Question

For an LTID system described by the difference equation y[k + 2]-0.5ylk +1]-0.24 3flk + 2lFind the zero-input response if yI-1] 0 and y[-2] 4) 100 72"

Answer 1

kindly see the above image for the solution.

Given Y[K + 2] - 0.5y [K + 1] - 0.24y [K] = 3f [K + 2] middot y [-1] = 0 middot y [-2] = 100/72 The system equation E operator form is [E^2 - 0.5E - 0.24] y [K] = 5E^2 f (k) (because E f (k) = f (K = 1) e^2 y [K] = y [K + 2] E y [K] = y [K + 1]) The characteristic equation obtained by making f (k) = 0 gamma^2 - 0.5 gamma - 0.24 = 0 gamma^2 - 0.8 gamma + 0.3 gamma - 0.24 = 0 gamma (gamma - 0.8) + 0.3 (gamma - 0.8) = 0 gamma = 0.8, - 0.3 The zero input response is, y_0 [K] = C_1 (- 0.3)^K + C_2 (0.8)^K rightarrow (1) substituting y [-1] = 0 0 = C_1 (-10/3) + C_2 (10/8) C_1/3 = C_2/8 Substituting y [-2] = 100/72 C_1 [100/9] + C_2 [25/16] = 100/72 implies 3C_2/8 times 100/9 + C_2 [25/126] = 100/72 implies [300/72 + 25/16] C_2 = 100/72