Fuzzy fault detection of conic-type nonlinear systems within the finite frequency domain

https://doi.org/10.1016/j.amc.2020.125181Get rights and content

Highlights

  • •

    Fault detection scheme for conic nonlinear systems in finite frequency domain is designed.

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    Sufficient conditions to realize H_ fault sensitivity and H_ disturbance attenuation.

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    The fuzzy fault detection problem is transformed into an optimization algorithm.

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    A tunnel diode-circuit is given to confirm the validity of the designed scheme.

Abstract

In this paper, we focused on the issue of fault detection subject to a class of conic-type nonlinear systems in the finite frequency domain (FFD). Applying Takagi Sugeno (T-S) fuzzy models, the conic-type dynamic error system is established. To prove that the residual vector is robust to external disturbances and sensitive to faults, we provide some conditions to realize H− fault sensitivity performance and H∞ disturbance attenuation performance within the FFD. Then, utilizing linear matrix inequalities techniques, the fuzzy fault detection observer design problem is transformed into an optimization algorithm. Finally, a numerical example related to a tunnel diode-circuit is given to confirm the validity of the designed scheme.

Introduction

For some complex dynamic processes, it will seriously affect the safety of the control systems by faults. Hence, it is crucial to detect faults in time when faults occur and take necessary measures to overcome the influence. In fact, fault detection (FD) has been concerned for a long time and many publications are available. For some related results, we refer to [1], [2], [3], [4], [5], [6], [7] and the references therein. As an effective FD method, the observer-based FD [8], [9], [10], [11] has been always viewed as a robustness method. Observer-based FD is primarily employed to build a residual vector, during which we need to select an appropriate residual evaluation function (REF). The initiative of the proposed method is that the residual vector can depict the robustness to external disturbances and sensitivity to faults and nonlinearities. In view of these, some research have been issued for the robust FD problems for uncertain and nonlinear/linear systems [12], [13], [14], [15], [16].

More recently, as the capability of Takagi-Sugeno (T-S) fuzzy approach to describe nonlinearity, it has drawn more and more attention [17], [18], [19], [20], [21], [22]. To get a T-S model, the first step is to linearize the system dynamics in the state space at a series of operating points, followed by applying optimization algorithms to mitigate system error. Lastly, we can employ the T-S fuzzy model to denote the linear relationship associated with nonlinear systems. By applying the fuzzy membership function, an available structure for representing nonlinear systems with a range of linear models is presented [23], [24], [25], [26], [27].

In prior research, most of the supposed fault detection observer (FDO) design and T-S fuzzy analysis approaches are studied within the full frequency domain. Nevertheless, faults usually emerge in a finite frequency domain (FFD) in practice. Therefore, it is more credible to research FDO within an FFD. In [28], the author used a certain weighting function to design an FDO within an FFD, but it is usually difficult to seek for the weighting functions [29]. In response to this, the authors in [30] offered a valid generalized Kalman–Yakubovich–Popov (GKYP) lemma [31], [32] to handle the H− analysis within an FFD. Comparing with linear dynamics, the FDO design for nonlinear systems is a more general problem. In practice, a wide variety of engineering nonlinearities may be converted into conic-type nonlinearities, such as Lipschitz nonlinearities, locally sinusoidal nonlinearities, diodes and amplifiers with dead zone nonlinearities, etc. Recently, some significant development about this kind of nonlinearities has been made in robust H∞ control [33], finite-time control [34], sliding mode control [35], etc. As far as we know, the FD problems for conic-type nonlinear systems have yet been comprehensively investigated. However, we can not be straight-forward to apply the GKYP lemma because of the existence of conic-type nonlinearities.

In this study, we attempt to study the FD problem for conic-type nonlinear systems via T-S fuzzy models. Subsequently, the conic-type dynamic error system is established by the FDO. Then, sufficient conditions are given based on the conic-type dynamic error system, which shows sensitivity to fault by H− performance index and robustness to external disturbances by H∞ performance index. The FDO design problem is transformed into an optimization algorithm and converted as linear matrix inequalities (LMIs) problem by MATLAB tools. In the end, a numerical simulation is presented to demonstrate the validity of the method designed.

Section snippets

Problem formulation

Consider the following T-S fuzzy models-based nonlinear systems described by :

System Rule μ:

IF ϖ1(t) is νμ1, ϖ2(t) is νμ2, and ..., ϖJ(t) is νμJ, THEN{x˙(t)=A¯μx(t)+Bμd(t)+Mμf(t)+g(t)y(t)=C¯μx(t)+Dμd(t)+Nμf(t)where x(t) ∈ Rk denotes the state, y(t) ∈ Rr denotes the measured output, f(t)∈L2l[0,∞] denotes the detectable fault signal, d(t)∈L2q[0,∞] denotes the external disturbance. ϖ1(t), ϖ2(t), ... ϖJ(t) are premise variables. νμj, μ=1,2,…,ϝ, j=1,2,…,J are fuzzy sets, J and ϝ denote the number of

Main results

Theorem 1

The conic-type dynamic error system (10) (setting f(t) ≡ 0) is asymptotically stable and satisfies the given H∞ index in (12), if for given scalars ξ1 > 0 and α1 > 0, there have positive matrices P1=P1T, P2=P2T, matrices Jμ and Kμ satisfying the following LMIs:ℶμμ<0,μ=1,2,…ϝℶμj+ℶjμ<0,μ<j,μ,j=1,2,…ϝwhere

ℶμj=[Π1ϰ1GμTGdμ+ϰ2GμTGdμP1Bμ0P10Cμ−CjP1M1μ0*Π2ξ1(JμBμ−KμBμ)Π30P2CjTξ1JμM1μ−ξ1KμM2μ**−χ2IΠ400Ddj00***−α1Jμ−α1JμT000α1JμM1μ−α1KμM2μ****−ϰ1I0000*****−ϰ2I000******−IM2μ0*******−κ1I0********−κ2I]<0with

Numeral example

Consider a tunnel diode-circuit system established in [41]. Let x1(t)=vC(t), x2(t)=iL(t), where vC(t) is the capacitor voltage and iL(t) is the inductance current. Then, the tunnel diode circuit system is described by:{Cx˙1(t)=−1.25x1(t)−0.01x13+x2(t)Lx˙2(t)=−x1(t)−Rx2(t)+d(t)y(t)=Sx(t)+d(t)where x(t)=[x1T(t)x2T(t)]T is the state variable, d(t) and y(t) are the external disturbances and the measured output respectively. S=[10] is the sensor matrix. In the circuit model, the relevant parameters

Conclusions

Throughout this work, the FD problem of conic-type nonlinear systems is tackled. To make the residual vector be robust to external disturbances and be sensitive to faults, we get the sufficient condition by H−fault sensitivity performance as well as H∞ disturbance attenuation performance. The obtained conditions are converted to LMIs such that the fuzzy FDO can be designed by Matlab tools. Finally, a simulation result related to a tunnel diode-circuit demonstrates the effectiveness of the

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China (No. 61673001, 61722306), the State Key Program of National Natural Science Foundation of China (No. 61833007), the Foundation for Distinguished Young Scholars of Anhui Province (No. 1608085J05), the Key Support Program of University Outstanding Youth Talent of Anhui Province (No. gxydZD2017001) and the open fund for Discipline Construction, Institute of Physical Science and Information Technology, Anhui

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