Sensorless control of induction motors

Transcription

1 CODEN:LUTEDX/(TEIE-5227)/1-36/(26) Industial Electical Engineeing and Automation Sensoless contol of induction motos Simulating the application of an exted Kalman filte togethe with a quadatic filte Mai Lod Dept. of Industial Electical Engineeing and Automation Lund Univesity

2 Pontifícia Univesidade Católica do Rio de Janeio Depatment of Electical Engineeing Lund Institute of Technology Industial Electical Engineeing Sensoless contol of induction motos - Simulating the application of an exted Kalman filte togethe with a quadatic filte Mai Lod Rio de Janeio, 26

3 Abstact Title Autho Sensoless contol of induction motos Simulating the application of an exted Kalman filte togethe with a quadatic filte. Mai Lod, electical engineeing, Lund s Institute of Technology, Sweden. Supevisos Macos Azevedo da Silveia, Depatment of Electical Engineeing, PUC-Rio, Rio de Janeio, Bazil. Mats Alaküla, Depatment of Industial Electical Engineeing and Automation, Lund s Institute of Technology, Sweden. Repot Pupose Method Conclusion Keywods Maste thesis at the depatment of Industial Electical Engineeing and Automation, Lund s Institute of Technology. Pefomed at univesity PUC-Rio, Rio de Janeio, Bazil, fom Decembe 25 to Apil 26. The pupose is to ceate an algoithm that will make it possible to contol induction motos without sensos. The idea is based on fome eseach that consists of a method whee the autho of a doctoal thesis uses estimation instead of measuing to find out the speed of the oto. Cetain delimitations will be used in ode to save memoy and simplify the Matlab code. Afte intepetation of the doctoal thesis a pogam fo the algoithm is witten in the Matlab language. This algoithm is late to be implemented in Simulink via a so called S-function to be able to simulate and evaluate the esults. The filtes tun out to show unstable esults with a oto speed that keeps on gowing instead of stabilize when eaching its expected value. Since the theoy shows that this filte set-up should be stable, it is inteesting to keep on woking on the algoithm in ode to impove the pefomance of the filtes. Induction moto, sensoless contol, exted Kalman filte, oto speed 2

4 Contents Abstact...2 Contents Intoduction Backgound Pupose Method Delimitations Results Thesis contents Contolling the induction machine The induction machine The moto model Sensoless contol Specification of the actual poblem Filteing The Kalman filte The Kalman filte algoithm The exted Kalman filte The sub-optimal filte The quadatic filte The filte algoithm Building the algoithm Initial values The p-index Simulations and esults Results Conclusions...2 Refeences...21 Appix

5 1 Intoduction 1.1 Backgound Induction motos ae electomechanical systems suitable fo a lage spectum of industial applications. It is necessay to be able to contol the speed of these moto dives and the most common way of doing this is by using vecto contol. This method equies a speed senso which is usually placed on the oto shaft of the machine. The speed senso has some disadvantages though, since it - besides fom being costly - also educes the obustness and eliability of the induction moto. Consequently this has opened a new inteesting aea fo eseach and duing the last few yeas a vaiety of diffeent solutions has eached the maket and sensoless contol has become industial standad fo medium and low pefomance applications. Atificial intelligence and neual netwoks ae two examples of this new technology but they both show weak pefomance unde speed changes and they also need offline calculations to wok coectly. Since the induction moto is epesented by a fifth ode, nonlinea model with unknown state vaiables and extenal inputs, sensoless contol is a challenging theoetical poblem 1. One of the esults of all the eseach that has been made within this aea is a doctoal thesis called Moto speed estimation with sensoless vecto contol, employing an exted Kalman filte with estimation of the covaiance of the noises, witten by Jaime Antonio Gonzalez Castellanos 2. In this thesis the autho pesents a solution of contol of an induction moto without sensos whee he uses an exted Kalman filte togethe with a quadatic filte in ode to estimate the noise covaiance matices. These matices ae necessay fo the calculations in the Kalman filte. 1.2 Pupose Good esults wee obtained in the doctoal thesis and the aim of this maste thesis is to take that wok a little bit futhe. The main change is that the oto speed will no longe be consideed constant, which is a petty ough appoximation, but will instead be estimated in a completely closed system. 1.3 Method Afte sping time on undestanding the theoy of the doctoal thesis, the wok basically consists in ceating an algoithm based on this theoy but with modifications to suit the new idea. The algoithm is to be witten in Matlab language and then implemented, with help of an S-function, in Simulink fo simulations Gonzalez Castellanos J. A., (24) 4

6 1.4 Delimitations When witing the pogam in Matlab it is necessay to make some modifications in ode not to occupy too much memoy. Othewise the memoy would keep on gowing and in the equie vey much space, which patly would be vey costly but at the same time it would also take a lot of time to execute the pogam. 1.5 Results The filtes do not manage to give desied esults. It tuns out to be difficult to contol the system and the speed keeps on gowing instead of stabilize in one expected value. In othe wods the filtes act like they wee unstable and appaently thee is something in the algoithm that does not wok satisfactoy. 1.6 Thesis contents The epot stats with a theoetic pat, explaining the backgound and fome eseach. Then follows a pat of algoithm ceation, simulations and evaluations and finally a discussion with sum up, conclusions and poposals on futhe eseach. Chapte 2: Contolling the induction machine gives a pesentation of the induction machine and sensoless contol in geneal and in Chapte 3: Specification of the actual poblem a shot explanation of the specific poblem being handled in this thesis is pesented. Chapte 4: Filteing teats filte theoy, especially talking about the Kalman filte and its exted vesion, followed by an explanation of the seconday filte in Chapte 5: The quadatic filte. Futhe on, in Chapte 6: Building the algoithm, the actual algoithm is shown befoe eaching the simulations in Chapte 7: Simulations and esults whee some simulation issues ae discussed befoe pesenting the esults. Finally in Chapte 8: Conclusions the thesis is wounded up with a discussion aound the esults. In the an appix containing the algoithm witten in Matlab code is to be find. 5

7 2 Contolling the induction machine 2.1 The induction machine As mentioned in the intoduction the induction machine, also known as asynchonous machine, is the most common type of electical machine seen today in pactice. It is used in high powe as well as low powe applications and obviously being so popula because of its cheap and simple constuction The moto model To undestand the basic theoy of this wok it is necessay to have some knowledge of the induction moto itself. The dynamic model of the induction moto can be descibed in stato coodinates by the following equations: i i λ λ ω sd sq d q 1 ad = dd 1- ad dd - cω ω bd 1- ed D D cω - ω bd D D 1- ed i i λ λ 1 ω sd sq d q fd + fd u u sd sq whee the fist vecto is the state x(k), the last vecto is the input u(k) and the output 1 y(k) is given by C x(k), whee C =. The output consists in othe wods 1 of the cuents isd and i sq. In this desciption the oiginal moto model has been discetized and is to be sampled with an inteval denoted by D. Thee has also been added a new state, ω, which is the state wished to be estimated. As a consequence of adding this new state, the moto model is no longe linea. Futhe on: R s 1 σ M M a = +, b =, c =, σ Ls σ T T σ Ls L σ Ls L 2 M 1 1 M L d =, e =, f =, σ = 1-, T = T T σ L L L R s s 1 Alaküla M., (21) 6

8 R s = the oto esistance Ls = the oto inductance M = the mutual inductance whee. T = the oto time constant R = the stato esistance s σ = the total linkage facto 2.2 Sensoless contol The aim is to apply sensoless contol to this machine and to do so by using two filtes. An exted Kalman filte will wok togethe with a quadatic filte in ode to fist lineaize the non-linea system of this moto dive, then use the Kalman filte itself and eventually calculate the noise covaiance matices which ae needed fo the Kalman filte calculations. The equation above can be witten as x(k) = A(, ) x() + B u() with k epesenting a new iteation, the time. Then, as we wish to apply Kalman filte theoy, the noises ae simply added to the equations as follows: ω x(k) = A(, ) x() + B u() + G v() y(k) = C x(k) + w(k) Knowing that G is a weighting matix fo the noise of the system and can be chosen easily, these equations give us two unknowns: v() and w(k). They ae noise sequences which can be epesented by thei covaiance matices Q and R and consequently need to be calculated o estimated. In this case a seconday filte, the quadatic filte, will be used to estimate the optimal values of the noise covaiance matices. Obseving the ode of pefomance, the initial values of the unknown covaiance matices ae obviously impotant 1. The pupose of the sensoless contol is to estimate the moto speed, ω, instead of measuing it. Thus, an exted Kalman filte will afte each iteation, fo evey new k, give a new value of ω the oto speed. The value of the oto speed is needed in ode to pefom vecto contol. Vecto contol is the name of a goup of methods that ae based on the moto model. The methods consist in contolling the toque without being depent on the cuents that poduce the flux and the toque. A Simulink model gives a bette pictue of how the induction machine woks without sensoless contol: ω 1 Gonzalez Castellanos J. A., (24) 7

9 Figue 1. Simulink model of an induction moto with vecto contol. To sum up the ideas of this geneal poblem, the goal is to estimate the oto speed with the help of two filtes an exted Kalman filte and a quadatic filte. The quadatic filte, also efeed to as the seconday filte, is used in ode to estimate optimal values of the noise covaiance matices Q and R which ae needed fo the pefomance of the Kalman filte calculations. 8

10 3 Specification of the actual poblem The ideas pesented in the pevious chapte undelie the doctoal thesis, Moto speed estimation with sensoless vecto contol, employing an exted Kalman filte with estimation of the covaiance of the noises, by Jaime Antonio Gonzalez Castellanos, mentioned in the intoduction. In this wok the autho managed to obtain vey good estimates of the oto speed. Fist of all it will theefoe be necessay to ceate a pogam (which will be witten in Matlab) fo the algoithm that will give these estimates. Thus, the estimation of the oto speed will be made in a way simila to the method used in the doctoal thesis. Then, in ode to poceed with the eseach made in this doctoal thesis, the aim is to make the whole system wok as a completely closed system. This idea is explained in Figue 2: Figue 2. Simulink model of an induction moto with vecto contol and sensoless contol. Obseve the impotant diffeence in the INDUCTION MOTOR -box and the output om that efes to omega, ω, in othe wods the oto speed. This paamete is no longe the input to the omega in the AC-DC CONVENTION -box whee it has been eplaced with the ω estimated in THE ESTIMATOR, the box that pefoms the sensoless contol. 9

11 4 Filteing Thee exist many methods fo speed estimation in induction machines. One categoy that shows good pefomance is the kind of methods based on vecto contol whee the moto model makes it possible to estimate the speed. Among these methods the exted Kalman filte is one of the moe successful. To be able to undestand the implementation of the methods used in this poject will in this chapte be given a moe detailed filte theoy desciption. 4.1 The Kalman filte The Kalman filte and its exted vesion ae efficient and obust speed estimatos fo linea and non-linea systems espectively. The filtes use knowledge of the dynamic system, its statistic chaacteistics and noise souces in ode to poduce an optimal estimate of the state and at the same time minimize the covaiance eo. The filte is fomed in tems of the state vaiables of the system and its solution is ecusively computed. This means that evey time an estimated state is updated, the only infomation that is needed in ode to compute this new state is the peviously estimated values and the new infomation given fom the system at the cuent time. It is theefoe necessay to stoe only the infomation of the value estimated most ecently because at evey new instant the new estimation is a pojection on the fome estimations 1. But the Kalman filte needs cetain infomation to be able to wok in this way. Fist of all the filte has to have some knowledge of the basic paametes of the system. Then it is also necessay to know the values of the noise covaiance matices of the system as well as of the obsevations. Thee ae special methods to obtain the optimal values of these matices. If thee is lacking infomation about one o moe of these matices, the filte will be called sub-optimal. The pupose of the Kalman filte is to poduce an algoithm that makes it possible to compute an optimal estimate and the eo of the covaiance (which in this epot will be efeed to as P). The non-linea system on which this is applied can be descibed by the following equation: x(k) = A(, k 1 ω ) x() + B u() + G v() whee: x(k) = the state vecto of length n at time k, A = a non-singula matix fo state tansition of size n x n, deping on theefoe non-linea. u() = the input vecto at time, G = a weighting matix fo the noise of the system of size n x n, v = a vecto to descibe the noise sequence of length. k 1 ω and 1 Luenbege D. G., (1969) 1

12 PP The noise is of so called white Gaussian type, o nomal distibuted in othe wods, which means that its values ae andom, Gaussian (nomal distibuted) vaiables, uncoelated and with zeo mean when time goes towads infinity. When this is the case the noise can be totally epesented by its covaiance. Consequently the expected value E{v k v m Q fo m = k } = fo m k The obsevation system, o the output, is epesented by: y(k) = C x(k) + w(k) and its noise chaacteistics given by E{w k w m R fo m = k } = fo m k The Kalman filte algoithm As descibed above the Kalman filte uses a ecusive way to solve the poblem. This can be seen clealy in the algoithm which is pefomed in the following steps: Pediction of the state x f (k) = A(, ω ) x a () + B u() Estimation of the matix of the covaiance eo f (k) = Ψ() P a () Ψ T () + G Q() G T Calculation of the gain of the Kalman filte K KB (k) = P f (k) C T [C P f (k) C T + R()] -1 Estimation of the state x a (k) = x f (k) + K KB (k) [y(k) C x f (k)] Updating the matix of the estimation covaiance eo PPa (k) = [I K KB (k) C] P f (k) whee Ψ() is the deivative of the matix A(, ω ) with espect to x(). Obseve that when calculating x a (k) (the new state value) the Kalman filte gain is multiplied with the eo of the output the innovation. The innovation pocess has an impotant pat of the solution in this wok and the innovations ae defined by: 11

13 η(k) = y(k) C x f (k) whee x f (k) equals the pedicted x at time k, knowing the value of x(). The innovation pocess consists in ceating a quadatic output, using estimatos fo the second ode innovation moments E{η(k) η T (k-m)}, whee m k. In othe wods the new innovation, at time k, is multiplied with the old innovations, at time k m. One impotant quality of the innovations is that when m =, they ae othogonal one to anothe, fom which follows that E{η(k) η T (k)} = The exted Kalman filte Since we ae woking with a system that is non-linea, the odinay Kalman filte is not a filte that will solve ou poblems. But it is possible, though a pocess of lineaization, to ext the Kalman filte in ode to use it on a non-linea system. This exted Kalman filte fist lineaizes the non-linea state fom time and then in its next step, at time k it uses this lineaized state in the nomal Kalman filte. Hee, it is this exted Kalman filte that is called the pimay filte and is epesented by the fomulas given above in the algoithm. As mentioned in the beginning of this chapte it is necessay to know some of the basic paametes and to have knowledge of the noise covaiance matices. In this case the noise covaiance matices of the system and of the obsevation, Q and R espectively, ae unknown and the filte will in othe wods be sub-optimal. 4.3 The sub-optimal filte When one o moe matices ae not fully known they will be eplaced with a coesponding matix and the filte will be called sub-optimal. In this case the S- matices will epesent the optimal P-matices fom the exted Kalman filte, pincipally in ode to ceate the obsevation matix F. The S-matices ae defined by: a S (k) = A() S i () A T () + G Q i G T S (k) = [I K KB f (k) C] (k) [I-K KB (k) C] T + K KB (k) R i K KB T (k) f i a i and S i S a (k) N a = αi (k)si, N f S (k) = i= 1 i= 1 α i (k)s f i whee α is a vecto of length N and will be teated moe in detail in Chapte 5. The F-matices ae then constucted in the following way: 1 Dee D. P., (1983) 12

14 F i (k,) = C f S i (k) C T + R i (k) F i (k,1) = C Ψ() [I - K KB () C] f S i f S i F i (k,m) = C Ψ() [I - K KB () C] (k) C (k) C T - K KB R i (k) T fo m>1 Since the matices used in this sub-optimal filte ae not exact, the estimated state will diffe slightly fom the eal state and the estimation will obviously not be optimal. It has been shown though that P f (k) = S f (k) + eo(k) and that the eo exponentially conveges to zeo 1. Theefoe, with a k that is big enough, the filte will give optimal esults afte a sufficient time. 1 Dee D. P., (1983) 13

15 5 The quadatic filte The pupose of the quadatic filte is to obtain the optimal values of the noise covaiance matices Q and R. To estimate these values, an algoithm will be poposed that is based on the innovation pocess of the pimay filte. 5.1 The filte algoithm Fist the noise covaiance matices ae descibed as linea combinations of aleady known matices, Q i and R i, accoding to a method pesented by Bélange in Q(k) N = αi (k) Q i and N R(k) = i= 1 i= 1 α (k) i R i whee α is a vecto of length N. It is necessay to find an obsevation model fo this vecto and in ode to do so the estimato of α will be fomed as a seconday filte of the sub-optimal exted Kalman filte. The equation that descibes the obsevation model can be witten as: 2 η(k) η (k) N T = i= 1 α (k) F (k,m) + ξ(k, m) i i whee η(k), as mentioned in chapte 4.1 is the innovation y(k) C x f (k). ξ(k,m) is a double sequence of andom vaiables. In the same way as in the case of the noise sequences, these vaiables ae consideed uncoelated, Gaussian and of zeo mean why in othe wods they can be totally epesented by thei covaiance matix W(k,m). Since ξ(k,m) is a double sequence, W(k,m) will be a matix of foth ode moments. F i (k,m) is a matix, ecusively calculated fom the sub-optimal matices (k) and f S i (k) and used as an obsevation matix, thus coesponding to the C-matix when compaing with the exted Kalman filte. Futhe on the gain of the exted Kalman filte K KB will be epesented by K and the actual covaiance eo matix (in the Kalman filte efeed to as P) is in the seconday filte named θ. Next step is to fom the obsevations, consisting of the innovations η(k) η T (k-m) and the F i -matices, in a special way in ode to be coheent with the vecto α, (that is to be obseved). The idea is to stack the columns of the matices and in this way ceate a vecto (vec). If the matix is symmetic the epeated tems ae ignoed (Tvec): a S i 1 Bélange P. and Caew B., (1973) 2 Dee D. P., (1983) 14

16 T Tvec[ηvec η (k)] + ξ(k,) z = T vec[ ηvec η (k m)] + ξ(k, m) if m = if m > The F i (k,m)- and the W(k,m)-matices ae constucted in the same way as the obsevations, η(k) η T (k-m), in the sense that they ae all multiplied with thei fome values at time k-m. The same technique of stacking the matix columns is theefoe used to fom the F- and the W-matices and the quadatic filte can then be descibed with the following fomulas: K(p) = θ(p-1) F T (p) [F(p) θ(p-1) F T (p) + W(p)] -1 θ(p) = [I K(p) F(p)] θ(p-1) α(p) = α(p-1) + K(p) [z(p) F(p) α(p-1)] A new index, p, is intoduced hee specially fo the quadatic filte and each p simply coesponds to each couple (k,m) in the pimay filte. Thus, at evey time k, the quadatic filte will make k iteations, since m uns fom to k. The quadatic filte passes on the values given fom the last of these iteation, that is to say when m = k. 15

17 PP 6 Building the algoithm Afte obtaining enough knowledge of the theoy pesented in the pevious chaptes it is possible to stat woking on the algoithm. This one is supposed to be witten in Matlab language and ceated based on the following fomulas: The states x(k) = [i sd i sq λ d λ q ω ] T u(k) = [u sd u sq ] T 1 y(k) = C x(k) whee C =. 1 The pimay filte x f (k) = A() x a () + B u() f (k) = Ψ() P a () Ψ T () + G Q() G T K KB (k) = P f (k) C T [C P f (k) C T + R()] -1 x a (k) = x f (k) + K KB (k) [y(k) C x f (k)] PPa (k) = [I K KB (k) C] P f (k) whee Q and R ae the noise covaiance matices which can be descibed as a linea combination of simple matices: Q(k) N = αi (k) Q i and N R(k) = i= 1 i= 1 α (k) i R i The seconday filte K(p) = θ(p-1) F T (p) [F(p) θ(p-1) F T (p) + W(p)] -1 θ(p) = [I K(p) F(p)] θ(p-1) α(p) = α(p-1) + K(p) [z(p) F(p) α(p-1)] whee p is a new index epesenting all (k,m)-couples in each iteation k. 6.1 Initial values In the beginning, fo the vey fist otation of the pogam in Matlab, thee ae vaious values that ae unknown in the algoithm. Assuming fo example that the iteation will stat at time k = 1, the following paametes will have to be initialized with a tustwothy value: x a (), P a (), α(), S a () (which is used to calculate the F-matix) and θ(). The matix G also has to be set but this is a constant matix and its values ae taken fom the ones used in fome eseach 1. 1 Gonzalez Castellanos J. A., (24) 16

18 1 G = In the othe cases diffeent values ae tested and x a (), P a (), α(), θ() wok pefectly fine with simple initializations such as: x a () = a vecto of zeos, PPa () and θ() = the identity matix, I, α() = a vecto of ones, while S a () tuns out to be moe difficult though, since in a lot of liteatue S a () is said to be a matix of zeos 1. Howeve, this did not wok in a pleasant way in this algoithm and afte tying out some diffeent values, S a () is set to the identity matix with two exta values in ode two facilitate the calculations. 1.1 Sa() = The p-index The p-index is intoduced fo the quadatic filte and seves in ode to make this seconday filte pefom k iteations fo each iteation of the pimay filte. Thus, fo evey step k+1 the seconday filte has to make one moe iteation than the time befoe and it is easy to undestand that this amount of iteations afte a cetain time will occupy a lot of memoy and a lot of execution time. It is theefoe convenient to pogam the algoithm in a way to limit this gowth. The assumption can be made that it is sufficient to look at ten (k,m)-couples at each iteation k and this means that p in othe wods uns fom to 9. Since it is inteesting to include only the ten most ecent obsevations in the calculations, p coesponds in this case to m fom which follows that p = is equivalent to (k,) and p = 9 to (k,9). Consequently when using this system fo the algoithm, p can be eplaced by m. In the initialization of the pogam, it has to be taken into consideation that when k<1 the pogam does not wok as in the geneal case, when moe time has passed and k>=1. 1 Dee D. P., (1983) 17

19 7 Simulations and esults When the pogam fo the algoithm woks satisfactoy in Matlab, it needs to be implemented in Simulink in ode to make simulations of the pogam. Only by simulating it is possible to see how the pogam fo the algoithm woks togethe with the actual induction moto. This is evident since the only inputs to the algoithm, u() and y(k), ae the stato voltages (u = [u sd u sq ]) and the stato cuents (y = [i sd i sq ]) of the moto. The implementation is made though a so called S-function which constitutes a pat of Simulink that makes it possible to add you own blocks in a Simulink model. These blocks can be ceated fo example in Matlab. The S-function block is hidden inside the block called THE ESTIMATOR (see figue 2, chapte 3). This is, as mentioned in the text, the box that epesents the algoithm that pefoms the sensoless contol and inside it looks like this: Figue 3. A Simulink model of the pat that concludes the S-function. When fist executing the simulation it is necessay to look at the signal ômega coming fom the estimation box and compae this signal with the one poduced fom the actual induction moto, om. In this way it is possible to see if the algoithm woks satisfactoy befoe tying to eplace the eal measued omega, om, with ou new estimated omega, ômega. One impotant thing to conside befoe stating the simulations is the mix of analog and digital signals appeaing in the moto model and the estimato. Since the estimato is woking in disceet time the inputs need to be digital signals and ae theefoe taken fom appopiate places in the moto model (see figue 2, chapte 3). It is also impotant that the sample times in diffeent blocks, as well as between the estimato and the moto model, ae coheent. 7.1 Results In the beginning it was difficult to make the pogam otate and it stopped almost immediately with the message that the vecto α had obtained values that wee not consideed values, (the eo code is NaN in Matlab language which has the signification Not a Numbe). The eason of this tuned out to be a numeical poblem since the matix that was to be inveted in the seconday filte became so small that the esult of the invesion gave infinite numbes. Changing the ode of the fomulas and 18

20 using new ideas fo how to bette ceate the matices W and F, made the pogam advance slowly. Late occued new poblems when the F-matices stated to gow too fast and θ at the same time was going vey fast towads. When obseving the algoithm and the fomulas in the seconday filte it is undestood that with a θ that equals the seconday filte is not using any new infomation when pefoming its calculations. A θ that equals means that the covaiance eo is and the filte would with othe wods be pefect and stop making coections. Theefoe a tempoay solution was intoduced and θ was given a new value, bigge than, in the of evey iteation. With this coection the filte stated to show bette pefomance and the oto speed, omega, stated to gow with an acceptable velocity towads its desied value. Once eaching this value the oto speed did not decline as expected but kept on gowing. It is obseved that the poblem with fast gowing F-matices still emained which esults in the gain of the filte, K, becoming vey big. The gains of the two filtes, K and K KB, ae supposed to gow a lot in the beginning and then, afte some time, slowly decease and become smalle. If this gowth is vey big, as will be the case in the seconday filte if the F-matices become too big, it is likely that the numbes each a size that is not within the Matlab limit. Then the poblem is a numeical question and it would be necessay to use a calculation pogam that can handle numbes of this size. 19

21 8 Conclusions The system tuns out to be petty difficult to contol. The method being used when ceating the S-function consists in saving all the conditions in one long vecto and it is theefoe difficult to get an oveview of what is happening with each state at evey iteation. A bette method would pobably be to ceate some kind of libay whee it is possible to get and stoe each one of the conditions and in this way get a bette oveview and moe contol of how the situation is developing. Anothe, o an additional, suggestion is to ewite the fomulas, maybe in anothe ode, but thee is also the possibility of witing them in a diffeent way in ode to moe likely avoid the numeical poblems that we have seen hee. Duing the last month the solution was getting close and close fo each week and with moe time it would cetainly be possible to solve this poblem and obtain a lot bette esults. The time dedicated to this job still has its value since it will be passed on to anothe student to keep on woking on the solution. Thee ae a lot of things that ae inteesting to continue woking on, not only with the pogam pesented hee in this thesis but also with altenative solutions in ode to ealize and develop this idea. Pehaps it is possible fo example to use anothe set of filtes that has a moe simple constuction and doesn t use foth ode moments. 2

22 Refeences Alaküla M., (21) Powe electonic contol, Lund s univesity, Lund Alaküla M., Getma L., Samuelsson O.(22) Elenegiteknik, KFS, Lund Bélange P. and Caew B., (1973) Identification of optimum filte steady state gain fo systems with unknown noise covaiances, IEEE Tans. on automatic contol, Vol. AC18, N 6, page Dee D. P., (1983) Computational aspects of adaptive filteing and applications to numeical weathe pediction, Doctoal thesis CI-6-83, Couant institute of mathematics, New Yok univesity, New Yok Gonzalez Castellanos J. A., (24) Estimação de velocidade do moto com contole vetoial sem senso, utilizando filto estido de Kalman com estimação da covaiância dos uídos, Doctoal thesis (5716), PUC-Rio, Rio de Janeio Hanselman D., Littlefield B., (23) MATLAB 6 Cuso completo, Peason Education Inc., São Paulo. (Bazilian vesion) Luenbege D. G., (1969) Optimization by vecto space methods, John Wiley and Sons Inc., Stanfod univesity, Califonia Peteson B., (1996) Induction machine speed estimation, obsevation on obseves, Depatment of Industial Electical Engineeing and Automation, Lund s Institute of Technology, Lund Intenet souces Montanai M. (23) "Sensoless Contol of Induction Motos: Nonlinea and Adaptive Techniques", found on 28 feb 26 21

23 Appix 1. The algoithm witten in Matlab: function x = TheEstimato(t,x,Vsd,Vsq,Isd,Isq) % The moto L = 252.e-3; Ls = 252.e-3; R = 1.87; Rs = 3.88; M = 236.3e-3; T = L/R; sig1 = 1-(M^2)/(Ls*L); %oto inductivity %stato inductivity %oto esistance %stato esistance %main inductivity a = (Rs/(sig1*Ls)+(1-sig1)/(sig1*T)); b = M/(T*sig1*Ls*L); c = M/(sig1*Ls*L); d = M/T; e = 1/T; f = 1/(sig1*Ls); delta =.1; % sampling inteval % The states xa = [x(1) x(2) x(3) x(4) x(5)]'; Pa = [x(6) x(7) x(8) x(9) x(1); x(11) x(12) x(13) x(14) x(15); x(16) x(17) x(18) x(19) x(2); x(21) x(22) x(23) x(24) x(25); x(26) x(27) x(28) x(29) x(3)]; Sa = [x(31) x(32) x(33) x(34) x(35); x(36) x(37) x(38) x(39) x(4); x(41) x(42) x(43) x(44) x(45); x(46) x(47) x(48) x(49) x(5); x(51) x(52) x(53) x(54) x(55); x(56) x(57) x(58) x(59) x(6); x(61) x(62) x(63) x(64) x(65); x(66) x(67) x(68) x(69) x(7); x(71) x(72) x(73) x(74) x(75); x(76) x(77) x(78) x(79) x(8); x(81) x(82) x(83) x(84) x(85); x(86) x(87) x(88) x(89) x(9); x(91) x(92) x(93) x(94) x(95); x(96) x(97) x(98) x(99) x(1); x(11) x(12) x(13) x(14) x(15); x(16) x(17) x(18) x(19) x(11); x(111) x(112) x(113) x(114) x(115); x(116) x(117) x(118) x(119) x(12); x(121) x(122) x(123) x(124) x(125); x(126) x(127) x(128) x(129) x(13); x(131) x(132) x(133) x(134) x(135); x(136) x(137) x(138) x(139) x(14); x(141) x(142) x(143) x(144) x(145); x(146) x(147) x(148) x(149) x(15); x(151) x(152) x(153) x(154) x(155); x(156) x(157) x(158) x(159) x(16); x(161) x(162) x(163) x(164) x(165); x(166) x(167) x(168) x(169) x(17); x(171) x(172) x(173) x(174) x(175); x(176) x(177) x(178) x(179) x(18); x(181) x(182) x(183) x(184) x(185); x(186) x(187) x(188) x(189) x(19); x(191) x(192) x(193) x(194) x(195); x(196) x(197) x(198) x(199) x(2); x(21) x(22) x(23) x(24) x(25)]; Inov = [x(26) x(27) x(28) x(29) x(21) x(211) x(212) x(213) x(214) x(215); x(216) x(217) x(218) x(219) x(22) x(221) x(222) x(223) x(224) x(225)]; Kkb = [x(226) x(227); x(228) x(229); x(23) x(231); x(232) x(233); x(234) x(235)]; U = [x(236) x(237)]'; Theta = [x(238) x(239) x(24) x(241) x(242) x(243) x(244); x(245) x(246) x(247) x(248) x(249) x(25) x(251); x(252) x(253) x(254) x(255) x(256) x(257) x(258); x(259) x(26) x(261) x(262) x(263) x(264) x(265); x(266) x(267) x(268) x(269) x(27) x(271) x(272); x(273) x(274) x(275) x(276) x(277) x(278) x(279); x(28) x(281) x(282) x(283) x(284) x(285) x(286)]; alfa = [x(287) x(288) x(289) x(29) x(291) x(292) x(293)]'; M = [x(294) x(295) x(34) x(35) x(314) x(315) x(324) x(325)...; x(296) x(297) x(36) x(37) x(316) x(317) x(326) x(327)...; x(298) x(299) x(38) x(39) x(318) x(319) x(328) x(329)...; x(3) x(31) x(31) x(311) x(32) x(321) x(33) x(331)...; 22

24 x(32) x(33) x(312) x(313) x(322) x(323) x(332) x(333)...]; V = [x(994) x(995) x(996) x(997) x(998) x(999) x(1) x(11) x(12) x(13) x(14) x(15) x(16) x(17) x(18) x(19) x(11) x(111) x(112) x(113); x(114) x(115) x(116) x(117) x(118) x(119) x(12) x(121) x(122) x(123) x(124) x(125) x(126) x(127) x(128) x(129) x(13) x(131) x(132) x(133)]; k = x(134); % Constant values and matices L = 1; % Iteations in the secunday filte B = [f*delta ; f*delta; ; ; ]; C = [1 ; 1 ]; G = [1e-6 ; 1e-6 ; 1e-6 ; 1e-6 ; 1e-3]; Po5 = spdiags(ones(5,1),,5,7); % Ceates a 5x7-matix with 5 ones in the fist "diagonal" Po2 = spdiags(ones(2,1),5,2,7); % Ceates a 2x7-matix with 2 ones in the 6th "diagonal" eps = zeos(5,5); mu = zeos(2,2); SF = zeos(5,5); MA = zeos(5,16); % Inputs u = [Vsd,Vsq,Isd,Isq]'; U = [u(1) u(2)]'; y = [u(3) u(4)]'; % The pimay filte: omega = x(5); A = [1-a*delta b*delta c*delta*omega ; 1-a*delta -c*delta*omega b*delta ;d*delta 1-e*delta - delta*omega ; d*delta delta*omega 1-e*delta ; 1]; Psi = [1-a*delta b*delta c*delta*omega c*delta*xa(4); 1-a*delta -c*delta*omega b*delta - c*delta*xa(3);d*delta 1-e*delta -delta*omega -delta*xa(4); d*delta delta*omega 1-e*delta delta*xa(3); 1]; xf = A*xa + B*U; Pf = ((Psi*Pa*Psi' + G*diag(Po5*alfa)*G')+((Psi*Pa*Psi' + G*diag(Po5*alfa)*G')'))/2; Kkb = Pf*C'*((((C*Pf*C' + diag(po2*alfa))^-1)+((c*pf*c' + diag(po2*alfa))^-1)')/2); inov = y - C*xf; xa = xf + Kkb*inov; Pa = (((eye(5) - Kkb*C)*Pf)+((eye(5) - Kkb*C)*Pf)')/2; U = [Vsd Vsq]'; % Modification fo the fist ten iteations if k<1 L=k; % Calculating the Fi-matices fo i=1:7 j=8; =1; 23

25 if i<6 eps(i,i)=1; else mu(i-5,i-5)=1; Sai = Sa((1+(i-1)*5):(5+(i-1)*5),1:5); Sf = ((Psi*Sai*Psi'+G*eps*G')+(Psi*Sai*Psi'+G*eps*G')')/2; Sai = (((eye(5)-kkb*c)*sf*(eye(5)-kkb*c)'+kkb*mu*kkb')+((eye(5)-kkb*c)*sf*(eye(5)- Kkb*C)'+Kkb*mu*Kkb')')/2; x(31+25*(i-1):35+25*(i-1)) = Sai(1,:); x(36+25*(i-1):4+25*(i-1)) = Sai(2,:);x(41+25*(i- 1):45+25*(i-1)) = Sai(3,:);x(46+25*(i-1):5+25*(i-1)) = Sai(4,:);x(51+25*(i-1):55+25*(i-1)) = Sai(5,:); SF = SF + Sf; fo m=1:l if m==1 Ma1 = Sf*C'; F = C*Ma1 + mu; F1 = F(1,1); F2 = F(2,1); F4 = F(2,2); vecf(1,i) = F1; vecf(2,i) = F2; vecf(3,i) = F4; elseif m==2 KKB = [x(226) x(227); x(228) x(229); x(23) x(231); x(232) x(233); x(234) x(235)]; Mi = M(1:5,(1+(i-1)*2):(2+(i-1)*2)); Ma2 = Psi*(eye(5)-KKB*C)*Mi-KKB*mu; F = C*Ma2; F1 = F(1,1); F2 = F(1,2); F3 = F(2,1); F4 = F(2,2); vecf(4,i) = F1; vecf(5,i) = F3; vecf(6,i) = F2; vecf(7,i) = F4; else KKB = [x(226) x(227); x(228) x(229); x(23) x(231); x(232) x(233); x(234) x(235)]; Mi = M(1:5,(3+(i-1)*2+(m-3)*2):(4+(i-1)*2+(m-3)*2)); Ma = Psi*(eye(5)-KKB*C)*Mi; MA(1:5,:+1) = Ma; F = C*Ma; F1 = F(1,1); F2 = F(1,2); F3 = F(2,1); F4 = F(2,2); vecf(j,i) = F1; vecf(j+1,i) = F3; vecf(j+2,i) = F2; vecf(j+3,i) = F4; j = j+4; =+2; if k==1 x(294+1*(i-1)) = Ma1(1,1); x(295+1*(i-1)) = Ma1(1,2); x(296+1*(i-1)) = Ma1(2,1); x(297+1*(i-1)) = Ma1(2,2); x(298+1*(i-1)) = Ma1(3,1); x(299+1*(i-1)) = Ma1(3,2); x(3+1*(i-1)) = Ma1(4,1); x(31+1*(i-1)) = Ma1(4,2); x(32+1*(i-1)) = Ma1(5,1); x(33+1*(i-1)) = Ma1(5,2); elseif k==2 x(294+1*(i-1)) = Ma1(1,1); x(295+1*(i-1)) = Ma1(1,2); x(296+1*(i-1)) = Ma1(2,1); x(297+1*(i-1)) = Ma1(2,2); x(298+1*(i-1)) = Ma1(3,1); x(299+1*(i-1)) = Ma1(3,2); x(3+1*(i-1)) = Ma1(4,1); x(31+1*(i-1)) = Ma1(4,2); x(32+1*(i-1)) = Ma1(5,1); x(33+1*(i-1)) = Ma1(5,2); x(34+1*(i-1)) = Ma2(1,1); x(35+1*(i-1)) = Ma2(1,2); x(36+1*(i-1)) = Ma2(2,1); x(37+1*(i-1)) = Ma2(2,2); x(38+1*(i-1)) = Ma2(3,1); x(39+1*(i-1)) = Ma2(3,2); x(31+1*(i-1)) = Ma2(4,1); x(311+1*(i-1)) = Ma2(4,2); x(312+1*(i-1)) = Ma2(5,1); x(313+1*(i-1)) = Ma2(5,2); else x(294+1*(i-1)) = Ma1(1,1); x(295+1*(i-1)) = Ma1(1,2); x(296+1*(i-1)) = Ma1(2,1); x(297+1*(i-1)) = Ma1(2,2); x(298+1*(i-1)) = Ma1(3,1); x(299+1*(i-1)) = Ma1(3,2); x(3+1*(i-1)) = Ma1(4,1); x(31+1*(i-1)) = Ma1(4,2); 24

26 x(32+1*(i-1)) = Ma1(5,1); x(33+1*(i-1)) = Ma1(5,2); x(34+1*(i-1)) = Ma2(1,1); x(35+1*(i-1)) = Ma2(1,2); x(36+1*(i-1)) = Ma2(2,1); x(37+1*(i-1)) = Ma2(2,2); x(38+1*(i-1)) = Ma2(3,1); x(39+1*(i-1)) = Ma2(3,2); x(31+1*(i-1)) = Ma2(4,1); x(311+1*(i-1)) = Ma2(4,2); x(312+1*(i-1)) = Ma2(5,1); x(313+1*(i-1)) = Ma2(5,2); o=; fo s=1:8 x(314+1*(i-1)+(s-1)*1) = MA(1,1+o); x(315+1*(i-1)+(s-1)*1) = MA(1,2+o); x(316+1*(i-1)+(s-1)*1) = MA(2,1+o); x(317+1*(i-1)+(s-1)*1) = MA(2,2+o); x(318+1*(i-1)+(s-1)*1) = MA(3,1+o); x(319+1*(i-1)+(s-1)*1) = MA(3,2+o); x(32+1*(i-1)+(s-1)*1) = MA(4,1+o); x(321+1*(i-1)+(s-1)*1) = MA(4,2+o); x(322+1*(i-1)+(s-1)*1) = MA(5,1+o); x(323+1*(i-1)+(s-1)*1) = MA(5,2+o); o=o+2; if i<6 eps(i,i)=; else mu(i-5,i-5)=; % Ceating the innovations if k<1 p=k; fo m=2:l Inov(1:2,p)=Inov(1:2,p-1); p=p-1; Inov(1:2,1)=inov else p=1; fo m=2:l Inov(1:2,p)=Inov(1:2,p-1); p=p-1; Inov(1:2,1)=inov; % Ceating the V:s (the moments of the W-matices) v = C*SF*C'+diag(Po2*alfa); if k<1 p=2*; fo m=2:l V(1:2,p:p+1)=V(1:2,p-2:p-1); p = p-2; V(1:2,1:2) = v; else 25

27 p=19; fo m=2:l V(1:2,p:p+1)=V(1:2,p-2:p-1); p = p-2; V(1:2,1:2) = v; a = V(1,1); b = (V(1,2)+V(2,1))/2; c = V(2,2); if a<1e-3 a = 1e-3; elseif c<1e-3 c = 1e-3; % Ceating the z:s and the F:s s = 3; % index fo W j = 4; fo m=1:l z = inov*inov(1:2,m)'; z1 = z(1,1); z2 = z(1,2); z3 = z(2,1); z4 = z(2,2); if m==1 z = [z1 z2 z4]'; F = vecf(1:3,1:7); else z = [z1 z2 z3 z4]'; F = vecf(j:j+3,1:7); j=j+4; % Ceating the W:s if m==1 W = [2*a*a 2*a*b 2*b*b;2*a*b b*b+a*c 2*b*c;2*b*b 2*b*c 2*c*c]; else A = V(1,s); B = (V(1,s+1)+V(2,s))/2; C = V(2,s+1); if A<1e-3 A = 1e-3; elseif C<1e-3 C = 1e-3; W = [a*a b*a a*b b*b;b*a c*a b*b c*b;a*b b*b a*c b*c;b*b c*b b*c c*c]; s = s+2; % The secunday filte Mat = F*Theta*F' + W; MatI = ((Mat^-1)+(Mat^-1)')/2 26

28 K = Theta*F'*MatI; %Theta = (((eye(7) - K*F)*Theta)+(((eye(7) - K*F)*Theta)'))/2; Theta alfa = alfa + K*(z - F*alfa); alfa k=k+1; % New values and matices Theta = 1*(eye(7)); x = [xa(1) xa(2) xa(3) xa(4) xa(5) Pa(1,1) Pa(1,2) Pa(1,3) Pa(1,4) Pa(1,5) Pa(2,1).k]; 2. The S-function witten in Matlab function [sys,x,st,ts] = The_Estimato_sf(t,x,u,flag) switch flag, % Initialization case, [sys,x,st,ts] = mdlinitializesizes; % Update case 2, sys = mdlupdate(t,x,u); % Output case 3, sys = mdloutputs(t,x,u); % Teminate case 9, sys = []; othewise eo(['unhandled flag = ',num2st(flag)]); % Retun the sizes, initial conditions, and sample times fo the S-function. function [sys,x,st,ts]=mdlinitializesizes sizes = simsizes; sizes.numcontstates = ; sizes.numdiscstates = 134; sizes.numoutputs = 1; sizes.numinputs = 4; sizes.difeedthough = ; sizes.numsampletimes = 1; sys = simsizes(sizes); t = 1; ny =.1/7; my = 1/7; 27

29 x = [,,,,,1,,,,,,1,,,,,,1,,,,,,1,,,,,,1,my,ny,,,,ny,my,,,,,,my,,,,,,my,,,,,,my,my,ny,,,,ny,my,,,,,,my,,,,,,my,,,,,,my,my,ny,,,,ny,my,,,,,,my,,,,,,my,,,,,,my,my,ny,,,,ny,my,,,,,,my,,,,,,my,,,,,,my,my,ny,,,,ny,my,,,,,,my,,,,,,my,,,,,,my,my,ny,,,,ny,my,,,,,,my,,,,,,my,,,,,,my,my,ny,,,, ny,my,,,,,,my,,,,,,my,,,,,,my,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,t,,,,,,,,t,,,,,,,,t,,,,,,,,t,,,,,,,,t,,,,,,,,t,,,,,,,,t,1,1,1,1,1,1,1,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,1]; st = []; ts = [.1 ]; % Sample peiod of.1 seconds (1 khz) % Handle discete state updates, sample time hits, and majo time step equiements. function sys = mdlupdate(t,x,u) Vsd = u(1); Vsq = u(2); Isd = u(3); Isq = u(4); sys = The_Estimato(t,x,Vsd,Vsq,Isd,Isq); % Retun the output vecto fo the S-function function sys = mdloutputs(t,x,u) q = x(5) sys = q; 28