Design of high-gain observers based on sampled measurements via the interval arithmetic

Abstract The main objective of this paper is to implement a high-gain “practical” observer with the interval arithmetic in the case of sampled measurements. In order to pursue this objective, two closed-form expressions are proposed to compute certified bounds on the solution of a single-input linear system, with distinct, real, and negative eigenvalues, when both the initial condition and the control input lie in given intervals. Furthermore, it is shown how such formulas can be extended in the case of sampled input measurements. These results are used to design an interval high-gain observer for non-linear systems in observer normal form. In particular, it is shown how to implement a high-gain “practical” observer with the interval arithmetic, both in continuous-time and in discrete-time, so to compute guaranteed interval estimates of the state of the system.

[1]  Nedialko S. Nedialkov,et al.  Validated solutions of initial value problems for ordinary differential equations , 1999, Appl. Math. Comput..

[2]  M. H. van Emden,et al.  Interval arithmetic: From principles to implementation , 2001, JACM.

[3]  Wilfrid Perruquetti,et al.  On interval observer design for time-invariant discrete-time systems , 2013, 2013 European Control Conference (ECC).

[4]  L. Shampine,et al.  Computer solution of ordinary differential equations : the initial value problem , 1975 .

[5]  Denis V. Efimov,et al.  Design of interval observers for uncertain dynamical systems , 2016, Automation and Remote Control.

[6]  Carl D. Meyer,et al.  Matrix Analysis and Applied Linear Algebra , 2000 .

[7]  Benoît Chachuat,et al.  Unified framework for the propagation of continuous-time enclosures for parametric nonlinear ODEs , 2015, J. Glob. Optim..

[8]  Olivier Bernard,et al.  Interval observers for linear time-invariant systems with disturbances , 2011, Autom..

[9]  Wolfgang Kuehn,et al.  Rigorously computed orbits of dynamical systems without the wrapping effect , 1998, Computing.

[10]  René Thomas,et al.  DETERMINISTIC CHAOS SEEN IN TERMS OF FEEDBACK CIRCUITS: ANALYSIS, SYNTHESIS, "LABYRINTH CHAOS" , 1999 .

[11]  J. Gauthier,et al.  Deterministic Observation Theory and Applications , 2001 .

[12]  Olivier Bernard,et al.  Interval observers for linear systems with delay , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[13]  Laura Menini,et al.  A “practical” observer for nonlinear systems , 2017, 2017 IEEE 56th Annual Conference on Decision and Control (CDC).

[14]  Denis V. Efimov,et al.  Interval estimation for continuous-time switched linear systems , 2018, Autom..

[15]  Olivier Bernard,et al.  Near optimal interval observers bundle for uncertain bioreactors , 2007, 2007 European Control Conference (ECC).

[16]  E. Adams,et al.  The Reliability Question for Discretizations of Evolution Problems , 1993 .

[17]  Olivier Bernard,et al.  Robust interval observers for global Lipschitz uncertain chaotic systems , 2010, Syst. Control. Lett..

[18]  Denis V. Efimov,et al.  Interval State Estimation for a Class of Nonlinear Systems , 2012, IEEE Transactions on Automatic Control.

[19]  Martin Berz,et al.  Performance of Taylor Model Methods for Validated Integration of ODEs , 2004, PARA.

[20]  Lawrence F. Shampine,et al.  Solving ODEs with MATLAB , 2002 .

[21]  J. Grizzle,et al.  Observer design for nonlinear systems with discrete-time measurements , 1995, IEEE Trans. Autom. Control..

[22]  Laura Menini,et al.  A dynamical interval Newton method , 2021, Eur. J. Control.

[23]  Daniele Astolfi,et al.  Design of local observers for autonomous nonlinear systems not in observability canonical form , 2019, Autom..

[24]  Hans J. Stetter,et al.  Validated Solution of Initial Value Problems for ODE , 1990, Computer Arithmetic and Self-Validating Numerical Methods.

[25]  P. Eijgenraam.  The Solution of Initial Value Problems Using Interval Arithmetic , 1981 .

[26]  Nedialko S. Nedialkov,et al.  On Taylor Model Based Integration of ODEs , 2007, SIAM J. Numer. Anal..

[27]  Hend Dawood,et al.  Theories of Interval Arithmetic: Mathematical Foundations and Applications , 2011 .

[28]  R. Baker Kearfott,et al.  Introduction to Interval Analysis , 2009 .

[29]  Olivier Bernard,et al.  Asymptotically Stable Interval Observers for Planar Systems With Complex Poles , 2010, IEEE Transactions on Automatic Control.

[30]  Carlos Vázquez,et al.  Interval differentiators: On-line estimation of differentiation accuracy , 2016, 2016 European Control Conference (ECC).

[31]  C. Desoer,et al.  Feedback Systems: Input-Output Properties , 1975 .

[32]  Alfredo Germani,et al.  A New Approach to Design Interval Observers for Linear Systems , 2015, IEEE Transactions on Automatic Control.

[33]  J. Gouzé,et al.  Interval observers for uncertain biological systems , 2000 .

[34]  Laurent Bako,et al.  Interval-valued state estimation for linear systems: The tightest estimator and its relaxations , 2019, Autom..

[35]  Yi Shen,et al.  Interval Estimation Methods for Discrete-Time Linear Time-Invariant Systems , 2019, IEEE Transactions on Automatic Control.

[36]  A. Tornambè.  High-gain observers for non-linear systems , 1992 .

[37]  Karl Nickel.  How to Fight the Wrapping Effect , 1985, Interval Mathematics.