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pi and pid controllers ( see , _ e.g. _ , @xcite ) are still by far the most popular feedback design in industry . to the best of our knowledge
, there is no clear - cut explanation of their strange ubiquity for a wide range of systems .
remember that , from a purely mathematical standpoint , they are only fully justified until now for first and second order linear differential equations with constant coefficients !
we solve here this long - standing and quite irritating open problem via the newly introduced _
intelligent _ pids ( @xcite ) , which have already been utilized quite successfully in several concrete situations ( see , _ e.g. _ , @xcite ) .
the proof relies on a crude time - sampling of both types of regulators .
it shows that the gains in a classic pi or pid take into account , if they are properly tuned , the estimated `` structural '' part of the intelligent controllers .
thus the efficiency of these intelligent controllers with respect to arbitrary nonlinear plants ( @xcite ) is enough for fulfilling our purpose .
let us nevertheless emphasize that the classic tuning rules are quite intricate whereas their counterparts for intelligent controllers are obvious .
only few references ( see , _
e.g. _ , @xcite ) in the huge literature on pids exhibit some connections with our viewpoint .    our paper is organized as follows .
section [ mfc ] is devoted to a brief review of _ model - free _ control and of the corresponding intelligent pid controllers .
section [ connect ] , which establishes our results by comparing the sampling of classical and intelligent controllers , gives a table for the connections between classic and intelligent gains .
the computer simulations in section [ applications ] confirms the superiority of the intelligent controllers ( see , also , @xcite for other examples ) .
some concluding remarks are given in section [ conclusion ] .
the input - output behavior of the system , which for simplicity s sake is assumed to be monovariable , is `` approximatively '' governed within its operating range by an * unknown * finite - dimensional ordinary differential equation , which is not necessarily linear , @xmath0 replace equation by the following `` phenomenological '' model , which is only valid during a very short time interval , @xmath1 the derivation order @xmath2 , which is in general equal to @xmath3 or @xmath4 , and the constant parameter @xmath5 are chosen by the practitioner .
it implies that @xmath2 is not necessarily equal to the derivation order @xmath6 of @xmath7 in equation ( [ e ] ) .
the numerical value of @xmath8 at any time instant is deduced from those of @xmath9 and @xmath10 , thanks to our quite efficient numerical differentiators , which are moreover real - time .
the desired behavior is obtained by implementing , if , for instance , @xmath11 , the _ intelligent pid controller _ ( _ i - pid _ ) @xmath12 where    * @xmath13 is the output reference trajectory , which is determined e.g. via the rules of flatness - based control ; * @xmath14 is the tracking error ; * @xmath15 , @xmath16 , @xmath17 are the usual tuning gains .
let us consider the following special cases :    * if again @xmath11 , we may use an _ intelligent pd controller _
( _ i - pd _ ) @xmath18 * if @xmath19 , we can restrict ourselves to * * an _ intelligent pi controller _ ( _ i - pi _ ) @xmath20 * * or even to an _ intelligent p controller _ ( _ i - p _ ) @xmath21    if @xmath11 ( resp .
@xmath3 ) , plugging equations or ( resp . or ) in equation
yields the control of a pure double ( resp .
simple ) integrator .
this is why tuning the gains of our intelligent controllers is quite straightforward
.    it should be emphasized , if @xmath11 ( resp .
@xmath3 ) , that equation ( resp . ) is mathematically sufficient for ensuring stability around the reference trajectory . the integral term @xmath22 in equation ( resp . )
nevertheless adds some well known robustness properties .
consider the classic continuous - time pi controller @xmath23 a crude sampling of the integral @xmath24 through a riemann sum @xmath25 leads to @xmath26 where @xmath27 is the sampling interval .
the corresponding discrete form of equation reads : @xmath28 combining the above equation with @xmath29 yields @xmath30    a trivial sampling of the `` velocity form '' of equation @xmath31 yields @xmath32 which is equivalent to equation .
utilize , if @xmath19 in equation , the i - p , which may be rewritten as @xmath33 replace , according to the computer implementation in @xcite , @xmath8 by @xmath34 and therefore by @xmath35 it yields @xmath36      * fact*.- equations and become * identical * if we set @xmath37    it should be emphasized that the above property , defined by equations , does not hold for continuous - time pis and i - ps .
this equivalence is strictly related to time sampling , _
i.e. _ , to computer implementation , as demonstrated by taking @xmath38 in equations .
extending the calculations of section [ 1 ] is quite obvious .
the velocity form of the pid @xmath39 reads @xmath40 .
it yields the obvious sampling @xmath41 if @xmath11 on the other hand , equation yields @xmath42 . from the computer implementation @xmath43
, we derive @xmath44    * fact*.- equations and become * identical * if we set @xmath45      equation becomes with the i - pid @xmath46 introduce the pii@xmath47d controller @xmath48 where a double integral appears . to its velocity form
@xmath49 corresponds the sampling @xmath50 which is identical to equation if one sets @xmath51 the connection between ipis and pii@xmath47s follows at once .
the previous calculations yield the following correspondence table between the gains of our various controllers :    [ table ]    [ cols="<,<,<,<,<,<",options="header " , ]     due to the form of equation ( [ f ] ) , it should be noticed that the tuning gains of the classic regulators ought to be negative .      the previous calculations and table 1
explain why sampled classic pi and pid controllers take into account , if their gains are properly tuned , the structural term @xmath52 , which contains all the structural information of the unknown nonlinear systems , in equations , , , .
the superiority of intelligent controllers , which was already noted in @xcite , is however confirmed :    1 .   tuning the gains of intelligent controllers is straightforward whereas it is complex and painful for classic pids in spite of all the numerous existing rules in the literature ( see , _ e.g. _ , @xcite ) .
contrarily to intelligent controllers , a correctly tuned classic pi or pid controller is unable to take into account heat effects , ageing processes , characteristic dispersions due to mass production ,  .
fault tolerant control is much better handled by intelligent controllers than by classic ones .
for the nonlinear system @xmath53 we deduce a classic pi controller thanks to a method due to broda and dindeleux @xcite which improves the well - known ziegler - nichols rules ( see , _ e.g. _ , @xcite ) .
note however that the open loop response of system , with @xmath54 , is somehow difficult to exploit as shown by figure [ x1bo ] .
it yields    * a delay system @xmath55 where @xmath56 , @xmath57 , @xmath58 ; * a pi where @xmath59 , @xmath60 .
figures [ x1pi ] and [ x1ipi ] , which depict the simulation results for the above pi and an i - pi , do not show any significant difference .
remember however that the i - pi , where @xmath61 , @xmath62 , @xmath63 , does not necessitate any cumbersome identification procedure .    without any new calibration of the pi for another operating range figure [ x1pi5 ]
shows a deterioration of the performances , whereas the performances of the i - pi , which are depicted in figure [ x1ipi5 ] , remain good .
introduce now a fault accommodation via a control power loss @xmath64 , @xmath65 , where the sampling time @xmath66 .
the i - pi behaves then much better ( figure [ x1ipipp ] ) than the pi ( figure [ x1pipp ] ) .
note nevertheless a small deviation of the i - pi controller when the power loss becomes quite important ( figure [ x1ipipp]-(b ) ) .
the above numerical simulations as well as many existing experimentations ( see @xcite , and @xcite ) demonstrate that intelligent pid controllers yield better performances than classic ones .
this is achieved moreover thanks to a quite straightforward and natural gain tuning , which contrasts with the numerous complex rules for classic pids .
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the ubiquity of pid controllers in the industry has remained mysterious until now . we provide here a mathematical explanation of this strange phenomenon by comparing their sampling with the the one of `` intelligent '' pid controllers , which were recently introduced . 
 some computer simulations nevertheless confirm the superiority of the new intelligent feedback design . 
 +   + * _ keywords_* pid , model - free control , intelligent pid , sampling .