Observers for Lipschitz non-linear systems

In an interesting paper R. Rajamani and Y. M. Cho have proposed a systematic methodology to design observers. They have introduced a new problem: relation between distance to unobservability and convergence of 'Luenberger-like' observers. A result for the convergence of the observer has also been given. They have presented a quantity denoted by i, claimed as the distance to unobservability. We show that this number is not actually the distance from the system to the set of unobservable systems. Moreover, the result for the observer is incorrect. We provide a counterexample for the result of convergence. In this paper results for convergence are obtained, with additional strengthened hypothesis, to correct Rajamani and Cho's result. The results are used to design an observer for a, now classical, single-link flexible robot joint. The behaviour of the observer to noisy output is quite satisfactory.

[1]  J. Willems.  Least squares stationary optimal control and the algebraic Riccati equation , 1971 .

[2]  V. Kučera.  A contribution to matrix quadratic equations , 1972 .

[3]  F. Thau.  Observing the state of non-linear dynamic systems† , 1973 .

[4]  Tzyh Jong Tarn,et al.  Exponential Observers for Nonlinear Dynamic Systems , 1975, Inf. Control..

[5]  S. Liberty,et al.  Linear Systems , 2010, Scientific Parallel Computing.

[6]  Mark A. Shayman,et al.  Geometry of the Algebraic Riccati Equation, Part II , 1983 .

[7]  Gene H. Golub,et al.  Matrix computations , 1983 .

[8]  D. Bestle,et al.  Canonical form observer design for non-linear time-variable systems , 1983 .

[9]  F Rikus Eising,et al.  Between controllable and uncontrollable , 1984 .

[10]  J. Gauthier,et al.  Observability and observers for non-linear systems , 1986, 1986 25th IEEE Conference on Decision and Control.

[11]  M. Spong.  Modeling and Control of Elastic Joint Robots , 1987 .

[12]  S. Żak,et al.  State observation of nonlinear uncertain dynamical systems , 1987 .

[13]  S. Żak,et al.  Comparative study of non-linear state-observation techniques , 1987 .

[14]  P. Khargonekar,et al.  State-space solutions to standard H2 and H∞ control problems , 1988, 1988 American Control Conference.

[15]  R. Byers.  A Bisection Method for Measuring the Distance of a Stable Matrix to the Unstable Matrices , 1988 .

[16]  P. Khargonekar,et al.  State-space solutions to standard H/sub 2/ and H/sub infinity / control problems , 1989 .

[17]  S. H. Zak,et al.  On the stabilization and observation of nonlinear/uncertain dynamic systems , 1990, IEEE 1989 International Conference on Systems Engineering.

[18]  Stephen P. Boyd,et al.  A bisection method for computing the H∞ norm of a transfer matrix and related problems , 1989, Math. Control. Signals Syst..

[19]  S. Bittanti,et al.  The Riccati equation , 1991 .

[20]  B. Anderson,et al.  A first prin-ciples solution to the nonsingular H control problem , 1991 .

[21]  J. Gauthier,et al.  A simple observer for nonlinear systems applications to bioreactors , 1992 .

[22]  J. Hedrick,et al.  Observer design for a class of nonlinear systems , 1994 .

[23]  David J. N. Limebeer,et al.  Linear Robust Control , 1994 .

[24]  Leiba Rodman,et al.  Algebraic Riccati equations , 1995 .

[25]  H. Hammouri,et al.  On uniform observation of nonuniformly observable systems , 1996 .

[26]  Vladimir L. Kharitonov,et al.  On stability of a weighted diamond of real quasi-polynomials , 1997, IEEE Trans. Autom. Control..

[27]  R. Rajamani.  Observers for Lipschitz nonlinear systems , 1998, IEEE Trans. Autom. Control..

[28]  R. Rajamani,et al.  Existence and design of observers for nonlinear systems: Relation to distance to unobservability , 1998 .

[29]  Olivier Bernard,et al.  Nonlinear observers for a class of biological systems: application to validation of a phytoplanktonic growth model , 1998, IEEE Trans. Autom. Control..

[30]  H. Trentelman.  When does the algebraic Riccati equation have a negative semi-definite solution? , 1999 .

[31]  H. Shim,et al.  A Simple Observer for Nonlinear Systems in x-coordinate using Lipschitz Extension Technique , 2000 .