Exercise 1.0

Exercise 1.0#

Analysis ∙ Linear dynamical systems

Consider a simple mechanical system described by a mass \(m\), a spring constant \(k\), and a damping coefficient \(c\). The system is subjected to an external force \(u(t)\), and its position \(y(t)\) evolves according to the following linear ordinary differential equation (ODE):

\[ \ddot{y}(t) + 3 \dot{y}(t) + 2 y(t) = u(t) \]

  1. Compute the transfer function \(G(s)\) between the input \(u\) and the output \(y\).

  2. Compute the poles and zeros of the system.

  3. What is the dominant pole and its corresponding time-constant?

  4. Given this dominant pole, how long does it take before the output is in a 1% band of its final value?

  5. Compute the static gain of the system by setting \(s=0\).