1.4 KiB
+++ title = "Reference Tracking" author = ["Dehaeze Thomas"] draft = false +++
Tags :
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).
G = tf(1); % Plant
K = 2*pi*10/s; % Controller
The transfer function from the reference to the output is: \[ T(s) = \frac{G(s)K(s)}{1 + G(s)K(s)} \]
T = G*K/(1 + G*K); % Transmissibility
The reference signal is a ramp with a "velocity" \(r_v = 1\) unit/sec.
% 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
{{< figure src="/ox-hugo/reference_tracking_ramp_one_int.png" caption="<span class="figure-number">Figure 1: 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.