Relaxed state estimation of discrete-time nonlinear control systems via an improved fuzzy partition-based switching observer

Abstract

The problem of relaxed state estimation of a class of discrete-time nonlinear plants is studied by developing an improved fuzzy partition-based switching observer. By introducing a pair of weighted scalars, the whole interval spanned by all the normalized fuzzy weighting functions at the present sampling instant is delimited as some non-overlapping sub-intervals. Then, an available fuzzy switching observer can be designed with all kinds of possible switching modes. Because much more information for each partition sub-interval could be considered in its activated switching mode, the obtained method can be less conservative than the existing methods reported in recent references. Finally, two illustrative simulations are given for testing the superiority of the proposed method over previous ones.

Introduction

The great majority of the real world plants exhibit strong nonlinear dynamics and this makes it very challenging for system modelling and control synthesis. Indeed, it has been verified that Takagi-Sugeno (T–S) fuzzy systems [1] are capable of capturing bounded nonlinearities information and thus a great deal of T–S fuzzy-model-based methods have been designed in order to cope with strong nonlinear dynamics [2], [3], [4], [5], [6]. Because T-S fuzzy systems belong to universal approximators and a class of nonlinear functions can be smoothly approximated, many difficult problems about nonlinear dynamics have been successfully solved in the past ten years, e.g., Xie et al. [7], [8], [9], [10], [11], [12], [13], [14], [15]. However, it must be worthwhile pointing out that actual system state vector is partially measurable on most occasions. Hence, how to reconstruct accurate system state vector becomes very meaningful in practice [16], [17], [18]. Therefore, there have been considerable quantity of T–S fuzzy-model-based state estimation topics reported in recent literature, for example, fuzzy estimation of continuous-time nonlinear plants [19], [20], [21], [22], non-fragile robust filtering [23], [24], [25], fuzzy filtering of discrete-time nonlinear plants [26], [27], fuzzy filtering of nonlinear plants with time delays [28], [29], [30], and fuzzy model reduction [31], [32].

Nevertheless, there exists a very tricky problem on the topic of fuzzy state estimation, i.e., the existing methods may be too conservative to be applied in some harsh scenes. The main reason is that they were designed under the framework of the simple parallel distributed compensation (PDC) theory, in which its extended fuzzy filtering/observer could be exclusively dependent on the information of present sampling instant (single-step) normalized fuzzy weighting functions [33]. Recently, an effective multi-steps method has been proposed in [34] and a different kine of fuzzy observers that are parameter-dependent on multi-steps normalized fuzzy weighting functions have been developed, so that much more relaxed conditions of state estimations have been provided in [34]. More recently, this topic is further carried forward via proposing a so-called maximum-priority-based fuzzy observer in [35]. Even to this day, it is important to point out that there still much improvement room to be promoted if we can give a more advanced way to make use of additional systems information. Therefore, the problem of more relaxed results ought to be addressed, which motivates the authors to give this work.

The problem of relaxed state estimation of the underlying discrete-time nonlinear plants will be addressed by developing an improved fuzzy partition-based switching observer. By introducing a pair of weighted scalars, the whole interval spanned by all the normalized fuzzy weighting functions at the present sampling instant is delimited as some non-overlapping sub-intervals. Then, an available fuzzy switching observer can be designed with all kinds of possible switching modes. Because much more information for each partition sub-interval could be considered in its activated switching mode, the obtained method can be less conservative than existing methods reported in [34], [35]. Finally, two illustrative simulations are given for testing the superiority of the developed approach over previous ones.

The remains of the paper are organized as follows: Preliminaries are provided in Section 2 and our developed main result is accomplished in Section 3. Two illustrative simulations are carried in Section 4. Finally, this paper is ended with necessary conclusions in Section 5.

Applied notations. $\mathbb{N}$ represents the set of natural numbers, $\mathbb{R}$ expresses the set of real numbers and ${\mathbb{Z}}_{+}$ means the set of positive integers. s! is defined as the factorial, i.e., $s!=s\times \cdots \times 1$ for any $s\in {\mathbb{Z}}_{+}$. The left-hand side for any underlying relation is represented by Left( · ) and $V+V^T$ is represented by He(V).