As you know the P stands for proportional action, the I stands for Integral action and the D stands for derivative action. Now let's consider the Laplace transformation of the Integral action, it will be: Ki/s * L{e(t)} where Ki is the integral gain, L{e(t)} is the Lapalce transformation of the error. On the contrary the Laplace transformation of the Derivative action will be: Kd*s*L{e(t)} where Kd is the derivative gain and L{e(t)} has the same meaning.

At this point if we look at the Bode diagram of the Integral action it is traslated as a pole in the origin, while the Derivative action is a zero. For this reason if we try to implement a Derivative action and/or a PD we are creating a system with an infinite band (the Bode magnitude plot never go down). As you know any system is characterized by a finite band.

..........any system is characterized by a finite band.

Yes - and by a finite gain as well.

@Chitta, systems that are realizable we call PROPER. You can find the answer about the number of poles and zeroes in any control book, in wikipedia..., in both time and complex domain. Consider transfer function and its nominator and denominator, and You gonna get the answer (meaning of poles and zeroes). There are some examples in the following link: http://en.wikibooks.org/wiki/Control_Systems/Poles_and_Zeros

Regards,

Ljubomir

I disagree that PD controllers are "practically realizable". Maybe you're talking about "ideal PD controllers". Yes, ideal PD controllers cannot be realized due to the reasons mentioned by our colleagues here. However, there are already many methods that can be employed in order to create a practically-suitable PD controller for engineering applications. But, these PD controllers will only operate on a certain range of operation that is normally specified during the design stage.

"I disagree that PD controllers are NOT...."

....knowing the terms is important.

Thus, I think: PD is PD, full stop.

In reality, because analog PD is not realizable we always use PD-T1.

I am not very familiar with digital control - thus my question: Is there digital implementation of PD?

This discussion suffers from the fact that "practically realizable" is an undefined term. For all sensible definitions/explanations I can think of, D is practically realizable to the same degree as any linear system is.

A transfer function is called realizable when we are able to write a state space realization for it. Note that we are not able to write a state space representation for any improper transfer function, therefore these types of transfer functions (ideal PD included) are not realizable

... the question refered to "practically realizable". If you look at, e.g. electrical networks containing inductor cut-sets or capacitor loops (which are just perfectly realizable in practice) you see that "practically realizable" is certainly different from "realizable by a state space system".

In practice, improper systems aren't realizable. An easy way to check is by noting that they imply in infinite gain for infinite frequency (or very large gains for large frequencies).

Note that not being able to write a state space realization for a system, means that it cannot be described a set of ordinary differential equations and I thought (maybe in a wrong way) that could not be "physically realizable".

Victor, I infer from your comments that you call a system described by linear state equations realizable even though you cannot realize linearity for very large signal values. At the same time you don't want to call an improper system realizable, just because you cannot realize very large gains for large frequencies. I don't see the difference.

Electrical networks containing inductor cut-sets or capacitor loops can be described by descriptor systems (ODEs of the form Ex' = Ax + Bu, with E not the identity). They apprear quite often in practice, see e.g. the filter design handbook by Saal and Entenmann.

Thank you for the clarification. I stand corrected about the physically realizable comment I made.

From the theoretical point of view, the D part of the PID controller is not realizable because we cannot calculate the D part (that is, the derivative of the error) for the given present and past error (note that "realization" is defined under the condition that the derivative is not available and only the present and past error are available). But, from practical point of view, the D part of the PID controller can be realized by backward numerical derivative with the sampling time of the PID controller. Of course, using the numerical derivative with the nonzero sampling time is an approximation of the D part.

Even though the D part of the PID controller is approximately realizable, the ideal PID controller should not used if the sampling time is small because the output of the PID controller severely fluctuates, resulting in shortening the life of actuators such as valves because the sensitivity of the numerical derivative to noises is inversely proportional to the magnitude of the sampling time. So, commercial PID controllers usually suppress the effects of noises by adding a low-pass filter to the D part.

For details, please, refer to the following textbook.

Sung, S. W., Lee, J., & Lee, I. B. (2009). Process Identifi cation and PID Control, Wiley-IEEE Press.

because (Derivative) function is very sensitive to noise so that cant use alone but with proportional controller (P), also the nature of all physical system seems to be integral not derivative

PD controller cannot eliminate offset errors.....................

Controller design depends on the application one wants to design the controller algorithm for. In the case of PD, indeed if one desires to "realise" it using analog components this cannot be done and an approximate P+\tilde{D} (i.e. instead of D, get a HP filter; or get PD+LP filter) must be followed.

With the advances introduced by using computers/microprocessors and digital control, a PD could be realised in a computer (using numerical methods). Note though, that in case of noisy signals... noise will be largely amplified by the D term...

from a theoretical point of view: D-parts would be acausal, because we needed *now* to already have computed how *now* the signal is changing...

D-parts with lag time are realizable, though.

Chitta Behira- The PD controller is of the form G(s)=kd(s+kp/kd), which yields only a zero at -kp/kd. The transfer function is improper (zero in excess of poles), and anti causal, which is not implementable.

The derivative mode is neglected when the input error is constant.

Any transfer function which has only zeros is not realizable , realizability requires less zeros than poles (relative degree of order= pole/zero excess of at least zero). It is better to deal with noise filtering if the relative degree is greater than one. If the realtive degree is positive ( more zeros than poles) then the system is not realizable since the derivative with respect to time effect is dominant , also noise is transmitted andalso the output would depend on future inputs which is impossible to synthesize. Note that derivatives need to fix an open ball centred around the point where the time - derivative is computed and this would require to have a local definition of the signal ( to be derived with respect to time ) along a future time interval.

Dear Chitta Behera:

Effect cannot precede the cause in any physical system. The effect is felt once there is a cause to it. When the number of zeros are greater that the number of poles, it implies that the effect is felt before the cause occurred. This is the case with an ideal PD Controller. So, it is not physically realizable. To make it physically realizable, we have to introduce a derivative filter in its denominator, thus making the number of zeros equal to number of poles. Ideally number of poles should be more than number of zeros in a physically realizable system. Hope this clarifies. 

Hi Mr Behera:

It is possible to implement practically when derivative term is cascading with a first-order low pass filter [1/(N*Td*s +1 )], where N=0.1 or 0.01 and Td is derivative time constant.