Finite-horizon quantized H∞ filter design for a class of time-varying systems under event-triggered transmissions

doi:10.1016/j.sysconle.2017.02.011

Systems & Control Letters

Volume 103, May 2017, Pages 38-44

https://doi.org/10.1016/j.sysconle.2017.02.011 Get rights and content

Abstract

This paper is concerned with the finite-horizon quantized $H_{\infty}$ filter design problem for a class of time-varying systems with quantization effects and event-triggered measurement transmissions. A componentwise event-triggered transmission strategy is put forward to reduce the unnecessary communication burden for the purpose of energy efficiency. The transmitted measurements triggered according to prespecified events are quantized by a logarithmic quantizer. Special attention is paid to the design of the filter such that a prescribed $H_{\infty}$ performance can be guaranteed over a given finite horizon in the presence of nonlinearities, quantization effects and event-triggered transmissions. Two sets of Riccati difference equations are introduced to ensure the $H_{\infty}$ estimation performance of the designed filter. The filter design algorithm is recursive and thus suitable for online computation. A simulation example is illustrated to show the effectiveness of the proposed algorithm applied to the fault detection problem.

Introduction

In the past decades, the filtering problem has been receiving persistent research attention from both control and signal processing communities. Among existing filtering approaches, the $H_{\infty}$ filtering has been extensively studied to guarantee a bound of the worst-case estimation error where the information on the disturbances statistics is not required. To deal with the $H_{\infty}$ filtering/control problems for time-varying systems over a finite horizon, differential/difference linear matrix inequality (LMI) and recursive linear matrix inequality have been adopted in [1], [2], and the backward recursive Riccati difference equation (RDE) approach has also recently drawn specific interests [3], [4].

The event-triggered transmission mechanism has recently gained much research attention owing to the need for reducing energy consumptions and extending the lifetimes of the services [5], [6], [7]. Compared with the conventional clock-driven strategy, the outputs/inputs in an event-triggered scheme are released only when some conditions are violated in the event generator. As such, the event-triggered transmission strategy can reduce energy consumptions and extend the lifetimes of the services. The event-triggered minimum-variance filter has been studied in [8], [9], where the probability density functions (PDFs) of the states conditional on measurements have been approximated with Gaussian distributions. Modified Kalman filter for linear systems with event-triggered transmissions has been designed in [10], [11], [12], where the difference between the current measurement and the previously transmitted one has been assumed to be uniformly distributed. The event-triggered $H_{\infty}$ filtering problem has also stirred some initial research attention in [13], [14] for linear time-invariant systems.

It should be pointed out that, in almost all existing literature concerning event-triggered filtering problems, the time-varying and nonlinear behaviors, which are ubiquitous in practice, have not yet been taken into account despite their engineering significance. Moreover, in most reported results, the Euclidean norm (or the weighed Euclidean norm) of the measurement output has been utilized in the event generator. In practical, however, it is often the case that certain components of the measurement vector deserve more attention, which should be evaluated and transmitted separately/individually [10], [11], [15]. In such a case, the transmitted output signals are more sensitive to the variations of measurements and this enables us to monitor the addressed systems more accurately with individual componentwise thresholds. Therefore, there is a practical need to address the $H_{\infty}$ filtering problem for time-varying nonlinear systems with componentwise event-triggered transmissions.

Measurement outputs of practical systems are often subject to quantization effects due to limited bandwidth in a networked environment, and there has been a rich body of literature focusing on the filtering problem with quantization effects [16], [17], [18]. Unfortunately, when it comes to the event-triggered scenario, the corresponding filtering problem with quantization effects has not gained adequate research attention yet especially when the system is time-varying and nonlinear. In [15], the event-triggered $H_{\infty}$ filtering with output quantization has been investigated for linear time-invariant systems by using the LMI method. Therefore, it is of significance to consider the event-triggered $H_{\infty}$ filter design for time-varying nonlinear systems with quantization effects over a finite horizon.

In this paper, we aim to deal with the $H_{\infty}$ filtering problem for a class of time-varying systems subject to event-triggered measurement transmissions and quantization effects. A componentwise event-triggered mechanism is put forward and a logarithmic quantizer is taken into account. Sufficient conditions are established to guarantee the prescribed $H_{\infty}$ performance over a finite horizon. The filter gain is designed based on an $H_{2}$ -type requirement, and the filter is capable of estimating the states over a finite horizon with a locally minimized cost. An illustrative example is provided to show the effectiveness of the proposed method in detecting possible faults in the system. The main novelty of the paper lies in the following aspects: (1) a comprehensive model is considered which covers nonlinearities, componentwise event-triggered measurement transmissions, and quantization effects; (2) an $H_{\infty}$ filtering performance is considered over a finite horizon in response to the time-varying nature of the addressed system; and (3) the finite-horizon $H_{\infty}$ filter is designed via solving backward RDEs and the recursive algorithm is suitable for online computation.

Notations. The notation used in the paper is fairly standard except where otherwise stated. $R^{n}$ and $R^{n \times m}$ denote, respectively, the $n$ -dimensional Euclidean space and the set of all $n \times m$ real matrices. $l_{2} [0, N - 1]$ is the space of square-summable vector functions over $[0, N - 1]$ . The superscript “T” denotes the transpose, and the superscript “ $†$ ” denotes the Moore–Penrose pseudo inverse. $I$ is the identity matrix with compatible dimension. $E {x}$ stands for the expectation of the stochastic variable $x$ . $‖ A ‖$ denotes the spectral norm of matrix $A$ , and $‖ x ‖$ refers to the Euclidean norm of vector $x$ . $diag {\dots}$ stands for a block-diagonal matrix.

Section snippets

Problem formulation

Consider the following class of discrete-time nonlinear systems: $\{\begin{matrix} x_{k + 1} & = & A_{k} x_{k} + g (x_{k}) + D_{k} w_{k} + E_{k} f_{k}, \\ y_{k} & = & C_{k} x_{k} + F_{k} v_{k}, \end{matrix}$ where $x_{k} \in R^{n}$ is the state; $y_{k} \in R^{m}$ is the measurement output; $f_{k} \in R^{t}$ is the additive fault; $w_{k} \in R^{p}$ and $v_{k} \in R^{q}$ are the process noise and measurement noise, respectively. It is also assumed that $f_{k}, w_{k}, v_{k} \in l_{2} [0, N - 1]$ . $A_{k}$ , $C_{k}$ , $D_{k}$ , $E_{k}$ , and $F_{k}$ are known time-varying matrices with appropriate dimensions. The nonlinear function $g (\cdot)$ satisfies $g (0) = 0$ and the Lipschitz condition as follows: $‖ g (x_{1}) - g (x_{2}) ‖^{2} \leq l_{g} ‖ x_{1}$

Filter design

Let us rewrite (13) as follows: $ξ_{k + 1} = {\tilde{C}}_{k} ξ_{k} + {\tilde{D}}_{k} {\hat{w}}_{k}$ where ${\hat{w}}_{k} = {[ϖ_{k}^{T}, ε_{1 k} {\tilde{g}}^{T} (ξ_{k}), ε_{2 k} {({\tilde{Δ}}_{k} F_{k} H_{2} ϖ_{k})}^{T}, ε_{3 k} {({\tilde{Δ}}_{k} C_{k} H_{1} ξ_{k})}^{T}, ε_{4 k} {({\tilde{Δ}}_{k} s_{k})}^{T}, ε_{5 k} s_{k}^{T}]}^{T},$ ${\tilde{D}}_{k} = [{\tilde{E}}_{k}, ε_{1 k}^{- 1} I, ε_{2 k}^{- 1} {\tilde{F}}_{k} \bar{Δ}, ε_{3 k}^{- 1} {\tilde{F}}_{k} \bar{Δ}, ε_{4 k}^{- 1} {\tilde{F}}_{k} \bar{Δ}, ε_{5 k}^{- 1} {\tilde{F}}_{k}],$ and $ε_{i k} (i = 1, \dots, 5)$ are known positive scalars.

For system (15), define the following performance requirement: $\bar{J} = \sum_{k = 0}^{N - 1} (∥ e_{k} ∥^{2} - γ^{2} ∥ {\hat{w}}_{k} ∥^{2}) - γ^{2} ξ_{0}^{T} R ξ_{0} + γ^{2} \sum_{k = 0}^{N - 1} (l_{g} ∥ ε_{1 k} \times ξ_{k} ∥^{2} + ‖ ε_{2 k} F_{k} H_{2} ϖ_{k} ‖^{2} + ‖ ε_{3 k} C_{k} H_{1} ξ_{k} ‖^{2} + ∥ ε_{4 k} Λ \times y_{k} ∥^{2} + ‖ ε_{5 k} Λ y_{k} ‖^{2}) < 0 ({ϖ_{k}}, ξ_{0} \neq 0),$ where $Λ = diag {ς_{1}, \dots, ς_{m}}$ .

Theorem 1

Considering requirements (14), (16) , we have $J \leq \bar{J}$

An illustrative example

Consider the following time-varying nonlinear system: $\{\begin{matrix} x_{k + 1}^{(1)} = & 0.5 x_{k}^{(1)} sin k - 0.7 x_{k}^{(2)} - 1.5 u_{k} + 0.01 w_{k}^{(1)} + f_{k}, \\ x_{k + 1}^{(2)} = & 0.15 sin x_{k}^{(1)} + 0.4 x_{k}^{(2)} + 0.75 x_{k}^{(3)} cos k + 0.01 w_{k}^{(2)}, \\ x_{k + 1}^{(3)} = & 0.2 x_{k}^{(1)} + 0.7 u_{k} x_{k}^{(2)} + 0.4 x_{k}^{(3)} + 0.01 w_{k}^{(3)}, \\ y_{k}^{(1)} = & x_{k}^{(1)} + 0.01 v_{k}^{(1)}, \\ y_{k}^{(2)} = & x_{k}^{(2)} + 0.01 v_{k}^{(2)}, \\ u_{k} = & 0.2 {\tilde{y}}_{k}^{(1)} + 0.4 {\tilde{y}}_{k}^{(2)}, \end{matrix}$ where $x_{k} = {[x_{k}^{(1)}, x_{k}^{(2)}, x_{k}^{(3)}]}^{T} \in R^{3}$ is the system state, $y_{k} = {[y_{k}^{(1)}, y_{k}^{(2)}]}^{T} \in R^{2}$ is the measurement output, ${\tilde{y}}_{k} = {[{\tilde{y}}_{k}^{(1)}, {\tilde{y}}_{k}^{(2)}]}^{T} \in R^{2}$ is the transmitted output, $u_{k} \in R$ is the control input, and $f_{k} \in R$ is the additive fault.

The disturbances $w$

Conclusion

The filtering problem has been considered for a class of time-varying systems with quantization effects and event-triggered measurement transmissions. A componentwise event-triggered transmission scheme has been put forward to reduce unnecessary signal transmissions and improve the energy efficiency. The output measurements have been assumed to be quantized by a logarithmic quantizer. An $H_{\infty}$ requirement over a given finite horizon has been investigated. Two sets of Riccati-like matrix equations

Acknowledgments

This work was supported by National Natural Science Foundation of China (NSFC) under Grants 61490701, 61290324, 61210012, China Postdoctoral Science Foundation under Grant 2016M600546, Qingdao Postdoctoral Applied Research Projects under Grant 2016112, and Research Fund for the Taishan Scholar Project of Shandong Province of China under Grant LZB2015-162. The work of Zidong Wang was also supported by NSFC under Grant 61273156. The work of Xiao He was also partially supported by NSFC under

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Finite-horizon quantized H∞ filter design for a class of time-varying systems under event-triggered transmissions

Abstract

Introduction

Section snippets

Problem formulation

Filter design

An illustrative example

Conclusion

Acknowledgments

Automatica

Automatica

Automatica

J. Franklin Inst.

Automatica

Int. J. Electr. Power Energy Syst.

H∞ Control and Estimation of State-Multiplicative Linear Systems

Finite-horizon H∞ filtering with missing measurements and quantization effects

IEEE Trans. Automat. Control

A survey of event-based strategies on control and estimation

Syst. Sci. Control Eng.: An Open Access J.

Event-triggered filtering for nonlinear networked discrete-time systems

IEEE Trans. Ind. Electron.

On set-valued Kalman filtering and its application to event-based state estimation

IEEE Trans. Automat. Control

Finite-horizon quantized $H_{\infty}$ filter design for a class of time-varying systems under event-triggered transmissions

$H_{\infty}$ Control and Estimation of State-Multiplicative Linear Systems

Finite-horizon $H_{\infty}$ filtering with missing measurements and quantization effects