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\hyphenation{op-tical net-works semi-conduc-tor}


\begin{document}
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\title{A Root-Mean-Square-based Measurement Method to Optimize a Parameter in the Control Systems Design}
\author{Rhiza~S.~Sadjad,~\IEEEmembership{Member,~IEEE}}
\maketitle

\begin{abstract}
In the classical control systems design, a parameter is typically determined by the system's time response or frequency response. A standard unstable double-integrator plant with a spring constant K and a mass M is a perfect illustrative example. A stabilizing damper B is determined by setting the time-response to be qualitatively fast and non-oscillatory, which is basically setting the damping-ratio of the second order system equal to 1. The time-response is the system's response to a certain input, such as an impulse-function, a step-function, etc., not its response to an arbitrary disturbance, which is more likely to occur in the field. The root-mean-square-(rms)-based approach introduced here allows a designer to estimate a better parameter to minimize an objective function. The objective function is a combined quadratic function of the "error" and the "effort" rms values, both to be minimized by selecting the damper B accurately for each of the 3 (three) different schemes of disturbances. After successfully applied to the standard double-integrator plant given as an illustrative example, the similar approach is implemented for an armature-controlled dc motor speed-control design with PID controller.
\end{abstract}

\begin{IEEEkeywords}
root mean square, rms, objective function, double-integrator, disturbance, dc-motor, PID. 
\end{IEEEkeywords}
\IEEEpeerreviewmaketitle

\section{Introduction}

Classical methods to determine a parameter in the control systems design have been developed for almost a hundred years~\cite{phillips} since the steam engine's governor was invented. The  most common methods are derived analytically from the system's responses to a certain input. The time response of a system to a step-input is one of the most common ways to characterize the performance of a control system [2]. The frequency response is also commonly used such as in methods based on the Nyquist criteria [3], or other methods. 
Those methods based on the time response or the frequency response or both are usually valid and analytically verified, but in the field they - at least most of them - are not easily implemented due to a couple of reasons, among others for instance: (1) the methods require the use of sophisticated equipment such as an oscilloscope with the capability of displaying one-shot signals or a spectrum-analyzer for the frequency response, (2) the inputs should be a certain kind that is not easy to generate  (an “ideal” step-function or an impulse-function d(t) never actually exists),  (3) even if the specific inputs may be approximated for some cases, they are not always applicable, for example: a step input of armature voltage cannot be actually applied to a large DC motor.   
Measuring equipments - particularly used in electrical engineering - mostly display the measurement results in rms values - also called “effective” values - especially when the signal being measured varies with time considerably [4]. Other physical units are typically converted into electrical quantities using sensors, and then their rms values are measured.

\section{The Basic Concept}
This method is proposed to be applied in a practical situation in the field, so that all quantities are supposed to be physically - or to be more specific: electrically - measureable. All calculations involving the measured quantities should be made available either in analog or digital forms, or both, at the real time during the system operation. The following defined variables are in scalar form for single-input single-output systems, however, they can be generalized into vector and matrix forms for multi-input multi-output systems or systems with multi-variables.
Firstly, we define the error e(t) as the deviation of the  actual output y(t) as compared to the desired output y'(t):
                                (1)
                       This error is caused by an arbitrary disturbance, which is supposed to be repeatable, most preferably periodic, so that   we can measure the rms value of e(t), E, as follows [5]:
                                                      (2)
                       where T is the time period of the disturbance if it is periodic. If the disturbance is not periodic, then we determine T as a finite period of time when the disturbance still considerably affecting the output. 
                       Secondly, we define the control effort u(t) as a designed variable to reduce the rms error E:
                                                 (3)
                       where C ={c1, c2, c3, ......, cn} is a set of control parameters to optimize the system and f(*) is the control algorithm.  
                       The rms value control effort u(t), U, is measured as follows:
                                                     (4)
                        A control system is said to be “correctly designed” if increasing the effort is effectively decreasing the error, and vice versa. This characteristic is very important to guarantee the existence of optimum design obtained by minimizing an objective function J, subject to the variations of control parameters C ={c1, c2, c3, ......, cn} formulated as the following standard unconstrained optimization problem [6]:
                       minimize:                                    (5)
                                subject to the variations of  C ={c1, c2, c3, ......, cn} 
                       where p and q are coeficients to give a freedom to the designer  to emphasize the weight of either the error or the effort, or both, of their contributions to the objective function J. Those coeficients are also useful to equalize the physical units of E and U if they are obtained from different measurement systems. The square-root is optional, just in case the numerical results of squaring the quantities are either too large or too small. 
                       A well established case of control design in the classical control theory is the unstable double-integrator plant model of a spring-mass-damper system with a mass M connected to a reference wall by a spring with the constant K stabilized by a damper with the constant B as represented by the block-diagram shown in Fig. 1. We use this case as an illustrative example of the basic concept described previously. 

\begin{figure}
 \begin{center}
  \end{center}
 \caption{Simulation Model 1}
 \label{sim1} 
\end{figure}

\begin{table}[t]
\begin{center}
\caption{Simulation Parameters}
\begin{tabular}{|l||c|}
\hline
VoIP Codec & G.711 \\
\hline
WLAN Standard & IEEE802.11g \\
\hline
Supported Data Rate & 6 9 12 18 24 36 48 54Mbps \\
\hline
Fading Model & Nakagami Ricean K = 4.84 \\
\hline
\hline
\hline
CWmin, CWmax & 15, 1023 \\
\hline
\end{tabular}
\label{simParameter}
\end{center}
\end{table}

\section{The Practical Application}

\begin{figure}[t]
 \begin{center}
 \end{center}
 \caption{Relationship between RTS Retry and MOS over Distances}
 \label{rts} 
\end{figure}


\begin{figure}[t]
 \begin{center}
 \end{center}
 \caption{Relationship among \# of MNs, AP queue length and MOS}
 \label{QL-MOS_MNs} 
\end{figure}

\begin{figure}[t]
 \begin{center}
 \end{center}
 \caption{AP Queuing Delay and RTT between AP and MN}
 \label{rtt} 
\end{figure}


\begin{figure}[t]
 \begin{center}
 \end{center}
 \caption{Relationship among AP queue length, RTT, and MOS}
 \label{RTT-MOS_QL} 
\end{figure}


\subsection{................}

\begin{equation}\label{rtsRetryRate}
 \begin{split}
 {RTS\ Retry\ Ratio = \frac{Number\ of\ RTS\ Frame \ Retries}{Total\ Transmitted\ Frames }} 
 \end{split}
\end{equation}

\subsection{....................}


\begin{figure}[t]
 \begin{center}
 \end{center}
 \caption{Switching to Single/Multi-Path Transmission}
 \label{fig:spt} 
\end{figure}

\begin{figure}[t]
 \begin{center}
 \end{center}
 \caption{Switching to Single-Path Transmission}
 \label{fig:mpt} 
\end{figure}


\begin{figure}[t]
 \begin{center}
 \end{center}
 \caption{HO based on RTS Frame Retry Ratio}
 \label{fig:retry} 
\end{figure}


\begin{figure}[t]
 \begin{center}
 \end{center}
 \caption{HO based on Transmission Rate}
 \label{fig:transrate} 
\end{figure}


\section{.............................}

\begin{figure*}[t]
 \begin{center}
 \end{center}
 \caption{Calculating RTT from captured probe packet and obtaining the right to send probe packets}
\label{redundant} 
\end{figure*}

\subsection{................................}


\subsection{......................}


\subsection{...................}


\section{..........}
\begin{figure}[t]
 \begin{center}
 \end{center}
 \caption{Simulation Model 2}
 \label{sim2}
\end{figure}

Fig.~\ref{sim2} and Table~\ref{simParameter} 

\begin{figure}[t] %[hbtp]
\begin{center}
%\hspace{1in}
\end{center}
\caption{Variation of AP queue length and MOS}
%\caption{Two Shapes: \protect\subref{fig:prev} A Rectangle and
%\protect\subref{fig:new} A Circle}
%\label{fig:shapes2}
\label{QL_Prev-New}
\end{figure}


\section{Conclusion}
 

\section*{Acknowledgment}


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 \bibitem{phillips}
 C.~L.~Phillips~and~R.~D.~Harbor, \emph{Feedback Control Systems},~Englewood Cliffs, NJ, USA, Prentice Hall, Inc.,1988.
 \bibitem{Koh}
 
 \bibitem{John}
 
  \bibitem{Tsukamoto}
 
 \bibitem{velayos}
 
 \bibitem{SKashihara}
  \bibitem{Yuzo}
 \bibitem{G.107}
 \bibitem{snt}

\end{thebibliography}

\begin{IEEEbiography}[{\includegraphics[width=1.0in,clip,keepaspectratio]{rhiza2019.eps}}]{Rhiza~S.~Sadjad} 
was born in 1957, completed his elementary and secondary education at his home town Bogor, Indonesia. He received his first college degree of Ir. from the Department of Electrical Engineering, Bandung Institute of Technology (ITB), Bandung, Indonesia in 1981, then received the M.S.E.E. (1989) and the Ph.D. (1994) majoring in Automatic Control Systems from the Department of Electrical and Computer Engineering, University of Wisconsin, Madison WI, USA. In 1981 he took a teaching position at the Faculty of Electrical Engineering, Satya Wacana Christian University in Salatiga, Central Java, Indonesia, then moved to the Department of Electrical Engineering, Hasanuddin University, Makassar, South Sulawesi, Indonesia in 1983 and has been with this department until now.. He founded the Control Systems and Instrumentation Laboratory in 1995 and has been the head of the laboratory since then. He teaches almost all courses in automatic control systems area, and has recently been interested in the process control systems and technology. He has supervised a Ph.D. dissertation (completed in 2010) to develop a process control mini-plant for industrial solid materials, and now is advising undergraduate and masters students whose final projects and thesis are related to the field of process control technology. 
\end{IEEEbiography}
\end{document}
