\documentclass[11pt]{vjmech} \usepackage{graphicx,subfigure} \usepackage{amsmath,amsxtra,amssymb,latexsym,amscd} \usepackage[mathscr]{eucal} \usepackage[subfigure,caption2]{ccaption} \usepackage{float} \usepackage{multirow} \usepackage{bm} \usepackage[colorlinks]{hyperref} \hypersetup{pdfborder=000, linkcolor=blue} \usepackage{cite} \usepackage{booktabs} \usepackage[section]{placeins} \usepackage{xcolor} \newcommand{\rev}[1]{#1} % \usepackage{draftwatermark} % --------------------------------------------------------------- % Convenience macros % --------------------------------------------------------------- \newcommand{\Phib}{\bm{\Phi}} \newcommand{\Fb}{\mathbf{F}} \newcommand{\pb}{\mathbf{p}} \newcommand{\Ib}{\mathbf{I}} \newcommand{\lndet}{\ln\det} \begin{document} \setlength{\baselineskip}{13truept} \title{\Large MAC-Penalised Fisher Information Criterion for Optimal Sensor Placement in Structural Health Monitoring of Lattice Tower Structures} \author{% \textbf{Zijun Yin$^{1,*}$}\\ $^1$\textit{Qidi College, Tongji University, Shanghai 200092, China}\\ \small{$^*$E-mail: \href{mailto:2552840@tongji.edu.cn}{2552840@tongji.edu.cn},\ Tel: +86\,15245072783}\\ } \maketitle \markboth{Z. Yin}{MAC-Penalised FIM for Optimal Sensor Placement} % --------------------------------------------------------------- \begin{abstract} Optimal sensor placement (OSP) is a critical step in structural health monitoring (SHM) of large civil structures, where the number of available measurement channels is limited relative to the structural degrees of freedom. Classical Effective Independence (EFI) methods maximise the Fisher Information Matrix (FIM) determinant but ignore inter-modal correlation, leading to spatially clustered sensors that cause severe modal aliasing in higher-frequency modes. This paper proposes a MAC-penalised FIM criterion that jointly maximises the $\ln\det(\Fb)$ information objective and suppresses the maximum off-diagonal Modal Assurance Criterion (MAC) coefficient. A step-wise Min-Max normalisation scheme resolves the multi-objective scale mismatch, and a Woodbury rank-1 downdate reduces the per-step cost to $\mathcal{O}(m^2)$. Numerical experiments on a 60-m three-dimensional lattice tower (44 nodes, 164 elements, 8 extracted modes) demonstrate that the proposed method reduces $\max_{i\neq j}|\mathrm{MAC}_{ij}|$ from 0.9974 (classical EFI) to \rev{0.7543 (a 24.4\,\% relative reduction), at the cost of a 15.8\,\% reduction in $\ln\det\Fb$ relative to EFI. At low-to-moderate SNR\,(10--20\,dB), MAC-FIM reconstruction accuracy is comparable to EFI; at high SNR\,(30\,dB) the Fisher-information sacrifice produces a higher mean RMSE, though with a substantially smaller inter-quartile range (IQR\,=\,0.0021 versus EFI's 0.0057), indicating greater trial-to-trial consistency. Moreover, MAC-FIM's 95th-percentile (worst-case) RMSE is 3.7\,\% lower than EFI's at SNR\,=\,10\,dB and 3.7\,\% lower at SNR\,=\,20\,dB, indicative of improved tail reliability at operationally relevant noise levels.}\bigskip\\ \textit{Keywords}: optimal sensor placement, Fisher information matrix, modal assurance criterion, structural health monitoring, lattice tower, Woodbury identity. \end{abstract} % =============================================================== \section{INTRODUCTION} % =============================================================== Structural health monitoring (SHM) of civil infrastructure relies on vibration measurements to track natural frequencies, mode shapes, and damping ratios over the operational life of a structure~\cite{Farrar2007}. The accuracy of modal identification is fundamentally constrained by the spatial information content of the sensor array; poorly placed sensors yield linearly dependent or near-degenerate measurement channels that inflate modal reconstruction error and confound closely spaced modes \rev{\cite{Kammer1991,Allemang2003}}. Optimal sensor placement (OSP) has therefore attracted sustained research attention over the past three decades. Kammer~\cite{Kammer1991} introduced the Effective Independence (EFI) method, which sequentially eliminates candidate degrees of freedom (DOFs) that contribute least to the independence of the mode-shape matrix as measured by the Fisher Information Matrix (FIM). Variants based on eigenvalue decomposition~\cite{Papadopoulos1998}, genetic algorithms~\cite{Worden1996}, and convex relaxation~\cite{Joshi2009} have since been proposed\rev{; dimensionality-reduction approaches based on Principal Component Analysis (PCA) have also been applied to select sensor subsets that capture the dominant variance directions of the mode-shape matrix~\cite{Papadopoulos1998}}. \rev{Despite this breadth of methods,} the fundamental tension between information maximisation and spatial diversity remains. The Modal Assurance Criterion (MAC)~\cite{Allemang2003} quantifies the cross-correlation between estimated mode shapes; off-diagonal values close to unity indicate that two modes are indistinguishable from the measurement set, rendering modal separation ill-conditioned. Classical EFI does not explicitly penalise such coupling, and on symmetric or near-symmetric structures the selected DOFs tend to cluster in regions where dominant modes share similar spatial patterns. \rev{Existing practice typically runs an EFI-based placement first and then evaluates MAC as a post-hoc quality diagnostic, leaving no mechanism to re-optimise the sensor configuration in response to excessive inter-modal coupling. The proposed method closes this gap by incorporating the MAC penalty \emph{directly into the greedy elimination score}, so that both the information objective and the modal-independence objective are driven towards their optima within a single forward pass. This joint optimisation is structurally distinct from two-step approaches and, as shown in Section~\ref{sec:results}, leads to configurations that no amount of post-hoc MAC checking can recover from an EFI solution.} \rev{Table~\ref{tab:novelty} contrasts the proposed method with the most closely related approaches, clarifying where each method optimises and whether information and modal-independence objectives are handled jointly or in sequence.} \begin{table}[htbp] \caption{\rev{Comparison of sensor placement methods by optimisation objective and strategy. ``Joint'' means both objectives enter the same elimination score; ``Sequential'' means one criterion is applied after the other.}} \label{tab:novelty} {\renewcommand\arraystretch{1.25} \setlength{\tabcolsep}{1.5mm} \begin{tabular}{p{3.0cm}p{1.6cm}p{2.2cm}p{2.0cm}p{1.7cm}} \toprule Method & FIM & MAC & Strategy & Update \\ \midrule EFI~\cite{Kammer1991} & $\ln\det\Fb$ & --- & Single-pass & Proj.\ matrix \\ MAC-only & --- & $\min$ MAC & Single-pass & --- \\ PCA~\cite{Papadopoulos1998} & Variance & --- & Single-pass & --- \\ EFI+post-hoc MAC & $\ln\det\Fb$ & diagnostic & Sequential & --- \\ \textbf{Proposed} & $\ln\det\Fb$ & $\min$ MAC & \textbf{Joint} & \textbf{Wood\-bury} \\ \bottomrule \end{tabular} } \end{table} This paper makes the following contributions: \begin{enumerate} \item A unified objective function $J(\mathcal{E}) = \ln\det(\Fb) - \lambda\,\max_{i\neq j}|\mathrm{MAC}_{ij}|$ that \rev{jointly} maximises Fisher information and suppresses inter-modal coupling \rev{in a single optimisation pass, in contrast to post-hoc MAC evaluation applied to a pre-computed EFI solution.} \item A step-wise Min-Max normalisation \rev{scheme} that makes $\lambda$ a dimensionless relative weight, resolving the order-of-magnitude scale mismatch between the two objectives\rev{, so that $\lambda\in[0,1]$ interpolates continuously from pure EFI ($\lambda=0$) to pure MAC minimisation ($\lambda=1$)}. \item A $\mathcal{O}(m^2)$ greedy sequential elimination algorithm employing the Woodbury matrix identity for efficient FIM inverse updates. \item Quantitative validation on a 3-D lattice tower model under Gaussian measurement noise at multiple SNR levels, including \rev{worst-case (95th-percentile) RMSE analysis indicating improved tail reliability at operationally relevant noise levels.} \end{enumerate} The remainder of the paper is organised as follows. Section~\ref{sec:method} presents the theoretical framework. Section~\ref{sec:setup} describes the finite element model and simulation protocol. Section~\ref{sec:results} reports and discusses the numerical results. Section~\ref{sec:conclusion} concludes. % =============================================================== \FloatBarrier \section{METHODOLOGY} \label{sec:method} % =============================================================== \subsection{Problem formulation} \subsubsection{Measurement model and Fisher Information Matrix} Consider a linear measurement model in which the response at the selected sensor DOFs is a linear combination of $m$ mode shapes corrupted by independent Gaussian noise: \begin{equation}\label{eq:meas_model} \tilde{\mathbf{y}} = \Phib_{\mathcal{E}}\,\bm{q} + \bm{\eta}, \qquad \bm{\eta}\sim\mathcal{N}(\mathbf{0},\,\sigma^2\mathbf{I}_{n_s}), \end{equation} where $\bm{q}\in\mathbb{R}^m$ is the unknown vector of modal coordinates, $\Phib_{\mathcal{E}}\in\mathbb{R}^{n_s\times m}$ is the mode-shape sub-matrix restricted to the active sensor set $\mathcal{E}\subseteq\{1,\ldots,N\}$, and $\sigma^2$ is the noise variance. Under this model the log-likelihood is \begin{equation}\label{eq:log_lik} \ell(\bm{q}) = -\frac{n_s}{2}\ln(2\pi\sigma^2) -\frac{1}{2\sigma^2}\|\tilde{\mathbf{y}}-\Phib_{\mathcal{E}}\bm{q}\|^2. \end{equation} The Fisher Information Matrix (FIM) for $\bm{q}$ is \begin{equation}\label{eq:FIM} \Fb(\mathcal{E}) = -\mathbb{E}\!\left[\frac{\partial^2\ell}{\partial\bm{q}\,\partial\bm{q}^{\top}}\right] = \frac{1}{\sigma^2}\,\Phib_{\mathcal{E}}^{\top}\Phib_{\mathcal{E}} \;\in\;\mathbb{R}^{m\times m}. \end{equation} Since $\sigma^2$ is \rev{the noise variance, which is assumed identical for all candidate sensor configurations and acts only as a common positive scalar multiplier on $\Fb$; it} does not affect the ranking of sensor configurations\rev{.} \rev{Without} loss of generality we \rev{therefore} set $\sigma^2=1$ and write $\Fb(\mathcal{E}) = \Phib_{\mathcal{E}}^{\top}\Phib_{\mathcal{E}}$. \rev{Note that $\bm{q}$ does not cancel from $\ell(\bm{q})$ in Eq.~\eqref{eq:log_lik}; it is $\partial^2\ell/\partial\bm{q}\,\partial\bm{q}^{\top} = -\Phib_{\mathcal{E}}^{\top}\Phib_{\mathcal{E}}/\sigma^2$ that is independent of $\bm{q}$, so the FIM in Eq.~\eqref{eq:FIM} is a constant matrix that depends only on the sensor configuration~$\mathcal{E}$.} Let $\bm{\Phi}\in\mathbb{R}^{N\times m}$ be the full mass-normalised mode-shape matrix with $N$ unconstrained degrees of freedom. By the Cramér--Rao lower bound, the covariance matrix of any unbiased estimator $\hat{\bm{q}}$ satisfies $\mathrm{Cov}(\hat{\bm{q}})\succeq\Fb^{-1}$. Maximising $\ln\det\Fb$ is therefore equivalent to minimising the geometric mean of the Cramér--Rao variances, or equivalently minimising the volume of the confidence ellipsoid in modal-coordinate space. This is the D-optimality criterion in the optimal experimental design literature, and it is the basis of the classical EFI method~\cite{Kammer1991}: \begin{equation} \mathcal{E}^{*}_{\mathrm{EFI}} = \arg\max_{\mathcal{E},\,|\mathcal{E}|=n_s}\ln\det\Fb(\mathcal{E}). \end{equation} \subsubsection{Modal Assurance Criterion} Although D-optimality controls the overall estimation uncertainty, it does not prevent the mode-shape columns of $\Phib_{\mathcal{E}}$ from becoming nearly collinear. Such collinearity is quantified by the Modal Assurance Criterion~\cite{Allemang2003}. The $(i,j)$-th MAC coefficient is defined as \begin{equation}\label{eq:MAC} \mathrm{MAC}_{ij}(\mathcal{E}) = \frac{\bigl(\bm{\phi}_{\mathcal{E},i}^{\top} \bm{\phi}_{\mathcal{E},j}\bigr)^{2}} {\bigl(\bm{\phi}_{\mathcal{E},i}^{\top} \bm{\phi}_{\mathcal{E},i}\bigr) \bigl(\bm{\phi}_{\mathcal{E},j}^{\top} \bm{\phi}_{\mathcal{E},j}\bigr)}, \quad i,j = 1,\ldots,m, \end{equation} where $\bm{\phi}_{\mathcal{E},i}$ denotes the $i$-th column of $\Phib_{\mathcal{E}}$. By the Cauchy--Schwarz inequality $\mathrm{MAC}_{ij}\in[0,1]$ for all $i,j$, and $\mathrm{MAC}_{ii}=1$. A value $\mathrm{MAC}_{ij}\approx 1$ for $i\neq j$ means that $\bm{\phi}_{\mathcal{E},i}\approx c\,\bm{\phi}_{\mathcal{E},j}$ for some scalar $c$, so the two modes produce virtually identical measurement patterns; any reconstruction procedure will conflate them regardless of the signal-to-noise ratio. Note that this collinearity is not captured by $\det\Fb$: it is entirely possible for $\Fb$ to be well-conditioned (large determinant) while two specific columns of $\Phib_{\mathcal{E}}$ are highly correlated, because the determinant reflects the product of all eigenvalues rather than the worst-case pairwise alignment. This motivates adding an explicit MAC penalty to the placement objective. \subsection{Proposed MAC-penalised FIM criterion} \subsubsection{Composite objective} We propose the composite objective \begin{equation}\label{eq:objective} J(\mathcal{E}) = \ln\det\Fb(\mathcal{E}) - \lambda\,\max_{i\neq j}|\mathrm{MAC}_{ij}(\mathcal{E})|, \end{equation} where $\lambda\geq 0$ is a penalty weight. This formulation explicitly couples the Fisher information objective with a MAC independence constraint, forcing the algorithm to trade a marginal gain in $\ln\det\Fb$ for a substantial reduction in inter-modal confusion. \subsubsection{Interpretation of $\lambda$} When $\lambda=0$, $J$ reduces to $\ln\det\Fb$ and the algorithm is identical to classical EFI. When $\lambda=0.5$, the normalised score in Eq.~\eqref{eq:score} gives equal weight to information retention and MAC reduction. For $\lambda>0.5$ the MAC term progressively dominates, and at $\lambda=1$ it becomes the sole objective; however, as shown in Sec.~\ref{sec:results}, $\ln\det\Fb$ degrades rapidly beyond $\lambda\approx 1$ without further MAC improvement, so practical operating points lie in $\lambda\in(0.5,\,1.0)$. The sensitivity of results to $\lambda$ is studied systematically in Sec.~\ref{sec:results}. \subsection{Sequential elimination with Min-Max normalisation} \subsubsection{Greedy backward elimination} Exact maximisation of Eq.~\eqref{eq:objective} over all $\binom{N_c}{n_s}$ subsets is NP-hard in general. We adopt a greedy backward elimination strategy: starting from the full candidate set of $N_c$ DOFs, we iteratively remove the DOF whose removal causes the smallest decrease in $J$, until $n_s$ sensors remain. \subsubsection{Closed-form log-determinant change} At elimination step $t$, let $\mathcal{A}^{(t)}$ denote the active set with current FIM $\Fb^{(t)}$. Removing DOF $k$ yields $\Fb^{(t+1)} = \Fb^{(t)} - \bm{\phi}_k\bm{\phi}_k^{\top}$, a rank-1 downdate. The change in log-determinant follows from the matrix determinant lemma: for any invertible matrix $\mathbf{A}$ and vectors $\mathbf{u},\mathbf{v}$, \begin{equation} \det(\mathbf{A} + \mathbf{u}\mathbf{v}^{\top}) = \det(\mathbf{A})\,(1 + \mathbf{v}^{\top}\mathbf{A}^{-1}\mathbf{u}). \end{equation} Applying this with $\mathbf{A}=\Fb^{(t)}$ and $\mathbf{u}=-\bm{\phi}_k$, $\mathbf{v}=\bm{\phi}_k$: \begin{equation} \det\!\left(\Fb^{(t+1)}\right) = \det\!\left(\Fb^{(t)}\right)\,\bigl(1 - h_k\bigr), \qquad h_k = \bm{\phi}_k^{\top}\bigl(\Fb^{(t)}\bigr)^{-1}\bm{\phi}_k, \end{equation} where $h_k\in[0,1)$ is the leverage score of DOF $k$. Note that $h_k$ coincides with the diagonal element $E_{kk}$ of the projection matrix $\mathbf{E}=\Phib(\Phib^{\top}\Phib)^{-1}\Phib^{\top}$ introduced by Kammer~\cite{Kammer1991}; setting $\lambda=0$ thus recovers the classical EFI elimination order exactly. Taking logarithms gives the closed-form log-determinant change: \begin{equation}\label{eq:delta_ld} \Delta\ell_k = \ln\det\Fb^{(t+1)} - \ln\det\Fb^{(t)} = \ln\bigl(1 - h_k\bigr) \;\leq\; 0. \end{equation} The DOF with the largest $\Delta\ell_k$ (closest to zero) causes the smallest information loss and is therefore the best candidate for removal. \subsubsection{MAC metric at each step} For each candidate removal $k$, the post-removal MAC penalty is \begin{equation}\label{eq:mac_k} \mu_k = \max_{i\neq j} \bigl|\mathrm{MAC}_{ij}\!\left(\mathcal{A}^{(t)}\setminus\{k\}\right)\bigr|, \end{equation} evaluated by recomputing the MAC matrix of the surviving set $\mathcal{A}^{(t)}\setminus\{k\}$ using Eq.~\eqref{eq:MAC}. A small $\mu_k$ indicates that removing DOF $k$ leaves the remaining sensors well-separated in modal space. \subsubsection{Scale mismatch and Min-Max normalisation} The two raw metrics lie on incommensurable scales: $\Delta\ell_k$ is typically in the range $[-0.05, -10^{-4}]$ for well-conditioned configurations, whereas $\mu_k\in[0,1]$. Forming a weighted combination $\Delta\ell_k - \lambda\mu_k$ with a fixed $\lambda$ would therefore be dominated by whichever term happens to be larger in absolute value, making $\lambda$ impossible to interpret as a genuine trade-off weight. We resolve this by applying step-wise Min-Max normalisation within each elimination step, mapping both metrics to $[0,1]$: \begin{equation}\label{eq:normalise} \tilde{\ell}_k = \frac{\Delta\ell_k - \min_j\Delta\ell_j} {\max_j\Delta\ell_j - \min_j\Delta\ell_j + \varepsilon}, \qquad \tilde{\mu}_k = \frac{\mu_k - \min_j\mu_j} {\max_j\mu_j - \min_j\mu_j + \varepsilon}, \end{equation} where $j$ ranges over all currently valid candidates and $\varepsilon=10^{-12}$ prevents division by zero. After normalisation $\tilde{\ell}_k=1$ identifies the candidate that, if removed, retains the most Fisher information, and $\tilde{\mu}_k=0$ identifies the one whose removal gives the lowest off-diagonal MAC. The combined elimination score is \begin{equation}\label{eq:score} s_k = (1-\lambda)\,\tilde{\ell}_k - \lambda\,\tilde{\mu}_k, \end{equation} and the DOF with the highest $s_k$ is removed. With the normalisation in Eq.~\eqref{eq:normalise}, $\lambda\in[0,1]$ acts as a genuine interpolation weight: $\lambda=0$ recovers classical EFI and $\lambda=1$ minimises only MAC coupling regardless of information loss. \subsection{Woodbury rank-1 downdate} \subsubsection{Derivation} Recomputing $\Fb^{-1}$ from scratch after each elimination requires an $m\times m$ matrix inversion costing $\mathcal{O}(m^3)$ per step, and $\mathcal{O}(N_c m^3)$ in total. The elimination update has a rank-1 structure: \begin{equation} \Fb_{\mathrm{new}} = \Fb - \bm{\phi}_k\bm{\phi}_k^{\top}. \end{equation} The Sherman--Morrison identity states that for invertible $\mathbf{A}$ and vectors $\mathbf{u},\mathbf{v}$ with $1+\mathbf{v}^{\top}\mathbf{A}^{-1}\mathbf{u}\neq 0$: \begin{equation} (\mathbf{A}+\mathbf{u}\mathbf{v}^{\top})^{-1} = \mathbf{A}^{-1} - \frac{\mathbf{A}^{-1}\mathbf{u}\,\mathbf{v}^{\top}\mathbf{A}^{-1}} {1 + \mathbf{v}^{\top}\mathbf{A}^{-1}\mathbf{u}}. \end{equation} Substituting $\mathbf{A}=\Fb$, $\mathbf{u}=-\bm{\phi}_k$, $\mathbf{v}=\bm{\phi}_k$, and defining $\mathbf{v}^{(k)}=\Fb^{-1}\bm{\phi}_k$: \begin{equation}\label{eq:woodbury} \Fb_{\mathrm{new}}^{-1} = \Fb^{-1} + \frac{\mathbf{v}^{(k)}\bigl(\mathbf{v}^{(k)}\bigr)^{\top}} {1 - h_k}, \qquad h_k = \bm{\phi}_k^{\top}\mathbf{v}^{(k)}. \end{equation} The vector $\mathbf{v}^{(k)}\in\mathbb{R}^m$ is computed with one matrix-vector product (cost $\mathcal{O}(m^2)$), and the outer product update is also $\mathcal{O}(m^2)$, reducing the per-step cost from $\mathcal{O}(m^3)$ to $\mathcal{O}(m^2)$ and the total algorithm complexity from $\mathcal{O}(N_c m^3)$ to $\mathcal{O}(N_c m^2)$. \rev{Table~\ref{tab:flops} gives the exact floating-point operation counts for each step, confirming the $\mathcal{O}(m^2)$ per-step bound.} \begin{table}[htbp] \caption{\rev{Floating-point operation (flop) count per elimination step for the na\"ive inversion and the Woodbury rank-1 downdate. Both operations on the right column cost $m^2$ flops, giving a total per-step cost of $2m^2 + \mathcal{O}(m)$ versus $m^3 + \mathcal{O}(m^2)$ for direct inversion.}} \label{tab:flops} {\renewcommand\arraystretch{1.25} {\tabcolsep = 4mm \begin{tabular}{lll} \toprule Operation & Na\"ive inversion & Woodbury update \\ \midrule Compute $\mathbf{v}^{(k)} = \Fb^{-1}\bm{\phi}_k$ & $m^2$ (mat-vec) & $m^2$ (mat-vec) \\ Update $\Fb^{-1}$ & $m^3$ (LU / Cholesky) & $m^2$ (rank-1 outer product) \\ Leverage score $h_k = \bm{\phi}_k^\top\mathbf{v}^{(k)}$ & --- & $m$ (dot product) \\ \midrule \textbf{Total per step} & $\boldsymbol{m^3 + \mathcal{O}(m^2)}$ & $\boldsymbol{2m^2 + \mathcal{O}(m)}$ \\ \textbf{Total ($N_c$ steps)} & $\mathcal{O}(N_c m^3)$ & $\mathcal{O}(N_c m^2)$ \\ \bottomrule \end{tabular}}} \end{table} \subsubsection{Numerical validity condition} The denominator $1-h_k$ in Eq.~\eqref{eq:woodbury} vanishes when $h_k\to 1$, which occurs when $\bm{\phi}_k$ lies in the column space of the remaining mode-shape rows. In this case, removing DOF $k$ would render $\Fb_{\mathrm{new}}$ singular (loss of rank), meaning that DOF $k$ is the unique provider of information about at least one mode and must not be eliminated. In the implementation, any candidate with $h_k\geq 1-10^{-10}$ is excluded from the elimination pool at that step. % =============================================================== \FloatBarrier \section{SIMULATION SETUP} \label{sec:setup} % =============================================================== \subsection{Three-dimensional lattice tower model} A tapered four-leg X-braced steel lattice tower is modelled in MATLAB using three-dimensional Euler--Bernoulli beam elements with 12 DOFs per node (three translations, three rotations at each end). The geometric and material parameters are summarised in Tab.~\ref{tab:fem_params}. The tower comprises 44 nodes and 164 elements, yielding 264 total DOFs; after clamping the four base nodes (24 fixed DOFs), $N = 240$ free DOFs remain. \begin{table}[htbp] \caption{FEM model parameters of the 3-D lattice tower.} \label{tab:fem_params} {\renewcommand\arraystretch{1.3} {\tabcolsep = 5mm \begin{tabular}{lll} \toprule Parameter & Value & Unit \\ \midrule Height & 60 & m \\ Number of storeys & 10 & --- \\ Base width (half) & 5.0 & m \\ Top width (half) & 2.0 & m \\ Young's modulus $E$ & $2.10\times10^{11}$ & Pa \\ Mass density $\rho$ & 7850 & kg/m$^3$ \\ Leg cross-section $A_{\ell}$ & $50\times10^{-4}$ & m$^2$ \\ Brace cross-section $A_{b}$ & $20\times10^{-4}$ & m$^2$ \\ Extracted modes $m$ & 8 & --- \\ \bottomrule \end{tabular}}} \end{table} The eigenvalue problem $\mathbf{K}\bm{\Phi} = \mathbf{M}\bm{\Phi}\bm{\Omega}^2$ is solved by dense \texttt{eig} decomposition; the first eight mass-normalised mode shapes are retained. \rev{The natural frequencies and modal classifications of all eight extracted modes are summarised in Tab.~\ref{tab:modal_freqs}. Modes $M_1$--$M_2$ form a closely spaced bending pair ($\Delta f = 0.0032$\,Hz), $M_3$--$M_4$ form a second bending pair with moderate frequency separation, and $M_6$--$M_8$ are three tightly clustered bending modes ($\Delta f < 0.011$\,Hz) near 4.65\,Hz. Mode $M_5$ is a distinct torsion-coupled mode at 3.98\,Hz.} \begin{table}[htbp] \caption{\rev{Natural frequencies and mode classifications of the first eight modes.}} \label{tab:modal_freqs} {\renewcommand\arraystretch{1.3} {\tabcolsep = 3mm \begin{tabular}{clll} \toprule Mode & Freq.\ (Hz) & Classification & Remarks \\ \midrule $M_1$ & 2.1498 & \rev{1st bending ($x$)} & \rev{Closely spaced pair ($\Delta f=0.003$\,Hz)} \\ $M_2$ & 2.1530 & \rev{1st bending ($y$)} & \\ $M_3$ & 3.2211 & \rev{2nd bending ($x$)} & \rev{Dominant EFI coupling pair} \\ $M_4$ & 3.3150 & \rev{2nd bending ($y$)} & \rev{max$|\mathrm{MAC}_{34}|=0.9974$ under EFI} \\ $M_5$ & 3.9800 & \rev{Torsion-coupled} & \rev{Isolated mode} \\ $M_6$ & 4.6520 & \rev{3rd bending ($x$)} & \rev{Tightly spaced triple ($\Delta f<0.011$\,Hz)} \\ $M_7$ & 4.6612 & \rev{3rd bending ($y$)} & \\ $M_8$ & 4.6619 & \rev{3rd bending (mixed)} & \\ \bottomrule \end{tabular}}} \end{table} Candidate DOFs are restricted to the $x$- and $y$-translational components (local indices 1 and 2 within each node block), giving $N_c = 80$ candidate measurement channels for 44 free-node levels after excluding base nodes. \subsection{Compared algorithms} Four placement strategies are evaluated: \begin{enumerate} \item \textbf{Random}: $n_s$ DOFs drawn uniformly at random (seed fixed for reproducibility). \item \textbf{Uniform}: DOFs sampled at equal spacing along the sorted candidate index list. \item \textbf{Classical EFI}: standard sequential elimination ($\lambda = 0$) as in Kammer~\cite{Kammer1991}. \item \textbf{MAC-FIM (Proposed)}: sequential elimination with $\lambda = 0.8$ and step-wise Min-Max normalisation (Eqs.~\eqref{eq:normalise}--\eqref{eq:score}). \end{enumerate} All iterative methods start from the full $N_c = 80$ candidate set and eliminate down to $n_s = 12$ sensors. \subsection{Noise simulation} Modal reconstruction under noise is assessed via Monte Carlo simulation. For each SNR level $\rho_{\mathrm{dB}}\in\{10,20,30\}\,\mathrm{dB}$, the noiseless observation $\mathbf{y} = \Phib_{\mathcal{E}}\bm{q}$ is corrupted by additive white Gaussian noise: \begin{equation}\label{eq:noise} \tilde{\mathbf{y}} = \mathbf{y} + \bm{\eta},\qquad \bm{\eta}\sim\mathcal{N}\!\left(\mathbf{0},\, \sigma^2\mathbf{I}\right),\qquad \sigma^2 = \frac{\|\mathbf{y}\|^2}{n_s\cdot 10^{\rho_{\mathrm{dB}}/10}}. \end{equation} The modal coordinate vector $\bm{q}$ is recovered by least-squares: $\hat{\bm{q}} = \Fb^{-1}\Phib_{\mathcal{E}}^{\top}\tilde{\mathbf{y}}$, and the reconstruction RMSE is \begin{equation}\label{eq:rmse} \mathrm{RMSE} = \frac{1}{m}\sum_{i=1}^{m} \min\bigl(|\hat{q}_i - q_i|,\,|\hat{q}_i + q_i|\bigr), \end{equation} where the $\min(\cdot)$ accounts for mode-shape sign ambiguity. Results are averaged over $T = 100$ independent trials. % =============================================================== \FloatBarrier \section{RESULTS AND DISCUSSION} \label{sec:results} % =============================================================== \subsection{Sensor placement topology} Figure~\ref{fig:topology} shows the 3-D sensor locations for all four algorithms on the lattice tower. \rev{Classical EFI selects 12 DOFs across 10 unique nodes spanning four height levels ($z = 6, 18, 30, 60$\,m), with two nodes at the tower apex (60\,m) capturing the apex mode contributions. In contrast, MAC-FIM concentrates all 12 DOFs within the lowest three instrumented storeys ($z = 6, 12, 24$\,m), with no sensors above 24\,m (i.e.\ the lower 40\,\% of the tower height). Both methods select 10 \emph{unique} nodes for the 12 DOFs: in each case, two nodes each contribute both their $x$- and $y$-translational DOFs to the selected set, which is why the topology plots for panels (c) and (d) show 10 marker positions rather than 12. The MAC-FIM clustering arises because the MAC penalty drives the algorithm to retain sensors at the storeys where the mode-shape vectors of the critical mode pair ($M_3$, $M_4$) are most linearly independent in the sensor subspace; for this symmetric tower geometry, that maximum independence occurs in the lower storeys. As a consequence, MAC-FIM sacrifices spatial coverage of the upper tower and the torsion-coupled mode $M_5$ (3.98\,Hz), a trade-off that is quantified in the subsequent reconstruction accuracy analysis. \rev{From an engineering standpoint, the bottom-40\,\% clustering means that structural damage or stiffness changes occurring in the upper tower ($z > 24$\,m) would be detectable only indirectly through their influence on the lower-storey response, potentially reducing sensitivity to localised upper-chord damage. Practitioners operating in low-to-moderate noise environments where worst-case modal reconstruction reliability is the primary concern may accept this spatial trade-off, whereas applications requiring full-height damage localisation should introduce an explicit height-diversity constraint (e.g.\ a minimum-height-spread penalty or a hard constraint prescribing at least one sensor above a threshold elevation), which is left as a direction for future work.}} \begin{figure}[htbp] \centering \includegraphics[width=\textwidth]{Fig1_Tower_Topology} \caption{\textit{Sensor placement topology for the four algorithms on the 60-m three-dimensional lattice tower. Large filled circles: selected $n_s = 12$ sensors; small grey circles: candidate nodes. \rev{Panels~(a,b) (Random, Uniform): 12~DOFs at 12 distinct nodes (one DOF per node, so 12 visible markers). Panels~(c,d) (EFI, MAC-FIM): 12~DOFs at 10 unique nodes---two nodes each contribute both their $x$- and $y$-translational DOFs, so only 10 marker positions appear.}}} \label{fig:topology} \end{figure} \FloatBarrier \subsection{Modal independence: MAC analysis} Figure~\ref{fig:mac} presents the MAC matrices for all four placements with diagonal entries zeroed to enhance off-diagonal contrast. Under the classical EFI placement, the maximum off-diagonal coefficient reaches $\max_{i\neq j}|\mathrm{MAC}_{ij}| = 0.9974$, \rev{which occurs at the mode pair $(M_3, M_4)$ (the second paired bending modes at 3.221 and 3.315\,Hz). Additional notable cross-mode coupling under EFI appears at $(M_1, M_8)$ ($|\mathrm{MAC}_{18}| = 0.6323$) and $(M_2, M_7)$ ($|\mathrm{MAC}_{27}| = 0.4774$). Modes $M_1$--$M_2$ and $M_6$--$M_8$ have near-zero off-diagonal MAC values under EFI despite their tight frequency spacing, because the EFI-selected sensors happen to spatially distinguish those particular pairs.} The proposed MAC-FIM method reduces \rev{$|\mathrm{MAC}_{34}|$} from \rev{0.9974} to $0.7543$, a 24.4\% relative reduction, while the Uniform and Random baselines remain above 0.994. \begin{figure}[htbp] \centering \includegraphics[width=\textwidth]{Fig2_MAC_Heatmaps} \caption{\textit{MAC matrices (diagonal zeroed) for the four sensor placements. \rev{All four panels share a common colourbar range $[0,\,0.9974]$ (the global maximum off-diagonal MAC value, achieved by classical EFI at the $M_3$--$M_4$ pair), so colours are directly comparable across panels.}}} \label{fig:mac} \end{figure} The spatial origin of this improvement is illustrated in Fig.~\ref{fig:delta_mac}(a), which plots the difference $\Delta|\mathrm{MAC}_{ij}| = |\mathrm{MAC}_{ij}^{\mathrm{EFI}}| - |\mathrm{MAC}_{ij}^{\mathrm{MAC\text{-}FIM}}|$ for all mode pairs. Positive values (warm colours) indicate pairs for which MAC-FIM achieves better decoupling than EFI. \rev{The only mode pair showing a positive improvement is $(M_3, M_4)$ ($\Delta|\mathrm{MAC}_{34}| = +0.243$), the dominant coupling pair under EFI. All other pairs show zero or negative $\Delta|\mathrm{MAC}_{ij}|$: by concentrating sensors in the lower storeys to decouple $M_3$--$M_4$, MAC-FIM introduces modest new coupling at pairs that EFI leaves near zero (e.g.\ $|\mathrm{MAC}_{35}|$: $0.011\to 0.606$; $|\mathrm{MAC}_{14}|$: $0.000\to 0.315$). This is an inherent cost of the sensor concentration strategy.} Figure~\ref{fig:delta_mac}(b) summarises the $\max_{i\neq j}|\mathrm{MAC}_{ij}|$ values across all four algorithms. MAC-FIM is the only method that breaks the 0.80 barrier, \rev{achieved by specifically reducing the dominant $M_3$--$M_4$ coupling at the cost of concentrating sensors in the lower tower storeys.} \begin{figure}[htbp] \centering \subfigure[Delta-MAC heatmap]{% \includegraphics[width=0.52\textwidth]{Fig2b_DeltaMAC}} \hfill \subfigure[Maximum off-diagonal MAC per algorithm]{% \includegraphics[width=0.44\textwidth]{Fig2c_MAC_Bar}} \caption{\textit{(a) Difference heatmap $\Delta|\mathrm{MAC}_{ij}| = |\mathrm{MAC}^{\mathrm{EFI}}_{ij}| - |\mathrm{MAC}^{\mathrm{MAC\text{-}FIM}}_{ij}|$. Positive (red) cells: mode pairs where MAC-FIM achieves superior decoupling. (b) Maximum off-diagonal MAC coefficient per algorithm (lower is better); MAC-FIM reduces $\max|\mathrm{MAC}_{ij}|$ by 24.4\% relative to EFI.}} \label{fig:delta_mac} \end{figure} \FloatBarrier \subsection{FIM convergence} Figure~\ref{fig:convergence} tracks $\ln\det\Fb$ as DOFs are eliminated from the full 80-candidate set down to the target 12 sensors. Both EFI and MAC-FIM follow the same descending trajectory for most of the elimination path. The right panel zooms into the final 15 elimination steps: MAC-FIM sacrifices \rev{$\Delta\ln\det\Fb = 18.3$ (15.8\,\% relative to EFI's terminal value of $-115.96$, yielding $-134.24$) in exchange for the MAC reduction documented above---a deliberate and controlled trade-off governed by $\lambda$.} \begin{figure}[htbp] \centering \includegraphics[width=\textwidth]{Fig3_LogDet_Convergence} \caption{\textit{Evolution of $\ln\det(\Fb)$ during sequential elimination. Left: full elimination trajectory; right: tail zoom for the final 15 steps near the target sensor count (dashed vertical line).}} \label{fig:convergence} \end{figure} \FloatBarrier \subsection{Modal reconstruction accuracy} \rev{Figure~\ref{fig:rmse} shows the RMSE distributions from the Monte Carlo noise simulation ($T = 100$ trials per SNR level). The relative RMSE improvement of MAC-FIM over each baseline is defined as $\Delta\mathrm{RMSE}\% = (\overline{\mathrm{RMSE}}_{\mathrm{baseline}} - \overline{\mathrm{RMSE}}_{\mathrm{MAC\text{-}FIM}}) / \overline{\mathrm{RMSE}}_{\mathrm{baseline}} \times 100$, where $\overline{\mathrm{RMSE}}$ denotes the mean over the $T = 100$ trials. At SNR\,=\,10\,dB, noise dominates and all four methods perform similarly. MAC-FIM achieves a mean RMSE of 0.0806, marginally better than EFI ($+1.0\%$) and Random ($+0.7\%$), though slightly worse than Uniform ($-0.3\%$). At SNR\,=\,20\,dB the picture is similar; differences among methods remain within ${\pm}1\%$. At SNR\,=\,30\,dB, MAC-FIM records a \emph{higher} mean RMSE (0.0787) than all baselines: $-1.7\%$ relative to Random (0.0774), $-2.6\%$ relative to Uniform (0.0767), and $-5.3\%$ relative to classical EFI (0.0747). This high-SNR degradation is the direct and expected consequence of the 15.8\,\% Fisher-information sacrifice: when noise is small, the sensor information content dominates reconstruction accuracy, and EFI's higher $\ln\det\Fb$ translates into lower mean RMSE. It is therefore not a flaw of the proposed criterion but a predictable trade-off that practitioners must weigh against the 24.4\,\% MAC improvement obtained. However, MAC-FIM's inter-quartile range at SNR\,=\,30\,dB (IQR\,=\,0.0021) is substantially narrower than EFI's (0.0057), Random's (0.0055), and Uniform's (0.0059). This reduced spread indicates that MAC-FIM produces more consistent reconstruction quality across trials and across mode indices, which can be valuable in practice when worst-case rather than average performance is the design criterion.} \rev{To understand \emph{why} MAC reduction narrows the RMSE distribution, consider the least-squares estimator $\hat{\bm{q}} = \Fb^{-1}\Phib_{\mathcal{E}}^{\top}\tilde{\mathbf{y}}$. Under the noise model in Eq.~\eqref{eq:noise}, the estimation error $\bm{\varepsilon} = \hat{\bm{q}} - \bm{q}$ satisfies $\mathrm{Cov}(\bm{\varepsilon}) = \sigma^2 \Fb^{-1}$. When two mode-shape columns are nearly collinear ($\mathrm{MAC}_{ij} \approx 1$), the corresponding rows of $\Fb^{-1}$ are large and strongly positively correlated, so the per-mode errors $\varepsilon_i$ and $\varepsilon_j$ covary; a single noise realisation that inflates $|\varepsilon_i|$ simultaneously inflates $|\varepsilon_j|$, creating occasional extreme values in the aggregate RMSE. Reducing MAC coupling pushes the off-diagonal elements of $\Fb^{-1}$ towards zero, making the per-mode errors more statistically independent. Approximately, \begin{equation} \mathrm{Var}\!\left(\frac{1}{m}\sum_i|\varepsilon_i|\right) = \frac{1}{m^2}\!\left[\sum_i \mathrm{Var}|\varepsilon_i| + 2\sum_{i1.0$ provides diminishing MAC benefit at disproportionate information cost.} \begin{table}[htbp] \caption{\rev{Mean and 95th-percentile (p95) RMSE for all four methods across SNR levels ($T=100$ trials). Percentage in parentheses: MAC-FIM improvement over classical EFI (positive = MAC-FIM lower error).}} \label{tab:p95} {\renewcommand\arraystretch{1.25} {\tabcolsep = 3mm \begin{tabular}{llcccc} \toprule Metric & SNR & Random & Uniform & EFI & MAC-FIM \\ \midrule Mean RMSE & 10\,dB & 0.0812 & 0.0804 & 0.0815 & 0.0806 ($+1.0\%$) \\ & 20\,dB & 0.0807 & 0.0794 & 0.0801 & 0.0800 ($+0.1\%$) \\ & 30\,dB & 0.0774 & 0.0767 & 0.0747 & 0.0787 ($-5.3\%$) \\ \midrule p95 RMSE & 10\,dB & 0.0863 & 0.0859 & 0.0867 & \textbf{0.0835} ($+3.7\%$) \\ & 20\,dB & 0.0862 & 0.0852 & 0.0854 & \textbf{0.0822} ($+3.7\%$) \\ & 30\,dB & 0.0829 & 0.0826 & 0.0818 & 0.0818 ($0.0\%$) \\ \bottomrule \end{tabular}}} \end{table} % =============================================================== \FloatBarrier \section{CONCLUSION} \label{sec:conclusion} % =============================================================== This paper has presented a MAC-penalised FIM criterion for optimal sensor placement in structural health monitoring of 3-D lattice tower structures. The key methodological contributions are: (i) a composite objective function that balances Fisher information maximisation with inter-modal independence; (ii) step-wise Min-Max normalisation that makes $\lambda$ a dimensionless, scale-free penalty weight; and (iii) a Woodbury rank-1 downdate that keeps the per-step computational cost at $\mathcal{O}(m^2)$ \rev{by updating $\Fb^{-1}$ via a matrix--vector product followed by an outer-product correction (Eq.~\eqref{eq:woodbury}), reducing total complexity from $\mathcal{O}(N_c m^3)$ to $\mathcal{O}(N_c m^2)$}. \rev{Experiments on a 60-m four-leg X-braced steel tower with eight extracted modes and 12 sensors reveal the following findings. MAC-FIM reduces $\max_{i\neq j}|\mathrm{MAC}_{ij}|$ from 0.9974 (classical EFI) to 0.7543 (a 24.4\,\% relative reduction), specifically decoupling the dominant $M_3$--$M_4$ mode pair, at the cost of a 15.8\,\% reduction in $\ln\det\Fb$ relative to EFI. This MAC improvement comes at the cost of spatial coverage: MAC-FIM concentrates all sensors in the lower 40\,\% of the tower ($z \leq 24$\,m), and at high SNR (30\,dB) this Fisher-information sacrifice translates to a 5.3\,\% higher mean RMSE than classical EFI. At low-to-moderate SNR (10--20\,dB), where noise dominates over the information loss, MAC-FIM is comparable to or marginally better than EFI ($\leq+1.0\%$). Across all SNR levels, MAC-FIM's inter-quartile range is substantially narrower (IQR\,=\,0.0021 at 30\,dB) than EFI's (0.0057), indicating greater trial-to-trial consistency. \rev{Importantly, MAC-FIM's 95th-percentile (worst-case) RMSE is 3.7\,\% lower than EFI's at SNR\,=\,10\,dB and 3.7\,\% lower at SNR\,=\,20\,dB, indicative of improved tail reliability at operationally relevant noise levels; at high SNR (30\,dB) both methods converge to near-identical worst-case performance.} The $\lambda$-sensitivity study identifies a clear inflection point near $\lambda = 0.7$, providing actionable guidance for parameter selection in practice: operating points in $(0.7, 1.0)$ achieve the steepest MAC reduction for the least additional Fisher-information cost.} Future work will extend the algorithm to time-varying structures, incorporate model uncertainty via Bayesian Fisher information, and validate the placement strategy on experimental data from physical tower models. % --------------------------------------------------------------- \section*{ACKNOWLEDGMENT} The author thanks the Engineering Intelligence Fundamentals course at Qidi College, Tongji University, whose Engineering Internet domain project inspired this research. % --------------------------------------------------------------- \bibliographystyle{vjmech} \bibliography{references} \end{document}