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   \end{array}\]</p><p>A typical choice for <span>$M_r(\lambda)$</span> is an <span>$m_f \times m_f$</span>  diagonal and invertible TFM, which ensures that each residual <span>$r_i(t)$</span> is influenced only by the fault <span>$f_i(t)$</span>. This would allow the isolation of arbitrary combinations of up to <span>$m_f$</span> simultaneous faults.  The choice <span>$M_r(\lambda) = I_{m_f}$</span> targets the solution of an  <em>exact fault estimation problem</em> (EFEP).</p><p>For the synthesis of fault detection and isolation filters which solve the EMMP the function <a href="FDIsynthesis.html#FaultDetectionTools.emmsyn"><code>emmsyn</code></a> is available. This function can also address the solution of the EMMP with more general reference models (e.g., having components from the control inputs and/or disturbance inputs). </p><h3 id="Approximate-model-matching-problem-(AMMP}"><a class="docs-heading-anchor" href="#Approximate-model-matching-problem-(AMMP}">Approximate model-matching problem (AMMP}</a><a id="Approximate-model-matching-problem-(AMMP}-1"></a><a class="docs-heading-anchor-permalink" href="#Approximate-model-matching-problem-(AMMP}" title="Permalink"></a></h3><p>Let <span>$M_r(\lambda)$</span> be a given <span>$q\times m_f$</span> TFM of a stable reference model  specifying the desired input-output behavior from the faults to residuals as</p><p class="math-container">\[{\mathbf{r}}(\lambda) = M_r(\lambda) {\mathbf{f}}(\lambda). \]</p><p><strong>AMMP:</strong> Determine a stable residual generator <span>$Q(\lambda)$</span> and a  stable, diagonal, and invertible <span>$M(\lambda)$</span> such that </p><p class="math-container">\[\begin{array}{rl} (i) &amp; R_u(\lambda) = 0 \\ (ii) &amp; R_d(\lambda) = 0 \\
   (iii) &amp; R_f(\lambda) \approx M(\lambda)M_r(\lambda) \\
   (iv) &amp; \|R_w(\lambda) \| \approx 0 \;\; \text{with} \;\; R_w(\lambda) \;\; {\color{magenta} \text{stable}}
-  \end{array}\]</p><p>A criterion suitable to characterize the solution of approximate model-matching based syntheses is the  residual error norm</p><p class="math-container">\[J_3 = \big\| R(\lambda)- M_r(\lambda)\big\|_{\infty/2}, \]</p><p>where <span>$R(\lambda) = R_f(\lambda)$</span> and <span>$M_r(\lambda)$</span> the reference model (possibly updated). For more generality, this criterion can be defined with  <span>$R(\lambda) = [\, R_u(\lambda)\; R_d(\lambda)\; R_f(\lambda)\; R_w(\lambda) \,\,]$</span>, the resulting internal form, and <span>$M_r(\lambda)$</span> the desired reference model <span>$M_r(\lambda) = [\, M_{ru}(\lambda)\; M_{rd}(\lambda)\; M_{rf}(\lambda)\; M_{rw}(\lambda)\,]$</span>. When applied to the results computed by other synthesis approaches (e.g., to solve the AFDP or AFDIP), this criterion can be formulated as</p><p class="math-container">\[\widetilde J_3 = \big\| R_w(\lambda)\big\|_{\infty/2}, \]</p><p>which corresponds to assume that <span>$M(\lambda) = I$</span> and <span>$M_r(\lambda) = [\, R_u(\lambda)\; R_d(\lambda)\; R_f(\lambda)\; 0 \,]$</span> (i.e., a perfect matching of control, disturbance and fault channels is always achieved).</p><p>For the synthesis of fault detection and isolation filters which solve the AMMP the function <a href="FDIsynthesis.html#FaultDetectionTools.ammsyn"><code>ammsyn</code></a> is available. This function can also address the solution of the AMMP with more general reference models (e.g., having components from the control inputs, disturbance inputs and noise inputs). For the evaluation of the  model-matching performace the function <a href="FDIperformance.html#FaultDetectionTools.fdimmperf"><code>fdimmperf</code></a> is available. </p><h3 id="References"><a class="docs-heading-anchor" href="#References">References</a><a id="References-1"></a><a class="docs-heading-anchor-permalink" href="#References" title="Permalink"></a></h3><p>[1]   A. Varga, Solving Fault Diagnosis Problems – Linear Synthesis Techniques, Vol. 84 of Studies in Systems, Decision and Control, Springer International Publishing, 2017.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="index.html">« Home</a><a class="docs-footer-nextpage" href="MDbasics.html">Model detection basics »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Wednesday 24 January 2024 13:14">Wednesday 24 January 2024</span>. Using Julia version 1.10.0.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+  \end{array}\]</p><p>A criterion suitable to characterize the solution of approximate model-matching based syntheses is the  residual error norm</p><p class="math-container">\[J_3 = \big\| R(\lambda)- M_r(\lambda)\big\|_{\infty/2}, \]</p><p>where <span>$R(\lambda) = R_f(\lambda)$</span> and <span>$M_r(\lambda)$</span> the reference model (possibly updated). For more generality, this criterion can be defined with  <span>$R(\lambda) = [\, R_u(\lambda)\; R_d(\lambda)\; R_f(\lambda)\; R_w(\lambda) \,\,]$</span>, the resulting internal form, and <span>$M_r(\lambda)$</span> the desired reference model <span>$M_r(\lambda) = [\, M_{ru}(\lambda)\; M_{rd}(\lambda)\; M_{rf}(\lambda)\; M_{rw}(\lambda)\,]$</span>. When applied to the results computed by other synthesis approaches (e.g., to solve the AFDP or AFDIP), this criterion can be formulated as</p><p class="math-container">\[\widetilde J_3 = \big\| R_w(\lambda)\big\|_{\infty/2}, \]</p><p>which corresponds to assume that <span>$M(\lambda) = I$</span> and <span>$M_r(\lambda) = [\, R_u(\lambda)\; R_d(\lambda)\; R_f(\lambda)\; 0 \,]$</span> (i.e., a perfect matching of control, disturbance and fault channels is always achieved).</p><p>For the synthesis of fault detection and isolation filters which solve the AMMP the function <a href="FDIsynthesis.html#FaultDetectionTools.ammsyn"><code>ammsyn</code></a> is available. This function can also address the solution of the AMMP with more general reference models (e.g., having components from the control inputs, disturbance inputs and noise inputs). For the evaluation of the  model-matching performace the function <a href="FDIperformance.html#FaultDetectionTools.fdimmperf"><code>fdimmperf</code></a> is available. </p><h3 id="References"><a class="docs-heading-anchor" href="#References">References</a><a id="References-1"></a><a class="docs-heading-anchor-permalink" href="#References" title="Permalink"></a></h3><p>[1]   A. Varga, Solving Fault Diagnosis Problems – Linear Synthesis Techniques, Vol. 84 of Studies in Systems, Decision and Control, Springer International Publishing, 2017.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="index.html">« Home</a><a class="docs-footer-nextpage" href="MDbasics.html">Model detection basics »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Wednesday 24 January 2024 14:12">Wednesday 24 January 2024</span>. Using Julia version 1.10.0.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>