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\title{OPEN CHANNEL TRANSIENT FLOW CONTROL BY\\ DISCRETE TIME LQR METHODS}

\author{A. GARCIA\thanks{University of North Carolina at Chapel Hill,
Department of Computer Science, Chapel Hill, North Carolina, USA}, M.
HUBBARD\addtocounter{footnote}{-1}\footnotemark
~~and J.J. DE VRIES\thanks{University of Rochester, Computer
Science Department, Rochester, NY 14627, USA}
}

\abstract{A real-time compensation scheme for multipool canals is
developed using linear quadratic methods. A special response of the
simple wave equation is used as a basis for developing the performance
index which is minimized for a linear method of characteristic flow
model (discrete time). State estimation, using only depths adjacent to
underflow gates, is shown always to be possible. Fixed compensation
for a large flow transition is demonstrated in an example where the
controlled objet is a realistic, nonlinear numerical flow model.}

\keywords{Distributed parameter systems; hydraulic systems; water
supply; flow control; partial differential equations; civil
engineering}

\maketitle
\thispagestyle{empty}\pagestyle{empty}

\section{INTRODUCTION}

The operation of modern canals is quite complex - a hierarchical
control system is typically employed to minimize operational expenses
and schedule and regulate the actual flow of water. The main concern
here is with the lowest level control measure - regulation (see Fig.
\ref{f:canal}). 

Disturbances caused by wind, rain, inaccurately predicted usageat
turnouts, etc. and imperfect knowledge of the system parameters (e.g.
gate discharge and Manning coefficients) are reasons why 
actual flows may deviate from scheduled flows and hence provide the
motivation for employing feedback control in canal operations.
Accurate feedback controllers can reduce the waste in delivering
water. Canal regulation is revised.


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\caption{Canal terms and parameters}
\label{f:canal}
\end{figure}

Canal regulation is receiving renewed interest; a variety of methods
have been proposed for approaching the problem. Early work on the
design of feedback controllers for check gates was done by \citeasnoun{Abl:45}
using classical control theory. \citeasnoun{AbTaRu:54}, also using
classical frequency response methods, investigated the stability of
closed-loop level controllers. \citeasnoun{AbTaRu:54} and
\citeasnoun{ChaRou:66} applied the linear quadratic regulator (LQR)
technique to open channel flow control using a linearized, spatially
discretized version of the St Venant equations. Predictive control
strategies based on simplified flow models have also been investigated
\cite{Abl:45,Bak:63a,Dog:58,Keo:58,Pow:85,Sol:89}.\nocite{Bak:63b}
The latter works do not address the problem of transient wave
magnitude control. 

In this paper, three major issues are addressed in the application of
the LQR theory to the regulation of large flow transitions in
multipool canal systems. These are: the development of an accurate and
simple linear model of the flow dynamics, a physically meaningful
method of performance index selection, and the generation of reference
inputs which allow control over transient wave magnitude.
The models used to develop the regulation algorithm is a time and
space discretized approximation of the St Venant equations. The
dynamic response of the wave equation, which is easily developed
analytically, is used as a guideline for developing the penalty
function coefficients. The wave equation is also used to develop
reference inputs for large flow transitions so that transient wave
magnitudes  can be controlled. This affects the dynamic response of the
wave.

\section{OPEN CHANNEL TRANSIENT MODELS}


The one-dimensional equations for gradually varied, unsteady flow in
a prismatic channel are:
\begin{equation}
X_{k+1}=jx_k + Gu_{k+1}
\label{eq:1}
\end{equation}
The nonstandard form of (\ref{eq:1}) is a consequence of the fact that
the boundary condition can immediately affect states adjacent to the
boundary. The nonstandard form is a consequence of the fact that the
boundary condition can affect the model.\footnote{Intersystem linkages
do occur on return.}

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\caption{Basic structure of the development-environ\-ment interface}
\label{f:impact}
\end{figure}

Together these policy variables comprise a policy system that is
capable of responding to a set of impacts that affect that system.
One of the reasons the development system has grown so large is the
set of positive development impacts that have created an propelled
that system over time.

\section{ENVIRONMENTAL IMPACT}

One of the reasons the environment system has evolved is the set of
negative environmental impacts from development activities that have
given rise to the creation of a system to offset development
pressures (see Fig. \ref{f:impact}).

\subsection{Core Actors}
This group has continuous and intensive involvment in the
technological program. It is usually the core actors who initiate a
program via one or more fundamental decisions.

\subsubsection*{Allied supporting actors}
Independent Central Actors: Actors of this type have a degree of
independence of autonomy from both the proponents and adversaries of a
given development program because of research and the resulting effect
of the degree of autonomy. More research is needed on this point.

\begin{table}[hbt]
\caption{Results of systems analysis}
\label{table}
\begin{center}
\begin{tabular}{lcccccc}\hline\\[-2mm]
&Jan&Mar&May&Jul&Sep&Nov\\[2mm]\hline\\[-2mm]
Day\\
1& 4& 22\\
2\\
3& & &31 &86\\
4&&&&&107\\
5&&&&&&189\\[2mm]\hline\\
\end{tabular}
\end{center}
\end{table}

\section{CONCLUSION}
Environmental adequacy then is the joint outcome of a truly
comprhensive and integrated environmental system. By adequacy is meant
the matching if the environment and development responses to the
challenges posed by the development process. These challenges are in
the form of environmental problems emanating fom the development. The
concern hereis for a set of satisfactory or adequate solutions for all
actors and impactees for resolving the conflicts associated with the
development program. An environmental management system that is
comprehensive in its approach and integratedinto the development
decision makeing process should be adequate in meeting the
environmental problems stemming from the development program.

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