The reason behind transfer function representation valid for only LTI system


One of the major advantages of state space representation is the fact that it can represent non-LTI system as well. I tried searching a lot to find a reason but couldn't get a reason explicitly stating why tf method is not valid for non-LTI systems.

Best Answer

The reason why transfer functions work so well for linear time-invariant (LTI) systems (and don't for non-linear systems) is that they are the Laplace transform (or, in discrete time) the Z-transform of the system's impulse response, which completely characterizes the behavior of such systems. I.e., the impulse response, or, equivalently, the transfer function is all you need to know to compute the response of the system to any input signal. The reason for this is that any input signal can be written as an integral (or a sum in discrete time) of scaled and shifted delta impulses, and due to linearity and time invariance, the response to this integral/sum equals the integral/sum of scaled and shifted impulse responses:

$$x(t)=\int_{-\infty}^{\infty}x(\tau)\delta(t-\tau)d\tau\quad\Longrightarrow\quad y(t)=\int_{-\infty}^{\infty}x(\tau)h(t-\tau)d\tau\tag{1}$$

where \$x(t)\$ is the input signal, \$y(t)\$ is the system's response, and \$h(t)\$ is the impulse response. Equation (1) is the convolution integral, which completely describes the system's behavior. In the transform domain (i.e., Laplace transform, Fourier transform, or Z-transform), convolution becomes multiplication.

For non-linear systems you can of course compute or measure the system's response to an impulse, but this function does not tell you anything about the system's response to other input signals, or even to a scaled impulse. This is why the impulse response and its transform do not have any significance for non-linear systems.