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# Transfer Function Question

## Transfer Function Question

(OP)
Why we don't need the initial conditions for the transfer function ?

Thanks
Replies continue below

### RE: Transfer Function Question

As commonly used the transfer function is basically the frequency response of a single input/single output system.

You can extend the concept to generalized time response including initial conditions, but that is more for specific applications and performance modeling, as the transfer function is now a matrix.

### RE: Transfer Function Question

(OP)
I mean when we convert the differential equation of a system into a transfer function form by Laplace transform we always put the initial conditions of the states equal to zero, the question why we always assume they are zeros in the transfer function.

Thanks

### RE: Transfer Function Question

How would you possibly assign values to states of a system that you haven't yet analyzed? The possibly plausible state that you can assign is one where everything is zero.

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### RE: Transfer Function Question

Use of null initial conditions is only common when considering the dynamic steady-state response, whether Laplace, Heaviside, Z-transforms, or the sinusoidal steady-state where the effect of the initial conditions have damped out.

Ignoring them is easier for the student and suitable for a broad range of practical problems, but hardly a complete solution to your problem.

### RE: Transfer Function Question

As far as a single-input, single-output system's output signal is concerned, the complete output response can be thought of as the sum of two terms: the zero-state response and the zero-input response. The zero-state response (i.e., the response when all initial conditions are zero) is the product of the transfer function and an arbitrary input signal. The zero-input response (i.e., the response when the input signal is zero) is due to the stored energy in the non-zero initial conditions. Thus, transfer functions have no dependence on initial conditions, therefore transfer functions may be defined as the ratio of the output of an LTI system to its input in the complex frequency domain with zero initial conditions.

Given a linear time-invariant differential equation model, any non-zero initial conditions must be included when taking the Laplace transform of any derivative terms. This is how they end up in the system output signal as the zero-input response term. However, they end up dropping out of the response equation if set to zero, and only the transfer function and its accompanying arbitrary input signal remain.

Zero initial conditions can be created by using deviation variables, finding the zero-state response, then adding the initial conditions back to the zero-state response. Deviation variables can be created by subtracting the steady-state equation from the differential equation, and are often created by linearization of nonlinearities that are differentiable around the linearization point.

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### RE: Transfer Function Question

(OP)
Thank you for all replies.

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