The integration of a constant value U creates a time variable output k∙U∙t, increasing (if U >0) or decreasing (if U

It means the action gradually increased to eliminate the steady state error 

Static(random) error, means a constant error in sample, and would be seen larger, for a sample,  could be reduced to elimination, if samples are integrated for a time interval, mainly used in measurement of energy ( integral 00 to..t..Wdt ).

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I havent information about it

Integral action is used to overcome the steady state error .

Static error is intrinsic of Proportional controller operation. Look at the transfer function of a Proportional-controlled process.

The PI (proportional-integral) control remove the static error, asintotically, but may nedd stability analysis to check for instability 

Using feedback to control a process, the control signal is typically (a) proportional to the error, (b) proportional to the time integral of the error, or (c) proportional to the derivative of the error.  [Error is the difference between the desired value of the process and the actual output of the process.]  A small steady state error may persist in (a), but in (b) the integral of a constant error value eventually leads to a large control input to correct it.  Note a steady error gets ignored by (c).

The integration of a constant value U creates a time variable output k∙U∙t, increasing (if U >0) or decreasing (if U

(1) Integral action enables PI controllers to eliminate the deviation between the measured system output and the set point (such deviation is also regarded as the steady-state error), while P-only controller can NOT cancel the steady-state error. Thus, PI controllers provide a balance of complexity and capability that makes them by far the most widely used algorithm in the process control applications.

Let us explore PI controller in the continuous-time domain form

C(t)=K_P*e(t)+K_I*integral e(t)dt

e(t) is the controller error, the deviation between the measured system output and the set point

K_P is proportial gain, and K_I is integral gain.

If the controller error e(t) does NOT reduce to zero, integral action will continue to update (i.e., increase or decrease) the control signal, depending on the positive (or negative) value of the controller error e(t), until the controller error e(t) reaches zero.

(2) The tutorial [Copper08] provides the description of Function of the Integral Term

While the Proportional term considers the current size of e(t) only at the time of the controller calculation, the integral term takes into account the history of the error, or how long and how far the measured process variable has been from the set point over time.

Integration is a continual summing. Integration of error over time means that we sum up the complete controller error history up to the present time, starting from when the controller was first switched to automatic.

There are challenges in employing the PI algorithm:

(I) The two tuning parameters interact with each other, and their influence must be balanced by the designer.

(II) The integral term tends to increase the oscillatory or rolling behavior of the process response.

[Copper08] D.J. Copper "Integral Action and PI Control", Practical Process Control E-Textbook, 2008.

http://www.controlguru.com/wp/p69.html

(3) The paper [HY09] uses OPNET network simulator to demonstrate the magic of integral action which solves the issue of network bandwidth under-utilization in a multi-bottleneck network.

[HY09] Y. Hong and O.W.W. Yang,  "Can API-RCP Achieve Max-Min Fair Bandwidth Allocation in a Multi-Bottleneck Network?" Proceedings of 43rd Annual Conference on Information Sciences and Systems (IEEE CISS), Johns Hopkins University, Baltimore, MD, U.S.A., March 2009, pp. 723-728. Available from the following RG Link.

https://www.researchgate.net/publication/220991920_Can_API-RCP_Achieve_Max-Min_Fair_Bandwidth_Allocation_in_a_Multi-Bottleneck_Network

(4) PID Control and Smith Predictor were listed in the “Leaders of the Pack” InTech’s 50 most influential industry innovators since the year 1774.

http://archive.today/2RoSK

PID Control was listed twice (the dominant control method in the industrial applications) -- (1) John G. Ziegler and Nathaniel B. Nichols and classical PID Control; (2) Karl Johan Åström and modern PID Control (IEEE Medal of Honor, 1993)

http://en.wikipedia.org/wiki/IEEE_Medal_of_Honor

Discussions on control system design

"What are trends in control theory and its applications in physical systems (from a research point of view)?"

https://www.researchgate.net/post/What_are_trends_in_control_theory_and_its_applications_in_physical_systems_from_a_research_point_of_view2

Conference Paper Can API-RCP Achieve Max-Min Fair Bandwidth Allocation in a M...

We use an integrator  to to  eliminate the steady state error.

All the above answers are correct. I am just adding same answer in other way. Actually, the integral term considers the history of the error, or how long and how far the measured process variable has been from the set point over the total time. Continually sum up the controller error.

Linking the integral term to the error's history is a very good point. In such perspective the derivative effect represents an estimation of the future, while the proportional is what remains: the present. Past, present, future, what more can you ask from a poor controller?

Thanks for all your answers, they are very useful.

Sincerely

An integral action can diminish fluctuation of collected data before statistical analysis. Integrals appear in many practical situations. Using definite integral given a function f of a real variable x and an interval [a, b] of the real line, is defined informally as the signed area of the region in the xy-plane that is bounded by the graph of f, the x-axis and the vertical lines x = a and x = b. The result is average or mean of values between x=a and x=b. Therefore, data variation become more smooth and statistical error of analyzing these data can be reduces. However, the uncertain error due to smoothing data fluctuation remains.

in my opinion for more accuracy

Hi

I advice you to see these documents. You will find what you need.

I hope that I helped you, let us know if you have another questions or you need more details.

Best regards

https://www.google.fr/url?sa=t&rct=j&q=&esrc=s&source=web&cd=6&cad=rja&uact=8&ved=0CEIQFjAFahUKEwjs19SUvZjJAhXEPBoKHToEAU0&url=http%3A%2F%2Farxiv.org%2Fpdf%2F1310.8620&usg=AFQjCNFjTClYTae7aAP1nkFZBcmudtjK7g&bvm=bv.107467506,bs.1,d.d24

https://www.google.fr/url?sa=t&rct=j&q=&esrc=s&source=web&cd=4&cad=rja&uact=8&ved=0CDYQFjADahUKEwjs19SUvZjJAhXEPBoKHToEAU0&url=http%3A%2F%2Fwww.cds.caltech.edu%2F~murray%2Fcourses%2Fcds101%2Ffa04%2Fcaltech%2Fam04_ch8-3nov04.pdf&usg=AFQjCNEj_zk0FFC4sOpz0OPgbYjRwa3gkQ&bvm=bv.107467506,bs.1,d.d24

https://web.stanford.edu/~boyd/ee102/int-ctrl.pdf

It change the type of the system, if your system belongs to type 0, it change s to type 1, the static error constants automatically get changed as per the new system. For example, type 0 system static error constant is Kp for the step input, but for the type 1 system it becomes zero.

In other words, adding a pole at zero will drag the system to reach the reference input. But sometimes only integral compensation makes the closed system unstable as well.