@@ -25,7 +25,7 @@ Block diagrams are useful graphical instruments to describe, study and build a c
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@@ -25,7 +25,7 @@ Block diagrams are useful graphical instruments to describe, study and build a c
\end{figure}
\end{figure}
\noindent
\noindent
Every variable under interest at the output of each block can be evaluated by \textit{solving} the diagram. Referring to Fig. \ref{blockB}, solving a block diagram means solving the system of equations involving the variable under exam and each block. The product of the components of the block diagram give the total gain of the loop (see following sections).
Every variable of interest at the output of each block can be evaluated by \textit{solving} the diagram. Referring to Fig. \ref{blockB}, solving a block diagram means solving the system of equations involving the variable under exam and each block. The product of the components of the block diagram give the total gain of the loop (see following sections).
\section{Control analysis}
\section{Control analysis}
Once the control loop has been schematically drafted, it needs to be finalized: the software section implies instructions. These are given by a computation of the transfer functions of the whole system, which gives the response in the frequency domain of the output to a given input. The computed (and measured) transfer function will then be modified with suitable filters to make the output adjust to the reference setpoint.
Once the control loop has been schematically drafted, it needs to be finalized: the software section implies instructions. These are given by a computation of the transfer functions of the whole system, which gives the response in the frequency domain of the output to a given input. The computed (and measured) transfer function will then be modified with suitable filters to make the output adjust to the reference setpoint.
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@@ -64,7 +64,7 @@ where z and p are the $i$th zeros and poles of the polynomial, of order $j$ and
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@@ -64,7 +64,7 @@ where z and p are the $i$th zeros and poles of the polynomial, of order $j$ and
\end{itemize}
\end{itemize}
\noindent
\noindent
The gain of the system is defined as the radio between the amplitudes of the input and output, i.e. it's the absolute value of the transfer function:
The gain of the system is defined as the ratio between the amplitudes of the input and output, i.e. it's the absolute value of the transfer function:
\begin{equation}
\begin{equation}
\centering
\centering
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@@ -84,7 +84,7 @@ The Bode plot is a graph representing the response in frequency of the magnitude
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@@ -84,7 +84,7 @@ The Bode plot is a graph representing the response in frequency of the magnitude
\end{equation}
\end{equation}
\noindent
\noindent
The phase is expressed in degrees (deg) and it is comoputed as:
The phase is expressed in degrees (deg) and it is computed as:
where G$_{OL}$ is the open-loop gain and also a pole for this relation. This means that if G$_{OL}$ = -1, G$_{CL}$ diverges and the loop is unstable. On the phase plot, we will have that$\varphi$ = -180$^{\circ}$. In general, when the trace on the phase plot approaches this value at certain frequencies, it means that the loop that we are building is unstable in that region.
where G$_{OL}$ is the open-loop gain and also a pole for this relation. This means that if G$_{OL}$ = -1, G$_{CL}$ diverges and the loop is unstable. On the phase plot, this corresponds to$\varphi$ = -180$^{\circ}$. In general, when the trace on the phase plot approaches this value at certain frequencies, it means that the loop that we are building is unstable in that region.
\subsection{Spectral density}
\subsection{Spectral density}
Spectral densities are views of a signal in a frequency spectrum. It is a useful instrument to detect effects on the signal during processing, like peaks due to harmonics, or resonances. The physical parameter used in this study is the power spectral density, which measures the power of a signal as a function of frequency and has units of W/Hz$^{-1/2}$. When there is no direct power associated to the measurement (like in case of Volts) the units are in terms of the square of the signal per Hz. In some cases, an Amplitude Spectral Density (ASD), defined as the square root of the power spectral density, is used when the shape of the signal is quite constant; in this case the units are in the form of 1/Hz$^{-1/2}$ and the variations in the ASD will then be proportional to the variations of the signal itself.
Spectral densities are views of a signal in a frequency spectrum. It is a useful tool to detect effects on the signal during processing, like peaks due to harmonics, or resonances. The physical parameter used in this study is the power spectral density, which measures the power of a signal as a function of frequency and has units of W/Hz$^{-1/2}$. When there is no direct power associated to the measurement (like in case of Volts) the units are in terms of the square of the signal per Hz. In some cases, an Amplitude Spectral Density (ASD), defined as the square root of the power spectral density, is used when the shape of the signal is quite constant; in this case the units are in the form of 1/Hz$^{-1/2}$ and the variations in the ASD will then be proportional to the variations of the signal itself.
\subsection{Coherence}
\subsection{Coherence}
The coherence is a statistic relation between two signals or data sets x and y. It is defined as the ratio between the cross spectral density of the two functions and the product of the spectral densities of each function:
The coherence is a statistic relation between two signals or data sets x and y. It is defined as the ratio between the cross spectral density of the two functions and the product of the spectral densities of each function:
