Physical interpretation of transfer function in control theory

Question

I'm learning about transfer functions in control theory. I'm struggling to find a physical interpretation for the input and output of a transfer function, both of which may be complex numbers. In the time domain, the physical interpretation of the system solution is clear: the input is time and the output is a vector of physical state variables.

Is there a physical interpretation for the complex input and output of a transfer function?

Answer 1

Unlike Fourier transform*, Laplace transforms bears very little practical or physical insight. Actually the only reason you learn partial fraction expansion in the undergrad courses is because of Laplace Transforms.

The original insight of taking the Laplace transforms was to solve differential equations much more efficiently and in a more humane manner. But this got stuck and we are using it extremely blindly. 80% of control engineering students learn it as replace the number of dots with powers of s. It is very ill-taught and frankly you need to know lots of mathematics to start this type of queries if you want to do justice. For example, you stop worrying about the domain of convergence, analytic continuation and many many interesting features of this transform just after you learn about the definitions. If we are interested in stable systems and if s is not defined on the left half plane how is it that we discuss negative real part poles etc. See it immediately gets confusing because of the terrible motivation of such concepts. It is the same problem with Dirac's delta function. It only makes sense under the integral sine but we keep on multiplying with time functions etc. as if it is a real function. Hence, things become pretty tricky if you are not exposed to these concepts.

Another shortcoming of such thinking is that it forces control engineers think in terms of artificial causalities that bears no value in the original system. Take a clamped mass-spring-damper system, the equations that govern the motion is:

$$ m\ddot x(t) + b\dot x(t) + kx(t) = F(t) $$ now, if you follow the typical route, you get $$ \frac{X(s)}{F(s)} = \frac{1}{ms^2+bs+k} $$

here you must think that because of force we obtained a position output and hence the causality is position due to force. But obviously there is no such thing, the position and the force satisfy the same differential equation simultaneously. There is no such thing as force comes in and creates displacement.

My suggestion is that you keep this transform as a tool for solving/modeling differential equations as systems. Also, laplace transforms (or other integral kernels) are a much richer elements of mathematics. Hence, one should not overload them with artificial physical meanings.

* Fourier Transforms at least give some (and emphasis on some towards little) understanding of the steady state(!) behavior when the input is assumed to be pure sine. They are by no means indicative if you have a problematic transient regime.

Answer 2

They are the Laplace transforms of the time-domain input and output signals. In the frequency domain, the input and output signals are complex functions of a complex variable $s$. In the time domain they are real functions of a real variable $t$, different ways of representating the same signals.

It may be easier to think in terms of the Fourier transform, which is a generalization of the Fourier series to signals that are not necessarily periodic. There exist the sine series, the cosine series, and the complex exponential series, which is most convenient and leads to the complex Fourier transform. The Fourier transform represents a time-domain signal in terms of its frequency components. The transform is defined to be a function of a complex variable, to take advantage of the properties of the complex exponential. The transfer function relates the Fourier transforms of the input and the output.

The transfer function concept is the same, whether you use the Fourier or the Laplace transform. The latter is more practical in the study of control systems, because it takes care of initial conditions problems more easily, and is convenient in stability analysis.

The transfer function maps a complex function of the complex variable $s$ into another complex function - not a complex variable into another. The functions represent the input and the output in the frequency domain - as compared to the more usual representation of the input and output as functions of the time $t$.