State estimation for discrete-time systems with generalized Lipschitz nonlinear dynamics
- Dazhong Wang^{1},
- Fang Song^{2, 3} and
- Wei Zhang^{1, 2}Email author
https://doi.org/10.1186/s13662-015-0645-x
© Wang et al. 2015
Received: 4 August 2015
Accepted: 24 September 2015
Published: 6 October 2015
Abstract
This paper considers the state estimation problem for a class of discrete-time systems with generalized Lipschitz nonlinear dynamics. Under the assumption that the system nonlinearities satisfy a quadratically inner-boundedness condition, we design both the full-order observer and the reduced-order observer for the discrete-time nonlinear system. Sufficient conditions ensuring the existence of full-order observers as well as reduced-order observers for such systems are established and formulated in terms of linear matrix inequality (LMI). Compared with some existing results, we remove the one-sided Lipschitz restrict and extend the classical Lipschitz observer design to a larger class of discrete-time nonlinear systems. A numerical example is included to illustrate the effectiveness of the proposed design.
Keywords
1 Introduction
During the past two decades, the state estimation or observer design problem for nonlinear dynamic systems has received extensively research attention; see [1–14] and the references therein. This is partly due to the fact that knowledge of the state of a dynamic system plays a key role in many control problems. It is well known that state estimation can be used for control design, diagnosis or synchronization and unknown input recovery. However, designing a state observer for a general nonlinear system is not easy or even impossible. Many current research efforts are focused on some specialized classes of nonlinear systems. For instance, Arcak et al. [1, 2] developed a circle-criterion approach to design observer for sector nonlinear systems. For Lipschitz nonlinear systems, the existence conditions of the full-order as well as the reduce-order observers were established in Rajamani [3] and Zhu and Han [4], respectively. Robust observers for Lipschitz nonlinear systems subject to disturbances were proposed in [5, 6]. Nonlinear observer for neutral uncertain time-delay systems was addressed in [7]. Very recently, the classical Lipschitz nonlinear observer design has been extended to the one-sided Lipschitz case; see e.g. [8–14].
It should be noted that most of the above-mentioned works are concerned on continuous-time nonlinear systems. Generally, the state estimation problem for discrete-time nonlinear systems has received little attention. Moreover, in the existing literature there have been several useful observer design approaches for some specialized classes of discrete-time nonlinear systems [15–23]. For example, Ibrir [15] proposed the circle-criterion approach to discrete-time nonlinear observer design. In [16] and [17], the authors considered the observer design for discrete-time Lipschitz nonlinear systems. Motivated by the Arcak-type observer design [1, 2], Zemouche and Boutayeb [18] provided a unified observer design method for discrete-time Lipschitz systems and extended it to \(H_{\infty}\) synchronization and unknown input recovery. An LMI approach was proposed by Wang et al. [19] to design state observer for discrete-time Lipschitz descriptor systems. In [20] the authors considered an observer design for discrete-time epidemic models. A new reduced-order observer normal form for nonlinear discrete-time systems was provided in [21].
Very recently, several authors have considered the observer design for one-sided Lipschitz nonlinear systems in the discrete-time case. Both full-order and reduced-order observer designs were studied in Benallouch et al. [22]. In fact, they have developed an LMI-based design approach to deal with the state estimation problem of one-sided Lipschitz discrete-time systems. Zhang et al. [23] investigated the same problem and proposed a simple observer synthesis condition to ensure the asymptotic convergence. It should be emphasized that the systems considered by Benallouch et al. [22] and Zhang et al. [23] are actually a subset of one-sided Lipschitz nonlinear systems (see Figure 1 below). More precisely, the systems are assumed to simultaneously satisfy the one-sided Lipschitz condition and the quadratically inner-bounded condition. This assumption may lead to more conservative results and bring additional restrictions on the system model. How to reduce the conservatism in the existing results of observer design of nonlinear systems is still an open problem. This motivates our present research.
In this paper, we focus on state observer design for a general class of nonlinear discrete-time systems that satisfies the quadratically inner-bounded condition only. The main contributions of this paper are two folds. First, we remove the one-sided Lipschitz restriction and only need the assumption of quadratically inner-bounded condition. Note that the quadratically inner-bounded condition includes the classical Lipschitz condition as a special case; see e.g. Figure 1 below. Therefore, we extend the state observer design to a larger class of discrete-time nonlinear systems. Second, some simple stability conditions are obtained for both full-order and reduced-order observer designs. In our approach, the observer designs are formulated as an LMI feasible problem, which is easily solved by standard convex optimization algorithms. An example on the single-link flexible joint robot is given to illustrate the effectiveness of the proposed design.
Notations: \(\mathbb {R}^{n}\) denotes the n-dimensional real Euclidean space. \(\langle\cdot, \cdot\rangle\) represents the inner product in \(\mathbb {R}^{n}\), i.e., for given \(x,y \in \mathbb {R}^{n}\), then \(\langle x,y\rangle= x^{T}y\), where \(x^{T}\) is the transpose of the column vector \(x \in \mathbb {R}^{n}\). \(\Vert { \cdot} \Vert \) denotes the Euclidean norm on \(\mathbb {R}^{n}\). For a symmetric matrix P, \(P > 0\) (\(P < 0\)) means that the matrix is positive definite (negative definite). In symmetric block matrices, we use an asterisk ∗ to represent a term induced by symmetry. I represents an identity matrix with appropriate dimension.
2 Problem statement and preliminaries
Assumption 1
For the purpose of comparison, we introduce the following two assumptions, which are commonly used in the recent literature for observer design of nonlinear systems. For further details, we refer the interested reader to [8, 9, 22].
Assumption 2
(see e.g. [9])
Assumption 3
Notice that Assumption 2 is the well-known Lipschitz condition, while Assumption 3 is the so-called one-sided Lipschitz condition. It is worth mentioning that the one-sided Lipschitz condition has been frequently employed in the study of synchronization of complex networks [25, 26]. Moreover, as shown in [8] and [9], the one-sided Lipschitz condition implies the Lipschitz condition but the converse is not true. Figure 1 shows the relation between the Lipschitz, one-sided Lipschitz, and quadratically inner-bounded function sets [24].
We end this section by introducing a useful lemma.
Lemma 1
(The Schur complement lemma; see e.g. [27])
- (1)
\(\Sigma:=\bigl [ {\scriptsize\begin{matrix}{} \Sigma_{11} & \Sigma_{12} \cr \Sigma_{12}^{T} & \Sigma_{22} \end{matrix}} \bigr ]<0\).
- (2)
\(\Sigma_{11} < 0\), and \(\Sigma_{22} - \Sigma_{12}^{T}\Sigma_{11}^{ - 1}\Sigma_{12} < 0\).
- (3)
\(\Sigma_{22} < 0\), and \(\Sigma_{11} - \Sigma_{12} \Sigma_{22} ^{-1}\Sigma_{12}^{T} < 0\).
3 Full-order observer design
Now, we have the following conclusion.
Theorem 1
Proof
Since the quadratically inner-bounded condition include the Lipschitz condition as a special case, we immediately have Corollary 1.
Corollary 1
4 Reduced-order observer design
Now, we have the following theorem.
Theorem 2
Proof
Therefore, if the matrix inequality (21) has a feasible solution, we have \(\Delta V_{k} < 0\) for all \(\varepsilon(k) \ne0\). By the standard Lyapunov theorem, we know that the estimation error system is asymptotically stable, which means (18) is an asymptotic reduced-order observer for system (1). This completes the proof. □
Remark 1
Compared with the full-order or the reduced-order observer design in [22], the paper removes the one-sided Lipschitz restriction, which significantly reduces the conservatism and complexity of the designs. In fact, in Theorems 1 and 2, we only assume that f satisfies the quadratically inner-bounded condition (2) and do not employ the one-sided Lipschitz condition (3).
Remark 2
Similarly, we have Corollary 2, since the quadratically inner-bounded condition includes the Lipschitz condition as a special case.
Corollary 2
5 Illustrative example
6 Conclusion
We have addressed the state estimation problem for a general class of nonlinear discrete-time systems that satisfies the quadratically inner-bounded condition. The system under consideration need not satisfy the one-sided Lipschitz restriction, which is a common assumption in some recent literature on observer design for nonlinear discrete-time systems. We considered both the full-order and the reduced-order observer designs and formulated the observer synthesis condition as an LMI formulation. Finally, we used an example on the single-link flexible joint robotic system to illustrate the effectiveness of the proposed design.
Declarations
Acknowledgements
This work was supported in part by the National Natural Science Foundation of China under Grant 51505273, the State Key Laboratory of Robotics and System (HIT) under Grant SKLRS-2014-MS-10, the Jiangsu Provincial Key Laboratory of Advanced Robotics Fund Projects under Grant JAR201401, and the Foundation of Shanghai University of Engineering Science under Grant nhky-2015-06.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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