H DESIGN OF ROTOR FLUX ORIENTED CONTROLLED INDUCTION

Transcription

1 H DESIGN OF ROTOR FLUX ORIENTED CONTROLLED INDUCTION MOTOR DRIVES: SPEED CONTROL, STABILITY ROBUSTNESS AND NOISE ATTENUATION João C. Bailio,, Joé A. Silva Jr.,, Jr., and Lui G. B. Rolim, Member, IEEE, DEE, UFRJ, Cidade Univeritária, Rio de Janeiro, Brazil, COPPE, UFRJ, Cidade Univeritária, Rio de Janeiro, Brazil, alencar@pee.coppe.ufrj.br Abtract In thi paper the deign of a control ytem for rotor flux oriented controlled induction motor drive uing H optimal control theory i carried out. Two H problem are conidered: (i) a -block problem with the view to maximizing the ytem performance; (ii) a -block problem with the weight initially choen for maximizing both the ytem performance (tranient repone) and the ytem tolerance to uncertainty in the plant model (robut performance), and in the equel the weight are modified to account for tranient performance and noie attenuation. All the deigned controller have been implemented in a real ytem, and the reult are hown in the paper. KEYWORDS Vector control, H control theory, optimal control, feedback ytem. I. INTRODUCTION Rotor flux oriented controlled induction motor drive have recently received great attention in the literature [], [], and the reference therein. Thi ha led to the o-called vector control method which i baed on the tranformation of the nonlinear model of the induction motor in a linear time-invariant firt order model. In order to obtain an exact firt order model, it i neceary to pre-multiply the ytem by a matrix whoe element are function of the rotor flux angle. However, thi angle i not known exactly, being etimated from the rotor time contant (the ratio between the rotor inductance and reitance), and the haft velocity. Since, up to now, there i no reliable method for the exact determination of the rotor time contant, the tranformation doe not lead to an exact firt order model. The difference between the exact firt order model (which in thi paper will be referred to a the nominal model) and the real model can be een a a model uncertainty. Thi ugget that robut deign technique can be employed to obtain a controller for thi ytem. One of the deign technique that can be ued i the o-called H optimal control theory [], [4]. Thi approach ha been deployed in previou work, [5], [6], in which -block H problem have been formulated and olved uing the o-called DGKF approach [7]. The main difference between thee work i the choice of weight: in [5], calar weight have been ued while in [6], although table rational function have been ued a weight, they have been choen according to H dogma, i.e. the weight for robutne i a high pa function, a precribed in the H theory, without checking if thi i what actually happen when the etimated rotor time contant i uppoed to be different from the real one. Partially upported by CNPq Partially upported by CAPES In thi paper two H problem are conidered: (i) a - block problem, with the view to maximizing the ytem performance (auming the haft velocity a the variable to be controlled); (ii) a -block problem with the weight initially choen for maximizing both the ytem performance (tranient repone) and the ytem tolerance to uncertainty in the plant model (robut performance), and, in the equel, the weight are modified to account for tranient performance and noie attenuation. Differently from [5], [6], the olution here i obtained following the tandard 984 approach [4], [8]. The main advantage of uing uch an approach over DGKF olution i that the deigner ha a clear view on how to change the weight in order to improve the control objective. Furthermore, the ue of 984 approach ha led to an intereting reult: the H controller, olution to the -block problem i a PI controller, whoe parameter are tuned according to the o-called internal model control [9]. Thi unexpected reult jutifie, from the H theory point of view, why PI controller have been uccefully ued in field oriented control of induction motor. Another point which differ thi paper from [6], i that here the weight for robutne i obtained from frequency repone experiment carried out in the real ytem by perturbing the etimated rotor time contant. It i well known that the objective of robutne and noie attenuation are expreed in term of the ame -block H optimization problem; the only difference i that the weight mut now be a high pa rational function. Thi problem ha alo been tackled in thi paper, and the reult obtained from practical implementation are hown. II. A LINEAR MODEL FOR ROTOR FLUX ORIENTED CONTROL OF CURRENT-FED INDUCTION MOTORS A. Theoretical background Auming a input, the tator current vector component in field coordinate, i Sd (t) (the tator current component in the direction of the magnetiing current vector, uually referred to a the direct component) and i Sq (t) (the quadrature component of the tator current, which i perpendicular to i Sd (t)), then current-fed induction motor can be modeled a []: d T R dt i mr(t) i mr (t) = i Sd (t) d ρ(t) = ω(t) i Sq(t) dt T R i mr (t) J d () dt ω(t) fω(t) = ki mr(t)i Sq (t) ω(t) = d dt ε(t) where T R = L R /R R denote the rotor time contant, L R

2 I Sqref ˆω I Sdref ˆT R Î mr ˆTR ˆω mr ˆρ k ab τ ω ε I S I S I S I Sq I Sd ω ω mr ρ Coordinate tranformation Coordinate tranformation T R T R I mr Fig.. Block diagram of an ideal inverter in cacade with a current fed induction motor, and with a rotor flux angle etimator. and R R are, repectively, the rotor inductance and reitance, i mr (t) i the magnetiing current, ρ(t) i the rotor flux angle with repect to the tator axi, J i the motor inertia, f i the vicou friction, ω(t) i the intantaneou angular velocity of the rotor, k i the coupling factor, which i a function of the total leakage factor of the motor and of the tator inductance, and ε(t) the angle of rotation of the rotor, meaured in the tator frame of coordinate. Notice that, if in Eq. (), the direct component of the tator current i made contant, i.e., i Sd (t) = I Sd, then, after a brief tranient, which i dictated by the rotor time contant T R, i mr (t) become equal to I Sd. When thi happen, the electrical torque become a function of i Sq (t), only, and, thu, the model of a current-fed induction motor become analogou to that of a contant field dc-motor controlled by the armature current. However, i Sq (t) and i Sd (t), are not acceible; they are actually function of the line current i S (t), i S (t) and i S (t), a follow: [ ] isd (t) i Sq(t) = co ρ(t) in ρ(t) in ρ(t) in ρ(t) is(t) i S(t). co ρ(t) co ρ(t) i S(t) () Therefore, in order to have a input the direct and quadrature component of the tator current, one ha to carry out an invere tranformation of Eq. (), given a: is(t) co ρ(t) in ρ(t) [ ] isd (t) i S(t) = in ρ(t) co ρ(t). () i S(t) i Sq(t) in ρ(t) co ρ(t) The device which caue the motor current to follow the reference given in () i called a current-controller PWM inverter (in thi work it will be imply referred to a inverter). It i clear from Eq. () that the inverter input are the deired value for the direct and quadrature component of the tator current, here denoted a i Sdref (t) and i Sqref (t) and a output the line current i S (t), i S (t) and i S (t) neceary to make i Sd (t) and i Sq (t) (the actual value) equal their reference value. Auming i Sdref (t) = I Sdref (contant), then the tranfer function for the ytem formed with an ideal inverter and a current-fed induction motor which relate i Sqref (t) and ω(t) i given by: G() = Ω() I Sqref () = ki Sdref J f = k abi Sdref (4) τ where τ = J/f and k ab = k/f. Although appealing from the theoretical point of view, thi approach ha the following drawback: ince the rotor flux angle ρ(t) cannot be meaured, it hould be etimated. However, a hown in Fig., it etimate depend on the knowledge of the rotor time contant T R, whoe value cannot be determined preciely. If by chance the value of the etimated rotor time contant ˆT R i exactly equal to T R then the behavior of the ytem etimatorinverterinduction motor can be decribed by Eq. (4). However, in general ˆT R T R, and thu Eq. (4) i no longer a reliable model for the ytem. One way of taking into account model uncertainty i to ue, intead of the tranfer function given in Eq. (4), a o-called perturbed tranfer function G P (). In thi work, multiplicative untructured perturbation [] will be ued, which i given by: G P () = [ W ()]G(), (5) where W () i a table tranfer function and will be determined from frequency repone experiment carried out at the real ytem by perturbing ˆT R. Thi provide the neceary information for dealing with controller deign within the H control theory. B. Parameter identification and model validation The motor ued to validate the theory preented in thi paper i a cla A three-phae two-pole -connected induction motor with quirrel-cage rotor. It nominal value are: (i) Power: 6W ; (ii) Line voltage: V, (iii) Frequency: 6Hz; (iv) Line current:.8a and (v) Rotor inertia: J =.57Kg.m. According to the model given in Fig., the only parameter to be determined are T R, k ab and τ. Let u conider, initially, the identification of T R. An immediate choice for the etimated value of T R ( ˆT R ) i that obtained by performing tandard tet for determining the teady-tate circuit model parameter []. For the induction motor ued in thi work, it ha been obtained ˆT R =.59. A better etimation for the rotor time contant can be obtained if we notice that the ytem inverteretimatorinduction motor only behave a

3 d() Angular peed (rpm) r() e() u() K() G() y() η() 6 5 Fig.. Block diagram of a tandard negative unit feedback ytem. Angular peed (rpm) (a) (b) Fig.. Step repone for the real ytem (olid line) and for the Simulink model (dotted line). and ideal firt order one when ˆT R i exactly equal T R (the actual value of the rotor time contant). A better etimation of T R can then be obtained by trial-and-error, i.e., by changing the value of ˆT R in the real etimator until the ytem repone to an tep input matche that of a firt order one. Thi ha actually been done in thi work, leading to a value of ˆT R approximately 5% maller than that obtained previouly, i.e. ˆT R =.49. Conider, now, the etimation of k ab and τ. Since the value of ˆT R i uch that the ytem now behave a a firt order one, uual tet for etimation of the parameter of firt order ytem can be deployed to find k ab and τ. Among the available technique, the o-called Method of Area [] eem more appropriate due to the noie introduced in the ytem repone by the enor. At thi point, it i worth remarking that, before applying a tep ignal in i Sqref (t), it i neceary to apply a contant input to i Sdref (t), i.e., i Sdref (t) = I Sdref. For the experiment carried out in thi work, it ha been adopted I Sdref =.8A. Finally, average value for k ab and τ have been obtained after performing thee experiment for everal tep ignal with different amplitude, leading to k ab = and τ =.. Fig. how the comparion between the imulation reult (dotted line) and thoe obtained experimentally (olid line) by applying to a Simulink model, equivalent to the block diagram of Fig. (with the above etimated value of ˆT R, k ab and T R = ˆT R ), and to the real ytem tep ignal of amplitude.46a (Fig..a) and 7.999A (Fig..b). It i worth remarking that in both cae, tep of amplitude.58a were initially applied to i Sqref (t) and that I Sdref =.8A. III. CHARACTERIZATION OF CONTROL OBJECTIVES IN TERMS OF THE MINIMIZATION OF H NORMS OF TRANSFER FUNCTIONS Conider the block diagram of Fig. where r(), e(), d(), y() and η() denote, repectively, the Laplace tran- form of the reference, error, external diturbance, output and enor noie ignal, and G() and K() are the tranfer function of the plant and the controller, repectively. Stability robutne (or ytem tolerance to plant uncertainty), tracking of quare integrable reference ignal, tranient performance, external diturbance rejection and noie attenuation are uual control objective and can be expreed in term of the minimization of the H norm of tranfer function a follow:. Robut tability: min W T, (6) where W () i a table rational weighting function defined according to Eq. (5) and T() = G()K()[ G()K()] i the tranfer function between r() and y().. Tracking of quare integrable reference ignal and tranient performance: min W S, (7) StabilizingK() where W () i a table rational weighting function ued to place more penalty on the relevant frequencie and S() = [G()K()] = T() i the well known enitivity function.. Rejection of quare integrable external diturbance ignal: min W SG. (8) 4. Noie attenuation: min W 4T. (9) where W 4 () i a table rational weighting function ued to place more penalty on the noie dominant frequencie. H optimization problem can alo be formulated to take into account two or more control objective; for example, robut tability and tracking or noie attenuation and tracking can be addreed imultaneouly, a follow: min W it W S, () where i = for robut tability and i = 4 for noie attenuation. It i important to remark that ince T()S() =, then control objective addreed in Eq. () are conflicting, in the ene that it i not poible to have at the ame time tability robutne and tranient performance. The only exception i when the weighting function W (), W () and W 4 () place more penalty on different frequencie, a follow: (i) in linear ytem, frequency repone identification uually lead to more imprecie decription at high frequencie, and thu, in thi cae, W () mut be a high pa tranfer function; (ii) ignal to be tracked have uually a pre-defined frequency, and thu, W () mut be a low pa tranfer function; (iii) finally, The H -norm of a table tranfer function H() i defined a H = max ω R H(jw), where. denote abolute value.

4 enor noie have uually high frequency component, which implie that W 4 () mut alo be a high pa tranfer function. In order to obtain the olution of H optimization problem poed above, the firt tep i to guarantee that the reulting controller tabilize the nominal tranfer function. Thi i done through the o-called Youla-Kucera parametrization, in which the controller i parametrized in term of a table and proper rational tranfer function Q() a follow: Y () M()Q() K() = () X() N()Q() where N(), M(), X() and Y () are table rational function uch that G() = N() () M() and atify the Bezout equation X()M() Y ()N() =. () It can be hown [4] that all the optimization problem decribed by Eq. (6) to () can be tranformed into the following problem min T T Q, (4) where T () and T () are rational function and depend on the optimization problem which i being conidered. Thi optimization problem i uually referred to, in the literature, a a model matching problem [4]. IV. A -BLOCK H CONTROLLER FOR SPEED CONTROL A. Problem formulation and main reult The control problem to be conidered initially ha a unique control objective, i.e. tracking and tranient performance of a reference ignal, which in thi cae i the rotor angular velocity. Within the H framework, thi control object lead to the optimization problem given by Eq. (7). Therefore, uing the Youla-Kucera parametrization () in Eq. (7), and after ome traightforward calculation, one may write: min W S = min W (X NQ)M = min T T Q, (5) where, in thi cae, T () = W ()X()M() and T () = W ()N()M(). Notice that, ince the plant tranfer function (4) i already table, then an immediate choice for N() and M() which atify Eq. () i given a: N() = G() and M() =. (6) It i therefore eay to ee that X() and Y () olution to Eq. () will be given by: X() = and Y () =. (7) Thu, it i not hard to ee that the olution to the optimization problem (5) i trivial and independent of W (), being given by: Q() = G() = τ. (8) k ab I Sdref However thi olution i improper and, therefore, doe not atify the requirement that Q() RH. In order to circumvent thi problem, what i uually done [] i to approximate Such a function i uually referred to a an RH function or to belong to RH. isqref (A) Angular peed (rpm) τ = τ/5 τ = 5τ τ = τ/5 τ = 5τ (a) (b) Fig. 4. Cloed loop repone for a tep reference ignal of 4rpm (a) and control ignal i Sqref (t) (b) obtained from the real ytem (olid line) and by imulation (dotted line). thi function by a rational one. Thi i carried out by introducing a polynomial factor τ on the denominator polynomial of Q(), i.e. Q P () = τ Q() = τ (9) k ab I Sdref ( τ ) where τ i choen with the view to approximating Q P () and Q() at the frequency range of interet. Direct ubtitution of N(), M(), X(), Y () and Q P () given by Eq. (6), (7) and (9) in the controller expreion () and after ome traightforward calculation, reult in: where K() = ( τ τ = K p ) k ab I dref τ T i K p = () τ k ab I dref τ and T i = τ. () Eq. () above how that the H controller which optimize tracking and tranient performance i a PI controller whoe parameter are tuned according to the o-called internal model principle applied to PID controller [9]. Thi i an amazing reult and explain, from the H point of view, why PI controller have been ued uccefully in vector control. Another important contribution of thi reult i that it preent the correct way of tuning the PI controller, a hown in Eq. (). Furthermore, notice from Eq. (), that the proportional gain K p increae when τ decreae. The importance of thi fact i that it doe not contradict the control ytem theory for which the increae in the gain i ued to peed up the ytem repone and the H control theory, for which the bet controller for performance i obtained when τ approache zero. B. Experimental reult With the view to howing the validity of the theoretical reult preented in thi ection, two real H PI controller have been ued to control the rotor velocity of the real

5 induction motor whoe parameter are given in Subection II-B. In all experiment, I Sdref =.8A, and thu, according to Eq. (), the controller parameter mut be tuned a: K p =.49 and T i =.. () τ Experimental reult are hown in Fig. 4 for τ = 5τ and τ = τ/5. Fig. 4(a) how the rotor peed for a tep reference ignal of 4rpm and Fig. 4(b) how the behavior of the quadrature component of the tator current; the olid line have been acquired from the real ytem while the dotted line have been obtained from imulation. Notice that the repone rie time and ettling time have decreaed, repectively, from approximately.9 and.5 for τ = 5τ to 5m and 4m for τ = τ/5, producing the expected reult. V. -BLOCK H CONTROLLERS According to Eq. () when other control objective, uch a robut tability or noie attenuation, are to be conidered in addition to ytem performance, it i neceary to formulate a -block H problem. Furthermore, ubtituting Eq. () in (), the following problem, equivalent to that given in () i obtained: [ min Wi Y Ñ ] [ ] Wi N W X M MQ W N () where i = for robut tability and i = 4 for noie attenuation. Defining [ Wi Y T () = Ñ ] [ ] Wi N W X M and T () = M (4) W N then Eq. () i equivalent to: min T T Q. (5) A pointed out in Section III, the weighting function W () and W 4 () hould be, repectively, low and high pa tranfer function, and are choen by the deigner in order to place more penalty at the deired frequencie. On the other hand, weight W () take into account uncertaintie on the mathematical model and, for thi reaon, hould be obtained experimentally; thi will be the ubject of the next ubection. In both cae, firt order weight will be deployed, having, therefore, the following tranfer function W i () = a i( b i ) ( c i ) and W () = a ( b ) ( c ) (6) where a i, b i, c i >, i =, 4, b > c and b 4 < c 4. The relationhip between b and c will be defined from frequency repone experiment. Expreion for T () and T () can be obtained by ubtitution of Eq. (6), (7) and (6) in (4), being given a: T () = a ( b ), T () = ( c ) ai( bi) ( c i) a ( b k abi Sdref ) τ. ( c ) (7) Differently from the -block problem, it i not poible to obtain here a cloed olution. It i well known that the olution for the -block H problem i obtained by iterative method. The reader are referred to [4] for more detail on the olution of problem () for T () and T () given by Eq. (7). Magnitude Angular frequency (rad/ec) Fig. 5. Experimental reult obtained for ˆT R =.49 ( o ) and by perturbing ˆT R in 5% ( o ) and 5% ( ) and G(jω) ± W (jω) (olid line). A. Influence of the inexact knowledge of the rotor time contant on the induction motor linear model A point out in Subection II-A the inexact knowledge of the rotor time contant will be conidered a a model uncertainty. In thi work it ha been ued the o-called model with multiplicative perturbation given in Eq. (5). It i clear from Eq. (5) that, for each frequency ω k G P (jω k ) G(jω k ) = W (jω k ) (8) and, thu, it i traightforward to ee that G P (jω k ) G(jω k ) W (jω k ) G P (jω k ) G(jω k ). (9) It i worth noting that G(jω k ) will be obtained for ˆT R nominal while G P (jω k ) will be obtained from the ytem frequency repone by perturbing ˆT R. Fig. 5 how the reult obtained experimentally by perturbing ˆT R in 5% (cro-dotted line) and 5% (tar-dotted line). From thee data, a firt order weight, defined according to Eq. (6), i given by:.( ) W () =. () Notice that, thi weight atifie Eq. (9) for each ω k, a can be hown in Fig. 5 (olid line). Furthermore, it i clear that the rational function given by Eq. () place more penalty at low than at high frequencie. Thi contradict the uual aumption of H control theory that W () hould be a high pa tranfer function. Therefore, the weight ued in [6] to addre robutne bear no relationhip with practice, ince it ha been choen a a high pa tranfer function. The main implication of the weight given in () i the fact that both weight W () and W () place penalty at the ame frequency range (low frequencie) and thu the -block H problem cannot be ued to improve robutne without evere degradation on ytem performance. B. -block H controller for tranient performance and noie attenuation Conider again the minimization problem given in Eq. (4) for i = 4, i.e., with the objective of noie attenuation at the plant output being incorporated to the controller deign. It i important to remark that ince the tachometer i the main ource of noie, then the meaure of the noie attenuation at the plant output will be made in an indirect way, namely, through the meaure of the plant input ignal, which in the preent work i i Sqref (t).

6 Percent noie in isqref isqref (A) 4 6 (a) Percent noie in isqref Fig. 7. Noie level in i Sqref (t) for -block (top plot) and -block (bottom plot) H controller Angular peed (rpm) (b) Fig. 6. Cloed loop repone for a tep reference ignal of 4rpm (a) and control ignal i Sqref (t) (b) obtained from the real ytem (olid line) and by imulation (dotted line) for the -block H controller An important tep in the deign of H controller i the choice of weight. Although the deigner know what their frequency repone hould look like, better weight are choen in a trial-and-error bae. Indeed, uing a weight W 4 () = and W.( ) () =., () then the following H controller ha been obtained:.45( )( )(.9) K() = ( )(.6664)(.7). () The ytem cloed-loop repone for a tep reference ignal i hown in Fig. 6(b) and the correponding control ignal (i Sqref (t)) i hown in Fig. 6(a). Notice that the rie time i now 9m and the ettling time 5m, approximately, which i lightly wore than the correponding performance indexe for the -block H controller for τ = τ/5, whoe repone i hown in Fig. 4.b. In addition, ince the controller ha no pole at the origin, the repone exhibit a teady-tate error of approximately.9%; it i important to remark that a maller teady-tate error could be obtained by increaing the dc-gain of W (). A far a noie attenuation i concerned, notice from Fig. 7 that the noie level in i Sqref (t) ha decreaed from approximately - to 5% for the -block H controller (top plot) to -5 to 6% for the -block H controller (bottom plot). VI. CONCLUSION In thi paper, H optimal control ha been uccefully applied for peed control and noie attenuation in rotor flux oriented controlled induction motor drive. Another contribution of the paper i that the influence of the inexact knowledge of the rotor time contant on the linear model of induction motor and peed performance cannot be imultaneouly conid- ered in an H deign ince the former give rie to a low pa weighting function; therefore contradicting the uual H requirement of high pa weighting function for robut tability. REFERENCES [] W. Leonhard, Control of Electrical Drive, Springer- Verlag, Berlin, Germany, nd edition, 996. [] J. A. Santiteban and R. M. Sthephan, Vector control method for induction machine: an overview, IEEE Tranaction on Education, vol. 44, pp. 7 75,. [] G. Zame and B. A. Franci, Feedback, minimax enitivity, and optimal robutne, IEEE Tranaction on Automatic Control, vol. 8, pp , 98. [4] B. A. Franci, A coure in H control theory, in Lecture Note in Control and Information Science, Berlin : Germany, 987, vol. 88, Springer-Verlag. [5] C. Attaianee, G. Tomao, and A. Perfetto, Field oriented control of induction motor by mean of a H controller, in Proceeding IEEE Sympoium on Indutrial Electronic, Bled : Slovenia, 999. [6] A. Makouf, M. E. H. Benbouzid, D. Diallo, and N. E. Bouguechal, Induction motor robut control: an H control approach with field orientation and input-output linearizing, in Proceeding of the 7th Annual Conference of the IEEE Indutrial Electronic Society, Denver : USA,. [7] J. C. Doyle, K. Glover, P. P. Khargonekar, and B. A. Franci, State-pace olution to tandard H and H control problem, IEEE Tranaction on Automatic Control, vol. 4, pp , 989. [8] J. C. Doyle, Advance in multivariable control, in ONR/Honeywell Workhop, Minneapoli : USA, 984. [9] D. E. Rivera, M. Morari, and S. Skogetad, Internal model control. 4. PID controller deign, Ind. Eng. Chem. Proce Deign Development, vol. 5, pp. 5 65, 986. [] M. Vidyaagar, Control Sytem Synthei: a factorization approach, MIT Pre, Cambridge, USA, 985. [] S. J. Chapman, Electric Machinery Fundamental, McGraw-Hill Inc., New York, USA, nd edition, 99. [] K. J. Atrom and T. Haglund, Automatic Tuning of PID controller, Intrument Society of America, Reearch Triangle Park, NC, t edition, 988. [] J. C. Doyle, B. A. Franci, and A. Tannenbaum, Feedback control theory, Macmillan Publihing Company, New York, 99.