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Work on decoupling strategies

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Thomas Dehaeze 2025-04-04 22:49:15 +02:00
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% Created 2025-04-04 Fri 11:38
% Created 2025-04-04 Fri 19:20
% Intended LaTeX compiler: pdflatex
\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
@ -32,7 +32,6 @@ Several considerations:
\item Section \ref{sec:detail_control_optimization}: How to design the controller
\end{itemize}
\chapter{Multiple Sensor Control}
\label{sec:orgf67d083}
\label{sec:detail_control_multiple_sensor}
\textbf{Look at what was done in the introduction \href{file:///home/thomas/Cloud/work-projects/ID31-NASS/phd-thesis-chapters/A0-nass-introduction/nass-introduction.org}{Stewart platforms: Control architecture}}
@ -135,7 +134,6 @@ In this section, we wish to study if sensor fusion can be an option for multi-se
\item optimize the dynamical uncertainty
\end{itemize}
\section{Sensor fusion - Introduction}
\label{sec:org119298d}
Measuring a physical quantity using sensors is always subject to several limitations.
First, the accuracy of the measurement is affected by several noise sources, such as electrical noise of the conditioning electronics being used.
@ -183,13 +181,11 @@ Although many design methods of complementary filters have been proposed in the
Fortunately, both the robustness of the fusion and the super sensor characteristics can be linked to the magnitude of the complementary filters \cite{dehaeze19_compl_filter_shapin_using_synth}.
Based on that, this work introduces a new way to design complementary filters using the \(\mathcal{H}_\infty\) synthesis which allows to shape the complementary filters' magnitude in an easy and intuitive way.
\section{Sensor Fusion and Complementary Filters Requirements}
\label{sec:org8c6ec99}
\label{ssec:detail_control_sensor_fusion_requirements}
Complementary filtering provides a framework for fusing signals from different sensors.
As the effectiveness of the fusion depends on the proper design of the complementary filters, they are expected to fulfill certain requirements.
These requirements are discussed in this section.
\paragraph{Sensor Fusion Architecture}
\label{sec:org1e5ae2a}
\subsubsection{Sensor Fusion Architecture}
A general sensor fusion architecture using complementary filters is shown in Figure \ref{fig:detail_control_sensor_fusion_overview} where several sensors (here two) are measuring the same physical quantity \(x\).
The two sensors output signals \(\hat{x}_1\) and \(\hat{x}_2\) are estimates of \(x\).
@ -210,8 +206,7 @@ That is, unity magnitude and zero phase at all frequencies.
\begin{equation}\label{eq:detail_control_comp_filter}
H_1(s) + H_2(s) = 1
\end{equation}
\paragraph{Sensor Models and Sensor Normalization}
\label{sec:org429b48b}
\subsubsection{Sensor Models and Sensor Normalization}
In order to study such sensor fusion architecture, a model for the sensors is required.
Such model is shown in Figure \ref{fig:detail_control_sensor_model} and consists of a linear time invariant (LTI) system \(G_i(s)\) representing the sensor dynamics and an input \(n_i\) representing the sensor noise.
@ -253,8 +248,7 @@ The super sensor output \(\hat{x}\) is therefore described by \eqref{eq:detail_c
\includegraphics[scale=1]{figs/detail_control_fusion_super_sensor.png}
\caption{\label{fig:detail_control_fusion_super_sensor}Sensor fusion architecture with two normalized sensors.}
\end{figure}
\paragraph{Noise Sensor Filtering}
\label{sec:org8410a77}
\subsubsection{Noise Sensor Filtering}
First, suppose that all the sensors are perfectly normalized \eqref{eq:detail_control_perfect_dynamics}.
The effect of a non-perfect normalization will be discussed afterwards.
@ -290,8 +284,7 @@ This is the simplest form of sensor fusion with complementary filters.
However, the two sensors have usually high noise levels over distinct frequency regions.
In such case, to lower the noise of the super sensor, the norm \(|H_1(j\omega)|\) has to be small when \(\Phi_{n_1}(\omega)\) is larger than \(\Phi_{n_2}(\omega)\) and the norm \(|H_2(j\omega)|\) has to be small when \(\Phi_{n_2}(\omega)\) is larger than \(\Phi_{n_1}(\omega)\).
Hence, by properly shaping the norm of the complementary filters, it is possible to minimize the noise of the super sensor.
\paragraph{Sensor Fusion Robustness}
\label{sec:orgda97814}
\subsubsection{Sensor Fusion Robustness}
In practical systems the sensor normalization is not perfect and condition \eqref{eq:detail_control_perfect_dynamics} is not verified.
@ -351,15 +344,13 @@ For instance, the phase \(\Delta\phi(\omega)\) added by the super sensor dynamic
As it is generally desired to limit the maximum phase added by the super sensor, \(H_1(s)\) and \(H_2(s)\) should be designed such that \(\Delta \phi\) is bounded to acceptable values.
Typically, the norm of the complementary filter \(|H_i(j\omega)|\) should be made small when \(|w_i(j\omega)|\) is large, i.e., at frequencies where the sensor dynamics is uncertain.
\section{Complementary Filters Shaping}
\label{sec:orgc4a4ceb}
\label{ssec:detail_control_hinf_method}
As shown in Section \ref{ssec:detail_control_sensor_fusion_requirements}, the noise and robustness of the super sensor are a function of the complementary filters' norm.
Therefore, a synthesis method of complementary filters that allows to shape their norm would be of great use.
In this section, such synthesis is proposed by writing the synthesis objective as a standard \(\mathcal{H}_\infty\) optimization problem.
As weighting functions are used to represent the wanted complementary filters' shape during the synthesis, their proper design is discussed.
Finally, the synthesis method is validated on an simple example.
\paragraph{Synthesis Objective}
\label{sec:orgec6a952}
\subsubsection{Synthesis Objective}
The synthesis objective is to shape the norm of two filters \(H_1(s)\) and \(H_2(s)\) while ensuring their complementary property \eqref{eq:detail_control_comp_filter}.
This is equivalent as to finding proper and stable transfer functions \(H_1(s)\) and \(H_2(s)\) such that conditions \eqref{eq:detail_control_hinf_cond_complementarity}, \eqref{eq:detail_control_hinf_cond_h1} and \eqref{eq:detail_control_hinf_cond_h2} are satisfied.
@ -372,8 +363,7 @@ This is equivalent as to finding proper and stable transfer functions \(H_1(s)\)
& |H_2(j\omega)| \le \frac{1}{|W_2(j\omega)|} \quad \forall\omega \label{eq:detail_control_hinf_cond_h2}
\end{align}
\end{subequations}
\paragraph{Shaping of Complementary Filters using \(\mathcal{H}_\infty\) synthesis}
\label{sec:orgc16ac7f}
\subsubsection{Shaping of Complementary Filters using \(\mathcal{H}_\infty\) synthesis}
The synthesis objective can be easily expressed as a standard \(\mathcal{H}_\infty\) optimization problem and therefore solved using convenient tools readily available.
Consider the generalized plant \(P(s)\) shown in Figure \ref{fig:detail_control_h_infinity_robust_fusion_plant} and mathematically described by \eqref{eq:detail_control_generalized_plant}.
@ -418,8 +408,7 @@ Therefore, applying the \(\mathcal{H}_\infty\) synthesis to the standard plant \
Note that there is only an implication between the \(\mathcal{H}_\infty\) norm condition \eqref{eq:detail_control_hinf_problem} and the initial synthesis objectives \eqref{eq:detail_control_hinf_cond_h1} and \eqref{eq:detail_control_hinf_cond_h2} and not an equivalence.
Hence, the optimization may be a little bit conservative with respect to the set of filters on which it is performed, see \cite[,Chap. 2.8.3]{skogestad07_multiv_feedb_contr}.
\paragraph{Weighting Functions Design}
\label{sec:org81bbe63}
\subsubsection{Weighting Functions Design}
Weighting functions are used during the synthesis to specify the maximum allowed complementary filters' norm.
The proper design of these weighting functions is of primary importance for the success of the presented \(\mathcal{H}_\infty\) synthesis of complementary filters.
@ -450,8 +439,7 @@ The typical magnitude of a weighting function generated using \eqref{eq:detail_c
}\right)^n
\end{equation}
\end{minipage}
\paragraph{Validation of the proposed synthesis method}
\label{sec:org4c438a2}
\subsubsection{Validation of the proposed synthesis method}
The proposed methodology for the design of complementary filters is now applied on a simple example.
Let's suppose two complementary filters \(H_1(s)\) and \(H_2(s)\) have to be designed such that:
@ -498,7 +486,6 @@ The \(\mathcal{H}_\infty\) norm is here found to be close to one which indicates
This is confirmed by the bode plots of the obtained complementary filters in Figure \ref{fig:detail_control_hinf_filters_results}.
This simple example illustrates the fact that the proposed methodology for complementary filters shaping is easy to use and effective.
\section{Synthesis of a set of three complementary filters}
\label{sec:org6c33bf8}
\label{sec:detail_control_hinf_three_comp_filters}
Some applications may require to merge more than two sensors \cite{stoten01_fusion_kinet_data_using_compos_filter,fonseca15_compl}.
@ -602,7 +589,6 @@ Filter \(H_1(s)\) is defined using \eqref{eq:detail_control_h1_compl_h2_h3} thus
Figure \ref{fig:detail_control_three_complementary_filters_results} displays the three synthesized complementary filters (solid lines) which confirms that the synthesis is successful.
\section*{Conclusion}
\label{sec:org0d8cdf5}
A new method for designing complementary filters using the \(\mathcal{H}_\infty\) synthesis has been proposed.
It allows to shape the magnitude of the filters by the use of weighting functions during the synthesis.
This is very valuable in practice as the characteristics of the super sensor are linked to the complementary filters' magnitude.
@ -614,15 +600,28 @@ Future work will aim at developing a complementary filter synthesis method that
\begin{itemize}
\item Talk about the possibility to use H2 to minimize the RMS value of the super sensor noise? (or maybe make a section about that?)
There is a draft paper about that.
\item For the NASS, it was shown that the HAC-IFF strategy works fine and is easy to understand and tune
\item It would be very interesting to see how sensor fusion (probably between the force sensor and the external metrology) compares in term of performance and robustness
\end{itemize}
\chapter{Decoupling}
\label{sec:orgbbaf347}
\label{sec:detail_control_decoupling}
When dealing with MIMO systems, a typical strategy is to:
\begin{itemize}
\item[{$\square$}] \href{file:///home/thomas/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/bibliography.org}{Decoupling Strategies}
\item First decouple the plant dynamics (discussed in this section)
\item Apply SISO control for the decoupled plant (discussed in section \ref{sec:detail_control_optimization})
\end{itemize}
Another strategy would be to apply a multivariable control synthesis to the coupled system.
Strangely, while H-infinity synthesis is a mature technology, it use for the control of Stewart platform is not yet demonstrated.
From \cite{thayer02_six_axis_vibrat_isolat_system}:
\begin{quote}
Experimental closed-loop control results using the hexapod have shown that controllers designed using a decentralized single-strut design work well when compared to full multivariable methodologies.
\end{quote}
\begin{itemize}
\item[{$\square$}] Review of \href{file:///home/thomas/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/bibliography.org}{Decoupling Strategies} for stewart platforms
\item[{$\square$}] Add some citations about different methods
\end{itemize}
@ -647,34 +646,17 @@ Voice Coil & Geophone + Eddy Current (Struts, collocated) & Decentralized (Sky H
Piezoelectric & Force, Position & Vibration isolation, Model-Based, \textbf{Modal control}: 6x PI controllers & \cite{yang19_dynam_model_decoup_contr_flexib}\\
\midrule
PZT & Geophone (struts) & \textbf{H-Infinity} and mu-synthesis & \cite{lei08_multi_objec_robus_activ_vibrat}\\
Voice Coil & Force sensors (strus) + accelerometer (cartesian) & Decentralized Force Feedback + Centralized H2 control based on accelerometers & \cite{xie17_model_contr_hybrid_passiv_activ}\\
Voice Coil & Force sensors (struts) + accelerometer (cartesian) & Decentralized Force Feedback + Centralized H2 control based on accelerometers & \cite{xie17_model_contr_hybrid_passiv_activ}\\
Voice Coil & Accelerometers & MIMO H-Infinity, active damping & \cite{jiao18_dynam_model_exper_analy_stewar}\\
\bottomrule
\end{tabularx}
\end{table}
When dealing with MIMO systems, a typical strategy is to:
\begin{itemize}
\item first decouple the plant dynamics
\item apply SISO control for the decoupled plant
\end{itemize}
Another strategy would be to apply a multivariable control synthesis to the coupled system.
From \cite{thayer02_six_axis_vibrat_isolat_system}:
\begin{quote}
Experimental closed-loopcontrol results using the hexapod have shown that controllers designed using a decentralized single-strut design work well when compared to full multivariable methodologies.
\end{quote}
Assumptions:
\begin{itemize}
\item parallel manipulators
\end{itemize}
Review of decoupling strategies for Stewart platforms:
\begin{itemize}
\item \href{file:///home/thomas/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/bibliography.org}{Decoupling Strategies}
\end{itemize}
The goal of this section is to compare the use of several methods for the decoupling of parallel manipulators.
It is structured as follow:
@ -687,7 +669,6 @@ It is structured as follow:
\item Conclusions are drawn on the three decoupling methods
\end{itemize}
\section{Test Model}
\label{sec:orgf9db9a6}
\label{ssec:detail_control_decoupling_comp_model}
Let's consider a parallel manipulator with several collocated actuator/sensors pairs.
@ -705,9 +686,19 @@ We will note:
\end{itemize}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/detail_control_model_test_decoupling.png}
\caption{\label{fig:detail_control_model_test_decoupling}Model use to compare decoupling techniques}
\begin{subfigure}{0.58\textwidth}
\begin{center}
\includegraphics[scale=1,scale=1]{figs/detail_control_model_test_decoupling.png}
\end{center}
\subcaption{\label{fig:detail_control_model_test_decoupling}Geometrical parameters}
\end{subfigure}
\begin{subfigure}{0.38\textwidth}
\begin{center}
\includegraphics[scale=1,scale=1]{figs/detail_control_strut_model.png}
\end{center}
\subcaption{\label{fig:detail_control_strut_model}Strut model}
\end{subfigure}
\caption{\label{fig:detail_control_model_test_decoupling_detail}3DoF model used to study decoupling strategies}
\end{figure}
The magnitude of the coupled plant \(G\) is shown in Figure \ref{fig:detail_control_coupled_plant_bode}.
@ -718,9 +709,7 @@ The magnitude of the coupled plant \(G\) is shown in Figure \ref{fig:detail_cont
\caption{\label{fig:detail_control_coupled_plant_bode}Magnitude of the coupled plant.}
\end{figure}
\section{Decentralized Plant / Control in the frame of the struts}
\label{sec:orgf446497}
\section{Jacobian Decoupling}
\label{sec:orgfcc4afb}
\label{ssec:detail_control_comp_jacobian}
The Jacobian matrix can be used to:
@ -754,8 +743,10 @@ The inputs and outputs of the decoupled plant \(\bm{G}_{\{O\}}\) have physical m
\end{itemize}
It is then easy to include a reference tracking input that specify the wanted motion of the payload in the frame \(\{O\}\).
\subsubsection{Center Of Mass}
\subsubsection{Center Of Stiffness}
\section{Modal Decoupling}
\label{sec:org896946d}
\label{ssec:detail_control_comp_modal}
Let's consider a system with the following equations of motion:
@ -861,7 +852,6 @@ The obtained system on the diagonal are second order resonant systems which can
Using this decoupling strategy, it is possible to control each mode individually.
\section{SVD Decoupling}
\label{sec:org40b06bc}
\label{ssec:detail_control_comp_svd}
Procedure:
@ -894,10 +884,8 @@ The inputs and outputs are ordered from higher gain to lower gain at the chosen
\item[{$\square$}] Can we take advantage of the fact that U and V are unitary?
\end{itemize}
\section{Comparison}
\label{sec:org766237f}
\label{ssec:detail_control_decoupling_comp}
\paragraph{Jacobian Decoupling}
\label{sec:orgfc21963}
\subsubsection{Jacobian Decoupling}
Decoupling properties depends on the chosen frame \(\{O\}\).
Let's take the CoM as the decoupling frame.
@ -907,8 +895,7 @@ Let's take the CoM as the decoupling frame.
\includegraphics[scale=1]{figs/detail_control_jacobian_plant.png}
\caption{\label{fig:detail_control_jacobian_plant}Plant decoupled using the Jacobian matrices \(G_x(s)\)}
\end{figure}
\paragraph{Modal Decoupling}
\label{sec:org124e143}
\subsubsection{Modal Decoupling}
For the system in Figure \ref{fig:detail_control_model_test_decoupling}, we have:
\begin{align}
\bm{x} &= \begin{bmatrix} x \\ y \\ R_z \end{bmatrix} \\
@ -962,8 +949,7 @@ And the plant in the modal space is defined below and its magnitude is shown in
\end{figure}
Let's now close one loop at a time and see how the transmissibility changes.
\paragraph{SVD Decoupling}
\label{sec:org3f3f14e}
\subsubsection{SVD Decoupling}
\begin{table}[htbp]
\caption{\label{}Real approximate of \(G\) at the decoupling frequency \(\omega_c\)}
\centering
@ -986,7 +972,6 @@ Let's now close one loop at a time and see how the transmissibility changes.
\caption{\label{fig:detail_control_svd_plant}Svd plant \(G_m(s)\)}
\end{figure}
\section*{Conclusion}
\label{sec:org79124be}
The three proposed methods clearly have a lot in common as they all tend to make system more decoupled by pre and/or post multiplying by a constant matrix
However, the three methods also differs by a number of points which are summarized in Table \ref{tab:detail_control_decoupling_strategies_comp}.
@ -1041,7 +1026,6 @@ Conclusion about NASS:
\end{tabularx}
\end{table}
\chapter{Closed-Loop Shaping using Complementary Filters}
\label{sec:org5dd9a93}
\label{sec:detail_control_optimization}
Performance of a feedback control is dictated by closed-loop transfer functions.
@ -1077,8 +1061,9 @@ It will be experimentally demonstrated with the NASS.
\begin{itemize}
\item frequency domain techniques
\item manual loop-shaping - key idea: modification of the controller such that the open-loop is made according to specifications \cite{oomen18_advan_motion_contr_precis_mechat}.
\end{itemize}
This works well because the open loop transfer function is linearly dependent of the controller.
Different techniques for open loop shaping \cite{lurie02_system_archit_trades_using_bode}
\end{itemize}
However, the specifications are given in terms of the final system performance, i.e. as closed-loop specifications.
@ -1102,10 +1087,8 @@ In this paper, we propose a new controller synthesis method
\item direct translation of requirements such as disturbance rejection and robustness to plant uncertainty
\end{itemize}
\section{Control Architecture}
\label{sec:org34a8e50}
\label{ssec:detail_control_control_arch}
\paragraph{Virtual Sensor Fusion}
\label{sec:org95786c4}
\subsubsection{Virtual Sensor Fusion}
Let's consider the control architecture represented in Figure \ref{fig:detail_control_sf_arch} where \(G^\prime\) is the physical plant to control, \(G\) is a model of the plant, \(k\) is a gain, \(H_L\) and \(H_H\) are complementary filters (\(H_L + H_H = 1\) in the complex sense).
The signals are the reference signal \(r\), the output perturbation \(d_y\), the measurement noise \(n\) and the control input \(u\).
@ -1143,8 +1126,7 @@ y &= \frac{1}{1+G^{\prime} K H_L} dy &&+ \frac{G^{\prime} K}{1+G^{\prime} K
u &= \frac{-K H_L}{1+G^{\prime} K H_L} dy &&+ \frac{K}{1+G^{\prime} K H_L} r &&- \frac{K H_L}{1+G^{\prime} K H_L} n
\end{alignat}
with \(K = \frac{k}{1 + H_H G k}\)
\paragraph{Asymptotic behavior}
\label{sec:org96ac3b2}
\subsubsection{Asymptotic behavior}
We now want to study the asymptotic system obtained when using very high value of \(k\)
\begin{equation}
\lim_{k\to\infty} K = \lim_{k\to\infty} \frac{k}{1+H_H G k} = \left( H_H G \right)^{-1}
@ -1177,7 +1159,6 @@ We obtain a sensitivity transfer function equals to the high pass filter \(S = \
Assuming that we have a good model of the plant, we have then that the closed-loop behavior of the system converges to the designed complementary filters.
\section{Translating the performance requirements into the shapes of the complementary filters}
\label{sec:org5196a79}
\label{ssec:detail_control_trans_perf}
The required performance specifications in a feedback system can usually be translated into requirements on the upper bounds of \(\abs{S(j\w)}\) and \(|T(j\omega)|\) \cite{bibel92_guidel_h}.
The process of designing a controller \(K(s)\) in order to obtain the desired shapes of \(\abs{S(j\w)}\) and \(\abs{T(j\w)}\) is called loop shaping.
@ -1185,8 +1166,7 @@ The process of designing a controller \(K(s)\) in order to obtain the desired sh
The equations \eqref{eq:detail_control_cl_system_y} and \eqref{eq:detail_control_cl_system_u} describing the dynamics of the studied feedback architecture are not written in terms of \(K\) but in terms of the complementary filters \(H_L\) and \(H_H\).
In this section, we then translate the typical specifications into the desired shapes of the complementary filters \(H_L\) and \(H_H\).\\
\paragraph{Nominal Stability (NS)}
\label{sec:org6c13475}
\subsubsection{Nominal Stability (NS)}
The closed-loop system is stable if all its elements are stable (\(K\), \(G^\prime\) and \(H_L\)) and if the sensitivity function (\(S = \frac{1}{1 + G^\prime K H_L}\)) is stable.
For the nominal system (\(G^\prime = G\)), we have \(S = H_H\).
@ -1194,8 +1174,7 @@ For the nominal system (\(G^\prime = G\)), we have \(S = H_H\).
Nominal stability is then guaranteed if \(H_L\), \(H_H\) and \(G\) are stable and if \(G\) and \(H_H\) are minimum phase (to have \(K\) stable).
Thus we must design stable and minimum phase complementary filters.\\
\paragraph{Nominal Performance (NP)}
\label{sec:org4d66e50}
\subsubsection{Nominal Performance (NP)}
Typical performance specifications can usually be translated into upper bounds on \(|S(j\omega)|\) and \(|T(j\omega)|\).
Two performance weights \(w_H\) and \(w_L\) are defined in such a way that performance specifications are satisfied if
@ -1218,15 +1197,13 @@ The translation of typical performance requirements on the shapes of the complem
\end{itemize}
We may have other requirements in terms of stability margins, maximum or minimum closed-loop bandwidth.\\
\paragraph{Closed-Loop Bandwidth}
\label{sec:org3d13ca8}
\subsubsection{Closed-Loop Bandwidth}
The closed-loop bandwidth \(\w_B\) can be defined as the frequency where \(\abs{S(j\w)}\) first crosses \(\frac{1}{\sqrt{2}}\) from below.
If one wants the closed-loop bandwidth to be at least \(\w_B^*\) (e.g. to stabilize an unstable pole), one can required that \(|S(j\omega)| \le \frac{1}{\sqrt{2}}\) below \(\omega_B^*\) by designing \(w_H\) such that \(|w_H(j\omega)| \ge \sqrt{2}\) for \(\omega \le \omega_B^*\).
Similarly, if one wants the closed-loop bandwidth to be less than \(\w_B^*\), one can approximately require that the magnitude of \(T\) is less than \(\frac{1}{\sqrt{2}}\) at frequencies above \(\w_B^*\) by designing \(w_L\) such that \(|w_L(j\omega)| \ge \sqrt{2}\) for \(\omega \ge \omega_B^*\).\\
\paragraph{Classical stability margins}
\label{sec:org7217eab}
\subsubsection{Classical stability margins}
Gain margin (GM) and phase margin (PM) are usual specifications on controlled system.
Minimum GM and PM can be guaranteed by limiting the maximum magnitude of the sensibility function \(M_S = \max_{\omega} |S(j\omega)|\):
\begin{equation}
@ -1240,8 +1217,7 @@ For the nominal system \(M_S = \max_\omega |S| = \max_\omega |H_H|\), so one can
|H_H(j\omega)| \le 2 \quad \forall\omega
\end{equation}
and thus obtain acceptable stability margins.\\
\paragraph{Response time to change of reference signal}
\label{sec:orge54cf0a}
\subsubsection{Response time to change of reference signal}
For the nominal system, the model is accurate and the transfer function from reference signal \(r\) to output \(y\) is \(1\) \eqref{eq:detail_control_cl_performance_y} and does not depends of the complementary filters.
However, one can add a pre-filter as shown in Figure \ref{fig:detail_control_sf_arch_class_prefilter}.
@ -1258,8 +1234,7 @@ Typically, \(K_r\) is a low pass filter of the form
K_r(s) = \frac{1}{1 + \tau s}
\end{equation}
with \(\tau\) corresponding to the desired response time.\\
\paragraph{Input usage}
\label{sec:org2711dd4}
\subsubsection{Input usage}
Input usage due to disturbances \(d_y\) and measurement noise \(n\) is determined by \(\big|\frac{u}{d_y}\big| = \big|\frac{u}{n}\big| = \big|G^{-1}H_L\big|\).
Thus it can be limited by setting an upper bound on \(|H_L|\).
@ -1267,8 +1242,7 @@ Thus it can be limited by setting an upper bound on \(|H_L|\).
Input usage due to reference signal \(r\) is determined by \(\big|\frac{u}{r}\big| = \big|G^{-1} K_r\big|\) when using a pre-filter (Figure \ref{fig:detail_control_sf_arch_class_prefilter}) and \(\big|\frac{u}{r}\big| = \big|G^{-1}\big|\) otherwise.
Proper choice of \(|K_r|\) is then useful to limit input usage due to change of reference signal.\\
\paragraph{Robust Stability (RS)}
\label{sec:orge7938b5}
\subsubsection{Robust Stability (RS)}
Robustness stability represents the ability of the control system to remain stable even though there are differences between the actual system \(G^\prime\) and the model \(G\) that was used to design the controller.
These differences can have various origins such as unmodelled dynamics or non-linearities.
@ -1307,8 +1281,7 @@ Robust stability is then guaranteed by having the low pass filter \(H_L\) satisf
\end{equation}
To ensure robust stability condition \eqref{eq:detail_control_nominal_perf_hl} can be used if \(w_L\) is designed in such a way that \(|w_L| \ge |w_I| (2 + |w_I|)\).\\
\paragraph{Robust Performance (RP)}
\label{sec:orge9f7a36}
\subsubsection{Robust Performance (RP)}
Robust performance is a property for a controlled system to have its performance guaranteed even though the dynamics of the plant is changing within specified bounds.
For robust performance, we then require to have the performance condition valid for all possible plants in the defined uncertainty set:
@ -1345,13 +1318,10 @@ Robust performance is then guaranteed if \eqref{eq:detail_control_robust_perf_a}
One should be aware than when looking for a robust performance condition, only the worst case is evaluated and using the robust stability condition may lead to conservative control.
\section{Analytical formulas for complementary filters?}
\label{sec:org6512345}
\label{ssec:detail_control_analytical_complementary_filters}
\section{Numerical Example}
\label{sec:org16118e8}
\label{ssec:detail_control_simulations}
\paragraph{Procedure}
\label{sec:org56ef466}
\subsubsection{Procedure}
In order to apply this control technique, we propose the following procedure:
\begin{enumerate}
@ -1365,8 +1335,7 @@ If one does not want to use the \(\mathcal{H}_\infty\) synthesis, one can use pr
\item Design a pre-filter \(K_r\) if requirements on input usage or response to reference change are not met
\item Control implementation: Filter the measurement with \(H_L\), implement the controller \(K\) and the pre-filter \(K_r\) as shown on Figure \ref{fig:detail_control_sf_arch_class_prefilter}
\end{enumerate}
\paragraph{Plant}
\label{sec:orgdce9917}
\subsubsection{Plant}
Let's consider the problem of controlling an active vibration isolation system that consist of a mass \(m\) to be isolated, a piezoelectric actuator and a geophone.
We represent this system by a mass-spring-damper system as shown Figure \ref{fig:detail_control_mech_sys_alone} where \(m\) typically represents the mass of the payload to be isolated, \(k\) and \(c\) represent respectively the stiffness and damping of the mount.
@ -1392,8 +1361,7 @@ Its bode plot is shown on Figure \ref{fig:detail_control_bode_plot_mech_sys}.
\includegraphics[scale=1]{figs/detail_control_bode_plot_mech_sys.png}
\caption{\label{fig:detail_control_bode_plot_mech_sys}Bode plot of the transfer function \(G(s)\) from \(F\) to \(x\)}
\end{figure}
\paragraph{Requirements}
\label{sec:org450d439}
\subsubsection{Requirements}
The control objective is to isolate the displacement \(x\) of the mass from the ground motion \(w\).
The disturbance rejection should be at least \(10\) at \(\SI{2}{\hertz}\) and with a slope of \(-2\) below \(\SI{2}{\hertz}\) until a rejection of \(10^4\).
@ -1430,8 +1398,7 @@ All the requirements on \(H_L\) and \(H_H\) are represented on Figure \ref{fig:d
\end{subfigure}
\caption{\label{fig:detail_control_spec_S_T_obtained_filters}Caption with reference to sub figure (\subref{fig:detail_control_spec_S_T}) (\subref{fig:detail_control_hinf_filters_result_weights})}
\end{figure}
\paragraph{Design of the filters}
\label{sec:org571449a}
\subsubsection{Design of the filters}
\textbf{Or maybe use analytical formulas as proposed here: \href{file:///home/thomas/Cloud/research/papers/dehaeze20\_virtu\_senso\_fusio/matlab/index.org}{Complementary filters using analytical formula}}
@ -1451,8 +1418,7 @@ After the \(\hinf\text{-synthesis}\), we obtain \(H_L\) and \(H_H\), and we plot
H_H &= \frac{0.9936 (s+111.1) (s^2 + 0.3988s + 0.08464)}{(s+57.99) (s^2 + 65.77s + 2981)}
\end{align}
\end{subequations}
\paragraph{Controller analysis}
\label{sec:org418c661}
\subsubsection{Controller analysis}
The controller is \(K = \left( H_H G \right)^{-1}\).
A low pass filter is added to \(K\) so that it is proper and implementable.
@ -1481,8 +1447,7 @@ It is implemented as shown on Figure \ref{fig:detail_control_mech_sys_alone_ctrl
\end{subfigure}
\caption{\label{fig:detail_control_bode_Kfb_loop_gain}Caption with reference to sub figure (\subref{fig:detail_control_bode_Kfb}) (\subref{fig:detail_control_bode_plot_loop_gain_robustness})}
\end{figure}
\paragraph{Robustness analysis}
\label{sec:orge5d85d1}
\subsubsection{Robustness analysis}
The robust stability can be access on the nyquist plot (Figure \ref{fig:detail_control_nyquist_robustness}).
The robust performance is shown on Figure \ref{fig:detail_control_robust_perf}.
@ -1503,12 +1468,10 @@ The robust performance is shown on Figure \ref{fig:detail_control_robust_perf}.
\caption{\label{fig:fig_label}Caption with reference to sub figure (\subref{fig:detail_control_nyquist_robustness}) (\subref{fig:detail_control_robust_perf})}
\end{figure}
\section{Experimental Validation?}
\label{sec:org23a68bd}
\label{ssec:detail_control_exp_validation}
\href{file:///home/thomas/Cloud/research/papers/dehaeze20\_virtu\_senso\_fusio/matlab/index.org}{Experimental Validation}
\section*{Conclusion}
\label{sec:orge845c83}
\begin{itemize}
\item[{$\square$}] Discuss how useful it is as the bandwidth can be changed in real time with analytical formulas of second order complementary filters.
Maybe make a section about that.
@ -1522,7 +1485,6 @@ Maybe give analytical formulas of second order complementary filters in the digi
\end{itemize}
\end{itemize}
\chapter*{Conclusion}
\label{sec:org08e0307}
\label{sec:detail_control_conclusion}
\printbibliography[heading=bibintoc,title={Bibliography}]
\end{document}