an embodiment of the present invention will now be described in more detail . before this , however , a theoretical introduction to backlash will be given . after the description of the automatic backlash estimation of the disclosed embodiment , certain industrial tests will be briefly referred to . finally , backlash compensation — which is advantageously performed based on the estimated backlash as made available thanks to the present invention — will be discussed . reference is again made to the control loop and deadband illustration shown in fig1 and 2 . the describing function y n of a backlash is : where a is the input amplitude and d is the backlash given in fig2 . the negative inverse of the describing function of the backlash is shown in fig4 ( solid line ). fig4 also shows the nyquist plots of two loop transfer functions obtained when the integrating process p 1 = e − 0 . 2s /( s ( 1 + 0 . 8 s )) ( dashed line ) and the stable process p 2 = 1 /( 1 + s ) 4 ) ( dash - dotted line ), respectively , are controlled by pid controllers . from this figure it can be concluded that integrating processes controlled by controllers with an integrator result in limit cycles . on the other hand , stable processes , reasonably tuned , will not give limit cycles . since d is divided by a at every position where it appears in ( 1 ), the shape of the describing function is independent of d . this has an interesting consequence . it means that the magnitude d of the backlash will influence the oscillation amplitudes , but since the intersection with the nyquist plot occurs at the same position , the oscillation period will remain the same independent of the magnitude of the backlash . throughout this document , it is assumed that the controller c in fig1 is a pid controller . this is most often the case in practice . however , the results presented herein can quite easily be modified to other controllers having integral action . where u is the controller output , y sp is the setpoint , y f is the filtered process output , and the controller parameters are gain k , integral time t i , and derivative time t d . the controller has setpoint weights equal to zero in both the proportional term and the derivative term . this is common in industrial controllers . the process output is filtered through a second - order low - pass filter gjghjghj where y and y f are the laplace transforms of the process output and the filtered process output , respectively . a second - order filter is used to guarantee high - frequency roll off in the controller , and the filter - time constant is t f = t d / 5 . if a pi controller is used , it is suggested to use the filter - time constant t f = t i / 10 . the following two examples illustrate the problems caused by backlash in the feedback loop . is controlled by a pid controller of the form ( 2 ) with parameters the controller parameters are derived using a design method known as migo . a backlash of 5 % ( d = 0 . 05 ) is introduced in the control loop . fig4 shows the nyquist plot of the loop transfer function and the negative inverse describing function of the backlash . the curves intersect , which indicates that a limit cycle will occur . the describing function analysis predicts a limit cycle with an amplitude in the process output signal of 4 . 4 % and an oscillation period of 7 . 7 s . fig5 shows the results of the simulations , where a setpoint change is made at t = 0 and a load disturbance is applied at the process input at t = 100 . as seen in the figure , the control loop oscillates . the amplitude of the process output is 3 . 2 %, and the oscillation period is 5 . 7 s . this is fairly close to what was predicted by the describing function analysis . is controlled by a pid controller of the form ( 2 ) with parameters the controller parameters are derived using the migo design method . a backlash of 5 % ( d = 0 . 05 ) is introduced in the control loop . fig4 shows the nyquist plot of the loop transfer function and the negative inverse describing function of the backlash . the curves do not intersect , which indicates that no limit cycle will occur . fig6 shows the results of the simulations , where a setpoint change is made at t = 0 and a load disturbance is applied at the process input at t = 100 . furthermore , noise with a standard deviation of 1 % is added to the process output . the figure shows that even though there is no limit cycle as in the previous example , there is a severe deterioration of the control caused by the backlash . because of the noise , the control error will never settle . the control signal has to pass the dead band every time the rate of change of the process input is to be reversed . this means that there will be low - frequency disturbances of the process output . the describing function analysis and the examples illustrate the control problems that arise when backlash is introduced in the control loop . control loops where integrating processes are controlled with controllers having integral action will go into a limit - cycle oscillation . these oscillations may be detected by oscillation detection procedures known per se . except for extremely lag - dominant processes , control loops for stable processes will normally not go into limit - cycle oscillations . however , the control performance is deteriorated even in these cases . this is illustrated in fig7 . the figure shows how the iae (“ integrated absolute error ”) value ( left diagram ) and the peak error e max ( right diagram ) increase at load disturbances caused by backlash appearing in the control loop presented in example 2 above . the solid line corresponds to a load change of 10 %, and the dashed line to a load change of 20 %. both the iae value and e max increase as the backlash d increases , even though e max is very noise sensitive . the increase is dependent on the magnitude of the load disturbances . the results agree well with those that are previously known . the backlash introduces a dead time in the control loop . the length of this dead time is dependent on several states and parameters . the dead time appears only when the control signal action is reversed . the dead time is the time it takes for the control signal to pass the dead band . a low integral gain k / t i gives a long dead time . the dead time becomes short if the control error is large . it means that the dead time is shorter for large load disturbances than for shorter . this explains the results shown in fig7 . stable loops with backlash are normally not detected by oscillation detection procedures , since the oscillation amplitude is quite small . a new automatic detection and estimation procedure for these processes will now be presented . a method for automatic backlash estimation according to one embodiment will now be described . the method treats only stable processes . as mentioned before , integrating processes with backlash will result in an oscillating control loop , which can be detected by procedures known in the art . the automatic backlash estimation method generally operates on a control loop as shown in fig1 . the controller c is a pid controller ( or any other type of controller with integral action ), operable to generate a control signal fig8 shows a part of the simulation given in fig6 . the process output signal y has been filtered through the filter ( 3 ). this means that the process output signal presented in fig8 is the signal that enters the pid algorithm . the signals show the typical pattern obtained when a stable process is controlled by a controller having integral action and when there is backlash in the control loop . the process output signal y is a distance δy from the setpoint while the control signal u drifts through the dead band caused by the backlash . when the control signal has changed an amount δu , the process output signal is moved towards the setpoint y sp . the time instances when the process output signal y crosses the setpoint are marked at t = 40 and t = 60 in fig8 . the time between these zero crossings is δt = t i + 1 − t i . the change δu of the control signal is mainly caused by the integral part of the controller c . this means that if the signals change slowly , the process dynamics can be neglected , and the relation between the process output signal y and the control signal u is mainly determined by the static process gain k p . in more detail , k p is the relation , after transients , between a change in y and a change in u , the change in the former being caused by a change in the latter . thus , the relation is δy = k p δu true , ( 9 ) where δu true is the part of δu where the backlash is closed and the valve moves . this means that from equations ( 7 ) to ( 10 ), the following equation for estimating the backlash is obtained : the backlash estimator ( 11 ) assumes that the signals change slowly . a convenient way to check this is to see if δt is long compared to the closed - loop time constant of the entire control system ( including the controller c and the controlled process p ). since the closed - loop time constant of the entire control system is typically not known , beneficial use is made of the fact that t i , which is one of the controller parameters of the pid controller c and thus is known , is closely related to the closed - loop time constant . therefore , the disclosed embodiment involves a step of verifying that both the control signal u and the process output signal y are slowly changing signals by testing whether δt ≧ 5t i . estimation is only performed if this criterion is met . the information required to determine the backlash on - line is the controller parameters k and t i , and the static process gain k p . further , it is necessary to measure δy from ( 8 ), i . e . to integrate the control error e between zero crossings , and the time δt between these zero crossings . the fact that the process gain k p is used in the algorithm needs some consideration , since this gain is typically not known . on the other hand , the estimate { circumflex over ( d )} is quite insensitive to errors in the estimate of k p . to see this , rewrite ( 11 ) to the first term inside the brackets is always greater than 5 , since it is assumed that δt ≧ 5t i . for well - tuned controllers applied to processes that are not delay - dominated , the product kk p is normally larger than 0 . 5 . this means that the first term in ( 12 ) dominates and that the backlash estimator ( 11 ) is insensitive to errors in k p . since industrial controllers normally work with normalized signals , k p = 1 is conveniently used as a default value . it is important that the noise does not cause zero - crossings . therefore , the process output signal y is not only filtered by the second - order filter ( 3 ), but an additional second - order filter is applied before the signal is treated in the estimation procedure . in the examples presented in this document , the time constant of this latter filter is t i / 2 . on - line procedures like this should have a security net . in the disclosed embodiment , one element of such a security net is load disturbance prevention , since such disturbances may deteriorate the backlash estimation . to check that the process output signal has a form similar to the one in fig8 , estimation is only performed when e max ≦ 2δy , where e max is the absolute value of the largest control error in the interval [ t 1 , t i + 1 ]. other elements of this security net may have to be developed during industrial field tests . thus , the automatic backlash estimation method according to this embodiment comprises the following steps , which are also shown in the flowchart diagram of fig9 : in a first step 100 , necessary variables and counters are initialized . then , the process output signal y ( filtered as described above ) is monitored in step 110 for a first zero crossing with respect to the set value y sp . once the first zero crossing has been detected , the execution proceeds to step 120 , where the control error e is integrated until a second zero crossing has been detected in step 130 . then , in step 140 , the time interval δt between the first and second zero crossings is determined . the afore - described check for slowly varying signals y , u is performed in the following step 150 . the execution ends if the test fails ; otherwise the execution proceeds to step 160 , where δy is calculated as explained in ( 8 ). step 170 is the load disturbance prevention step which was explained above . if the load disturbance is deemed to be small enough , i . e . if e max ≦ 2δy , the backlash estimation { circumflex over ( d )} is calculated in step 180 ; otherwise the execution ends . a skeleton code describing the backlash estimator is given in fig1 . the functionality of the automatic backlash estimation method may be performed by any suitable equipment which is available in an actual implementation . for instance , as will be seen later with reference to the industrial tests and fig1 , the functionality may be performed by appropriate programming of a processor or other logic device in the hardware equipment that implements the pid controller c ( as realized by a pid1 controller 32 of a control loop 30 in fig1 , the pid1 controller having a programmable cpu 33 with associated memory 34 capable of storing program code and work data ). alternatively , again as seen in fig1 , the functionality of the automatic backlash estimation method may be performed by a conventional computer 50 ( such as a personal computer , workstation , laptop computer or hand - held computer ), when appropriately programmed and coupled to the controller c ( pid controller 32 ) of the controlled process p . the result of the automatic backlash estimation method , i . e . the backlash estimation d , may be presented in any suitable manner to a control system operator ( see 52 in fig1 ), for instance by way of software running in the computer 50 , and / or by way of a user interface of controller c ( pid1 controller 32 in fig1 ). the result may be used in different ways , as desired in an actual implementation . it may be used as a simple indication of detected backlash ( for instance when { circumflex over ( d )} exceeds a threshold value indicative of noticeable backlash ), or an estimation of the amount of backlash present , or as an alarm trigger to service / maintenance personnel that the controllable member ( e . g . valve 38 in fig1 ) needs service , repair or replacement , or as input data to compensation steps of a backlash compensation method performed in association with the backlash estimation method . see the last section of this description for further information on backlash compensation . thus , to conclude the above , a new automatic method ( on - line procedure ) for backlash estimation has been presented . it is given by equation ( 12 ), and some further features are summarized in fig9 and 10 . the estimation method is automatic , which is believed to be beneficial for its acceptance in the process control industry . the estimation method can be used in many ways . first of all , it can be used as a detection procedure in a way similar to oscillation detection procedures known per se . according to the method , the control performance between zero crossings is studied . it can be concluded that backlash is present in the loop if the rate of backlash detections gets high . if backlash is detected , and if the estimated backlash values are close , then one can also draw a conclusion about the amount of backlash . this is necessary if the goal is not only to detect backlash , but also to compensate for it . if there is stiction present in the control loop , the backlash estimated by the estimation method is the sum of the backlash and the dead band caused by stiction . this is believed to be a good feature , since a backlash compensator will then not only compensate for the backlash , but also for the stiction . in the derivation of the backlash estimator above , it was assumed that a pid controller was to be used . however , it is straightforward for a skilled person to modify the method to other controllers having integral action . the backlash estimation method has been tested on a flow control loop in a paper mill . this environment is schematically illustrated in fig1 , which has already been referred to above . the process section is a pipe 12 where pulp is transported from a recycling - pulp tower 10 to a tank 60 . a pid controller ( pid 1 ) 32 controls the pulp flow through a valve 38 and therefore forms one control loop 30 . the process output y is the pulp flow , measured in the range 0 - 900 m 3 / h by a flow detector ( f ) 36 , and the controller output u is in the range 0 - 100 %. the setpoint y sp is external and is given by a level controller ( pid 2 ) 42 at the tank 60 downstream . this means that the flow controller 42 forms a slave control loop 40 in a cascade configuration . the pulp flow is driven by a pump 28 which is controlled by a pressure controller ( pid 0 ) 22 involved in another control loop 20 and using a pressure detector 26 . the flow and the pressure controllers interact . to reduce this interaction , the bandwidth of the flow loop , which normally is quite fast , has been reduced by introducing a low - pass filter with a time constant of 20 s in the loop . a manual test was performed to check the amount of backlash in the valve . the result is shown in fig1 . the controller output is first increased to ensure that the gap is closed . since the flow increases , the gap is closed when the control signal is at the final value u = 39 %. the controller output is then reversed and decreased in steps of 1 %. the first steps do not result in any flow decrease , indicating that the control signal is inside the dead band . however , the step made from the value u = 36 % gives a flow decrease , showing that the gap is closed near this value of the control signal . the test shows that the backlash is around d = 3 %. the flow controller is a pi controller with parameters k = 0 . 6 and t i = 28 s . the signals used in the controller are normalized to the range [ 0 , 1 ]. the static process gain was estimated to k p = 1 . 3 . fig1 shows the result of a recording made for about 4000 s . the upper graph shows the external setpoint ( noisy signal ) and the process output ( flow ). the lower graph shows the control signal . the estimated backlash values are indicated in the upper graph . the loop is oscillating because of the oscillating setpoint . the setpoint oscillations are probably generated by the flow variations caused by the backlash . fig1 shows that backlash was detected five times during the test , with backlash estimates ranging from 2 . 5 % to 3 . 1 %. these estimates are close to those obtained from the manual tests in fig1 . setpoint variations may disturb the backlash estimator . if the test e max ≦ 2δy were not present , fifteen detections would have been obtained during the test , and especially the last two major setpoint changes would have given backlash estimates that are far too large . to get rid of the disturbances caused by the external setpoint , experiments with a fixed internal setpoint were also performed . fig1 shows the results of such a test . the upper graph shows the constant internal setpoint and the process output ( flow ). the lower graph shows the control signal . the estimated backlash values are indicated in the upper graph . comparing fig1 and 14 , it is clear that the oscillations caused by the external setpoint have disappeared . fig1 shows that there are some low - frequency disturbances present . they are probably caused by interaction from the pressure control loop . three backlash detections were made during the experiments , with the estimates 1 . 6 %, 2 . 4 %, and 2 . 4 %, respectively . these values are slightly lower than those obtained in fig1 . this is expected , since the setpoint variations in the previous example amplifies the effect of the backlash . furthermore , already the derivation of the method and simulation examples conducted have shown that the backlash estimates are expected to be conservative . to summarize , the industrial tests have shown that the backlash estimation method works also in an industrial environment with difficult low - frequency disturbances . to obtain a robust method for an actual implementation that is automatic in the sense that no user interaction is needed , additional elements of the security net may have to be added , as is well within reach of the skilled person . when it is discovered that a control valve has got so much backlash that the control is deteriorated , the best action to take is , of course , to replace or repair the valve . the fact that the amount of backlash normally increases with time makes this even more important . however , to replace or repair a valve normally means that the production has to be stopped . for this reason , and for the economical reason that it is of interest to use a valve for as long time as possible , it is of interest to compensate for the backlash . a control valve will normally not move on its own or when the control signal is constant , unless the actuator is undersized or the positioner is unstable . therefore , the position of the control signal with respect to the backlash is given by the control signal and its history . this means that the backlash is an invertible nonlinearity . one way to compensate for the backlash is to make the control signal jump through the backlash every time the control action is reversed . the compensation can be seen as a feedforward compensation u = u fb + u ff , ( 13 ) where u is the controller output , u fb is the feedback term , e . g . the output from the pid controller ( 2 ), and u ff is the term compensating for the backlash . this compensation is not realizable in a noisy environment . a possible modification is to filter the control signal before taking the derivative . it gives the following compensation where u f is the filtered control signal . note that the gain of the compensator is changed from the true backlash d to a value δ , where δ ≦ d . the filtering of the control signal will introduce a delay in the detection of the sign changes of the control signal rate . this means that the control signal has already started its way through the backlash . therefore , the compensation must be smaller than in the ideal case . there are other possibilities to perform the backlash compensation . in ( 15 ), the control signal u has been passed trough a low - pass filter to reduce the noise introduced in the controller by the process output y . inside the controller , the measurement signal is fed through a high - pass filter because of the derivative term . so , the noise level is first amplified and then reduced by the low - pass filter . a more direct way is to base the feedforward on the measurement signal directly . one approach that will be used in this document is where the control error is e = y sp − y f and y f is the filtered process output given by ( 3 ). when the control error e changes sign , so does the rate of the integral term in the controller . therefore , the feedforward ( 16 ) can be seen as an approach where only the noise - insensitive integral part of the controller is considered , and the noise - sensitive proportional and derivative parts are excluded from the compensation . the backlash compensation will now be illustrated for the two examples from a previous section of this document . consider the control problem in example 1 . a backlash compensator of the form ( 15 ) is added to the controller . the filtered control signal is generated as this is a filter with relatively high bandwidth . on the other hand , the process output is noise free in this example . because of this , the gain of the compensator was chosen equal to the backlash , i . e . δ = d = 0 . 05 , so the compensator coincides with the ideal compensator ( 14 ). the results of the simulations are given in fig1 . comparing fig5 and 15 it is obvious that the backlash compensator gives an almost ideal compensation in this noise - free case . consider the control problem in example 2 . a backlash compensator of the form ( 16 ) is added to the controller . in this example , the compensation is more complicated than in the previous example , since the process output is corrupted with noise . fig1 shows the result when a compensator with δ = d = 0 . 05 is used . it is clear from the figure that the gain of the compensator is too high , and that the compensator causes the loop to oscillate . reducing the compensator gain to δ = 0 . 4 gives the results shown in fig1 . this compensator gives a process output that is almost unaffected by the backlash . the control signal has some high - frequency shifts at certain periods . this could have been avoided by adjusting the filtering of the process output . on the other hand , these variations do not cause any valve movements because of the backlash . fig7 shows that a backlash of 5 % gives an increased iae value of about 45 % when the load changes 20 %. with the compensator , this increase is reduced to about 15 %. backlash ( and stiction ) in control valves is a major problem at the loop level in process control plants . there are two aspects of the problem . first of all , the nonlinearities deteriorate the control performance . but , an additional problem is that the loops facing these problems often remain undiscovered by the personnel in process control plants . procedures for detection of stiction and for compensation for stiction have been available for a number of years , and they are used in many industrial plants today . compensation for backlash is simple as such , but the known procedures are seldom used in process control plants . the major reason for this is that no backlash detection and backlash on - line estimation method have been presented . this document has presented an on - line method for detection and estimation of backlash in control loops . the method is automatic in the sense that no information has to be provided from the user . the only information needed except for the signals in the control system are the controller parameters . the effectiveness of the method has been demonstrated through simulations and industrial field tests . the invention has mainly been described above with reference to a few embodiments . however , as is readily appreciated by a person skilled in the art , other embodiments than the ones disclosed above are equally possible within the scope of the invention , as defined by the appended patent claims .