Modified Euler-Frobenius Polynomials With Application to Sampled Data Modelling

The broad class of polynomials generally known as Eulerian or Euler-Frobenius polynomials has a rich history in Mathematics and Engineering. They have been generalised in several directions and find application in diverse areas ranging from sampled-data modelling and polynomial interpolation to wavelets. Here we describe a modified class of Euler-Frobenius polynomials which are functions of two independent variables. We show that the sampling zero (dynamics) of general linear and nonlinear systems can be described using this new class of polynomials when considering several non-standard sampled-data modelling problems, including when generalised hold functions are used to generate the input and when the continuous system has a time delay.

[1]  M. Ishitobi.  Stability of zeros of sampled system with fractional order hold , 1996 .

[2]  M. Araki,et al.  Stability of the limiting zeros of sampled-data systems with zero-and first-order holds , 1993 .

[3]  D. D. Perlmutter,et al.  Stability of time‐delay systems , 1972 .

[4]  Cishen Zhang,et al.  Robustness analysis of control systems using generalized sample hold functions , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[5]  S. Monaco,et al.  Zero dynamics of sampled nonlinear systems , 1988 .

[6]  J. Lambert.  Numerical Methods for Ordinary Differential Equations , 1991 .

[7]  CHENG ZENG.  Sampled-Data Models for Nonlinear Systems via the Taylor-Lie Series and Multirate Input , 2014 .

[8]  Nadine Gottschalk,et al.  Computer Controlled Systems Theory And Design , 2016 .

[9]  Er-Wei Bai,et al.  A note on generalized hold functions , 1990 .

[10]  Jean-Pierre Richard,et al.  Time-delay systems: an overview of some recent advances and open problems , 2003, Autom..

[11]  Graham C. Goodwin,et al.  Control System Design , 2000 .

[12]  B. Wahlberg.  Limit results for sampled systems , 1988 .

[13]  Graham C. Goodwin,et al.  Vector Measures of Accuracy for Sampled Data Models of Nonlinear Systems , 2013, IEEE Transactions on Automatic Control.

[14]  K. Åström,et al.  Zeros of sampled systems , 1980 .

[15]  P. Kloeden,et al.  Numerical Solution of Stochastic Differential Equations , 1992 .

[16]  D. Normand-Cyrot,et al.  Minimum-phase nonlinear discrete-time systems and feedback stabilization , 1987, 26th IEEE Conference on Decision and Control.

[17]  Graham C. Goodwin,et al.  Sampled Data Models for Nonlinear Stochastic Systems: Truncation Errors and Sampling Zero Dynamics , 2016, IEEE Transactions on Automatic Control.

[18]  S. L. Sobolev,et al.  On the roots of Euler polynomials , 2006 .

[19]  Vladimir L. Kharitonov,et al.  Robust stability analysis of time delay systems: A survey , 1999 .

[20]  Shinji Hara,et al.  Properties of zeros in digital control systems with computational time delay , 1989 .

[21]  Claude H. Moog,et al.  On the Existence of the Normal Form for Nonlinear Delay Systems , 2016 .

[22]  Thomas Kailath,et al.  Linear Systems , 1980 .

[23]  Graham C. Goodwin,et al.  On sampled-data models for nonlinear systems , 2005, IEEE Transactions on Automatic Control.

[24]  Dominique Foata,et al.  Eulerian Polynomials: From Euler’s Time to the Present , 2010 .

[25]  Graham C. Goodwin,et al.  Sampling Zeros of Discrete Models for Fractional Order Systems , 2013, IEEE Transactions on Automatic Control.

[26]  G. Goodwin,et al.  Generalized sample hold functions-frequency domain analysis of robustness, sensitivity, and intersample difficulties , 1994, IEEE Trans. Autom. Control..

[27]  S. Monaco,et al.  Multirate Sampling and Zero Dynamics: from linear to nonlinear , 1991 .

[28]  P. Kabamba.  Control of Linear Systems Using Generalized Sampled-Data Hold Functions , 1987, 1987 American Control Conference.

[29]  Dorothée Normand-Cyrot,et al.  Issues on Nonlinear Digital Control , 2001, Eur. J. Control.

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

[31]  Serge Lang.  Linear Algebra third edition undergraduate text in mathematics. , 1987 .

[32]  Marian J. Blachuta,et al.  On zeros of pulse transfer functions of systems with first-order hold , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[33]  Vladimir A. Bondarko,et al.  Limiting zeros of sampled systems with time delay , 2015, Autom. Remote. Control..

[34]  S. Liang,et al.  Asymptotic properties of zero dynamics for nonlinear discretized systems with time delay via Taylor method , 2015 .

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

[36]  M. Blachuta.  On fast sampling zeros of systems with fractional-order hold , 2001, Proceedings of the 2001 American Control Conference. (Cat. No.01CH37148).

[37]  S. Weller,et al.  Sampling zeros and the Euler-Frobenius polynomials , 1997, Proceedings of the 36th IEEE Conference on Decision and Control.

[38]  G. Goodwin,et al.  Generalised hold functions for fast sampling rates , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[39]  Marian J. Blachuta,et al.  On zeros of pulse transfer functions , 1999, IEEE Trans. Autom. Control..

[40]  Charles K. Chui,et al.  An Introduction to Wavelets , 1992 .

[41]  CHENG ZENG,et al.  Asymptotic Properties of Zero Dynamics of Multivariable Sampled-Data models with Time Delay , 2014 .

[42]  L. Schumaker.  Spline Functions: Basic Theory , 1981 .

[43]  Juan I. Yuz,et al.  Sampled-Data Models for Linear and Nonlinear Systems , 2013 .

[44]  Mitsuaki Ishitobi.  Stable zeros of a discrete system obtained by sampling a continuous-time plant with a time delay , 1994 .

[45]  D.G. Dudley,et al.  Dynamic system identification experiment design and data analysis , 1979, Proceedings of the IEEE.

[46]  W. Marsden.  I and J , 2012 .

[47]  Leonard Carlitz,et al.  Eulerian Numbers and Polynomials , 1959 .