Exponential State Observers for Nonlinear Systems with Incremental Quadratic Constraints and Output Nonlinearities
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Exponential State Observers for Nonlinear Systems with Incremental Quadratic Constraints and Output Nonlinearities Younan Zhao1,2 · Wei Zhang1,2
· Wei Guo1,2 · Su Yu1 · Fang Song1
Received: 4 September 2017 / Revised: 24 December 2017 / Accepted: 23 January 2018 © Brazilian Society for Automatics–SBA 2018
Abstract This paper considers the exponential observer design for a class of nonlinear systems with output nonlinearities. The nonlinear terms in the systems are assumed to satisfy incremental quadratic constraints which include many commonly encountered nonlinearities in existing literature as some special cases. We construct a circle-criterion-based observer by injecting both the linear and the nonlinear output error terms into the observer system. Sufficient conditions ensuring the exponential stability of the proposed observer are established and formulated in terms of linear matrix inequalities. Finally, the advantages and effectiveness of the proposed design approach are illustrated through two examples. Keywords Observer design · Nonlinear systems · Exponential observer · Incremental quadratic constraints (δ QC) · Linear matrix inequalities (LMIs )
1 Introduction It is known that not all the state variables of a dynamic system are available in engineering practice, while most of the unknown ones are very important in many engineering applications (Diab 2014; Phanomchoeng et al. 2011; Raghavan and Hedrick 1994; Bernard et al. 2017; Khalil 2002; Astolfi and Marconi 2015; Dong et al. 2014). Consequently, a fundamental problem in system analysis and control design is that of determining the state of a system from its measured inputs and outputs (Khalil 2002). The state estimation problem for linear systems has been well studied. However, till now, the observer design for general nonlinear systems is still a challenging problem. Because in practice, there are so many kinds of nonlinearities, and it is quite difficult to handle all kinds of nonlinearities in a single unified framework. Therefore, specialized state observers are designed for different kinds of nonlinearities in nonlinear systems (Khalil
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Wei Zhang [email protected]
1
Laboratory of Intelligent Control and Robotics, Shanghai University of Engineering Science, Shanghai 201620, China
2
MOE Key Laboratory of Image Processing and Intelligence Control, Huazhong University of Science and Technology, Wuhan 430074, China
2002; Astolfi and Marconi 2015; Dong et al. 2014; Zhang et al. 2016). Thau (1973) initiated observers for traditional Lipschitz nonlinear systems, which has been further developed by Rajamani (1998) for full-order observers and Zhu and Han (2002) for reduced-order observers. Howell and Hedrick (2002) studied nonlinear observer design via the convex optimization approach, which gives how the observer gain matrices can be optimally chosen. Abbaszadeh and Marquez (2008) developed asymptotic observer synthesis methods for systems with globally Lipschitz nonlinearities. Sliding-mode observers are investigated by Edwards and other
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