Thomas Decomposition and Nonlinear Control Systems

This paper applies the Thomas decomposition technique to nonlinear control systems, in particular to the study of the dependence of the system behavior on parameters. Thomas' algorithm is a symbolic method which splits a given system of nonlinear partial differential equations into a finite family of so-called simple systems which are formally integrable and define a partition of the solution set of the original differential system. Different simple systems of a Thomas decomposition describe different structural behavior of the control system in general. The paper gives an introduction to the Thomas decomposition method and shows how notions such as invertibility, observability and flat outputs can be studied. A Maple implementation of Thomas' algorithm is used to illustrate the techniques on explicit examples

Yalta: A Matlab Toolbox For The H∞-stability Analysis Of Classical And Fractional Systems With Commensurate Delays

In this paper we describe YALTA, a Matlab toolbox dedicated to the H ∞-stability analysis of classical and fractional systems with commensurate delays given by their transfer function. Delay systems of both retarded and neutral type are considered. The asymptotic position of high modulus poles is given. For a fixed known delay, poles of small modulus of standard delay systems are approximated through a Padé-2 scheme. For a delay varying from zero to a prescribed positive value, stability windows as well as root loci are given. We deeply describe how we have circumvented the numerical issues of algorithms developed in Fioravanti et al. [2010a, 2012] as well as the limitations of this toolbox. Finally, several examples are given. © 2013 IFAC.839844IFAC - Technical Committee on Linear Control Systems,Technical Committee on Control Design,Technical Committee on Non-Linear Control Systems,Technical Committee on Robust Control,Technical Committee on Distributed Parameter SystemsBellman, R., Cooke, K., (1963) Differential-Difference Equations, , Academic PressBonnet, C., Partington, J.R., Analysis of fractional delay systems of retarded and neutral type (2002) Automatica, (38), pp. 1133-1138Bonnet, C., Fioravanti, A., Partington, J.R., Stability of neutral systems with multiple delays and poles asymptotic to the imaginary axis (2011) SIAM J. Control Opt., 49 (2), pp. 498-516Engelborghs, K., Luzyanina, T., Samaey, G., DDE-BIFTOOL v. 2.00: A matlab package for bifurcation analysis of delay differential equations (2001) Technical Report TW-330, Department of Computer Science, , K.U. Leuven, Leuven, BelgiumFioravanti, A.R., Bonnet, C., Özbay, H., Niculescu, S.-I., A numerical method to find stability windows and unstable poles for linear neutral time-delay systems (2010) 9th IFAC Workshop on Time Delay Systems, , Prague Czech Republic, June 7-9Fioravanti, A.R., Bonnet, C., Özbay, H., Stability of fractional neutral systems with multiple delays and poles asymptotic to the imaginary axis (2010) 49th IEEE Conference on Decision and Control, , Atlanta, USA, December 15-17Fioravanti, A.R., Bonnet, C., Ozbay, H., Niculescu, S.I., A numerical method for stability windows and unstable root-locus calculation for linear fractional time-delay systems (2012) Automatica, 48 (11), pp. 2824-2830Gu, K., Kharitonov, V.L., Chen, J., (2003) Stability of Time-delay Systems, , BirkhauserGumussoy, S., Michiels, W., Root-locus for SISO dead-time systems: A continuation based approach (2012) Automatica, 43 (3), pp. 480-489Hilfer, R., (2000) Applications of Fractional Calculus in Physics, , World ScientificHwang, C., Cheng, Y.C., A note on the use of the lambert W function in the stability analysis of time-delay systems (2005) Automatica, 41 (11), pp. 1979-1985Hwang, C., Cheng, Y.C., A numerical algorithm for stability testing of fractional delay systems (2006) Automatica, 42 (5), pp. 825-831Lee, E.B., Gu, G., Khargonekar, P.P., Misra, P., Finite-dimensional approximation of infinite dimensional systems (1990) Conference on Decision and ControlLee, E.B., Gu, G., Khargonekar, P.P., Misra, P., Approximation of infinite dimensional systems (1989) IEEE Trans. Automatic Control, 34, pp. 610-618Mäkilä, M., Partington, J.R., Shift operator induced approximations of delay systems (1999) SIAM J. Control Optimiz., 37 (6), pp. 1897-1912Maset, S., Vermiglio, R., Pseudospectral differencing methods for characteristic roots of delay differential equations SIAM J. Sci. Comput., 27 (2), pp. 482-495Matignon, D., Stability properties for generalized fractional differential systems ESAIM: Proceedings, 5, pp. 145-158Nguyen, L.H.V., Fioravanti, A.R., Bonnet, C., Stability of neutral systems with commensurate delays and many chains of poles asymptotic to same points on the imaginary axis (2012) 10th IFAC Workshop on Time Delay Systems, , TDS 12, Boston, JuneNiculescu, S.-I., (2001) Delay Effects on Stability. A Robust Control Approach, , Springer-VerlagNiculescu, S.-I., Gu, K., (2004) Advances in Time-Delay Systems, , SpringerOlgac, N., Sipahi, R., An exact method for the stability analysis of time delayed LTI systems (2002) IEEE Transactions on Automatic Control, 47 (5), pp. 793-797Olgac, N., Sipahi, R., A practical method for analyzing the stability of neutral type LTI-time delayed systems (2004) Automatica, 40, pp. 847-853Partington, J.R., (2004) Linear Operators and Linear Systems: An Analytical Approach to Control Theory, , Cambridge University PressPontryagin, L.S., On the zeros of some elementary transcendental functions (1955) Amer. Math. Soc. Transl, 2 (1), pp. 95-110Richard, J.P., Time-delay systems: An overview of some recent advances and open problems (2003) Automatica, 39, pp. 1667-1694Seydel, R., (2010) Practical Bifurcations and Stability Analysis, , SpringerVyhlídal, T., Zítek, P., Quasipolynomial mapping based rootfinder for analysis of time delay systems (2003) Proc. of IFAC Workshop on Time-Delay Systems, , TDS'03, RocquencourtWalton, K., Marshall, J.E., Direct method for TDS stability analysis (1987) IEE Proceedings, (134 PART D), pp. 101-10

Some recent results on direct delay-dependent stability analysis: Review and open problems

This contribution focuses an overview of selected results on time-delay systems stability analysis in the delay space, recently published in outstanding high-impacted journals and top conferences and meetings. A numerical gridding algorithm solving this problem designed by the first author is included as well. The theoretical background and a concise literature overview are followed by the list of practical and software applications. Unsolved tasks and open problems stemming from the analysis of presented methods and results concisely conclude the paper. The reader is supposed to use this survey to follow some of the presented techniques in his/her own research or engineering practice. © 2019, Springer International Publishing AG, part of Springer Nature.MSMT-7778/2014; LO130