Question

  

Example: Consider the system Y(s) =G(s) =s2 d'y dt y-u()U(s) and determine the feedback gain to place the closed-loop poles at s--1ti. Therefore, we require that α2-2. With xrV and x-dy/dt, the matrix equation for the system and αι G(s) is dy d2y 2dt2 Dorf and Bishop, Modern Control System

Answer 1

G(S) = partial differential y/partial differential t^2 = U(t) implies ut xi = dy/dt = x_2 x_2^. = partial differential^2 y/ partial differential + 2 = Thus [x_1^. x_2^.] = [0 1 0 0] [x_1 x_2] + [0 1] uc +) Let state feedback gain, k = [k_1 k_2] such that, U = -k x is added to place holes at S = -1 plusminus I \r\n x^. = A X + B U U = -k x x^. = A X + B X - K X x^. = (A - B K) X A_CL rightarrow close Loop matrix A_CL = [A - B k] A_CL = [0 1 0 0] - [0 1]^T [k_1 k_2] = [0 1 0 0] - [0 1] [k_1 k_2] = [0 1 0 0] - [0 0 k_1 k_2] = [0 7 - k_1 - k_2] These are the close aoad pole location \r\n Thus - k_1 = -1 notequal I k_1 = 1 - I - k_2