Comparative study between the conventional regulators and fuzzy logic controller: application on the induction machine

Transcription

1 International Journal of Sciences and Techniques of Automatic control & comuter engineering IJ-STA, Volume 1, N, December 007, Comarative study between the conventional regulators and fuzzy logic controller: alication on the induction machine Chaari Abdelkader 1, Soltani Moez, Gossa Moncef 1 1 Unité Commande Surveillance et sûreté des systèmes (C3S) Ecole Suérieure des Sciences et Techniques de Tunis (ESSTT), 5 av. Taha Hussein BP Tunis {Assil.chaari,Moncef.Gossa}@esstt.rnu.tn Unité Commande Surveillance et sûreté des systèmes (C3S) Ecole Suérieure des Sciences et Techniques de Tunis (ESSTT), 5 av. Taha Hussein BP Tunis soltani.moez@yahoo.fr Abstract. This aer rooses a new alication aroach for robust seed control induction machine using the equivalence modal rincile. Indeed, the majority of the classic methods of regulation are ineffective in front of the variation of the arameters and uncertainties which can affect the model of the asynchronous machine. In order to get rid of this draw backs, an association of the fuzzy logic regulators with a strategy of vector control, alied to the asynchronous machines, is resented in this aer. Indeed, this law of control lends itself well to the regulation and the controlling of rocess with variable arameters during the time. A comarative study of the indirect stator flux orientation control of induction motor drive by classic regulators (Integral Proortional (IP) and Proortional Integral (PI)) and fuzzy logic controller (FLC) is resented. The robustness between these two regulators was tested and validated under simulations with the resence of variations of the arameters of the machine, in articular the rotor time-constant and face of disturbances of load torque. In the objective of the real-time imlementation of the regulator FLC, a neuro-fuzzy version is also resented. Keywords. Induction machine, stator flux-oriented control, Proortional Integral, fuzzy logic. 1 Introduction AC motors, articularly the squirrel-cage induction motor (SCIM), enjoy several inherent advantages like simlicity, reliability, low cost and virtually maintenancefree electrical drives [1]. However, for high dynamic erformance industrial alications, their control remains a challenging roblem it is very comlex due to the couling of the hysical arameters and they exhibit significant non-linearties. This aer was recommended for ublication in revised form by the editor Staff. Edition: CPU of Tunis, Tunisia, ISSN:

2 Comarative study between the conventional regulators and fuzzy A. Chaari et al. 197 Several methods of control are used to control the induction motor among which the vector control or field orientation control that allows a decouling between the torque and the flux, in order to obtain an indeendent control of torque and the flux like DC motors. The overall erformance of field-oriented-controlled induction motor drive systems is directly related to the erformance of current control []. Therefore, decouling the control scheme is required by comensation of the couling effect between q-axis and d-axis current dynamics [3] [4]-[5]. The simulation results studied in [], [6] and [7] show that the conventional regulators do not make it ossible to solve roblems variations of the machine as well as the changes of load torque. Advanced strategies of control have drawn the attention of the control engineers in the last few years. A solution had been roosed on using a sohisticated method such as fuzzy logic controller (FLC) which lends itself very well to regulation and control to the understandable rocess using the aroriate conventional classic methods. Fuzzy logic control has excelled in dealing with systems that are comlex, ill-defined, non-linear, or time-varying [8]. FLC is relatively easy to imlement, as it usually needs no mathematical model of the controlled system. This is achieved by converting the linguistic control strategy of human exerience or exerts' knowledge into an automatic control strategy [9]. In this aer, we treat indirect stator flux orientation control (ISFOC) of induction machine with two tyes of regulators, conventional regulators (IP and PI) and the fuzzy logic controller (FLC). Indeed [10] to develo and aly this technique to a system of regulation of level in a refinery tank [11]. Our idea had enlarged the study to controlling of an asynchronous machine, which made it ossible to develo a technique of control in order to obtain satisfactory results in site of the arametric variations or other tyes of disturbances. Which conform the originality of this work. The aer is organized as follows. Section resents the induction motor model emloyed. Section3 describes the basic indirect stator field-oriented control (ISFOC) of induction motor. The synthesis of the conventional regulators is resented in Section 4. The fuzzy logic based roortional integral controller (FLPI) and using the modal equivalences is resented in Section 5. These controllers are evaluated and comared under simulations for a variety of oerating conditions of the drive machine in Section5. Section 6 resents a comarative study between the two strategies of control. Finally, Section 7 resents a version of controller based in both fuzzy and neural networks techniques and a conclusion will be resented at the end of this aer. Induction machine model Using the dynamic model of an induction machine as a controlled lant may be exressed in terms of the d-q axes comonents in a synchronous rotating frame resented in [1], the voltage equations in terms of stator current and rotor flux linkage

3 198 IJ-STA, Volume 1, N, December 007. can be restated in matrix form as [1]: d φ ds dt d φ qs φ ds dt 1 φ qs 0 0 v ds di σ L ds s [ A ] i ds + v qs dt 1 iqs 0 0 C r di qs σ L s ω r dt n d ω 0 0 r J dt (1) Where: [ A] 0 ωs Rs 0 0 ωs 0 0 Rs 0 1 ωr Tr + T s ωsl 0 σtl r s σls σtt r s ωr 1 Tr + Ts ωsl 0 σls σtrls σtt r s n n f iqs ids 0 0 J J J L L σ 1 ; ; ; ω ω ω r LL R R M T s r s T r sl s s r s r 3 Indirect field-oriented control of induction motor The main objective of the vector control of induction motors is, as in DC machines, to indeendently control the torque and the flux; this is done by using a d-q rotating reference frame synchronously with the stator flux sace vector [13]. Then, the d-axis is aligned with the stator flux vector and the stator flux linkage to be constant, which means [1]: dφqs φds φs ; φqs 0 () dt Thus by taking into account these new conditions and emloying () on the (1), the

4 dynamic model of an induction machine became: Rs( Ts + Tr) σ T (1 rts φ ds ) s V + ids σls ωsl iqs Tr Ts + Tr Tr (3) Rs( Ts + Tr) σ T (1 rts Vqs + ) iqs + σ Ls ωsl ids + ωφs Tr Ts + Tr It can be seen that the voltage equations include two terms of couling between d-axis and q-axis. These terms are considered as disturbances and are cancelled by using a decouling method that utilizes nonlinear feedback of the couling voltages [1]. Then, defined two new intermediate variables of decouling whose exressions are as follows: Vds1 Vds + Ed (4) V V + E With: Comarative study between the conventional regulators and fuzzy A. Chaari et al. 199 qs1 qs q φ E σ L ω i + s d s sl qs Tr Eq σlsωslids ωφs After modeling the induction motor, we will be interested in the synthesis of the conventional regulators using of PI controller and IP controller. (5) 4 Synthesis of the conventional regulators 4.1 Seed regulator tye IP The closed-loo seed transfer function is [1]: Ω ( ) 1 Ω *( ) kk τ kk 1 i 1 i With : k1 k /( f + k) Where k and ki denote roortional and integral gains of IP seed controller. (6)

5 00 IJ-STA, Volume 1, N, December 007. Since, the choice of the arameters of the regulator is selected according to the choice of the daming ratio (ξ ) and natural frequency ( ω ): o 1 ξ kk 1 i ωo τ 1 kk 1 i ωo (7) 4. Seed regulator PI The closed-loo seed transfer function is [1]: ( k + ki ) ( ) Ω ( ) Ω *( ) J f n k n ki (8) It can be seen that the motor seed is reresented by second order differential equation: + ξω + ω 0 (9) o o By identification, we obtain the following arameter: k ξωoj ki Jωo f (10) 4.3 Currents regulator with PI controller The transfer functions of the stator currents are obtained from the system equation (3) and by canceling E d and Eq by the feed forward comensation term [1]: ( ) V V E R T + T σ TT i s s r s r ds1 ds + d 1+ ds Tr Ts + Tr ( ) V V E R T + T σ TT i s s r s r qs1 qs + q 1+ qs Tr Ts + Tr (11)

6 Comarative study between the conventional regulators and fuzzy A. Chaari et al. 01 From (11), we obtain: Where k c kc Gd ( ) G q( ) 1 + (1) τ T r ( + ) R T T s s r c σ TT s r are is a gain and τ c T + T Then the closed-loo current transfer function is: BF ω k Hi ( ) This allows us to write finally: n i ξωn ωn kii 1 s r is a time constant. (13) k i k ii ξω nτ c 1 k ω n τ k c c c (14) 5 Fuzzy logic controller Historically, the first fuzzy logic regulator which aeared is that Mamdani. Then, other tyes of fuzzy controllers were defined; in articular that of Sugeno in 1985 [14]. When an exert understands qualitatively the rocess dynamics, it is ossible to secify a qualitative control strategy using linguistic rules, named fuzzy rules. A classical examle of those fuzzy control configurations is shown in fig. 1[15]. Fig. 1. Synotic diagram of a seed fuzzy controller We find in the inut and in the outut of the fuzzy controller gains named factors of scale which allow changing the sensitivity of the fuzzy regulator without changing its structure [16].

7 0 IJ-STA, Volume 1, N, December 007. In the continuation, we are interested in another form of fuzzy regulator using the concets of the rinciles modal equivalences. This regulator takes as a starting oint the arameters of the regulator roortional integral (PI). 5.1 Synthesis of a fuzzy regulator by alication of the rincile of modal equivalences The PI transfer function connecting the error ε to the control signal u exressed in z is: u ( z ) z k + k (15) I ε z z ( ) 1 Where k and k I, are the gains of the PI controller. From (15), we obtain the following equation: 1 1 ( )( 1 ) (1 ) ε( ) ε( ) u z z k z z + k z (16) If we note resectively δε the change of the error ε and δ u the change of the signal control u then (16) becomes: δ δε + ε (17) u k k I From fig. 4, it is noticed, that the outut of regulator PI controller is according to the error ε and of its changeδε. Then, it aears comletely natural to reserve the same inuts and outut for the fuzzy controller equivalent. I 5. Universes of discourse artition To avoid at most introducing nonlinearities into the fuzzy controller, a regular triangular artitioning is carried out on each universe of discourse [10]. The Fig. illustrates the artition of the fuzzy subsets on the universe of discourse associated to the error. To simlify the calculation, the distribution of the membershi functions will be suosed symmetric with regard to zero. What returns us to take ε δε δ. 0 0 u0 0 The distribution of the various membershi functions is obtained according to the following formula: Where: ε iδ a+ ε (18) i Δa is the difference between two consecutive membershi functions. 0

8 Comarative study between the conventional regulators and fuzzy A. Chaari et al. 03 δ u Fig.. Fuzzy artition of the universe of discourse associated to the error In the same way for the change of the errorδε and the change of the control signal, we obtain: δε j jδ b+ δε0 (19) u kδ c+ u δ k With Δb is the difference between two consecutive membershi functions relating to the change of the error and c is the difference between two consecutive membershi functions relating to the change of control signal. δ The fuzzy rule base model Once the inuts and outut of the fuzzy controller for whom we try to synthesize and their oerating range on any universe of discourse were defined. It remains now to determine the rules base which is, in the case of a controller of tye Mamdani, can be written: If ε is Ei and δε is de j then δ u est du f ( ij, ) (0) The alication of the rincile of modal equivalences leads to the following constraint: δu k δε + k εi (1) ( ) f i, j j I Substitution of (18) and (19) in (1), gives Δb Δa f ( i, j) jk + i ki Δc Δc ()

9 04 IJ-STA, Volume 1, N, December 007. If we ut: Δ a α k I Δ c α and β b and Δb β k, we obtain : Δc f ( i, j ) i α + j β (3) ( ) With: eing integers verifying BCD α, β 1. Finally, the distribution of the membershi functions is carried out in order to check the following relation: Δ c Δ c k I and (4) Δa α Δb The ratio α k β is a Δ roortionality factor between the ratio and a ki Δ b. In that, it thus joins in a method directed synthesis and so differs from existing works, directed analyze. The ractical imlementation of the synthesized fuzzy controller now requires to give an actual value to each arameter Δa, Δb, Δc, α and β. If no constraint is imosed on the distribution of the membershi functions, in other term if the finale number of rules is not limited, a simle choice consists in fixingα β 1. We obtain then the rules base summarized in the table. 1 [10]. k β Table. 1. The control rules for FLC We can notice that we have arrived at the table of inferencee that we will be able to obtain it by alying the method of Mac Vicar-Whelan [18]. For the adjustment of the fuzzy controller, we can lay on one of these arameters Δa, Δb and Δc or th e coefficients α and β so that the equality (4) is satisfied. Since the values of the regulatorr roortional integral regulator (K and K I ) are suosed to be known, it now remains to determine Δa, Δb, Δc, α and β.

10 Comarative study between the conventional regulators and fuzzy A. Chaari et al. 05 k For: ki Δ a Δ b 3 et Δ c and () 1 α 1, we find these various values of Δa, Δb and Δc: β 10 Δ a Δ b 0.5 et Δ c 0.03 ( ) Δ a Δ b 0.5 et Δ c 0.03 ( 3) Δ a Δ b 0.5 et Δ c 0.03 ( 4) The simulation results obtained with these various values are given in fig. 3. Fig. 3. Simulation results obtained for the various values of Δa, Δb and Δc It is noted that the use of 5 or 7 symbol for ε and δε allows obtaining a relatively satisfactory resonse. We also note that, when we vary the number of rule or the distance between the various symbols, the machine resonse also varies (resonse time and the rejection of the disturbance of the load). It thus is imossible to obtain a Mamdani controller of size restricted by simly exloiting the basic adjustments on Δa, Δband Δc. The other disadvantage is the significant number of rule used for the regulation, but the interest of the FLC is that only a small number of rules are necessary. In site of that, the synthesized fuzzy regulator imorted imrovement on the dynamics of the system esecially oint of view it is not so sensitive to the variation of load torque contrary to target PI regulator (see fig. 3).

11 06 IJ-STA, Volume 1, N, December Imlementation and results Simulations were erformed using the conventional controllers (PI and IP) and the fuzzy logic controllers (FLPI and FL using the rincile of modal equivalences (FLME)). These controllers are evaluated and comared under simulations using the Matlab-Simulink software. Next, the robustness of each controller against system arameters variations are evaluated by the settling time, overshoot and rejection of load disturbances. These values are calculated by alication of load disturbances at 5.5 sec, while the system arameters are varied from -80% to +80% of the nominal values. First for a variation from -80% to 80% of rotor resistance Rr, second for a variation from -80% to 80% and of moment of inertia J. The comarison results are indicated in Figs. 4 and 5. Next, these controllers are evaluated and comared under simulations for a variety of oerating conditions of the drive machine. In this study, we evaluate these three criteria: Settling time: it corresonds to time uts by the seed to reach 999, 98 rm for reference seed from to 1000 rm. Overshooting: it corresonds to the difference between the reference seed and real maximum seed to reach at the time of this level. Time of rejection of the disturbance: it corresonds to time that uts the seed to return in the reach [999,98, 1000,0] rm at the time of alication of load torque of 10Nm with reference seed of 1000 rm. According to the simulation results, the variation of the rotor resistance R r has a big influence on the settling time of the system using a PI and IP regulator, whether it is at the settling time, the time of rejection of disturbance and overshooting. In fig 4. (a), the FLPI is faster comared to the fuzzy FLME what esecially causes a overshooting during a big variation of Rr (figure 4. (c)). From fig 4. (b), we notice that the FLME has the best time of rejection of disturbance with regard to the other regulators. According to figure 5. (b), we observe a better settling time of PI regulator comared to the other regulators. But this imrovement is accomanied by a great overshooting towards the reference seed during a great variation of J (figure 5. (c)). According to the results simulations results, the fuzzy regulator obtained by the alication of the rincile of modal equivalences (FLME), with license to obtain the best erformances wished for imlementation in the loo of seed regulation. The fuzzy logic regulators (FLPI and FLME) are less sensitive to the moment inertia variation; since it reaches seed command with a overshooting much lower comared to PI and IP regulators (fig 5. (c)).

12 Comarative study between the conventional regulators and fuzzy A. Chaari et al. 07 Fig. 4. Reaction of the various seed regulators for a variation from -80% to +80 of R r Fig. 5. Reaction of the various seed regulators for a variation from -80% to +80 of J

13 08 IJ-STA, Volume 1, N, December 007. We as notice, as the fuzzy logic regulator obtained using the rincile of modal equivalence, kee the same erformances (settling time, time of rejection of disturbance and overshoot) for big variations of J. 7 Neuro-fuzzy regulator In our case, the fuzzy regulator obtained by the rincile of modal equivalence has a very imortant base of rules what makes its real time imlementation heavy and difficult. It is suitable to use instead a more sohisticated controller based on both fuzzy and neural networks techniques. In fact, this neuro-fuzzy controller is based on the training of a secified feedforward neural network which will relace overall stes of the Mamdani fuzzy standard controller [19]. The figure 7 shows the training bloc diagram stes of both controllers. Fig. 6. Neural-network learning. 7.1 Neural networks model structure We shall train a feedforward neural network of three layers (1 inut layer, 1 hidden layer and 1 outut layer). The inut layer has 6 inut neurons as shown in figure 7. While the outut layer exhibits the corresonding fuzzy controller outut. The hidden layer has 9 hidden neurons.

14 Comarative study between the conventional regulators and fuzzy A. Chaari et al. 09 Fig. 7. Neural-network structure We define the rediction error e(k) as: * * ( ) ω ( ) ωˆ ( ) e k sl k sl k (5) The neural network is then trained to reresent the non-linear function f [.]: * * * ωsl ( k + 1) f ωsl ( k) ωsl ( k 1 ) ε( k) ε( k 1) dε( k) dε( k 1) (6) The training rocedure consists in sequentially adjusting the network weight vectors, so that the mean squared error between the desired resonse (the values from the target vector) and the network outut is minimized [19],[0]: k ω * * sl sl (7) i 1 J ( k) ( ) ( ) k ωˆ k The activation functions are hyerbolic tangent sigmoid transfer function ψ (x) for the first layer and linear for the second layer, let: ψ ( x ) (1 + ex( * x )) 1 The neural-model for the fuzzy logic controller can be exressed as: * ω sl ( k 1 ) w [ ψ ( w1ϕ ( k) b1) ] T * * With: ( ) ( ) (8) b (9) ϕ ( k) ωsl k ωsl k 1 ε( k) ε( k 1) dε( k) dε( k 1) Where w1, w, b1 and b are the weights and biases matrices of the neural network. The neural network with the constant values for the weight vectors, obtained after the training, reresents the non-linear model of the FLC. Thus when a certain sto criteria is satisfied the training algorithm gives a set of otimal values of the weight vectors. The corresonding neural network training error is shown on the figure 8.

15 10 1 Sum square error 10 IJ-STA, Volume 1, N, December Error goal Fig. 8. Neural-network training 7. Simulation results After the training hase, we have relaced the fuzzy controller by the neural network based one in order to test its generalization behaviour. With this new controller we find out that its erformances are accetable in both tracking and regulation settings and it is also robust in resence of arametric variations. The related simulation results are given in Figure 9. Fig. 9. Resonse of the neural network regulator

16 8 Conclusion Comarative study between the conventional regulators and fuzzy A. Chaari et al. 11 In this aer, a stator field orientation of induction motor has been resented. The simulation results have enabled to us to judge the robustness of fuzzy controller comaring with the classical control. We notice that the obtained results by using regulators with fuzzy logic of oint of view insensitivity towards the variations of the external and internal disturbances (that is arametric variations) are very satisfactory. Which confirmed our aroach. It is to indicate also the good behavior of the neuro-fuzzy regulator during the existence of the arametric variations. References 1. Mouloud D., SID Ahmed A., Fuzzy and neurol control of induction motor, Int. J. Al. Math. Comut. Sci., vol.1, No.,. 1 33, 00.. M.Boussak, Jarray K A high-erformance sensorless indirect stator flux orientation control of induction motor drive, IEEE Trans. Ind. Electron., vol.53, no.1, , February Jung J., Nam K., A dynamic decouling control scheme for high seed oeration of induction motors, IEEE Trans. Ind. Electron., vol. 46, no. 1, , February Lin F. J., Wai R. J., Lin C. H., Liu D. C., Decouling stator-flux oriented induction motor drive with fuzzy neural network uncertainly observer, IEEE Trans. Ind. Electron., vol. 47, no., , Aril Suwankawin S., Sangwongwanich S., A seed sensorless IM drive with decouling control and stability analysis of seed estimation, IEEE Trans. Ind. Electron., vol. 49, no., , Aril Larabi A., Boucherit M., Commande robuste de vitesse ar orientation du flux rotorique de la machine asynchrone en utilisant des régulateurs flous, CD-JTEA, Hammamet, Tunisie, Mai Saravuth P., Issarachai N., Suwan R., Prinya T. Design of Otimal Fuzzy Logic based PI Controller using Multile Tabu Search Algorithm for Load Frequency Control, International Journal of Control, Automation, and Systems, vol. 4, no., , Aril Kim Chwee N., Yun L., Design of sohisticated fuzzy logic controllers using genetic algorithms, Proc. 3rd IEEE Int. Conf. on Fuzzy Systems, Orlando, Abdeldjebar H., B. Ismail Khalil, K. Mokhtar, Design of a Fuzzy Sliding Mode Controller by Genetic Algorithms for Induction Machine Seed Control, International Journal of Emerging Electric Power Systems, Volume 1, Issue,. 1-17, Sylvie G., Contrôle flou: de l interolation numérique au codage de l exertise, Habilitation à diriger des recherches en électronique, électrotechnique et automatique, Université de Savoie, Décembre Galichet S., Foulloy L., Chebre M., BeaucheneJ., Fuzzy logic control of a floating level in rafinery tank, Proc. Of the 3 rd IEEE conf. on Fuzzy System (FUZZ-IEEE 94), Orlando, USA, , June Jarray K., Laakam M., Sbita L. Robust seed control for stator flux oriented controlled induction motor drive», CD- JTEA, 1-, Hammamet, (Tunisie), Mai 004.

17 1 IJ-STA, Volume 1, N, December Lorenz R. D., Lawson D. B., A Simlified Aroach to Continuous On-Line Tuning of Field-Oriented Induction Machine Drives, IEEE Trans. Ind.Al, vol. 6, Issue.3, May- June Min T. C., Commande numérique de machines asynchrones ar logique floue, Thèse de doctorat de l Université Laval Québec, Alejandro A., Joseh A., A simlified version of Mamdani s fuzzy controller : the natural logic controller, IEEE transactions on fuzzy systems, vol,14, no.1,.16-30, February Baghli L., Contribution à la commande de la machine asynchrone, utilisation de la logique floue, des réseaux de neurones et des algorithmes génétiques, Thèse de doctorat de l Université Henri Poincaré, Nancy-I, Janvier Mary Margaret B., Self-learning redictive control using relational based fuzzy logic, Thesis of the University of Alberta, Mac Vicar-Whelan P.J., Fuzzy sets for man-machine interaction, Int. J. Man-Machine Studies, vol.8, , Yaïch. A., A. Chaari. A. and M. Kamoun., Real-time fuzzy and neuro-fuzzy regulation of current motor. JTEA, Nabeul, Tunisia, vol. 1, May Mircea L., Octavian P., A neural redictive controller for non-linear systems, Mathematics and Comuters in Simulation, Chaari A.,Soltani M., Gossa M., Comarative study between the conventional regulators and fuzzy logic controller, CD-STA-ACS-95, Monastir (Tunisie), November 007. Aendix vds, qs ids, iqs Φds, qs : d, q-axis stator voltage comonents. : d, q-axis stator current comonents. : d, q-axis stator flux comonents. Rs, Rr : Stator and rotor winding resistance. Ls, Lr : Stator and rotor self-inductance. M : Mutual inductance. n : Number of ole airs. : Differential oerator. ωs, ωr : Synchronous and rotor angular seed. Ce, Cr : Electromagnetic and load torque. J : Moment of inertia. f : Friction constant. σ : Total leakage constant BCD : The Biggest Common Divisor ω : Sli angular seed sl