0 1. Consider an autonomous system (t) = [ O (a) Determine the eigenvalues and eigenvectors of A. (b) Show that the state transition matrix of the system is et [ 2. Consider the * (t) * (t) *3 (t) = 1 0 system i(t) = [ ] z(1) + [ ] u(t) x(t) (a) Compute the controllability gramian as a function of a where t is the last digit of your student number. If the last digit of your student number is 0, you will take t as 0.5. (b) For which values of a, is the system controllable? (Use controllability gramian) 3. The dynamical equations of a system are given as = = r(t) where a and w are two real numbers. cos(wt) sin(wt) x(t)- x(t) + u(t) x (t) + x3 (t) 2 x2(t) x (t)- x3(t) (a) Calculate the state transition matrix of the system. (b) Analyze the stability and controllability of the system.