Reduced-order observer design for a class of generalized Lipschitz nonlinear systems with time-varying delay☆
Introduction
Observer design problem for nonlinear systems has been an important research topic in control theory [1], [2], [6], [7]. It is well known that when not all the internal states of the dynamic system are available, a state observer is usually required for implementation of the control design. Observer design is also significantly employed in many engineering applications, such as process monitoring, fault detection and isolation, neural network [1], and chaos synchronization-based secure communication [2]. The nonlinear observer design is relatively difficult compared with the observer design theory for linear systems. Generally speaking, there is no general approach for the nonlinear observer design problem although most physical and engineering systems are inherently nonlinear.
Most of the existing nonlinear observer design methods focus on certain special class of nonlinearities. The Lipschitz system is such a popular class of nonlinear systems investigated by the researchers in the past decades [7], [8], [19]. Many of the physical system models satisfy the Lipschitz condition globally or at least locally. The drawback of existing observer methods for Lipschitz system is that they fail to deal with large values of Lipschitz constant. Hu [9] firstly introduced the one-sided Lipschitz condition of nonlinear systems to overcome this limitation. Recently, Abbaszadeh and Marquez [11] extended Hu’s one-sided Lipschitz condition and meanwhile introduced another new concept named quadratically inner-bounded condition to make the design tractable. It is known that both the one-sided Lipschitz systems [10], [12], [13], [14] and quadratically inner-bounded systems encompass the traditional Lipschitz systems and then these kinds of systems exhibit inherent advantages with respect to conservativeness. Thereafter, observer design for nonlinear systems satisfying one-sided Lipschitz condition and quadratically inner-bounded condition simultaneously are commonly investigated in [15], [17], [18], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29]. Nonlinear H∞ observer for one-sided Lipschitz nonlinear systems was proposed in [15]. The observer design for discrete-time one-sided Lipschitz nonlinear systems was investigated in [18], [24]. Observer design for one-sided Lipschitz descriptor systems was presented in [23]. The observer-based control problem for one-sided Lipschitz systems was studied in [21]and [22]. Robust observer design for uncertain one-sided Lipschitz systems with disturbances was researched in [28].
Indeed, the nonlinear systems considered in the aforementioned reference are only a subset of quadratically inner-bounded nonlinear systems, which are also one-sided Lipschitz. In [16], the authors presented the improved exponential observer design for nonlinear systems satisfying one-sided Lipschitz condition only. A natural question will be similarly put forward that can we consider the quadratically inner-bounded condition only? This will make it possible to reduce the conservatism and complexity of the design since checking these two conditions is not so easy for many nonlinear functions. In addition, time delay is also a challenging problem which may cause unsatisfactory performance and instability of dynamical systems [4]. Nonlinear observer design for time-delay nonlinear system is more complicated since extra effort should be made to deal with the time-delay issue. Nonlinear reduced-order observers were proposed to one-sided Lipschitz discrete-time and continuous-time systems with time-varying delay quite recently in [25], [26], [27]. LMI conditions for the observer synthesis problems of one-sided Lipschitz nonlinear systems with interval time-varying delay and uncertainties were presented in [20]. In a word, the problem of designing observers for time-delay nonlinear systems is not fully investigated. This motivates the present work of this paper. In this paper, we stress on reduced-order observer design for a general class of delayed nonlinear systems that satisfies the quadratically inner-bounded condition only. The main contributions of the paper are summarized below:
(1) A novel H∞ reduced-order observer design methodology for a class of nonlinear systems with time-varying delay is proposed.
(2) Unlike some existing works dealing with nonlinear systems satisfying one-sided Lipschitz condition and the quadratically inner-bounded condition simultaneously, the present work removes the constraint of one-sided Lipschitz condition and considers quadratically inner-bounded condition only, which will significantly reduce the conservatism and extend the observer design to a larger class of nonlinear systems.
(3) Some new and less conservative synthesis conditions of observer design which can be tackled by Matlab tool box directly are presented for nonlinear systems with interval time-varying delay by employing an extended reciprocal convexity inequality quite recently proposed by Zhang et al. [3].
This paper is organized as follows. Section 2 gives the problem formulation and preliminaries. Section 3 presents the main results. Simulation results showing the efficiency and high performance of proposed observer design approach are provided in Section 4. Section 5 draws conclusions of the study.
Notation: The notation throughout this paper is standard. AT denotes the transpose of matrix A. I and 0, respectively, are the identity matrix and zero matrix with appropriate dimensions. The symbol * represents symmetric terms in a symmetric matrix and for any we define is the Euclidean norm of vector x. is the inner product of vectors x, y. is the the norm of vector x, where is the set defined by .
Section snippets
Problem formulation and preliminaries
We consider the following nonlinear system with time-varying delay where is the state vector, is the control input, is the measurement output, is the disturbance input ; are known real constant matrices with C being of full row rank; ϕ(t) is the initial condition; The delay d(t) is a known
Main results
In this section, we will present our main results.
We will study the stability problem of the error dynamics (15) by the Lyapunov–Krasovskii theory. An important inequality in Lemma 2 which is recently proved to be tighter than reciprocally convexity inequality and the free-weighting matrix technique will be employed to derive less conservative conditions. The nonlinear delay-range-dependent synthesis conditions for the H∞ reduced-order observer design and determination of the observer gain N
Numerical examples
To demonstrate the effectiveness of our results, two numerical examples are given in this section.
Example 1 Consider the following nonlinear system (1) borrowed from the well-known single-link flexible joint robot system in the literature [27].
It has already been
Conclusion
The H∞ reduced-order observer design problem for a class of generalized nonlinear systems subject to interval time-varying delay and disturbances is investigated in this paper. The considered nonlinear system satisfies quadratically inner-bounded condition only and encompasses the classical Lipschitz system. Employing the quadratically inner-bounded condition, extended reciprocal convexity inequality, together with free-weighting matrix technique, several new and less conservative synthesis
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This work is supported in part by the National Natural Science Foundation of China (61673227, 61473160, 61573204) and in part by the Natural Science Foundation of Shandong Province, China (ZR2016FM06). Qing-Guo Wang acknowledges the financial support of the National Research Foundation of South Africa (Grant number: 113340) and Oppenheimer Memorial Trust grant, South Africa, which partially funded his research on this work.