IFAC Digital Control: Past,Present and Future of PlO Control.Terrassa.Spain.2000This paper presents a methodology for tuning PIDs considering the nominal performance and the robustness as control specifications. The synthesis procedure is similar to the Ziegler-Nichols method for PID controllers and can be easily used for industrial processes. As a workbench for testing the PID controller a mobile robot has been used. The path tracking problem of a mobile robot has been used as a workbench for testing the PID controller

Alacalá, Ismael

Camacho, Eduardo F.

Gómez Ortega, Juan

Normey Rico, Julio Elías

English

idUS. Depósito de Investigación Universidad de Sevilla

Copyright @ IFAC Digital Control: Past. Present andFuture of PlO Control. Terrassa. Spain. 2000ROBUST PID TUNING. APPLICATION TO AMOBILE ROBOT PATH TRACKING PROBLEMJulio E. Normey-Rico· Ismael Alcahi ••Juan G6mez-Ortega·· Eduardo F. Camacho··• Dpto. de Automa~ao e Sistemas, Federal Univ. of SantaCatarina, Brazil, e-mail:julio@lcmi.ufsc.br•• Dpto. Ingenieria de Sistemas y Automatica, Univ. of Seville,Spain, e-mail:{ismael.juango.eduardoJ@cartuja.us.esAbstract: This paper presents a methodology for tuning PIDs considering the nominalperformance and the robustness as control specifications. The synthesis procedure is~imilar .to the Ziegler-Nichols method for PID controllers and can be easily used formdustnal processes. As a workbench for testing the PID controller a mobile robot hasbeen used. The path tracking problem of a mobile robot has been used as a workbenchfor testing the PID controller. Copyright © 2000 IFACKeywords: PlO, Robust Control, Time delay, Mobile robots, Path tracking1. INTRODUCTIONIn spite of the fact that the most frequently con-trollers used in industry are PIDs, studies pre-sented by many authors have shown that a greatamount of control problems are caused by inap-propriate tuning of the PID parameters. Thus,study of PID tuning methods is of great impor-tance.Many methods have been proposed in literatureto tune PIDs. One of the most known meth-ods for tuning PID controllers when the processcan be modeled as a first order transfer functionwith a delay, either stable (Pe) or integrative(Pi) is the tuning method proposed by Zieglerand Nichols (1942). This method generates ac-ceptable responses in stable processes with lowratio between the dead time (L) and time con-stant (T) of the process, but the responses arevery oscillating or very slow with high ratio LITor integrative processes. In this paper, a PID istuned approximating the dead time by a transferfunction (Leonard, 1998) and taking into accountrobustness considerations.559As a workbench for testing the proposed PIDcontroller a mobile robot has been used. Theproblem treated in this work is known as PathTracking (PT) which is that of driving a mobilerobot to follow a previously calculated referencepath, defined as a set of consecutive points. Asynchro drive mobile robot is used and a robusttuned PID controller is proposed to solve thePT problem. This controller has been tuned forintegrative processes and allows the use of simplemodels.The paper is organized as follows: In section 2the description of the proposed tuning for thePID controller is presented for stable plants andintegrative plants are described in section 3. Therobustness of the proposed PID is discussed insection 4. The model considered for the mobilerobot kinematics is presented in section 5 andthe way in which the PID is applied to thePT problem is shown in section 6. In section 7,experimental results on a Nomad 200 mobile robotare shown. The paper ends with the conclusions.In the following two sections, the PID tuning isdiscussed for stable and integrative plants mod-2. PID TUNING FOR STABLE PROCESSESeled with Pe(s) and Pi(s) defined as:P. (s) =~e-L. R(8) = K ve- L• (1)e 1 + Ts ,. 8A simple and frequently used (Leonard, 1998) wayfor tuning a PlO controller for Pe plants is usinga rational first order function instead of the deadtime, which provides a good approximation in thefrequency domain:e-·L / 2 1 - 8L/2e-·L= e.L/2 - 1 + sL/2 (2)This approximation allows a low frequencies sec-ond order transfer function model given by:K p (1- sL/2)PI (s) = (1 + Ts)(1 + sL/2) (3)A PlO controller with a low pass filter (whichreduces high frequency derivative gain) has atransfer function:1 1C(s) =Kc (1 + -T + Tds)-lT (4)is + JSThe parameters of the controller are tuned usingthe root locus method, looking for quick responsewithout oscillations (see figure 1)T 2LT• d = L+2T• Ti = T + L/2• TJ is chosen to reduce high frequency gain.A simple way is to chose TJ = oL, witho< 0 < T / L , that allocates filter poles tothe left of -l/T (see figure 1).• Kc is chosen in such a way that exist twoequal closed loop real poles in s = -1/To:L+2T /Kc =~(40 + 1- 4(02 + 0/2)1 2)p(5)..,quick responses (low To values). If the model isequal to the process, this tuning guarantees quickresponses without oscillations and governed by8 = -l/To real poles. But in real processes, low Tomeans allocate poles in complex plane where can-not be used the approximation given by equation2. In these cases, errors between the model andthe process are important and the time responsehas high frequency components that generate os-cillations. If 0 > T / L a slow response is obtaineddue to low pass PID filter that cancels derivativeeffects. Also the system performance is limited bymodel errors for small 0 values. This problem issimilar to that of high order plants modeled withlow order models, where closed loop poles mustbe allocated considering unmodeled dynamic.In real practice, dynamic model errors and param-eters estimation uncertainties must be considered.As will be proved in this work, parameter 0 isrelated to the controller robustness. This create acompromise between performance and robustness.Once a PlO is well tuned for perturbation re-jection (with an adequate 0) the responses showa higher overshoot than the one specified. Thiscorresponds to the zeros allocated in the closedloop and can be corrected introducing a referencefilter (as shown in figure 2) that reduces the ef-fect of these zeros. The filter can be tuned in asimple way because zeros are introduced by thecontroller. This problem is known as two degreesof freedom control and is analyzed in some works(Morari and Zafiriou, 1989) (Astrom and Witen-mark, 1984). Also, a study of the zeros influencein system response based on basic tools (timeresponse and frequency domain) is presented in(Normey-Rico, 1995).Fig. 2. System controlled with PlO and referencefilter(6)'1'_ 0 L.Io - -(02 + 0/2)1/2 - 0 2with:o.:~':-,-.7,-.-::',.':----:.7",---:r-:;;';:'--:1a-x-:iso;----:o~,-7----:"Fig. 1. Root locus diagram for PlO tuningFrom these relations and from root locus dia-gram it can be seen that low 0 values generate(7)W(s) 1 + OATrs1 +sTrand the zeros effect can be canceled with Tr .The slow PlO zero is allocated in s = -2/L,and then Tr can be chosen equal to L/2. Thus,the number of parameters remains the same andthe controller simplicity is also maintained. Abetter performance is shown in figure 3 wherethe simulation conditions and PlO parameters areheld.In particular, the reference filter can be chosen asfollows:·'IT-1lTl..........560(13)TT =0.75Li,02I· ,',.~' 0.1: .', .'''"Y = 0.5I .'..0." I;• i·... 0.8~Q,~ 0.6-020 20 40 60 80 100 120 140 160 180 200Time (0)Fig. 5. System responses in closed loop with theproposed PID for an integrative plantFrom these results, it can be concluded that thisPID can give good results if very short time re-sponse is not required. In these cases, the approx-imation used for time delay modelling is no longervalid and the performance is deteriored. In prac-tice, the error in the time delay approximation isnot the only one difference between the real plantand its model and all of them must be consideredin the controller design in order to guarantee the1.2It is clear that the scheme presented in figure4 can be transformed into the scheme of figure2. The relation between W'(s), Ko and C'(s)and C(s) and W(s) is given by: C(s) = Ko +C'(s), W(s) = w'~~~'(8) where can be observedthat although C(s) is a PID controller, W(s) is nomore a first order filter and the simple procedureused with C'(s) cannot be repeated. In figure 5it is shown responses to different "I values for theproposed C'(s) e W'(s).• Ti = 1.5L e Td =2L/3,• TI proportional to L: TI = "IL, with 0 < "I <0.75 (for the allocation of the filter poles tothe left to the P2 (s) poles)• Kc for real poles is a function of "I and isgiven by:Kc = ;:v (4"1 + 1- 4("/2 + "1/2)1/2) (11)that allocates the poles in s = -11T1 with:T1 = "I !:. (12)("/2 + "112)1/2 - "I 2The obtained relations are similar to that of thestable case expressions. Again, a new parameter("/) defines robustness characteristics of the sys-tem and also, it is used a reference filter W'(s) =1t~'i::;" for better transient responses in referencechanges, and it is tuned as in the stable case.In this case, C'(s) has complex conjugate zeroswhose real part is used for tuning TT; then:eo ';~e (~ 140 160 180 200,..,. :1l Q == 0.02, ..', ., 2 , .·'. , ·."0 ,· '. ..-, : . . ,~ , .~ 0 ~ .... ,...- ..\.~,Q, 0 0~oa 0 0 ,1,,/ ...,)0.o. 0 ~'O.l0 Q00,04 0 .0.020 ,i Q = 0.5,,Fig. 4. Integrative process controlled by Ko gainand PID with reference filterdelay, Ko must be carefully chosen in order toavoid oscillatory responses in Pi modeled pro-cesses. In this work, is proposed to use the samedelay approximation than in the stable case:Pi(s) = K v (l - sL/2) (8)s(l + sL/2)From the root locus diagram it is possible tocalculate the Ko value which makes that the in-ner closed loop poles generate responses with anovershoot less than 5%. Thus, the Ko final value isKo = ~.5L and the new transfer function betweencontrol 'and output (with the delay approxima-tion) is given by:p. (s) =2Kv / L(l - sL/2) (9)2 1 + 1.5Ls + L2 S 2In the second step, the PID controller C'(s) istuned using the same ideas presented for thestable case:C'(s) = K c(1 + T1 + Td S )(-1IT) (10)is + ISand the following values are selected:Dead time and integrative behavior makes PIDtuning more difficult. As was mentioned, Ziegler-Nichols method is not good enough in these cases.Alternative methods have been proposed (Morariand Zafiriou, 1989) (Chien and Fruehauf, 1990)(Luyben, 1996). In this work, simplicity is main-tained with two steps based method. In the firststep a proportional control Ko (see figure 4) istuned. This transforms integrative system dy-namic into a stable process. Due to the systemq(l)u(l) +1+Fig. 3. Closed loop system responses with PIDand reference filter for stable plant3. PID TUNING FOR INTEGRATIVEPROCESSES4. ROBUST TUNING OF PID CONTROLLERrobustness. This will be analyzed in the followingsection.The robust stability in closed loop can be analyzedin a simple way taking into account that P canbe written as P :;;; Pn + OP, where OP are themodeling errors and Pn represents the nominalplant. This type of uncertainties model is notstructured because they are not linked to specificparameters (gain, delay, etc). In this case, Pn(s)is the transfer function used to tune the PlO:Pn(s) :;;; Pe(s) in the stable case, and Pn(s) :;;;Pi (s) in the integrative case. The closed loopcharacteristic equation is: 1 + C(s)P(s) = 1 +C(s)(Pn + oP) = 0, where C(s) is the totalcontrol action in cascade with the process. A polardiagram of the system is shown in figure 6.-r = 0.5'0'10'.---------~------__.dP(w) coincides with the inverse of the systemclosed loop transfer function for W (s) = 1. Thisshows the relation between performance and ro-bustness. Using expressions of Pn(s) and C(s)with the proposed tuning for the stable case:l:,Pe(w) = 1(1 + Tojw)2 I Vw (17)11- jwL/21in integrative case is:l:,Pi(w) = I (1 + Tdw)2 I Vw (18)11- jwL/2\As To is a function of (} and T l is a function off, these parameters can be used for tuning thesystem robustness in closed loop. Greater (} (()values, mean greater To (Tl ) values, and therefore,greater dP(w) in high frequencies. On the otherhand, a robustness improvement implies slowerresponses and performance losses. For instance,in figure 7 dPi(w) is shown for different values off·1...&Real-I-1-ePnC(Pn-oliP) cp,l.\ .,:;~ 10~ r----==~====--~...::~_J-r=-_ ..... --_ .... .. ... modeling errorsFig. 6. Polar diagram for robust stability study ofPlOFirst, it is calculated the polar diagram in openloop and nominal case CPn(jw). With the tuningchosen in the nominal case, the -1 point is notincluded. Then, the distance between diagrampoints CPn(jw)and -1 point are computed by11-1 - CPn(jw)ll. This is a measure of the closedloop robustness. From the characteristic equationof the system it is obtained:1-1 - CPn 1=11 + CPn 1=1 CoP I (14)Now it is computed the polar diagram for C(Pn +oP). If the -1 point is not included in the newdiagram, then the real system is stable and thedistance between real and nominal curves arealways greater than 11 +CPn I for all frequencies.The maximum distance between nominal and realplants (l:,P) that holds robust stability in closedloop is calculated by:IoP(jw) I~ l:,P(w) = 11 + fg~~n,(jW) I Vw.(15)or relative to Pn :I oP(~w) 1< dP(w) = 11 + ~(jw)~~(j)wl) I Vw.Pn(Jw) - I C(Jw)Pn JW(16)'0~'L-------~------,""""",1~ 1~ 1~FrequencyFig. 7. PID robustness with different f values.In the same figure, the modeling error ~ isdepicted (dotted line). With this method it ispossible to tune (} (() parameter.Also it is possible to chose (} (() as the parameterthat guarantees a frequency response module inclosed loop (the inverse of ~) without a peak.It can be proved that this is guaranteed with(} = f = 0.13 (note that in the proposed cases,the function is the same for stable and integra-tive cases). Chosing (} = f = 0.13, the tuningparameters can be written in a tabular form likeZiegler-Nichols known rules. Table 1 shows thetuning for PID(s) = K c(1 + i.. + Tds)(l+h.),reference filter Filter(s) - l+O.4Tr B and K (in- l+TrB 0the integrative case).5. THE MOBILE ROBOT PATH TRACKINGPROBLEMAs a application for testing the proposed robusttuned PID controller a Nomad 200 mobile robot is562Model Kc T; Td T( Ko TT~e-L- U.~(~IL.+' T+L/2 LT O.13L 0 L/21+_ K»L L+2T~e-L- ~i! 1.5L 2L/3 O.13L 2# L O.75L•Table 1. PID parameters value for the proposed tuning7. EXPERIMENTAL RESULTSThen, for the mobile robot, the reduced model wasdefined with Kv = 1, L = 0.2 s and 'Y was chosenequal to 0.5 for compensate uncertainties in thesimplified model given by equation 19.Using 'Y = 0.5 the parameters for the referencefilter W(s), C(s) and Ko are computed. Thesetransfer functions were calculated for discrete timeand the final control law programmed in the robotmobile main processor.For testing the behaviour of the robust tunedPID, the block diagram that appears in figure 10was implemented in a Nomad 200 mobile robot.The reference steering angle 8r is computed in theapproximation point block from the robot position(xg,yg) and the points that define the path. Inorder to tune the proposed PID, a linear model ofthe process is needed. The linear model has beenobtained from the hypothesis that the incrementsin the reference steering velocity are small and thedynamic of the rate limiter block is not present.In this case the model can be simplified by anintegrative process with dead-time given by thefollowing equation:e-O.2sPi(s) =-- (19)s6. PATH TRACKING WITH THE PROPOSEDPIDFor the experimental tests the controller parame-ters have been chosen as follows: >. = 0.6 meters,V = 0.4 m/ s (90% of the maximum robot veloc-ity),8max = 30 degrees/s2 and the sampling timeT was chosen equal to 0.2 secondsFigure 11 shows the performance of the proposedcontroller for two different real dead times. Thesteering velocity reference 8r has been chosen, anda constant linear velocity V of the mobile robothas been used. The model of the robot for thesteering angle is given by a kinematic equationand a non linear dynamics related to a rate limiterin the steering velocity. As was mentioned, thecontrol variable used has been the steering veloc-ity reference 8r . The steering velocity 8(t) followsits reference with a non linear dynamics. Whena change in 8r occurs, the steering velocity has asmall dead time L = 0.2 seconds and is limitedby the maximum acceleration (8ma.,) given by thesteering motors.85.1 Synchro-drive steering angle modelFig. 9. Approximation points and their respectivereference steering angles in two different po-sitionsFig. 8. Global reference frame for the mobile robotomad 200used. This robot has a synchro-drive type locomo-tion system which consists of three drive wheelswhich turning speed and orientations vary simul-taneously. In figure 8 it is shown the synchro-driveconfiguration and the steering angle 8 referencedto the global axis.The robot provides a position estimation systembased on odometry. It is well known that odom-etry is a technique which has an accumulativeerror which implies the need for update the es-timation from environmental data provided fromsensor systems with a predetermined frequency.This problem is considered to be decoupled withthe PT problem and has not been accomplishedin this work.X g XglobalY',}--+-~The mobile robot steering angle {) must follow asteering angle reference 8r defined by an approx-imation point located at a fixed distance, >., inthe path (see figure 9). As the control variable theYglobalXglobal563Path trackin with estimation dela errorsPath tracking without ~til1lation delay I?frors~_I .if\vi. ~ :1~ : \ ~ ~ !~~~ i v0\:: .\'S : :::=-.. ....,,~_ .• '<) ,(b)·~ ,~-.,(a)'~ •(a) 'I. .':"Fig. 10. Block diagram of the proposed PlOcontroller implemented in the mobile robotfor path trackingPath tracking without estimation delay errorsI !':f( 000 000 1,:(~~'l Th':f _.~IIrltOOOOOO~~Pat!}. tracking with estimation delay errors,! 0 1ut ~j ~:_-'''''') I" -~ 1,,[ Jo'CX)ooo~O'O~ I(b) \, 1 " J ,':,. ) UFig. 11. Path tracking for the mobile robot: (a)with real delay L, (b) with real delay 1.4£initial position for the mobile robot was Xo =0.6, Yo = 1.13 and orientation 00 = 0 (parallelto x g axis). Note that the reference path chosenhas small curvatures, which makes more difficultto follow the reference. The path tracking in thetwo cases of figure 11 are very similar withoutdelay errors (figure l1(a)) and with a 40% in thedelay error (figure l1(b)). The difference appearsin the very first sample times, where the pathfollowing with estimation delay errors (b) is moreoscillating.In figure 12 also there are differences when thecontrol demands more control effort in path track-ing with estimation error (figure l1(b)).Note the satisfactory performance obtained inthe reference following (see figure 11) althoughthe linear velocity V is 90% of the Nomad'stop velocity and the reference steering angle Orchanges continuously.8. SUMMARY AND CONCLUSIONSA method for robust tuning of PID controllers hasbeen proposed based on classical control conceptsFig. 12. ~teering velocities 6r (continuous line)and 0 (dashed line): (a) with real delay L (b)with real delay 1.4£and can be applied to stable and integrative pro-cesses that are modeled like first order transferfunctions with a delay. The synthesis procedureis similar to the Ziegler-Nichols method for PIDcontrollers. The robustness of the controller canbe changed with parameters (0 and 'Y for sta-ble and integrative plants) and allows a goodperformance with uncertainties in the dead time.Finally the proposed PID has been implementedfor a mobile robot path tracking problem andsome experimental results have shown the goodperformance obtained in spite of delay estimationuncertainties.9. REFERENCESAstrom, K.J. and Witenmark (1984). ComputedControlled Systems. Prentice Hall, New York.Chien, I.L. and Fruehauf (1990). Consider imctuning to improve controller performance.Ind. Eng. Chem. Res. 86(10), 33-41.Leonard, F. (1998). Delay approximation com-parison in a cacsd context. IFAC LTDS'98pp. 99-104.Luyben (1996). Tuning proportional-integral-derivative controllers for integrator dead-time processes. Ind. Eng. Chem. Res.35(10),3480-3483.Morari, M. and E. Zafiriou (1989). Robust ProcessControl. Prentice-Hall.Normey-Rico, J. (1995). Teaching classical con-troller design in a basic control course. Tech-nical report. Proc. III LASAC, Chile.Ziegler, J.G. and N.B. Nichols (1942). Optimumsettings for automatic controllers.. Trans.ASME 64 pp. 759-768.564

Robust PID tuning. Application to a Mobile Robot Pathtraking problem.

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Robust PID tuning. Application to a Mobile Robot Pathtraking problem.

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