Question

can somebody please help me with this problem !

Problem 2 (25 Points). Let asys state and output equation matrices: ete followin -3 0 A2 For this system, answer the following imperatives. (a) Find the eigenvalue matrix Λ and comment on the stability of the system (justify your comment). Use the convention that 1 2 and order Л accordingly. (b) Find the eigenvectors and the modal matrix M. (c) Find the state transition matrix Φ(t). Hint: first find the "diagonalized" state transition matrix Φ"(t). ld) Using the state transition matrix, find the output homogeneous solution for initial condition x(o)1

Answer 1

Matrix A = [-3 1 0 -2] B = [0 1] C = [0 1] a) Eigen value Vertical-bar SI - A Vertical-bar = 0 Vertical-bar s 0 0 s Vertical-bar - [-3 1 0 -2] = [s + 3 -1 0 s + 2] (s + 3) (s + 2) = 0 s = -3, -2 Eigen values are = -3, -2 since all eigen value lies on the negative side of real axis, so system is (b) Eigen vector (A - lambda_i I)x = 0 where lambda_i = Eigen vectors for Eigen value lambda = -3 (A + 3I)X = 0 {[-3 1 0 -2] + [3 0 0 3]} [x_1 x_2] = [0 0] [0 1 0 1] [x_1 x_2] = [0 0] \r\n for Eigen vector Ax = lambda_1 x [-3 1 0 -2][x_1 x_2] = -3[x_1 x_2] [-3x_1 x_1 - 2x_2] = [-3x_1 -3x_2] x_1 - 2x_2 = -3x_2 x_1 = -x_2 rightarrow x_1 /-1 = x_2 /1 Eigen vectors = [-1 1] for Eigen vector lambda = -2 Ax = lambda x [-3 1 0 -2][x_1 x_2] = [-2x_1 -2x_2]