how do I approach this?

Consider the following third order system * = Ax + Bu, x(t) ER, uER! Assume that the characteristic polynomial is given by p(1) = det ()I - A) = 13 + a2s' +ais + ao and consider the 3 x 3 matrix 1 a2 a1 P := C(A, B) 0 1 a2 O 0 1 where C(A, B) is the controllability matrix. a) Show that the following equality holds B = P b) Show that the following equality holds - a2 - a1 - AP = P 1 O oo O[Hintz Compute separately the left and the right hand side of the equations and show that the two matrices are equal with the help of CayleyHamilton theorem. c) Show that if the system is controllable then P is a non-singular matrix. d) Combining parts (a)-(c), obtain a coordinate transformation to express the system in the controllable canonical form 9": = Iii + Bu 02 a1 a0 1 where A = 1 0 0 and B = 0 . 0 1 0 0 (1) Write a program in MATLAB or Python to transform the following pair into the controllable canonical form 6 4 1 1 A: 5 4 0 B: 1 4 3 1 1 [Hint: Use MATLAB functions poly(A) to compute the characteristic polynominal of A and ctrb(A,B) to compute the controllability matrix.]