Question

For the LTI ODE x(t) + 6x(t) + 11x(t) + 6x(t) = u(t)

Find the transfer function if y (t) = 3x(t) + 9x(t)

Answer 1

The transfer function H(s) for the given LTI ODE system is calculated using the Laplace transform of the input and output relations, which yields H(s) = (3s + 9)X(s) / (s^3 + 6s^2 + 11s + 6).

Step-by-step explanation:

Finding the Transfer Function of an LTI System

To find the transfer function of an LTI (Linear Time-Invariant) system, we first express the system's differential equation in terms of the Laplace transform variables. The given ordinary differential equation (ODE) is:

x(t) + 6·x(t) + 11·x(t) + 6·x(t) = u(t)

Let's denote the Laplace transform of x(t) by X(s) and the Laplace transform of u(t) by U(s). Implementing the Laplace transform and using the linearity property yields:

X(s)\(s^3\) + 6\(s^2\)X(s) + 11sX(s) + 6X(s) = U(s)

This can be simplified to:

\((s^3 + 6s^2 + 11s + 6)X(s) = U(s)\)

The output y(t) = 3x(t) + 9·x(t) in the Laplace domain becomes:

Y(s) = 3X(s)s + 9X(s)

Combining these we can obtain the transfer function H(s) = Y(s)/U(s) by dividing by the input transform expression:

H(s) = Y(s)/U(s) = (3s + 9)X(s) / (s^3 + 6s^2 + 11s + 6)

This represents the transfer function of the system for the given ODE and the output relation.