2. Consider the system \ x(t) Ax(t) + Bu(t) 1 yt) Cx(t) i) How would one check that the system is fully controllable? [2 marks] ii) An engineer states that the controllability criterion only applies when the D matrix is singular. Is the engineer correct? Explain your reasoning. [2 marks] iii) A second order system modelled as the system has state-space matrices 2 A-[ * ] 8-[] = = -4 0 B C = [ 10] 4 Is the system completely controllable? Explain your answer. [4 marks] iv) In order to improve the stability of the above system, a state feedback control law must be designed so that the closed-loop system has a natural frequency of on=4 radians per second and a damping ratio of G =0.8. Calculate a feedback matrix F = [fi f2] which will achieve this. [8 marks] v) Due to a fault in the system, the matrices abruptly change to 2 A= -4 0 3] B= [8] = C = [10] Is the system after the fault occurs controllable? Is it stabilisable? Justify your answers. [4 marks]