Question

The transfer function model of a system is the reshaping of the differential equations of which type of system?

• A. linear time varying
• B. linear time invariant
• C. non-linear time invariant
• D. non-linear time varying

Hint:

Transfer functions are restricted to Linear Time-Invariant (LTI) systems. For non-linear or time-varying systems, state-space representations or numerical simulation techniques must be used instead.

Answer 1

The Correct Option is B

Concept: The transfer function of a continuous-time system is defined as the mathematical ratio of the Laplace transform of the output variable to the Laplace transform of the input variable, assuming all initial conditions are exactly equal to zero. The system must satisfy two fundamental conditions to be modeled by a standard transfer function: * Linearity: The system must obey the principles of superposition and homogeneity. * Time-Invariance: The parameters of the system must be constant over time, meaning a time shift in the input produces an identical time shift in the output.

Step 1: Analyze the requirements of the Laplace transform.
Consider a general $n^{\text{th}}$-order differential equation relating an input $u(t)$ to an output $y(t)$: \[ a_n \frac{d^ny}{dt^n} + a_{n-1} \frac{d^{n-1}y}{dt^{n-1}} + \dots + a_0 y(t) = b_m \frac{d^mu}{dt^m} + \dots + b_0 u(t) \] To take the Laplace transform and factor out $Y(s)$ and $U(s)$ into an algebraic ratio: 1. The coefficients ($a_i, b_j$) must be constants independent of time. If they varied with time (a time-varying system), the transformation would produce complex frequency-domain convolutions instead of simple algebraic products. 2. The differential terms must be linear combinations. If non-linear terms like $y^2(t)$ or $\sin(y)$ were present, they could not be converted into standard polynomial functions of the complex frequency variable $s$.

Step 2: Formulate the standard Transfer Function.
Applying the Laplace transform with zero initial conditions yields the algebraic form: \[ G(s) = \frac{Y(s)}{U(s)} = \frac{b_m s^m + b_{m-1} s^{m-1} + \dots + b_0}{a_n s^n + a_{n-1} s^{n-1} + \dots + a_0} \] This formulation is only possible for Linear Time-Invariant (LTI) systems. Therefore, Option (B) is the correct choice.