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Write introduction and conclusion

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@ -939,32 +939,42 @@ Prefixes:
** DONE [#A] Finish writing "multiple sensor" control section ** DONE [#A] Finish writing "multiple sensor" control section
CLOSED: [2025-04-09 Wed 13:55] SCHEDULED: <2025-04-08 Tue> CLOSED: [2025-04-09 Wed 13:55] SCHEDULED: <2025-04-08 Tue>
** TODO [#A] Rework table that compares decoupling strategies
SCHEDULED: <2025-04-13 Sun>
** TODO [#B] Review of control for Stewart platforms? ** TODO [#B] Review of control for Stewart platforms?
[[file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/bibliography.org::*Control][Control]] [[file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/bibliography.org::*Control][Control]]
Or html version: https://research.tdehaeze.xyz/stewart-simscape/docs/bibliography.html Or html version: https://research.tdehaeze.xyz/stewart-simscape/docs/bibliography.html
** TODO [#C] Discuss different strategies? ** CANC [#C] Discuss different strategies?
CLOSED: [2025-04-13 Sun 10:40]
- State "CANC" from "TODO" [2025-04-13 Sun 10:40]
- Robust control - Robust control
- Adaptive control - Adaptive control
- etc... - etc...
* Introduction :ignore: * Introduction :ignore:
When controlling a MIMO system (specifically parallel manipulator such as the Stewart platform?) Three critical elements for the control of parallel manipulators such as the Nano-Hexapod were identified: effective utilization and combination of multiple sensors, appropriate plant decoupling strategies, and robust controller design for the decoupled system.
- [ ] *Should the quick review of Stewart platform control be here?* During the conceptual design phase of the NASS, pragmatic approaches were implemented for each of these elements.
In that case it should be possible to highlight three areas:
- use of multiple sensors
- decoupling strategy
- control optimization
Several considerations: The High Authority Control-Low Authority Control (HAC-LAC) architecture was selected for combining sensors.
- Section ref:sec:detail_control_sensor: How to most effectively use/combine multiple sensors Control was implemented in the frame of the struts, leveraging the inherent low-frequency decoupling of the plant where all decoupled elements exhibited similar dynamics, thereby simplifying the Single-Input Single-Output (SISO) controller design process.
- Section ref:sec:detail_control_decoupling: How to decouple a system For these decoupled plants, open-loop shaping techniques were employed to tune the individual controllers.
- Section ref:sec:detail_control_cf: How to design the controller
While these initial strategies proved effective in validating the NASS concept, this work explores alternative approaches with the potential to further enhance the performance.
Section ref:sec:detail_control_sensor examines different methods for combining multiple sensors, with particular emphasis on sensor fusion techniques that utilize complementary filters.
A novel approach for designing these filters is proposed, which allows optimization of the sensor fusion effectiveness.
Section ref:sec:detail_control_decoupling presents a comparative analysis of various decoupling strategies, including Jacobian decoupling, modal decoupling, and Singular Value Decomposition (SVD) decoupling.
Each method is evaluated in terms of its theoretical foundations, implementation requirements, and performance characteristics, providing insights into their respective advantages for different applications.
Finally, Section ref:sec:detail_control_cf addresses the challenge of controller design for decoupled plants.
A method for directly shaping closed-loop transfer functions using complementary filters is proposed, offering an intuitive approach to achieving desired performance specifications while ensuring robustness to plant uncertainty.
* Multiple Sensor Control * Multiple Sensor Control
:PROPERTIES: :PROPERTIES:
@ -2274,7 +2284,7 @@ exportFig('figs/detail_control_sensor_three_complementary_filters_results.pdf',
:UNNUMBERED: t :UNNUMBERED: t
:END: :END:
A new method for designing complementary filters using the $\mathcal{H}_\infty$ synthesis has been proposed. A new method for designing complementary filters using the $\mathcal{H}_\infty\text{-synthesis}$ has been proposed.
This approach allows shaping of the filter magnitudes through the use of weighting functions during synthesis. This approach allows shaping of the filter magnitudes through the use of weighting functions during synthesis.
This capability is particularly valuable in practice since the characteristics of the super sensor are directly linked to the complementary filters' magnitude. This capability is particularly valuable in practice since the characteristics of the super sensor are directly linked to the complementary filters' magnitude.
Consequently, typical sensor fusion objectives can be effectively translated into requirements on the magnitudes of the filters. Consequently, typical sensor fusion objectives can be effectively translated into requirements on the magnitudes of the filters.
@ -3269,27 +3279,24 @@ SVD decoupling can be implemented using measured data without requiring a model,
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". 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".
Several ways to design a controller to obtain a given performance while ensuring good robustness properties can be implemented. Several ways to design a controller to obtain a given performance while ensuring good robustness properties can be implemented.
# Add reference In some cases [[cite:&furutani04_nanom_cuttin_machin_using_stewar;&du14_piezo_actuat_high_precis_flexib;&yang19_dynam_model_decoup_contr_flexib]], "fixed" controller structures are utilized, such as PI and PID controllers, whose parameters are manually tuned.
In some cases, "fixed" controller structures are utilized, such as PI and PID controllers [[cite:&furutani04_nanom_cuttin_machin_using_stewar;&du14_piezo_actuat_high_precis_flexib;&yang19_dynam_model_decoup_contr_flexib]].
In such cases, the controller coefficients are manually tuned to obtain acceptable performance and robustness.
Another popular method is Open-Loop shaping, that was used during the conceptual phase after the plan was decoupled in the frame of the struts. Another popular method is Open-Loop shaping, which was used during the conceptual phase after the plan was decoupled in the frame of the struts.
The idea of open-loop shaping is to tune the controller (using a series of standard leads, lags, notches, low pass filters) such that the open-loop transfer function $G(s)K(s)$ is made according to specification (i.e. Open-loop shaping involves tuning the controller through a series of "standard" filters (leads, lags, notches, and low-pass filters) to shape the open-loop transfer function $G(s)K(s)$ according to desired specifications, including bandwidth, gain and phase margins, and gain at specific frequencies [[cite:&schmidt20_desig_high_perfor_mechat_third_revis_edition, chapt. 4.4.7]].
bandwidth, gain and phase margins, gain at a specific frequency, etc...) [[cite:&schmidt20_desig_high_perfor_mechat_third_revis_edition, chapt. 4.4.7]]. 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.
Open-Loop shaping is very popular because the open-loop transfer function depends linearly on the controller, making it relatively straightforward to tune the controller to achieve desired open-loop characteristics. Another key advantage is that controllers can be tuned directly from measured frequency response functions of the plant without requiring an explicit model.
Another key advantage is that controllers can be tuned directly from measured frequency response functions without requiring an explicit plant model.
However, the behavior (i.e. performance) of a feedback system is a function of closed-loop transfer functions [[cite:&skogestad07_multiv_feedb_contr, chapt. 3]]. However, the behavior (i.e. performance) of a feedback system is a function of closed-loop transfer functions.
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. 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:&skogestad07_multiv_feedb_contr, chapt. 3]].
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. 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.
In order to synthesize a controller that directly shapes the closed-loop transfer functions (and therefore the performance metric), $\mathcal{H}_\infty$ loop-shaping may be used [[cite:&skogestad07_multiv_feedb_contr]]. 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]].
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]]. 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]].
$\mathcal{H}_{\infty}$ synthesis has been applied for the Stewart platform [[cite:&jiao18_dynam_model_exper_analy_stewar]], but comparative studies with more simple decentralized controllers did not show large improvements [[cite:&thayer02_six_axis_vibrat_isolat_system;&hauge04_sensor_contr_space_based_six]]. $\mathcal{H}_{\infty}$ 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 negligible [[cite:&thayer02_six_axis_vibrat_isolat_system;&hauge04_sensor_contr_space_based_six]].
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). 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).
In Section ref:ssec:detail_control_cf_control_arch, the proposed control architecture including the complementary filters is presented. In Section ref:ssec:detail_control_cf_control_arch, the proposed control architecture is presented.
In Section ref:ssec:detail_control_cf_trans_perf, typical performance requirements are translated into the shape of the complementary filters. In Section ref:ssec:detail_control_cf_trans_perf, typical performance requirements are translated into the shape of the complementary filters.
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. 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.
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. 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.
@ -3322,10 +3329,9 @@ freqs = logspace(-1, 3, 1000);
<<ssec:detail_control_cf_control_arch>> <<ssec:detail_control_cf_control_arch>>
**** Virtual Sensor Fusion **** Virtual Sensor Fusion
The concept of using complementary filters in control architecture originates from sensor fusion techniques [[cite:&collette15_sensor_fusion_method_high_perfor]], where two sensors are combined using complementary filters. 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.
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. 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.
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 $H_L(s) + H_H(s) = 1$.
The 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 $H_L(s) + H_H(s) = 1$.
In this arrangement, the physical plant is controlled at low frequencies, while the plant model is utilized at high frequencies to enhance robustness. In this arrangement, the physical plant is controlled at low frequencies, while the plant model is utilized at high frequencies to enhance robustness.
#+begin_src latex :file detail_control_cf_arch.pdf #+begin_src latex :file detail_control_cf_arch.pdf
@ -3432,14 +3438,14 @@ Consequently, this structure is mathematically equivalent to the single-loop arc
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. 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.
\begin{equation}\label{eq:detail_control_cf_high_k} \begin{equation}\label{eq:detail_control_cf_high_k}
\lim_{k\to\infty} K(s) = \lim_{k\to\infty} \frac{k}{1+H_H(s) G(s) k} = \left( H_H(s) G(s) \right)^{-1} \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}
\end{equation} \end{equation}
If the resulting $K$ is improper, a low-pass filter with sufficiently high corner frequency can be added to ensure its causal realization. If the resulting $K$ is improper, a low-pass filter with sufficiently high corner frequency can be added to ensure its causal realization.
Furthermore, for $K$ to be stable, both $G$ and $H_H$ must be minimum phase transfer functions. Furthermore, for $K$ to be stable, both $G$ and $H_H$ must be minimum phase transfer functions.
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. 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.
The dynamics of this closed-loop system are described by eqref:eq:detail_control_cf_sf_cl_tf_K_inf. 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.
#+begin_src latex :file detail_control_cf_arch_class.pdf #+begin_src latex :file detail_control_cf_arch_class.pdf
\tikzset{block/.default={0.8cm}{0.6cm}} \tikzset{block/.default={0.8cm}{0.6cm}}
@ -3484,7 +3490,7 @@ The dynamics of this closed-loop system are described by eqref:eq:detail_control
\end{align} \end{align}
\end{subequations} \end{subequations}
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 described by eqref:eq:detail_control_cf_sf_cl_tf_K_inf_perfect. 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.
\begin{subequations}\label{eq:detail_control_cf_sf_cl_tf_K_inf_perfect} \begin{subequations}\label{eq:detail_control_cf_sf_cl_tf_K_inf_perfect}
\begin{alignat}{5} \begin{alignat}{5}
@ -3494,13 +3500,13 @@ At frequencies where the model accurately represents the physical plant ($G^{-1}
\end{subequations} \end{subequations}
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 $T = \frac{y}{n} = H_L$. 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 $T = \frac{y}{n} = H_L$.
Hence, when the plant model closely approximates the actual system, the closed-loop behavior becomes fully determined by the designed complementary filters, enabling direct translation of performance requirements into filter design. Hence, when the plant model closely approximates the actual system, the closed-loop transfer functions converge to the designed complementary filters, allowing direct translation of performance requirements into complementary filter design.
** Translating the performance requirements into the shapes of the complementary filters ** Translating the performance requirements into the shapes of the complementary filters
<<ssec:detail_control_cf_trans_perf>> <<ssec:detail_control_cf_trans_perf>>
**** Introduction :ignore: **** Introduction :ignore:
Performance specifications in feedback systems can be expressed as upper bounds on the magnitudes of closed-loop transfer functions such that the sensitivity $|S(j\omega)|$ and complementary sensitivity $|T(j\omega)|$ transfer functions [[cite:&bibel92_guidel_h]]. 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]].
The design of a controller $K(s)$ to achieve desired shapes of these closed-loop transfer functions is known as closed-loop shaping. The design of a controller $K(s)$ to obtain the desired shapes of these closed-loop transfer functions is known as closed-loop shaping.
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 $K(s)$. 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 $K(s)$.
Therefore, performance requirements must be translated into constraints on the shapes of these complementary filters. Therefore, performance requirements must be translated into constraints on the shapes of these complementary filters.
@ -3515,6 +3521,7 @@ Consequently, stable and minimum phase complementary filters must be employed.
**** Nominal Performance (NP) **** Nominal Performance (NP)
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. 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.
The weighting functions define the maximum magnitude of the closed-loop transfer functions as a function of frequency, effectively determining their "shape."
\begin{subequations}\label{eq:detail_control_cf_weights} \begin{subequations}\label{eq:detail_control_cf_weights}
\begin{align} \begin{align}
@ -3523,7 +3530,7 @@ Performance specifications can be formalized using weighting functions $w_H$ and
\end{align} \end{align}
\end{subequations} \end{subequations}
For the nominal system, where $S = H_H$ and $T = H_L$, nominal performance is ensured by satisfying eqref:eq:detail_control_cf_nominal_performance. 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.
\begin{equation}\label{eq:detail_control_cf_nominal_performance} \begin{equation}\label{eq:detail_control_cf_nominal_performance}
\Aboxed{\text{NP} \Longleftrightarrow {\begin{cases*} \Aboxed{\text{NP} \Longleftrightarrow {\begin{cases*}
@ -3532,23 +3539,20 @@ For the nominal system, where $S = H_H$ and $T = H_L$, nominal performance is en
\end{cases*}}} \end{cases*}}}
\end{equation} \end{equation}
Typical performance requirements can therefore be translated into constraints on the complementary filters.
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. 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.
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. 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.
The closed-loop bandwidth can be effectively limited by ensuring that $|T(j\omega)|$ remains below $\frac{1}{\sqrt{2}}$ at frequencies above the maximum desired bandwidth.
By carefully selecting the shapes of these complementary filters, nominal performance specifications can be directly addressed in an intuitive manner.
Classical stability margins (gain and phase margins) are also related to the maximum amplitude of the sensitivity transfer function. Classical stability margins (gain and phase margins) are also related to the maximum amplitude of the sensitivity transfer function.
Typically, maintaining $|S|_{\infty} \le 2$ ensures a gain margin of at least 2 and a phase margin of at least $\SI{29}{\degree}$. Typically, maintaining $|S|_{\infty} \le 2$ ensures a gain margin of at least 2 and a phase margin of at least $\SI{29}{\degree}$.
Therefore, by carefully selecting the shapes of the complementary filters, nominal performance specifications can be directly addressed in an intuitive manner.
**** Robust Stability (RS) **** Robust Stability (RS)
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. 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.
These discrepancies may arise from unmodeled dynamics or nonlinearities. These discrepancies may arise from unmodeled dynamics or nonlinearities.
To represent these model-plant differences, input multiplicative uncertainty as illustrated in Figure ref:fig:detail_control_cf_input_uncertainty is employed. To represent these model-plant differences, input multiplicative uncertainty as illustrated in Figure ref:fig:detail_control_cf_input_uncertainty is employed.
The set of possible plants $\Pi_i$ is described by eqref:eq:detail_control_cf_multiplicative_uncertainty. 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 $\Pi_i$.
With the weighting function $w_I$ selected such that all possible plants $G^\prime$ are contained within the set $\Pi_i$.
\begin{equation}\label{eq:detail_control_cf_multiplicative_uncertainty} \begin{equation}\label{eq:detail_control_cf_multiplicative_uncertainty}
\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 \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
@ -3616,8 +3620,7 @@ After algebraic manipulation, robust stability is guaranteed when the low-pass c
**** Robust Performance (RP) **** Robust Performance (RP)
Robust performance ensures that performance specifications eqref:eq:detail_control_cf_weights are met even as plant dynamics varies within specified bounds. 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.
This requires the performance condition to be valid for all possible plants in the defined uncertainty set $\Pi_i$:
\begin{equation}\label{eq:detail_control_cf_robust_perf_S} \begin{equation}\label{eq:detail_control_cf_robust_perf_S}
\text{RP} \Longleftrightarrow |w_H(j\omega) S(j\omega)| \le 1 \quad \forall G^\prime \in \Pi_I, \ \forall\omega \text{RP} \Longleftrightarrow |w_H(j\omega) S(j\omega)| \le 1 \quad \forall G^\prime \in \Pi_I, \ \forall\omega
@ -3706,7 +3709,7 @@ This real-time tunability allows rapid testing of different control bandwidths t
For many practical applications, first order complementary filters are not sufficient. For many practical applications, first order complementary filters are not sufficient.
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. 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.
For these cases, the second-order complementary filters presented in Equation eqref:eq:detail_control_cf_2nd_order are proposed. For these cases, the complementary filters analytical formula in Equation eqref:eq:detail_control_cf_2nd_order are proposed.
\begin{subequations}\label{eq:detail_control_cf_2nd_order} \begin{subequations}\label{eq:detail_control_cf_2nd_order}
\begin{align} \begin{align}
@ -3716,12 +3719,9 @@ For these cases, the second-order complementary filters presented in Equation eq
\end{subequations} \end{subequations}
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. 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.
The parameter $\alpha$ primarily affects the damping characteristics near the crossover frequency, while $\omega_0$ determines the frequency at which the transition between high-pass and low-pass behavior occurs. 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.
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. 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.
The presented analytical formulations offer an attractive balance between design simplicity and performance.
This capability to tune parameters in real-time is particularly valuable during commissioning of the controller.
#+begin_src matlab :exports none :results none #+begin_src matlab :exports none :results none
%% Analytical Complementary Filters - Effect of alpha %% Analytical Complementary Filters - Effect of alpha
freqs_study = logspace(-2, 2, 1000); freqs_study = logspace(-2, 2, 1000);
@ -3800,17 +3800,17 @@ exportFig('figs/detail_control_cf_analytical_effect_w0.pdf', 'width', 'half', 'h
<<ssec:detail_control_cf_simulations>> <<ssec:detail_control_cf_simulations>>
**** Procedure :ignore: **** Procedure :ignore:
To systematically apply the proposed control technique, the following procedure is recommended: To implement the proposed control architecture in practice, the following procedure is proposed:
1. Identify the plant to be controlled to obtain the plant model $G$. 1. Identify the plant to be controlled to obtain the plant model $G$.
2. Design the weighting function $w_I$ such that all possible plants $G^\prime$ are contained in the uncertainty set $\Pi_i$. 2. Design the weighting function $w_I$ such that all possible plants $G^\prime$ are contained within the uncertainty set $\Pi_i$.
3. Translate performance requirements into upper bounds on the complementary filters as explained in Section ref:ssec:detail_control_cf_trans_perf. 3. Translate performance requirements into upper bounds on the complementary filters as explained in Section ref:ssec:detail_control_cf_trans_perf.
4. 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. 4. 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.
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 $w_I$. 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 $w_I$.
For simpler cases, the analytical formulas for complementary filters presented in Section ref:ssec:detail_control_cf_analytical_complementary_filters can be employed. For simpler cases, the analytical formulas for complementary filters presented in Section ref:ssec:detail_control_cf_analytical_complementary_filters can be employed.
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. 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.
**** Plant **** Plant :ignore:
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). 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).
In this model, a payload with mass $m$ is positioned on top of a stage. In this model, a payload with mass $m$ is positioned on top of a stage.
@ -3823,11 +3823,9 @@ The positioning stage itself is characterized by stiffness $k$, internal damping
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. 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.
\begin{equation}\label{eq:detail_control_cf_test_plant_tf} \begin{equation}\label{eq:detail_control_cf_test_plant_tf}
G(s) = \frac{1}{m s^2 + c s + k} 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)}
\end{equation} \end{equation}
The parameter values are set to $m = \SI{20}{\kg}$, $k = 1\si{\N/\mu\m}$, and $c = 10^2\si{\N\per(\m\per\s)}$.
The plant dynamics include uncertainties related to limited support compliance, unmodeled flexible dynamics, payload dynamics, and other factors. The plant dynamics include uncertainties related to limited support compliance, unmodeled flexible dynamics, payload dynamics, and other factors.
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: 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:
@ -4173,6 +4171,23 @@ It will be experimentally validated with the NASS during the experimental phase.
:END: :END:
<<sec:detail_control_conclusion>> <<sec:detail_control_conclusion>>
In order to optimize the control of the Nano Active Stabilization System, several aspects of control theory were studied in this section.
Different approaches to combine sensors were compared in Section ref:sec:detail_control_sensor.
While High Authority Control-Low Authority Control (HAC-LAC) was successfully applied during the conceptual design phase, the focus of this work was extended to sensor fusion techniques where two or more sensors are combined using complementary filters.
It was demonstrated that the performance of such fusion depends significantly on the proper design of these complementary filters.
To address this challenge, a synthesis method based on $\mathcal{H}_\infty\text{-synthesis}$ was proposed, allowing for intuitive shaping of the complementary filters through weighting functions.
This approach enabled the translation of sensor fusion objectives directly into requirements on filter magnitudes.
For the NASS, while HAC-LAC remains a natural way to combine sensors, the potential benefits of sensor fusion merit further investigation.
Various decoupling strategies for parallel manipulators were examined in Section ref:sec:detail_control_decoupling, including decentralized control, Jacobian decoupling, modal decoupling, and Singular Value Decomposition (SVD) decoupling.
The main characteristics of each approach were highlighted, providing valuable insights into their respective strengths and limitations.
Among the examined methods, Jacobian decoupling was determined to be most appropriate for the NASS, as it provides straightforward implementation while preserving the physical meaning of inputs and outputs.
With the system successfully decoupled, attention shifts to designing appropriate SISO controllers for each decoupled direction.
A method for directly shaping closed-loop transfer functions is proposed, based on complementary filters that can be designed using either the $\mathcal{H}_\infty$ approach described earlier or through analytical formulas.
This straightforward approach enables intuitive parameter tuning while maintaining design simplicity.
Experimental validation of this method on the NASS will be conducted during the experimental tests on ID31.
* Bibliography :ignore: * Bibliography :ignore:
#+latex: \printbibliography[heading=bibintoc,title={Bibliography}] #+latex: \printbibliography[heading=bibintoc,title={Bibliography}]

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@ -1,4 +1,4 @@
% Created 2025-04-11 Fri 14:30 % Created 2025-04-13 Sun 11:32
% Intended LaTeX compiler: pdflatex % Intended LaTeX compiler: pdflatex
\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt} \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
@ -23,24 +23,23 @@
\tableofcontents \tableofcontents
\clearpage \clearpage
When controlling a MIMO system (specifically parallel manipulator such as the Stewart platform?) Three critical elements for the control of parallel manipulators such as the Nano-Hexapod were identified: effective utilization and combination of multiple sensors, appropriate plant decoupling strategies, and robust controller design for the decoupled system.
\begin{itemize} During the conceptual design phase of the NASS, pragmatic approaches were implemented for each of these elements.
\item[{$\square$}] \textbf{Should the quick review of Stewart platform control be here?}
In that case it should be possible to highlight three areas:
\begin{itemize}
\item use of multiple sensors
\item decoupling strategy
\item control optimization
\end{itemize}
\end{itemize}
Several considerations: The High Authority Control-Low Authority Control (HAC-LAC) architecture was selected for combining sensors.
\begin{itemize} Control was implemented in the frame of the struts, leveraging the inherent low-frequency decoupling of the plant where all decoupled elements exhibited similar dynamics, thereby simplifying the Single-Input Single-Output (SISO) controller design process.
\item Section \ref{sec:detail_control_sensor}: How to most effectively use/combine multiple sensors For these decoupled plants, open-loop shaping techniques were employed to tune the individual controllers.
\item Section \ref{sec:detail_control_decoupling}: How to decouple a system
\item Section \ref{sec:detail_control_cf}: How to design the controller While these initial strategies proved effective in validating the NASS concept, this work explores alternative approaches with the potential to further enhance the performance.
\end{itemize} Section \ref{sec:detail_control_sensor} examines different methods for combining multiple sensors, with particular emphasis on sensor fusion techniques that utilize complementary filters.
A novel approach for designing these filters is proposed, which allows optimization of the sensor fusion effectiveness.
Section \ref{sec:detail_control_decoupling} presents a comparative analysis of various decoupling strategies, including Jacobian decoupling, modal decoupling, and Singular Value Decomposition (SVD) decoupling.
Each method is evaluated in terms of its theoretical foundations, implementation requirements, and performance characteristics, providing insights into their respective advantages for different applications.
Finally, Section \ref{sec:detail_control_cf} addresses the challenge of controller design for decoupled plants.
A method for directly shaping closed-loop transfer functions using complementary filters is proposed, offering an intuitive approach to achieving desired performance specifications while ensuring robustness to plant uncertainty.
\chapter{Multiple Sensor Control} \chapter{Multiple Sensor Control}
\label{sec:detail_control_sensor} \label{sec:detail_control_sensor}
@ -555,7 +554,7 @@ Filter \(H_1(s)\) is defined using \eqref{eq:detail_control_sensor_h1_compl_h2_h
Figure \ref{fig:detail_control_sensor_three_complementary_filters_results} displays the three synthesized complementary filters (solid lines), confirming the successful synthesis. Figure \ref{fig:detail_control_sensor_three_complementary_filters_results} displays the three synthesized complementary filters (solid lines), confirming the successful synthesis.
\section*{Conclusion} \section*{Conclusion}
A new method for designing complementary filters using the \(\mathcal{H}_\infty\) synthesis has been proposed. A new method for designing complementary filters using the \(\mathcal{H}_\infty\text{-synthesis}\) has been proposed.
This approach allows shaping of the filter magnitudes through the use of weighting functions during synthesis. This approach allows shaping of the filter magnitudes through the use of weighting functions during synthesis.
This capability is particularly valuable in practice since the characteristics of the super sensor are directly linked to the complementary filters' magnitude. This capability is particularly valuable in practice since the characteristics of the super sensor are directly linked to the complementary filters' magnitude.
Consequently, typical sensor fusion objectives can be effectively translated into requirements on the magnitudes of the filters. Consequently, typical sensor fusion objectives can be effectively translated into requirements on the magnitudes of the filters.
@ -1104,26 +1103,24 @@ SVD decoupling can be implemented using measured data without requiring a model,
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''. 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''.
Several ways to design a controller to obtain a given performance while ensuring good robustness properties can be implemented. Several ways to design a controller to obtain a given performance while ensuring good robustness properties can be implemented.
In some cases, ``fixed'' controller structures are utilized, such as PI and PID controllers \cite{furutani04_nanom_cuttin_machin_using_stewar,du14_piezo_actuat_high_precis_flexib,yang19_dynam_model_decoup_contr_flexib}. In some cases \cite{furutani04_nanom_cuttin_machin_using_stewar,du14_piezo_actuat_high_precis_flexib,yang19_dynam_model_decoup_contr_flexib}, ``fixed'' controller structures are utilized, such as PI and PID controllers, whose parameters are manually tuned.
In such cases, the controller coefficients are manually tuned to obtain acceptable performance and robustness.
Another popular method is Open-Loop shaping, that was used during the conceptual phase after the plan was decoupled in the frame of the struts. Another popular method is Open-Loop shaping, which was used during the conceptual phase after the plan was decoupled in the frame of the struts.
The idea of open-loop shaping is to tune the controller (using a series of standard leads, lags, notches, low pass filters) such that the open-loop transfer function \(G(s)K(s)\) is made according to specification (i.e. Open-loop shaping involves tuning the controller through a series of ``standard'' filters (leads, lags, notches, and low-pass filters) to shape the open-loop transfer function \(G(s)K(s)\) according to desired specifications, including bandwidth, gain and phase margins, and gain at specific frequencies \cite[, chapt. 4.4.7]{schmidt20_desig_high_perfor_mechat_third_revis_edition}.
bandwidth, gain and phase margins, gain at a specific frequency, etc\ldots{}) \cite[, chapt. 4.4.7]{schmidt20_desig_high_perfor_mechat_third_revis_edition}. 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.
Open-Loop shaping is very popular because the open-loop transfer function depends linearly on the controller, making it relatively straightforward to tune the controller to achieve desired open-loop characteristics. Another key advantage is that controllers can be tuned directly from measured frequency response functions of the plant without requiring an explicit model.
Another key advantage is that controllers can be tuned directly from measured frequency response functions without requiring an explicit plant model.
However, the behavior (i.e. performance) of a feedback system is a function of closed-loop transfer functions \cite[, chapt. 3]{skogestad07_multiv_feedb_contr}. However, the behavior (i.e. performance) of a feedback system is a function of closed-loop transfer functions.
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. 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}.
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. 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.
In order to synthesize a controller that directly shapes the closed-loop transfer functions (and therefore the performance metric), \(\mathcal{H}_\infty\) loop-shaping may be used \cite{skogestad07_multiv_feedb_contr}. 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}.
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}. 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}.
\(\mathcal{H}_{\infty}\) synthesis has been applied for the Stewart platform \cite{jiao18_dynam_model_exper_analy_stewar}, but comparative studies with more simple decentralized controllers did not show large improvements \cite{thayer02_six_axis_vibrat_isolat_system,hauge04_sensor_contr_space_based_six}. \(\mathcal{H}_{\infty}\) 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 negligible \cite{thayer02_six_axis_vibrat_isolat_system,hauge04_sensor_contr_space_based_six}.
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). 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).
In Section \ref{ssec:detail_control_cf_control_arch}, the proposed control architecture including the complementary filters is presented. In Section \ref{ssec:detail_control_cf_control_arch}, the proposed control architecture is presented.
In Section \ref{ssec:detail_control_cf_trans_perf}, typical performance requirements are translated into the shape of the complementary filters. In Section \ref{ssec:detail_control_cf_trans_perf}, typical performance requirements are translated into the shape of the complementary filters.
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. 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.
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. 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.
@ -1131,10 +1128,9 @@ Finally, in Section \ref{ssec:detail_control_cf_simulations}, a numerical exampl
\label{ssec:detail_control_cf_control_arch} \label{ssec:detail_control_cf_control_arch}
\paragraph{Virtual Sensor Fusion} \paragraph{Virtual Sensor Fusion}
The concept of using complementary filters in control architecture originates from sensor fusion techniques \cite{collette15_sensor_fusion_method_high_perfor}, where two sensors are combined using complementary filters. 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.
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. 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.
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 \(H_L(s) + H_H(s) = 1\).
The 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 \(H_L(s) + H_H(s) = 1\).
In this arrangement, the physical plant is controlled at low frequencies, while the plant model is utilized at high frequencies to enhance robustness. In this arrangement, the physical plant is controlled at low frequencies, while the plant model is utilized at high frequencies to enhance robustness.
\begin{figure}[htbp] \begin{figure}[htbp]
@ -1160,14 +1156,14 @@ Consequently, this structure is mathematically equivalent to the single-loop arc
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}. 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}.
\begin{equation}\label{eq:detail_control_cf_high_k} \begin{equation}\label{eq:detail_control_cf_high_k}
\lim_{k\to\infty} K(s) = \lim_{k\to\infty} \frac{k}{1+H_H(s) G(s) k} = \left( H_H(s) G(s) \right)^{-1} \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}
\end{equation} \end{equation}
If the resulting \(K\) is improper, a low-pass filter with sufficiently high corner frequency can be added to ensure its causal realization. If the resulting \(K\) is improper, a low-pass filter with sufficiently high corner frequency can be added to ensure its causal realization.
Furthermore, for \(K\) to be stable, both \(G\) and \(H_H\) must be minimum phase transfer functions. Furthermore, for \(K\) to be stable, both \(G\) and \(H_H\) must be minimum phase transfer functions.
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. 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.
The dynamics of this closed-loop system are described by \eqref{eq:detail_control_cf_sf_cl_tf_K_inf}. 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}.
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
@ -1182,7 +1178,7 @@ The dynamics of this closed-loop system are described by \eqref{eq:detail_contro
\end{align} \end{align}
\end{subequations} \end{subequations}
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 described by \eqref{eq:detail_control_cf_sf_cl_tf_K_inf_perfect}. 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}.
\begin{subequations}\label{eq:detail_control_cf_sf_cl_tf_K_inf_perfect} \begin{subequations}\label{eq:detail_control_cf_sf_cl_tf_K_inf_perfect}
\begin{alignat}{5} \begin{alignat}{5}
@ -1192,11 +1188,11 @@ At frequencies where the model accurately represents the physical plant (\(G^{-1
\end{subequations} \end{subequations}
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 \(T = \frac{y}{n} = H_L\). 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 \(T = \frac{y}{n} = H_L\).
Hence, when the plant model closely approximates the actual system, the closed-loop behavior becomes fully determined by the designed complementary filters, enabling direct translation of performance requirements into filter design. Hence, when the plant model closely approximates the actual system, the closed-loop transfer functions converge to the designed complementary filters, allowing direct translation of performance requirements into complementary filter design.
\section{Translating the performance requirements into the shapes of the complementary filters} \section{Translating the performance requirements into the shapes of the complementary filters}
\label{ssec:detail_control_cf_trans_perf} \label{ssec:detail_control_cf_trans_perf}
Performance specifications in feedback systems can be expressed as upper bounds on the magnitudes of closed-loop transfer functions such that the sensitivity \(|S(j\omega)|\) and complementary sensitivity \(|T(j\omega)|\) transfer functions \cite{bibel92_guidel_h}. 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}.
The design of a controller \(K(s)\) to achieve desired shapes of these closed-loop transfer functions is known as closed-loop shaping. The design of a controller \(K(s)\) to obtain the desired shapes of these closed-loop transfer functions is known as closed-loop shaping.
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 \(K(s)\). 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 \(K(s)\).
Therefore, performance requirements must be translated into constraints on the shapes of these complementary filters. Therefore, performance requirements must be translated into constraints on the shapes of these complementary filters.
@ -1209,6 +1205,7 @@ Consequently, stable and minimum phase complementary filters must be employed.
\paragraph{Nominal Performance (NP)} \paragraph{Nominal Performance (NP)}
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. 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.
The weighting functions define the maximum magnitude of the closed-loop transfer functions as a function of frequency, effectively determining their ``shape.''
\begin{subequations}\label{eq:detail_control_cf_weights} \begin{subequations}\label{eq:detail_control_cf_weights}
\begin{align} \begin{align}
@ -1217,7 +1214,7 @@ Performance specifications can be formalized using weighting functions \(w_H\) a
\end{align} \end{align}
\end{subequations} \end{subequations}
For the nominal system, where \(S = H_H\) and \(T = H_L\), nominal performance is ensured by satisfying \eqref{eq:detail_control_cf_nominal_performance}. 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}.
\begin{equation}\label{eq:detail_control_cf_nominal_performance} \begin{equation}\label{eq:detail_control_cf_nominal_performance}
\Aboxed{\text{NP} \Longleftrightarrow {\begin{cases*} \Aboxed{\text{NP} \Longleftrightarrow {\begin{cases*}
@ -1226,22 +1223,19 @@ For the nominal system, where \(S = H_H\) and \(T = H_L\), nominal performance i
\end{cases*}}} \end{cases*}}}
\end{equation} \end{equation}
Typical performance requirements can therefore be translated into constraints on the complementary filters.
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. 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.
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. 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.
The closed-loop bandwidth can be effectively limited by ensuring that \(|T(j\omega)|\) remains below \(\frac{1}{\sqrt{2}}\) at frequencies above the maximum desired bandwidth.
By carefully selecting the shapes of these complementary filters, nominal performance specifications can be directly addressed in an intuitive manner.
Classical stability margins (gain and phase margins) are also related to the maximum amplitude of the sensitivity transfer function. Classical stability margins (gain and phase margins) are also related to the maximum amplitude of the sensitivity transfer function.
Typically, maintaining \(|S|_{\infty} \le 2\) ensures a gain margin of at least 2 and a phase margin of at least \(\SI{29}{\degree}\). Typically, maintaining \(|S|_{\infty} \le 2\) ensures a gain margin of at least 2 and a phase margin of at least \(\SI{29}{\degree}\).
Therefore, by carefully selecting the shapes of the complementary filters, nominal performance specifications can be directly addressed in an intuitive manner.
\paragraph{Robust Stability (RS)} \paragraph{Robust Stability (RS)}
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. 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.
These discrepancies may arise from unmodeled dynamics or nonlinearities. These discrepancies may arise from unmodeled dynamics or nonlinearities.
To represent these model-plant differences, input multiplicative uncertainty as illustrated in Figure \ref{fig:detail_control_cf_input_uncertainty} is employed. To represent these model-plant differences, input multiplicative uncertainty as illustrated in Figure \ref{fig:detail_control_cf_input_uncertainty} is employed.
The set of possible plants \(\Pi_i\) is described by \eqref{eq:detail_control_cf_multiplicative_uncertainty}. 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 \(\Pi_i\).
With the weighting function \(w_I\) selected such that all possible plants \(G^\prime\) are contained within the set \(\Pi_i\).
\begin{equation}\label{eq:detail_control_cf_multiplicative_uncertainty} \begin{equation}\label{eq:detail_control_cf_multiplicative_uncertainty}
\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 \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
@ -1276,8 +1270,7 @@ After algebraic manipulation, robust stability is guaranteed when the low-pass c
\end{equation} \end{equation}
\paragraph{Robust Performance (RP)} \paragraph{Robust Performance (RP)}
Robust performance ensures that performance specifications \eqref{eq:detail_control_cf_weights} are met even as plant dynamics varies within specified bounds. 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}.
This requires the performance condition to be valid for all possible plants in the defined uncertainty set \(\Pi_i\):
\begin{equation}\label{eq:detail_control_cf_robust_perf_S} \begin{equation}\label{eq:detail_control_cf_robust_perf_S}
\text{RP} \Longleftrightarrow |w_H(j\omega) S(j\omega)| \le 1 \quad \forall G^\prime \in \Pi_I, \ \forall\omega \text{RP} \Longleftrightarrow |w_H(j\omega) S(j\omega)| \le 1 \quad \forall G^\prime \in \Pi_I, \ \forall\omega
@ -1328,7 +1321,7 @@ This real-time tunability allows rapid testing of different control bandwidths t
For many practical applications, first order complementary filters are not sufficient. For many practical applications, first order complementary filters are not sufficient.
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. 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.
For these cases, the second-order complementary filters presented in Equation \eqref{eq:detail_control_cf_2nd_order} are proposed. For these cases, the complementary filters analytical formula in Equation \eqref{eq:detail_control_cf_2nd_order} are proposed.
\begin{subequations}\label{eq:detail_control_cf_2nd_order} \begin{subequations}\label{eq:detail_control_cf_2nd_order}
\begin{align} \begin{align}
@ -1338,12 +1331,9 @@ For these cases, the second-order complementary filters presented in Equation \e
\end{subequations} \end{subequations}
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}. 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}.
The parameter \(\alpha\) primarily affects the damping characteristics near the crossover frequency, while \(\omega_0\) determines the frequency at which the transition between high-pass and low-pass behavior occurs. 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.
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. 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.
The presented analytical formulations offer an attractive balance between design simplicity and performance.
This capability to tune parameters in real-time is particularly valuable during commissioning of the controller.
\begin{figure}[htbp] \begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth} \begin{subfigure}{0.48\textwidth}
\begin{center} \begin{center}
@ -1361,18 +1351,17 @@ This capability to tune parameters in real-time is particularly valuable during
\end{figure} \end{figure}
\section{Numerical Example} \section{Numerical Example}
\label{ssec:detail_control_cf_simulations} \label{ssec:detail_control_cf_simulations}
To systematically apply the proposed control technique, the following procedure is recommended: To implement the proposed control architecture in practice, the following procedure is proposed:
\begin{enumerate} \begin{enumerate}
\item Identify the plant to be controlled to obtain the plant model \(G\). \item Identify the plant to be controlled to obtain the plant model \(G\).
\item Design the weighting function \(w_I\) such that all possible plants \(G^\prime\) are contained in the uncertainty set \(\Pi_i\). \item Design the weighting function \(w_I\) such that all possible plants \(G^\prime\) are contained within the uncertainty set \(\Pi_i\).
\item Translate performance requirements into upper bounds on the complementary filters as explained in Section \ref{ssec:detail_control_cf_trans_perf}. \item Translate performance requirements into upper bounds on the complementary filters as explained in Section \ref{ssec:detail_control_cf_trans_perf}.
\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}. \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}.
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 \(w_I\). 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 \(w_I\).
For simpler cases, the analytical formulas for complementary filters presented in Section \ref{ssec:detail_control_cf_analytical_complementary_filters} can be employed. For simpler cases, the analytical formulas for complementary filters presented in Section \ref{ssec:detail_control_cf_analytical_complementary_filters} can be employed.
\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. \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.
\end{enumerate} \end{enumerate}
\paragraph{Plant}
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}). 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}).
In this model, a payload with mass \(m\) is positioned on top of a stage. In this model, a payload with mass \(m\) is positioned on top of a stage.
@ -1385,11 +1374,9 @@ The positioning stage itself is characterized by stiffness \(k\), internal dampi
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}. 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}.
\begin{equation}\label{eq:detail_control_cf_test_plant_tf} \begin{equation}\label{eq:detail_control_cf_test_plant_tf}
G(s) = \frac{1}{m s^2 + c s + k} 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)}
\end{equation} \end{equation}
The parameter values are set to \(m = \SI{20}{\kg}\), \(k = 1\si{\N/\mu\m}\), and \(c = 10^2\si{\N\per(\m\per\s)}\).
The plant dynamics include uncertainties related to limited support compliance, unmodeled flexible dynamics, payload dynamics, and other factors. The plant dynamics include uncertainties related to limited support compliance, unmodeled flexible dynamics, payload dynamics, and other factors.
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: 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:
@ -1502,5 +1489,23 @@ The control architecture has been presented for SISO systems, but can be applied
It will be experimentally validated with the NASS during the experimental phase. It will be experimentally validated with the NASS during the experimental phase.
\chapter*{Conclusion} \chapter*{Conclusion}
\label{sec:detail_control_conclusion} \label{sec:detail_control_conclusion}
In order to optimize the control of the Nano Active Stabilization System, several aspects of control theory were studied in this section.
Different approaches to combine sensors were compared in Section \ref{sec:detail_control_sensor}.
While High Authority Control-Low Authority Control (HAC-LAC) was successfully applied during the conceptual design phase, the focus of this work was extended to sensor fusion techniques where two or more sensors are combined using complementary filters.
It was demonstrated that the performance of such fusion depends significantly on the proper design of these complementary filters.
To address this challenge, a synthesis method based on \(\mathcal{H}_\infty\text{-synthesis}\) was proposed, allowing for intuitive shaping of the complementary filters through weighting functions.
This approach enabled the translation of sensor fusion objectives directly into requirements on filter magnitudes.
For the NASS, while HAC-LAC remains a natural way to combine sensors, the potential benefits of sensor fusion merit further investigation.
Various decoupling strategies for parallel manipulators were examined in Section \ref{sec:detail_control_decoupling}, including decentralized control, Jacobian decoupling, modal decoupling, and Singular Value Decomposition (SVD) decoupling.
The main characteristics of each approach were highlighted, providing valuable insights into their respective strengths and limitations.
Among the examined methods, Jacobian decoupling was determined to be most appropriate for the NASS, as it provides straightforward implementation while preserving the physical meaning of inputs and outputs.
With the system successfully decoupled, attention shifts to designing appropriate SISO controllers for each decoupled direction.
A method for directly shaping closed-loop transfer functions is proposed, based on complementary filters that can be designed using either the \(\mathcal{H}_\infty\) approach described earlier or through analytical formulas.
This straightforward approach enables intuitive parameter tuning while maintaining design simplicity.
Experimental validation of this method on the NASS will be conducted during the experimental tests on ID31.
\printbibliography[heading=bibintoc,title={Bibliography}] \printbibliography[heading=bibintoc,title={Bibliography}]
\end{document} \end{document}