Question

Person A said that given a transfer function H(s) the system was unique as the impulse function h(t) is fixed.   Person B said that two different systems can have the same transfer function. Then he produced the following two ODEs:

(a) System A that is given by dx dt dt2 dt (b) System B that is given by dy dt Can you verify that the transfer functions for the two systems are the same but prove that the systems are nevertheless different. Hint: Find the ZIR for both systems. Are they the same?

Answer 1

For System A d^2y/dt^2 + 3 dy/dt + 2y = dx/dt + 2x Applying laplace transform on both sides. s^2 y (s) + 3sy (s) + 2y (s) = s times (s) + 2x (s) y (s)/x (s) = s + 2/s^2 + 3s + 2 y (s)/x (s) = s + 2/(s + 1) (s + 2) y (s)/x (s) = 1/s + 1 (b) For system (B) dy/dt + y = x s y (s) + y (s) = x (s) y (s)/x (s) = 1/s + 1 System A and System B has same Transfer function. The zero input response of a system is obtained when the input is made zero and all initial conditions are present. For System A, when input x is zero, the differential equation reduces to d^2y/dt^2 + 3dy/dt + 2y = 0 let d/dt = D (D^2 + 3D + 2) y = 0 The roots are D = -1 and D = -2.