@@ -25,7 +25,7 @@ Block diagrams are useful graphical instruments to describe, study and build a c
\end{figure}
\noindent
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).
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.
\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.
In the frame of control loops, the transfer function is given by the gain contributions of all the subsystems of the loop.
In the frame of control loops, the transfer function is given by the gain contributions of all the subsystems of the loop. So, recalling the previous block diagram, if we want to know the transfer function of the system, we need to solve the system of equations:
\begin{equation*}
\centering
\left\{\begin{split}
&X_{err} = IN+OUT\\
&x_f = C \cdot X_{err}\\
&X_a = A \cdot X_f\\
&X = P \cdot X_a\\
&OUT = S \cdot X
\end{split}\right.
\end{equation*}
\begin{equation}
\centering
OUT = S \cdot P \cdot A \cdot C \cdot (IN+OUT) = G \cdot (IN+OUT),
\end{equation}
\noindent
where G = S P A C. The transfer function is then:
\begin{equation}
\centering
\dfrac{OUT}{IN} = \dfrac{G}{1-G},
\end{equation}
\noindent
which is also called the \textit{open loop gain}. Since normally the variable we want to control is $X_{err}$, we want to solve the closed loop:
\begin{equation}
\centering
X_{err} = IN + G \cdot X_{err} = \dfrac{IN}{1-G}.
\end{equation}
\noindent
The term 1/(1+G) is called \textit{closed loop gain}.
\subsection{Phase and magnitude interpretation: the Bode plot}
The Bode plot is a graph representing the response in frequency of the magnitude and phase of the system under exam. It is largely used to define the marginal conditions for the stability of the loop. The magnitude is expressed in dB = 20$\log_{10}(x)$ and it is computed as the absolute value of the transfer function:
...
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@@ -92,15 +127,15 @@ The phase is expressed in degrees (deg) and it is computed as:
\end{equation}
\noindent
In the frame of loops, the closed-loop gain is given by:
In the frame of loops, the closed-loop gain was:
\begin{equation}
\begin{equation*}
\centering
G_{CL} = \dfrac{G_{OL}}{1+G_{OL}},
\end{equation}
G_{CL} = \dfrac{1}{1-G},
\end{equation*}
\noindent
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.
where G is a pole for this relation. This means that if G = 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}
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.
