2. Consider the system x(t) ly(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 A = = -4 2 A-[] B-[8] = = 0 4 Ax(t) + Bu(t) Cx(t) -4 2 [4 marks] Is the system completely controllable? Explain your answer. 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 n = 4 radians per second and a damping ratio of = 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 0 -4 ] [8] C = [10] Is the system after the fault occurs controllable? Is it stabilisable? Justify your answers. C = [10] B = [4 marks]