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\title{Optimal Control of Nonlinear Time-Delay Systems by Equal Delay in State and Control Vector's}
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%% Group authors per affiliation:
\author[M. Gerivani]{Mahin Gerivani\corref{mycorrespondingauthor}}
\cortext[mycorrespondingauthor]{Corresponding author}
\ead{ma_ge506@stu.um.ac.ir}


\author[S. Effati]{Sohrab Effati}
\ead{s-effati@um.ac.ir}

\address[mymainaddress]{mathematics, Ferdowsi University of Mash-had, Iran}



\begin{abstract}
In this paper, we consider the optimal control problem of nonlinear time-delay systems affected by an external persistent disturbance with known dynamic characteristics. By using the successive approximation approach (SAA, Ref. 4), we obtain the FFOC law. A sequence of nondelay inhomogeneous TPBV problems is constructed, which converges uniformly to the original nonlinear TPBV problem with time-delay. Therefore, the optimal control for the original system is transformed into a sequence of nondelay inhomogeneous TPBV problems. 
\end{abstract}

\begin{keyword}
\texttt{Nonlinear time-delay systems, successive approximation approach, optimal control, feedforward control.}
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\end{frontmatter}

\linenumbers

\section{Introduction}
In this paper, we consider the optimal control problem of nonlinear time-delay systems affected by an external persistent disturbance with known dynamic characteristics. By using the successive approximation approach (SAA, Ref. 4), we obtain the FFOC law. A sequence of nondelay inhomogeneous TPBV problems is constructed, which converges uniformly to the original nonlinear TPBV problem with time-delay. Therefore, the optimal control for the original system is transformed into a sequence of nondelay inhomogeneous TPBV problems. The obtained FFOC law consists of analytical linear feedforward and feedback terms and the limit of a compensation sequence. By using a finite-step iteration of the compensation sequence, a feedforward and feedback suboptimal control law is obtained. A disturbance observer is designed such that the feedforward control law is physically realizable. The effectiveness of the proposed approach is demonstrated by simulation studies. The rest of this paper is organized as follows. In Section 2, we present the model of the system with an external persistent disturbance and state its background andsignificance. Section 3serves as presentation of theSAAand proof of theexistenceanduniquenessofthecontrollaw.InSection4,thephysicallyrealizable problem of the feedforward control law is considered by using a disturbance observer. Simulations are presented in Section 5 to illustrate the effectiveness of the proposed controller. Conclusions are in Section 6.

\section{Problem statement}
Consider the nonlinear time-delay system described by
In this paper, we consider the optimal control problem of nonlinear time-delay systems affected by an external persistent disturbance with known dynamic characteristics. By using the successive approximation approach (SAA, Ref. 4), we obtain the FFOC law. A sequence of nondelay inhomogeneous TPBV problems is constructed, which converges uniformly to the original nonlinear TPBV problem with time-delay. Therefore, the optimal control for the original system is transformed into a sequence of nondelay inhomogeneous TPBV problems. The obtained FFOC law consists of analytical linear feedforward and feedback terms and the limit of a compensation sequence. By using a finite-step iteration of the compensation sequence, a feedforward and feedback suboptimal control law is obtained. A disturbance observer is designed such that the feedforward control law is physically realizable. The effectiveness of the proposed approach is demonstrated by simulation studies. The rest of this paper is organized as follows. In Section 2, we present the model of the system with an external persistent disturbance and state its background andsignificance. Section 3serves as presentation of theSAAand proof of theexistenceanduniquenessofthecontrollaw.InSection4,thephysicallyrealizable problem of the feedforward control law is considered by using a disturbance observer. Simulations are presented in Section 5 to illustrate the effectiveness of the proposed controller. Conclusions are in Section 6.

\section{ Design of the FFOC Law}
Introduce a sequence of TPBV problems as follows:
In this paper, we consider the optimal control problem of nonlinear time-delay systems affected by an external persistent disturbance with known dynamic characteristics. By using the successive approximation approach (SAA, Ref. 4), we obtain the FFOC law. A sequence of nondelay inhomogeneous TPBV problems is constructed, which converges uniformly to the original nonlinear TPBV problem with time-delay. Therefore, the optimal control for the original system is transformed into a sequence of nondelay inhomogeneous TPBV problems. The obtained FFOC law consists of analytical linear feedforward and feedback terms and the limit of a compensation sequence. By using a finite-step iteration of the compensation sequence, a feedforward and feedback suboptimal control law is obtained. A disturbance observer is designed such that the feedforward control law is physically realizable. The effectiveness of the proposed approach is demonstrated by simulation studies. The rest of this paper is organized as follows. In Section 2, we present the model of the system with an external persistent disturbance and state its background andsignificance. Section 3serves as presentation of theSAAand proof of theexistenceanduniquenessofthecontrollaw.InSection4,thephysicallyrealizable problem of the feedforward control law is considered by using a disturbance observer. Simulations are presented in Section 5 to illustrate the effectiveness of the proposed controller. Conclusions are in Section 6.

\section{ Simulation Examples}

\begin{example}
Consider the 2nd order nonlinear time-delay system (\ref{E1}) , where

\section{Conclusions}
This paper presents a systematic method for

\newpage

\begin{thebibliography}{9}

\bibitem{bib10}
RAY,W. H.,and SOLIMAN,M. A.,The optimal control of processes containing pure time delays - I Necessary conditions for an optimum,Chemical Engineering Science, Vol. 25, pp. 1911-1925,1970
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\bibitem{bib9} 
TANG,G.Y.,Suboptimal Conrol for Nonlinear Systems:A Successive Approximation Approach, Systems and Control Letters,Vol.8,pp. 429-434,2005.
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