Position Control of Induction Motors by Exact Feedback Linearization *

Transcription

1 BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume 8 No Sofia 008 Position Control of Induction Motors by Exact Feedback Linearization * Kostadin Kostov Stanislav Enev Farhat Fnaiech Assen Todorov Technical University - Sofia 8 Kl. Ohridski Str. 000 Sofia Bulgaria sta_enev@yahoo.com Université de Tunis ESSTT 5 Av. Taha Hussein 008 Tunis Tunisia Abstract: Inut-outut linearization aroach to induction motor osition control roblem is investigated. Linearizing transformation and control law for the full sixth-order induction motor model are resented. The effects of variations in mechanical arameter values on the linearized system are derived and discussed. Simulation results with the discrete-time version of the obtained overall controller are resented. Keywords: feedback linearization induction motor osition control.. Introduction Motion control alications are mostly based on DC and PM motors. Induction motors on the other hand are known with their ruggedness and reliability due to their simle construction much lower cost lack of commutating elements better ower to mass ratio comared to the DC motors which make them an attractive alternative in these alications. However the advantages above mentioned come with the very comlicated strongly couled nonlinear dynamics which requires utting in lace sohisticated control algorithms in order to obtain good erformance. A good overview of the state of the art in electric servo drives can be found in [5]. Different aroaches to induction motors osition control are available in the scientific literature mostly based on assivity theory and sliding-mode designs for the current-command mode with different adative features in the control scheme * This work is suorted by the R&D Sector at the Technical University Sofia PhD Research Project contract No ПД

2 [0 3 4] in order to comensate for mechanical arameter variations. In [] a feedback linearization-like technique is used to decoule rotor osition and flux. An adative osition tracking control algorithm is resented in [9]. The alication of inut-outut feedback linearizing techniques to induction motor control design [ ] enables the exact linearization of rotor seed/osition and flux dynamics. Unlike the case of field-oriented control where only asymtotic decouling is achieved i.e. when the flux is constant here this restriction is eliminated and both system oututs are comletely decouled. This feature enables the otimization of a motor torque [] without degrading mechanical outut regulation which makes feedback linearization an attractive aroach to induction motor osition control. Paers [ 6 7] reort exerimental results of an induction motor osition control system based on inut-outut linearization of the current-fed field-oriented model showing good osition tracking and ability to indeendently control the flux magnitude. In this aer inut-outut linearization aroach to induction motor osition control roblem is investigated. Linearizing transformation and control law for the full sixth-order induction motor model are resented. The effects of variations in mechanical arameter values on the linearized system are derived and discussed. Outer-loo controllers are roosed and simulation results with the discrete-time version of the obtained overall controller are resented.. Dynamic modeling of the induction motor The induction motor considered here is a three-hase stator three-hase short circuited rotor machine. The following considerations are valid for the case of a squirrel-cage rotor since it is equivalent to a three-hase short-circuited one through a simle transformation. The common assumtions are adoted for the modeling i.e. symmetrical construction sinusoidal distribution of the field in the air-ga and linearity of magnetic circuits. Remark: The rotor flux magnitude can be ket away from the saturation zone by an aroriate control action thus forcing the assumtion for linear magnetic circuits. Writing the equations describing the motor dynamic behavior in the two-hase stator-fixed α β frame and eliminating stator fluxes and rotor currents the following equivalent two-hase model is obtained: () ω = µ ψ ψ ω ( RαiSβ RβiSα) cj J L ψ = ηψ n ωψ + ηmi Rα Rα Rβ Sα ψ = ηψ + n ωψ + ηmi Rβ Rβ Rα Sβ i i n l u i i n l u θ = ω Sα = γ Sα + ηζψ Rα + ζ ωψ Rβ + ( σ S) Sα Sβ = γ Sβ + ηζψ Rβ ζ ωψ Rα + ( σ S) Sβ 35

3 where: i i Sα Sβ are stator currents ψ ψ rotor fluxes ω is rotor seed Rα θ rotor osition us u voltage inuts to the motor α Sβ l S( R) hase stator (rotor) winding inductances r S( R) hase stator (rotor) winding resistances m0 = /3m mutual inductance n number of ole-airs J rotor moment of inertia; c viscous friction coefficient η = r / l Rβ R R σ = ( ll R S m)/( ll R S) γ = ( lr R S + mlrη)/( σll R S) µ = nm /( lj R ) ζ = m/( σlrls ) L load torque. The comlete derivation of the model can be found in [ 3]. The induction motor model () is ut in the following form: x = f( x) + gu + g u + f () x = [ x x x x x x ] = [ ωψ ψ i i θ] u = u u = u T T Rα Rβ Sα Sβ Sα Sβ f( x) µ ( xx 5 xx 3 4) cj x J L ηx nxx3 + ηmx4 ηx + n x x + ηmx γx5 + ηζ x3 ζnxx x 3 5 = γx4 + ηζ x + ζnxx g = ( σls ) 0 0 g = 0 ( σls ) 0 f J L 0 0 = Inut-outut feedback linearization of the induction motor Basically feedback linearization consists of alying a nonlinear transformation on system variables i.e. exressing them in a new suitable coordinate system which will enable the introduction of a feedback so that an inut-outut or state linearization in the new coordinates is achieved. Theoretical foundations and a systematic rocedure for finding these can be found in [4 5 ]. For the induction motor case by choosing the outut functions as the rotor osition and flux square resectively y = h ( x) = x (3) 6 and alying the following transformation: y = h ( x) = x + x 3 36

4 (4) ξ = h ( x) = x 6 ξ = Lh( x) = x f ξ = L h ( x) = µ ( x x x x ) cj x 3 f ξ = h ( x) = x + x 3 ξ = Lh( x) = η( x + x) + ηmxx ( + xx) f x 3 χ = arctg x where Lh denotes the Lie derivative of the scalar function h with resect to (or f h along) the vector field f and reresents a scalar function defined by Lh f = f x n n (iteratively L h= L L h) the system () is transformed in the following normal form: (5) f f f ξ = Lh f ( x) ξ = Lf h( x) + Lf Lfh( x) 3 ξ3 = Lf h( x) + Lg L f h( x) u + Lg L f h( x) u + Lf Lf h( x) ξ = Lh f ( x) ξ = Lf h( x) + Lg L fh( x) u + Lg L fh( x) u χ = L χ ( x). f The transformation (4) is basically the one given in [8] only osition coordinate is added. The corresondant Lie derivatives are given by the following exressions: (6) L h ( x) = µ ( x x x x )( cj + η + γ) + 3 f cj x µ nx( xx + xx) µξnx( x + x) L h ( x) = η ( + mξ)( x + x ) + η m ( x + x ) + f ηmn x ( x x x x ) ηm(3 η + γ)( x x + x x ) L L h ( x) = µ x /( σ l ) L L h ( x) = µ x /( σl ) L L h ( x) = g f 3 S g f = ηmx /( σl ) L L h ( x) = ηmx /( σl ) S g f 3 S L L h ( x) = J L L h ( x) = cj L χ ( x) = f f L f f L f S g f xx xx r = nx ηm = n ξ + ( Jξ + cξ ) R 3 x + x3 n ξ Choosing the linearizing control law in the form 37

5 (7) 3 u v Lf h( x) Ax ( ) u = v Lf h( x) the linearized system is ut in the following form: ξ = ξ ξ = ξ3 J L (8) ξ3 = v + cj L ξ = ξ ξ = v rr (9) χ = n ξ + ( J ξ 3 + c ξ). n ξ (0) The matrix A( x ) called decouling is given by L L Ax ( ) = h ( x) L L h ( x) = x x g f g f µ 3 µ L g L 3 fh( x) Lg L fh( x) σls ηmx ηmx As seen from (0) the decouling matrix is invertible only when the singularity condition is verified i.e. when flux magnitude is not zero: () det( Ax ( )) = η mµ ( x + x3) / ( σl S ) 0. By choosing the control law as (7) the dynamics of the original nonlinear system is decomosed into two arts: a linear inut-outut ma given by (8) and a nonlinear unobservable through the outut internal art (9). The stability roerties of this internal dynamics a general limitation of feedback linearization control are not an issue in this case since χ reresents an angle by definition. In Fig. the resulting linear decouled inut-outut system is shown. It is seen that the roblem of controlling rotor osition and flux is rendered to controlling a trile integrator for the osition loo and a double integrator for the flux loo.. 38 Fig.. Inut-outut linearized sysytem

6 To study the effects of variations in mechanical arameter values let us assume uncertainties on these arameters formalized in the following form: J = kj c = c+ c with J = J. System () can be rewritten as () x = f( x) + f( x) + f( x) + gu + gu + f where T f ( x) = [( k )( µ ( x x x x ) cjx ) ] (3) (4) T ( ) = [ ] T f x kj cx f ( x) = [ kj L ]. Exression (5) takes the following form with variables defined as in (4): ξ = Lh f ( x) ξ = Lf h( x) + Lf L fh( x) + Lf L fh( x) + Lf Lfh( x) 3 ξ = L h ( x) + L L h ( x) + L L h ( x) + 3 f f f f f + L L h ( x) u + L L h ( x) u + L L h ( x) g f g f f f ξ = Lh f ( x) ξ = Lf h( x) + Lg L fh( x) u + Lg L fh( x) u χ = Lf χ( x). The resective Lie derivatives are given by exressions L L h ( x) = ( k )( µ ( x x x x ) cjx ) = ( k ) L h ( x) = ( k ) ξ f f f 3 f f ( ) = = f ( ) = ξ f f ( ) = ( )( µ ( 5 3 4) ) = = cj ( k ) L f h( x) = cj ( k ) ξ3 f f ( ) = ( ) = cl fh( x) = kj c c ξ f f ( ) = L f f ( ) = L. L L h x kj cx kj cl h x kj c L L h x cj k x x x x cjx L L h x cj kj cx kj c L L h x kj L L h x kj c Alying the same linearizing control law (7) the system is ut in the following form: ξ = ξ ξ = k ξ3 kj c ξ kjl ξ3 = cj( k ) ξ3 + kj c c ξ + kj cl + v (5) ξ = ξ ξ = v r χ = n ξ + ( J ξ + c ξ ). R 3 n ξ By comaring (5) with (8) and (9) it is seen that the internal feedback connections in the osition loo aear due to the mechanical arameter 39

7 uncertainties and the double integration is lost. Also additional disturbances enter this subsystem. The flux loo is not affected by these variations neither couling between the subsystems is restored. Fig. visualizes the resulting erturbed linear decouled inut-outut system. Fig.. Perturbed inut-outut linearized sysytem 4. Outer control loos and flux observer The block diagram of the roosed control scheme is given in Fig. 3. The osition control loo is realized by using a cascade rincile and consists of a fast inner seed control loo using a PID controller also reducing the effects of mechanical arameter variations and osition feedback control loo with a P controller. The flux control loo is realized by using a PID controller determining fast outut resonse thus giving the ossibility of setting a desired dynamics of the flux value by an adequate re-filter. Since generally the flux vector comonents are not measurable a simle flux observer reresenting a simulation of the second and third equations in () is used to estimate their values. Assuming ω = const kts < t < ( k + ) TS and is* = const kts < t < ( k + ) TS a discrete-time version of the oen-loo simulator is given by: ψ ( k + ) ψ ( k) i ( k) (6) Rα = A ( ) R ( ) S d k α + Bd k α ψ Rβ( k + ) ψ Rβ( k) isβ( k) where cos( nω( k) TS) sin( nω( k) T ) ηt S S Ad ( k) = e sin( nω( k) TS) cos( nω( k) TS) Bd( k) = ηmad ( k)( Ad( k) I). 40

8 Fig. 3. Overall control system 5. Simulation setu and results The motor arameters used for simulation uroses chosen as in [6] are given below: r = 0.3 Ω r = 3 Ω l =.05 H l =.33 H m = H S R S J = N.m.s c = N.m.s n =. The resective gain values for both PID controllers in the osition and flux control loos resectively are set to: 3 5 K P = 4.0 K I = 48.0 K D = 300. The gain value of the P controller is set to K = 00. R Fig. 4. Transient resonses 4

9 In order to assess more realistically the feasibility of the roosed control system both linearizing and outer loos as well as the flux observer have been simulated with their discrete-time realizations with samling eriod T S = s. Also a delay of one samling eriod is added to control inuts. The transient resonses of the system are shown in Fig. 4. First the flux is regulated to its nominal value and then the motor is required to attain a osition of 90 rads in one second along a half-wave sinusoidal seed trajectory. At t = 0.5 s a N.m load torque unknown to the controller is alied. The uncertainties on the mechanical arameter values are set to c= 0.5cand k = 0.66 i. e. J =.5J. 6. Conclusion In this aer an induction motor osition control system based on inut-outut feedback linearization is roosed. Simulation results show the system ability to track the desired osition trajectory. The main advantage consists in the exact decouling between mechanical outut and flux obtained as a result of the linearization a result that can be used for torque otimization. The influence of ossible uncertainties on mechanical arameters in the setu of this aroach is derived. It is shown that both subsystems remain indeendent even in their resence though some internal feedback connections and additional disturbances aear in the osition control subsystem. However due to errors introduced by the discrete-time realization of the flux observer and the linearizing control loo certain couling is re-instated as the flux value is affected by the rotor seed which is seen on the transients. Fast outer control loos with high-gain PID controllers for both subsystems are roosed in order to enable setting the dynamics of their resonses as desired by using refilters and to rovide robustness against arameter variations. Simulations have shown that the control system can track reference ositions achieving seeds much greater than the nominal seed of the motor though in those cases the additional delay accounting for control value calculation becomes critical. Currently an exerimental setu is ut in lace to confirm the feasibility and racticability of the roosed control system. References. Leonhard W. Control of Electrical Drives. nd Edition. Berlin Sringer C h i a s s o n J. Modeling and High-Performance Control of Electric Machines. Hoboken New Jersey John Wiley & Sons Inc D e l a l e a u E. J. P. L o u i s R. O r t e g a. Modeling and Control of Induction Motors. Int. J. Al. Math. Comut. Sci. Vol. 00 No K h a l i l H. Nonlinear Systems. nd Edition. Englewood Cliffs Prentice Hall S l o t i n e J. J. E. W. L i. Alied Nonlinear Control. Englewood Cliffs NJ Prentice Hall B o d s o n M. J. C h i a s s o n. Differential-Geometric Methods for Control of Electric Motors. Int. J. Robust Nonlinear Control

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