Adaptive Internal Model Control for Mid-Ranging of Closed-Loop Systems with Internal Saturation

Transcription

1 213 IEEE/RSJ International Conerence on Intelligent Robot and Sytem (IROS) November 3-7, 213. Tokyo, Japan Adaptive Internal Model Control or Mid-Ranging o Cloed-Loop Sytem with Internal Saturation Olo Sörnmo, Björn Oloon, Ander Roberton and Rol Johanon Abtract Thi paper conider the problem o perorming mid-ranging control o two cloed-loop controlled ytem that have internal aturation. The problem originate rom previou work in machining with indutrial robot, where an external compenation mechanim i ued to compenate or poition error. Becaue o the limited workpace and the coniderably higher bandwidth o the compenator, a midranging control approach i propoed. An adaptive, modelbaed olution i preented, which i veriied through imulation and experiment, where a cloe correpondence o the obtained reult i achieved. Comparing the IAE o experiment uing the propoed controller to previouly etablihed method, a perormance increae o up to 56 % i obtained. I. INTRODUCTION An experimental etup or perorming high-preciion milling ha previouly been developed and teted in milling cenario [1], [2]. The etup conit o a conventional indutrial robot and an external high-dynamic compenation mechanim [3], which i to compenate or the error in the robot arm-ide poition that occur becaue o, e.g., trong proce orce and poor robot motion accuracy. However, ince the external compenation mechanim only ha a workpace o approximately.5 mm, the mechanim may reach it actuation limit when perorming advanced milling tak. Thu, we propoe a mid-ranging approach or controlling the relative poition between the robot and the milling tool held by the compenation mechanim. Mid-ranging i a control trategy that i deigned or the cae when two actuator ytem control the ame variable, uch a low or poition, and one o the ytem i ater and poibly more accurate, but ha a limited working range. The idea i then to utilize both ytem to control the deired variable, making ue o the higher bandwidth o the at ytem, while keeping it poition cloe to the midpoint o it working range, o a not to reach the limit [4]. In order to achieve a ytem that i robut to proce parameter variation, which may occur becaue o the trong proce orce o the milling proce and varying cutting condition, it i deirable to employ an adaptive control tructure. However, in the cenario conidered in thi paper, the two actuator are already cloed-loop ytem which contain internal input aturation. Thi doe not render the O. Sörnmo, B. Oloon, A. Roberton, and R. Johanon are with the Department o Automatic Control, LTH, Lund Univerity, SE 221 Lund, Sweden. E mail: Olo.Sornmo@control.lth.e. The reearch leading to thee reult ha received unding rom the European Union eventh ramework program (FP7/27-213) under grant agreement COMET (Re. #258769) and SMErobotic (Re. #287787). The author are member o the LCCC Linnaeu Center and the ELLIIT Excellence Center at Lund Univerity. deign o an adaptive mid-ranging controller traightorward. Motivated by thi, an adaptive internal model control cheme or mid-ranging control i preented in thi paper, with adaptive dynamic reerence governor or handling o internal aturation, making the control approach poible. The utilization o a robot in combination with an additional manipulator in a cloed kinematic chain wa invetigated in [5], where the concept o macro and micro manipulator were introduced to decribe the robot and the additional compenation mechanim, repectively. The term macro and micro manipulator are adopted in thi paper. A et o dierent mid-ranging control trategie were evaluated in [4], baed on, e.g., Valve Poition Control (VPC) and Model Predictive Control (MPC). Deign and tuning guideline o VPC and Modiied VPC (MVPC) controller were preented in [6]. Anti-windup cheme or VPC controller were introduced in [7]. Internal Model Control (IMC) wa reviewed and compared with imilar control trategie in [8], where alo everal IMC tability theorem were proven and practical tuning guideline were provided. The problem o having a control ignal aturation or an IMC controller wa conidered in [9]. Deign and tability analyi o Adaptive Internal Model Control (AIMC) wa preented in [1], and the dicrete-time counterpart wa decribed in [11]. Nonlinear approache to AIMC were invetigated in [12], a well a in [13], where neural network were utilized. The application o mid-ranging control uing IMC wa invetigated in [14], where deign rule were preented and veriied through imulation tudie. Dicrete-time Dynamic Reerence Governor (DRG) or contrained nonlinear ytem were conidered in [15], and reerence governor or ytem with input and tate aturation were preented in [16]. The method preented in thi paper i baed on [14], which i here extended by introducing adaptivity to the IMC mid-ranging tructure, inpired by [1] and [11]. Further, the control cheme i modiied to account or internal aturation, by introducing a dynamic reerence governor baed on the concept o [16], but derived uing a dierent approach. Further, in order to maintain good perormance under parameter variation, the DRG i alo made adaptive. II. METHOD Conider two table, dicrete-time cloed-loop ytem on the tandard eedback orm, ee Fig. 1, denoted H cl (z) and Hcl (z), repreenting the micro and macro manipulator, repectively. Conequently, the bandwidth o H cl (z) i /13/$ IEEE 4893

2 r H c u H p Fig. 1. Block diagram or the tandard eedback orm, with aturation on the input to the proce. In thi igure, H cl (z) i diplayed. r r C r C H cl H cl Fig. 2. Block diagram or the VPC and MVPC mid-ranging control tructure. igniicantly higher than that o Hcl (z). The controller in the cloed-loop ytem are aumed known, and the output ignal o the cloed-loop ytem, denoted y and y, are meaured. The midpoint o the micro manipulator workpace i zero. The objective to perorm mid-ranging control o the two cloed-loop ytem can be met by tandard method uch a VPC control, a decribed in [4] and [6]. The block diagram or the VPC cheme i contructed a diplayed in Fig. 2, where y r i the relative poition o the manipulator, r r the deired relative poition, and r the deired etpoint o the mid-ranged input. It i to be noted that in thi paper, ince the midpoint o the micro manipulator i zero, the input r i alo zero and will hence be diregarded. The tructure o the controllerc (z) and C (z) can be choen arbitrarily, but are commonly elected a PI controller. Experimental tuning o the controller i tediou work and even with accurate model o the procee, arbitrary pole placement i not alway poible. Internal Model Control i an appealing olution which ha been proven to yield atiactory reult in mid-ranging cenario [14]. However, a mentioned earlier, the proce parameter may change over time and it i thereore deirable to adapt the IMC controller in order to correct or the proce change. The block diagram or the mid-ranging IMC i diplayed in Fig. 3, where C and C are the controller, Ĥ cl and Ĥ cl the internal model o H cl and Hcl, repectively. It i to be noted that the notation C and C repreent Q 1 and Q 2, repectively, in the Youla parametrization o the IMC according to [14]. The r r Fig. 3. w w Block diagram or the IMC mid-ranging control tructure. y y y y r yr complementary enitivity unction T (z) i introduced a the deired repone o the ytem rom r r to y r, and T (z) i the deired repone o the ytem rom r r to y r with no input to H cl. In order to achieve the deired mid-ranging eect, the ollowing condition or the controller mut be ulilled: T (z) = C (z)h cl (z) C (z)h cl(z) (1) T (z) = C (z)hcl (z). (2) The controller are thu calculated a C (z) = T (z)+t (z) H cl (z) (3) C (z) = T (z) H cl (z) (4) where the parameter o H cl (z) and H cl (z) hould be updated in order to adapt the controller. To thi purpoe, a Recurive Leat Square (RLS) algorithm [17] with orgetting actor λ < 1 i introduced, becaue o it at convergence or input ignal o proper excitation, which i aumed to be preent. For ytem with low excitation, a Kalman ilter might exhibit better perormance. The propoed method can eaily be modiied to the utilization o a dierent etimator. The etimator i ued to continuouly etimate the proce parameter o H cl and H cl, and update the internal model Ĥ cl and Ĥ cl, a well a the controller C and C. However, a diplayed in Fig 1, H cl contain an internal aturation o the inner control ignal u (denotedu,k in thi ection, where k i current ample), with a given aturation level at ±u at. The ytem Hcl i aumed not to have an internal aturation. Once the control ignal aturate, i.e., when the ytem leave it linear region, the linear internal model can no longer accurately decribe the proce. Further, the etimation o H cl will be corrupted a a reult o the aturation, ince the input/output relation o the plant i no longer linear. Thi will lead to a ale etimate o the ytem parameter and in turn unexpected behavior, in the wort cae intability. Thi problem may be olved by implementing a nonlinear model and applying nonlinear etimation technique, which will become intricate, epecially i H cl i implemented with anti-windup. Intead, a DRG i introduced to modiy the input to the ytem, o that the ytem i never allowed to enter aturation. Thi approach make linear modeling till eaible. The dynamic o antiwindup cheme poibly implemented in the cloed-loop ytem can be diregarded, ince the ytem i deigned to never enter aturation. Conider the control tructure oh cl a diplayed in Fig. 1, where the controller and proce dynamic are given by the rational, dicrete-time traner unction H c (z) = l +l 1 z 1 + +l a 1 z (a 1) +l a z a n +n 1 z 1 + +n b 1 z (b 1) +n b z b (5) H p(z) = q +q 1 z 1 + +q c 1 z (c 1) +q c z c p +p 1 z 1 + +p d 1 z (d 1) +p d z d (6) 4894

3 where [a,b,c,d] Z. The objective o the DRG i to dynamically modiy the reerence input r,k to H cl, o that u,k u at, k. Since y,k i aumed to be meaured and Hc(z) i known, thi can be olved by imply calculating the current control ignal u,k and modiying the input to H cl accordingly. However, ince the mid-ranging controller i meant to control an exiting cloed-loop ytem, the output o H cl may be delayed rom the network connection between the controller. Thereore, a model-baed olution i propoed, that i independent o meaurement o y,k. The modiied input to the ytem i denoted w,k, and the dynamic relation between w,k and r,k i given by w,k = w,k 1 +α k (r,k w,k 1 ), (7) which i a irt order low-pa ilter with a time-varying parameter α k. When α k = 1, the ilter doe not aect the input and w,k = r,k hold true, and converely when α k =, w,k = w,k 1. Obviouly, it i required that the inner proce Hp (z) contain at leat one integrator, in order or the ytem to be able to reach any deired et-point. When u,k u at, k, i atiied, the control ignalu,k i given by e q iz i H c u,k = 1+HcH p w,k = i= w,k (8) p iz i i= where e = a + d and = max(d + b,a + c). In order to determine α k, the predicted control ignal with unaltered reerence i denoted by û,k and deined a û,k = u,k αk =1. I û,k u at, there i no need to alter the input, and thu α k = 1. Otherwie, the deired control ignal hould be a large a poible, i.e., ±u at. The deired control ignal in the current time-tep k i denoted u d,k and deined a u d,k = gn(û,k)u at which together with (7) and (8) give the expreion or α k : { p u d,k +χ(u,k,w,k ) α k = q (r,k w,k 1 ), û,k > u at (9) 1, û,k u at where χ(u,k,w,k ) = p iz i u,k i=1 e q iz i w,k q w,k 1 i=1 (1) It i clear rom (9) that q mut be ulilled, i.e., that Hc(z) mut have a direct eedthrough path, which can eaily be atiied by a proportional part in the controller. Since the ytem H cl (z) i likely to be time-varying, and the inner controller Hc(z) i ixed, the inner proce Hp (z) mut be etimated in order to adapt the DRG to the proce change. Since meaurement o u,k are not available, and it i only etimated baed on time-invariant rr r Fig. 4. Block-diagram or the mid-ranging adaptive internal model control, with internal aturation compenation. model, the proce dynamic Hp(z) cannot be determined baed on the etimation. However, under the aumption that u,k u at, k, hold true, the inner proce o the ytem can be expreed in term o H cl (z), which i already etimated to adapt the IMC controller, and Hc(z); w Hp(z) H cl = (z) Hc(z)(1 H (11) cl (z)). Since Hc (z) i known, pat meaured value o y can be ued to calculate the real value o pat control ignal, in order to improve the prediction o u,k, reducing error that occur becaue o proce variation. Beore activating the AIMC controller, an initial etimation phae i perormed, uing a quare wave a input to the ytem, until the etimated model have reached the deired accuracy. During thi phae, the requirement u,k u at, k, i unlikely to be ulilled by the DRG ince it prediction model i being etimated. Thereore it i important to chooe the excitation ignal uch that the ytem doe not aturate during thi phae. The inal control architecture or the mid-ranging adaptive internal model control with compenation or internal aturation i diplayed in Fig. 4. Analogouly, it i to be noted that the propoed approach can be extended to ytem where internal aturation appear in both H cl and H cl. In order to evaluate the propoed control cheme, a comparion to exiting method, uch a the MVPC tructure, i perormed. Following the tuning rule given in [6], the controller C and C in Fig. 2, are choen a PI controller, and deigned uing the ame deired cloed-loop ytem a or the propoed controller. Since H cl contain an internal aturation, and the current control ignal i not available, the PI controller will undoubtedly uer rom integrator windup problem. However, auming that the current control ignal i available to the controller, a tracking anti-windup algorithm [7] can be implemented. Both controller, with and without anti-windup, denoted MVPC and MVPC+AW, are evaluated in imulation and experiment. III. SIMULATION RESULTS The propoed control cheme in Fig. 4 wa implemented and teted in MATLAB Simulink, uing u at = 1 and the Hc u Hp y y d yr 4895

4 u (mm/) αk y, y (mm) Fig. 5. Simulation reult o the AIMC controller ubject to a ramped quare wave with a low-requency ine wave uperimpoed a reerence poition (red curve in top plot). The relative poition repone i diplayed in blue in the top panel, and the actuator poition in the bottom panel, where y i green and y i magenta. At time 18, the ytem i aected by a tep poition diturbance with an amplitude o 5 mm. ollowing ytem: Hc(z) = 5, Hp(z) = 2hz 1 1 z 1, H 1 e h cl (z) = 1 e h z 1 wherehi the ample time o the imulation, in thi cae h =.4. The deired complementary enitivity unction are et to T (z) = H cl (z), T (z) = Hcl (z) (12) which correpond to preerving the bandwidth o the cloed-loop ytem. Thi choice i motivated by the aumption that the ytem are well-controlled cloed-loop ytem, ideally having a high bandwidth a poible. An initial gue i provided to the etimator, and the ytem are excited uing a low amplitude quare wave. In the irt imulation, a ramped quare wave with a uperimpoed low requency ine wave i ent a relative poition reerence r r and the relative poition y r i ubject to a tep poition diturbance d. The reult o the imulation i diplayed in Fig. 5. The ramped input i ued to demontrate the midranging eect o the micro manipulator ytem poition y, which i clearly viible rom the bottom panel in Fig. 5, where the green curve i kept cloe to it midpoint. Further, it can be concluded that H cl never enter aturation, ince the control ignal u, which i the unaturated control ignal, i kept within the aturation bound. It i alo noted that the poition diturbance at 18 i attenuated rapidly, imilar to the repone o the cloed-loop ytem. Thi i expected ince the diturbance d on y r can be een a a diturbance on r r, thu exhibiting the ame dynamic a the cloed-loop ytem rom r r to y r. The econd imulation ocue on teting the adaptivity o the control, i.e., it robutne to proce variation. The imulation i perormed uing the ame input ignal a the irt imulation but without the ramp, and alo increaing the gain o the plant Hp(z) by 5 % at time 8, and u (mm/) αk Fig. 6. Simulation reult o the AIMC controller ubject to a quare wave with a ine wave uperimpoed a reerence poition (red curve in top plot). The relative poition repone i diplayed in blue in the top panel. The dahed green line indicate change o proce gain. ubequently decreaing the gain by 6 % at time 16. The reult o the imulation i diplayed in Fig. 6. It i noted that the gain change are only viible in the repone o the relative poition or one period o the quare wave. Further, ince the gain o H p(z) increae, the ytem become ater and conequently lower control ignal i needed to achieve the deired repone. Thi lead to le time in aturation and a higher value o α k. The imulation reult or the MVPC controller compared to the propoed controller are preented together with the experimental reult in Section V, or coheivene. IV. EXPERIMENTAL SETUP The experimental etup ued to evaluate the propoed control cheme i deigned to be a mall cale verion meant to emulate the etup decribed in [1]. The etup conit o an ABB IRB24 robot with an S4CPlu controller, which act a the macro manipulator, and an ABB IRB12 robot with an IRC5 controller, which correpond to the micro manipulator with high bandwidth. Naturally, both robot have aturation limit on velocity, but or proo o concept, the micro manipulator i et to have an input aturation at ±8 mm/, and the macro manipulator i aumed to be low enough to not reach the velocity aturation limit. In addition to the poition meaurement provided by the robot joint reolver, the IRB12 robot i equipped with a Heidenhain linear encoder o model ST378 [18], which meaure the relative ditance between the two robot end-eector, with a working range o 26 mm at an accuracy o 2 µm. Thi meaurement i eential in order to be able to compenate or arm-ide poition diturbance, that the motor-ide robot joint reolver are unable to meaure. The IRB24 robot i attached rigidly to the ground, while the IRB12 robot i attached to a bae that can move in one direction, in order to introduce diturbance in the poition, which requently appear in the real milling etup. The robot are interaced uing an open robot control extenion o the conventional robot controller called ORCA [19], running at 25 Hz. MATLAB Simulink model are tranlated to C-code uing Real-Time Workhop and 4896

5 Fig. 7. Experimental etup or perorming mid-ranging control. The IRB12 robot (micro manipulator) holding the Heidenhain linear encoder i een to the let, and the IRB24 robot (macro manipulator) to the right. y, y (mm) Fig. 8. Etimation phae o the two ytem, where y i green and y magenta. The etimate o thee ignal are denoted ŷ and ŷ, and are hown in dahed blue and black, repectively. compiled in order to run them on the robot ytem. A picture o the experimental etup i hown in Fig. 7. V. EXPERIMENTAL RESULTS Prior to perorming experiment, dynamic model o the two robot, with Carteian velocity reerence a input and Carteian poition a output, were identiied in one axi uing the Prediction Error Method [17]. In order to provide excitation or the identiication algorithm, a quare wave wa ued a reerence, which i converted to joint motor angle velocity reerence, uing the invere Jacobian o the robot. The reulting meaured Carteian poition o the robot wa calculated uing orward kinematic, and ued a ytem output. Both robot exhibit imilar dynamic, and the control loop that orm H cl and H cl, were cloed uing proportional controller uch that the micro manipulator ytem had ive time higher bandwidth than the macro manipulator ytem. New model o the cloed-loop ytem were calculated, reulting in third-order model, which were ued a initial guee in the RLS etimator. The online etimation o the model wa evaluated beore initiating the ull AIMC control tructure, by ending quare wave a poition reerence to the two robot. The reult o the etimation procedure i diplayed in Fig. 8, where the bandwidth dierence o the two ytem i clearly illutrated. The deired complementary enitivity unction T and T were choen a irt-order ytem with bandwidth matchingh cl andh cl, repectively. The irt experiment perormed wa deigned to reemble the imulation in Fig. 5, but ince the linear encoder ha u (mm/) y, y (mm) Fig. 9. Experimental reult o the AIMC controller ubject to a ramped quare wave with a ine wave uperimpoed a reerence poition (red curve in top plot). The relative poition y r repone i diplayed in blue in the top panel, and the robot poition in the bottom panel, where y i green and y i magenta. u (mm/) Fig. 1. Zoomed view o the tep ater 4 in Fig. 9. The dahed green line how the deired repone o the ytem, i.e., the repone o T (z). limited meauring range, a ramp ignal a input would leave that range quickly. Thu, the relative poition o the robot wa intead calculated rom the motor-ide poition meaurement o the robot, o that a ramp ignal could be ued a input. The obtained reult i diplayed in Fig. 9. It i evident rom the igure that the deired mid-ranging eect i achieved, a well a that the control ignal u i kept within the given boundarie. A zoomed view o a tep repone rom Fig. 9 i diplayed in Fig. 1, where the deired repone i alo hown. It i noted that the repone or the relative poition y r i cloe to the deired repone. There i however an initial dicrepancy, which appear becaue o the act that the at ytem ha an input aturation. A diplayed in the lower panel o Fig. 1, the upper boundary on the control ignal u ha been reached, limiting the achievable bandwidth o the cloed-loop ytem. It i to be noted that, given a perect model o the ytem, the repone o the ytem with and without the DRG or any input ignal, would look the ame. The control ignal beore the aturation, i.e., u, would however, not be the ame. Additional experiment were perormed in order to tet how well the ytem handle poition diturbance. For thi purpoe, the linear encoder wa put into operation, replacing the reolver meaurement or the relative poition, o that diturbance in poition can be meaured and compenated or. The experiment wa deigned uch that once the eti- 4897

6 d (mm) u (mm/) y, y (mm) Fig. 11. Experimental reult o the AIMC controller with a quare wave a reerence poition (red curve in top plot), ubject to poition diturbance d a diplayed in the econd panel. The relative poition y r repone i diplayed in blue in the top panel, and the robot poition in the third panel, where y i green and y i magenta. mation phae inihe, the macro manipulator i controlled to move until the linear encoder i in the middle o it meaurement range, which i et to be the zero poition. The AIMC controller i then activated, with a quare wave a reerence ignal, while imultaneouly moving the bae with the micro manipulator, in order to introduce poition diturbance. A mentioned earlier, the linear encoder only ha a meaurement range o 26 mm, and thu the amplitude o the relative poition reerence r r wa choen to be 5 mm. Since a maller amplitude o the reerence reult in lower control ignal, the deired bandwidth o T could be increaed by a actor o 5. The reult o the experiment are hown in Fig. 11. It i noted that under no diturbance, the ytem repond rapidly in a well-damped manner. When ubject to continuou diturbance, the macro manipulator ha to deviate urther rom it deired poition, in order to cancel the diturbance. It doe, however, eventually return to it midpoint. In order to evaluate the perormance o the propoed control cheme, the average Integrated Abolute Error (IAE) over everal tep repone wa choen to quantiy the perormance or the dierent controller. The dicrete-time approximation o the IAE i deined a k max IAE = h r r,k y r,k. (13) k= The MVPC controller were tuned a decribed in Sec. II, uing model-order reduced verion o the previouly identiied model o H cl and H cl. The reult o a erie o imulation and experiment are preented in Table I, where the IAE value have been normalized by the AIMC controller IAE value, in order to impliy comparion. The irt etup wa deigned uch that r r i mall enough to ulill u u at, wherea the ollowing our etup were deigned or the er (mm) ed (mm) Fig. 12. Experimental tep repone with r r = 3, where the AIMC, MVPC+AW, and MVPC controller are hown in blue, magenta, and green, repectively. The deired tep repone i hown in dahed black. TABLE I NORMALIZED IAE VALUE OF STEP RESPONSES. Setup AIMC MVPC+AW MVPC 1. Sim, r r = Sim, r r = Sim, r r = 5, noie Exp, r r = Exp, r r = oppoite. In the third etup, meaurement noie wa added in the imulation. Experiment to tet the robutne to proce variation were perormed by attaching a weight to the micro manipulator while running the controller. Even though the weight wa cloe to the maximum payload o the robot, however, no igniicant change in the proce dynamic wa oberved. Intead, an artiicial proce variation wa introduced by changing the gain o the input to H cl. In a imilar manner to the imulation tudie, the gain wa increaed by 25 % and ubequently decreaed by 3 %. The reult o the experiment are hown in Fig. 13. It i evident rom the igure that the perormance i deteriorated once the proce change, but in approximately three period o the input ignal, the ytem adapted and the deired repone wa achieved. It i alo noted that the DRG wa eective in keeping the contraint, even though coniderable model error occur during the adaptation to the proce change. u (mm/) Fig. 13. Experimental reult o the AIMC controller ubject to a ramped quare wave with a ine wave uperimpoed a reerence poition (red curve in top plot). The relative poition y r repone i diplayed in blue in the top panel. The dahed green line indicate change o proce gain. 4898

7 VI. DISCUSSION AND CONCLUSIONS It wa hown in imulation and veriied through experiment that the propoed mid-ranging control tructure or a macro and micro manipulator etup, perorm atiactory with a repone cloe to the peciication, while maintaining deired propertie in the preence o internal aturation, proce variation, and poition diturbance. It i alo noted that the obtained reult rom the imulation and experiment exhibit cloe correpondence. A hown in Table I, the perormance uing the propoed controller compared to the MVPC controller in the cae o time-invariant procee, i increaed by a much a 56 %. In the cae where proce variation are preent, the propoed controller will naturally perorm igniicantly better than the MVPC controller, due to it adaptive propertie. Furthermore, looking at the error between the ytem repone and the deired repone, the achieved IAE wa up to a actor 16 lower than that o the MVPC controller, a i clearly diplayed in Fig. 12. The perormance o the MVPC controller depend on whether it i poible to implement anti-windup cheme, but according to the original problem ormulation in thi paper, it would not be poible. The MVPC controller without anti-windup will naturally perorm wore or high-amplitude tep and high-requency reerence, ince the controller will be aturated more requently. It could be poible to implement a DRG or the MVPC control cheme, imilar to the propoed control cheme, in order to etimate the control ignal or the anti-windup cheme. Since the purpoe wa to compare the propoed control to previouly etablihed method, thi i not conidered. Further, it i noted that the perormance o the propoed control cheme i deteriorated when ubject to ubtantial meaurement noie, ince it corrupt the etimation o the model. Thi can be improved by increaing the orgetting actor to a value cloer to 1, but will conequently reult in lower adaptation to proce variation. The propoed controller doe, however, till perorm 38 % 47 % better than the compared controller. When proce variation were introduced, the ytem adapt quickly to the new parameter, and the eect o the change can only be een or a ew period o the input ignal. It i, however, noted that the experimental reult exhibit lightly lower adaptation with more pronounced tranient than in imulation. Thi i caued by the act that noie i preent in the experiment, and a dicued earlier, the orgetting actor hould be et to a higher value in order to reduce noie enitivity. Additionally, tranient appear becaue o the increaed complexity in etimating the parameter or a third-order model, a compared to the irt-order model that are ued in imulation. The adaptation o the ytem i dependent on the excitation o the input ignal, with an input ignal o high degree o excitation, the ytem will adapt ater. Converely, i the input ignal i o low excitation, the ytem will adapt lowly and the tranient perormance will be poor. The propoed adaptive DRG wa proven to be eective, a een in Fig. 6, 1, and 13, the control ignal i kept within the contraint, even under igniicant proce variation. Even i no variation in the proce dynamic are preent, the adaptivity o the controller i till beneicial. Thi wa demontrated in Fig. 8, where it i noted that the identiied model are improved throughout the etimation procedure, thu increaing the perormance o the cloedloop ytem. REFERENCES [1] B. Oloon, O. Sörnmo, U. Schneider, A. Roberton, A. Puzik, and R. Johanon, Modeling and control o a piezo-actuated high-dynamic compenation mechanim or indutrial robot, in Proc. IEEE/RSJ Int. Con. on Intelligent Robot and Sytem, San Francico, CA, USA, September 211, pp [2] O. Sörnmo, B. Oloon, U. Schneider, A. Roberton, and R. Johanon, Increaing the milling accuracy or indutrial robot uing a piezo-actuated high-dynamic micro manipulator, in IEEE/ASME Int. Con. on Advanced Intelligent Mechatronic, Kaohiung, Taiwan, July 212, pp [3] A. Puzik, A. Pott, C. Meyer, and A. Verl, Indutrial robot or machining procee in combination with an additional actuation mechanim or error compenation, in 7th Int. Con. on Manuacturing Reearch (ICMR), Univerity o Warwick, UK, September 29. [4] B. Allion and A. Iakon, Deign and perormance o mid-ranging controller, J. o Proce Control, vol. 8, no. 5, pp , [5] A. Sharon, N. Hogan, and D. E. Hardt, The macro/micro manipulator: An improved architecture or robot control, Robotic & Computer- Integrated Manuacturing, vol. 1, no. 3, pp , [6] B. Allion and S. Ogawa, Deign and tuning o valve poition controller with indutrial application, Tranaction o the Intitute o Meaurement and Control, vol. 25, no. 1, pp. 3 16, 23. [7] S. Haugwitz, M. Karlon, S. Velut, and P. Hagander, Anti-windup in mid-ranging control, in Proc. 44th IEEE Con. on Deciion and Control and European Control Con., Seville, Spain, Dec. 25, pp [8] C. Garcia and M. Morari, Internal model control. a uniying review and ome new reult, Indutrial & Engineering Chemitry Proce Deign and Development, vol. 21, no. 2, pp , [9] A. Zheng, V. Mayureh, and M. Morari, Anti-windup deign or internal model control, Int. J o Control, vol. 6, no. 5, pp , [1] A. Datta and J. Ochoa, Adaptive internal model control: deign and tability analyi, Automatica, vol. 32, no. 2, pp , [11] G. Silva and A. Datta, Adaptive internal model control: the dicretetime cae, in Proc. Am. Control Con., vol. 1. San Diego, CA, USA: IEEE, 1999, pp [12] Q. Hu and G. Rangaiah, Adaptive internal model control o nonlinear procee, Chemical Engineering Science, vol. 54, no. 9, pp , [13] K. Hunt and D. Sbarbaro, Neural network or nonlinear internal model control, in IEE Proc. D o Control Theory and Application, vol. 138, no. 5. IET, 1991, pp [14] S. Gayadeen and W. Heath, An internal model control approach to mid-ranging control, IFAC Advanced Control o Chemical Procee, vol. 7, no. Part 1, pp , 29. [15] A. Bemporad, Reerence governor or contrained nonlinear ytem, IEEE Tran. on Automatic Control, vol. 43, no. 3, pp , [16] E. Gilbert, I. Kolmanovky, and K. Tan, Dicrete-time reerence governor and the nonlinear control o ytem with tate and control contraint, Int. J. o Robut and Nonlinear Control, vol. 5, no. 5, pp , [17] R. Johanon, Sytem Modeling and Identiication. Englewood Cli, New Jerey: Prentice Hall, [18] Heidenhain, 213, URL: [19] A. Blomdell, I. Dreler, K. Nilon, and A. Roberton, Flexible application development and high-perormance motion control baed on external ening and reconiguration o ABB indutrial robot controller, in Proc. Workhop o Innovative Robot Control Architecture or Demanding (Reearch) Application How to Modiy and Enhance Commercial Controller, IEEE Int. Con. Robotic and Automation, Anchorage, AK, USA, June 21, pp