The theory of controls is a branch of science and technology which studies how to drive a given dynamical system. This is generally characterized by inputs and outputs and the former can be manipulated to obtain a desired output, which is chosen by a reference setpoint. The design and the technology involved depends on the system, but in general they imply a sensing section, a software section which can modify some features of the input and a feedback section to check that the manipulation of the input signal gave the output as set by the reference.
The theory of controls is a branch of science and technology which studies how to drive a given dynamical system. This is generally characterized by inputs and outputs and the former can be manipulated to obtain a desired output, which is chosen by a reference setpoint. The design and the technology involved depends on the system, but in general they imply a sensing section, a software section which can modify some features of the input and a feedback section to check that the manipulation of the input signal gives the output as set by the reference.
\subsection{Principles: control loops}
\subsection{Principles: control loops}
The collection of all the sections forms a control loop and controls the behaviour of a given variable under exam. Control loops can be:
The collection of all the sections forms a control loop and manages the behaviour of a given variable under exam. Control loops can be:
\begin{itemize}
\begin{itemize}
\item{Open-loop}: the control action is independent from the output.
\item{Open-loop}: the control action is independent from the output.
\item{Closed-loop}: the control action depends on the desired output conditions. This kind of loop uses feedback loops, that assure the process is correctly going on, i.e. the value of the variable under exam is that of setpoint.
\item{Closed-loop}: the control action depends on the desired output conditions. This kind of loop uses feedback loops, that assure that the process is correctly going on, i.e. the value of the variable under exam is that of the setpoint.
\end{itemize}
\end{itemize}
\noindent
\noindent
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@@ -15,16 +15,17 @@ In order to design a control loop, we need to build all the subsystems up: the m
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@@ -15,16 +15,17 @@ In order to design a control loop, we need to build all the subsystems up: the m
\subsection{How to: block diagrams}
\subsection{How to: block diagrams}
Block diagrams are useful graphical instruments to describe, study and build a control loop. Each element of the control system is represented by a block and each block is joined by lines with arrows showing the sequence of controls. Following the logical steps stated before, we can then draw the block diagram of a basic system to be controlled:
Block diagrams are useful graphical instruments to describe, study and build a control loop. Each element of the control system is represented by a block and each block is joined by lines with arrows showing the sequence of controls. Following the logical steps stated before, we can then draw the block diagram of a basic system to be controlled as in Fig. \ref{blockB}.
\begin{figure}[h!]
\begin{figure}[h!]
\centering
\centering
\includegraphics[scale=0.8]{images/block.png}
\includegraphics[scale=0.8]{images/block.png}
\caption[Basic block diagram of a control loop.]{Basic block diagram of a control loop. The setpoint is injected as an input.}
\caption[Basic block diagram of a control loop.]{Basic block diagram of a control loop. The setpoint is injected as an input.}
\label{blockB}
\end{figure}
\end{figure}
\noindent
\noindent
Every variable under interest at the output of each block can be evaluated by \textit{solving} the diagram.
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).
\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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@@ -46,7 +47,7 @@ T(s) = \dfrac{Y(s)}{X(s)}
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@@ -46,7 +47,7 @@ T(s) = \dfrac{Y(s)}{X(s)}
\end{equation}
\end{equation}
\noindent
\noindent
and so the output is characterized by the product between the transfer function and the input signal in the Laplace domain, which is the Laplace transform of the convolution of the two functions in the time domain\footnote{An interesting demonstration of this statement can be found ....}.\\
and so the output is characterized by the product between the transfer function and the input signal in the Laplace domain, which is the Laplace transform of the convolution of the two functions in the time domain\footnote{An interesting demonstration of this statement can be found in \cite{wiki}}.\\
Since, in general, a function can be written as a product of polynomials, the transfer function is also in the form:
Since, in general, a function can be written as a product of polynomials, the transfer function is also in the form:
\begin{equation}
\begin{equation}
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@@ -75,7 +76,7 @@ and the phase is $\varphi$ = arg(T(s)).\\
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@@ -75,7 +76,7 @@ and the phase is $\varphi$ = arg(T(s)).\\
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.
\subsection{Phase and magnitude interpretation: the Bode plot}
\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 in expressed in dB = 20$\log_{10}(x)$ and it is computes as the absolute value of the transfer function:
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:
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, 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.
\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$^{-1}$. 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 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.
\subsection{Coherence}
\subsection{Coherence}
The coherence is a statistic relation between two signals or data sets x and y. It is defined as the ration 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:
@@ -112,6 +112,7 @@ RIN = Relative Intensity Noise\\
...
@@ -112,6 +112,7 @@ RIN = Relative Intensity Noise\\
SC = Sensor Correction\\
SC = Sensor Correction\\
SR = Signal Recycling\\
SR = Signal Recycling\\
SRCL = Signal Recycling Cavity Length\\
SRCL = Signal Recycling Cavity Length\\
STS = \\
TEC = Thermo-Electric Controller\\
TEC = Thermo-Electric Controller\\
UoB = University of Birmingham\\
UoB = University of Birmingham\\
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@@ -214,31 +215,40 @@ Beginning of Gravitational Wave Astronomy}
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@@ -214,31 +215,40 @@ Beginning of Gravitational Wave Astronomy}
%cpsdiff
%cpsdiff
\bibitem{biscans} S. Biscans et al., \textit{Control strategy to limit duty cycle impact of earthquakes on the LIGO gravitational-wave detectors}, arXiv:1707.03466, 2017
\bibitem{proposal} C. Di Fronzo et al., \textit{Proposal for an experiment at LHO: Locking PRCL to IMCL}, proposal, 2019, DCC T1900656-v2
\bibitem{lantztalk} B. Lantz, \textit{System-wide upgrades to improve the Seismic Isolation and control of detectors beyond A+}, talk, 2020
\bibitem{technote1} C. Di Fronzo et al., \textit{Reducing differential motion using CPS sensors}, technical note, 2019, DCC T1900777-v1
\bibitem{kisseltalk} J. Kissel, \textit{Advanced LIGO Active Seismic Isolation}, talk, 2011
\bibitem{chiatalk} C. Di Fronzo, \textit{Reducing differential motion of Advanced LIGO seismic platforms to improve interferometer control signals}, talk, LVK September 2020
\bibitem{lantztech} B. Lantz et al., \textit{Estimates of HAM-ISI motion for A+}, technical note, 2018, DCC T1800066-v2
\bibitem{biscans} S. Biscans et al., \textit{Control strategy to limit duty cycle impact of earthquakes on the LIGO gravitational-wave detectors}, arXiv:1707.03466, 2017
\bibitem{hammodel} S. Cooper et al., \textit{Ham ISI Model}, technical note, 2018
\bibitem{kisseltalk1} J. Kissel, \textit{On Relaxing Our Demand for Single IFO Duty Cycle}, talk, 2019, DCC G1901125-v1
\bibitem{biscanstalk} S. Biscans, \textit{Global seismic control}, talk, GWADW 2019
\bibitem{biswas} A. Biswas et al., \textit{New methods to assess the impact of seismic events on LIGO detector duty cycle}, 2019, arXiv:1910.12143
\bibitem{kisselthesis} J. Kissel, \textit{Calibrating and improving the sensitivity of the LIGO detectors}, PhD thesis, 2010
\bibitem{kisselthesis} J. Kissel, \textit{Calibrating and improving the sensitivity of the LIGO detectors}, PhD thesis, 2010
\bibitem{proposal} C. Di Fronzo et al., \textit{Proposal for an experiment at LHO: Locking PRCL to IMCL}, proposal, 2019, DCC T1900656-v2
\bibitem{technote1} C. Di Fronzo et al., \textit{Reducing differential motion using CPS sensors}, technical note, 2019, DCC T1900777-v1
\bibitem{kisseltalk2} J. Kissel, \textit{Advanced LIGO Active Seismic Isolation}, talk, 2011
\bibitem{technote2} C. Di Fronzo et al., \textit{Reducing differential motion of Advanced LIGO seismic platforms to improve interferometer control signals: block diagrams and maths}, technical note, 2020, DCC T2000108-v1
\bibitem{technote2} C. Di Fronzo et al., \textit{Reducing differential motion of Advanced LIGO seismic platforms to improve interferometer control signals: block diagrams and maths}, technical note, 2020, DCC T2000108-v1
\bibitem{technote3} C. Di Fronzo et al., \textit{Reducing differential motion of Advanced LIGO seismic platforms to improve interferometer control signals:analysis of feasibility}, technical note, 2020, DCC T2000365-v2
\bibitem{technote3} C. Di Fronzo et al., \textit{Reducing differential motion of Advanced LIGO seismic platforms to improve interferometer control signals:analysis of feasibility}, technical note, 2020, DCC T2000365-v2
\bibitem{chiatalk} C. Di Fronzo, \textit{Reducing differential motion of Advanced LIGO seismic platforms to improve interferometer control signals}, talk, LVK September 2020
\bibitem{lantztalk} B. Lantz, \textit{System-wide upgrades to improve the Seismic Isolation and control of detectors beyond A+}, talk, 2020
\bibitem{jenne} J. Driggers, \textit{Noise Cancellation for Gravitational Wave Detectors}, PhD thesis, 2016
\bibitem{llo} A. Pele' et al., \textit{ECR: Differential CPS and cavity offload}, proposal, 2019, DCC E1900330-v1
\bibitem{llo} A. Pele' et al., \textit{ECR: Differential CPS and cavity offload}, proposal, 2019, DCC E1900330-v1
\bibitem{jenne} J. Driggers, \textit{Noise Cancellation for Gravitational Wave Detectors}, PhD thesis, 2016
\bibitem{lantztech} B. Lantz et al., \textit{Estimates of HAM-ISI motion for A+}, technical note, 2018, DCC T1800066-v2
\bibitem{hammodel} S. Cooper et al., \textit{Ham ISI Model}, technical note, 2018
%6D
%6D
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@@ -258,6 +268,14 @@ Beginning of Gravitational Wave Astronomy}
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@@ -258,6 +268,14 @@ Beginning of Gravitational Wave Astronomy}
\bibitem{freise} Andreas Freise, \textit{The Next Generation of Interferometry: Multi-Frequency Optical Modelling, Control Concepts and Implementation}, Appendix B \textit{Control loops}, PhD thesis, 2003
