Adaptive adjustable dimension observer based fault estimation for switched fuzzy systems with unmeasurable premise variables☆
Introduction
In recent decades, the requirements of the safety and reliability for modern industrial systems are much higher than ever before. However, due to the increase in equipment operating time and changes in the operating environment, various faults inevitably occur in practice. Under such background, more and more attention is paid to fault diagnosis and fault tolerance control [1], [2], [3], [4], [5], [6], [7].
Fault estimation is an important part of fault diagnosis, and it can obtain specific fault information, which is very helpful for making targeted decisions about the occurred fault. Generally speaking, fault estimation can be achieved by structuring the fault estimation observer (or named fault estimator) to reconstruct the possible fault in the system. In the last dozen years, fault estimation problem has become a hot issue, and various types of fault estimation observers have been structured to reconstruct various faults, such as the proportional-integral observer [8], [9], sliding-mode observer [10], [11], descriptor observer [12], [13] and so on. Specially, in the descriptor observer design process, the plant was rewritten as an augmented descriptor system by treating the sensor fault as an augmented state, and then the descriptor observer was designed, which could avoid the sensor fault being amplified by the observer gain.
It should be noted that many plants exhibit switching behaviors. Switched systems, especially switched nonlinear systems, are very effective tools to model such plants. Recently, this type of systems have attracted the attention of scholars [14], [15], [16], [17]. While, the system analysis and design for switching nonlinear systems are not easy, due to the combination of nonlinear dynamics and switching rules. Fortunately, T-S fuzzy model provides an effective strategy to describe the nonlinear dynamic. 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], [19], [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], [22], [23], [24], where fuzzy models are used to approximate the nonlinear dynamics. While, most of existing results focused on the system analysis and design, and few results considered fault diagnosis and fault tolerance control for switched fuzzy systems. In [25], [26], [27], the fault detection filters and fault detection observers have been considered for switched fuzzy systems, and LMIs have been introduced to calculate the corresponding parameter matrices. In [28], dynamic observer based fault estimation has been considered for switched nonlinear systems, where T-S fuzzy models have been introduced to model the nonlinear switched subsystems. The problem of actuator and sensor faults simultaneous estimation problem has been discussed in [29], [30] for switched fuzzy systems. In [31], [32], the state feedback and output feedback based fault-tolerant control problems have been reported for switched fuzzy systems, respectively.
It should be noted that in the existing results of switched fuzzy systems, the premise variables of each fuzzy rule were assumed to be measurable. While, these were not easy to be achieved in some practical cases [33], [34], [35]. In fact, the unmeasurable premise variables may lead to unknown fuzzy rule weight, thus, this case is more challenging than the case that premise variables are measurable. In the classical fuzzy observer design results, there were two common ways to deal with the unmeasurable premise variables. One method was to consider the fuzzy weight and its estimation simultaneously, then constructed the augmented systems, which contained the error dynamic and the plant [33], [36], [37]. Compared with the measurable premise variables case, the number of LMIs to be solved by this method was increased by r times, where r represented the number of fuzzy rules. And the problem of computational complexity is more serious in switched fuzzy systems, since there is more than one switched model. Recently, the descriptor redundancy method, which was proposed in [38], has been applied in this case. Based on this technique, which could avoid the multiplication of the system matrices, observer gains and the controller gains, some effective results have been reported in [39], [40]. The other method was with the same computation complexity as the measurable premise variables case, while introduced a Lipschitz nonlinear term to represent the error of the unknown fuzzy weight and its estimation, and the Lipschitz nonlinear observer design method was considered for this case [41], [42], [43]. While, it was not easy to calculate the Lipschitz constant, and introducing the Lipschitz constant into LMIs might lead to infeasible LMIs.
So far, to the best of our knowledge, the fault estimation problem for switched fuzzy systems with unmeasurable premise variables has not been fully considered and remains to be important and challenging. This motivates our study. In this paper, a novel adaptive adjustable dimension fault estimation observer design method is proposed for switched fuzzy systems, and under arbitrary switching signals, the sensor fault and system state can be reconstructed simultaneously. The main contributions can be summarized as follows.
- 1)
The premise variables of the fuzzy models are assumed to be unmeasurable, and the Lipschitz nonlinear term, which is caused by the unmeasurable premise variables, is divided into two parts. Based on selecting appropriate parameter matrices and adding adaptive compensator, the influence of these two parts can be decoupled and compensated, respectively.
- 2)
In the proposed method, the Lipschitz constant of the nonlinear team in the error dynamic is not needed. Compared with the existing results, the proposed method can avoid calculating the Lipschitz constant and can reduce the conservatism caused by introducing the constant into LMIs.
- 3)
The observer dimension is not fixed, and it can be selected in a certain range. As it pointed in [1], both the estimation accuracy and cost increase with the increased observer order. This means that the proposed method is with more design freedom for different conditions.
This paper is organized as follows. Section 2 is the system preliminaries. Section 3 provides the main results, where the system transformation, observer design, observer parameter selection and the stability analysis are provided. Section 4 verifies the proposed method by two examples, and Section 5 concludes this paper.
Section snippets
System preliminaries
In this paper, consider the following switched nonlinear systems: where , denote the system state and output, and are the control input and disturbance, and represent the actived switching subsystem labels. For any fixed , denotes the nonlinear function of the switching model, and are constant output matrices. In the rest of this paper, we omit the needless t.
For each switched
Coordinate transformation
Let , then we have where , , , , , , .
Based on (5), it can be found that . Thus,
Adding to both sides of it, we have where . It can be found
Simulation results
In this section, two examples are provided.
Example 1 Consider the systems (1)–(2) with two switching subsystems, where each switching subsystem includes two fuzzy rules. Assume that the parameter matrices in them are: The corresponding fuzzy rules are assumed to be
Conclusions
In this paper, the adaptive observer-based fault estimation problem has been addressed for switched fuzzy systems. The novel observers with adaptive compensator have been designed to reconstruct the sensor fault and system state, simultaneously. The premise variables of the fuzzy models are assumed to be unmeasurable, and the adaptive compensator has been added in the designed observers to eliminate the effect of the Lipschitz nonlinear term caused by the unmeasurable premise variables. It is
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.
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This work was supported by the National Natural Science Foundation of China (61903173, 61973149, 61903174), and Shandong Provincial Natural Science Foundation, China (ZR2019BF016, ZR2019PF014, ZR2019PF006), and the Open Project Program of Shandong Marine Aerospace Equipment Technological Innovation Center, Ludong University (Grant No. MAETIC2021-05).