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+title = "Reference Tracking"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+:
+
+
+## Following Ramp inputs with one integrator {#following-ramp-inputs-with-one-integrator}
+
+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).
+
+```matlab
+G = tf(1); % Plant
+K = 2*pi*10/s; % Controller
+```
+
+The transfer function from the reference to the output is:
+\\[ T(s) = \frac{G(s)K(s)}{1 + G(s)K(s)} \\]
+
+```matlab
+T = G*K/(1 + G*K); % Transmissibility
+```
+
+The reference signal is a ramp with a "velocity" \\(r\_v = 1\\) unit/sec.
+
+```matlab
+% Time domain simulation
+Ts = 1e-4; % Sampling Time [s]
+t = 0:Ts:0.4; % Time vector [s]
+r = zeros(size(t)); % Sepoint
+r(t>0.1) = t(t>0.1)-0.1;
+y = lsim(T, r, t); % Output
+```
+
+<a id="figure--fig:reference-tracking-ramp-one-int"></a>
+
+{{< 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" >}}
+
+The error converges to a constant equal to \\(\frac{r\_v}{\omega\_c} \approx 0.016\\).
+
+The output "lags" behind the reference by \\(\frac{1}{\omega\_c}\\) in seconds.
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+</div>
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