Fault detection in finite frequency domain for T-S fuzzy systems with partly unmeasurable premise variables
Introduction
Due to the wide existence of nonlinearity in practical applications, considerable research effort has been devoted to the investigation of nonlinear systems [4], [5], [6]. Particularly, T-S fuzzy model [1], which is represented by locally time-invariant subsystems connected by fuzzy rules, provides a general framework to deal with nonlinear dynamics. Consequently, diverse meaningful results on the T-S fuzzy systems have been published. For example, the problem of designing controller for the fuzzy systems was studied in [7]; the stabilization and stability analysis of the fuzzy systems were investigated in [8], [9], [10]; the authors in [11] investigated the fuzzy filter design problem. Note that, the aforementioned references mainly consider the T-S fuzzy systems in the fault-free case.
However, with the increasing complexity of practical systems, faults usually occur and may result in system performance degradation [12], [43]. Therefore, it is important to detect faults in time so as to decrease the influences of them. Among various FD methods, the model-based FD technique has attracted considerable attention, and based on which some remarkable results have been published, such as [13], [14], [15], and the references therein. It should be mentioned that the above FD schemes are all considered in entire frequency range, which cannot always meet the requirements of practical systems, because the frequency domains of some external inputs may be known beforehand [16], [17], [18]. Specially, the generalized Kalman-Yakubovich-Popov (GKYP) lemma was proposed in [19] to characterize frequency range inequalities with (semi)finite frequency domains in the formulation of linear matrix inequalities (LMIs), and based on which a great number of effective FD schemes have been developed, for instance [21], [22], [23], [24], [25].
The aforementioned full frequency and finite frequency FD methods have made significant progress in the study of FD for T-S fuzzy systems, however, only the T-S fuzzy systems with measurable premise variables were considered. Indeed, it is not the case that the premise variables of the fuzzy systems are always available. For instance, they may include unmeasurable system states [38], [39]. In this case, the so-called PDC strategy based FD methods [18], [21], [23], [24] cannot be directly applied. Then, numerous valuable results have been reported to solve this problem. For example, in [31], the Lipschitz method is used to solve the observer design problem for T-S fuzzy systems with unmeasurable premise variables, where the premise variables of the designed observer depend on the estimations of the ones of the fuzzy model. A linear filter with fixed gains was designed in [32] to detect the sensor faults. Based on a sliding mode observer, the adaptive control problem for T-S fuzzy systems with semi-Markov switching was considered in [42]. For a class of uncertain T-S fuzzy systems, the robust stability problem was studied in [44]. However, the partly available premise variables are not sufficiently used in these works, which may result in some conservative results. With that in mind, a switching-type FD filter was designed in [33], which has a promising feature that the bounds of the unknown weight functions can be used in the FD filter. Recently, the problem of sensor FD for the fuzzy systems with partly available premise variables was considered in [34], where the T-S fuzzy systems are described with the help of some concepts of the set, and the measurable premise variables are fully used for fault detection. However, the FD scheme in [34] cannot be directly applied to detect actuator or process faults due to the non-convex problem included in the performance. Furthermore, the frequency ranges of external inputs were not considered, which might be conservative in some degree.
Motivated by the above discussions, the problem of finite frequency FD for T-S fuzzy systems with partly unmeasurable premise variables is studied in this paper. First, a description method based on some concepts of the set [26] is employed to describe the considered fuzzy systems. Within this framework, the measured premise variables are explicitly separated from the unmeasurable ones, then the available information is fully used in the designed observer to decrease the conservatism of the existing results [27], [28], [29], [30], [31]. Furthermore, inspired by [35], the asynchronization of the membership functions is coped with the differential mean value theorem, and some slack variables are introduced to further reduce the conservatism of the proposed methods.
This paper is organized as follows. The preliminaries and system description are given in Section 2. The main design conditions of the FD observer are shown in Section 3. Some simulation results are provided in Section 4 to illustrate the validity of the theoretical results. Finally, the conclusions are derived in Section 5.
Notation is the n-dimensional Euclidean space. “⁎” represents the symmetric term in a symmetric matrix and is a block-diagonal matrix. and respectively denote the transpose and the orthogonal complement of the matrix A, and . I represents the identity matrix with suitable dimension. For a symmetric matrix , and denote positive (semi)definiteness and negative (semi)definiteness, respectively. “∧” denotes a classic logical operator “conjunction”. Some notations about the sets and the subscripts used throughout this paper are summarized in Table 1.
Section snippets
Preliminaries and problem statement
Consider a nonlinear system described by the following T-S fuzzy model:
Plant Rule :
IF is , is , …, and is where , denotes the system state, represents the input signal, indicates the system output, represents the external disturbance which is assumed to belong to , and denotes the fault signal which includes actuator fault and process fault. ,
Fault detection observer design
Because the premise variables () are measurable and () are unmeasurable, therefore, the following unknown input observer with the measurable premise variables and the estimations of the unmeasurable ones is designed in this paper:
Plant Rule :
IF is is and is , is where is the estimation of and represent the estimations of (). , , and R are the observer gain
Simulation examples
In this section, some simulation results are provided to illustrate the advantages and effectiveness of the proposed FD scheme.
Example 1 Consider the nonlinear mass-spring-damper system [8] described by where denotes the measurable displacement from a reference position, denotes the input force, and , , , , , , . It is assumed that and . Define , and assume that the nonlinear
Conclusion
The problem of finite frequency FD for T-S fuzzy systems with partly unmeasurable premise variables has been investigated in this paper. By employing the description scheme based on some concepts of the set, the premise variables of the considered fuzzy system are divided into the measurable and unmeasurable two parts. Within this framework, combining performance, a new FD observer with measurable premise variables and the estimations of the unmeasurable ones of the fuzzy system has been
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work was supported in part by the Funds of the National Natural Science Foundation of China (Grant Nos. 61621004 and U1908213), and the Research Fund of State Key Laboratory of Synthetical Automation for Process Industries (Grant no. 2018ZCX03).
References (45)
Finite time command filtered adaptive fault tolerant control for a class of uncertain nonlinear systems
Automatica
(2019)- et al.
Delay-dependent stability and control for a class of fuzzy descriptor systems with time-delay
Fuzzy Sets Syst.
(2009) - et al.
An LMI approach to index and mixed / fault detection observer design
Automatica
(2007) - et al.
An LMI approach to design robust fault detection filter for uncertain LTI systems
Automatica
(2003) filtering for fuzzy systems with immeasurable premise variables: an uncertain system approach
Fuzzy Sets Syst.
(2009)- et al.
Fault detection for T-S fuzzy systems with partly unmeasurable premise variables
Fuzzy Sets Syst.
(2018) - et al.
Survey of robust residual generation and evaluation methods in observer-based fault detection systems
J. Process Control
(1997) - et al.
Stabilization of Takagi-Sugeno models with non-measured premises: input-to-state stability approach
Fuzzy Sets Syst.
(2017) - et al.
A robust observer-based controller design for uncertain T-S fuzzy systems with unknown premise variables via LMI
Fuzzy Sets Syst.
(2013) - et al.
Fuzzy identification of systems and its applications to modeling and control
IEEE Trans. Syst. Man Cybern. Syst.
(1985)
Calculus: Early Transcendentals
A Unified Algebraic Approach to Linear Control Design
Event-based filter design for a class of nonlinear time-varying systems with fading channels and multiplicative noises
IEEE Trans. Signal Process.
Adaptive fault estimation for T-S fuzzy interconnected systems based on persistent excitation condition via reference signals
IEEE Trans. Cybern.
Control synthesis of discrete-time T-S fuzzy systems via a multi-instant homogenous polynomial approach
IEEE Trans. Cybern.
Robust stabilization of a class of uncertain nonlinear systems via fuzzy control: quadratic stabilizability, control theory, and linear matrix inequalities
IEEE Trans. Fuzzy Syst.
Stability analysis and stabilization for discrete-time fuzzy systems with time-varying delay
IEEE Trans. Syst. Man Cybern., Part B, Cybern.
filter design for a class of networked control systems via T-S fuzzy-model approach
IEEE Trans. Fuzzy Syst.
Adaptive fault diagnosis for T-S fuzzy systems with sensor faults and system performance analysis
IEEE Trans. Fuzzy Syst.
Observer-based fault detection for nonlinear systems with sensor fault and limited communication capacity
IEEE Trans. Autom. Control
A finite frequency domain approach to fault detection observer design for linear continuous-time systems
Asian J. Control
Finite frequency fault detection for networked systems with access constraint
Int. J. Robust Nonlinear Control
Cited by (11)
Extended dissipative analysis of affine transformed IT2 fuzzy control systems with time delay and disturbances
2023, Journal of the Franklin InstituteAdaptive adjustable dimension observer based fault estimation for switched fuzzy systems with unmeasurable premise variables
2023, Fuzzy Sets and SystemsCitation Excerpt :Based on T-S fuzzy model, the nonlinear function can be approximated by a set of linear subsystems, and the linear subsystems are connected by corresponding nonlinear weights. In this way, some classic tools for handling linear systems, such as linear matrix inequality technique, can be applied to nonlinear systems [18–20]. In recent years, in order to deal with the nonlinear dynamics in switched systems, switched fuzzy system models have attracted extensive attention from scholars [21–24], where fuzzy models are used to approximate the nonlinear dynamics.
Simultaneous fault detection and robust control for a dynamic observer-based switched time delay systems with car roll dynamic application
2024, International Journal of General SystemsA new switching control for T-S fuzzy systems with mixed time delays
2023, International Journal of Robust and Nonlinear ControlFault detection for Takagi–Sugeno fuzzy systems using multiple observers and ellipsoidal analysis
2023, International Journal of Robust and Nonlinear ControlCompensation-Based Output Feedback Control for Fuzzy Markov Jump Systems With Random Packet Losses
2022, IEEE Transactions on Cybernetics