+++
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
```
{{< figure src="/ox-hugo/reference_tracking_ramp_one_int.png" caption="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.
## Bibliography {#bibliography}