Chapter 6.P2, Problem 6P

Find the rank of the controllability matrix for the four mass system in all of the cases that a single control is applied to each of the four masses and compare your results with the conclusions obtained in Problem 4. (Note that by symmetry considerations, it is only necessary to consider one of the masses on the end and one of the interior masses.)

Repeat Problem 2 for a system of four masses connected by springs. Give a physical interpretation of the normal modes of vibration. Does the normal mode representation rule out complete controllability for the case of a scalar control applied to any single mass? Explain.

$\frac{d^{2}x}{d{t}^2}=\text{G}x+\text{u}(t),$ (2)