Nonlinear Observer Design for One-Sided Lipschitz Generalized System

State estimation of systems satisfying some special nonlinearities have been important topics in nonlinear system theory. In this paper, we discuss the problem ofobserver design for one-sided Lipschitz nonlinear generalized systems basedon generalized quadratic stability and Lyapunov function, by using the linear matrix inequality (LMI) andlinear matrix equality (LME) approach, we propose some sufficient conditions for thegeneralized quadratic stability of one-sided Lipschitz generalized systems, which ensure that the observererror dynamics is generalized quadratically stable. Simulation results on one example are given to illustratethe effectiveness and advantages of the proposed design. Introduction During the last decades, enormous efforts are put into the study of nonlinear system[1-2], and the observer design problem for nonlinear systems satisfying a Lipschitz conditionhas motivated plenty of interests in nonlinear system theory. In [3], Thau presented asufficient condition which ensures the asymptotical stability of observer error dynamics.Raghavan and Hedrick in [4] derived a constructive design by iteratively solving a series of Riccati equations. What’s more, a lot of researchers studied observer design forLipschitz systems with different methods. It is interesting that the Lipschitz constants of such functions are often region based,with the operating region enlarged, the Lipschitz constants usually increase dramatically.However, most of the existing approaches can only deal with the situation when Lipschitzconstants are small, in other words, these techniques can not find solutions with largeLipschitz constants. For the purpose of overcoming this drawback, Lipschitz continuityis generalised to a less restrictive condition known as one-sided Lipschitz continuity,which has been extensively applied to the stability analysis and numerical analysis ofordinary differential equations. Compared with the Lipschitz constant, the one-sidedLipschitz constant is much more suited for estimating the influence of nonlinear partbecause it possesses inherent advantages with respect to conservativeness. Inspired bythe advances of the one-sided Lipschitz constant, various methods are considered andapplied to design observers of nonlinear systems, more results on this problem can befound in [5-6]. In this paper, we extend the nonlinear observer design method on generalized systems. We present some sufficient conditions which ensure thatthe observer error dynamics isgeneralized quadratic stability stable. A simulative example is included to illustrate the effectiveness and advantages of the proposed methods.The remainder of the paper is organized as follows: Section 2 states the problem we deal with and introduce some basicdefinitions. Section 3, whichcontains the main results, presents LMI/LME-based observer design approachesfor one-sided Lipschitz generalized systems. Then section 4 provides an illustrative example. In the end section 5 gives the conclusion of this paper. Problem statement In this section, we consider the following nonlinear generalised system , , (1) 2nd International Conference on Machinery, Materials Engineering, Chemical Engineering and Biotechnology (MMECEB 2015) © 2016. The authors Published by Atlantis Press 790 where ∈ is a singular matrix known as the state matrix, ∈ is the state, ∈ is the control input, ∈ is the measured output, ∈ , ∈ are const matrices. The nonlinear function , is continuous with respect to both and . Definition1[7].The nonlinear function , issaid to be ’one-sided Lipschitz’ in a region if there exists ∈ such that∀ , ∈ , , ∗ , ∗ , ‖ ‖ . Definition2[7].The nonlinear function , issaid to be ’quadratically inner-bounded’ in a region if there exists , ∈ such that∀ , ∈ , ‖ , , ‖ ‖ ‖ , , , . Definition3[8].System (1) issaid to be ’ generalized quadratically stable’if there exists a matrix such that 0, , , 0. For system (1), we consider a full-order observer of the following form , , (2) where is a gain matrix for the observer. Denote , , , , ,Thenthe observer error dynamics is given by , (3) assuming that is one-sided Lipschitz and quadratically inner-bounded. Let ,with definition2 and definition 3, we obtain , (4) . (5) Our design goal is to find an observer gain matrix for system (1), with satisfying conditon (4) and (5), such that the observer error dynamics (3) is generalized quadraticallystable. Main Results Theorem1. Assume that in system (1) satisfies the conditions (4) and (5), the state observer for system (1) holds the form of (2), then the observer error dynamics (3) is generalized quadraticallystable, if there existsan invertible matrix , a matrix and 2 scalars 0, 0,such that the following LMI and LME are feasible: 0, (6) 0. (7) If (6) and (7) are feasible, then the gain matrixis given by . Proof. Suppose there existsan invertible matrix satisfying 0, select Lyapunov function ,then 0 .(8) For any positive scalars , , with condition (4) and (5), we obtain 0 0, (7) Thus, adding the left terms of (7) to (6), we obtain . (8)