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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

This paper is concerned with the reliable finite frequency filter design for networked control systems (NCSs) subject to quantization and data missing. Taking into account quantization, possible data missing and sensor stuck faults, NCSs are modeled in the framework of discrete time-delay switched systems, and the finite frequency _{2} gain is adopted for the filter design of discrete time-delay switched systems, which is converted into a set of linear matrix inequality (LMI) conditions. By the virtues of the derived conditions, a procedure of reliable filter synthesis is presented. Further, the filter gains are characterized in terms of solutions to a convex optimization problem which can be solved by using the semi-definite programme method. Finally, an example is given to illustrate the effectiveness of the proposed method.

In recent years, there has been a growing interest in networked control systems (NCSs), which is a class of systems in which sensors, controllers and plants are connected over the network media [

On the other hand, filtering problem has been playing an important role in control engineering and signal processing that has attracted constant research attention, (see e.g., [

It should be noted that disturbances considered in those papers are all considered in full frequency domain. However, practical industry systems often employ large, complex, or lightweight structures, which include finite frequency fundamental vibration modes. Thus, it is more reasonable to design reliable filters in finite frequency domain. However, to the best of the authors' knowledge, reliable filtering problems for NCSs subject to packet loss and quantization have not been fully investigated, especially in finite frequency domain where faults occur frequently. This motivates the investigation of this work.

In response to the above discussions, in this paper, the reliable finite frequency filtering problem for NCSs subject to packet loss and quantization is investigated in finite frequency domain against sensor stuck faults. Specifically, with consideration of quantization, possible packet losses and possible sensor stuck faults, NCSs are modeled in a framework of discrete time-delay switched system. Then, the definition of finite frequency _{2} gain is given and an analysis condition to capture such a performance for discrete time-delay switched system is derived. With the aid of the derived conditions, a reliable filter is designed and the conclusions are presented in terms of linear matrix inequalities (LMIs). Finally, an example is given to illustrate the effectiveness of the proposed method.

The reminder of the paper is organized as follows. The problem of system modeling for NCSs with packet losses and quantization is presented in Section 2. Section 3 provides sufficient conditions for the design of reliable filters. In Section 4, an example is given to illustrate the effectiveness of the proposed method. Finally, some conclusions are presented in Section 5.

Throughout the paper, the superscript _{2} denotes the Hilbert space of square integrable functions. In block symmetric matrices or long matrix expressions, we use * to represent a term that is induced by symmetry; The sum of a square matrix ^{T}^{T}

The NCS under consideration is setup in ^{n}^{m}^{p}^{d}_{2}[0, ∞).

In this paper, we make the following assumption:

System (1) is stable.

Assumption 1 is required to get stable dynamics of the filter system. If this assumption is not satisfied, a stabilizing output feedback controller is required.

When the sensors in the NCSs experience faults, we consider the following sensor stuck fault model similar to [_{sij}_{i}

It is also assumed that, as shown in _{q}_{q},δ_{q}_{q}

Therefore, the faulty measurements together with quantization and the data transmission in the networks can be described by
_{q}^{Fi}

The description of data transmission (8) was introduced in [_{q}^{Fi}_{q}^{Fi}

In this paper, the following reliable filter is constructed:
_{f}, B_{f}_{f}

Denoting ^{T}^{T}^{T}

_{1}: No packet dropout occurs.

_{2}: Packet dropout occurs.

Due to packet drop-out, the filtering error system can be seen as combined by subsystem _{1} and _{2}, which can be lumped into the following discrete time-delay switched system:
_{k}_{k}

Next, we will discuss how to design the filter parameters _{f}, B_{f}_{f}

System (10) is said to be asymptotical stable under switching signal σ_{k}, if the solution satisfies

Ding _{2} gain _{2} such that the following hold

For the low-frequency range |_{l}

For the middle-frequency range _{1} ≤ _{2}
_{w}_{2} − _{1})/2.

For the high-frequency range |_{h}

Now, the reliable filtering problem to be addressed in this paper can be formulated as follows:

Design a stable reliable filter (9) such that, for the quantization error, possible data missing and sensor faults, the filtering error system (10) is asymptotical stable, and with a prescribed finite-frequency _{2} gain _{1} from _{2} gain _{2} from _{si}_{w}_{s}

Before ending this section, the following lemmas will be first given to help us to prove our main results.

For ^{n}^{n}^{×}^{n}^{n}^{×}^{m}^{⊥} be any matrix such that
^{⊥}

^{T}^{T}x

^{⊥}
^{⊥}^{T}

∃^{n}^{T}

∃
^{m}^{×}^{n}^{T}^{T}

Given the matrices ^{T}

In this section, the reliable filtering problem proposed in the above section will be investigated.

Consider system (10) for _{1} < 0, then _{2} gain _{1},if there exist matrices

For the low-frequency range |_{w}_{wl}

For the middle-frequency range _{1} ≤ _{w}_{2}
_{c}_{2} + _{1})/2, _{w}_{2} − _{1})/2.

For the high-frequency range |_{w}_{wh}

We first consider the middle-frequency case for system (10) with _{si}^{T}^{T}_{σki}_{σki}Z

Similarly, by choosing _{1}:= −_{wl}_{2}:= _{wl}_{1}:= _{wh}_{2}:= 2_{wh}

Lemma 8 presents an analysis condition for finite frequency _{2} gain of system (10), where less conservatism is introduced compared with the existing full frequency conditions when frequency ranges of disturbances are known.

The sufficient condition, which guarantees a prescribed low frequency _{2} gain from y_{si}(

In this section, sufficient conditions to capture the finite frequency performance from

Consider system (10) in fault free and faulty cases (_{wl}_{2} gain _{1} from _{1} > 0, matrices _{f}, ℬ_{f}, C_{f}

It is shown, from Lemma 8, that the condition (16) can be reached if

Exploiting Lemma 6 and explicit null space bases calculations on ^{⊥} = [−_{κi}_{w}A_{κdi}^{T}

Performing Schur's complement on

In Theorem 11, by introducing a variable

The previous Theorem 11 presented the condition to capture the low frequency performance. Similarly, conditions for middle frequency and high frequency performance are presented in the following two corollaries.

Consider system (10) in fault free and faulty cases (_{1} ≤ |_{w}_{2},which is with a prescribed _{2} gain _{1} from _{1} > 0, matrices _{f}, ℬ_{f}, C_{f}_{c}_{2} + _{1})/2, _{w}_{2} − _{1})/2,

By following the same lines of Theorem 1, it is immediate.

Consider system (10) in fault-free and faulty cases (_{w}_{wh}_{2} gain _{1} from _{1} > 0,matrices _{f}, ℬ_{f}, C_{f}

By following the same lines of Theorem 1, it is immediate.

In this subsection, sufficient conditions to capture the low frequency performance (17) for system (10) will be deduced.

Consider system (10) in faulty cases (_{s}_{sl}_{2} gain _{2} for nonzero _{si}_{2} > 0, matrices _{f}, ℬ_{f}, C_{f}

It is easily derived, from Lemma 8 and Remark 10, that the condition (17) holds if
_{d}

Following the same process in Theorem 11, we first rewrite the inequality (33) to the following form,

Exploiting Lemma 6 and explicit null space bases calculations on it, we have ^{⊥} = [−_{κi}A_{κdi}_{fi}

By applying Lemma 7 on

Since Theorems 1 and 2 presented in the above section cannot guarantee the stability of the system (10), in this subsection, asymptotical stability conditions for system (10) will be presented.

Consider system (10) in fault free and faulty cases (_{si}_{3} > 0,matrices _{f}, ℬ_{f}, C_{f}

Consider the following Lyapunov functional candidate for _{si}

The forward difference of the Lyapunov functional _{i}^{T}^{T}^{T}

By following an opposite direction to the proof for Theorem 11 and exploiting Schur's complement and Lemma 7 on _{κi}_{κi}

Based on the above analysis, a set of optimal solutions _{f}, ℬ_{f}_{f}_{q}

Then the dynamic output feedback controller gains can be computed by the following equalities:

On the other hand, we can obtain a coarser quantizer through solving the following optimization problem for given finite-frequency _{2} gains _{2} and _{2}

In this section, an application and simulations are given to illustrate the effectiveness of the proposed methods.

The utilized model is the F-404 aircraft engine described by the following state space model [

Assume the sampling period is

For given _{wl}_{sl}_{q}_{1} = 0.0825 and _{2} = 0.0378 with the corresponding reliable filter parameters

In the following, the system is simulated in low frequency domain, where the faults always occur, under the following two fault modes with zero initial condition and the disturbance input

The first sensor being stuck at 0, _{si}

The second sensor being stuck at 0, _{si}

The controlled outputs and the corresponding estimations for both the two fault modes are shown in

It is easily seen from these figures that all the expected system performance requirement are well achieved, which shows the effectiveness of the proposed method.

In this paper, the reliable finite frequency filtering problem for NCSs subject to quantization and packet have been studied with finite frequency specifications when sensor fault would occur. The considered NCSs have been first modeled as a discrete time-delay switched system. Subsequently, a sufficient condition to characterize the finite frequency _{2} gain has been presented. Then by virtues of the derived condition, a procedure of reliable filter synthesis has been derived in terms of LMIs. Finally, an example has been given to illustrated the effectiveness of the proposed method.

The authors greatly appreciate the editor, associate editor and anonymous referees for their insightful comments and suggestions that have significantly improved this article. This research has been supported by the Funds of National Science of China (Grant No. 61004061, 60975065).

_{∞}control for networked systems with random communication delays

_{∞}control of systems with uncertainty

_{∞}control for network-based systems with time-varying delay and packet disording

_{∞}filter design of a class of switched systems with sensor failures

_{∞}filtering against sensor failures

_{∞}filtering with sensor failure

_{∞}filter design for discrete-time systems with sector-bounded nonlinearities: An LMI optimization

_{∞}fuzzy filtering of nionlinear system with intermittent measurements

_{∞}filtering for discrete time-delay systems with randomly occured nonlinearities via delay-partitioning method

Structure of the Networked Control Systems.

The number of the lost packet.

The disturbance input

The

The