410 lines
32 KiB
TeX
410 lines
32 KiB
TeX
% Created 2025-11-27 Thu 21:31
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% Intended LaTeX compiler: pdflatex
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\documentclass[lettersize,journal]{IEEEtran}
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\input{preamble.tex}
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\input{preamble_extra.tex}
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\author{ \IEEEauthorblockN{Dehaeze Thomas} \IEEEauthorblockA{\textit{European Synchrotron Radiation Facility} \\ Grenoble, France\\ \textit{Precision Mechatronics Laboratory} \\ \textit{University of Liege}, Belgium \\ thomas.dehaeze@esrf.fr }\and \IEEEauthorblockN{Verma Mohit} \IEEEauthorblockA{\textit{BEAMS Department}\\ \textit{Free University of Brussels}, Belgium\\ \textit{Precision Mechatronics Laboratory} \\ \textit{University of Liege}, Belgium \\ mohit.verma@ulb.ac.be }\and \IEEEauthorblockN{Collette Christophe} \IEEEauthorblockA{\textit{BEAMS Department}\\ \textit{Free University of Brussels}, Belgium\\ \textit{Precision Mechatronics Laboratory} \\ \textit{University of Liege}, Belgium \\ ccollett@ulb.ac.be }}
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\title{Closed-Loop Shaping using Complementary Filters}
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\begin{document}
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\maketitle
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\begin{abstract}
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This document describes the most common article elements and how to use the IEEEtran class with \LaTeX \ to produce files that are suitable for submission to the IEEE. IEEEtran can produce conference, journal, and technical note (correspondence) papers with a suitable choice of class options.
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\end{abstract}
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\begin{IEEEkeywords}
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Article submission, IEEE, IEEEtran, journal, \LaTeX, paper, template, typesetting.
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\end{IEEEkeywords}
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Once the system is properly decoupled using one of the approaches described in Section \ref{sec:detail_control_decoupling}, SISO controllers can be individually tuned for each decoupled ``directions''.
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Several ways to design a controller to obtain a given performance while ensuring good robustness properties can be implemented.
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In some cases ``fixed'' controller structures are utilized, such as PI and PID controllers, whose parameters are manually tuned \cite{furutani04_nanom_cuttin_machin_using_stewar,du14_piezo_actuat_high_precis_flexib,yang19_dynam_model_decoup_contr_flexib}.
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Another popular method is Open-Loop shaping, which was used during the conceptual phase.
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Open-loop shaping involves tuning the controller through a series of ``standard'' filters (leads, lags, notches, low-pass filters, \ldots{}) to shape the open-loop transfer function \(G(s)K(s)\) according to desired specifications, including bandwidth, gain and phase margins \cite[, chapt. 4.4.7]{schmidt20_desig_high_perfor_mechat_third_revis_edition}.
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Open-Loop shaping is very popular because the open-loop transfer function is a linear function of the controller, making it relatively straightforward to tune the controller to achieve desired open-loop characteristics.
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Another key advantage is that controllers can be tuned directly from measured frequency response functions of the plant without requiring an explicit model.
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However, the behavior (i.e. performance) of a feedback system is a function of closed-loop transfer functions.
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Specifications can therefore be expressed in terms of the magnitude of closed-loop transfer functions, such as the sensitivity, plant sensitivity, and complementary sensitivity transfer functions \cite[, chapt. 3]{skogestad07_multiv_feedb_contr}.
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With open-loop shaping, closed-loop transfer functions are changed only indirectly, which may make it difficult to directly address the specifications that are in terms of the closed-loop transfer functions.
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In order to synthesize a controller that directly shapes the closed-loop transfer functions (and therefore the performance metric), \(\mathcal{H}_\infty\text{-synthesis}\) may be used \cite{skogestad07_multiv_feedb_contr}.
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This approach requires a good model of the plant and expertise in selecting weighting functions that will define the wanted shape of different closed-loop transfer functions \cite{bibel92_guidel_h}.
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\(\mathcal{H}_{\infty}\text{-synthesis}\) has been applied for the Stewart platform \cite{jiao18_dynam_model_exper_analy_stewar}, yet when benchmarked against more basic decentralized controllers, the performance gains proved small \cite{thayer02_six_axis_vibrat_isolat_system,hauge04_sensor_contr_space_based_six}.
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In this section, an alternative controller synthesis scheme is proposed in which complementary filters are used for directly shaping the closed-loop transfer functions (i.e., directly addressing the closed-loop performances).
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In Section \ref{ssec:detail_control_cf_control_arch}, the proposed control architecture is presented.
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In Section \ref{ssec:detail_control_cf_trans_perf}, typical performance requirements are translated into the shape of the complementary filters.
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The design of the complementary filters is briefly discussed in Section \ref{ssec:detail_control_cf_analytical_complementary_filters}, and analytical formulas are proposed such that it is possible to change the closed-loop behavior of the system in real time.
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Finally, in Section \ref{ssec:detail_control_cf_simulations}, a numerical example is used to show how the proposed control architecture can be implemented in practice.
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\section{Control Architecture}
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\label{ssec:detail_control_cf_control_arch}
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\subsubsection{Virtual Sensor Fusion}
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The idea of using complementary filters in the control architecture originates from sensor fusion techniques \cite{collette15_sensor_fusion_method_high_perfor}, where two sensors are combined using complementary filters.
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Building upon this concept, ``virtual sensor fusion'' \cite{verma20_virtual_sensor_fusion_high_precis_contr} replaces one physical sensor with a model \(G\) of the plant.
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The corresponding control architecture is illustrated in Figure \ref{fig:detail_control_cf_arch}, where \(G^\prime\) represents the physical plant to be controlled, \(G\) is a model of the plant, \(k\) is the controller, and \(H_L\) and \(H_H\) are complementary filters satisfying \(H_L(s) + H_H(s) = 1\).
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In this arrangement, the physical plant is controlled at low frequencies, while the plant model is utilized at high frequencies to enhance robustness.
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\begin{figure}[htbp]
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\begin{subfigure}{0.48\textwidth}
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\begin{center}
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\includegraphics[scale=1,scale=0.9]{figs/detail_control_cf_arch.png}
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\end{center}
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\subcaption{\label{fig:detail_control_cf_arch}Virtual Sensor Fusion}
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\end{subfigure}
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\begin{subfigure}{0.48\textwidth}
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\begin{center}
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\includegraphics[scale=1,scale=0.9]{figs/detail_control_cf_arch_eq.png}
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\end{center}
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\subcaption{\label{fig:detail_control_cf_arch_eq}Equivalent Architecture}
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\end{subfigure}
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\caption{\label{fig:detail_control_cf_arch_and_eq}Control architecture for virtual sensor fusion (\subref{fig:detail_control_cf_arch}). An equivalent architecture is shown in (\subref{fig:detail_control_cf_arch_eq}). The signals are the reference signal \(r\), the output perturbation \(d_y\), the measurement noise \(n\) and the control input \(u\).}
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\end{figure}
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Although the control architecture shown in Figure \ref{fig:detail_control_cf_arch} appears to be a multi-loop system, it should be noted that no non-linear saturation-type elements are present in the inner loop (containing \(k\), \(G\), and \(H_H\), all numerically implemented).
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Consequently, this structure is mathematically equivalent to the single-loop architecture illustrated in Figure \ref{fig:detail_control_cf_arch_eq}.
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\subsubsection{Asymptotic behavior}
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When considering the extreme case of very high values for \(k\), the effective controller \(K(s)\) converges to the inverse of the plant model multiplied by the inverse of the high-pass filter, as expressed in \eqref{eq:detail_control_cf_high_k}.
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\begin{equation}\label{eq:detail_control_cf_high_k}
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\lim_{k\to\infty} K(s) = \lim_{k\to\infty} \frac{k}{1+H_H(s) G(s) k} = \big( H_H(s) G(s) \big)^{-1}
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\end{equation}
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If the resulting \(K\) is improper, a low-pass filter with sufficiently high corner frequency can be added to ensure its causal realization.
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Furthermore, for \(K\) to be stable, both \(G\) and \(H_H\) must be minimum phase transfer functions.
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With these assumptions, the resulting control architecture is illustrated in Figure \ref{fig:detail_control_cf_arch_class}, where the complementary filters \(H_L\) and \(H_H\) remain the only tuning parameters.
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The dynamics of this closed-loop system are described by equations \eqref{eq:detail_control_cf_cl_system_y} and \eqref{eq:detail_control_cf_cl_system_y}.
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/detail_control_cf_arch_class.png}
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\caption{\label{fig:detail_control_cf_arch_class}Equivalent classical feedback control architecture}
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\end{figure}
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\begin{subequations}\label{eq:detail_control_cf_sf_cl_tf_K_inf}
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\begin{align}
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y &= \frac{ H_H dy + G^{\prime} G^{-1} r - G^{\prime} G^{-1} H_L n }{H_H + G^\prime G^{-1} H_L} \label{eq:detail_control_cf_cl_system_y}\\
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u &= \frac{ -G^{-1} H_L dy + G^{-1} r - G^{-1} H_L n }{H_H + G^\prime G^{-1} H_L} \label{eq:detail_control_cf_cl_system_u}
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\end{align}
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\end{subequations}
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At frequencies where the model accurately represents the physical plant (\(G^{-1} G^{\prime} \approx 1\)), the denominator simplifies to \(H_H + G^\prime G^{-1} H_L \approx H_H + H_L = 1\), and the closed-loop transfer functions are then described by equations \eqref{eq:detail_control_cf_cl_performance_y} and \eqref{eq:detail_control_cf_cl_performance_u}.
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\begin{subequations}\label{eq:detail_control_cf_sf_cl_tf_K_inf_perfect}
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\begin{alignat}{5}
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y &= H_H dy &&+ r &&- H_L n \label{eq:detail_control_cf_cl_performance_y} \\
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u &= -G^{-1} H_L dy &&+ G^{-1} r &&- G^{-1} H_L n \label{eq:detail_control_cf_cl_performance_u}
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\end{alignat}
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\end{subequations}
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The sensitivity transfer function equals the high-pass filter \(S = \frac{y}{dy} = H_H\), and the complementary sensitivity transfer function equals the low-pass filter \(T = \frac{y}{n} = H_L\).
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Hence, when the plant model closely approximates the actual dynamics, the closed-loop transfer functions converge to the designed complementary filters, allowing direct translation of performance requirements into the design of the complementary.
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\section{Translating the performance requirements into the shape of the complementary filters}
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\label{ssec:detail_control_cf_trans_perf}
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Performance specifications in a feedback system can usually be expressed as upper bounds on the magnitudes of closed-loop transfer functions such as the sensitivity and complementary sensitivity transfer functions \cite{bibel92_guidel_h}.
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The design of a controller \(K(s)\) to obtain the desired shape of these closed-loop transfer functions is known as closed-loop shaping.
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In the proposed control architecture, the closed-loop transfer functions \eqref{eq:detail_control_cf_sf_cl_tf_K_inf} are expressed in terms of the complementary filters \(H_L(s)\) and \(H_H(s)\) rather than directly through the controller \(K(s)\).
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Therefore, performance requirements must be translated into constraints on the shape of these complementary filters.
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\subsubsection{Nominal Stability (NS)}
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A closed-loop system is stable when all its elements (here \(K\), \(G^\prime\), and \(H_L\)) are stable and the sensitivity function \(S = \frac{1}{1 + G^\prime K H_L}\) is stable.
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For the nominal system (\(G^\prime = G\)), the sensitivity transfer function equals the high-pass filter: \(S(s) = H_H(s)\).
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Nominal stability is therefore guaranteed when \(H_L\), \(H_H\), and \(G\) are stable, and both \(G\) and \(H_H\) are minimum phase (ensuring \(K\) is stable).
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Consequently, stable and minimum phase complementary filters must be employed.
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\subsubsection{Nominal Performance (NP)}
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Performance specifications can be formalized using weighting functions \(w_H\) and \(w_L\), where performance is achieved when \eqref{eq:detail_control_cf_weights} is satisfied.
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The weighting functions define the maximum magnitude of the closed-loop transfer functions as a function of frequency, effectively determining their ``shape''.
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\begin{subequations}\label{eq:detail_control_cf_weights}
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\begin{align}
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|w_H(j\omega) S(j\omega)| &\le 1 \quad \forall\omega\\
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|w_L(j\omega) T(j\omega)| &\le 1 \quad \forall\omega
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\end{align}
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\end{subequations}
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For the nominal system, \(S = H_H\) and \(T = H_L\), hence the performance specifications can be converted on the shape of the complementary filters \eqref{eq:detail_control_cf_nominal_performance}.
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\begin{equation}\label{eq:detail_control_cf_nominal_performance}
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\Aboxed{\text{NP} \Longleftrightarrow {\begin{cases*}
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|w_H(j\omega) H_H(j\omega)| \le 1 & \forall\omega \\
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|w_L(j\omega) H_L(j\omega)| \le 1 & \forall\omega
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\end{cases*}}}
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\end{equation}
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For disturbance rejection, the magnitude of the sensitivity function \(|S(j\omega)| = |H_H(j\omega)|\) should be minimized, particularly at low frequencies where disturbances are usually most prominent.
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Similarly, for noise attenuation, the magnitude of the complementary sensitivity function \(|T(j\omega)| = |H_L(j\omega)|\) should be minimized, especially at high frequencies where measurement noise typically dominates.
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Classical stability margins (gain and phase margins) are also related to the maximum amplitude of the sensitivity transfer function.
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Typically, maintaining \(|S|_{\infty} \le 2\) ensures a gain margin of at least 2 and a phase margin of at least \(\SI{29}{\degree}\).
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Therefore, by carefully selecting the shape of the complementary filters, nominal performance specifications can be directly addressed in an intuitive manner.
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\subsubsection{Robust Stability (RS)}
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Robust stability refers to a control system's ability to maintain stability despite discrepancies between the actual system \(G^\prime\) and the model \(G\) used for controller design.
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These discrepancies may arise from unmodeled dynamics or nonlinearities.
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To represent these model-plant differences, input multiplicative uncertainty as illustrated in Figure \ref{fig:detail_control_cf_input_uncertainty} is employed.
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The set of possible plants \(\Pi_i\) is described by \eqref{eq:detail_control_cf_multiplicative_uncertainty}, with the weighting function \(w_I\) selected such that all possible plants \(G^\prime\) are contained within the set \(\Pi_i\).
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\begin{equation}\label{eq:detail_control_cf_multiplicative_uncertainty}
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\Pi_i: \quad G^\prime(s) = G(s)\big(1 + w_I(s)\Delta_I(s)\big); \quad |\Delta_I(j\omega)| \le 1 \ \forall\omega
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\end{equation}
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\begin{figure}[htbp]
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\begin{subfigure}{0.48\textwidth}
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\begin{center}
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\includegraphics[scale=1,scale=1]{figs/detail_control_cf_input_uncertainty.png}
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\end{center}
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\subcaption{\label{fig:detail_control_cf_input_uncertainty}Input multiplicative uncertainty}
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\end{subfigure}
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\begin{subfigure}{0.48\textwidth}
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\begin{center}
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\includegraphics[scale=1,scale=1]{figs/detail_control_cf_nyquist_uncertainty.png}
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\end{center}
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\subcaption{\label{fig:detail_control_cf_nyquist_uncertainty}Nyquist plot - Effect of multiplicative uncertainty}
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\end{subfigure}
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\caption{\label{fig:detail_control_cf_input_uncertainty_nyquist}Input multiplicative uncertainty to model the differences between the model and the physical plant (\subref{fig:detail_control_cf_input_uncertainty}). Effect of this uncertainty is displayed on the Nyquist plot (\subref{fig:detail_control_cf_nyquist_uncertainty})}
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\end{figure}
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When considering input multiplicative uncertainty, robust stability can be derived graphically from the Nyquist plot (illustrated in Figure \ref{fig:detail_control_cf_nyquist_uncertainty}), yielding to \eqref{eq:detail_control_cf_robust_stability_graphically}, as demonstrated in \cite[, chapt. 7.5.1]{skogestad07_multiv_feedb_contr}.
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\begin{equation}\label{eq:detail_control_cf_robust_stability_graphically}
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\text{RS} \Longleftrightarrow \left|w_I(j\omega) L(j\omega) \right| \le \left| 1 + L(j\omega) \right| \quad \forall\omega
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\end{equation}
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After algebraic manipulation, robust stability is guaranteed when the low-pass complementary filter \(H_L\) satisfies \eqref{eq:detail_control_cf_condition_robust_stability}.
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\begin{equation}\label{eq:detail_control_cf_condition_robust_stability}
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\boxed{\text{RS} \Longleftrightarrow |w_I(j\omega) H_L(j\omega)| \le 1 \quad \forall \omega}
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\end{equation}
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\subsubsection{Robust Performance (RP)}
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Robust performance ensures that performance specifications \eqref{eq:detail_control_cf_weights} are met even when the plant dynamics fluctuates within specified bounds \eqref{eq:detail_control_cf_robust_perf_S}.
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\begin{equation}\label{eq:detail_control_cf_robust_perf_S}
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\text{RP} \Longleftrightarrow |w_H(j\omega) S(j\omega)| \le 1 \quad \forall G^\prime \in \Pi_I, \ \forall\omega
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\end{equation}
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Transforming this condition into constraints on the complementary filters yields:
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\begin{equation}\label{eq:detail_control_cf_robust_performance}
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\boxed{\text{RP} \Longleftrightarrow | w_H(j\omega) H_H(j\omega) | + | w_I(j\omega) H_L(j\omega) | \le 1, \ \forall\omega}
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\end{equation}
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The robust performance condition effectively combines both nominal performance \eqref{eq:detail_control_cf_nominal_performance} and robust stability conditions \eqref{eq:detail_control_cf_condition_robust_stability}.
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If both NP and RS conditions are satisfied, robust performance will be achieved within a factor of 2 \cite[, chapt. 7.6]{skogestad07_multiv_feedb_contr}.
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Therefore, for SISO systems, ensuring robust stability and nominal performance is typically sufficient.
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\section{Complementary filter design}
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\label{ssec:detail_control_cf_analytical_complementary_filters}
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As proposed in Section \ref{sec:detail_control_sensor}, complementary filters can be shaped using standard \(\mathcal{H}_{\infty}\text{-synthesis}\) techniques.
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This approach is particularly well-suited since performance requirements were expressed as upper bounds on the magnitude of the complementary filters.
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Alternatively, analytical formulas for complementary filters may be employed.
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For some applications, first-order complementary filters as shown in Equation \eqref{eq:detail_control_cf_1st_order} are sufficient.
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\begin{subequations}\label{eq:detail_control_cf_1st_order}
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\begin{align}
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H_L(s) &= \frac{1}{1 + s/\omega_0} \\
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H_H(s) &= \frac{s/\omega_0}{1 + s/\omega_0}
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\end{align}
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\end{subequations}
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These filters can be transformed into the digital domain using the Bilinear transformation, resulting in the digital filter representations shown in Equation \eqref{eq:detail_control_cf_1st_order_z}.
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\begin{subequations}\label{eq:detail_control_cf_1st_order_z}
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\begin{align}
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H_L(z^{-1}) &= \frac{T_s \omega_0 + T_s \omega_0 z^{-1}}{T_s \omega_0 + 2 + (T_s \omega_0 - 2) z^{-1}} \\
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H_H(z^{-1}) &= \frac{2 - 2 z^{-1}}{T_s \omega_0 + 2 + (T_s \omega_0 - 2) z^{-1}}
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\end{align}
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\end{subequations}
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A significant advantage of using analytical formulas for complementary filters is that key parameters such as \(\omega_0\) can be tuned in real-time, as illustrated in Figure \ref{fig:detail_control_cf_arch_tunable_params}.
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This real-time tunability allows rapid testing of different control bandwidths to evaluate performance and robustness characteristics.
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/detail_control_cf_arch_tunable_params.png}
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\caption{\label{fig:detail_control_cf_arch_tunable_params}Implemented digital complementary filters with parameter \(\omega_0\) that can be changed in real time}
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\end{figure}
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For many practical applications, first order complementary filters are not sufficient.
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Specifically, a slope of \(+2\) at low frequencies for the sensitivity transfer function (enabling accurate tracking of ramp inputs) and a slope of \(-2\) for the complementary sensitivity transfer function are often desired.
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For these cases, the complementary filters analytical formula in Equation \eqref{eq:detail_control_cf_2nd_order} is proposed.
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\begin{subequations}\label{eq:detail_control_cf_2nd_order}
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\begin{align}
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H_L(s) &= \frac{(1+\alpha) (\frac{s}{\omega_0})+1}{\left((\frac{s}{\omega_0})+1\right) \left((\frac{s}{\omega_0})^2 + \alpha (\frac{s}{\omega_0}) + 1\right)}\\
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H_H(s) &= \frac{(\frac{s}{\omega_0})^2 \left((\frac{s}{\omega_0})+1+\alpha\right)}{\left((\frac{s}{\omega_0})+1\right) \left((\frac{s}{\omega_0})^2 + \alpha (\frac{s}{\omega_0}) + 1\right)}
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\end{align}
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\end{subequations}
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The influence of parameters \(\alpha\) and \(\omega_0\) on the frequency response of these complementary filters is illustrated in Figure \ref{fig:detail_control_cf_analytical_effect}.
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The parameter \(\alpha\) primarily affects the damping characteristics near the crossover frequency as well as high and low frequency magnitudes, while \(\omega_0\) determines the frequency at which the transition between high-pass and low-pass behavior occurs.
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These filters can also be implemented in the digital domain with analytical formulas, preserving the ability to adjust \(\alpha\) and \(\omega_0\) in real-time.
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\begin{figure}[htbp]
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\begin{subfigure}{0.48\textwidth}
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\begin{center}
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\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_control_cf_analytical_effect_alpha.png}
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\end{center}
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\subcaption{\label{fig:detail_control_cf_analytical_effect_alpha}Effect of $\alpha$}
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\end{subfigure}
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\begin{subfigure}{0.48\textwidth}
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\begin{center}
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\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_control_cf_analytical_effect_w0.png}
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\end{center}
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\subcaption{\label{fig:detail_control_cf_analytical_effect_w0}Effect of $\omega_0$}
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\end{subfigure}
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\caption{\label{fig:detail_control_cf_analytical_effect}Shape of proposed analytical complementary filters. Effect of \(\alpha\) (\subref{fig:detail_control_cf_analytical_effect_alpha}) and \(\omega_0\) (\subref{fig:detail_control_cf_analytical_effect_w0}) are shown.}
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\end{figure}
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\section{Numerical Example}
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\label{ssec:detail_control_cf_simulations}
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To implement the proposed control architecture in practice, the following procedure is proposed:
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\begin{enumerate}
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\item Identify the plant to be controlled to obtain the plant model \(G\).
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\item Design the weighting function \(w_I\) such that all possible plants \(G^\prime\) are contained within the uncertainty set \(\Pi_i\).
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\item Translate performance requirements into upper bounds on the complementary filters as explained in Section \ref{ssec:detail_control_cf_trans_perf}.
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\item Design the weighting functions \(w_H\) and \(w_L\) and generate the complementary filters using \(\mathcal{H}_{\infty}\text{-synthesis}\) as described in Section \ref{ssec:detail_control_sensor_hinf_method}.
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If the synthesis fails to produce filters satisfying the defined upper bounds, either revise the requirements or develop a more accurate model \(G\) that will allow for a smaller \(w_I\).
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For simpler cases, the analytical formulas for complementary filters presented in Section \ref{ssec:detail_control_cf_analytical_complementary_filters} can be employed.
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\item If \(K(s) = H_H^{-1}(s) G^{-1}(s)\) is not proper, add low-pass filters with sufficiently high corner frequencies to ensure realizability.
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\end{enumerate}
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To evaluate this control architecture, a simple test model representative of many synchrotron positioning stages is utilized (Figure \ref{fig:detail_control_cf_test_model}).
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In this model, a payload with mass \(m\) is positioned on top of a stage.
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The objective is to accurately position the sample relative to the X-ray beam.
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The relative position \(y\) between the payload and the X-ray is measured, which typically involves measuring the relative position between the focusing optics and the sample.
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Various disturbance forces affect positioning stability, including stage vibrations \(d_w\) and direct forces applied to the sample \(d_F\) (such as cable forces).
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The positioning stage itself is characterized by stiffness \(k\), internal damping \(c\), and a controllable force \(F\).
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The model of the plant \(G(s)\) from actuator force \(F\) to displacement \(y\) is described by Equation \eqref{eq:detail_control_cf_test_plant_tf}.
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\begin{equation}\label{eq:detail_control_cf_test_plant_tf}
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G(s) = \frac{1}{m s^2 + c s + k}, \quad m = \SI{20}{\kg},\ k = 1\si{\N/\mu\m},\ c = 10^2\si{\N\per(\m\per\s)}
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\end{equation}
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The plant dynamics include uncertainties related to limited support compliance, unmodeled flexible dynamics and payload dynamics.
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These uncertainties are represented using a multiplicative input uncertainty weight \eqref{eq:detail_control_cf_test_plant_uncertainty}, which specifies the magnitude of uncertainty as a function of frequency.
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\begin{equation}\label{eq:detail_control_cf_test_plant_uncertainty}
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w_I(s) = 10 \cdot \frac{(s+100)^2}{(s+1000)^2}
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\end{equation}
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Figure \ref{fig:detail_control_cf_bode_plot_mech_sys} illustrates both the nominal plant dynamics and the complete set of possible plants \(\Pi_i\) encompassed by the uncertainty model.
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\begin{figure}[htbp]
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\begin{subfigure}{0.3\textwidth}
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\begin{center}
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\includegraphics[scale=1,scale=1]{figs/detail_control_cf_test_model.png}
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\end{center}
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\subcaption{\label{fig:detail_control_cf_test_model}Test model}
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\end{subfigure}
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\begin{subfigure}{0.66\textwidth}
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\begin{center}
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\includegraphics[scale=1,scale=1]{figs/detail_control_cf_bode_plot_mech_sys.png}
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\end{center}
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\subcaption{\label{fig:detail_control_cf_bode_plot_mech_sys}Bode plot of $G(s)$ and associated uncertainty set}
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\end{subfigure}
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\caption{\label{fig:detail_control_cf_test_model_plant}Schematic of the test system (\subref{fig:detail_control_cf_test_model}). Bode plot of the transfer function \(G(s)\) from \(F\) to \(y\) and the associated uncertainty set (\subref{fig:detail_control_cf_bode_plot_mech_sys}).}
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\end{figure}
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\subsubsection{Requirements and choice of complementary filters}
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As discussed in Section \ref{ssec:detail_control_cf_trans_perf}, nominal performance requirements can be expressed as upper bounds on the shape of the complementary filters.
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For this example, the requirements are:
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\begin{itemize}
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\item track ramp inputs (i.e. constant velocity scans) with zero steady-state error: a \(+2\) slope at low frequencies for the magnitude of the sensitivity function \(|S(j\omega)|\) is required
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\item filtering of measurement noise above \(\SI{300}{Hz}\), where sensor noise is significant (requiring a filtering factor of approximately 100 above this frequency)
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\item maximizing disturbance rejection
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\end{itemize}
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Additionally, robust stability must be ensured, requiring the closed-loop system to remain stable despite the dynamic uncertainties modeled by \(w_I\).
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This condition is satisfied when the magnitude of the low-pass complementary filter \(|H_L(j\omega)|\) remains below the inverse of the uncertainty weight magnitude \(|w_I(j\omega)|\), as expressed in Equation \eqref{eq:detail_control_cf_condition_robust_stability}.
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Robust performance is achieved when both nominal performance and robust stability conditions are simultaneously satisfied.
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All requirements imposed on \(H_L\) and \(H_H\) are visualized in Figure \ref{fig:detail_control_cf_specs_S_T}.
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While \(\mathcal{H}_\infty\text{-synthesis}\) could be employed to design the complementary filters, analytical formulas were used for this relatively simple example.
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The second-order complementary filters from Equation \eqref{eq:detail_control_cf_2nd_order} were selected with parameters \(\alpha = 1\) and \(\omega_0 = 2\pi \cdot 20\,\text{Hz}\).
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There magnitudes are displayed in Figure \ref{fig:detail_control_cf_specs_S_T}, confirming that these complementary filters are fulfilling the specifications.
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\begin{figure}[htbp]
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\begin{subfigure}{0.48\textwidth}
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\begin{center}
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\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_control_cf_specs_S_T.png}
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\end{center}
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\subcaption{\label{fig:detail_control_cf_specs_S_T}Specifications and complementary filters}
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\end{subfigure}
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\begin{subfigure}{0.48\textwidth}
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\begin{center}
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\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_control_cf_bode_Kfb.png}
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\end{center}
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\subcaption{\label{fig:detail_control_cf_bode_Kfb}Bode plot of $K(s) \cdot H_L(s)$}
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\end{subfigure}
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\caption{\label{fig:detail_control_cf_specs_S_T_obtained_filters}Performance requirement and complementary filters used (\subref{fig:detail_control_cf_specs_S_T}). Obtained controller from the complementary filters and the plant inverse is shown in (\subref{fig:detail_control_cf_bode_Kfb}).}
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\end{figure}
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\subsubsection{Controller analysis}
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The controller to be implemented takes the form \(K(s) = \tilde{G}^{-1}(s) H_H^{-1}(s)\), where \(\tilde{G}^{-1}(s)\) represents the plant inverse, which must be both stable and proper.
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To ensure properness, low-pass filters with high corner frequencies are added as shown in Equation \eqref{eq:detail_control_cf_test_plant_inverse}.
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\begin{equation}\label{eq:detail_control_cf_test_plant_inverse}
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\tilde{G}^{-1}(s) = \frac{m s^2 + c s + k}{1 + \frac{s}{2\pi \cdot 1000} + \left( \frac{s}{2\pi \cdot 1000} \right)^2}
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\end{equation}
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The Bode plot of the controller multiplied by the complementary low-pass filter, \(K(s) \cdot H_L(s)\), is presented in Figure \ref{fig:detail_control_cf_bode_Kfb}.
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The frequency response reveals several important characteristics:
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\begin{itemize}
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\item The presence of two integrators at low frequencies, enabling accurate tracking of ramp inputs
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\item A notch at the plant resonance frequency (arising from the plant inverse)
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\item A lead component near the control bandwidth of approximately 20 Hz, enhancing stability margins
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\end{itemize}
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\subsubsection{Robustness and Performance analysis}
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Robust stability is assessed using the Nyquist plot shown in Figure \ref{fig:detail_control_cf_nyquist_robustness}.
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Even when considering all possible plants within the uncertainty set, the Nyquist plot remains sufficiently distant from the critical point \((-1,0)\), indicating robust stability with adequate margins.
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Performance is evaluated by examining the closed-loop sensitivity and complementary sensitivity transfer functions, as illustrated in Figure \ref{fig:detail_control_cf_robust_perf}.
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It is shown that the sensitivity transfer function achieves the desired \(+2\) slope at low frequencies and that the complementary sensitivity transfer function nominally provides the wanted noise filtering.
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\begin{figure}[htbp]
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\begin{subfigure}{0.49\textwidth}
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\begin{center}
|
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\includegraphics[scale=1,scale=0.8]{figs/detail_control_cf_nyquist_robustness.png}
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\end{center}
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\subcaption{\label{fig:detail_control_cf_nyquist_robustness}Robust Stability}
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\end{subfigure}
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\begin{subfigure}{0.49\textwidth}
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\begin{center}
|
|
\includegraphics[scale=1,scale=0.8]{figs/detail_control_cf_robust_perf.png}
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\end{center}
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\subcaption{\label{fig:detail_control_cf_robust_perf}Nominal and Robust performance}
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\end{subfigure}
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\caption{\label{fig:detail_control_cf_simulation_results}Validation of Robust stability with the Nyquist plot (\subref{fig:detail_control_cf_nyquist_robustness}) and validation of the nominal and robust performance with the magnitude of the closed-loop transfer functions (\subref{fig:detail_control_cf_robust_perf})}
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\end{figure}
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\section*{Conclusion}
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In this section, a control architecture in which complementary filters are used for closed-loop shaping has been presented.
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This approach differs from traditional open-loop shaping in that no controller is manually designed; rather, appropriate complementary filters are selected to achieve the desired closed-loop behavior.
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The method shares conceptual similarities with mixed-sensitivity \(\mathcal{H}_{\infty}\text{-synthesis}\), as both approaches aim to shape closed-loop transfer functions, but with notable distinctions in implementation and complexity.
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While \(\mathcal{H}_{\infty}\text{-synthesis}\) offers greater flexibility and can be readily generalized to MIMO plants, the presented approach provides a simpler alternative that requires minimal design effort.
|
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Implementation only necessitates extracting a model of the plant and selecting appropriate analytical complementary filters, making it particularly interesting for applications where simplicity and intuitive parameter tuning are valued.
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Due to time constraints, an extensive literature review comparing this approach with similar existing architectures, such as Internal Model Control \cite{saxena12_advan_inter_model_contr_techn}, was not conducted.
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Consequently, it remains unclear whether the proposed architecture offers significant advantages over existing methods in the literature.
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The control architecture has been presented for SISO systems, but can be applied to MIMO systems when sufficient decoupling is achieved.
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It will be experimentally validated with the NASS during the experimental phase.
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\bibliography{dehaeze26_control}
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\end{document}
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