\documentclass[12pt]{article} % please note that this paper uses the ams symbols so that the % amssymb.sty is included. If you don't use these then you can % exclude the next line. This paper requires the ams symbols to compile. % You might also use other standard style files such as amsmath, latexsym, etc % all of these should be included in the usepackage statement on the next line. \usepackage{amsmath,amsthm,amssymb,euscript,amscd} % The standard method of including graphics in Latex2e uses the graphicx style \usepackage{graphicx} % the most common useage to include a graphic scaled by .5 %(on a scale of 0 to whatever with 1 the actual size.) % % \centerline{ \includegraphics[scale=.5]{filename.eps} } % % there are of course many variation on this theme. %---------------------- %%%%% page size setting for United States %%%%%%%%%%%%% \setlength{\textwidth}{6.5in} \setlength{\textheight}{9in} % %\setlength{\hoffset}{-.5in} % \setlength{\topmargin}{-.5in} \setlength{\evensidemargin}{0in} \setlength{\oddsidemargin}{0in} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \renewcommand{\baselinestretch}{1.1} \parskip=0pt \parindent=10pt \newtheorem{example}{Example}[section] \newtheorem{corollary}{Corollary}[section] \newtheorem{lemma}{Lemma}[section] \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{definition}{Definition}[section] \newtheorem{convention}{Convention}[section] \newtheorem{notation}{Notation}[section] \newtheorem{assumption}{Assumption}[section] \newtheorem{remark}{Remark}[section] \newtheorem{problem}{Problem}[section] %%% For numbering of theorems, definitions, lemmas, etc -- \def\theequation{\thesection.\arabic{equation}} % here are examples of user defined expressions in the preamble % you can replace these with any of your favorite shortcuts % the information from here to NOTE are specific only to this % file and are not needed in you document. %---------------------------------------------- \newcommand{\eref}[1]{(\ref{#1})} \newcommand{\dst}{\displaystyle} \def\bbc{{\mathbb C}} \def\bbr{{\mathbb R}} \def\lag{\langle} \def\rag{\rangle} \def\nn{\nonumber} \def\lng{\langle} \def\rng{\rangle} \def\lp{\left(} \def\rp{\right)} \theoremstyle{plain} \newcommand{\script}[1]{\EuScript{#1}} \newcommand{\sA}{{\script A}} \newcommand{\sB}{{\script B}} \newcommand{\sC}{{\script C}} \newcommand{\sD}{{\script D}} \newcommand{\sE}{{\script E}} \newcommand{\sF}{{\script F}} \newcommand{\sG}{{\script G}} \newcommand{\sH}{{\script H}} \newcommand{\sI}{{\script I}} \newcommand{\sJ}{{\script J}} \newcommand{\sK}{{\script K}} \newcommand{\sL}{{\script L}} \newcommand{\sM}{{\script M}} \newcommand{\sN}{{\script N}} \newcommand{\sO}{{\script O}} \newcommand{\sP}{{\script P}} \newcommand{\sQ}{{\script Q}} \newcommand{\sR}{{\script R}} \newcommand{\sS}{{\script S}} \newcommand{\sT}{{\script T}} \newcommand{\sU}{{\script U}} \newcommand{\sV}{{\script V}} \newcommand{\sW}{{\script W}} \newcommand{\sX}{{\script X}} \newcommand{\sY}{{\script Y}} \newcommand{\sZ}{{\script Z}} \def\itb{\mathbf{B}} \def\itc{\mathbf{C}} \def\itd{\mathbf{D}} \def\itp{\mathbf{P}} \def\itphi{\mathbf{\Phi}} \def\fX{\mathfrak{X}} \def\fz{\mathfrak{z}} \def\fZ{\mathfrak{Z}} \def\fw{\mathfrak{w}} \def\fx{\mathfrak{x}} \def\d{\delta} \def\b{\beta} \def\g{\gamma} \def\G{\Gamma} \def\e{\epsilon} \def\a{\alpha} \def\ta{\theta} \def\i{\infty} \def\l{\lambda} \def\L{\Lambda} \def\o{\omega} \def\ra{\rightarrow} \def\vp{\varphi} \def\ve{\varepsilon} \def\wh{\widehat} \def\wt{\widetilde} \def\o{\omega} \def\O{\Omega} \def\s{\sigma} \def\cala{\mathcal{A}} \def\calg{\mathcal{G}} \def\calw{\mathcal{W}} \def\calf{\mathcal{F}} \def\calm{\mathcal{M}} \def\calb{\mathcal{B}} \def\calx{\mathcal{X}} \def\call{\mathcal{L}} \def\calp{\mathcal{P} } \def\cald{\mathcal{D}} \def\calh{\mathcal{H}} \def\calc{\mathcal{C}} \def\calr{\mathcal{R}} \def\cals{\mathcal{S}} \def\calq{\mathcal{Q}} \def\calz{\mathcal{Z}} \def\calt{\mathcal{T}} %---------------------------------------------- \title{The Regulator Equations for Retarded Delay Differential Equations} \author{Victor Shubov \thanks{Research supported by AFOSR grant} \\ Dept Math and Stat\\ TTU} \begin{document} \begin{abstract} In this work we present a simple formula for the solution for problems of output regulation (tracking and disturbance rejection) for systems governed by linear retarded functional differential equations. Our formulas are based on an analysis of the so-called regulator equations. In the geometric theory of output regulation the desired feedback gain is obtained from solution of these equations. For delay equations we show that the regulator equations reduce to a pair of finite dimensional linear systems of equations. For special exosystems we give an explicit formula for the solution. \end{abstract} \section{Introduction} \setcounter{equation}{0} Systems with delays have been studied by many authors and continue to be an important area of research by many authors in systems and control. Our interest lies in application of the geometric theory of output regulation as developed over many years in \cite{Francis77,knob,BI,poh3,shoe1,linreg}. Here we present some results on solution of problems of output regulation for delay systems from our recent work \cite{delay_linreg}. Following the notations set down in \cite{Cur_Zw}, we consider retarded delay differential systems in the form \begin{align} \dot{x}(t)&= A_0x(t) +\sum_{j=1}^p A_j x(t-h_j) +B_0u(t) +D_0(t), \label{delay_eq1}\\ x(0)&=r \in \bbr^n, \label{delay_eq2}\\ x(\ta)&=\vp \in L^2([-h_p,0],\bbr^n) \label{delay_eq3}\\ &0< h_1< \cdots 0$, and the output operator $C_0\in \call(\bbc^n,\bbc^m)$. It can be shown (see for example \cite{Cur_Zw}) that the solution to \eref{delay_eq1}-\eref{delay_eq3} with $u=0$ and $D_0=0$ can be expressed as \begin{equation}\label{delay_eq5} x(t)=e^{A_0 t} r + \sum_{j=1}^p \int_0^t e^{A_0(t-s)} A_jx(s-h_j)\, ds. \end{equation} We can also formulate this problem in a standard state space format in the infinite dimensional state space (cf, \cite{htb1}, \cite{htb3}, \cite{Cur_Zw}) \begin{equation}\label{delay_eq6} \sZ=\bbc^n\oplus L^2([-h_p,0],\bbc^n), \end{equation} with inner product in $\sZ$ is given by \begin{equation}\label{delay_eq7} \left \lag Z_1,Z_2 \right\rag =\lag r_1,r_2\rag_{\bbc^n} + \lag \vp_1,\vp_2\rag_{L^2}, \ \ \ Z_1=\begin{pmatrix}r_1\\\vp_1(\cdot)\end{pmatrix},\ \ Z_2=\begin{pmatrix}r_2\\\vp_2(\cdot)\end{pmatrix}. \end{equation} \begin{thebibliography}{1000} \bibitem{htb1} {\sc H.T. Banks}, {\em The representation of solutions of linear functional differential equations}, J. Diff. Eqs., 5, (1969), 399-410. \bibitem{htb3} {\sc H.T. Banks and J. Burns}, {\em An abstract framework for approximate solutions to optimal control problems governed by heriditary systems}, Proc. Int. Conf. Diff. Eqns., Univ. Southern Calif., 1974, Academic Press, 1975. \bibitem{delay_linreg} {\sc C.I. Byrnes, D.S. Gilliam and V.I. Shubov}, {\em Output regulation for linear delay systems}, Preprint Texas Tech University, 2002. \bibitem{linreg} {\sc C.I. Byrnes, D.S. Gilliam, I.G. Lauk\'o and V.I. Shubov}, {\em Output regulation for linear distributed parameter systems}, {\em IEEE Trans. Aut. Control}, Volume 45, Number 12, December 2000, 2236-2252. \bibitem{Cur_Zw} {\sc R.F. Curtain and H.J. Zwart}, {\em An Introduction to Infinite-Dimensional Linear Systems,} Springer-Verlag, New York, 1995. \bibitem{Francis77} {\sc B.~A. Francis}, {\em The linear multivariable regulator problem}, SIAM Journal of Control and Optimization, 14:486--505, 1977. \bibitem{BI} {\sc A.Isidori, C.I.Byrnes}, {\em Output regulation of nonlinear systems, } in IEEE Trans. Autom. Control, AC-35: 131--140, 1990. \bibitem{isidori} A. Isidori, {\em Nonlinear Control Systems,} third edition, Springer-Verlag, Berlin, 1995. \bibitem{knob} {\sc H.W. Knobloch, A. Isidori and D. Flockerzi}, {\em Topics in Control Theory,} Birkh\"auser Verlag, Basel, 1993. \bibitem {poh3} {\sc S. A. Pohjolainen}, {\em On the asymptotic regulation problem for distributes parameter systems}, Proc. Third Symposium on Control of Distributed Parameter Systems, Toulouse, France (July 1982). \bibitem{shoe1} {\sc J.M. Schumacher}, {\em Finite-dimensional regulators for a class of infinite dimensional systems}, Systems and Control Letters, 3 (1983), 7-12. \end{thebibliography} \end{document}