Exercise 2.4

Analysis ∙ Feedback control

An astronaut working in space needs to control their position using thrusters, which generate force by expelling gas. A control system is designed to achieve this, based on the following control law:

\[ u = K_1 (r - y) - K_1 K_2 \frac{dy}{dt}, \]

where:

• \(u\) is the thrust force generated by the thrusters (control input),

• \(r\) is the desired position of the astronaut (setpoint),

• \(y\) is the current position of the astronaut (measured output). \end{itemize}

The astronaut, including equipment, has a total mass of 100 kg. Your task is to determine suitable values for the controller parameters \(K_1 > 0\) and \(K_2 > 0\) such that the following requirements are met:

1. If the desired position is given as \(r(t) = \textrm{ramp}(t)\), there should be a time \(T_0\) such that the tracking error satisfies \(|r(t) - y(t)| < 1\) for all \(t \geq T_0\).

2. The damping ratio of the closed-loop system is 0.7. This ensures smooth and stable motion without excessive oscillations.