Fractional-Order PI Speed Control of a Two-Ma Drive Sytem with Elatic Coupling Mohammad Amin Rahimian, Mohammad Saleh Tavazoei, and Farzad Tahami Electrical Engineering Department, Sharif Univerity of Technology Tehran, Iran (e-mail: tavazoei@ harif.edu). Abtract: In thi paper a method for tabilization of a two-ma drive ytem with flexible coupling i uggeted. The propoed method i baed on the extra degree of freedom provided by the fractional-order of integration in PI λ controller and the adopted deign technique ue the tability region in the controller parameter pace to enure cloed-loop tability. The deigned control tructure ha the advantage of not requiring any additional feedback meaurement and it i implementable within the framework of traditional PI control tructure. Keyword: Motor drive, Elatic coupling, Two-ma ytem, Speed control, Fractional-order control, Fractional-order PI controller, Stability region, Symmetrical optimum tuning.. INTRODUCTION Motion control in two-ma drive ytem with elatic haft poe a particularly challenging problem that ha attracted much attention amongt reearcher. The control tructure of electrical drive in indutrial application are uually baed on linear P I controller. Szabat and Orlowka-Kowalka (27) preent a ytematic analyi of the peed control tructure for the two-ma ytem with a PI peed controller that i upported by different additional feedback and compare the dynamic propertie of uch tructure. However, the additional feedback loop that are ued in the tructure of uch controller, incur additional cot in term of meaurement enor and increaed complexity. The pat decade ha een a thriving interet in the adoption of fractional calculu by the control ytem community. The effectivene of fractional calculu technique in the modeling and control of a divere range of dynamical ytem i well recognized and widepread ue of fractional-order controller ha been jutified by their uperior performance over claical control technique epecially for the control of fractional-order ytem. Podlubny (999) propoed an extenion of the ordinary P ID controller to the cae, where the integrator and the differentiator ue nonnegative real number λ and µ a their order, repectively. He further demontrated the uperiority of hi o-called PI λ D µ controller, when it come to controlling plant that are decribed by fractional-order dynamic. The generalization of ordinary P ID controller to the fractional-order cae called PI λ D µ controller and their analyi, deign and tuning have been invetigated by everal reearcher ince then, Valrio and da Cota (26); Monje et al. (28). In general the extra degree of freedom provided by the extenion of derivation and integration order from a fixed integer, +, to poitive real number, µ and λ, reult in a more flexible tuning trategy and therefore an eaier way to achieve control requirement a compared to the claical controller. The two extra degree of freedom alo allow for a better adjutment of controller dynamic propertie. Thi will in turn increae the cloed-loop ytem robutne againt parameter uncertaintie. Other etablihed fractional-order control technique include TID (Tilt-Integral Derivative) controller, CRONE (Contrôle Robute d Ordre Non Entier) controller and fractional lead-lag compenator. In thi paper we exploited the fractional order of integration in a PI λ controller to deign imple control tructure that would enure cloed-loop tability of a flexible twoma drive ytem. Our deign method i baed upon plotting the tability region in the controller parameter pace and the deigned control tructure doe not require additional feedback loop. The remainder of thi paper i organized a follow. In Section 2 we dicu the baic tructure and modeling for a two-ma drive with elatic haft. Next in Section 3, we explain the claic peed control technique for the two-ma drive ytem that are baed on linear PI controller. We then go on to elaborate on the hortcoming of uch control technique in the preence of elaticity and in Section 4 we ugget an alternative method uing fractional-order control technique that would enure cloed-loop tability of uch ytem. The imulation reult in Section 5 confirm the uperiority of the propoed control technique and Section 6 conclude thi paper. 2. TWO-MASS DRIVE SYSTEMS A typical drive ytem uch a the one hown in Fig. i compried of a motor which i connected to a load machine through a haft. When coupling i flexible, part of the torque that i applied to the haft during acceleration will be diipated at deforming the haft. Thi torque will be determined by
Fig.. Schematic diagram of an electromechanical ytem with flexible coupling. The motor i coupled to the load machine by an elatic haft. the difference between the angular poition and peed at the two end of the haft and can be formulated a follow: T Shaft = c( β) + d( β), () where c and d are the tiffne and damping coefficient, repectively and β denote the torional angle of the coupling. It i deirable for a good coupling to dampen the torional vibration a much a poible. With the haft torque given a in (), the equation of motion for the motor and load axe can be written a: T el T Shaft = J m βm, (2) T Shaft T L = J L βl, (3) where T el and T L denote the motor and load torque and J m and J L are the moment of inertia on the motor and load ide, repectively. Fig. 2 how the relation between the motor and load torque, T el and T m, and their angular velocitie, ω m = β m and ω L = β L, repectively. Fig. 2. Block diagram of the two-ma ytem. The ramp block denote integration ( ). If the load torque doe not change fat, it can be modeled a a low frequency diturbance. Hence, the following inputoutput tranfer function for the block diagram of Fig. 2 i obtained: ω L T el = (J m + J L ) + d c + d c +. (4) JmJL c(j 2 m+j L) It i worth highlighting that the firt term in (4) correpond to the peed-torque relation of a two-ma ytem with rigid coupling and the econd term denote the effect of elaticity. The ditributed nature of flexible haft ha prompted ome reearch aimed at delivering more accurate model for the two-ma ytem with elatic coupling. Ma and Hori (27) have modeled the elatic two-ma drive a a ditributed parameter ytem and derived a fractionalorder modeling that include exponent of the Laplace variable,, in the denominator of the torque-peed characteritic. In effect the partial differential equation that decribe a ditributed parameter ytem lead to the emergence of the fractional-order operator in the Laplace domain. The ditributed and fractional-order nature of the model propoed by Ma and Hori (27) hed light on the potential benefit of fractional-order control trategie for two-ma drive ytem with elatic coupling. 3. CLASSICAL P I CONTROL STRUCTURE In addition to an electric motor that i coupled to a load machine through a flexible haft, a typical electromechanical two-ma drive ytem would include power electronic converter, a torque regulator and a peed controller a well a current and peed enor. The mot popular trategy for peed control in electric drive i to employ a cacade tructure of the from depicted in Fig. 3. The inner control loop perform motor torque regulation and conit of a power converter, the electromagnetic part of the motor, a well a a current enor and a current or torque controller. Thi o-called torque control loop i encloed by a econd loop that control the drive peed. Thi outer control loop conit of the mechanical part of the drive, peed enor, and peed controller, and i cacaded to the inner torque control loop. The outer control loop provide peed control according to it reference value. If required, a third loop, which encloe the previou two, can provide the two-ma drive ytem with poition control. Feedback from armature current are exploited for the electromagnetic torque control, wherea peed and poition enor help realize the latter two control loop. In general, the inherent integral relationhip between poition, peed, and torque implie that the cacade tructure of Fig. 3 doe not require an offet-compenating integrator to enure teady tate accuracy. Nonethele, the preence of load torque (T L ), which wa modeled a a tep diturbance input in Section 2, mean that an integrator i in effect needed for the ake of diturbance rejection. The deign of the cacade tructure entail a tepwie procedure, which tart from the innermot loop. At each tage of the deign the controller parameter can be determined independently of the previou or remaining level. The proper operation of thi tructure, however, depend on the relative bandwidth of each control loop. In the claical deign cheme, which i adopted in thi ection, the torque control loop bandwidth i aumed to be much higher than the peed control loop. In other word, the inner loop i deigned to provide ufficiently fat torque control, which can be approximated by an equivalent firt-order term. Fig. 4 how the reult when the torque control loop i replaced with a firt-order model. In the cae of the induction motor, it could be a field-oriented or direct torque control method and in
* _ Speed Controller _ Torque Controller Power Converter (PWM) Field and Rotor Electromagnetic Dynamic Two-ma Sytem m L Fig. 3. Cacade peed control tructure for a two-ma drive ytem. The inner loop i reponible for torque control, while the ultimate peed control i provided by the outer loop. Fig. 4. Simplified peed control loop for a one-ma drive ytem. the ytem with DC motor, it i uually a PI current controller tuned with the help of modulu criterion. If thi control i enured, then it can be approximated by an equivalent firt-order term and the type of the driven machine make no difference for the outer peed control loop, Szabat and Orlowka-Kowalka (27). We will now determine the deign parameter for a PI controller, given by: G I () = K C( + T C ) T C. (5) Our deign i aimed at the peed control of a two-ma drive ytem, auming a rigid joint that connect motor and load. Uing ymmetrical optimum tuning for the P I peed controller and with the torque control loop dynamic approximated by a firt order function, a in Fig. 4, the controller parameter K C and T C can be computed a: T C = 4T oi, K C = K ij, 2T oi where J = J m + J L. (6a) (6b) Next we hall ee how flexible coupling and elaticity of the haft joining motor to the load can affect the tability of the cloed-loop ytem and caue undeirable ocillation or degrade the ytem tranient behavior. If elatic coupling i taken into account, then the block diagram of Fig. 4 can be redrawn with (4) ued a the model of the two-ma drive ytem. Fig. 5 depict the reult. Fig. 5. Simplified peed control loop for a two-ma drive ytem with elatic haft. The only difference between the open-loop tranfer function of the rigid ytem of Fig. 4 and the elatic power tranfer linkage of Fig. 5 i in the lat term of the elatic drive model, which i a econd-order dynamic with the following undamped natural frequency ω, and damping ratio ζ: ω = c(j m + J L ) J m J L, (7) ζ = d 2 (J m + J L ). (8) c J m J L The decreae in the phae margin caued by the introduction of the elatic term in Fig. 5 may detabilize the cloed-loop ytem. The latter effect can be compenated for by increaing T C at the expene of increaed ettling time. The Bode diagram of Fig. 6 illutrate the effect of flexibility on the phae and gain margin of the cloed-loop ytem and how increaing T C can tabilize the flexible ytem. The bode diagram are plotted uing the parameter given in Table. Table. Parameter value Parameter Value Decription J m 3.5 moment of inertia on the motor ide J L 3 moment of inertia on the load ide c 5 tiffne coefficient (flexible haft) d 75 damping ratio (flexible haft) K i torque control loop gain T oi. torque control loop time contant Since, the increaed ytem repone time caued by increaing T C in Fig. 6(c) i a highly undeirable factor, alternative method that provide cloed-loop tability, while preerving the tranient repone quality are very appealing. The cloed-loop ytem with the PI controller i of the fourth order. Since the claical P I controller ha only two parameter, it i not poible to locate all the pole of the control tructure without additional feedback independently. To increae the degree of freedom in the control of uch a ytem, more of the ytem data hould be exploited in the feedback. Therefore, in order to improve the dynamical characteritic of the ytem, the application of additional feedback from elected tate variable i neceary. The additional feedback can be upplied from a variety of ource including electromagnetic, load and haft torque, a well a motor and load peed or their combination. Szabat and Orlowka-Kowalka (27) ha claified uch feedback into three different group and characterize their dynamic propertie. Fig. 7 depict an example, where additional feedback from haft torque i ued to provide uitable ocillation damping for the twoma drive ytem.
Gm = Inf, Pm = 9.9 deg (at 5e+3 rad/ec) 8 6 4 2 2 5 6 7 8 9 Frequency (rad/ec) 2 3 4 5 5 5 35 8 225 27 5 5 5 9 (a) Rigid Coupling Gm = 7.83 db (at 26.5 rad/ec), Pm = 9.76 deg (at 42.8 rad/ec) 2 3 4 35 8 225 27 Frequency (rad/ec) (b) Flexible Coupling Gm = 6.64 db (at 6 rad/ec), Pm = 2.4 deg (at 39.2 rad/ec) 2 3 4 Frequency (rad/ec) (c) Flexible Coupling with Increaed T C Fig. 6. Bode diagram howing the effect of flexibility on the cloed-loop ytem tability margin. Fig. 7. Ue of additional feedback from haft torque in the cacade control tructure. Uing extra ytem data ha it own drawback. Senor that are needed to meaure the tate feedback ignal incur additional cot and complexity. Etimation method, on the other hand, may have ome implication on the cloedloop tability of the ytem and are prone to error. In the next ection, we how how fractional-order control can provide a imple yet effective alternative for the aforementioned technique. 4. STABLIZING PI λ CONTROLLERS The P I-controller in the peed-control loop can be formulated a: C() = +. (9) The primary aim of the control ytem i to enure cloedloop tability. The et of parameter (, ) that would lead to a table cloed-loop ytem can be denoted a tability domain in the parameter pace P = {, }. Thee domain are ditinguihed by the boundarie that encloe them and can be formulated by locating the poition where a root of the characteritic polynomial croe the imaginary axi to tep from the left half plane to the right half plane and vice vera. Depending on whether the root croe the imaginary axi at infinity, origin or omewhere in between three type of boundarie may arie namely, infinite root boundarie (IRB), real root boundarie (RRB), and complex root boundarie (CRB). Hohenbichler and Ackermann (23) offer a method to plot uch boundarie for a given characteritic polynomial and thi method i then extended by Hamamci (28) and Hamamci and Kokal (2) to the cae of fractional-order characteritic polynomial. Thee method are exploited in thi paper to plot the tability region and draw concluion with regard to the effect of variou controller-parameter on the overall tability of the cloed-loop ytem. Fig. 8 how the tability region for the rigid ytem given by the block diagram in Fig. 4, uing the parameter in Table. Fig. 8 alo indicate the location of parameter and a determined by the ymmetrical optimum tuning method and equation (6a) and (6b). Next we plot the tability region for the ytem with a flexible haft decribed by the block diagram in Fig. 5. Fig. 9 depict the reult. A comparion of Fig. 8 and Fig. 9 reveal that the effect of flexibility i to change the hape of CRB from a traight line into a downward parabola. Moreover, the choice of parameter uggeted by the optimum tuning technique no longer fall within the tability region. The figure alo confirm that increaing
3 x 6 2.5 2.5 Rigid Coupling K C = K ij 2T oi T C = 4T oi the final cloed-loop ytem with the fractional-order integrator replaced by it integer-order approximation. Here we have λ ued CRONE method to derive a ixth order approximation of the term in the frequency.5 range [. rad rad, ]. It i poible that for ytem with reonance very cloe to each other, uch a in flexible mechanical ytem with electromechanical actuator, all reonant frequencie can be included in much a horter interval, Ukpai and Jayauriya (24). The reultant integerorder approximating filter i a follow:.5 CRB RRB Symmetrical Optimum Tuning 2 4 6 8 2 x 4 Fig. 8. Stability region for the controller parameter in the feedback control loop of Fig. 4. The RRB ha coincided with the = axi. T C can indeed bring the choen parameter bacnto the table region. The lower inclined line in Fig. 9 correpond to T C = 32T oi a oppoed to the T C = 4T oi uggeted by the ymmetrical optimum tuning in (6a). In thi ection, we ue the extra degree of freedom provided by the fractional integrator order λ to tabilize the cloed-loop ytem with a flexible haft. To thi end, the tability region for variou order of the fractional integrator, λ, are plotted in Fig. 9. Accordingly, increaing the order of integration from λ = can place the et of parameter given by equation (6a) and (6b) bacnto the tability region. 4 x 5 2 8 6 λ =.7 CRB RRB Sym. Opt. Tuning Flexible Coupling T C = 4T oi λ =.2 λ = C() = +.5 + 5 + 94.45 4 + 786 3 + 7976 2 + 848 + 778 ( 9.967 6 + 47.8 5 + 447 4 +.e4 3 + 5293 2 + 56.5 ). Rahimian and Tavazoei (2) have provided an extenion of the method ued by Hamamci (28) to plot the tability region for the integer-order approximation of PI λ and PD µ controller. Fig. how the tability region for the final high-order controller, and it overlapping ection with that of the original fractional-order controller. It i important for the deigned controller parameter (, ) to reide at the overlapping ection of the two tability region, Rahimian and Tavazoei (2). Finally, Fig. how the unit tep repone and bode diagram of the cloed ytem of Fig. 5 with a PI λ controller for which λ =.5 and (, ) are et according to equation (6a) and (6b). The approximating filter in equation () i ued to implement the fractional-order controller and ytem parameter are et according to Table. Fig. (b) alo include the deigned ytem phae and gain margin. Comparing the tability margin in Fig. (b) with thoe of Fig. 6(b) and 6(c) confirm the improvement that are brought about by the adoption of the fractional control tructure. Similar time-domain imulation reveal a % decreae in the unit tep-repone ettling time a compared to the ordinary PI controller with increaed T C in Fig. 6(c). () 4 6. CONCLUSION 2 2 increaed T C 4 2 2 4 6 8 x 4 Fig. 9. Stability region for the controller parameter in the feedback control loop of Fig. 5. 5. CONTROLLER DESIGN, IMPLEMENTATION AND SIMULATION RESULTS The reult of imulation reveal that with λ =.5 we can get a table cloed-loop ytem with reaonable teprepone ocillation. However, ince finite integer-order filter are to be ued to implement the fractional-order operator, it i neceary to invetigate the tability of It wa demontrated that the extra degree of freedom provided by the fractional integrator in a PI λ controller can be exploited to deign tabilizing controller for a twoma drive ytem with a flexible haft. The adopted deign method wa baed on plotting the tability region in the controller parameter pace. Accordingly, the fractionalorder of integration λ wa choen o that the choice of controller parameter and would enure cloed-loop tability. The propoed fractional control tructure ha the advantage that it doe not rely on the extra data provided through additional feedback loop. REFERENCES Hamamci, S.E. (28). Stabilization uing fractional-order PI and PID controller. Nonlinear Dynamic, 5(-2), 329 343. Hamamci, S.E. and Kokal, M. (2). Calculation of all tabilizing fractional-order PD controller for integrat-
6 x 6 5 Integer Order Approx. Frac. Order Controller Sym. Optimum Tuning.8.6 Step Repone 4.4 3.2 2 Amplitude.8.6.4.2 2.5.5.5 2 2.5 x 5 (a) Stability region for the fractional-order controller and it integer-order approximation given by equation (). x 6 4.4 4.2 4 3.8 3.6 3.4 Integer Approx. Fract. Controller..2.3.4.5.6.7.8 Time (ec) 2 5 5 5 35 (a) Step Repone Gm = 6.9 db (at 6 rad/ec), Pm = 2.3 deg (at 37.7 rad/ec) 3.2 3 2.8 The Overlapping ectrion of the two tability region 8 225 2.6 2 3 4 5 6 7 8 9 x 4 (b) Fig. (a) i zoomed to provide a better view of the boundarie of the overlapping ection. It i worth highlighting that while the overlapping ection coincide with the tability region of the integer-order function for the mot part, it i bounded on the bottom-left ide by the CRB of the fractional-order controller. Fig.. Stability region for the PI λ controller with λ =.5, and it integer-order approximation given by equation (). ing time delay ytem. Computer and Mathematic with Application, 59, 62 629. Hohenbichler, N. and Ackermann, J. (23). Synthei of robut PID controller for time delay ytem. In Proc. of the European Control Conference. Cambridge. Ma, C. and Hori, Y. (27). Fractional-order control: Theory and application in motion control [pat and preent]. Indutrial Electronic Magazine, IEEE, (4), 6 6. Monje, C.A., Vinagre, B.M., Feliu, V., and Chen, Y. (28). Tuning and auto-tuning of fractional order controller for indutry application. Control Engineering Practice, 6(7), 798 82. Podlubny, I. (999). Fractional-order ytem and PI λ D µ - controller. Automatic Control, IEEE Tranaction on, 27 2 2 3 Frequency (rad/ec) (b) Fig.. Bode diagram and unit tep repone of the final deigned ytem. The cloed-loop ytem i tabilized uing a PI λ controller with λ =.5, which i implemented uing the integer-order filter in equation (). The ytem parameter are et according to Table. 44(), 28 24. Rahimian, M.A. and Tavazoei, M.S. (2). Stabilizing fractional-order P I and P D controller: An integerorder implemented ytem approach. Journal of Sytem and Control Engineering, Accepted for publication. Szabat, K. and Orlowka-Kowalka, T. (27). Vibration uppreion in a two-ma drive ytem uing PI peed controller and additional feedback - comparative tudy. Indutrial Electronic, IEEE Tranaction on, 54(2), 93 26. Ukpai, U.I. and Jayauriya, S. (24). Uing controller reduction technique for efficient pid controller ynthei. Journal of Dynamic Sytem, Meaurement, and Control, 26(3), 692 696. Valrio, D. and da Cota, J.S. (26). Tuning of fractional PID controller with ziegler-nichol-type rule. Signal Proceing, 86(), 277 2784. Special Section: Fractional Calculu Application in Signal and Sytem.