it is easy to use. you don't require to know the plant model. just sending a sinusoidal signal, you get a sinusoidal output with a different gain and phase. this will give you the plant model in terms of gain and phase. its systematically explained in the introduction in ogata ( modern control engg.). frequency domain is a subset of time domain. a time domain signal in laplace domain (where t is turned to s), is represented by the variable s. s is nothing but a complex number (s = sigma + j omega). sigma translates to exponential term and omega to sinusoidal term. in frequency domain analysis, we just use j omega (sigma = 0). we are used to just substituting j omega in place of s in frequency response. if you consider a signals approach, a bounded input signal (sigma = -ve) for a stable strictly proper system gives bounded output signal and for unbounded signal (sigma = +ve), you get unbounded output. but a sinusoidal signal is neither (sigma = 0). so we calculate the gain and phase which denote how the system is responding to the input. if the gain is low, we add gain in the controller and so on. it gets complex further so only adding gain is not sufficient. 

As the most important aspect - the frequency domain is used for stability analyses. Each control system has negative feeedback - and, hence, tends to be unstable. For improving the system stability we are using compensator blocks with the aim to get more stabiliy margin. For dimensioning these blocks (definition of parameters like gain, poles, zeros) we NEED the frequency domain. For this purpose two basic diagrams are used to control and/or shape the LOOP GAIN correspondingly : BODE diagram and NYQUIST plot.

In the linear case, the frequency domain analysis provides  easy stability tests ( Bode plot, Nyquist criterion, root locus) and " relative stability " tests, cf. gain and frequencystability margins . In many cases , it is not necessary to have the detailed plots for cheking stability properties. Bode and  Nyquist plots  are  drawn from the open -loop system while they are informative for the closed -loop system  with unity feedback without the need of manipulating the " 1+" inherent to feedback.  For  cases under non-unity feedback the use of the equivalent  open-loop transfer function instead of the original one solves the problem in a close way. note that the frequency-domain description is  directly related to the models by transfer function/ matrices which can be also obtained experimentally , in an approximate way,  by injecting sinusoidal inputs of different frequencies to the system

In general system can represent in two forms, Transfer function model  (SISO & TITO) and State space model (MIMO). Both models can be developed for any physical system by conducting some tests. For this purpose we use some standard test signals namely, step, ramp, parabolic and sinusoidal.

Dear Colleagues,

Good Day,

Frequency Domain:

Frequency domain is a method used to analyze data. This refers to analyzing a mathematical function or a signal with respect to the frequency. Frequency domain analysis is widely used in fields such as control systems engineering, electronics and statistics. Frequency domain analysis is mostly used to signals or functions that are periodic over time. This does not mean that frequency domain analysis cannot be used in signals that are not periodic.

The most important concept in the frequency domain analysis is the transformation. Transformation is used to convert a time domain function to a frequency domain function and vice versa. The most common transformation used in the frequency domain is the Fourier transformations. Fourier transformation is used to convert a signal of any shape into a sum of infinite number of sinusoidal waves. Since analyzing sinusoidal functions is easier than analyzing general shaped functions, this method is very useful and widely used.

please, see the attached articles for more detail:

https://en.wikipedia.org/wiki/Frequency_domain

http://www.slideshare.net/sagar_kamble/ppt-on-time-domain-and-frequency-domain-analysis

https://www.youtube.com/watch?v=noycLIZbK_k

http://ctms.engin.umich.edu/CTMS/index.php?example=Introduction§ion=ControlFrequency

Both time and frequency domains permit to look at the same object, the system, from two different perspectives.

Even if what we observe in real life is time-domain, the importance of the frequency domain lies in the fact that it provides analysis and design tools which are, in many cases, more intuitive than the corresponding time-domain ones. Some examples are given by zeroes and poles, Nyquist criterion of stability, loop-shaping, robust and H infinity control, ...

Despite that, I would like to remark that the frequency-domain analysis possesses one major limitation: it holds for linear time-invariant systems only! Time-domain analysis, on the other hand, permits to treat in a rigorous way non-linear and time-varying plants as well. Nevertheless, the frequency domain tools are relatively simple and systematic to use, and thus they do not require the search for a Lyapunov function which usually makes the time-domain analysis of non-linear systems quite complex.

Hi Ankur,

The frequency analysis has two roots:

a)      The empiric root: when dealing with the electronic amplifiers instability, the Bell Laboratory team could not find any explanation by time analysis. They just noticed that unexpectedly, amplifiers begin to whistle under certain conditions. The explanation turned to be very logical when Hendrik Wade Bode used the frequency analysis, by means of his well-known frequency diagrams (see the Barkhausen criterion).

b)      The theoretical root: solving manually differential equations is difficult. These manipulations may be performed much easier if using the operational calculus (mainly Laplace). The complex variable s = σ + j ω happens to include the frequency ω! You can find an obvious physical significance to the transfer function (the main frequency analysis tool) if you compare it with the frequency diagrams (for σ = 0).

The complex generalization of the Bode frequency diagrams was proposed by his colleague Harry Nyquist, who extended the electronic amplifier frequency analysis over the whole System Theory. One perfect example for the utility of this approach is the Nyquist stability criterion.

You must think at the time analysis and the frequency analysis like at a stereoscopic mental sight. Each of them have pros and contras, but when you use them together, in an appropriate manner, you will understand much better the surrounding world.

I can give you and another comparison: time analysis stands for the clinical examination, while frequency analysis stands for radioscopy. A good doctor will rely on both of them.

Best regards,

Marius Balas

Hi Ankur,

Once you have the model of your system, there are methods to deal with it, either in time domain or in frequency domain.

However, how do you get the model, when the “plant” is just some combination of amplifiers, motors, controllers, etc.? The traditional way was to supply sinusoidal signals and measure the corresponding sinusoidal output. The ratio between them (gain and phase) gives you the frequency response, because it tells you how a sine is passed by the system.  If the gain is 1 and phase-shift is zero at some frequency, you know that this frequency is passed undisturbed. Other frequencies are amplified or attenuated, phase-shifted, etc. This gives you a “feeling” about “bandwidth” namely, what frequencies are more-or-less passed.

Now, once you have the gain and phase lines as functions of frequency, you start fitting a model. It can be a transfer function, with poles and zeros, which is frequency domain, or fit a state-space representation, which takes you to time domain.

In time domain, you deal with a differential equation, xdot=Ax+Bu, y=Cx, or with the convolution integral

y(t)=int{h(t-tau)u(tau)dtau }

while in frequency (Laplace) domain, it is just a product Y(s)=H(s)U(s). So, because people have already worked hard to prepare tables with Laplace transform of signals, sometimes is worth dealing with frequency domain instead of differential equations.

Now, why do we call the abstract H(s) frequency response, although it actually seems to exist only in some abstract s space?

Because the (genial) definition of Laplace transform is such that if the time domain signal is a simple exponential u(t)=exp(wt), the output of the convolution integral comes out to be y(t)=H(w)exp(wt) . Now, if instead of real exponentials we have imaginary  exponentials u(t)=exp(jwt) (which makes the sine function), then y(t)=H(jw)exp(jwt) and the complex H(jw) gives you the gain AND phase-shift of the real time-domain signal sin(wt).

Best regards,

Itzhak

Thanks everyone for your insightful answers!

 I think the response you got from Marius Balas is great and pretty complete.

I would only add that frequency response gives you the idea of bandwidth, which is important and is not that evident from time-domain analysis. Bandwidth tells you which frequencies your system can pass and which it cannot. This gives you a ”feeling” about your system, which is not that evident from time domain representation.

Sometimes, time domain analysis may proudly end with perfect following. It seems that the system can perfectly follow any input command. Then I ask “including white noise input?” So, even though instinctively we would like unlimited bandwidth, a second thought tells us that this is not necessarily so. On the other hand, getting the desired bandwidth requires higher loop gains and this creates stability problems, so there is always a trade-off between the two.

This is not that I prefer one over the other. It is good to understand them both, as each one has its advantages. In particular if you expect even any tiny nonstationary or nonlinear elements, time-domain is a must.

Going back to frequency domain analysis, even though Bode plots are the simplest and easiest to use, it is only in simple cases, minimum-phase and where the phase and gain do not move up-and-down too much. Otherwise, it is very difficult, if not impossible, to reach a conclusion.

Therefore, it is important to really understand Nyquist analysis and Nyquist Plots, which contain all information about any system. One problem with Nyquist plots is that it is difficult to see on a given plot what happens when the gain could change from 1 to, say, 100.

Therefore, the more useful and practical tool is Nichols Plot, which is sort of Nyquist Plot yet at logarithmic scale, or Bode Plot where, instead of plotting the gain and phase on two separate diagrams, you plot the gain vs. phase on one diagram. What is good about Nichols Plot is that the “good” properties of the plant are illustrated by the fact that the center of diagram remains untouched by the plots, which pass below this region. So, you may use one controller for 100 machines, which are similar yet not identical. After some practice, you learn that you may plot them all on one Nichols diagram and, if all curves pass below this central region, all are stable. Also, if they are not too far below the region (which would mean that the machines are slow), then you can also expect satisfactory performance.

Some practice is, of course, needed, before you can agree with the above and get a real  taste of it.

Best regards,

Itzhak

why delay independent stability criteria based on frequency domain methods are more preferred for time delay systems?