Exercise 7.1

Control design ∙ State feedback Analysis ∙ Linear dynamical systems

Fig. 60 Block diagram of the system

Consider the system described in Fig. 60.

1. Write the system in state-space form, where the state variables \(x_1\) and \(x_2\) according to Fig. 60.

2. Design the state feedback control law \(u = - \ell_1 x_1 - \ell_2 x_2 + \ell_0 r\), where \(r\) is the reference signal, such that the closed-loop eigenvalues are in \(-4\) and \(-4\)

3. Suppose now that the only measurable signal is the output \(y\). Design a state observer such that the state estimation error has eigenvalues \(-6\) and \(-6\). How is the control law modified?

4. Write the resulting controller in the general linear feedback form.