Add conclusion of the "detail design phase"
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@ -7412,7 +7412,7 @@ Based on this relationship, the present work introduces an approach to designing
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**** Sensor Fusion and Complementary Filters Requirements
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<<ssec:detail_control_sensor_fusion_requirements>>
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****** Sensor Fusion Architecture :ignore:
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***** Sensor Fusion Architecture :ignore:
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A general sensor fusion architecture using complementary filters is shown in Figure ref:fig:detail_control_sensor_fusion_overview, where multiple sensors (in this case two) measure the same physical quantity $x$.
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The sensor output signals $\hat{x}_1$ and $\hat{x}_2$ represent estimates of $x$.
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@ -7428,7 +7428,7 @@ The complementary property of filters $H_1(s)$ and $H_2(s)$ requires that the su
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H_1(s) + H_2(s) = 1
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\end{equation}
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****** Sensor Models and Sensor Normalization
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***** Sensor Models and Sensor Normalization
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To analyze sensor fusion architectures, appropriate sensor models are required.
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The model shown in Figure ref:fig:detail_control_sensor_model consists of a linear time invariant (LTI) system $G_i(s)$ representing the sensor dynamics and an input $n_i$ representing sensor noise.
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@ -7471,7 +7471,7 @@ The super sensor output $\hat{x}$ is therefore described by eqref:eq:detail_cont
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#+caption: Sensor fusion architecture with two normalized sensors.
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[[file:figs/detail_control_sensor_fusion_super_sensor.png]]
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****** Noise Sensor Filtering
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***** Noise Sensor Filtering
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First, consider the case where all sensors are perfectly normalized eqref:eq:detail_control_sensor_perfect_dynamics.
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The effects of imperfect normalization will be addressed subsequently.
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@ -7508,7 +7508,7 @@ However, sensors typically exhibit high noise levels in different frequency regi
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In such cases, to reduce the noise of the super sensor, $|H_1(j\omega)|$ should be minimized when $\Phi_{n_1}(\omega)$ exceeds $\Phi_{n_2}(\omega)$, and $|H_2(j\omega)|$ should be minimized when $\Phi_{n_2}(\omega)$ exceeds $\Phi_{n_1}(\omega)$.
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Therefore, by appropriately shaping the norm of the complementary filters, the noise of the super sensor can be minimized.
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****** Sensor Fusion Robustness
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***** Sensor Fusion Robustness
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In practical systems, sensor normalization is rarely perfect, and condition eqref:eq:detail_control_sensor_perfect_dynamics is not fully satisfied.
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To analyze such imperfections, a multiplicative input uncertainty is incorporated into the sensor dynamics (Figure ref:fig:detail_control_sensor_model_uncertainty).
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@ -7565,14 +7565,14 @@ As it is generally desired to limit the dynamical uncertainty of the super senso
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**** Complementary Filters Shaping
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<<ssec:detail_control_sensor_hinf_method>>
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****** Introduction :ignore:
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***** Introduction :ignore:
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As established in Section ref:ssec:detail_control_sensor_fusion_requirements, the super sensor's noise characteristics and robustness are directly dependent on the complementary filters' norm.
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A synthesis method enabling precise shaping of these norms would therefore offer substantial practical benefits.
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This section develops such an approach by formulating the design objective as a standard $\mathcal{H}_\infty$ optimization problem.
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The methodology for designing appropriate weighting functions (which specify desired complementary filter shape during synthesis) is examined in detail, and the efficacy of the proposed method is validated with a simple example.
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****** Synthesis Objective
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***** Synthesis Objective
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The primary objective is to shape the norms of two filters $H_1(s)$ and $H_2(s)$ while ensuring they maintain their complementary property as defined in eqref:eq:detail_control_sensor_comp_filter.
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This is equivalent to finding proper and stable transfer functions $H_1(s)$ and $H_2(s)$ that satisfy conditions eqref:eq:detail_control_sensor_hinf_cond_complementarity, eqref:eq:detail_control_sensor_hinf_cond_h1, and eqref:eq:detail_control_sensor_hinf_cond_h2.
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@ -7586,7 +7586,7 @@ Weighting transfer functions $W_1(s)$ and $W_2(s)$ are strategically selected to
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\end{align}
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\end{subequations}
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****** Shaping of Complementary Filters using $\mathcal{H}_\infty$ synthesis
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***** Shaping of Complementary Filters using $\mathcal{H}_\infty$ synthesis
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The synthesis objective can be expressed as a standard $\mathcal{H}_\infty$ optimization problem by considering the generalized plant $P(s)$ illustrated in Figure ref:fig:detail_control_sensor_h_infinity_robust_fusion_plant and mathematically described by eqref:eq:detail_control_sensor_generalized_plant.
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@ -7633,7 +7633,7 @@ Therefore, applying $\mathcal{H}_\infty\text{-synthesis}$ to the standard plant
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It should be noted that there exists only an implication (not an equivalence) between the $\mathcal{H}_\infty$ norm condition in eqref:eq:detail_control_sensor_hinf_problem and the initial synthesis objectives in eqref:eq:detail_control_sensor_hinf_cond_h1 and eqref:eq:detail_control_sensor_hinf_cond_h2.
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Consequently, the optimization may be somewhat conservative with respect to the set of filters on which it operates [[cite:&skogestad07_multiv_feedb_contr,Chap. 2.8.3]].
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****** Weighting Functions Design
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***** Weighting Functions Design
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Weighting functions play a crucial role during synthesis by specifying the maximum allowable norms for the complementary filters.
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The proper design of these weighting functions is essential for the successful implementation of the proposed $\mathcal{H}_\infty\text{-synthesis}$ approach.
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@ -7667,7 +7667,7 @@ The typical magnitude response of a weighting function generated using eqref:eq:
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\end{equation}
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#+end_minipage
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****** Validation of the proposed synthesis method
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***** Validation of the proposed synthesis method
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The proposed methodology for designing complementary filters is now applied to a simple example.
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Consider the design of two complementary filters $H_1(s)$ and $H_2(s)$ with the following requirements:
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@ -7979,7 +7979,7 @@ Depending on the symmetry present in the system, certain diagonal elements may e
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**** Jacobian Decoupling
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<<ssec:detail_control_decoupling_jacobian>>
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****** Jacobian Matrix
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***** Jacobian Matrix
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The Jacobian matrix $\bm{J}_{\{O\}}$ serves a dual purpose in the decoupling process: it converts strut velocity $\dot{\mathcal{L}}$ to payload velocity and angular velocity $\dot{\bm{\mathcal{X}}}_{\{O\}}$, and it transforms actuator forces $\bm{\tau}$ to forces/torque applied on the payload $\bm{\mathcal{F}}_{\{O\}}$, as expressed in equation eqref:eq:detail_control_decoupling_jacobian.
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@ -8007,7 +8007,7 @@ The transfer function from $\bm{\mathcal{F}}_{\{O\}$ to $\bm{\mathcal{X}}_{\{O\}
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The frame $\{O\}$ can be selected according to specific requirements, but the decoupling properties are significantly influenced by this choice.
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Two natural reference frames are particularly relevant: the center of mass and the center of stiffness.
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****** Center Of Mass
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***** Center Of Mass
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When the decoupling frame is located at the center of mass (frame $\{M\}$ in Figure ref:fig:detail_control_decoupling_model_test), the Jacobian matrix and its inverse are expressed as in eqref:eq:detail_control_decoupling_jacobian_CoM_inverse.
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@ -8065,7 +8065,7 @@ This phenomenon is illustrated in Figure ref:fig:detail_control_decoupling_model
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#+end_subfigure
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#+end_figure
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****** Center Of Stiffness
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***** Center Of Stiffness
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When the decoupling frame is located at the center of stiffness, the Jacobian matrix and its inverse are expressed as in eqref:eq:detail_control_decoupling_jacobian_CoK_inverse.
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@ -8122,7 +8122,7 @@ When a high-frequency force is applied at a point not aligned with the center of
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**** Modal Decoupling
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<<ssec:detail_control_decoupling_modal>>
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****** Theory :ignore:
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***** Theory :ignore:
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Modal decoupling represents an approach based on the principle that a mechanical system's behavior can be understood as a combination of contributions from various modes [[cite:&rankers98_machin]].
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To convert the dynamics in the modal space, the equation of motion are first written with respect to the center of mass eqref:eq:detail_control_decoupling_equation_motion_CoM.
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@ -8154,7 +8154,7 @@ The resulting decoupled system features diagonal elements each representing seco
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#+caption: Modal Decoupling Architecture
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[[file:figs/detail_control_decoupling_modal.png]]
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****** Example :ignore:
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***** Example :ignore:
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Modal decoupling was then applied to the test model.
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First, the eigenvectors $\bm{\Phi}$ of $\bm{M}_{\{M\}}^{-1}\bm{K}_{\{M\}}$ were computed eqref:eq:detail_control_decoupling_modal_eigenvectors.
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@ -8206,7 +8206,7 @@ Each of these diagonal elements corresponds to a specific mode, as shown in Figu
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**** SVD Decoupling
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<<ssec:detail_control_decoupling_svd>>
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****** Singular Value Decomposition
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***** Singular Value Decomposition
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Singular Value Decomposition (SVD) represents a powerful mathematical tool with extensive applications in data analysis [[cite:&brunton22_data, chapt. 1]] and multivariable control systems where it is particularly valuable for analyzing directional properties in multivariable systems [[cite:&skogestad07_multiv_feedb_contr]].
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@ -8219,7 +8219,7 @@ The SVD constitutes a unique matrix decomposition applicable to any complex matr
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where $\bm{U} \in \mathbb{C}^{n \times n}$ and $\bm{V} \in \mathbb{C}^{m \times m}$ are unitary matrices with orthonormal columns, and $\bm{\Sigma} \in \mathbb{R}^{n \times n}$ is a diagonal matrix with real, non-negative entries.
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For real matrices $\bm{X}$, the resulting $\bm{U}$ and $\bm{V}$ matrices are also real, making them suitable for decoupling applications.
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****** Decoupling using the SVD
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***** Decoupling using the SVD
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The procedure for SVD-based decoupling begins with identifying the system dynamics from inputs to outputs, typically represented as a Frequency Response Function (FRF), which yields a complex matrix $\bm{G}(\omega_i)$ for multiple frequency points $\omega_i$.
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A specific frequency is then selected for optimal decoupling, with the targeted crossover frequency $\omega_c$ often serving as an appropriate choice.
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@ -8244,7 +8244,7 @@ This information can be obtained either experimentally or derived from a model.
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While this approach ensures effective decoupling near the chosen frequency, it provides no guarantees regarding decoupling performance away from this frequency.
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Furthermore, the quality of decoupling depends significantly on the accuracy of the real approximation, potentially limiting its effectiveness for plants with high damping.
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****** Example
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***** Example
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Plant decoupling using the Singular Value Decomposition was then applied on the test model.
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A decoupling frequency of $\SI{100}{Hz}$ was used.
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@ -8388,7 +8388,7 @@ Finally, in Section ref:ssec:detail_control_cf_simulations, a numerical example
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**** Control Architecture
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<<ssec:detail_control_cf_control_arch>>
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****** Virtual Sensor Fusion
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***** 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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@ -8416,7 +8416,7 @@ In this arrangement, the physical plant is controlled at low frequencies, while
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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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****** Asymptotic behavior
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***** 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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@ -8455,21 +8455,21 @@ Hence, when the plant model closely approximates the actual dynamics, the closed
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**** Translating the performance requirements into the shape of the complementary filters
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<<ssec:detail_control_cf_trans_perf>>
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****** Introduction :ignore:
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***** Introduction :ignore:
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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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****** Nominal Stability (NS)
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***** 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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****** Nominal Performance (NP)
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***** 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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@ -8497,7 +8497,7 @@ Typically, maintaining $|S|_{\infty} \le 2$ ensures a gain margin of at least 2
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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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****** Robust Stability (RS)
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***** 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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@ -8539,7 +8539,7 @@ After algebraic manipulation, robust stability is guaranteed when the low-pass c
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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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****** Robust Performance (RP)
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***** 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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@ -8624,7 +8624,7 @@ These filters can also be implemented in the digital domain with analytical form
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**** Numerical Example
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<<ssec:detail_control_cf_simulations>>
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****** Procedure :ignore:
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***** Procedure :ignore:
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To implement the proposed control architecture in practice, the following procedure is proposed:
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@ -8636,7 +8636,7 @@ To implement the proposed control architecture in practice, the following proced
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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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5. 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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****** Plant :ignore:
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***** Plant :ignore:
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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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@ -8679,7 +8679,7 @@ Figure ref:fig:detail_control_cf_bode_plot_mech_sys illustrates both the nominal
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#+end_subfigure
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#+end_figure
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****** Requirements and choice of complementary filters
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***** 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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@ -8715,7 +8715,7 @@ There magnitudes are displayed in Figure ref:fig:detail_control_cf_specs_S_T, co
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#+end_subfigure
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#+end_figure
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****** Controller analysis
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***** 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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@ -8730,7 +8730,7 @@ The frequency response reveals several important characteristics:
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- A notch at the plant resonance frequency (arising from the plant inverse)
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- A lead component near the control bandwidth of approximately 20 Hz, enhancing stability margins
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****** Robustness and Performance analysis
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***** 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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@ -9384,6 +9384,33 @@ This rigorous methodology spanning requirement formulation, component selection,
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:END:
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<<sec:detail_conclusion>>
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In this chapter, a comprehensive approach to the detailed design of the nano-hexapod for the Nano Active Stabilization System has been presented.
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The design process was structured around four key aspects: geometry optimization, component design, control strategy refinement, and instrumentation selection.
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The geometry optimization began with a review of existing Stewart platform designs, followed by analytical modeling of the relationship between geometric parameters and performance characteristics.
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While cubic architectures are prevalent in the literature due to their purported advantages in decoupling and uniform stiffness, the analysis revealed that these benefits are more nuanced than commonly described.
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For the nano-hexapod application, struts were oriented more vertically than in a cubic configuration to address the stringent vertical performance requirements and to better match the micro-station's modal characteristics.
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For component optimization, a hybrid modeling methodology was used that combined finite element analysis with multi-body dynamics.
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This approach, validated experimentally using an Amplified Piezoelectric Actuator, enabled both detailed component-level optimization and efficient system-level simulation.
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Through this methodology, the APA300ML was selected as the optimal actuator, offering the necessary combination of stroke, stiffness, and force sensing capabilities required for the application.
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Similarly, the flexible joints were designed with careful consideration of bending and axial stiffness requirements, resulting in a design that balances competing mechanical demands.
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For control optimization, three critical challenges were addressed.
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First, the problem of optimally combining multiple sensors was investigated and was focused on the design of complementary filters for sensor fusion.
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A $\mathcal{H}_{\infty}\text{-synthesis}$ technique was formulated for designing complementary filters with precisely shaped magnitude responses.
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Second, various decoupling strategies for parallel manipulators were compared, filling a notable gap in current literature.
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Among the evaluated techniques (decentralized control, Jacobian decoupling, modal decoupling, and SVD decoupling), Jacobian decoupling was identified as the most suitable for the NASS due to its simplicity and ability to maintain physical interpretation of the decoupled plant's inputs and outputs.
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Third, a novel control architecture was developed that leverages complementary filters for direct shaping of closed-loop transfer functions.
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This framework, which will be validated during the experimental phase, offers an intuitive alternative to traditional methods by allowing designers to directly specify desired closed-loop characteristics in a simple and intuitive way.
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The instrumentation selection was guided by dynamic error budgeting, which established maximum acceptable noise specifications for each component.
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The selected components—including the IO131 ADC/DAC board, PD200 voltage amplifiers, and Vionic linear encoders—were then experimentally characterized to verify their performance.
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All components were found to meet or exceed their specifications, with the combined effect of all noise sources estimated to induce vertical sample vibrations of only $1.5\,\text{nm RMS}$, well below the $15\,\text{nm RMS}$ requirement.
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The outcome of this detailed design process is a nano-hexapod and associated instrumentation specifically tailored to the NASS applications.
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Following the completion of this design phase and the subsequent procurement of all specified components, the project progressed to the experimental validation stage, which forms the focus of the next chapter.
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* Experimental Validation
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<<chap:test>>
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\minitoc
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