Robust Anti-Windup Control Design for PID Controllers Theory and Experimental Verification

Transcription

1 Journal of Modern Procee n Manufacturng and Producton, Vol. 6, No. 3, Summer 17 Robut Ant-Wndu Control Degn for PID Controller Theory and Exermental Verfcaton Payam Kherkhahan 1* 1 Deartment of Electrcal Engneerng, Garmar branch, Ilamc Azad Unverty, Garmar, Iran *Emal of Correondng Author: kherkhahan_ayam@hotmal.com Receved: October3, 17; Acceted: December 1, 17 Abtract Th aer addree an aroxmaton-baed ant-wndu (AW) control trategy for ureng the wndu effect caued by actuator aturaton nonlnearty n roortonal ntegral dervatve (PID) controlled ytem. The effect of actuator contrant frtly regarded a a dturbance morted to the PID controller. The external dturbance can then be modeled by a lnear dfferental equaton wth unknown coeffcent. Ung Stone-Weertra theorem, t verfed that thee dfferental equaton are unveral aroxmator. An auxlary control gnal fnally degned to modfy the error gnal njected to the PID controller. The rooed AW control cheme mle, ytem ndeendent and alcable by dgtal or analog crcut. Analytcal tude a well a exermental reult ung MATLAB/SIMULINK external mode, demontrate hgh erformance of the rooed aroach. It hown that the rooed AW cheme render the erformance of the controlled ytem more robut toward the effect of wndu than conventonal PID AW cheme. The tablty analy rovded by Lyaunov' econd method. Keyword Back Calculaton, Condtonal Integraton, Lmted Integrator, Preloadng, Realzable Reference, Unveral Aroxmator 1. Introducton Practcal ytem have ome contrant owng to hycal retrcton of the actuator [1]. What ecally nvetgated here the hycal lmtaton aocated wth actuator aturaton. Becaue of th henomenon, the actual nut of the lant may dffer from the controller outut n ractce. If aturaton not roerly taken nto conderaton n the control degn, the reult can be datrou uch a erformance degradaton, overhoot, and underhoot a well a ntablty n the cloed-loo reone of control ytem due to a well-known henomenon called wndu effect []. Therefore, many aer have been ublhed and are related to the tablty of control nut aturaton [3-6]. The wnd-u roblem wa orgnally condered n ung PI and PID controller for controllng lnear ytem where an ntegrator n the error ath caue a ertent ncreae n the control gnal untl actuator aturaton occur. In order to weaken the effect of wndu, two bac oluton have been reorted n the lterature. One aroach to conder the lmtaton on actuator a art of the lant and ue nonlnear control theory for the degn rocedure. However, the control cheme obtaned n th manner unutable and moe many unneceary comlexte to the roblem, whle the mlcty of mlementaton hould be mantaned a a bac feature [7]. Another aroach to frt gnore actuator aturaton and degn a lnear controller that meet the 5

2 Robut Ant-Wndu Control Degn for PID Controller Theory and Exermental Vrfcaton, erformance ecfcaton, and then degn an ant-wndu comenator to handle advere effect of the nut aturaton. Th method wa wdely ued n control engneerng and there are many technque for degnng an ant-wndu comenator [8-1]. In order to avod the wndu henomena, ome ant-wndu tratege have already been develoed and gnfcant rogre ha been reorted n the lterature. Ant-wndu control methodology a oular aroach to aturaton control. The objectve of all ant-wndu control cheme to tablze the ytem, and to recover a much erformance a oble of the ytem n the reence of the aturaton. Th two-te degn rocedure ha everal dtnct advantage. Frt, t earate the requrement of tablty and erformance from aturaton control. When degnng the nomnal controller wthout regardng aturaton, the degner free to chooe dfferent control degn aroache a dered. If th rove to be nuffcent n the face of actuator aturaton, then a comenator can be added. Therefore, th aroach allevate the nherent comlcaton nvolved n the degn of aturaton comenator. Many oluton to the ant-wndu roblem have been gven n ome aer. It moble to lt all of thee, but comrehenve ummare of thee method can be found n [11-1] and reference theren. A general oluton wa decrbed n [13]. The dea to ue an oberver whch can take nto account ome aturaton on the control varable. A wndu wa orgnally oberved n Proortonal-Integral (PI) and PID controller degned for SISO control ytem wth a aturatng actuator, Ant-reet wndu whch ha alo been referred to a back-calculaton and ntegrator reettng wa ntroduced [14] to coe wth. A general framework that unfe a large cla of extng ant-wndu control cheme n term of two matrx arameter wa rooed n [15]. Th framework ueful for undertandng dfferent ant-wndu control cheme and motvate the develoment of ytematc rocedure for degnng ant-wndu controller that rovde guaranteed tablty and erformance. Poov tablty condton wa aled to the ant-wndu comenator degn roblem n [16]. However, the ynthe condton gven n couled Rccat equaton, whch dffcult to olve. Alternatvely, by modelng the aturaton a a ector-bounded nonlnearty, the ynthe condton of tatc and dynamc ant-wndu controller were formulated a LMI roblem n [17-] ung extended Crcle Crteron. Th aroach wa to recat the degn of ant-wndu comenator a a robut H roblem. Walgama and Sternby develoed an oberver-baed ant-wndu comenator [1]. Nu degned a robut ant-wndu controller baed on the Lyaunov aroach to accommodate the contrant and dturbance []. Another oluton namely, the condtonng technque wa derved n [3, 4]. The dea comutng the realzable reference varable that would jut have roduced the actual control varable. The orgnal veron of the condtonng technque retrcted to the cae where the controller ha no tme delay. In [5], a modfed veron of the condtonng technque wa ntroduced for a multvarable controller wth a general tme-delay tructure. One drawback n the condtonng technque wa addreed n [6] ung an adatve aroach. [7, 8] alo develoed an adatve ant-wndu comenator. In the cae of adatve control, t hould be noted that comutaton requrement for real-tme arameter dentfcaton and entvty to numercal accuracy and the extng noe ncreae n an underable form a the tate varable ncreae. 6

3 Journal of Modern Procee n Manufacturng and Producton, Vol. 6, No. 3, Summer 17 Moreover, they are unable to handle untructured uncertanty and external dturbance, whch are mng lnk n almot all the rooed aroache [9]. To tackle thee roblem, Neural-Network baed AW control degn ha been rooed. The unveral aroxmaton roerty ha been an mortant motvaton for th wderead alcaton of fuzzy ytem and neural network [3-31]. Many mrovement have been reorted, but the man dea whch etmatng and comenatng the uncertante ung the unveral aroxmaton roerty of neural network and fuzzy ytem ha been remaned unchanged. Fault-tolerance, arallelm and excellent learnng caablte are other benefcal charactertc of fuzzy ytem and neural network [3]. Although thee controller have been ractcally ucceful, ther degn rocedure not tran forward. In adatve fuzzy control, there are many tunng arameter uch a the center and wdth of the Gauan memberh functon and alo the weght of each rule. Uually, thee arameter are adjuted onlne ung the adataton law derved from tablty analy. Neverthele, the ntal value of thee arameter and ther convergence rate are mortant ue that conderably affect the controller erformance and hould be elected carefully [33]. Wderead alcaton of PID controller n ndutry and the fact that they are uually exerence wndu henomenon motvate u to dcu ome of ant-wndu aroache n th reearch. A an extenon n th feld, dfferental equaton utlzed to aroxmate actuator nonlnearty n the controller. An auxlary control gnal then degned and reented to modfy the error gnal njected to the PID controller. The unformly ultmately boundedne of all the control gnal guaranteed through rgorou Lyaunov analy. The exermental reult demontrate that the rooed controller can effectvely ure the actuator aturaton nonlnearty and offer ueror control erformance dete the extence of control nut contrant. The ret of th aer organzed a follow. In ecton, PID control degn revewed. Secton 3 reent ome of conventonal ant-wnd-u method conderng actuator voltage nut lmtaton. Secton 4 reent the rooed model-free ant-wndu control degn. Stablty analy and erformance evaluaton are dcued n ecton 5. Secton 6 dedcated to the exermental reult of a level control ytem whch ha been contructed n laboratory. Fnally, ecton 7 conclude the aer.. PID Control Degn Although many effcent aroache for controller degn have emerged n the lat decade, PID controller are tll the mot common controller n ndutry, nce the beneft-cot rato obtaned by thee controller not achevable ealy by the other controller [3, 34]. Contnuou tme PID controller have been tuded extenvely [35-37]. However, n ractcal alcaton, ther erformance lmted by actuator aturaton and thu, followed by ntegrator wnd-u. Let a dynamc ytem be controlled by a PID controller K ( ) a hown n Fgure 1: 7

4 Robut Ant-Wndu Control Degn for PID Controller Theory and Exermental Vrfcaton, Fgure1. Cloed loo ytem wth lmted nut Where k K ( ) = k + + k d (1) k, k, and k d are the roortonal, ntegral and dervatve contant of the controller, reectvely. The PID controller, K ( ) can be decrbed n a tme contant form a follow [3]: 1 K ( ) = k Td T () In whcht, andt d are ntegral tme contant, and dervatve tme contant, reectvely, or n a modfed form a 1 Td K ( ) = k T 1 + Td / N Where dervatve term relaced by a flter n order to attenuate the hgh gan effect nduced by hgh-frequency meaurement noe, and N uually between 7 and 15. It aumed that the nut lmtaton decrbed a (3) u max ; u > u max u = at ( u) = u ; umn u u u mn ; u < umn max (4) Where u the controller outut, u the actual roce nut and at ( ) denote the aturaton functon. A t obvou from Equaton (4), when actuator aturaton haen, the actual nut of the ytem dfferent from the controller outut. In th tuaton, f the controller contnue t oeraton baed on the ntal degn n lnear range, the erformance of the cloed-loo ytem devate from the roected lnear erformance and conequently the wnd-u henomenon occur. To decrbe th henomenon, t can be aumed that both the roce and the controller are n teady tate. A otve te change n reference gnal w ( ) caue a jum n the control gnal and lead to actuator aturaton at hgh lmt (f k > ). Thu, u become maller than u and y wll 8

5 Journal of Modern Procee n Manufacturng and Producton, Vol. 6, No. 3, Summer 17 change le lowly than the ytem reone wth unlmted nut. Snce y changng lowly e = w y reduce lowly and the ncreae n the ntegrator term greater than t ncreae for the unlmted cae. When y reache to w ( ), due to the large ntegral term, u reman aturated or cloe to the aturaton mode. After a uffcently long tme that the error remaned negatve for a long tme,u reduce. Thee behavor lead to a large ettlng tme and a large overhoot n the roce outut. To tackle th roblem, ome conventonal ant-wndu control tratege are revewed n the next ecton. 3. Conventonal Ant-Wndu Stratege 3.1 Lmted Integrator Lmted ntegrator a very mle aroach to reduce ntegrator wnd-u effect. In th aroach, a hown n Fgure, the ntegrator nut reduced by feedng back the ntegrator outut va a hgh-gan dead-zone. In order to utlze the whole lnear range of the actuator, the dead-zone regon and the lnear range of the actuator hould be the ame. When the ntegrator value out of deadzone band, a feedback gnal wth the amltude of Equaton 4 roduced to affect the ntegrator nut. f = b( ( t ) H ) (5) In (5), H a dead-zone band, b dead-zone gan and ( t ) the ntegrator value. If b elected large ( b > 1), the ntegrator outut lmted to H effectvely. 3. Back-Calculaton Back-calculaton (trackng ant-wndu) a conventonal method to revent ntegrator wnd-u. The TAW tandard tructure ued n the lterature hown n Fgure 3 n whch Tt the trackng tme-contant. When the controller outut exceed the actuator ermtted range, ntegrator nut reduced by feedng back the dfference between the aturated and unaturated control gnal. It hould be mentoned that the aturaton block hown n Fgure3 may rereent the real aturaton n actuator (f actuator-outut meaurable) or decrbe an aled model n the controller. If the 9

6 Robut Ant-Wndu Control Degn for PID Controller Theory and Exermental Vrfcaton, Fgure. PID control wth lmted ntegrator Fgure3. PID control wth Trackng Ant-Wndu Actuator decrbed by lnear dynamc wth aturaton art; the roblem that the controller outut and o the eed of actuator reone wll be lmted. To olve th roblem, the tructure hown n Fgure3 changed to the tructure n Fgure 4 where an unlmted control gnal aled to the roce and feedback gnal roduced ung dead-zone. It hould be mentoned that H denote the lnear range of the actuator and the relatonh betweent t and dead-zone gan decrbed a b = T / T (6) A rule of thumb for tunng Tt Tt t = T whch equvalent to b = 1. However, t ha been roven that the larger value of b n ecal condton reult n a more utable erformance. 3.3 Modfed Trackng Ant-Wndu Smulaton reult mly that trackng control entve to varaton n Tt and b. Choong very hgh value for b can reduce overhoot effectvely, whle t can lead to low reone. So n th ecton, the rooed tructure n [38] generalzed for PID controller (Fgure 5) 1

7 Journal of Modern Procee n Manufacturng and Producton, Vol. 6, No. 3, Summer 17 Fgure4. Equvalent form of Fgure3 Fgure5. Modfed Trackng Ant-Wndu The reaon of low reone that a very large ntal controller outut whch orgnated from large roortonal gan and dervatve oeraton, lead to a large feedback gnal for hgh gan b.th feedback gnal force the ntegrator to hgh negatve value, o the controller outut come back to lnear oeraton regon. A tme goe by, roortonal and dervatve term of the control gnal decreae. However, the ntegrator outut wll not ncreae radly to comenate th reducton. A a reult, the controller outut become very mall or even negatve. To avod th, an extra lmtaton aled on the roortonal-dervatve art of the control gnal to roduce antwndu feedback gnal whch make electon of hgh gan oble forb.e. b = 1. The effect of th addtonal aturaton actuator can be nterreted n th way that f the ntegrator value exceed from max( H u ( t ), H H ), a feedback gnal aled wth the amltude of PD PD { } f = b( ( t ) + mn H, u ( t ) H ) (7) PD PD Th feedback gnal reduce the ntegrator nut and o to ntegraton acton. Normal ntegraton done only when Equaton (8) atfed, { } { } H max H, u ( t ) < ( t ) < H mn H, u ( t ) (8) PD PD PD PD 11

8 Robut Ant-Wndu Control Degn for PID Controller Theory and Exermental Vrfcaton, In whch upd the roortonal-dervatve art of the control gnal. In other word, t lmted dynamcally to max( H u ( t ), H H ). By an extra lmtaton, a new degn arameter PD PD added. In mulaton, the rato r = H / H ued. The utable range for r [.5,1.5]. The PD choce r = 1can be nterreted a contnung ntegraton untl the controller gnal conted of the roortonal-dervatve art, u, come back to lnear oeratonal range and retartng the PD ntegraton. Conequently, ntegrator wll not go to negatve value and the low behavor of the te reone wth a hgh gan for dead-zone wll be revented. If the amltude of the te nut large n comaron wth the aturaton level (7% to 1%), the ytem reone not entve to varaton of the arameter r. However, for te nut wth mall amltude, choong mall value for r uch a r [.5,1] lead to fater reone wth large overhoot, and choong larger value for r n the nterval [1,1.5] yeld n reduced overhoot; although t make the controlled ytem reone luggh. 3.4 Realzable Reference In th method, controller oututu orgnated from alyng the realzable reference gnalw to the controller equal to real ytem nut, ( u ), whch obtaned from alyng reference gnalw to the controller. If thew decrbed n Fgure 3 aled, the aturaton block wll be deactvated ( u alway mlar to u ) and o the aturaton block can be elmnated accordng to Fgure 6. In addton, the gnal u and y llutrated n th fgure are equal to the gnalu and y llutrated n Fgure 3, reectvely. A dected n Fgure 6, any nonlnear term ha been excluded from the rooed cheme. In fact, the nonlnear term ha been ncluded n the reference gnalw. A hown n Fgure 6, the outut y trackw ntead ofw wth the exected lnear erformance. Ung realzable reference defnton, t obvou that 1 1 Td u = k 1+ w k 1+ + y T T 1 + Td / N Th can be rewrtten a 1 Td TTd 1+ + T N N T w = y + u Td k ( ) ( 1+ T ) 1+ T 1+ N Ung TAW algorthm, we have: 1 1 T d 1 u = k 1+ w k 1+ + y u u T T + T d 1+ Tt N ( ) (9) (1) (11) 1

9 Journal of Modern Procee n Manufacturng and Producton, Vol. 6, No. 3, Summer 17 Subtractng (11) from (9) and after ome manulaton, w obtaned a T 1+ T w = w + u u = w + G u u ( ) ( )( ) t w ktt 1+ T (1) Fgure6. Equvalent form of Fgure 3 Snce G ( ) a dynamc tranfer functon wth one zero and one ole, when the controller leave w the lmtaton andu whent t u =, w not mlar tow, unle G ( ) decreae to a tatc gan. In fact, = T, t follow from Eq. (1) that: Gw w 1 ( ) = (13) k u = w + u k Alo, when the controller leave the lmtaton, we wll havew trackng erformance. w (14) = w, whch reult n the bet 3.5 Condtonal Integraton In th method, the ntegrator term ncreaed only when ecal condton are met. Otherwe, t value ket contant. There are varou tye of condtonal ntegrator characterzed a below: 1- The ntegrator term lmted to a elected value. - The ntegraton toed when the ytem error become large ( e > e ) n whche a elected value. 3- The ntegraton toed when the controller aturated. 4- The ntegraton toed when actuator aturaton occur and the controller error and control gnalu have the ame gn ( e u > ). All of thee method have been comared n [39-4], and t ha been hown that the cheme 4 outerform the other. 13

10 Robut Ant-Wndu Control Degn for PID Controller Theory and Exermental Vrfcaton, Integrator Lmter (CI-ILIM) [41] Th method moe a hard lmtaton (aturaton) on the ntegrator valueη uch that: η [ ηmn, ηmax ] & e.( η η )>, & η = η ( ηmn + ηmax )/ (15) e ow. The ntal challenge of th method choong the ntegrator lmtη mn have elected them a mn max ( mn, max ) (, ) andη max. In th tudy, we η η = T u T u k k (16) 3.5. Condtonally Freeze Integrator (CI-CFRZ) In th method whenu move toward aturaton,η& frozen. It mean that: u u & e.( u u )> & η = (17) e o. w A common trategy related wth condtonal ntegraton, whch often aled n chemcal engneerng, freezng the ntegrator nut to zero whenu aturated. u u & η = (18) e ow. Although the ntegrator valueη held contant, leavng the aturaton regon not guaranteed n th method. 3.6 Preloadng In th method, when the controller outut aturated, the ntegrator value ( η) reet to the redefned valueη d. In the other word: α ( η ηd ) u u & η = (19) e ow. In whchα > determne the ntegrator delay rate durng aturaton of the control gnalu. Smlar to CI-ILIM method, choong the degn arameterη d a challengng roblem. Alo mlar to CI- FRZ, leavng the aturaton regon not guaranteed n th method. 14

11 Journal of Modern Procee n Manufacturng and Producton, Vol. 6, No. 3, Summer Varable Structure PID Ant-Wndu (VSPID) In th method, when the nomnal controlu n aturaton regon, the ntegrator term drven dynamcally uch that the control gnalu laced on the edge of the aturaton regon [4]. A a reult, th method remove the defect of revou method. In th trategy, η& gven by βt ( u u ) / k u u & e.( u u )>, & η = u ( umn + umax )/ () e ow.. Where β a otve contant and choen o thatu converge radly to the nearet edge of the nterval[ umn, u max ] (the ettlng tme n aturaton zone β = 1 T t T ettle 4 = ). It can be ealy verfed that f β and the wtch hown n Fgure 7 relaced wth an adder; VSPID wll be mlar to TAW [4]. Some ecfcaton of th method are a follow. 1- It oble for degner to aly ther knowledge n the feld of PID controller degn. Fgure7. Varable tructure PID ant-wndu - The degn arameter β can be determned ung the oeratonal ecfcaton of cloed-loo ytem. A a rule of thumb, β hould be elected uch that durng aturaton, ntegrator feedback loo erform -5 tme fater than the cloed-loo ytem ettlng tme. 3- Due to the wtchng behavor of the controller decrbed n () and feedng back the aturaton error ( u u ), the control gnalu tend to reman near the aturaton zone (durng actuator aturaton). A a reult, th controller turn to the lnear oeratonal zone fater than the other dcued controller. 3.8 Hybrd Method Some ant-wndu tratege by combnng condtonal ntegrator and back-calculaton method have been reented [4-43]. In [43], t ha been rooed to aly an extra lmtaton for roortonal-dervatve art n order to roduce an ant-wnd u feedback gnal. However, t 15

12 Robut Ant-Wndu Control Degn for PID Controller Theory and Exermental Vrfcaton, followed by roblem aocated wth tunng extra degn arameter. Suoe that a te change from the ntal value of y to the fnal value of y 1 requred. A mentoned before, ntegrator wndu occur when a te change nw lead to actuator aturaton. In th cae, the ytem error reduce lower than deal cae (wthout actuator aturaton) and o the ntegrator value become large. A a reult, even when the outut y reache tow, the controller tll n aturaton regon, due to the ntegrator term, whch lead to large overhoot and long ettlng tme. An mortant ont n thee technque ther defcency n the reence of dead-tme n rocee whch underable for ndutral regulator. The oluton combnaton of condtonal ntegraton and back-calculaton method. Back-calculaton aled when the controller aturated ( u u ), ytem error and the corrected value ofu have the ame gn ( e u > ), and the ytem outut ha converged to the new et ont. In other word: k 1 y > y f y > y e ( u u) f u u & e u & y y f y y 1 + > T Tt < 1 < & (1) η = k e T ow. Remark 1: The value oft t determne the rate of the ntegraton reet and alo the erformance of the total control cheme. Some of oton for electng T t are Tt = T and T t = T T d [39]. 4 Decrton of Control Structure In th ecton, the ubject of model free control ued to etablh an ant-wndu cheme. Controller wth actuator aturaton hown n Fgure 8. The actual nut of lant u, that related tou, control gnal, by the functon aturaton a hown n Equaton (4). Now, the crucal queton that the uncertanty belongng to the ytem model or controller equaton. A t known, one exlanaton of wndu henomena can be tated lke th: becaue of aturaton, the actual lant nut wll be dfferent from the outut of the controller, n turn, the controller outut doe not derve the lant and a a reult, the tate of the controller are wrongly udated. Therefore, when aturaton occur, the actual controller tate are dfferent from dered tate. Hence, we can nfer that the uncertanty aocated wth controller. It hould be mentoned that, by conderng uncertanty n controller, th aroach can be aled to other realzable lnear controller. By ung a roer comenator, the error between dered and actual tate of controller wll converge to zero aymtotcally and the cloed-loo reond retored. Equaton are rewrtten a follow x& = A x + B e () c c c c u = Ccx c + σ (3) Whereσ denote the actuator non-mlemented control gnal a σ = at ( u ) u (4) 16

13 Journal of Modern Procee n Manufacturng and Producton, Vol. 6, No. 3, Summer 17 In addton, the dered controller that yeld good trackng erformance n the abence of aturaton nonlnearty, gven by x& = A x + B e (5) d d c c c c d u = C x d d c c (6) Fgure8. Sytem wth actuator aturaton Th art of the degn rocedure qute traghtforward and carryng out ndeendent of aturaton nonlnearty. Subtractng (5) from () and (6) from (3) yeld the error dynamc equaton a E& = A E + B ℵ (7) c c ζ = C E + σ c (8) Where d E = x x, ℵ = e e, ζ = u u (9) c c d d Now, the roblem to degn an algorthm to mrove the oton error denoted bye o that t can mnmze the error of udatng tate of the controller. Wth th n mnd, the followng quantte are defned. ( ) ( j ) Ξ = E b E (3) j = 1 j ( ) ( j ) δ =ℵ b ℵ j = 1 j (31) Where b j ' are (real) calar. Dfferentatng both (7) and (8) wth reect to tme and ung (3) and (31) wth the aumton that σ can be modeled by a th order Ordnary Dfferental Equaton (ODE) a [44-47], 17

14 Robut Ant-Wndu Control Degn for PID Controller Theory and Exermental Vrfcaton, = ( ) ( j ) σ b σ j = 1 We have the followng reult Ξ & = A Ξ + B δ c ( ) ( j ) j = 1 j c ζ = b ζ + C Ξ + ( t ) j c (3) (33) (34) Where ( t ) the lumed aroxmaton error? It hould be noted that, the order reflect the dynamc tructure ofσ, whch n mot cae condered one or two. Now, we can defne a coordnate tranformaton rereented by ( 1) T & K (35) L = ζ ζ ζ Ξ Then, the controller tate equaton n new coordnate wll be L& = Λ L + Gδ + W ( t ) (36) That δ a new control nut vector defned a And ( j ) = L = Ξ (37) δ µ µ µ ζ j = 1 j 1 L 1 L M M M M M M Λ = L L 1 b b 1 L L b1 C c L L Ac T [ ] T G = B L c, W = 1 T (38) It mut be noted that f( A, B ) controllable, then( Λ,G ) alo controllable and fnally the error c can be forced to zero. Subttutng (37) nto (36) yeld L& = ( Λ G µ ) L + W ( t ) c The remanng tak here to adjut the control nute to account for the effect of actuator nonlnearty a follow: (39) 18

15 Journal of Modern Procee n Manufacturng and Producton, Vol. 6, No. 3, Summer 17 e = e d +ℵ (4) Whereℵ the auxlary control nut? The lat equaton ay that n order to coe wth the wndu henomenon, the error gnal njected to lnear controller mut be changed to (4). A block dagram of the cloed loo control ytem llutrated n Fgure 9. Fgure9. The block dagram of the rooed cheme Controller Degn Ste: In hort, the overall degn rocedure can be ummarzed a follow: 1. The control arameter matrce A c, B c, andc c are elected by tral and layed to acheve the bet oble tranent erformance wthout conderng the actuator nonlnearty.. The uncertanty modeled by a th order ODE. 3. Fnally, to mlement the rooed control method, the µ tate feedback vector calculated. Remark : A can be een, the Eq. (3) a order ODE. It can be ealy hown that the oluton of th Equaton a contnuou functon a: = 1 λ t ( ) σ ( t ) = c e Co ω t + θ (41) It nteretng to nvetgate the caablty of the lat aumton, Eq. (3), from a functon aroxmaton caablty ont of vew. Heren, we wll rove that equaton (41) ha the unveral aroxmaton caablty. In the followng, we uoe that the nut unvere of dcoure T a convex et nr. Prooton1. (Unveral Aroxmaton Theorem) Let σ ( t ) be a contnuou real functon on a convex et T nr, then for each arbtraryε >, there ext a functon n the form of 19

16 Robut Ant-Wndu Control Degn for PID Controller Theory and Exermental Vrfcaton, = 1 λ e t c Co t ( ω + θ ) (4) Such that t T = 1 λ t ( ) Su c e Co ω t + θ σ ( t ) < ε (43) Proof: let Z to be a et of contnuou functon on T convex et n the form of (41) and uoeσ 1 ( t ) andσ ( t ) are gven a [9] 1 = 1 j j = 1 λ t ( + ) σ ( t ) = c e Co ω t θ λ t ( j j ) j σ ( t ) = c e Co ω t + θ (44) We have λ t λ t ( ) ( j j ) (45) σ ( t ) + σ ( t ) = c e Co ω t + θ + c e j Co ω t + θ 1 j = 1 j = 1 1 σ ( t ). σ ( t ) = c c e Co 1 j = 1 j = = 1 j = 1 c c j ( ) ( ω ) j + ωj t λ + λ t + ( θ + θ j ) ( ) ( ω ) j t ω j t λ + λ Co + ( θ θ j ) e Hence, σ1( t ) + σ( t ) andσ 1 ( t ). σ ( t ) Z. Fnally, for any arbtraryγ R, = 1 λ t ( ) γ. σ ( t ) = γc e Co ω t + θ Whch alo n the form of (41) and hence Z algebra. Therefore, the frt condton of Weertra theorem true. We how that Z earate ont on T. We chooe the arameter of the σ ( t ) n the form of (41) a follow (46) c = 1, λ = 1, ω =, θ = (47) t1 t Sncet 1 t, thene e and therefore the econd condton alo verfed. To verfy the thrd condton of Weertra theorem, we mly oberve that any ytem n the form of (41) wth ω =, θ = andc > ha the roerty that

17 Journal of Modern Procee n Manufacturng and Producton, Vol. 6, No. 3, Summer 17 t T, σ ( t ) > (48) Hence, Z vanhe at no ont of T. Thu the three aforementoned condton are atfed. Therefore the reult follow Stone-Weertra Theorem. 5. Stablty and Performance Analy 5.1 Stablty Analy Here, we rove tablty of the rooed aroach whch ubject to untructured uncertanty. Toward th end, conder the followng Lyaunov functon canddate: V = T L PL (49) In T whch P = P a otve defnte matrx atfyng the Lyaunov equaton T ( Λ Gµ ) P + P( Λ Gµ ) = Q whereq a otve defnte matrx. Takng the tme dervatve of (49) along the trajectore of (39), we have: V& λ ( Q) L + λ ( P) L ( t ) mn max Where λ mn ( Q), and λ max ( P) denote the mnmum and maxmum Egen value ofq and P, reectvely. Remark 3: Suoe that the arorate model are ued and the aroxmaton error can be gnored. Then,V & negatve defnte, and the aymtotc tablty of L can be aroved. Remark 4: If the aroxmaton error cannot be gnored, after ome further manulaton of (5) we have (5) λmax 1 ( P) ( t ) V& λmn ( Q) L λmn ( Q) c (51) We would lke to relate (c) tov by conderng: V λ ( P) L max (5) Now, (51) can be rewrtten a λmax 1 ( P) ( t ) V& αv + αλ max ( P) λmn ( Q) L + λ ( Q) mn (53) λmn ( Q) Pckα <, then we have λ ( P) max 1

18 Robut Ant-Wndu Control Degn for PID Controller Theory and Exermental Vrfcaton, V& λ αv + max ( P) ( t ) λ A a reult, V & < whenever mn ( Q) (54) V λ ( P) > u ( τ ) (55) αλ max mn ( Q ) τ t ( 1) Th mle that ( ζ, & ζ,..., ζ, Ξ) unformly ultmately bounded. 5. Performance Analy Untl now, we have demontrated the boundedne of the error of the controller outut. It hould be mentoned that n ractcal mlementaton, the tranent erformance alo of great mortance. Solvng (54) yeld λ ( P) V ( t ) e V ( t ) + u ( τ ) (56) ( Q ) α ( t t ) max αλmn t τ t Therefore, accordng to the lower bound of Lyaunov functon, we have L α ( t t ) λ max V ( t ) ( P) e + u ( τ ) (57) λ ( P) αλ ( P) λ ( Q) mn mn mn t τ t That mean the trackng error bounded by a weghted exonental functon lu a contant. Th alo mle that by adjutng the controller arameter, we may mrove the trackng error convergence rate. A a conequence, λmax ( P) lm L u ( τ ) (58) αλ ( P) λ ( Q) τ t mn mn t t 6. Real-Tme Imlementaton 6.1 The Exermental Setu In order to evaluate the theoretcal method dcued n revou ecton, exermental reult on a tank manufactured n laboratory are reented. Frt, the element of the ytem are exlaned. In th reearch, the tuded roce a ytem cont of four nterconnected tank [48]. Only one of them ued for our exermental objectve. Water umed by a SAM-1118 DC um wth drvng voltage of 1V. The comuter command aled to the actuator by a PWM generator crcut and a PCL-818L card (Fgure 1). The lqud heght meaured ung a V441 1mBar reure enor. A current-to-voltage converter crcut ued n order to convert the current gnal obtaned from the enor nto voltage, for comuter calculaton. Alo, a oftware-mlemented low a flter wth a 5HZ cutoff frequency ued for roer enor noe attenuaton. For real-tme mlementaton, MATLAB Real-Tme Wndow Target (RTWT) aled.

19 Journal of Modern Procee n Manufacturng and Producton, Vol. 6, No. 3, Summer Exermental Reult To nvetgate the effect of ntegrator wndu, the tank decrbed n revou ecton wth PID controller condered. By ung the Zgler-Ncole method, controller arameter are obtaned a: k =6, T = 5, T = 1 (59) d Fgure1. The exermental etu In th reearch, we have focued on the et-ont roblem. The ntal value of the te reference nut y = 1.65 and t fnal value ha been et to y 1 = 1.9.Thee value have been elected baed on the range of the enor outut current and the dered reference nut. Note that for the emty and full tank, enor outut gnal after current-to-voltage converon 1.65 Volt and.5 Volt, reectvely. Pum aturaton voltage value are et onu mn = V andu max = 1V. The ytem reone to dered nut, um voltage gnal and ntegrator outut n the reence of actuator aturaton are llutrated n Fgure 11. A hown n th fgure, the underable reone tem from control gnal aturaton and the accumulated error n the ntegrator (ntegrator wndu). In more detal, durng aturaton, the feedback loo oened. A a reult, an ncrement n the control gnal amltude doe not yeld n a fater reone by the ytem. In th tuaton, f error ntegraton contnue, the ntegrator value ncreae wthout any effect on ytem outut. In t=46.1 ec, the ytem outut reache to et-ont value and the error gn change. However, due to ntegrator hgh value, the control gnal tll reman n aturaton regon. After th tme, ntegrator outut begn t reducton o that n t=98.8 ec, when the error become uffcently negatve, the control gnal leave the aturaton regon whch lead to a hgh overhoot and long ettlng tme. In order to olve th roblem and obtan a utable erformance n the reence of actuator aturaton, real tme mlementaton of eght conventonal ant-wndu mechanm ung PID controller comleted n MATLAB/Smulnk envronment and ther erformance are evaluated n th reearch. 3

20 Robut Ant-Wndu Control Degn for PID Controller Theory and Exermental Vrfcaton, Proce outut (y) voltage (volt) Integral value Tme(ec) Fgure11. Sytem reone, Control gnal, and Integrator outut for conventonal PID controller Lmted ntegrator [49]: The ytem outut and um voltage gnal ung 4 dfferent value ofb have been dected n Fgure 1. A can be een, by feedng back the ntegrator outut va a deadzone element wth hgh gan, t value lmted to the actuator lnear range. A a reult, the controlled ytem come out fater from aturaton regon and the reducton n ettlng tme and overhoot obvou. Trackng Ant-wndu method: The ytem reone, um voltage gnal and ntegrator outut n reence of actuator aturaton for dfferent value of the arametert t are hown n Fgure 13. Accordng to th fgure, maller value oft t obtan more utable erformance (le overhoot and fater ext from aturaton). Modfed Trackng ant-wndu control: A mentoned before, trackng control method entve to varaton of the arameter T t and gan b. In other word, choong mall value fort t reduce the overhoot effectvely. In th ecton, the modfed ant-wndu trackng controller tructure rooed n [38] nvetgated for PID controller. For better undertandng the role of added arameter n the degn rocedure, due to ntroducng an extra lmtaton, real-tme mlementaton reeated for valueb < 1, b > 1 and alo 3 value of r =.5, 1,

21 Journal of Modern Procee n Manufacturng and Producton, Vol. 6, No. 3, Summer 17 Proce outut (y) voltage (volt) Integral value Tme(ec) Fgure1. Sytem reone, Control gnal, and Integrator outut for lmted ntegrator b= b=.1 b=.5 b= Proce outut (y) voltage (volt) Integral value Tt=nf Tt=. -5 Tme(ec) Fgure13. Sytem reone, Control gnal, and Integrator outut for trackng ant-wndu Tt=5 Tt= 5

22 Robut Ant-Wndu Control Degn for PID Controller Theory and Exermental Vrfcaton, A. b < 1 : The ytem reone, um voltage gnal and ntegrator outut n reence of actuator aturaton for dfferent value of r are dected n Fgure14. A llutrated n th fgure, choong the arameter r between r = 1and r = 1.5obtan more atfactory reult. B. b > 1 : Fgure 15 how the ytem reone, um voltage gnal and ntegrator outut for 3 dfferent value of r and n reence of actuator aturaton, reectvely. Accordng to Fgure 15, a very large ntal value for controller outut (orgnated from large roortonal gan and dervatve oeraton) reult n a large feedback gnal wth a hgh gan forb. Th feedback gnal force ntegrator to hgh negatve value untl the controller outut come back to lnear oeratonal range. A a reult, for mall value of r, ( r.5k 1), we have a large overhoot, and choong larger value for r, ( r 1K 1.5), reduce the overhoot. Realzable reference: Fgure 16 llutrate the cloed-loo ytem reone to the realzable reference nut n reence of actuator aturaton and the ntegrator outut, reectvely. The mortant ont that th method comenate aturaton through changng the aled reference gnal to cloed-loo control ytem ( w ). Proce outut (y) voltage (volt) Integral value PID r=.5 r=1 r=1.5-5 Tme(ec) Fgure14. Sytem reone, Control gnal, and Integrator outut for modfed trackng ant-wndu (b<1) 6

23 Journal of Modern Procee n Manufacturng and Producton, Vol. 6, No. 3, Summer 17 Proce outut (y) voltage (volt) Integral value r=1.5-5 Tme(ec) Fgure15. Sytem reone, Control gnal, and Integrator outut for modfed trackng ant-wndu (b>1) PID r=.5 r=1 Proce outut (y) voltage (volt) Integral value 1.8 Realzable reference PID U U Tme(ec) PID PID Realzable reference Fgure16. Sytem reone, Control gnal, and Integrator outut for realzable reference Condtonal ntegrator: In th method, ntegraton wtched to one of the tuaton on or off n ecfc condton baed on the control gnal amltude or trackng error. In the off cae, the ntegrator ntal condton determne t value n that moment. In th ecton, three method namely CI-ILIM, CI-CFRZ and CI-FRZ are tuded n whch the controller arameter are choen 7

24 Robut Ant-Wndu Control Degn for PID Controller Theory and Exermental Vrfcaton, aη mn = andη max = 1. The cloed-loo ytem reone to dered nut, um voltage gnal and ntegrator term outut n the reence of actuator aturaton for thee method llutrated n Fgure 17. A hown n thee fgure, CI-CFRZ and CI-FRZ oerate ueror than CI-ILIM. It worthy to menton that n choong a ecfc method, the convergence of teady tate error to zero hould be guaranteed. It mean that atfactory teady tate hould not be obtaned n the abence of ntegrator. Preloadng: In th aroach, the controller arameter have been choen aη d = andα = 1. The cloed-loo ytem reone to dered nut, um voltage gnal and ntegrator outut hown n Fgure 18. Varable tructure PID ant-wndu control: In th ecton, exermental reult foru = 6 and three value of β =.1, β = 1and β = are hown n Fgure19. A llutrated n thee fgure, the degn arameter β conderably affect the tablty and erformance of th method. To be more rece, ncreang the gan β lead to decreae the ettlng tme and re tme. Proce outut (y) voltage (volt) Integral value PID CI-ILIM CI-CFRZ CI-FRZ -5 Tme(ec) Fgure17. Sytem reone, Control gnal, and Integrator outut for condtonal ntegraton 8

25 Journal of Modern Procee n Manufacturng and Producton, Vol. 6, No. 3, Summer 17 Proce outut (y) voltage (volt) Integral value 1.8 PID Preloadng Tme(ec) Fgure18. Sytem reone, Control gnal, and Integrator outut for reloadng Proce outut (y) voltage (volt) Integral value PID 5 β=.1 β=1 β= -5 Tme(ec) Fgure19. Sytem reone, Control gnal, and Integrator outut for varable tructure PID Hybrd method: Fgure rereent the ytem reone to dered nut and um voltage gnal for the rooed cheme by [34] n the reence of actuator aturaton. For better evaluaton of the reult, ntegrator outut for th cheme alo hown n Fgure. It follow from thee fgure 9

26 Robut Ant-Wndu Control Degn for PID Controller Theory and Exermental Vrfcaton, that the hybrd method wtht t = T T d yeld n a maller ettlng tme n the outut reone and rovde better erformance n comaron wtht t = T. Th concluon can be jutfed mly accordng to the fater ext of the control gnal from aturaton regon. Model free AW: Fgure1 how the caablty of the rooed aroach, under the ame control arameter. The non-comleted control gnal ha been modeled by an ordnary dfferental equaton of order. For better undertandng the control gnal are alo lotted. From Fgure1, t clear that the control gnal of the MFAW leave the aturaton regon more radly and wth a mooth change. One key feature of the rooed method robutne under arameter varaton of the model. Proce outut (y) voltage (volt) Integral value PID Tt=T Tt=qrt(T*Td) -5 Tme(ec) Fgure. Sytem reone, Control gnal, and Integrator outut for hybrd method 3

27 Journal of Modern Procee n Manufacturng and Producton, Vol. 6, No. 3, Summer 17 P roc e outut (y ) Integral v alue v oltage (v olt) 1.8 MFAW Tme(ec) Fgure1. Sytem reone, Control gnal, and Integrator outut for MFAW 7. Concluon PID controller are one of the mot convenent controller utlzed n ndutry. They can be mly tuned n order to acheve ecfc erformance requrement, robutne and zero teady tate error n the reence of contant dturbance. Alyng thee controller n rocee wth nut lmtaton and large et-ont varaton lead to an undered henomenon named ntegrator wndu whch yeld n actuator aturaton, erformance deteroraton, and even ntablty of the cloedloo ytem n ome cae. An aroxmaton-baed AW control trategy nvetgated n th aer. It ha been aumed that the actuator nonlnearty can be modeled by a lnear dfferental equaton wth unknown coeffcent. Ung Stone-Weertra theorem, t verfed that thee dfferental equaton are unveral aroxmator. Exermental reult evaluate the effcency of the rooed aroach. A can be een, the controlled ytem come out fater from aturaton regon and the reducton n ettlng tme and overhoot are obvou. Moreover, the unformly ultmately boundedne of teady tate error alo guaranteed. 8. Reference [1] Rehan, M., Qayyum Khan, A., Abd, M., Iqbal, N. and Huan, B. 13. Ant-wndu-baed dynamc controller ynthe for nonlnear Sytem under nut aturaton. Aled Mathematc and Comutaton. : [] Zaccaran, L. and Teel, A. R.. A Common Framework for Ant-wndu, Bumle Tranfer and Relable Degn. Automatca. 38: [3] Yang, Sh-K. 1. A new Ant-wndu Strategy for PID Controller wth Dervatve Flter, Aan Journal of Control. 14 (): PID 31

28 Robut Ant-Wndu Control Degn for PID Controller Theory and Exermental Vrfcaton, [4] Rehan, M. and Hong, K-Sh. 13. Decouled-archtecture-baed Nonlnear Ant-wndu Degn for a Cla of Nonlnear Sytem. Nonlnear Dyn. 73: [5] Ran, M., Wang, Q., N, M. and Dong, Ch. 15. Smulaton lnear and Ant-wndu controller ynthe: delayed actvaton cae. Aan Journal of Control. 17(3): [6] L, Y. and Ln, Z. 16. A Swtchng Ant-wndu Degn Baed on Parttonng of the Inut Sace. Sytem & Control Letter. 88: [7] Zaccaran, L. and Teel. A.R. 4. Nonlnear Scheduled Ant-wndu Degn for Lnear Sytem. IEEE Tranacton on Automatc Control. 49(11): [8] Cao, Y., Ln, Z. and Ward, D.G.. An Ant-wndu Aroach to Enlargng Doman of Attracton for Lnear Sytem Subject to Actuator Saturaton. IEEE Tranacton on Automatc Control. 47(1): [9] Mantz, R. J. and Battta, H. 4. Comment on Varable-Structure PID Control to Prevent Integrator Wndu. IEEE Tranacton on Indutral Electronc. 51(3): [1] Mehd, N., Rehan, M., Malk, F. M., Bhatt, A. I. and Tufal, M. 14. A Novel Ant-wndu Framework for Cacade Control Sytem: An Alcaton to Under Actuated Mechancal Sytem. ISA Tranacton. 53: [11] Tarbourech, S. and Turner, M. C. 9. Ant-wndu Degn: An Overvew of Some Recent Advance and Oen Problem. IET Control Theory and Alcaton. 3(1): [1] Galean, S., Tarbourech, S., Turner, M. C. and Zaccaran, L. 9. A Tutoral on Modern Ant- wndu Degn. Euroean Journal of Control. 15: [13] Åtröm, K.J. and Wrttenmark, B Comuter Control Sytem Theory and Degn, Englewood Clff: Prentce-Hall. [14] Åtröm, K.J. and Rundqwt, L Integrator Wndu and How to Avod t. Proceedng of the Amercan Control Conference: [15] Kothare, M.V., Camo, P. J., Morar, M. and Nett, C. N A Unfed Framework for the Ant-wndu Degn. Automatca. 3: [16] Tyan, B., and Bernteu, D. S Ant-wndu Comenator Synthe for Sytem wth Saturaton Actuator. Internatonal Journal of Robut Nonlnear Control. 5: [17] Mulder, E. F., Kothare, M.V. and Morar, M. 1. Multvarable Ant-wndu Controller Synthe Ung Lnear Matrx Inequalte. Automatca. 37: [18] Grmm, G., Hatfeld, J., Potlethwate, I., Teel, A. R., Turner, M.C. and Zaccaran, L. 3. Ant-wndu for Stable Lnear Sytem wth Inut Saturaton: An LMI-Baed Synthe. IEEE Tranacton on Automatc Control. 48: [19] Slva da., J.M.G. and Tarbourech., J.R. 5. Ant-wndu Degn Wth Guaranteed Regon of Stablty: An LMI-Baed Aroach. IEEE Tranacton on Automatc Control. 5: [] Grmma, G., Teel, A. R. and Zaccaran, L. 4. Lnear LMI-Baed External Ant-Wndu Augmentaton for Stable Lnear Sytem. Automatca. 4: [1] Walgama, K.S. and Sternby, J Inherent Oberver Proerty n a Cla of Ant-Wndu Comenator, Internatonal Journal of Control. 5: [] Nu, W., and Tomzuka, M A Robut Ant-Wndu Controller Degn for Aymtotc Trackng of Moton Control Sytem Subjected to Actuator Aaturaton. The 37th IEEE Decon and Control Conference. Tama: [3] Hanu, R A New Technque for Preventng Control Wndu. Automatca. 1: 15-. [4] Hanu, R., Knnaert, M. and Henrotte, J. L The Condtonng Technque, A General Ant-Wndu and Bumle Tranfer Method. Automatca. 1: [5] Hanu, R., and Peng, Y.199. Condtonng Technque for controller wth Tme Delay. IEEE Tranacton on Automatc Control. 37:

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