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static/ox-hugo/analog_filt_second_order_hpf.png create mode 100644 static/ox-hugo/analog_filt_second_order_lpf.png create mode 100644 static/ox-hugo/analog_filt_second_order_lpf_characteristics.png diff --git a/content/book/lyons11_under_digit_signal_proces.md b/content/book/lyons11_under_digit_signal_proces.md index b5ac02e..b552da8 100644 --- a/content/book/lyons11_under_digit_signal_proces.md +++ b/content/book/lyons11_under_digit_signal_proces.md @@ -5,10 +5,10 @@ draft = true +++ Tags -: [IRR and FIR Filters]({{}}), [Digital Filters]({{}}) +: [IRR and FIR Filters]({{< relref "irr_and_fir_filters.md" >}}), [Digital Filters]({{< relref "digital_filters.md" >}}) Reference -: +: (Lyons 2011) Author(s) : Lyons, R. @@ -43,13 +43,13 @@ Year ### Analyzing Linear Time-Invariant Systems {#analyzing-linear-time-invariant-systems} - + -{{< figure src="/ox-hugo/lyons11_lti_impulse_response.png" caption="Figure 1: LTI system unit impulse response sequences. (a) system block diagram. (b) impulse input sequence \\(x(n)\\) and impulse reponse output sequence \\(y(n)\\)." >}} +{{< figure src="/ox-hugo/lyons11_lti_impulse_response.png" caption="Figure 1: LTI system unit impulse response sequences. (a) system block diagram. (b) impulse input sequence \\(x(n)\\) and impulse reponse output sequence \\(y(n)\\)." >}} - + -{{< figure src="/ox-hugo/lyons11_moving_average.png" caption="Figure 2: Analyzing a moving average filter. (a) averager block diagram; (b) impulse input and impulse response; (c) averager frequency magnitude reponse." >}} +{{< figure src="/ox-hugo/lyons11_moving_average.png" caption="Figure 2: Analyzing a moving average filter. (a) averager block diagram; (b) impulse input and impulse response; (c) averager frequency magnitude reponse." >}} ## Periodic Sampling {#periodic-sampling} @@ -57,20 +57,20 @@ Year ### Aliasing: Signal Ambiguity In The Frequency Domain {#aliasing-signal-ambiguity-in-the-frequency-domain} - + -{{< figure src="/ox-hugo/lyons11_frequency_ambiguity.png" caption="Figure 3: Frequency ambiguity; (a) discrete time sequence of values; (b) two different sinewaves that pass through the points of discete sequence" >}} +{{< figure src="/ox-hugo/lyons11_frequency_ambiguity.png" caption="Figure 3: Frequency ambiguity; (a) discrete time sequence of values; (b) two different sinewaves that pass through the points of discete sequence" >}} ### Sampling Lowpass Signals {#sampling-lowpass-signals} - + -{{< figure src="/ox-hugo/lyons11_noise_spectral_replication.png" caption="Figure 4: Spectral replications; (a) original continuous signal plus noise spectrum; (b) discrete spectrum with noise contaminating the signal of interest" >}} +{{< figure src="/ox-hugo/lyons11_noise_spectral_replication.png" caption="Figure 4: Spectral replications; (a) original continuous signal plus noise spectrum; (b) discrete spectrum with noise contaminating the signal of interest" >}} - + -{{< figure src="/ox-hugo/lyons11_lowpass_sampling.png" caption="Figure 5: Low pass analog filtering prior to sampling at a rate of \\(f\_s\\) Hz." >}} +{{< figure src="/ox-hugo/lyons11_lowpass_sampling.png" caption="Figure 5: Low pass analog filtering prior to sampling at a rate of \\(f\_s\\) Hz." >}} ## The Discrete Fourier Transform {#the-discrete-fourier-transform} @@ -531,96 +531,8 @@ X(m) = \sum\_{n = 0}^{N-1} x(n) e^{-j2 \pi n m /N} ### 13.51 A Stable Goertzel Algorithm {#13-dot-51-a-stable-goertzel-algorithm} -## A: The Arithmetic Of Complex Numbers {#a-the-arithmetic-of-complex-numbers} +## Bibliography {#bibliography} - -### A.1 Graphical Representation Of Real And Complex Numbers {#a-dot-1-graphical-representation-of-real-and-complex-numbers} - - -### A.2 Arithmetic Representation Of Complex Numbers {#a-dot-2-arithmetic-representation-of-complex-numbers} - - -### A.3 Arithmetic Operations Of Complex Numbers {#a-dot-3-arithmetic-operations-of-complex-numbers} - - -### A.4 Some Practical Implications Of Using Complex Numbers {#a-dot-4-some-practical-implications-of-using-complex-numbers} - - -## B: Closed Form Of A Geometric Series {#b-closed-form-of-a-geometric-series} - - -## C: Time Reversal And The Dft {#c-time-reversal-and-the-dft} - - -## D: Mean,Variance, And Standard Deviation {#d-mean-variance-and-standard-deviation} - - -### D.1 Statistical Measures {#d-dot-1-statistical-measures} - - -### D.2 Statistics Of Short Sequences {#d-dot-2-statistics-of-short-sequences} - - -### D.3 Statistics Of Summed Sequences {#d-dot-3-statistics-of-summed-sequences} - - -### D.4 Standard Deviation (Rms) Of A Continuous Sinewave {#d-dot-4-standard-deviation--rms--of-a-continuous-sinewave} - - -### D.5 Estimating Signal-To-Noise Ratios {#d-dot-5-estimating-signal-to-noise-ratios} - - -### D.6 The Mean And Variance Of Random Functions {#d-dot-6-the-mean-and-variance-of-random-functions} - - -### D.7 The Normal Probability Density Function {#d-dot-7-the-normal-probability-density-function} - - -## E: Decibels (Db And Dbm) {#e-decibels--db-and-dbm} - - -### E.1 Using Logarithms To Determine Relative Signal Power {#e-dot-1-using-logarithms-to-determine-relative-signal-power} - - -### E.2 Some Useful Decibel Numbers {#e-dot-2-some-useful-decibel-numbers} - - -### E.3 Absolute Power Using Decibels {#e-dot-3-absolute-power-using-decibels} - - -## F: Digital Filter Terminology {#f-digital-filter-terminology} - - -## G: Frequency Sampling Filter Derivations {#g-frequency-sampling-filter-derivations} - - -### G.1 Frequency Response Of A Comb Filter {#g-dot-1-frequency-response-of-a-comb-filter} - - -### G.2 Single Complex Fsf Frequency Response {#g-dot-2-single-complex-fsf-frequency-response} - - -### G.3 Multisection Complex Fsf Phase {#g-dot-3-multisection-complex-fsf-phase} - - -### G.4 Multisection Complex Fsf Frequency Response {#g-dot-4-multisection-complex-fsf-frequency-response} - - -### G.5 Real Fsf Transfer Function {#g-dot-5-real-fsf-transfer-function} - - -### G.6 Type-Iv Fsf Frequency Response {#g-dot-6-type-iv-fsf-frequency-response} - - -## H: Frequency Sampling Filter Design Tables {#h-frequency-sampling-filter-design-tables} - - -## I: Computing Chebyshev Window Sequences {#i-computing-chebyshev-window-sequences} - - -### I.1 Chebyshev Windows For Fir Filter Design {#i-dot-1-chebyshev-windows-for-fir-filter-design} - - -### I.2 Chebyshev Windows For Spectrum Analysis {#i-dot-2-chebyshev-windows-for-spectrum-analysis} - -<./biblio/references.bib> +
+
Lyons, Richard. 2011. Understanding Digital Signal Processing. Upper Saddle River, NJ: Prentice Hall.
+
diff --git a/content/book/smith99_scien_engin_guide_digit_signal.md b/content/book/smith99_scien_engin_guide_digit_signal.md index ddc71f4..8d13fa5 100644 --- a/content/book/smith99_scien_engin_guide_digit_signal.md +++ b/content/book/smith99_scien_engin_guide_digit_signal.md @@ -1,15 +1,15 @@ +++ title = "The scientist and engineer's guide to digital signal processing - second edition" -author = ["Thomas Dehaeze"] +author = ["Dehaeze Thomas"] keywords = ["Signal Processing"] draft = true +++ Tags -: [Digital Signal Processing]({{< relref "digital_signal_processing" >}}) +: [Digital Signal Processing]({{< relref "digital_signal_processing.md" >}}) Reference -: ([Smith 1999](#org023917a)) +: (Smith 1999) Author(s) : Smith, S. W. @@ -18,7 +18,8 @@ Year : 1999 - ## Bibliography {#bibliography} -Smith, Steven W. 1999. _The Scientist and Engineer’s Guide to Digital Signal Processing - Second Edition_. California Technical Publishing. +
+
Smith, Steven W. 1999. The Scientist and Engineer’s Guide to Digital Signal Processing - Second Edition. California Technical Publishing.
+
diff --git a/content/phdthesis/babakhani12_activ_dampin_vibrat_high_precis_motion_system.md b/content/phdthesis/babakhani12_activ_dampin_vibrat_high_precis_motion_system.md index 4c7d66e..2cf391d 100644 --- a/content/phdthesis/babakhani12_activ_dampin_vibrat_high_precis_motion_system.md +++ b/content/phdthesis/babakhani12_activ_dampin_vibrat_high_precis_motion_system.md @@ -1,16 +1,16 @@ +++ title = "Active damping of vibrations in high-precision motion systems" -author = ["Thomas Dehaeze"] +author = ["Dehaeze Thomas"] draft = false ref_author = "Babakhani, B." ref_year = 2012 +++ Tags -: [Active Damping]({{}}) +: [Active Damping]({{< relref "active_damping.md" >}}) Reference -: ([Babakhani 2012](#orgf88b4c7)) +: (Babakhani 2012) Author(s) : Babakhani, B. @@ -19,7 +19,8 @@ Year : 2012 - ## Bibliography {#bibliography} -Babakhani, Bayan. 2012. “Active Damping of Vibrations in High-Precision Motion Systems.” University of Twente. . +
+
Babakhani, Bayan. 2012. “Active Damping of Vibrations in High-Precision Motion Systems.” University of Twente. doi:10.3990/1.9789036534642.
+
diff --git a/content/zettels/actuators.md b/content/zettels/actuators.md index 2102256..da6f0ce 100644 --- a/content/zettels/actuators.md +++ b/content/zettels/actuators.md @@ -7,12 +7,26 @@ draft = false Tags : -Links to specific actuators: + +## Short Stroke Actuators {#short-stroke-actuators} + +For short stroke and very high dynamic applications, mainly two types of actuators can be used: - [Voice Coil Actuators]({{< relref "voice_coil_actuators.md" >}}) - [Piezoelectric Actuators]({{< relref "piezoelectric_actuators.md" >}}) +## Long Stroke Actuators {#long-stroke-actuators} + +- Linear motors +- Piezoelectric Walking Drive ([PI](https://www.physikinstrumente.com/en/expertise/technology/piezoelectric-drives/piezowalk-piezo-motors/)) + +Rotational drives can be combined with ball-screw mechanisms for long (infinite) axial motion: + +- Brush-less DC Motor. See (Yedamale 2003) and this [working principle](https://www.electricaltechnology.org/2016/05/bldc-brushless-dc-motor-construction-working-principle.html). +- [Stepper Motor]({{< relref "stepper_motor.md" >}}) + + ## How to choose the correct actuator for my application? {#how-to-choose-the-correct-actuator-for-my-application} For vibration isolation: @@ -20,16 +34,6 @@ For vibration isolation: - In (Ito and Schitter 2016), the effect of the actuator stiffness on the attainable vibration isolation is studied ([Notes]({{< relref "ito16_compar_class_high_precis_actuat.md" >}})) -## Brush-less DC Motor {#brush-less-dc-motor} - -- (Yedamale 2003) - - - - -## [Stepper Motor]({{< relref "stepper_motor.md" >}}) {#stepper-motor--stepper-motor-dot-md} - - ## Bibliography {#bibliography}
diff --git a/content/zettels/continuous_transfer_functions.md b/content/zettels/continuous_transfer_functions.md new file mode 100644 index 0000000..f364223 --- /dev/null +++ b/content/zettels/continuous_transfer_functions.md @@ -0,0 +1,11 @@ ++++ +title = "Continuous Transfer Functions" +author = ["Dehaeze Thomas"] +draft = false ++++ + +Tags +: + + +## Typical Transfer Functions {#typical-transfer-functions} diff --git a/content/zettels/digital_filters.md b/content/zettels/digital_filters.md index 3c56dfa..8055f9e 100644 --- a/content/zettels/digital_filters.md +++ b/content/zettels/digital_filters.md @@ -1,6 +1,6 @@ +++ title = "Digital Filters" -author = ["Thomas Dehaeze"] +author = ["Dehaeze Thomas"] draft = false +++ @@ -9,4 +9,8 @@ Tags A nice open access book on digital filter is accessible here: -<./biblio/references.bib> + +## Bibliography {#bibliography} + +
+
diff --git a/content/zettels/discrete_transfer_functions.md b/content/zettels/discrete_transfer_functions.md new file mode 100644 index 0000000..4ee3828 --- /dev/null +++ b/content/zettels/discrete_transfer_functions.md @@ -0,0 +1,38 @@ ++++ +title = "Discrete Transfer Functions" +author = ["Dehaeze Thomas"] +draft = false ++++ + +Tags +: [Digital Filters]({{< relref "digital_filters.md" >}}) + + +## Typical Transfer functions {#typical-transfer-functions} + + +### Delay {#delay} + + +### First Order Low Pass {#first-order-low-pass} + + +### First Order High Pass {#first-order-high-pass} + + +### Integrator {#integrator} + + +### Derivator {#derivator} + + +### Second Order Low Pass {#second-order-low-pass} + + +### PID {#pid} + + +### Notch {#notch} + + +### Moving Average {#moving-average} diff --git a/content/zettels/electronic_passive_filters.md b/content/zettels/electronic_passive_filters.md index 473ea0b..7182c7d 100644 --- a/content/zettels/electronic_passive_filters.md +++ b/content/zettels/electronic_passive_filters.md @@ -7,39 +7,103 @@ draft = false Tags : -TODOS: - -- [X] Electronics circuits containing input voltage, output voltage, R L and C components -- [ ] Bode plot of the filter from input voltage to output voltage -- [ ] Equation of the transfer functions with nice parameters (\\(\omega\_c\\), \\(\xi\\)) - ## First Order Low Pass Filter {#first-order-low-pass-filter} - + -{{< figure src="/ox-hugo/elec_passive_first_order_low_pass_filter.png" caption="Figure 1: First Order Low Pass Filter using an RC circuit" >}} +{{< figure src="/ox-hugo/analog_filt_first_order_lpf.png" caption="Figure 1: First Order Low Pass Filter using an RC circuit" >}} + +\begin{equation} +\boxed{\frac{V\_o}{V\_i}(s) = \frac{1}{1 + \frac{s}{\omega\_0}}, \quad \omega\_0 = \frac{1}{RC}} +\end{equation} + +\begin{equation} +Z\_f(s) = \frac{V\_i}{i\_i}(s) = \frac{1 + RC \cdot s}{C \cdot s} +\end{equation} + +```matlab +%% First Order Low Pass Filter +R = 1e3; % [Ohm] +C = 1e-6; % [F] +``` + +```matlab +G = 1/(1 + R*C*s); % Filter Transfer Function +Z = (1 + R*C*s)/(C*s); % Filter Impedance +``` + + + +{{< figure src="/ox-hugo/analog_filt_first_order_lpf_characteristics.png" caption="Figure 2: First Order Low Pass Filter - Filter transfer function and filter impedance" >}} ## First Order High Pass Filter {#first-order-high-pass-filter} - + -{{< figure src="/ox-hugo/elec_passive_first_order_high_pass_filter.png" caption="Figure 2: First Order High Pass Filter using an RC circuit" >}} +{{< figure src="/ox-hugo/analog_filt_first_order_hpf.png" caption="Figure 3: First Order High Pass Filter using an RC circuit" >}} + +\begin{equation} +\boxed{\frac{V\_o}{V\_i}(s) = \frac{\frac{s}{\omega\_0}}{1 + \frac{s}{\omega\_0}}, \quad \omega\_0 = \frac{1}{RC}} +\end{equation} + +\begin{equation} +Z\_f(s) = \frac{V\_i}{i\_i}(s) = \frac{1 + RC \cdot s}{C \cdot s} +\end{equation} + +```matlab +%% First Order High Pass Filter +R = 1e3; % [Ohm] +C = 1e-6; % [F] +``` + +```matlab +G = R*C*s/(1 + R*C*s); % Filter Transfer Function +Z = (1 + R*C*s)/(C*s); % Filter Impedance +``` + + + +{{< figure src="/ox-hugo/analog_filt_first_order_hpf_characteristics.png" caption="Figure 4: First Order High Pass Filter - Filter transfer function and filter impedance" >}} ## Second Order Low Pass Filter {#second-order-low-pass-filter} - + -{{< figure src="/ox-hugo/elec_passive_second_order_low_pass_filter.png" caption="Figure 3: Second Order Low Pass Filter using an RLC circuit" >}} +{{< figure src="/ox-hugo/analog_filt_second_order_lpf.png" caption="Figure 5: Second Order Low Pass Filter using an RLC circuit" >}} + +\begin{equation} +\boxed{\frac{V\_o}{V\_i}(s) = \frac{1}{1 + 2 \xi \frac{s}{\omega\_0} + \frac{s^2}{\omega\_0^2}}, \quad \omega\_0 = \frac{1}{\sqrt{LC}},\quad \xi = \frac{1}{2R\sqrt{LC}}} +\end{equation} + +\begin{equation} +Z\_f(s) = \frac{V\_i}{i\_i}(s) = L \cdot s + \frac{R}{1 + RC \cdot s} +\end{equation} + +```matlab +%% Second Order Low Pass Filter +R = 1e2; % [Ohm] +C = 1e-5; % [F] +L = 1e-2; % [H] +``` + +```matlab +G = 1/(1 + L/R*s + C*L*s^2); % Filter Transfer Function +Z = L*s + (R)/(1 + R*C*s); % Filter Impedance +``` + + + +{{< figure src="/ox-hugo/analog_filt_second_order_lpf_characteristics.png" caption="Figure 6: Second Order Low Pass Filter - Filter transfer function and filter impedance" >}} ## Second Order High Pass Filter {#second-order-high-pass-filter} - + -{{< figure src="/ox-hugo/elec_passive_second_order_high_pass_filter.png" caption="Figure 4: Second Order High Pass Filter using an RLC circuit" >}} +{{< figure src="/ox-hugo/analog_filt_second_order_hpf.png" caption="Figure 7: Second Order High Pass Filter using an RLC circuit" >}} ## Bibliography {#bibliography} diff --git a/content/zettels/laplace_transform.md b/content/zettels/laplace_transform.md new file mode 100644 index 0000000..62cebdc --- /dev/null +++ b/content/zettels/laplace_transform.md @@ -0,0 +1,36 @@ ++++ +title = "Laplace Transform" +author = ["Dehaeze Thomas"] +draft = false ++++ + +Tags +: [Continuous Transfer Functions]({{< relref "continuous_transfer_functions.md" >}}) + + +## Definition {#definition} + +Laplace transform: + +\begin{equation} +\boxed{F(s) = \mathcal{L}{f(t)} = \int\_0^\infty e^{-st} f(t) dt} +\end{equation} + + +## Laplace transform of signals {#laplace-transform-of-signals} + + + + +## Laplace transform of functions {#laplace-transform-of-functions} + + + + +## Solving Linear Differential equations {#solving-linear-differential-equations} + + +### Mechanical System {#mechanical-system} + + +### Electrical System {#electrical-system} diff --git a/content/zettels/piezoelectric_actuators.md b/content/zettels/piezoelectric_actuators.md index 973f2e5..574500c 100644 --- a/content/zettels/piezoelectric_actuators.md +++ b/content/zettels/piezoelectric_actuators.md @@ -62,11 +62,13 @@ The Amplified Piezo Actuators principle is presented in (Lucinskis and Mangeot 2016). +A model of an amplified piezoelectric actuator is described in (Lucinskis and Mangeot 2016). + +Typical topology of mechanically amplified piezoelectric actuators are displayed in Figure [1](#figure--fig:ling16-topology-piezo-mechanism-types) (from (Ling et al. 2016)). -{{< figure src="/ox-hugo/ling16_topology_piezo_mechanism_types.png" caption="Figure 1: Topology of several types of compliant mechanisms " >}} +{{< figure src="/ox-hugo/ling16_topology_piezo_mechanism_types.png" caption="Figure 1: Topology of several types of compliant mechanisms" >}} | Manufacturers | Country | |----------------------------------------------------------------------------------------------------|-----------| @@ -208,5 +210,6 @@ Limitations of the electronics is discussed in [Design, modeling and control of
Claeyssen, Frank, R. Le Letty, F. Barillot, and O. Sosnicki. 2007. “Amplified Piezoelectric Actuators: Static & Dynamic Applications.” Ferroelectrics 351 (1): 3–14. doi:10.1080/00150190701351865.
Fleming, A.J. 2010. “Nanopositioning System with Force Feedback for High-Performance Tracking and Vibration Control.” Ieee/Asme Transactions on Mechatronics 15 (3): 433–47. doi:10.1109/tmech.2009.2028422.
-
Lucinskis, R., and C. Mangeot. 2016. “Dynamic Characterization of an Amplified Piezoelectric Actuator.”
+
Ling, Mingxiang, Junyi Cao, Minghua Zeng, Jing Lin, and Daniel J Inman. 2016. “Enhanced Mathematical Modeling of the Displacement Amplification Ratio for Piezoelectric Compliant Mechanisms.” Smart Materials and Structures 25 (7): 075022. doi:10.1088/0964-1726/25/7/075022.
+
Lucinskis, R., and C. Mangeot. 2016. “Dynamic Characterization of an Amplified Piezoelectric Actuator.”
diff --git a/content/zettels/stepper_motor.md b/content/zettels/stepper_motor.md index f3b67a3..a01121e 100644 --- a/content/zettels/stepper_motor.md +++ b/content/zettels/stepper_motor.md @@ -8,15 +8,32 @@ Tags : -## 2 phase VS 5 phase stepper motor {#2-phase-vs-5-phase-stepper-motor} +## Errors between steps (micro-stepping) {#errors-between-steps--micro-stepping} - +For a two phase stepper motor, there are (typically) **200 steps per revolution** (i.e. 1.8% per step). +Between each step, even when using some micro-stepping, there are some position errors that are due to non-perfect magnetic and electromagnetic fields. -## Errors {#errors} +The period of this error is corresponding to 200 period/revolution. -For a two phase stepper motor, there are (typically) 200 steps per revolution. -Errors with a period of 200 period/revolution can be expected. +Then scanning, this can lead to high frequency vibrations. + +This is what is typically limiting the accuracy of the stepper motor (usually specified in between 3% and 5% of the step increment). +This is approximately corresponding to **1mrad**. + +
+ +Consider a stepper motor with 200 steps by turn attached to a ball-screw with a pitch of 1mm per turn. +A rotation of 1 turn per second will induce vibrations at 200Hz with an amplitude of \\(1\\,\mu m\\). + +
+ +Note that this error is not a pure sine, it also has some harmonics with corresponding periods of 1/100 revolution and 1/50 revolution. + +This error should repeat every turn and can be calibrated provided it is repeatable over time. + +One way to reduce these errors is to use a ball-screw mechanism with a smaller pitch. +The price to pay is smaller velocity. ## Manufacturers {#manufacturers} @@ -25,3 +42,14 @@ Errors with a period of 200 period/revolution can be expected. |--------------------------------------------------------------------------|----------------| | [AML](https://arunmicro.com/) | United Kingdom | | [Sanyo](https://www.sanyodenki.com/catalogs/servo/stepping_systems.html) | | + + +## 2 phase VS 5 phase stepper motor {#2-phase-vs-5-phase-stepper-motor} + + + + +## Bibliography {#bibliography} + +
+
diff --git a/content/zettels/z_transform.md b/content/zettels/z_transform.md new file mode 100644 index 0000000..ffe498a --- /dev/null +++ b/content/zettels/z_transform.md @@ -0,0 +1,15 @@ ++++ +title = "Z-Transform" +author = ["Dehaeze Thomas"] +draft = false ++++ + +Tags +: [Discrete Transfer Functions]({{< relref "discrete_transfer_functions.md" >}}) + + +## From Continuous to Discrete transfer function {#from-continuous-to-discrete-transfer-function} + +A continuous transfer function (i.e. in the Laplace domain) can easily be converted to the discrete time domain (i.e. in the z-domain) using the `c2d` Matlab command ([doc](https://fr.mathworks.com/help/control/ref/lti.c2d.html;jsessionid=206bd0fc8950c5f8cb6b568d7393#mw_53fc4689-2099-41d0-93b3-de1e51a174c1)). + +Several methods can be used, each with some advantages and drawbacks, see [this document](https://fr.mathworks.com/help/control/ug/continuous-discrete-conversion-methods.html). diff --git 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