84 lines
2.5 KiB
Markdown
84 lines
2.5 KiB
Markdown
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title = "Time Delay"
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author = ["Dehaeze Thomas"]
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draft = false
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Tags
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:
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## Phase induced by a time delay {#phase-induced-by-a-time-delay}
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Having some time delay can be modelled by a transfer function having constant amplitude but a phase lag increasing with frequency.
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Such phase lag is linearly proportional to the time delay and to the frequency:
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\begin{equation}
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\phi(\omega) = -\omega \cdot T\_s
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\end{equation}
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with:
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- \\(\phi(\omega)\\) the phase lag in rad
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- \\(\omega\\) the frequency in rad/s
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- \\(T\_s\\) the time delay in s
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## Estimation of phase delay induced in sampled systems {#estimation-of-phase-delay-induced-in-sampled-systems}
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Consider a feedback controller implemented numerically on a system with a sampling frequency \\(F\_s\\).
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The time delay associated with the limited sampling frequency \\(F\_s\\) is:
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\begin{equation}
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\phi(\omega) = -\frac{\omega}{F\_s}
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\end{equation}
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with:
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- \\(\phi(\omega)\\) the phase lag in rad
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- \\(\omega\\) the frequency in rad/s
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- \\(F\_s\\) the sampling frequency in Hz
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Some values are summarized in Table [1](#table--tab:time-delay-phase-lag).
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<a id="table--tab:time-delay-phase-lag"></a>
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<div class="table-caption">
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<span class="table-number"><a href="#table--tab:time-delay-phase-lag">Table 1</a>:</span>
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Phase lag as a function of the frequency (relative to the sampling frequency )
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</div>
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| Frequency | Phase Delay [deg] |
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|----------------|-------------------|
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| \\(F\_s/100\\) | -3.6 |
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| \\(F\_s/10\\) | -36.0 |
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| \\(F\_s/2\\) | -180.0 |
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This is the main reason to have a sampling frequency much higher than the wanted feedback bandwidth is to limit the phase delay at the crossover frequency induced by the time delay.
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Having a sampling frequency a 100 times larger than the crossover frequency is a good objective.
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<div class="exampl">
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Take the example of a controller implemented with a sampling time of 0.1ms (10kHz sampling frequency).
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```matlab
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t_delay = 1e-4; % Delay [s]
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G_delay = exp(-t_delay*s);
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```
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The induced phase delay as a function of frequency is shown in Figure [1](#figure--fig:time-delay-induced-phase-lag).
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At the Nyquist frequency (5 kHz), the phase lag is 180 degrees.
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<a id="figure--fig:time-delay-induced-phase-lag"></a>
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{{< figure src="/ox-hugo/time_delay_induced_phase_lag.png" caption="<span class=\"figure-number\">Figure 1: </span>Phase lag induced by a time delay" >}}
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</div>
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## Bibliography {#bibliography}
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<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
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</div>
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