Consider the dynamical system = Ax(t) + Bu(t), x(0) = xo, where A: = [ ]. B =[8]. 0 1 -3 Obtain the response (t). t> 0, when u(t), t 0, is a unit-step function. Assume x(0) = 0. Problem 2. Let A Rnxn, P E Rxn, and RE Rxn be such that P > 0 and R> 0. Show that if ATP+ PA+R=0, then A is asymptotically stable, that is, Rex < 0 for all = spec (A). = Hint: Note that if A Espec (A), then An An, where n ER" is an eigenvector of A corresponding to the eigenvalue X. Problem 1. x (t) t 0, The nonlinear dynamics for the rigid spacecraft shown in Figure Problem 3. 1 are given by t0, (1) *(t) = I232 (t)x3(t), x1(0) = 10, *2(t) = 1313(t)x(t), x(0) = x20, *3(t) = 112x1(t)x(t), x3(0)= = 30, (2) (3) where 1, 2, and 23 represent the angular velocities of the spacecraft about the principal axes of rotation, I23 (12-13)/11, 131 = (13-11)/12, 112 (11-12)/13, and I1, I2, and 13 are the principal moments of inertia of the spacecraft such that I < 1 < 13. = = i) Compute all the equilibria of the rigid spacecraft and explain the type of motion for each case. ii) Linearize the nonlinear system (1)-(3) about the equilibrium point (1e, 2e, 3e) (0, 0, 0) and represent the system as Ar(t) = AAx(t), (4) f(x) where A = [1 - 1e, 2-X2, 3 - 3e] and A = [0,w.or. iii) Is the linearized system in part ii) Lyapunov stable, asymptotically stable, or unstable?