American Journal of Applied Sciences, 4 (4): 0-8, 007 ISSN 1546-939 007 Science Publications Fuzzy Anti-windup Scemes for NCTF Control of Point-to-point (PTP) Positioning Systems Wayudi, Riza Muida and Momo J.E. Salami Intelligent Mecatronics Systems Researc Group Department of Mecatronics Enginering, International Islamic University Malaysia P.O. Box, 10, 5078, Kuala Lumpur, Malaysia Abstract: Te positioning systems generally need a controller to acieve ig accuracy, fast response and robustness. In addition, ease of controller design and simplicity of controller structure are very important for practical application. For satisfying tese requirements, NCTF (nominal caracteristic trajectory following) controller as been proposed as a practical PTP positioning control. However, te effect of actuator saturation can not be completely compensated due to integrator windup because of plant parameter variations. Tis study presents a metod to improve te NCTF controller for overcoming te problem of integrator windup by adopting fuzzy anti-windup scemes. Two fuzzy antiwindup scemes based on Mamdani and Takagi-Sugeno fuzzy system are developed and evaluated teir effectiveness. Te improved NCTF controller wit te proposed fuzzy anti-windup scemes is evaluated troug simulation using dynamic model of a rotary positioning system. Te results sow tat te improved NCTF controller wit Takagi-Sugeno-based fuzzy windup is te best sceme to compensate for te effect of integrator windup. Keywords: Positioning, point-to-point, fuzzy, anti-windup, compensation, controller, robustness INTRODUCTION Motion control systems play important roles in industrial process suc as macine tools, semiconductor manufacturing systems and robot systems. One type of motion control systems is point-to-point (PTP) positioning system, wic is used to move a plant from one point to anoter point. Te positioning systems generally need a controller to satisfy suc requirements suc as ig accuracy, fast response and robustness. Up to now many types of controllers ave been proposed and evaluated for positioning systems; for example te model following type controller suc as controllers wit disturbance observer [1-4], time-optimal controllers [5-8] and sliding mode controllers [9,10]. Tese controllers will give good positioning performance if experts on motion control system do design te controller using te exact model and values of its parameters. It is well known tat exact modeling and parameter identifications are generally troublesome and time consuming tasks. In general, advanced controllers tend to be complicated and require deep knowledge concerning controller teory and design. However, in practical applications, engineers, wo are not experts in control systems, often need to design te controllers. Hence, ease of controller design and simplicity of controller structure are very important for practical applications. In order to overcome te above-mentioned problems, nominal caracteristic trajectory following (NCTF) controller as been proposed as a practical controller for point-to-point (PTP) positioning systems [11]. It as been sown tat te NCTF control system as a good positioning performance and robustness [1,13]. Te NCTF controller is also effective to compensate te effects of friction wic is te source of positioning inaccuracy [14]. However, te effect of actuator saturation can not be completely compensated due to integrator windup wen te plant parameters vary [15]. Te NCTF controller gives an excessive oversoot wen actuator saturation as well as parameter variations (especially inertia variation) occur in te positioning systems. Tis study describes a metod to improve te NCTF controller to overcome te degradation of te positioning performance due to integrator windup. Fuzzy Anti-windup scemes are introduced for te NCTF controller. Two fuzzy anti-windup scemes based on Mamdani and Takagi-Sugeno fuzzy system are developed and evaluated teir effectiveness troug Corresponding Autor: Wayudi, Intelligent Mecatronics Systems Researc Group, Department of Mecatronics Enginering, International Islamic University Malaysia, P.O. Box, 10, 5078, Kuala Lumpur Malaysia, Tel: +603-056-4469, Fax: +603-056-4433 0
simulation using dynamic model of a rotary positioning system. Te simulation results confirm tat te improved NCTF controller wit Takagi-Sugeno-based fuzzy windup is te best sceme to compensate for te effect of integrator windup. NCTF control system Basic concept of NCTF control system: Te structure of te NCTF control system is sown in Fig. 1. Te NCTF controller consists of a nominal caracteristic trajectory (NCT) and a compensator. Te NCTF controller works under te following two assumptions: * A DC or an AC servomotor is used as an actuator of te plant. * PTP positioning systems are cosen, so θ r is constant and θ r 0. Here, te objective of te NCTF controller is to make te plant motion follows te NCT and ends at te origin of te pase plane ( e, e ). Figure sows an example of a plant motion controlled by te NCTF controller. Te motion comprises two pases. Te first one is te reacing pase (RP) and te second is te following pase (FP). In te reacing pase, te compensator forces te plant motion to reac te NCT as fast as possible. However, in te following pase te compensator controls te plant motion so as to follow te NCT and end at te origin. Te plant motion stops at te origin, wic represents te end of te positioning motion. Tus, in te NCTF control system, te NCT governs te positioning response performance. Design of NCTF controller: Te NCTF controller is designed based on a simple open-loop experiment of te plant as follows: * Open-loop-drive te plant wit stepwise inputs and measure te displacement and velocity responses of te plant. Figure 3a sows te stepwise inputs and te velocity and displacement responses due to te stepwise inputs. In tis study, te rated input to te actuator u r is used as te eigt of te stepwise inputs. * Construct te NCT by using te plant responses. Te velocity and displacement responses are used to determine te NCT. Since te main objective of PTP system is to stop te plant at a certain position, a deceleration process is used, see curve A, Fig. 3(a). Te in Fig. 3 represents te maximum velocity. From te curve in te area A and in Fig. 3(a), te NCT in Fig. 3(b) is determined. Since te NCT is constructed based on te actual responses of te plant, te NCT includes nonlinearity effects suc as friction and saturation. Te Am. J. Appl. Sci., 4 (4): 0-8, 007 1 θr + - Fig. 1: e Error rate Fig. : Input, Displacement,Velocity e Error rate e Controller Nominal caracteristic trajectory (NCT) d dt e O P(e,e) u p θ e u p u Compensator NCTF control system NCT Plant motion O RP : Reacing pase FP : Following pase Error e NCT and plant motion u r A Input Velocity Displacement FP Plant RP Time (a) Stepwise inputs and responses A A O m Error e (b) Nominal caracteristics trajectory (NCT) Fig. 3 NCT determination A θ
important NCT information, wic will be used to design te compensator, are NCT inclination m near te origin and maximum error rate of. In tis case, from te relationsip between plant dynamics of Eq. (1) and Fig. 3(b), it is clear tat te inclination near origin m and te maximum error rate relate wit parameters of te plant as follows [1,15] : K = (1) u r α = m () * Design te compensator based on te NCT information. Here, te following PI compensator is adopted due to its simplicity: u = Kp u p + Ki u pdt (3) were K p and K i are proportional and integral gains respectively. Using te PI compensator parameters K p and K i and te NCT caracteristic near te origin (Fig. 3b), te transfer function of te closed-loop positioning system controlled by te NCTF controller can be approximated as follows [11-15] : Θ(s) = G(s) = G 1(s)G (s) (4) Θr (s) were α G 1 (s) = (5a) s + α ζωn + ωn G(s) = (5b) s + ζωn + ωn ζωn Kp = (5c) K α ωn Ki = (5d) K α were K and α are positive constants wic relate to te plant dynamics. Meanwile ζ and ω n are damping factor and natural frequency respectively. Wen ζ and ω n are large enoug, G(s) becomes nearly equal to G 1 (s), wic represent te condition wen te plant motion follows te NCT as te objective of te NCTF control system. Moreover, large ζ and ω n also make te closedloop system robust to friction or inertia variation of te plant in continuous systems [9]. Finally, by using ζ and ω n as design parameters and considering Eqs. () and (3), te PI compensator parameters are designed as follows: ζωnu r Kp = (6) m Am. J. Appl. Sci., 4 (4): 0-8, 007 ωnu r Ki = (7) m Here, ω n and ζ are design parameters wic sould be decided by te designer. Generally speaking, a iger ω and a larger ζ are preferable in te design of PI compensator parameters. However pysical constraint of te motor driver and digital implementation of te NCTF controller limits te design parameters to maintain te closed-loop stability as follows [1] : αks ω R n (8) ζω n (9) 3T were S R and T are motor driver slew rate and sampling time. Detailed discussion on te teoretical background of te NCTF control system can be seen in [1,14,15]. Due to te fact tat te NCT and te compensator are constructed from a simple open-loop experiment of te plant, te exact model including te friction caracteristic and te identification task of te plant parameters are not required to design te NCTF controller. Terefore, te controller design is simple and easy to implement in practical situation. Fuzzy anti-windup scemes: As te NCTF controller uses te PI compensator to force plant motion so tat it follows te NCT, te integrator windup up may occur in connection wit large position reference. As discussed in reference [15], in te case of no parameter variations, tere is no significant integrator windup due to te effect of te saturation. Te effect of te saturation is successfully compensated by using NCTF controller. However te integrator windup becomes a problem wen te parameters vary [15]. In order to overcome te problem of integrator windup, te PI compensator is improved by adopting a fuzzy anti-windup sceme. Hence an anti-windup PI compensator is proposed to be used instead of a pure PI compensator. Te purpose te FAW is to generate te appropriate output to adjust te value of te integrator part wen te saturation exists. Terefore, by using te proposed anti-windup PI compensator, it is expected tat once PI compensator output U(s) exceeds te actuator limits, te fuzzy ant-windup sceme will generate a signal to reduce te effect of te integrator windup. Furtermore, two fuzzy anti-windup scemes are proposed and evaluated teir effectiveness. Te first anti-windup system is Mamdani-based fuzzy antiwindup (MFA), wic is design based on Mamdani-
type of fuzzy system. Te second anti-windup system is Takagi-Sugeno based fuzzy anti-windup (TFA), wic is design based on Sugeno-type of fuzzy system. Am. J. Appl. Sci., 4 (4): 0-8, 007 Mamdani-based fuzzy anti-windup sceme: Te proposed Mamdani-based fuzzy anti-windup (MFA) for te PI compensator is sow in Fig. 4. Te purpose of te MFA is to generate te signal U MFA to reduce te integrate part if tere is input signal ( U). In te case of no saturation, te input U is equal to zero. Accordingly, te MFA will output signal U MFA of zero so tat it does not affect te performance of te system unless tere is saturation. Te first element of te proposed MFA is te fuzzy interface wic is used to convert crisp input of te control output into linguistic variables. To map te crisp inputs into related linguistic fuzzy sets, associated membersip functions ave to be constructed. Figure 5 sows te membersip functions of te MFA wic use simple and commonly used triangular membersip functions. Te membersip function parameters A i, B i and C i are determined based on te NCT parameter and te motor input rated as follows: C i = K P -u r (10) B i = 4 3 Ci (11) Fig. 4: Fig. 5: Mamdani-based fuzzy anti-windup sceme Membersip function of te MFA input A i = 1 Bi (1) were K p, and u r are proportional gain, maximum error of te NCT and te voltage input of te driver respectively. As sown in Fig. 5, te input of te MFA contains 4 fuzzy sets (linguistic variables) namely Negative Big (NB), Negative Small (NS), Zero (Z), Positive Small (PS) and Positive Big (PB). Te range of te fuzzy input depends only on NCTF controller parameters K p and and te saturated value of te control signal u r. Hence te design of te membersip function is simple and straigt forward. Te MFA output as also 4 fuzzy sets (linguistic variables) as sows in Fig. 6. Tey are Negative Big (NB), Negative Small (NS), Zero (Z), Positive Small (PS) and Positive Big (PB). Te range of te output is based on te range of te integrator term. Te output membersip function parameters A o, B o and C o are calculated based on te NCT parameter and te integrator gain of te PI compensator as follows: C o = K i (13) B o = 4 3 Co (14) A o = 1 Bo (15) Fig. 6: Membersip function of te MFA output were te K i is te integrator parameter of te PI compensator. Fuzzy rule base is te second element of te MFA. Te fuzzy rule-base of te Mamdani-based fuzzy system is composed by IF-THEN rules like IF antecedent 1 THEN Consequent 1 [16]. Te following fuzzy rules are derived and used to reduce te effect of te integrator windup: IF U is NB THEN U MFA is PB IF U is NS THEN U MFA is PS IF U is PS THEN U MFA is NS IF U is NB THEN U MFA is NB Furtermore, te fuzzy inference engine and te defuzzifier are used to carry out all te fuzzy operations and ence to determine te U(s) MFA on te activated rules along wit te membersip degrees of te associated fuzzy inputs. Mamdani fuzzy inference engine is used to decide te MFA output. Moreover, te Mamdani inference engine works base on Mamdani implication (Min operation) and disjunctive aggregator (Max operator), wile te MFA output is converted into crisp output based on te Centroid metod. 3
Takagi-sugeno-based fuzzy anti-windup sceme: Figure 7 sows te proposed Takagi-Sugeno-based fuzzy anti-windup (TFA). Te main objective of te fuzzy (TFA) Anti-windup is to generate te suitable signal to reduce te integrator part if tere is saturation oterwise te output from te Anti-windup is zero. Te first step of te TFA is to convert te crisp input to te membersip value of te fuzzy set. Similar wit te MFA, triangular membersip functions are adopted as sown in Fig. 8. Tere are tree linguistic variables represented by te triangular membersip function namely Negative Saturation (NS), Unsaturated (US) and Positive Saturation (PS). Unlike te MFA wis is signal U as te input, te control signal U is used as te input. Terefore te range of te fuzzy input depends on te range of U(s). Te membersip function parameters A and B are determined based on te NCT parameter and te PI parameters as follows: were A=u r (16) B= K P (17) Unlike te MFA wic te output is fuzzy variables, te output of te TFA is a crisp variable. In general, fuzzy rule-base of te Takagi-Sugeno fuzzy system is composed by If x 1 is A AND x is B THEN u 1 = f(x 1, x ), were x 1 and x are te antecedent and u 1 is te crisp consequence on x 1 and x [16]. Tree rules are used to cover te entire situation in te TFA, te rules are If U is NS THEN te output is U 1 If U is US THEN te output is U If U is PS THEN te output is U 3 were K 0.1 (A + 0.1)(K 0.1) U = U i + i 0.1 (18) 1 B A 0.1 B A 0.1 U = 0 (19) K 0.1 (A + 0.1)(K 0.1) U = U i i + 0.1 (0) 3 B A 0.1 B A 0.1 Finally, te overall output of te TFA U n is calculated based on te following equation: 3 µ U = i = U i U 1 i (1) n 3 µ i = U 1 i Am. J. Appl. Sci., 4 (4): 0-8, 007 Controller design Simulated plant description: Te NCTF controller wit anti-windup PI compensator is tested using a dynamic model of te experimental rotary positioning system as sown in Fig. 9. Te positioning system consists of an AC servomotor, a driver and an inertia 4 Fig. 7: Takagi-sugeno-based fuzzy anti-windup sceme Fig. 8: Takagi-Sugeno-based Fuzzy Anti-windup sceme Fig. 9: U( s) Experimental rotary positioning system Friction model Current controller + + - K 1 1 1 K v K ci + + Ω(s) Θ(s) sp Kcp + K t Ls + R Js + C - s - + s Velocity controller K b Fig. 10: Detailed model of rotary positioning system Error rate rad/s 00 100 0-100 -00 m = -67.4 = -40-15 -10-5 0 5 10 15 Error rad Fig. 11: Nominal caracteristic trajectory (NCT) mass (spindle). Te positioning performance was examined using te detailed model in Fig. 10. Its
Am. J. Appl. Sci., 4 (4): 0-8, 007 Table 1: Parameters of te plant Parameter Value Inertia load, J 1.17 x 10-3 kgm Motor resistance, R 1. Ω Motor inductance, L 8.7 mh Motor torque constant, K t 0.57 Nm/A Back-Emf constant, K b 0.57 Vs/rad Viscous friction, C 1.67 x 10-3 Nms/rad Frictional torque, τ f 0.15 Nm Proportional current gain, K cp 6. V/A Integral current gain, K ci 3.6 x 10 3 V/As Proportional velocity gain, K sp 8.60 x 10 - As/rad Input voltage range, u r ± 6 Volt by using and m of te NCT. For te PI compensator, design parameters ζ and ω n are cosen as 13 and 9 respectively [15]. Te values of te compensator parameters calculated using Eqs. (6) and (7) are K p = 0.79 and K i = 0.31. Anti-windup design: Bot te MFA and TFA are designed based on procedure discussed earlier. Tus, Fig. 1 sows te input and te output membersip functions of te MFA by considering te NCTF controller parameter discussed earlier wile tose of te TFA are depicted in Fig. 13. Te rules of te bot proposed anti-windup are simply designed by using te developed rules presented earlier. Fig. 1: Membersip function of te MFA Fig. 13: Membersip function of te TFA parameters are sown in Table 1. Te positioning performance is examined under two conditions, namely Normal Plant and Increased Inertia Plant. Normal Plant as te nominal plant parameters described in Table 1, wile Increased Inertia Plant as about 10 times spindle inertia tan tat of Normal Plant. Controller design: First, te NCTF controller was designed based on te Normal Plant. Fig. 11 sows te NCT as a result of a simulated experiment. Wit reference to Fig. 11, te inclination m and maximum error rate of te NCT are 67.4 rad and 40 rad/s respectively. Te compensator parameters are designed 5 RESULTS AND DISCUSSION Now, te performances of te positioning system controlled using te NCTF wit MFA (NCTF-MFA) and TFA (NCTF-TFA) are compared to tat wit tracking-anti-windup (NCTF-TAW). Tracking antiwindup sceme is classical anti-windup commonly used for PID controller [17]. Detail discussion on te implementation of te tracking anti-windup for te NCTF controller can found in reference [18]. Figure 14 sows te step responses to a 0.5 rad step input wen te controllers are used to control Normal Object and teir positioning performances are summarized in Table. As sown in Fig. 14, te 0.5-rad step input does not cause te saturation of te control signal. Here, all te controllers produce a similar response due to a similar bandwidt. Hence, in terms of oversoot and settling time, all of te controller give a similar performance. Furtermore, in order to estimate te robustness of te control systems to inertia variation, all te controllers are implemented on Increased Inertia Object. Fig. 15 sows te step responses to a 0.5-rad step input wen all te controllers are implemented for controlling Increased Inertia Object. Table sows te positioning performance resulting from all te controllers. All NCTF controllers give similar results since tere is no significant saturation of te actuator as sown Fig. 15(b). Te result confirms tat te use of anti-windup PI compensator does not affect te positioning performance wen tere is no saturation of te actuator. Next, simulation as been done for a larger step input so tat te actuator saturates. Figure 16 sows te step responses to a 5-rad step input wen all te controllers are implemented for controlling Increased Inertia Object. Table 3 sows te positioning performance resulting from all te controllers. Te saturation of te actuator occurs as sown in Fig. 16(b).
Table : Performance Comparison, 0.5 rad Step Input Object Controller Settling time Oversoot% Normal NTCF- TAW 0.059 0 NTCF- TFA 0.090 0 NTCF- MFA 0.060 0 Increased inertia NTCF- TAW 0. 14.8 NTCF- TFA 0.3 14 NTCF- MFA 0.195 14.8 Table 3: Performance Comparison, 5 rad Step Input and 50 rad Step Input Reference Controller Settling time Oversoot % 5 NTCF- TAW 0.413 9.8 NTCF- TFA 0.700.6 NTCF- MFA 0.31 9 50 NTCF- TAW 1.3 31.43 NTCF- TFA 1.04 6.5 NTCF- MFA 1.15 8.38 Am. J. Appl. Sci., 4 (4): 0-8, 007 Fig. 15: Step responses to a 0.5 rad step input, Increased Inertia Object Fig. 14: Step responses to a 0.5 rad step input, Normal Object However, Fig. 16(a) and Table 3 sow tat te positioning performance of te positioning system wit Fig. 16: Step responses to a 5 rad step input, Increased NCTF-TFA as less oversoot tan te rest, owever in Inertia Object terms of te settling time te NCTF-MFA is te better. In te same time, NCTF-TAW as te same oversoot and settling time is better tan NCTF-TAW. performances wen compared to te NCTF-MFA. On te oter and, NCTF-TFA still remains as te best Furtermore, te remarkable finding in te one. pervious simulation is tat te performances of te NCTF-TAW and NCTF-MFA seem to be similar in CONCLUSION terms of oversooting and settling time. To distinguis between bot of tem, a 50-rad step input is considered Tis study as documented te improvement of te for te next simulation study. As sown in Fig. 17 and NCTF controller to overcome te effects of integrator Table 3, te performance of te NCTF-MFA in terms of windup due to actuator saturation. Fuzzy anti-windup 6
Am. J. Appl. Sci., 4 (4): 0-8, 007 Fig. 17: Step responses to 50 rad step input, Increased Inertia Object PI compensator was used as compensator for te NCTF controller instead of a conventional PI compensator. Two fuzzy anti-windup scemes are designed namely Mamdani-based fuzzy anti-windup (MFA) and Takagi- Sugeno-based fuzzy anti-windup (TFA) scemes. Troug simulation using dynamic model of a rotary positioning system, te effectiveness of te NCTF controller wit te proposed fuzzy anti-windup PI compensator is evaluated. Te effectiveness of te proposed scemes are evaluated and compared wit te conventional tracking anti-windup sceme. Te results confirm tat te NCTF controller wit te Takagi- Sugeno-based fuzzy anti-windup PI compensator is better tan te oters. Te use of te Takagi-Sugenobased fuzzy anti-windup PI compensator is more effective to overcome te problem due to integrator windup compared to tat of te oters. Moreover, te simulation results also confirm tat te NCTF controller wit te Takagi-Sugeno-based anti-windup PI compensator, wic is design based a simple open loop experiment, gave te best positioning performance as well as performance robustness to inertia variations. ACKNOWLEDGMENTS Tis researc is financially supported by Ministry of Science, Tecnology and Innovation (MOSTI) under Sciencefund Grant 03-01-08-SF0036. 7 REFERENCES 1. Umeno, K., T. Kanoko and Y. Hori, 1993. Robust servosystem design wit two degree of freedom and its applications to novel motion control of robot manipulators. IEEE Trans. on Industrial Electronics, 40: 473-485.. Endo, S., H. Kobayasi, C.J. Kempf, S. Kobayasi, M. Tomizuka and Y. Hori, 1996. robust digital tracking controller design for ig-speed positioning systems. Control Engineering Practice, 4: 57-536. 3. Tomizuka, M., 1996. Robust digital motion controllers for mecanical systems. Robotics and Autonomous Systems, 19: 143-149. 4. Kempf, C. and S. Kobayasi, 1999. Disturbance observer and feedforward design for a ig-speed direct-drive positioning table. IEEE Trans. on Control Systems Tecnol., 7: 513-56. 5. Wu, S. and J. Fu, 1998. Time-optimal control of servo systems using PD algoritms. JSME Intl. J.: Series C, 41: 384-390. 6. Park, M.H. and C.Y. Won, 1991. Time optimal control for induction motor servo system. IEEE Trans. on Power Electronics, 6: 514-54. 7. Workman, M.L., R.L. Kosut and G.F. Franklin, 1987. Adaptive proximate time-optimal servomecanisms: Continuous time case. Proc. Am. Control Conf., pp: 589-594. Minneapolis, USA. 8. Kempf, C.J., 1996. Step and settle positioning algoritm for electro-mecanical system wit damping. Proc. of te 4t Intl. Worksop on Advanced Motion Control, pp: 47-5. Tsukuba, Japan. 9. Sankaranarayanan, S. and F. Korrami, 1997. Adaptive variable structure control and applications to friction compensations. Proc. of te 36t IEEE Conf. on Decision & Control, pp: 4159-4164. San Diego, USA. 10. Fujimoto, Y. and A. Kawamura, 1995. Robust servo-system based on two-degree-of-freedom control wit sliding mode. IEEE Trans. on Industrial Electronics, 4: 7-80. 11. Wayudi, 00. New practical control of PTP positioning systems. P.D. Dissertation. Dept. of Precision Macinery Systems, Tokyo Institute of Tecnology. 1. Wayudi, K. Sato and A. Simokobe, 001. Robustness evaluation of new practical control metod for PTP positioning systems. Proc. of 001 IEEE/ASME Intl. Conf. on Advanced Intelligent Mecatronics, pp: 843-848. Como, Italy.
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