐹, and 𝐸 with compatible dimensions and 𝐹^𝑇𝐹≤𝛿²𝐼, where 𝛿>0 is a scalar, then 𝑌+𝑀𝐹𝐸+(𝑀𝐹𝐸)^𝑇<0 (17) holds if and only if there exists a scalar 𝜀>0, such that 1𝑌+𝜀𝑀𝑀^𝑇+𝜀𝛿²𝐸^𝑇𝐸<0. (18)

3. Quantized FD Observer Design

In this section, an inequality for the stability condition (i) is given first. Then, considering the fault sensitivity problem, an inequality is given for the fault sensitivity condition (ii).

Firstly, considering the stability condition (i), we have the following lemma.

Lemma 6. Consider system (9) if there exists a matrix 𝑋=^𝑋0_0𝑋>0 such that the following inequality holds: −𝑋∗∗∗∗∗0−𝑋∗∗∗∗𝑋𝐴−Γ^𝑇𝐶0−𝑋∗∗∗−𝑋𝐵₂𝐾𝑋𝐴+𝑋𝐵₂𝐾0−𝑋∗∗00−Γ0−𝜀𝐼∗0𝛿_𝑎𝜀𝐶000−𝜀𝐼<0, (19) where Γ=𝐿^𝑇𝑋.

Proof. It is easy to know that (i) holds if there exists 𝑋>0, such that 𝐴^𝑇𝑋𝐴−𝑋<0. (20)
By using the Schur complement lemma, we have that −𝑋𝐴^𝑇𝑋𝑋𝐴−𝑋<0. (21) Equation (21) can be converted into −𝑋∗∗∗∗∗0−𝑋∗∗∗∗𝑋𝐴−Γ^𝑇𝐶−𝑋𝐿Δ(𝑘)𝐶−𝑋∗∗∗−𝑋𝐵₂𝐾𝑋𝐴+𝑋𝐵₂𝐾0−𝑋∗∗00−Γ0−𝜀𝐼∗0𝛿_𝑎𝜀𝐶000−𝜀𝐼<0. (22)
Obviously, (22) can be rewritten as −𝑋∗∗∗∗∗0−𝑋∗∗∗∗𝑋𝐴−Γ^𝑇𝐶0−𝑋∗∗∗−𝑋𝐵₂𝐾𝑋𝐴+𝑋𝐵₂𝐾0−𝑋∗∗00−Γ0−𝜀𝐼∗0𝛿_𝑎+000Δ(𝑘+000Δ(𝑘𝜀𝐶000−𝜀𝐼−Γ)0𝐶00−Γ)0𝐶00^𝑇<0. (23) By using Lemma 5, (19) holds, which shows that condition (i) holds if (19) holds. This completes the proof

In the following, the fault sensitivity problem is studied. Considering system (9), the following lemma is presented to give the fault sensitivity condition.

Lemma 7. Let real matrix 𝐴, 𝐵₁, 𝐶, 𝐷₁, a symmetric matrix Π=^−𝐼0_0𝛾²_𝐼, and scalars 𝜗₁, 𝜗₂ be given; consider system (9), then the following conditions are equivalent.
(i) The following finite frequency inequality 𝜎_min𝐺_𝑟𝑓𝑒^𝑗𝜃>𝛾,∀𝜃∈𝜗₁,𝜗₂ (24) holds, where 𝐺_𝑟𝑓(𝑒^𝑗𝜃)=𝐶(𝑒^𝑗𝜃𝐼−𝐴)⁻¹𝐵₁+𝐷₁ is the transfer function matrix from 𝑓(𝑘) to 𝑟(𝑘).
(ii) There exist 2𝑛×2𝑛 Hermitian matrices 𝑃 and 𝑄 satisfying 𝑄>0, and 𝐴𝐵𝐼0^∗Ξ𝐴𝐵₁+𝐼0𝐶𝐷₁0𝐼^∗Π𝐶𝐷₁0𝐼<0, (25) where Ξ=−𝑃𝑒^{𝑗(𝜈1+𝜈1)/2}𝑄𝑒^{−𝑗(𝜈1+𝜈1)/2}𝜈𝑄𝑃−2cos¹−𝜈₁2𝑄. (26)

Note that condition (25) in Lemma 7 can be rewritten as Υ𝐼𝑇𝑇Ξ00Π^∗Υ𝐼^∗<0, (27) where 𝐴Υ=^𝑇−𝐶^𝑇𝐿^𝑇−𝐾^𝑇𝐵^𝑇₂𝐶^𝑇−𝐶^𝑇Δ(𝑘)𝐿^𝑇𝐴^𝑇+𝐾^𝑇𝐵^𝑇₂Δ(𝑘)𝐶^𝑇𝐵^𝑇₁−𝐷^𝑇₁(𝐼+Δ(𝑘))^𝑇𝐿^𝑇𝐵^𝑇₁𝐷^𝑇₁(𝐼+Δ(𝑘))^𝑇, (28)

and 𝑇 is the permutation matrix defined as 𝑁₁𝑁₂𝑁₃𝑁₄𝑁₅𝑁₆𝑇=𝑁₁𝑁₂𝑁₅𝑁₃𝑁₄𝑁₆. (29)

By using Lemma 3, we have that (27) is equivalent to the existence of a multiplier 𝒳 such that 𝑇𝑇Ξ00Π^∗𝐼Υ<𝐻𝑒𝒳. (30)

Obviously, Υ can be rewritten as 𝐴Υ=^𝑇−𝐾^𝑇𝐵^𝑇₂𝐶^𝑇0𝐴^𝑇+𝐾^𝑇𝐵^𝑇₂Δ(𝑘)𝐶^𝑇𝐵^𝑇₁𝐵^𝑇₁𝐷^𝑇₁(𝐼+Δ(𝑘))^𝑇+−𝐶^𝑇−𝐶^𝑇Δ(𝑘)−𝐷^𝑇₁(𝐼+Δ(𝑘))^𝑇𝐿^𝑇𝐼00=𝒜+ℬ𝐿^𝑇𝒞. (31)

To facilitate dealing with the problem, restrict 𝒳 with the following structure: 𝒳=𝒞^†𝑋𝑅+𝐼−𝒞^†𝒞𝑉∣𝑋∈ℝ^𝑛×𝑛,det𝑋≠0, (32) where 𝑅 is a matrix to be specified later, 𝑋, 𝑉 are matrix variables.

Then, we have Υ𝒳=𝒜+ℬ𝐿^𝑇𝒞𝒞^†𝑋𝑅+𝐼−𝒞^†𝒞𝑉=𝒜𝒳+ℬΓ𝑅. (33)

Then the following lemma is given to show that the restriction of Υ𝒳 does not introduce conservatism if the matrix 𝑅 is specified appropriately.

Lemma 8. Consider system (9), let 𝑅 and Π be with appropriate dimensions, and let 𝑄>0; then the following statements are equivalent.
(i) There exists a gain matrix 𝐿 such that condition (25) and 𝑆𝑅^∗⟂𝑆𝑇𝑇Ξ00Π^∗𝑆^∗𝑆𝑅^∗⟂∗<0 (34) hold, where 𝐴𝑆=^𝑇−𝐶^𝑇𝐿^𝑇−𝐾^𝑇𝐵^𝑇₂𝐶^𝑇𝐼00−𝐶^𝑇Δ(𝑘)𝐿^𝑇𝐴^𝑇+𝐾^𝑇𝐵^𝑇₂Δ(𝑘)𝐶^𝑇𝐵0𝐼0_𝑇1−𝐷^𝑇₁(𝐼+Δ(𝑘))^𝑇𝐿^𝑇𝐵^𝑇₁𝐷^𝑇₁(𝐼+Δ(𝑘))^𝑇00𝐼𝐼000000𝐼0000. (35)
(ii) There exists matrix variable Γ=𝐿^𝑇𝑋 such that 𝑇𝑇Ξ00Π^∗𝒳<𝐻𝑒−𝒜𝒳−ℬΓ𝑅. (36)

Proof. The proof is similar to the proof of Lemma  5 in [20], it is omitted here.

Then, combining Lemmas 7 and 8, the following theorem is presented.

Denote 𝐹_𝑎1=0000000000000000𝐼−𝐶^𝑇0000−𝐷^𝑇₁−𝐷^𝑇₁,𝐹_𝑎2=𝐶^𝑇𝑉_𝑐1𝐶^𝑇𝑉_𝑐2𝐶^𝑇𝑉_𝑐3𝐶^𝑇𝑉_𝑐4𝐶^𝑇𝑉_𝑐5𝐶^𝑇𝑉_𝑐6ΓΓ00Γ−Γ𝐵₁𝑉_𝑐1𝑉_𝑐2𝑉_𝑐3𝑉_𝑐4𝑉_𝑐5𝑉_𝑐6ΓΓ00Γ−Γ𝐵₁,Λ=−𝑃₁₁−𝑋−𝑋^𝑇∗∗∗∗∗−𝑃₂₁−𝑉_𝑏1−𝑋^𝑇−𝑃₂₂−𝑉_𝑏2−𝑉^𝑇_𝑏2𝑒∗∗∗∗_{−𝑗𝜗𝑐}𝑄₁₁−𝑉_𝑐1𝑒^{−𝑗𝜗𝑐}𝑄₁₂−𝑉_𝑐2𝑉^𝑇_𝑏3ΦΦ∗∗∗₁Φ₂Φ₃Φ₄Φ∗∗₅Φ₆Φ₇Φ₈Φ₉∗Φ₁₀Φ₁₁Φ₁₂Φ₁₃Φ₁₄Φ₁₅, (37)(38) where Φ=𝑃₁₂−2cos𝜗_𝑤𝑄₁₂−𝑉_𝑐3−𝑉^𝑇_𝑐3,Φ₁=𝑒^{−𝑗𝜗𝑐}𝑄^𝑇₁₂+𝐴^𝑇𝑋−𝐾^𝑇𝐵^𝑇₂𝑉_𝑏1+𝐶^𝑇𝑉_𝑐1−𝐶^𝑇Γ−𝑋^𝑇,Φ₂=𝑒^{−𝑗𝜗𝑐}𝑄^𝑇₂₂+𝐴^𝑇𝑋−𝐾^𝑇𝐵^𝑇₂𝑉_𝑏2−𝐶^𝑇𝑉_𝑐3+𝐶^𝑇Γ−𝑉^𝑇_𝑏4,Φ₃=𝑃₁₂2cos𝜗_𝑤𝑄₂₂−𝐾^𝑇𝐵^𝑇₂𝑉_𝑏3+𝐶^𝑇𝑉₃+𝑉^𝑇_𝑐4,Φ₄=−𝐾^𝑇𝐵^𝑇₂𝑉_𝑏4+𝐶^𝑇𝑉_𝑐4−𝐾^𝑇𝐵^𝑇₂𝑉_𝑏4−𝐶^𝑇𝑉_𝑐4^𝑇,Φ₅=𝐴^𝑇+𝐾^𝑇𝐵^𝑇₂𝑉_𝑏1−𝑋^𝑇,Φ₆=𝐴^𝑇+𝐾^𝑇𝐵^𝑇₂𝑉_𝑏2−𝑉_𝑏5,Φ₇=𝐴^𝑇+𝐾^𝑇𝐵^𝑇₂𝑉_𝑏3−𝑉_𝑐5,Φ₈=−𝐼+𝐴^𝑇+𝐾^𝑇𝐵^𝑇₂𝑉_𝑏4+𝑉^𝑇_𝑐5+𝑋^𝑇𝐴−𝑉^𝑇_𝑏5𝐵₂𝐾−Γ^𝑇Φ𝐶,₉=𝐴^𝑇+𝐾^𝑇𝐵^𝑇₂𝑉_𝑏5−𝑉^𝑇_𝑏5𝐴+𝐵₂𝐾,Φ₁₀=𝐵^𝑇₁𝑋+𝐵^𝑇₁𝑉_𝑏1+𝐵^𝑇₁𝑋^𝑇−𝐷^𝑇₁𝑉_𝑐1−𝐷^𝑇₁ΦΓ,₁₁=𝐵^𝑇₁𝑋+𝐵^𝑇₁𝑉_𝑏2−𝑉^𝑇_𝑏6−𝐷^𝑇₁𝑉_𝑐2−𝐷^𝑇₁ΦΓ,₁₂=𝐵^𝑇₁𝑉_𝑏3−𝑉_𝑐6−𝐷^𝑇₁𝑉_𝑐3,Φ₁₃=𝐵^𝑇₁𝑉_𝑏4+𝐴^𝑇𝑋𝐵₁−𝐾^𝑇𝐵^𝑇₂𝑉_𝑏6+𝐶^𝑇𝑉_𝑐6+𝐶^𝑇Γ𝐵₁^𝑇−𝐷^𝑇₁𝑉_𝑐4,Φ₁₄=𝐵^𝑇₁𝑋+𝐵^𝑇₁𝑉_𝑏5+𝐴^𝑇+𝐾^𝑇𝐵^𝑇₂𝑉_𝑏6^𝑇−𝐷^𝑇₁𝑉_𝑐5−𝐷^𝑇₁ΦΓ,₁₅=𝐵^𝑇₁𝑋𝐵₁−𝐵^𝑇₁𝑉_𝑏6+𝐵^𝑇₁𝑋𝐵₁−𝐵^𝑇₁𝑉_𝑏6^𝑇−𝐷^𝑇₁𝑉_𝑐6+𝐷^𝑇₁Γ𝐵₁, (39) with 𝜗_𝑐=𝜗₁+𝜗₂2,𝜗_𝜔=𝜗₂−𝜗₁2. (40)

Theorem 9. Consider system (9); let 𝛾>0 and 𝛿_𝑎>0 be given constants and Π∶=−𝐼00𝛾²𝐼,𝑃,𝑄∈𝐻_𝑛 (41) and 𝑄>0. Provided that 𝑅=[𝐼𝐼00−𝐵₁], then 𝜎_min𝐺_𝑟𝑓𝑒^𝑗𝜃>𝛾,∀𝜃∈𝜗₂,𝜗₂, (42) holds if there exist matrix variables 𝑉_𝑏𝑖, 𝑉_𝑐𝑖, 𝑖=1,…,6,𝑋, and scalars 𝜗₁, 𝜗₂, and 𝜀 such that Λ𝐹_𝑎1𝛿_𝑎𝜀𝐹^𝑇_𝑎2∗−𝜀𝐼0∗∗−𝜀𝐼<0. (43)

Proof. By Lemmas 7 and 8, we have that (42) holds if inequality (36) holds. Similar to the proof of Theorem  1 in [20], since 𝒞=[𝐼00], we know that 𝒞^†=𝐼00. (44) Then from (32) we have 𝐼00𝒳=𝑋𝑅+0000𝐼000𝐼𝑉. (45) Partition 𝑉 as 𝑉=^𝑉𝑎𝑉_𝑏𝑉𝑐; then we get 𝒳=𝑉𝑋𝑅_𝑏𝑉_𝑐. (46) So, (36) can be written as 𝑇𝑇Ξ00Π^∗<𝐻𝑒𝐼0000𝐼0000𝐼0−𝐴^𝑇𝐾^𝑇𝐵^𝑇₂−𝐶^𝑇𝐶^𝑇0−𝐴^𝑇−𝐾^𝑇𝐵^𝑇₂−Δ(𝑘)𝐶^𝑇𝐶^𝑇Δ(𝑘)−𝐵^𝑇₁−𝐵^𝑇₁𝐷^𝑇₁(𝐼+Δ(𝑘))𝐷^𝑇₁((𝑘×𝑉𝐼+Δ))𝑋𝑅_𝑏𝑉_𝑐.Γ𝑅 (47) Let 𝑉_𝑏=[𝑉_𝑏1−𝑉_𝑏6], 𝑉_𝑐=[𝑉_𝑐1−𝑉_𝑐6], and partition 𝑃 and 𝑄 as 𝑃=^𝑃11𝑃12_𝑃21𝑃22,𝑄=^𝑄11𝑄12_{𝑄𝑇12𝑄22}>0; then we have Λ+𝐹_𝑎1diag{Δ(𝑘),Δ(𝑘),Δ(𝑘),Δ(𝑘)}𝐹_𝑎2+𝐹_𝑎1diag{Δ(𝑘),Δ(𝑘),Δ(𝑘),Δ(𝑘)}𝐹_𝑎2^𝑇<0, (48) where Δ(𝑘) satisfyies (7). By using Lemma 5, we can obtain that (43) holds. So, it concludes that (42) holds if (43) holds, which completes the proof.

Combining Lemma 6 and Theorem 9, we have the following theorem.

Theorem 10. Consider system (1), and let 𝛾>0 and 𝛿_𝑎>0 be given constants; there exists a quantized fault detection observer (2) such that error closed-loop system (9) is stable and with the finite frequency performance 𝜎_min𝐺_𝑟𝑓𝑒^𝑗𝜃>𝛾,∀𝜃∈𝜗₂,𝜗₂ (49) if there exist matrix variables Γ, 𝑉_𝑏𝑖, 𝑉_𝑐𝑖, 𝑖=1,…,6, 𝑋=^𝑋0_0𝑋>0, 𝑃=^𝑃11𝑃12_𝑃21𝑃22, and 𝑄=^𝑄11𝑄12_{𝑄𝑇12𝑄22}>0 and scalars 𝜗₁, 𝜗₂, 𝜀₁, and 𝜀₂ such that the following inequalities hold: −𝑋∗∗∗∗∗0−𝑋∗∗∗∗𝑋𝐴−Γ^𝑇𝐶0−𝑋∗∗∗−𝑋𝐵₂𝐾𝑋𝐴+𝑋𝐵₂𝐾0−𝑋∗∗00−Γ0−𝜀₁𝐼∗0𝛿_𝑎𝜀₁𝐶000−𝜀₁𝐼<0Λ𝐹_𝑎1𝛿_𝑎𝜀₂𝐹^𝑇_𝑎2∗−𝜀₂𝐼0∗∗−𝜀₂𝐼<0, (50) where 𝛿_𝑎=max{𝛿_𝑖,𝑖=1,…,𝑦} and Λ is defined by (38), and the observer gain 𝐿 is obtained as 𝐿^𝑇=Γ𝑋⁻¹.

Remark 11. Note that, due to the existence of the unknown controller gain, the conditions given in Theorem 10 are not convex. In order to solve this problem, we design a controller gain by state feedback method as follows: −𝑋𝑋𝐴^𝑇+𝐾^𝑇𝐵^𝑇₂𝐴𝑋+𝐵₂𝐾−𝑋<0, (51) for 𝑋>0, and the controller gain is given as 𝐾=𝐾𝑋⁻¹. Then use the state feedback controller gain as the initial value to obtain the observer gain 𝐿. So the following algorithm is given.

Algorithm 12. Let 𝛿_𝑎>0 be given scalars and 𝜁>0 a given small constant specifying a convergence criterion.

Step 1. By (51), we obtain the initial solutions 𝐾ⁱⁿⁱ; go to Step 2.

Step 2. Letting 𝐾=𝐾ⁱⁿⁱ, (),max𝛾s.t.50 (52) we obtain Γ, 𝑋, 𝑉_𝑏𝑖, 𝑖=1,…,6, and 𝛾_ini; then go to Step 3.

Step 3. Let 𝑋_ini=𝑋, 𝑉ⁱⁿⁱ_𝑏𝑖=𝑉_𝑏𝑖, 𝑖=1,…,6, ().max𝛾s.t.50 (53)𝛾 and 𝐾 are obtained. Then if ‖𝛾−𝛾_ini‖<𝜁, stop, and 𝐿^𝑇=Γ𝑋⁻¹, else, let 𝐾=𝐾 and 𝛾_ini=𝛾; return to Step 2.

Remark 13. 𝜀₁, 𝜀₂ in (50) can be obtained by searching method to guarantee that the performance 𝛾 is maximum.

4. Example

In this section, an example is given to illustrate the effectiveness of the developed theory. Consider a linear system of form (1) with ,𝐵𝐴=0.967300.120.02930.8763−0.40.025900.9032₁=0.100.50.06−0.20.15,𝐵₂=,𝐷10.0400.300.05₁=0.100.512,𝐷₂=,.100.50.1𝐶=0.100110 (54) For this example, set the quantization densities as 𝜌₁=𝜌₂=0.91. Assume that the frequency range of faults is known as 𝜃∈[0,0.5]. Let 𝜀₁=0.9899, 𝜀₂=1.1299; by Algorithm 12, we obtain the fault sensitivity performance index 𝛾=1.2131, and, correspondingly, the quantized fault detection observer gain matrix 𝐿_finite is obtained as 𝐿_finite=0.14520.0541−0.0145−0.2156−0.54210.3741. (55)

In order to study the effects of fault on residual of the quantized detection observer, the fault signal is selected as 𝑓(𝑘)=10.9^𝑇,𝑘≥20,00^𝑇,otherwise. (56)

Using the observer gain matrix 𝐿_finite given in (48), the two residual outputs are denoted by the solid lines of Figures 1 and 2, respectively, from which we can see that the faulty cases are well discriminated from the fault free cases in presence of the disturbance effects.

Figure 1 

Residual output signal.

Figure 2 

Residual output signal.

5. Conclusion

This paper considers the fault detection problem in the finite frequency domain for networked control systems with signal quantization. A quantized fault detection observer is designed by employing a performance index which is used to increase the fault sensitivity in finite frequency domain. By using the logarithmic quantizer method, the quantized measurement signals are dealt with by utilizing the sector bound method, in which the quantization error is treated as sector-bounded uncertainty. Further, By using the GKYP Lemma, an iterative LMI-based optimization algorithm is developed to design the quantized fault detection observer. Finally, a numerical example is given to illustrate the the effectiveness of the proposed method.

Conflict of Interests

One of the authors declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is partially supported by the Funds of National Science of China (Grant nos. 61104106 and 61104029), the Natural Science Foundation of Liaoning Province (Grant nos. 201202156 and 2013020144), and by Program for Liaoning Excellent Talents in University (LNET) (LJQ2012100).