Problem 3 : Controllability Gramian (a) Consider the following state equation: -0.5 0 x = - [ 85 9] = + [8]. x U 0 x(0) = [0:1] (i) (2 points) Show that there exists an input u(t) that drives x(0) to 0 in finite time. (ii) (3 points) Compute (by hand) the particular input u(t) that achieves this in 3 seconds. (iii) (3 points) Verify that the input achieves this by plotting y [1_0] x= x(t) and y = [0_1] x = x (t) versus t using the MATLAB command 1sim (sys,u,t, x0), where sys = ss (A,B,C,D) is an object representing the state-space model. (iv) (2 points) If possible, compute the controller canonical form (CCF) of the above state equation. If you cannot, why not? (b) (5 points) Given a controllable state equation, prove that the following input steers the state from x(to) = xo to x(t) = x for arbitrary x and x: u(t) = BT eAT (to-t) -t)W(to, t)(e(to-t) x xo), where W(to, tf) is the controllability gramian. (Hint: See proof of Theorem 3.2.)