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Observers for a class of Lipschitz systems with extension to $$H_{\infty }$$ performance analysis. (English) Zbl 1129.93006
Summary: In this paper, observer design for a class of Lipschitz nonlinear dynamical systems is investigated. One of the main contributions lies in the use of the differential mean value theorem (DMVT) which allows transforming the nonlinear error dynamics into a linear parameter varying (LPV) system. This has the advantage of introducing a general Lipschitz-like condition on the Jacobian matrix for differentiable systems. To ensure asymptotic convergence, in both continuous and discrete time systems, such sufficient conditions expressed in terms of linear matrix inequalities (LMIs) are established. An extension to $$H_{\infty }$$ filtering design is obtained also for systems with nonlinear outputs. A comparison with respect to the observer method of J. P. Gauthier, H. Hammouri and S. Othman [IEEE Trans. Autom. Control 37, No. 6, 875–880 (1992; Zbl 0775.93020)] is presented to show that the proposed approach avoids high gain for a class of triangular globally Lipschitz systems. In the last section, academic examples are given to show the performances and some limits of the proposed approach. The last example is introduced with the goal to illustrate good performances on robustness to measurement errors by avoiding high gain.

##### MSC:
 93B07 Observability 93C10 Nonlinear systems in control theory 93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, $$L^p, l^p$$, etc.) in control theory
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