DISTURBANCE INVARIANT SPEED CONTROLLED SERVO DRIVE WITH TESLA S INDUCTION MOTOR

Transcription

1 Septeber, 1 DISURBANCE INVARIAN SPEED CONROLLED SERVO DRIVE WIH ESLA S INDUCION MOOR Mili} R. Stoji}, Faculty of Electrical Engineering, University of Belgrade, Yugoslavia Milan S. Matijevi} Faculty of Mechanical Engineering, University of Kragujevac, Yugoslavia Abstract - In his patent [1], [] "Electro-Magnetic Motor" of May 1, 1888, esla wrote: "Be it known that I, Nikola esla, a subject of the Eperor of Austria, fro Siljan, Lika, border country of Austria-Hungary, now residing in New York, in the country and State of New York, has invented certain new Iproveents in Electro-Magnetic Motors, of which the following is a specification..." esla continued writing about the ode of operation, principles of construction, and advantages of the alternating current otor called today as the induction or asynchronous otor. Several original patents (see, for exaple, [3] for esla's induction otor, polyphase syste for the transission of the electrical power, and a further series of over 4 patents within the sae field fro 1887 to 1891 ake consistent esla's syste of the generation, transission, and various ways of application of alternative electric energy. Fro the late nineteenth century to these days, esla's otor was advantageously used in any applications where the torque capabilities, acceleration characteristics, power density, and high degree of reliability are indispensable. Nevertheless, in the design of high perforance electrical drives, the choice of DC otor was preferable, since its torque and flux are controlled independently through two separate terinals. o atch this suitable feature of DC otor in servo applications, the concept of vector control [4], [5] was developed to enable decoupled control of the flux and torque of induction otor; thus, esla's otor ay enable control characteristics of DC otor [6]. In this paper, it is shown how esla's induction otor can be applied in the design of speed digitally controlled servo drive in the presence of ieasurable arbitrary torque disturbances. For rejection of effects of torque disturbance on the steady-state value of otor speed the principle of absorption [7], [8] is applied in the design of disturbance estiator within the control portion of the syste controlling structure. Keywords - Speed controlled servo, esla's otor, Absorption principle, Ieasurable disturbance, IMPAC structure. I. INRODUCION he ain task in control syste design is the tracking of a reference signal with sufficiently sall or ero steadystate error in the presence of unknown external disturbance. In any applications, the designer knows an aount of apriori inforation about the class of disturbances and liits of interval changes or uncertainties of plant paraeters. It is known that the knowledge about isatches of plant paraeters is necessary for the robustness analysis in the design of control systes with internal odels [11], [1]. Most frequently an external disturbance ay be odeled as a solution of hoogenous differential or difference equation. In such case, it is possible to include the disturbance odel into the control portion of the syste in order to extract copletely influence of disturbance on the steady-state value of syste output [11]. Essentially, this design procedure is based upon the application of the IMP (Internal Model Principle. he idea of absorption principle has been priarily presented in the seinal work of Kulebakin [8] who exained the invariance conditions and introduced the concept of socalled selective invariance in the control syste theory. Johnson [9] used this principle in the servoechanis design. Davison [1] applied a siilar approach of using the internal disturbance odel for the rejection of disturbance. his ethod is based on so-called IMP (Internal Model Principle. Solving the regulator proble in ultivariable control systes, Francis and Wonha [11], [1] cae to a conclusion that for disturbance rejection it was necessary to include the disturbance odel into the control part of the syste. As a atter of fact, IPM was inspired by results of works by Francis and Wonha [11], [1]. Utiliing results of Kulebakin [8], sypkin and Holberg [16] expressed concisely the principle of absorption for digital control systes. At the sae tie, they defined the generalied disturbance and exained the possibility of its extraction [7]. he absorption principle has been advantageously applied in the design of a robust Sith predictor for a process with long dead tie [15]. he IMP and principle of absorption are based upon the sae fundaental idea of inclusion disturbance odel into the syste controlling structure. In this paper, the principle of absorption is forulated in a discrete for and then the possibility of its application in the design of disturbance invariant servoechaniss with esla's induction otor is exained. o enable extraction of an apriori unknown disturbance, the suitable odification of IMPAC (Internal Model Principle and Control ogether structure with the adaptation echanis within the internal odel of disturbance is proposed. he ability of the structure to absorb fast disturbances is approved by siulation. Although the idea of adaptation of the disturbance internal odel is not the novel one [17], it is first tie applied for the design of concrete control plant subjected to fast disturbances. he robust properties of the structure will not be analyed separately; however, it will be shown how the desired robustness and filtering properties of the syste can be achieved by tuning of single paraeter in the syste controlling structure. 9

2 Mikrotalasna revija Septebar 1. II. PRINCIPLE OF ABSORPION Suppose that kth saple of external disturbance f (t ay be deterined by finite nuber of previous saples. hen, the disturbance is regular and ay be described by extrapolation equation [7] f ( k = D f ( f (( k where D f ( is the prediction polynoial of order o. Relation (1 is called the equation of extrapolation or prediction [7] and it ay be rewritten as (1 (1 D f ( f ( = ( where f ( denotes the -transfor of disturbance. Relation ( is called copensation equation and FIR filter having the pulse transfer function (1 D f ( is the absorption filter or the copensation polynoial [7]. Absorption filter Φ f ( = 1 D f ( is designed for a known class of disturbances and its ipulse response becoes identically equal to ero after n sapling instants, where n o. Hence, the copensation equation ( ay be considered as the absorption condition of a given class of disturbances. he condition can be expressed as Φ f ( f ( =, for t = k (deg Φ f. (3 he extrapolation polynoial D f ( is deterined by an apriori inforation about disturbance f (t [7], [16]; nevertheless, it is siply resolved fro as wnu ( Φ f ( = wden (, fro f ( =. (4 wden ( In the case of a stochastic disturbance s(t, absorption filter (4 should suppress as uch as possible affects of disturbance on the syste output. hus, for a law frequency disturbance s(t, which can be generated by double integration of the white noise, an appropriate choice of absorption filter is Φ s ( = (1 that corresponds to absorption of linear (rap disturbance [16]. In [7], the absorption principle is generalied to enable the extraction of paraeter disturbances. Naely, changes of syste output due to differences between the real and noinal plants ay be treated as a paraeter disturbance ϕ p that influences the syste output. hus the generalied syste disturbance ϕ = ϕ w coprises the external disturbances p and uncertainties of plant paraeters. Hence, the real plant ay be considered as the noinal plant subjected to the generalied disturbance ϕ. If the syste is robustly stable, the paraeter disturbance is regular and its absorption is possible, i.e., with Φ ( 1 ϕ ( =, for t = k (deg Φ (5 p p p Φ p ( = 1 Dp( (6 where Φ ( and D ( denote the copensation and p p prediction polynoials of ϕ p, respectively. According to (3 and (5, the absorption condition of generalied disturbance ϕ = ϕ p f becoes Φ f ( Φ p( ϕ ( =, t = k (degφ f degφ p (7 or Φ( ϕ ( =, for t = k (deg Φ (8 where Φ ( denotes discrete absorption filter of generalied disturbance. he design of absorption filter greatly depends upon the aount of apriori inforation about the disturbance. If the class of disturbance is known in advance (constant, slow varying, rap, sinusoidal with known radial frequency, etc, the absorption filter can be designed exactly; otherwise, the design of absorption filter becoes ore difficult. If the class of disturbance is unknown, the disturbance is to be predicted by evaluating of disturbance signal in each sapling instant. Naely, the coefficient vector θ in prediction polynoial D (, θ is to be fitted by the on-line identification echanis based on estiated disturbance [11]. Consequently, the fitting of prediction polynoial coefficients requires the on-line estiation of disturbance. In the case of IMPAC control structure, the inforation about the generalied disturbance is obtained by including the noinal plant odel into the control portion of the syste. In a general case, let ( k be an estiated value of disturbance in sapling instant k. hen, analogously to (1, the following prediction relation is valid ( k = D(, θ (( k. (9 Relation (9 ay be rewritten as vector equation ( k = θ ψ ( k (1 where ψ ( k = [ (( k, (( k,..., (( k ]. (11 e he recurrence algorith for estiation of coefficient vector θ is given as θ ( k = θ (( k θ ( k, (1 θ ( k = Γ ( k( ( k θ (( k ψ ( k ψ ( k o where Γ o ( k is the gain atrix, which can be calculated in different ways [11]; the siplest one is based on the Kacar algorith Γ ( k = [ ψ ( k ψ ( k ] I (13 o where I denotes the unit atrix. In the presence of stochastic disturbances, it is convenient to estiate θ by using the least square ethod [11]. Indeed, the adaptation echanis should be chosen carefully not to be too coplicated. In ajority of practical applications an appropriate choice ight be D ( =. According to (4, this prediction polynoial rejects rap disturbances; but, it enables also the extraction of slow varying disturbances and even it suppresses effects of low frequency stochastic disturbances on the syste output, especially if the 3

3 Septeber, 1 sapling period is chosen sufficiently sall. It is to be noted, in the presence of adaptation echanis, the syste behavior becoes slightly nonlinear. herefore, before the identification of plant paraeters, the prediction polynoial should be reduced, as uch as possible, into an appropriate for. For exaple, in the case of a slow varying disturbance f (, t for which the first difference f ( k = f ( k f (( k is sall, an adequate prediction polynoial ight be the siplest one D ( = 1. f However, if the first difference is not sufficiently sall, but if the quantity f ( k α f (( k, which for α = 1 becoes equal to the second difference, is sall, then the appropriate prediction polynoial D ( = ( 1 α α (14 f ay be adopted. hen the fitting of adaptation loop is perfored by tuning paraeter α in (14. he successful ipleentation of the absorption principle greatly depends upon the quality of disturbance odel or of aount of apriori inforation about the disturbance and, in connection with that, on the adequate choice of prediction polynoial. Unlike other controlling structures that utilie the absorption principle, sypkin's IMPAC structure ipleents the absorption filter so that the requireents for an accurate disturbance copensation and syste stability conditions are not opposite to each other even in the case of unstable control plants. III. IMPAC SRUCURE Fig. 1 shows the IMPAC structure proposed by Ya.Z. sypkin [7], [16]. Actually, the structure in Fig. 1 represents the odified priarily IMPAC structure suited for speed controlled servo drive with esla's induction otor. he control portion of the structure is encircled by dotted lines. Fig. 1. IMPAC structure of digital control syste he noinal odel of the plant coprising the induction otor, vector-controlled invertor, and shaft encoder ay be approxiated at the low frequency band as o ω( W ( = = Z u( K Js K = = C J 1 1 (15 where K, J, and denote the electroagnetic torque coefficient, otor inertia, and sapling period, respectively. It is supposed that the pulse transfer function of the real plant ay be represented as o W( = W ( ( 1 δ W( (16 with the known noinal plant odel o P ( C K o u W ( = =, C = (17 o Q ( 1 J and its perturbation δ W ( liited by the ultiplicative bound of uncertainties ω [ jω δw ( e α( ω,, π (18 Paraeters of the noinal plant odel (17 deterine o o polynoials P ( and Q ( of the two-input internal u plant odel of the IMPAC structure in Fig. 1. he output of internal plant odel estiates the influence of external disturbance on the controlled variable ω (shaft speed [7], [16]. Consequently, signal u is a portion of anipulated variable u that should copensate effects of disturbance. In a particular case, when the plant doesn't include a dead tie, the internal odel of disturbance ay be ibedded into an adequate prediction polynoial D (. In addition, if the inially phase plant is under consideration, then, according to the standard procedure of IMPAC structure design, one ay assue R ( 1 = P( = C (19 u Polynoials P ( and P ( of the ain control loop r y are deterined by the specified closed-loop syste pole spectru or by the desired closed-loop syste transfer function. Since the closed-loop syste is of second order, the stability argin and speed of syste set-point response ay be specified by the relative daping coefficient ς and undaped natural frequency ω n of the doinant coplexconjugate pole pair inside the Nyquist frequency band. In doing so, one obtains the desired closed-loop syste transfer function as (1 ( 1 1 G de ( = 1 ( 1 1 ( where 1/ = exp( s1, with s1/ = ςω n ± jω n 1 ς. (1 Fro (, Polynoials P ( r P( = ( 1 ( r P ( = 1 ( y ] and P ( becoes y ( he choice of ς and ω n significantly affects the dynaic properties of closed-loop syste, robust stability, and filtering properties of the syste. Note that the robust stability condition plays an iportant role in the design of control syste with internal odels. Naely, if this condition is not satisfied, it doesn't ake sense to discuss the syste ability to extract external disturbances. he design of syste with prescribed degree of robustness with respect to uncertainties of plant paraeters has been outlined in [15]. In the absence of sufficient apriori inforation about the generalied disturbance, the on-line identification of disturbance and an appropriate adaptation of prediction polynoial coefficients are necessary. In a general case, regardless of the applied adaptation echanis, the IMPAC structure of the speed 31

4 Mikrotalasna revija Septebar 1. controlled servoechanis with asynchronous otor has the structure shown in Fig.. ω r u - L 1 1 ω Pr( D/A - - R ( Js Py( o Pu( D(, Θ K adaptation echanis ξ - - e Q o ( Fig.. IMPAC structure with adaptation of coefficients of prediction polynoial According to (9 and in virtue of Fig., it is evident that signal < estiates the portion of anipulated variable u which should copensate effects of generalied disturbance. In other words, if the syste absorbs the disturbance copletely, the steady-state value of the error of disturbance estiation ξ( 1 ( 1 = ( 1 (3 becoes equal to ero. his fact enables the use of an adaptive approach based upon the iniiation of ξ. hus, coordinates θ i of vector θ ˆ = [ θ1θ... θn ] are variable prediction polynoial coefficients that are to be deterined by iniiing the perforance index J( J(,,... 1 θ = θ θ θ = ξ( k (4 o 1 using the gradient ethod given in [18]. According to [18], the general recurrence forula of the on-line iterative process of adaptation is > θ ( k = > θ ( k γ θ ( k (5 i i i where θ i ( k, in the first place, deterines the direction of a change of paraeter > θ i at the sapling instant k, J( > θ, k θ i ( k = θ > ( k i (6 >( k 1 i = ξ ( k = ξ( k ( k θ> k i ( while the positive scalar γ regulates the step of iteration. Naely, the process of iteration converges to the proper paraeter vector if the constant γ satisfies condition < γ < γ c (7 where γ c is the upper liit of convergence step [18]. Practically, the positive constant γ is chosen by chance and then the convergence of iterative process is tested [18]. ω r u - L 1 1 ω P r( D/A - - R ( Js ξ - o K Pu( D(, Θ adapation echanis - c p Py( e o Q( 1-c p Fig. 3. IMPAC structure with predictive filter and adaptation of prediction polynoial o enable as fast as possible converge, the initial guess of coefficient vector of prediction polynoial D ( should be adopted in accordance with an available airport inforation about the disturbance. Fig. 3 shows the controlling structure obtained by odifying the structure of Fig. by introducing the siple prediction filter [19]. In the structure of Fig. 3, by decreasing of tuning paraeter c p one can enlarged the region of syste robust stability and iprove filtering properties of the syste. At the sae tie, the decrease of paraeter c p slightly derogates the speed (or dynaic of disturbance absorption. IV. ILLUSRAIVE EXAMPLE o investigate the ability of controlling structure of Fig. and in absorption of disturbance and to test the efficiency of the proposed adaptation echanis, several siulation runs were done. he paraeters of esla's induction otor used in the speed controlled servo were taken fro [6] as the total oent of inertia J =. 459 kg, electroagnetic torque constant K =. 1154, and sapling period = 1. he synthetic plant paraeter is calculated as C = he desired doinant pole pair of the closed-loop syste is specified by ς = 8. and ω n = 35 rad/s. In all siulation runs, the sae constant reference signal ω r ( t = 5. h( t 1. and load torque disturbance ( t = 15.sin( 46t h( t 5., L shown in Fig. 4, are applied siultaneously. Fig. 5 shows the transient responses of the syste in Fig. 1. races ω 1 ( t and ω ( t in Fig. 1 are obtained for D ( = 1 and D ( 1 =, respectively. In virtue of Fig. 5, the suppression of load torque disturbances is ore efficient when the absorption polynoial D ( = that corresponds to rap disturbances is eployed than in the case where D ( = 1 corresponding to constant disturbances is used. Nevertheless, in both cases the disturbance of Fig. 4 affects the steady-state value of the otor speed, i.e., the coplete rejection of disturbance is not achieved. 3

5 Septeber, L [N] ω 1(t ω (t Fig. 4. Load torque disturbance Fig. 6 visualies the operation of the syste in Fig. in which the proposed adaptation echanis with γ = 5 is applied. he trace ω 1 ( t is obtained by the initial prediction polynoial D ( = 1 which changes during the iteration t = process and converges to the prediction polynoial D ( = (8 ω 1(t ω (t Fig. 5. Operation of the syste in Fig. 1 ω 1 ( t <=> D ( = 1, ω ( t <=> D ( = 1 he trace ω ( t is obtained with the initial polynoial D ( = which converges to t = - -3 D ( = (9 Of course, in both cases (8 and (19 the sae structure of prediction polynoial D (, > > > > > 3 θ = θ θ θ θ has been adopted Fig. 6. Operation of the syste in Fig. in which the adaptation of prediction polynoial coefficients is applied he coparison of the related traces of Figs. 5 and 6, akes evident advantages of the use of prediction polynoial adaptation. Unlike the structure of Fig. 1, in the presence of adaptation echanis (Fig., the torque disturbance is copletely rejected in the steady-state ω 1(t ω (t Fig. 7. Operation of controlling structure of Fig. 1 with fixed prediction polynoial that corresponds to the periodical torque disturbance he disturbance in Fig. 4 is periodical and, according to the principle of absorption, the adequate prediction polynoial is D ( = -(n (3 where n denotes the nuber of sapling period within the disturbance period τ ( τ = n. In the exaple under consideration, n is not an integer but n = he syste responses ω 1 ( t and ω ( t are obtained for n = 14 and n = 13, respectively. In both cases the disturbance is significantly suppressed but still affects the steady-state value of the otor speed. 33

6 Mikrotalasna revija Septebar 1. Let us suppose an ideal case with the load disturbance ( t = 15.sin( t h( t 5. N and sapling L period = 1 s, where n is the integer 1 π n = = for which the adequate prediction polynoial becoes D ( = 3 he disturbance response in Fig. 8 deonstrates the efficiency of the controlling structure of Fig. in disturbance absorption in ideal cases; now, effects of disturbance on the steady-state value of otor speed is entirely rejected Fig. 8. Absorption of periodic disturbance with the absorption polynoial D ( = 3 V. CONCLUDING REMARKS One of the ain tasks of control syste is to track the reference signal, in the presence of external disturbances, without steady-state error. In ost practical applications, soe apriori inforation about the class of disturbances is available. In such cases, the extraction of entire disturbance is possible by using the principle of absorption and different controlling structures of the syste. he IMPAC structure, priarily developed by YaZ. sypkin, is especially useful for the ipleentation of absorption principle because in its application there are no opposite requireents between the syste stability and inclusion of the internal odel of expected disturbance into the control portion of the structure. Moreover, by using the on-line algorith of disturbance identification, it is possible to include the disturbance internal odel inside the IMPAC structure even in the case of a copletely unknown ieasurable arbitrary disturbance. In this paper, the IMPAC structure of the speed controlled electrical drive with asynchronous otor has been developed. We explained how probles concerning the drive design can be advantageously solved in the presence of load torque disturbance for which there is no any apriori inforation. It has been shown that even in this case the rejection of disturbance is possible by the on-line identification of disturbance and adaptation of disturbance internal odel. he results of siulation runs exposed exceptional possibilities of the novel approach to the use of absorption principle, proposed in this paper. o ephasie the efficiency of IMPAC structure with adaptation echanis, the relatively fast load torque disturbance and different kinds of prediction polynoial were applied ant it was shown that the best results are achieved by using the proposed procedure of adaptation of prediction polynoial coefficients. Nevertheless, results of this paper are opening new questions about the proper choice of prediction polynoial structure, since different prediction polynoials enable the entire extraction of disturbance. In addition, after the inclusion of adaptation echanis into the controlling structure, the syste becoes nonlinear and it ay cause the stability proble particularly in the presence of large reference signal and/or disturbances. REFERENCES [1] Nikola esla, "ELECRO-MAGNEIC MOOR," Patent No , dated May 1, 1888, Application filed Noveber 3, 1887, Serial No [] Nikola esla, "ELECRO-MAGNEIC MOOR," Patent No , dated May 1, 1888, Application filed October 1, 1887, Serial No [3] Nikola esla, "ELECRICAL RANSMISSION OF POWER," Patent No. 38.8, dated May 1, 1888, Application filed October 1, 1887, Serial No [4] F. Blaschke, "Das Princip der Feldorientierung, ein neues Verfahren ur Regelung der Asynchronaschine, Sieens- Forschungs-und Entwicklungsberichte 1 (1971 No.. [5] F. Blaschke, "Das Princip der Feldorientierung, die Grundlage für die RANSVEKOR Regelung von Drehfeld- Maschinen, Sieens Zeitschrift 45, Heft 1, pp , [6] M.R. Stoji}, S.N. Vukosavi}, "Design of icroprocessorbased syste for positioning servoechanis with induction otor, IEEE ransactions on Industrial Electronics, vol. IE- 38, No. 6, October [7] Ya.Z. sypkin, "Design of robust optial control systes having plants with bounded paraeter uncertainties" (in Russian, Avto.eleech., no. 9, pp , 199. [8] V.S. Kulebakin, "Behavior of continuous linear control systes subjected to external disturbances", (in Russian, Dokl. AN USSR, to 68, no. 5, pp , [9] C.D. Johnson, Accoodation of external disturbances in linear regulator and servoechanis probles, IEEE rans. A.C., vol. 16, pp , [1] E.J. Davison, he output control of linear tie-invariant systes with uneasurable arbitrary disturbances, IEEE rans. A.C., vol. 17, N o 5, pp , 197. [11] B.A. Francis and W.M. Wonha, he internal odel principle for linear ultivariable regulators, Appl. Math. Opt., vol., N o 3, pp. 1794, [1] B.A. Francis and W.M. Wonha, he internal odel principle of control theory, Autoatica, vol. 1, N o 5, pp , [13] G. Bengtsson, Output regulation and internal odels. A frequency doain approach, Autoatica, vol. 13, pp ,

7 Septeber, 1 [14] O.R. Gonale and P.J. Antsaklis, Internal odels in regulation, stabiliation and tracking, Int. J. Control, vol. 53, N o, pp , [15] M.R. Stoji}, M.S. Matijevi}, Lj.S. Draganovi}, "A robust Sith predictor odified by internal odels for integrating process with dead tie," IEEE rans. on Autoatic Control, vo.46, no.8, August 1. [16] Ya.Z. sypkin and U. Holberg, Robust stochastic control using the internal odel principle and internal odel control, Int. J. Control, vol. 61, N o 4, P 89-8, [17] Â.Z. C pkin,"adaptivno invari ntn e diskretn siste upravleni ", Avto. i eleeh. N o str. 961, [18] B. Dankovi}, D. Anti}, Z. Jovanovi}, Upravljanje procesia - identifikacija procesa, Elektronski fakultet u Ni{u, Ni{, [19] M.S. Matijevi}, Ravoj novih struktura digitalno upravljanih elektrootornih pogona i procesa, Doktorska disertacija, Ma{inski fakultet u Kragujevcu, 1. 35