Significant of S plane??

[sanaullah2015]

Hellow..

What is the significant of S plane in Control theory..
Being informed about the use of laplace for easier approach, what else mainly is concerned as we do our analysis on S plane?
My teacher told me , that For desiginning a system with your required specification in time domain, you can easily place poles in allowable regions and obtain a transfer funcation from that..

Knowing this,, i still want to know excluding this, How s plane help us in COntrol engineering??
What poles and Zeross shows??
the location actually?
The significant?? A detail answer is required...
Thanks

 

[snafflekid]

here is a page on filters that goes into some detail about the s-plane. The s-plane is shown in 3D which I found enlightening.

You will note that s is the complex-frequency variable σ + jω. A lot of the time engineers are only concerned about the sinusoid response of a system and in this case s in a transfer function is replaced with jω (which is equivalent to setting σ = 0), but systems also have a step response, σ.

If you do the Laplace transform of the transfer function, you will see something like e^st. This is equivalent to e^(σ + jω)t = (e^σt)(e^jωt). If the roots are negative (like they will be if in the LHP) σ is negative and the time response of (e^σt) is a decaying exponential waveform. The portion e^jωt is our friend the Euler identity.

 

[sanaullah2015]

Thanks for your post,your precious time and Concern.
I really appreciate...

The link is not mention in your reply

The answer still does not elaborate my question..!!!
I still have many questionss..
I m not gettiing a clear concept.
Kindly Help me .
Please send me links ,tutorials, books as well as a descriptive answer.

looking forward....

Regards

Sanaullah khan

 

[LvW]

Hi sanaullah2015

I suppose you didn't get a suitable answer up to now, because it's really not easy to formulate an answer in short. Nevertheless, I have tried it.

The following is based on a second order system. But it can be extended to higher order because each function can be split into several second order parts. Only poles are considered, for zeros some similar aspects can be formulated.

Some key aspects are listed in the following:

*The differential equation of a system with two energy storage elements (capacitors, inductances) always has a solution in time which includes an e-function like exp(s*t). It can be shown that very often the factor s has conjugate-complex properties, thus s=sigma+j*w. This happens when the system is able to perform some decaying oscillations. Think of a simple LRC system.

* Since exp(jwt) is identical to a sinusoidal wave, s can be called „complex frequency“ and the other part exp(sigma*t) is real and determines the amplitude properties (sigma negative for a decaying wave).

* It is a fundamental rule of system theory that the characteristic polynom corresponding to the above mentioned differential equ. is identical to the denumerator of the transfer function of the corresponding 2nd order system, if the normal frequency w is replaced by this new variable s >>> H(s).

* Therefore, the zeros of the characteristic equation are identical to the zeros of the denumerator; and these zeros are poles (infinite) of the transfer function H(s). In many (not in all) cases these poles are complex.

* Thus, the poles of this function H(s) play an important role as far as the system response is concerned. It is, therfore, interesting to see a graphical representation of these poles in a corresponding complex s-plane. They are either real or conjugate-complex.

* But the most important point comes now: It is easy to show that the position of these poles in the s-plane can be described by two parameters which appear also in the transfer function:
(1) The so called „pole frequency“ wp is the magnitude of the vector directed towards the pole position, and
(2) The pole Q (quality factor Qp) is identical to the inverse of 2*cos(phi) with phi=angle between the negativ-real axis of the s-plane and the mentioned vector.

* The great advantage: Both parameters (wp and Qp) appear in the denumerator of the transfer function if the function is written in ist „normal form“ which means: polynom in s and „1“ as the constant term:
N(s)=1+s/(wp*Qp)+s*s/wp*wp.

* That means, now we have two parameters which can be easily visualized in the s-plane and which can be used to describe the time response of a system as well as the frequency response. More than that, these parameters wp and Qp can be easily measured.

* Both parameters are extensively used to describe the response of second order filter functions.
For example: Rising Qp-values (poles get closer and closer to the Im axis) cause a step response which higher overshoot and peaking of the transfer function magnitude. If Qp>>>infinite (pole on the Im axis) we have an oscillator.

* This describes in short the relevance of the pole location in the complex s-plane.

(Fine)

 

[sanaullah2015]

Thank you very much Sir..!!
It really helped.

What are dominant poles?? and their significant in designing ?? and what mainly concern we have when we use different techniques of Performance index?
actually asking the concept of Performance index and its importance in diesgining?

Thank you again for your efforts.

Regards

Sanaullah Khan

 

[LvW]

.........
What are dominant poles?? and their significant in designing ??

Dominant poles are those poles (in fact a conjugate-complex pair) which mainly determine the circuit behaviour, for example like step response.
This is the pole pair with the largest time constant (smallest pole frequency) and, therefore, the pair which is located near the origin of the s-plane.

 

[snafflekid]

Here is the link I forgot:

**broken link removed**



This board is giving me a "no post mode specified" error! So frustrating!!!