Question

4. Splitting H into the sum of two first order transfer functions suggests the following parallel block diagram: Hi u(t) = est y(t) (Hi(s)H2(s)) est which suggests the following block diagram using a larger class of functions (not just ones proportional to est) and convolution,

Answer 1

Given that use have to prove h = h_1 + h_2 \r\n by laplace transform - or we have to prove H(s) = H_1 (s) + H_2 (s) given x_1 = h_1 times u by laplace - X_1 (s) = H(s) U(s) so H_1 (s) = X_1 (s)/U(s) by above block x_1 = lambda_1 x_1 + 21 v so by laplace - s X_1 (s) = R_1 X_1 (s) + 21 v(s) so H_1 (s) = X_1 (s)/U(s) = alpha_1/s - lambda_1 - - (1) by lower block - x_2 = lambda_2 x_2 + alpha_2 v by laplace - s X_2 (s) = lambda_2 X_2 (s) + alpha_2 2 v(s) so H_2 (s) = X_2 (s)/v(s) = alpha_2/s - lambda_2 - (2) by block diagram - y = x_1 + x_2 = h_1 times u + h_2 u c here Y(s)/v(s) = H(s) so by laplace - Y(s) = H_1 (s) - v(s) + H_2 (s) v(s) so H(s) = Y(s)/v(s) = H_1 (s) + H_2 (s)