51 lines
1.4 KiB
Markdown
51 lines
1.4 KiB
Markdown
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title = "Reference Tracking"
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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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## Following Ramp inputs with one integrator {#following-ramp-inputs-with-one-integrator}
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Let's suppose a static plant and a controller with one integrator with a crossover frequency of \\(\omega\_c = 10\cdot 2\pi\\) (i.e. 10Hz).
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```matlab
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G = tf(1); % Plant
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K = 2*pi*10/s; % Controller
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```
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The transfer function from the reference to the output is:
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\\[ T(s) = \frac{G(s)K(s)}{1 + G(s)K(s)} \\]
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```matlab
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T = G*K/(1 + G*K); % Transmissibility
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```
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The reference signal is a ramp with a "velocity" \\(r\_v = 1\\) unit/sec.
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```matlab
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% Time domain simulation
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Ts = 1e-4; % Sampling Time [s]
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t = 0:Ts:0.4; % Time vector [s]
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r = zeros(size(t)); % Sepoint
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r(t>0.1) = t(t>0.1)-0.1;
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y = lsim(T, r, t); % Output
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```
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<a id="figure--fig:reference-tracking-ramp-one-int"></a>
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{{< figure src="/ox-hugo/reference_tracking_ramp_one_int.png" caption="<span class=\"figure-number\">Figure 1: </span>Comparison of the setpoint and the plant output for a ramp with only one integrator in the loop" >}}
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The error converges to a constant equal to \\(\frac{r\_v}{\omega\_c} \approx 0.016\\).
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The output "lags" behind the reference by \\(\frac{1}{\omega\_c}\\) in seconds.
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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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