225 lines
12 KiB
Markdown
225 lines
12 KiB
Markdown
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title = "Mechatronic design of a magnetically suspended rotating platform"
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author = ["Thomas Dehaeze"]
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draft = false
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: [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting" >}})
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Author
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: Jabben, L.
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Year
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: 2007
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DOI
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:
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## Dynamic Error Budgeting {#dynamic-error-budgeting}
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### Introduction {#introduction}
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A large class of mechatronic machines have specifications based on their _standstill_ performance.
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The standstill performance is then limited by the (stochastic) disturbances action on the closed loop.
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The difficulty in calculation with stochastic signals and Bode plots, is that, instead of calculating with the complex response at one frequency, the **area** over a frequency range should be taken into account.
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The **error budgeting** is often used to estimate how much each component contributes to the total error
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Since many of the disturbances have a stochastic nature, they can be modelled with their **Power Spectral Densities**.
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The PSD of the performance measure in the closed loop system is the weigted sum of PSDs of the contributions of each disturbance to the performance channel.
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This approach allows frequency dependent error budgeting, which is why it is referred to as **Dynamic Error Budgeting**.
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### Common Mechatronics Disturbances {#common-mechatronics-disturbances}
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#### Ground vibrations {#ground-vibrations}
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#### Electronic Noise {#electronic-noise}
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**Thermal Noise** (or Johson noise).
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This noise can be modeled as a voltage source in series with the system impedance.
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The noise source has a PSD given by:
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\\[ S\_T(f) = 4 k T \text{Re}(Z(f)) \ [V^2/Hz] \\]
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with \\(k = 1.38 \cdot 10^{-23} \,[J/K]\\) the Boltzmann's constant, \\(T\\) the temperature [K] and \\(Z(f)\\) the frequency dependent impedance of the system.
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```text
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A kilo Ohm resistor at 20 degree Celsius will show a thermal noise of $0.13 \mu V$ from zero up to one kHz.
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```
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**Shot Noise**.
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Seen with junctions in a transistor.
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It has a white spectral density:
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\\[ S\_S = 2 q\_e i\_{dc} \ [A^2/Hz] \\]
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with \\(q\_e\\) the electronic charge (\\(1.6 \cdot 10^{-19}\, [C]\\)), \\(i\_{dc}\\) the average current [A].
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```text
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An averable current of 1 A will introduce noise with a STD of $10 \cdot 10^{-9}\,[A]$ from zero up to one kHz.
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```
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**Excess Noise** (or \\(1/f\\) noise).
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It results from fluctuating conductivity due to imperfect contact between two materials.
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The PSD of excess noise increases when the frequency decreases:
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\\[ S\_E = \frac{K\_f}{f^\alpha}\ [V^2/Hz] \\]
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where \\(K\_f\\) is dependent on the average voltage drop over the resistor and the index \\(\alpha\\) is usually between 0.8 and 1.4, and often set to unity for approximate calculation.
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**Signal to Noise Ration**
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Electronic equipment does most often not come with detailed electric schemes, in which case the PSD should be determined from measurements.
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In the design phase however, one has to rely on information provided by specification sheets from the manufacturer.
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The noise performance of components like sensors, amplifiers, converters, etc., is often specified in terms of a **Signal to Noise Ratio** (SNR).
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**The SNR gives the ratio of the RMS value of a sine that covers the full range of the channel through which the signal is propagating over the RMS value of the electrical noise.**
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Usually, the SNR is specified up to a certain cut-off frequency.
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If no information on the colouring of the noise is available, then the corresponding **PSD can be assumed to be white up to the cut-off frequency** \\(f\_c\\):
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\\[ S\_{snr} = \frac{x\_{fr}^2}{8 f\_c C\_{snr}^2} \\]
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with \\(x\_{fr}\\) the full range of \\(x\\), and \\(C\_{snr}\\) the SNR.
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#### AD and DA converters {#ad-and-da-converters}
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ADC and DAC add quantization noise to the signal.
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The variance can be calculated to be:
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\\[ \sigma^2 = \frac{q^2}{12} \\]
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with \\(q\\) the quantization interval.
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The corresponding PSD is white up to the Nyquist frequency:
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\\[ S\_Q = \frac{q^2}{12 f\_N} \\]
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with \\(f\_N\\) the Nyquist frequency [Hz].
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```text
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Let's take the example of a 16 bit ADC which has an electronic noise with a SNR of 80dB.
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Let's suppose the ADC is used to measure a position over a range of 1 mm.
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- ADC quantization noise: it has 16 bots over the 1 mm range.
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The standard diviation from the quantization is:
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\[ \sigma_{ADq} = \frac{1 \cdot 10^6/2^16}{\sqrt{12}} = 4.4\,[nm] \]
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- ADC electronic noise: the RMS value of a sine that covers to full range is $\frac{0.5}{\sqrt{2}} = 0.354\,[mm]$.
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With a SNR of 80dB, the electronic noise from the ADC becomes:
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\[ \sigma_{ADn} = 35\,[nm] \]
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Let's suppose the ADC is used to measure a sensor with an electronic noise having a standard deviation of $\sigma_{sn} = 17\,[nm]$.
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The PSD of this digitalized sensor noise is:
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\[ \sigma_s = \sqrt{\sigma_{sn}^2 + \sigma_{ADq}^2 + \sigma_{ADn}^2} = 39\,[nm]\]
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from which the PSD of the total sensor noise $S_s$ is calculated:
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\[ S_s = \frac{\sigma_s^2}{f_N} = 1.55\,[nm^2/Hz] \]
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with $f_N$ is the Nyquist frequency of 1kHz.
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```
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#### Acoustic Noise {#acoustic-noise}
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This can be a big error source in high precision machines, especially when the surface is big compare to the mass.
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The disturbance force acting on a body, is the **difference of pressure between the front and the back times the surface**.
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To have a pressure difference, the body must have a certain minimum dimension, depending on the wave length of the sound.
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For a body of typical dimensions of 100mm, only frequencies above 800 Hz have a significant disturbance contribution.
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```text
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Consider a cube with a rib size of 100 mm located in a room with a sound level of 80dB, distributed between one and ten kHz, then the force disturbance PSD equal $2.2 \cdot 10^{-2}\,[N^2/Hz]$
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```
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#### Brownian Noise {#brownian-noise}
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This is due to thermal effects and it notable where a small mass needs positioning.
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#### Turbulence {#turbulence}
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Rotation of the spindle introduces and air flow in which turbulence is cause by sharp angles on the rotor and stator.
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### Optimal Control {#optimal-control}
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#### The use of Optimal Control in DEB {#the-use-of-optimal-control-in-deb}
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Three factors influence the performance:
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- the disturbances: often a given value
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- the plant: can be costly to redesign
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- the controller
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The DEB helps identifying which disturbance is the limiting factor, and it should be investigated if the controller can deal with this disturbance before re-designing the plant.
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The modelling of disturbance as stochastic variables, is by excellence suitable for the optimal stochastic control framework.
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In Figure [1](#org30a4301), the generalized plant maps the disturbances to the performance channels.
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By minimizing the \\(\mathcal{H}\_2\\) system norm of the generalized plant, the variance of the performance channels is minimized.
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<a id="org30a4301"></a>
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{{< figure src="/ox-hugo/jabben07_general_plant.png" caption="Figure 1: Control system with the generalized plant \\(G\\). The performance channels are stacked in \\(z\\), while the controller input is denoted with \\(y\\)" >}}
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#### Using Weighting Filters for Disturbance Modelling {#using-weighting-filters-for-disturbance-modelling}
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Since disturbances are generally not white, the system of Figure [1](#org30a4301) needs to be augmented with so called **disturbance weighting filters**.
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A disturbance weighting filter gives the disturbance PSD when white noise as input is applied.
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This is illustrated in Figure [2](#org3b94947) where a vector of white noise time signals \\(\underbar{w}(t)\\) is filtered through a weighting filter to obtain the colored physical disturbances \\(w(t)\\) with the desired PSD \\(S\_w\\) .
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The generalized plant framework also allows to include **weighting filters for the performance channels**.
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This is useful for three reasons:
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- the performance channels might have different dimensions, which require scaling in order to compare
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- some performance channels may be of more importance than others
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- by using dynamic weighting filters, one can emphasize the performance in a certain frequency range
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<a id="org3b94947"></a>
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{{< figure src="/ox-hugo/jabben07_weighting_functions.png" caption="Figure 2: Control system with the generalized plant \\(G\\) and weighting functions" >}}
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The weighting filters should be stable transfer functions.
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**Obtaining the weighting filters**:
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If the PSD is given as a function \\(S\_x(j\omega)\\), the disturbance filter can be using **spectral factorization**:
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> Given a positive even function \\(S\_x(f)\\) of finite area, find a minimum-phase stable function \\(L(s)\\), such that \\(|L(j2\pi f)|^2 = S(s)\\)
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**Harmonic signals** can be approximately modeled by filtering white noise with a badly damped second order system, having a \\(+1\\) slope below the resonance frequency and a \\(-1\\) slope above the resonance frequency:
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\\[ V\_h = \frac{s}{s^2 + 2 \xi \omega\_h + \omega\_h^2} \\]
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with \\(\xi\\) the relative damping and \\(\omega\_h\\) the resonance frequency [rad/s].
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By making the \\(\mathcal{H}\_2\\) norm of \\(V\_h\\) equal to the RMS-value of the harmonic signal, the propagation of the disturbance to the performance channel can be well approximated.
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#### Balancing Control Effort vs Performance {#balancing-control-effort-vs-performance}
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IF only the output \\(y\\) are considered in the performance channel \\(z\\), the resulting optimal controller might result in very large actuator signals.
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So, to obtain feasible controllers, the performance channel is a combination of controller output \\(u\\) and system output \\(y\\).
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By choosing suitable weighting filters for \\(y\\) and \\(u\\), the performance can be optimized while keeping the controller effort limited:
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\\[ \\|z\\|\_{rms}^2 = \left\\| \begin{bmatrix} y \\ \alpha u \end{bmatrix} \right\\|\_{rms}^2 = \\|y\\|\_{rms}^2 + \alpha^2 \\|u\\|\_{rms}^2 \\]
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By calculation \\(\mathcal{H}\_2\\) optimal controllers for increasing \\(\alpha\\) and plotting the performance \\(\\|y\\|\\) vs the controller effort \\(\\|u\\|\\), the curve as depicted in Figure [3](#orgb0b1e78) is obtained.
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<a id="orgb0b1e78"></a>
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{{< figure src="/ox-hugo/jabben07_pareto_curve_H2.png" caption="Figure 3: An illustration of a Pareto curve. Each point of the curve represents the performance obtained with an optimal controller. The curve is obtained by varying \\(\alpha\\) and calculating an \\(\mathcal{H}\_2\\) optimal controller for each \\(\alpha\\)." >}}
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## Conclusion {#conclusion}
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> Using the DEB analysis during the design helped to formulate the specifications of the several subcomponents, such as:
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>
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> - The target bandwidth of the decentralized closed loops, which is very important for the mechanical design, as mechanical resonances can severely limit the bandwidth.
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> This value was also used to specify the current loop bandwidth of the custom designed power amplifiers for the RTAs and other components such as sensors and filters.
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> - The target value of the stiffness of the actuators was derived at 1000 N/m.
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> It was shown that the stiffness of a motor with back-iron is too much for the separated frame concept.
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> - The analysis pinpointed the most limiting component in the final design to be the Analogue-to-Digital Converter (ADC).
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<!--quoteend-->
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> In the DEB-framework there are three distinct factors which determine the performance.
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> These are the plant, the controller and the disturbances.
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> Synthesizing optimal controllers, such as H2-control, in the design helps to eliminate the controller out of the equation.
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> If the performance specifications are not met with an optimal controller, it is certain that a redesign of the system is required.
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> To use the measured PSDs in an optimal control design, such as H2-control, the disturbances must be modelled using linear time invariant models with multiple white noise input.
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> To derive such models, spectral factorization is used.
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> It is recommended to investigate which methods for spectral factorization are currently available and numerically robust.
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