Sliding mode observers for sensorless control of current-fed induction motors

Transcription

1 2 American Control Conference on O'Farrell Street, San Francisco, CA, USA une 29 - uly, 2 Sliding mode oservers for sensorless control of current-fed induction motors Daniele Bullo, Antonella Ferrara and Matteo Ruagotti Astract This paper presents the use of a higher order sliding mode scheme for sensorless control of induction motors. The second order su-optimal control law is ased on a reduced-order model of the motor, and produces the references for a current regulated PWM inverter. A nonlinear oserver structure, ased on Lyapunov theory and on different sliding mode techniques (first order, su-optimal and super-twisting generates the velocity and rotor flux estimates necessary for the controller, ased only on the measurements of phase voltages and currents. The proposed control scheme and oservers are tested on an experimental setup, showing a satisfactory performance. I. INTRODUCTION Most of the research in induction motors control during the last years has een on sensorless solutions. In such control schemes, the motor is controlled without measuring its speed, making the system less expensive and unaffected y sensor failures, maintaining at the same time the advantages that the induction motor has with respect to other electric machines []. Nonetheless, controlling an induction motor presents some difficulties, related with the strong nonlinearities and couplings of the system, and with the fact that some parameters can vary in a wide range (e.g. the rotor resistance varies with temperature up to 2% of the nominal value. Also, in a sensorless scheme it is necessary to oserve oth the rotor flux and the velocity, together with the unknown parameters, using only measurements of the electrical variales (phase voltages and currents. For all these reasons, control and oservation methodologies with strong roustness properties are required [2]. During the last years, many proposals for sensorless schemes have appeared in the literature (see, for instance [3], [4], [5], and the references therein. Some of the proposed strategies (e.g. [6], [7], [8], and [9] rely on sliding mode control, which guarantees roustness with respect to matched disturances and parameter variations []. The main drawack associated with the use of sliding mode techniques is the generation of the so-called chattering phenomenon [], [2], which consists in a high-frequency oscillation of the controlled or oserved variales when the sliding variales [] are steered to zero. To reduce the chattering, higher order sliding mode techniques (like the super-twisting and Daniele Bullo is with Terna S.p.A., Via Galileo Galilei 8, 26 Pero (MI, Italy. daniele.ullo@terna.it. Antonella Ferrara is with the Department of Computer Engineering and Systems Science, University of Pavia, Via Ferrata, 27 Pavia, Italy. antonella.ferrara@unipv.it Matteo Ruagotti is with the Department of Mechanical and Structural Engineering, University of Trento, Via Mesiano 77, 3823 Trento, Italy. matteo.ruagotti@ing.unitn.it. su-optimal algorithms have een introduced in recent years [3]. In this work, the sensorless control scheme (introduced in [4] for induction motors with speed measurement consists in a speed and rotor flux controller ased on the so-called su-optimal control law [5]. The rotor flux and speed, together with a parameter that takes into account the rotor resistance, are estimated using a nonlinear oserver which can e ased on first order sliding mode, similarly to [6], or on the super-twisting algorithm, like in [7], or on the suoptimal algorithm. The control scheme and the oservers are presented, and the staility properties of the whole system are discussed. Finally, some experimental results confirm the convergence properties of the sliding mode ased oservers. The contriution of this paper consists mainly in the proposal of the su-optimal algorithm as an alternative method for the design of speed and rotor flux oservers, comparing its performance with other similar approaches presented in the literature. The paper is organized as follows: Section II introduces the full-order and reduced-order models of the electric machine, together with the control and oservation ojectives. The control strategy is descried in Section III, while Section IV introduces the rotor flux and velocity oserver structure. Section V analyzes the staility properties of the overall sensorless control scheme, while some experimental results on the oserver performances are reported in Section VI. Finally, some conclusions are gathered in Section VII. II. PROBLEM FORMULATION In the fixed reference frame a-, the fifth order induction motor model is defined y [] ω = µ(ψ a i ψ i a K f ω Γ l ψ a = αψ a n p ωψ +Mαi a ψ = αψ +n p ωψ a +Mαi ( i a = γi a +αβψ a +βn p ωψ + ua i = γi +αβψ βn p ωψ a + u where α = R r L r, σ = M2 L s L r, β = M L r, γ = M2 R r σl 2 r L + s R s, µ = n pm L r, ω is the rotor speed, ψ a and ψ are the rotor fluxes, u a and u are the stator voltages, Γ l is the load torque (which is considered known in this work, n p is the numer of pole-pairs, is the moment of inertia, K f is the friction coefficient, R r, R s, L r, L s, and M are the rotor and stator windings resistances and inductances, and the mutual inductance, respectively //$26. 2 AACC 763

2 Assuming ψ 2 a +ψ 2 >, defining the new inputs v d = cos(ρu a +sin(ρu (2 v q = sin(ρu a +cos(ρu and applying the nonlinear coordinate transformation ω = ω ρ = atan( ψ i d i q ψ a = cos(ρi a +sin(ρi = sin(ρi a +cos(ρi = ψa 2 +ψ 2 to (, one otains the direct-quadrature model (d-q ω = µ i q K f ω Γ l = α +Mαi d ρ = n p ω +Mα i q i d = γi d + Mα σl r L s +n p ωi q +Mα i2 q + v d i q = γi q Mn pω σl r L s n p ωi d Mαi di q (3 + v q (4 If the currents are considered as inputs, the model of the motor can e simplified, reducing its order. In this way, ( ecomes ω = µ(ψ a i ψ i a K f ω Γ l ψ a = αψ a n p ωψ +Mαi a (5 ψ = αψ +n p ωψ a +Mαi where i a and i are now considered as inputs. Analogously, (4 can e replaced y ω = µ i q K f ω Γ l = α +Mαi d (6 ρ = n p ω +Mα iq The standard approach to simplify the fifth order model (e.g. (4 is to use an inner current loop. In this paper, a standard current regulated pulse width modulation (CRPWM inverter is used. The control is performed using relays, which compare the currents of each phase (i, i 2, i 3 with the reference currents (i, i 2, i 3 generated y the speed and rotor flux controllers (which generate two control variales i q and i d, and force the voltages of the phases (v, v 2, v 3 to e one of the two extreme values ± V. The commutation of the power electronics switches takes place when the difference etween the reference current and the actual one exceeds a tolerance (hysteresis value. The reference phase current is transformed from the a- or d-q axes to the -2-3 reference system of the three phases and vice-versa using the Park transformation matrix []. Using this voltage generation system, the current dynamics are neglected with respect to the flux and velocity ones, so one can consider i i, i 2 i 2, i 3 i 3, and then i d i d and i q i q. The ojective of the designed control strategy is to make the induction motor follow the generated references of rotor flux and velocity, defined as ψ d and ω r. The controller is designed in the d q reference system on system (6. Only the measurements of phase currents (output variale and voltages (input variale are availale. Then, the oserver, designed in the a reference system on system (, must provide a good estimate of ψ a, ψ, ω and α. III. THE PROPOSED SPEED AND FLUX CONTROLLER The design of the controller is ased on the method descried in [4] for a control scheme with oth velocity and rotor flux measurements. The reduced third-order model in the d-q reference frame reported in (6 is considered, ecause, in this reference frame, a natural decoupling is realized, i.e. the control variale i d acts on the flux, while the control variale i q acts on the velocity ω. Then, the following sliding variales are defined s ω = ω ω (7 s ψ = ψ d (8 where ω and ψd (generated such that their first and second order time derivatives ω, ω,, are ounded are the desired values of ω and, respectively. To solve the control prolem, the idea is to suitaly design i q and i d as discontinuous control variales, the values of i d and i q eing kept ounded, so that i d < I d and i q < I q, where I d and I q are positive values. More precisely, the method proposed in [4] relies on the so-called modified su-optimal second order sliding mode approach (the theoretical development of which has een presented in [5]. To this end, the values of f ω, f ψ, d ω, h ω, d ψ and h ψ must e ounded, that is, the oundedness of the variales on which they depend must e assured. A multi-input auxiliary control prolem can then e formulated. The control ojective is to steer to zero in a finite time the state variales s ω, s ψ and their first order derivatives in spite of the presence of the uncertain terms. This can e accomplished y designing the auxiliary control laws i q = sat [ Iq;+I q] i q (t + } t t i q i q = W q sign ( s ω 2 s (9 ω M i d = sat [ Id ;+I d ] i d (t + } t t i d i d = W d sign ( s ψ 2 s ( ψ M where i q (t and i q (t are the values of the currents i q and i d at the initial time instant t, W q and W d are positive constant values, s ωm and s ψm are the last extremal values of s ω and s ψ, respectively. The terms sat [ Iq ;+I q ] and sat [ Id ;+I d ] mean that the values of i q and i d can increase until i q I q and i d I d, then their values remain constant until a new switching in the sign of i q (or i q makes their asolute value decrease. IV. THE NONLINEAR OBSERVER Until this point, the whole state of the system was considered accessile for measurement. Actually, the rotor flux cannot e easily measured in practical applications, the coefficient α is unknown, and the velocity sensor can e unavailale in some applications. Then, a particular nonlinear oserver is introduced, in order to otain estimates of ψ a, ψ, ω and α. The complete scheme of the control system 764

3 ˆ ˆ is shown in Fig.. The whole structure of the oserver is ased on a sliding mode current oserver, defined as î a = β ˆψa + (u a R s i a +χ a ( î = β ˆψ + (u R s i +χ where ˆψ a, ˆψ, î a, and î are the oserved components of the rotor flux and stator currents, respectively. The terms χ a and χ can e defined, depending on the type of approach adopted, among First Order Sliding Mode (SM, Super Twisting and Su Optimal, as follows χ a = K a sign(ĩ a χa = z a k λa ĩ a 2 sign(ĩ a ż a = k αa sign(ĩ a χa = χ a χ a = µ a sign(ĩ a 2ĩa M First Order SM Super Twisting Su Optimal (2 χ = K sign(ĩ First Order SM χa = z k λ ĩ 2 sign(ĩ Super Twisting ż = k α sign(ĩ χ = χ χ = µ sign(ĩ Su Optimal 2ĩ M (3 where K a, K, k αa, k α, k λa, k λ, µ a, µ are positive constants. The variales ĩ a = î a i a and ĩ = î i are the estimation errors of the currents, while the extremal values i am and i M are defined according to the su-optimal algorithm in [8]. Now, take into account the dynamics of the estimation errors ĩ a = β ψa +χ a (4 ĩ = β ψ +χ If the first order SM oserver is used, it can e proved that, according to [], a sufficient large value of K a and K can steer the estimation error of the currents to zero in a finite time. As for the Super-Twisting strategy, choosing the terms k αa, k α, k λa, k λ large enough, one has that the convergence to zero of k αa, k α, k λa, k λ in a finite time is guaranteed [3]. As for the last oserver, the one designed according to a Su Optimal approach, it guarantees the convergence if suitale values of µ a and µ are chosen in order to dominate the system uncertainties (that is, the rotor flux errors. Of course, ĩ a and ĩ can e otained from measurements, while the estimation errors on the flux components is now assumed to e ounded, together with their first derivatives. This assumption will e verified in the sequel. Once guaranteed that the current estimation errors go to zero in a finite time (which is otained with all these sliding mode strategies a nonlinear oserver ased on the design of a suitale Lyapunov function design can e introduced, analogously to [2]. CRPWM INVERTER i i2 i 3 23 d-q id iq SLIDING MODE CONTROLLERS Fig.. d ˆd i i 2 i 3 COORDINATE CHANGE ˆa ˆ i a, 23 a- NONLINEAR OBSERVER INDUCTION MOTOR v a, Scheme of the induction motor control system Two auxiliary quantities are introduced as and their dynamics is LOAD TORQUE z a = ĩ a +β ψ a z = ĩ +β ψ (5 ż a = χ a (6 ż = χ and then, as in [6], it is possile to state that ψ a = za ĩ a β ψ = z ĩ β (7 to reconstruct the flux error estimates. Define now the nonlinear oserver as ˆψ a = ˆαˆψ a ˆωˆψ +Mˆαi a +f ψa ˆψ = ˆαˆψ ˆωˆψ a +Mˆαi +f ψ ˆω = M L r (ˆψ a i ˆψ i a K f ˆω Γ l ˆα = f α +f ω Choosing the functions f ψa, f ψ, f ω, f α as follows f ψa f ψ (8 = k ψ ψa = k ψ ψ f ω = γ ω ( ψa ˆψ ψ ˆψa M L ( ψa r i ψ i a f α = γ a ( ψa (ˆψa Mi a + ψ (ˆψ Mi (9 where k ψ, γ ω and γ α are strictly positive constants, it can e proven analogously to [6] that the asymptotic convergence of the estimation errors to zero is guaranteed. This is done introducing the Lyapunov function candidate V = 2 ( ψ 2a + ψ 2 + ω2 γ ω + α2 γ α (2 765

4 and proving that V = ψ a ψa + ψ ψ + ω ω = (α+k ψ ( ψ2 a + ψ 2 < γ ω + α α γ α K f ω 2 γ ω V. ANALYSIS OF THE SENSORLESS SCHEME (2 Now the staility properties of the overall sensorless scheme will e analyzed. The sliding mode ased current oserver, independently from the used technique, converges if the variales ˆψ a, ˆψ, ˆψ a and ˆψ are ounded, and this happens ecause it has een proved that the dynamics of the oservation errors ψ a. ψ, ω and α is asymptotically stale, and then the values of their time derivatives are ounded. Based on the same arguments used in [9], the persistency of excitation can e guaranteed for variales ˆα and ˆω, making the origin a gloally exponentially stale equilirium point for the estimation errors ψ a. ψ, ω and α. In particular, as for the estimation of α, it is possile to write that [ ] [ ] ψa ˆα = γ α ˆψa Mi a ˆψ Mi [ ] ψ (22 ψa = γ α Γ(t ψ and persistency of excitation is given if t+t t Γ(τΓ T (τdτ (23 is positive definite for any t and T >, which is exactly our case. Analogous considerations can e done for the velocity estimation. When using the oserved values instead of the actual ones in the control system descried in Section III, the sliding variales (7 and (8 now ecome s ω = ˆω ω (24 s ψ = ˆ ψ d (25 where ˆ = ˆψ2 a + ˆψ 2. Considering that the oservation errors ψ a. ψ, ω, α and their derivatives are ounded, sufficiently large values of I d and I q can guarantee the convergence to zero in a finite time of the sliding variales (24 and (25. Note that these variales will coincide asymptotically with (7 and (8 for the convergence properties of the nonlinear oserver. VI. EXPERIMENTAL RESULTS Though the proposed control scheme was validated in simulation also, we show here directly some results otained on an experimental setup. An induction motor with a nominal power of.75 kw has een directly coupled with a direct current machine, which acts as a time-varying load. The rotor speed is measured using an incremental optic encoder, while the voltages and currents of the different phases are measured y Hall-type sensors. The acquired analog signals are converted to 2-it digital with a sampling time of 4 s. A Pentium IV personal computer equipped with Simulink Real-Time Workshop is used to implement the proposed schemes. The simulation results otained in [4] were ased on the data of this motor, and the results relative to the tracking of the velocity reference here otained are not very different from them. In the sequel the performance of the oserver, which of course was not present in [4], is shown i a estimation error [A] i estimation error [A] Fig. 2. Estimation errors of the currents ĩ a and ĩ for the first order sliding mode scheme with equivalent control The first implemented oserver made use of the first order sliding mode control strategy, the signal generated y which has een filtered, according to the equivalent control method introduced in []. In Fig. 2 the time evolutions of the current estimation errors is shown, while Fig. 3 shows the time evolutions of the estimated rotor flux ˆ, the velocity estimate ˆω and the velocity estimation error ω. The supertwisting and the su-optimal controllers have also een implemented, and analogous results to those shown for the first order controller are depicted in Figs. 4, 5, 6 and 7. All the methods seem to have similar performances, in particular the su-optimal controller here proposed can significantly reduce the chattering amplitude on the current estimation errors. VII. CONCLUSIONS In this work a sensorless control scheme for induction motors is proposed. The flux and velocity controller are ased on the su-optimal second order sliding mode strategy, acting on a reduced order of the system. While the nonlinear oserver, ased on the full order model of the motor, exploit sliding mode techniques: all of them are shown to give appreciale results in order to estimate the rotor flux and the velocity. The proposed scheme is tested on an experimental setup, confirming its convergence properties. 766

5 estimation [W] estimation [W] ω estimation [rad/s] ω estimation [rad/s] velocity tracking error [rad/s] velocity tracking error [rad/s] Fig. 3. Estimated rotor flux ˆ, velocity estimate ˆω and velocity estimation error ω for the first order sliding mode scheme with equivalent control Fig. 5. Estimated rotor flux ˆ, velocity estimate ˆω and velocity estimation error ω for the super twisting scheme i a estimation error [A] i a estimation error [A] i estimation error [A] i estimation error [A] Fig. 4. scheme Estimation errors of the currents ĩ a and ĩ for the super twisting Fig. 6. scheme Estimation errors of the currents ĩ a and ĩ for the su-optimal 767

6 estimation [W] ω estimation [rad/s] velocity tracking error [rad/s] Fig. 7. Estimated rotor flux ˆ, velocity estimate ˆω and velocity estimation error ω for the su-optimal scheme REFERENCES [] W. Leonard, Control of electrical drives, 3rd ed. Berlin: Springer, 2. [2] R. Marino, S. Peresada, and P. Tomei, Output feedack control of current-fed induction motors withunknown rotor resistance, IEEE Transactions on Control Systems Technology, vol. 4, no. 4, pp , 996. [3]. Holtz, Sensorless control of induction motor drives, Proceedings of the IEEE, vol. 9, no. 8, pp , 22. [4] S. Gadoue, D. Giaouris, and. Finch, Sensorless control of induction motor drives at very low and zero speeds using neural network flux oservers, IEEE Transactions on Industrial Electronics, vol. 56, no. 8, pp , 29. [5] H. K. Khalil, E. G. Strangas, and S. urkovic, Speed oserver and reduced nonlinear model for sensorless control of induction motors, IEEE Transactions on Control Systems Technology, vol. 7, no. 2, pp , 29. [6] Z. Yan, C. in, and V. Utkin, Sensorless sliding-mode control of induction motors, IEEE Transactions on Industrial Electronics, vol. 47, no. 6, pp , 2. [7] A. Derdiyok, M. Guven, H. Rehman, N. Inanc, and L. Xu, Design and implementation of a new sliding-mode oserver for speed-sensorless control of induction machine, IEEE Transactions on Industrial Electronics, vol. 49, no. 5, pp , 22. [8] G. Bartolini, A. Damiano, G. Gatto, I. Marongiu, A. Pisano, and E. Usai, Roust speed and torque estimation in electrical drives y second-order sliding modes, IEEE Transactions on Control Systems Technology, vol., no., pp. 84 9, 23. [9] C. Lascu, I. Boldea, and F. Blaajerg, Direct torque control of sensorless induction motor drives: a sliding-mode approach, IEEE Transactions on Industry Applications, vol. 4, no. 2, pp , 24. [] V. Utkin, Sliding Mode in Control and Optimization. Berlin, Germany: Springer-Verlag, 992. [] I. Boiko and L. Fridman, Analysis of chattering in continuous slidingmode controllers, IEEE Transactions on Automatic Control, vol. 5, no. 9, pp , 25. [2] A. Levant, Chattering analysis, in Proceedings of the European Control Conference, Kos, Greece, uly 27. [3] L. Fridman and A. Levant, Higher order sliding modes, in Sliding mode control in engineering, ser. Control Engineering Series, W. Perruquetti and. P. Barot, Eds. London: CRC Press, 22. [4] A. Ferrara and M. Ruagotti, A su-optimal second order sliding mode controller for current-fed induction motors, in Proc. American Control Conference, St. Louis, MI, US, une 29. [5], A su-optimal second order sliding mode controller for systems with saturating actuators, IEEE Transactions on Automatic Control, vol. 54, no. 5, pp , 29. [6] C. Aurora, A. Ferrara, and M. Foscolo, A gloally stale adaptive second order sliding mode control scheme for sensorless current-fed induction motors, in Proceedings of the European Control Conference, Kos, Greece, uly 27. [7] C. Aurora and A. Ferrara, A sliding mode oserver for sensorless induction motor speed regulation, International ournal of Systems Science, vol. 38, no., pp , 27. [8] G. Bartolini, A. Ferrara, and E. Usai, Chattering avoidance y secondorder sliding mode control, IEEE Transactions on Automatic Control, vol. 43, no. 2, Feruary 998. [9] R. Marino, G. Santosuosso, and P. Tomei, Roust adaptive oservers for nonlinear systems with ounded disturances, IEEE Transactions on Automatic Control, vol. 46, no. 6, pp ,