The use of transfer functions in control theory

Question

I've gone through a control theory course, and after looking back at what we covered and how the subject was taught to us, I still have quite a few questions (which I unfortunately did not ask in class). As such, I thought I'd ask a few of them here, as separate posts, in order to clear up a few points that I don't understand.

Why are transfer functions used? They seem extremely cumbersome to derive from the differential equations that describe a system's dynamics. Why don't we apply our control laws directly to the differential equations themselves? Are transfer functions still used when designing modern control systems?

Answer 1

I think of transfer functions as the ratio of a system's output to its input in the frequency domain. They are not cumbersome to derive from a differential equation describing the system. In fact, the transfer function coefficients of the numerator are obtained from the coefficients of the input variable in the corresponding differential equation. The coefficients of the denominator are obtained from the coefficients of the output variable in the differential equation representation.

"Modern" control theory more often uses differential equations, after converting to what is known as the state space representation. State variables are the smallest set of linearly independent system variables that, along with system inputs, can be used to describe all other variables in the system.

The main advantage of the state variable representation is that it can handle multiple inputs and multiple outputs more easily, and that it can deal with nonlinear systems. Handling nonlinear systems is extremely important, as no system is linear. That being said, classical control techniques are still valuable as they allow us to simplify problems and design less complicated controllers (sometimes).

Answer 2

When the initial conditions of a linear system are zero, then the response to any input can be obtained by taking the convolution integral between the input and the impulse response of the system. However convolution in the time domain is identical to multiplication in the frequency domain. So by taking the Laplace transform of both the impulse response of the system and the input, then the response of the system can be obtained by multiplying these two and taking the inverse Laplace transform. Here the Laplace transform of the impulse response of the system is the same as the transfer function of the system. In practice one will usually never calculate the Fourier transform of the input, but it does show where it comes from.

I will only limit the rest of my answer to single-input single-output systems. When the transfer function of such a system is represented graphically, then usually $s$ is substituted by $j\,\omega$. This information is used for example both for a Bode and for a Nyquist plot. For physical/real systems a Bode plot is identical to the frequency response function (FRF) and can be measured relatively easy. Namely input a sine and measure the gain and phase shift of the sine wave of the same frequency in the output (this can only be done if the open loop system is stable). There are also more advance techniques, but what I am trying to say is that this information can be obtained without having a model of the system. Now when designing a controller the Nyquist plot could be used in order to determine the stability and margins of the closed loop system. For this one can multiplying the FRF of the controller by the measured FRF of the system. So without having a model of the system it is possible to design a controller which should give the closed loop system certain properties.

Yes, there are more advance techniques, but those often require having a good model of the system. This might require more experiments, which might cost more money then what would be saved by using a better controller. Also I have heard that in practice 90% of the controllers are PID, which can be tuned by just using FRF data.