A state observer for nonlinear delay systems

The problem of state reconstruction from input and output measurements for nonlinear time delay systems is studied in this paper and a state observer is proposed that is easy to implement and, under suitable assumptions on the system and on the input function, gives exponential observation error decay. The proposed observer is itself a delay system and can be classified as an identity observer, in that it is such that if at a given time instant the system and observer states coincide, on a suitable Hilbert space, the observation error remains zero in all following time instants. The computation of the feedback gain is very straightforward. A sketch of convergence proof is presented in the paper and simulations are reported that show the high effectiveness of the proposed observation algorithm.

[1]  A. Kumar,et al.  Delayless observers for systems with delay , 1986 .

[2]  Alfredo Germani,et al.  Approximation of the algebraic Riccati equation in the Hilbert space of Hilbert-Schmidt operators , 1993 .

[3]  Alfredo Germani,et al.  Numerical solution for optimal regulation of stochastic hereditary systems with multiple discrete delays , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[4]  D. Salamon.  Observers and duality between observation and state feedback for time delay systems , 1980 .

[5]  D. Salamon.  On controllability and observability of time delay systems , 1984 .

[6]  J. Gibson.  Linear-Quadratic Optimal Control of Hereditary Differential Systems: Infinite Dimensional Riccati Equations and Numerical Approximations , 1983 .

[7]  Erik I. Verriest,et al.  Vibrational control of nonlinear time lag systems with bounded delay: averaging theory, stabilizability, and transient behavior , 1994, IEEE Trans. Autom. Control..

[8]  Alfredo Germani,et al.  Local asymptotic stability for nonlinear state feedback delay systems , 2000, Kybernetika.

[9]  A. Olbrot.  Observability and observers for a class of linear systems with delays , 1981 .

[10]  A. Pearson,et al.  An observer for time lag systems , 1989 .

[11]  H. Banks,et al.  Spline approximations for functional differential equations , 1979 .

[12]  Alfredo Germani,et al.  A robust observer for discrete time nonlinear systems , 1995 .

[13]  Alfredo Germani,et al.  Observers for discrete-time nonlinear systems , 1993 .

[14]  Keiji Watanabe.  Finite spectrum assignment and observer for multivariable systems with commensurate delays , 1986 .

[15]  A. Germani,et al.  A Luenberger-like observer for nonlinear systems , 1993 .

[16]  A. Isidori.  Nonlinear Control Systems , 1985 .

[17]  G. Ciccarella,et al.  Asymptotic linearization and stabilization for a class of nonlinear systems , 1995 .

[18]  Alfredo Germani,et al.  The LQG control problem for a class of hereditary systems: a method for computing its approximate solution , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[19]  Jean-Pierre Richard,et al.  New results on constrained stabilizing control of time-delay systems , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[20]  Alfredo Germani,et al.  Linearization and decoupling of nonlinear delay systems , 1998, Proceedings of the 1998 American Control Conference. ACC (IEEE Cat. No.98CH36207).

[21]  I. G. Rosen.  Difference Equation State Approximations for Nonlinear Hereditary Control Problems , 1984 .