9th International Conference on Power Electronics and Motion Control - EPE-PEMC 2000 Košice
FLUX OBSERVERS BASED ON COMBINED VOLTAGE-CURRENT MODELS FOR CONTROL OF PMSM DRIVES Gheorghe Daniel Andreescu Politehnica University of Timisoara, Department of Automation and Industrial Informatics, Bd. Vasile Parvan 2, 1900 Timisoara, Romania tel.: +40 56 204333 / 680; fax: +40 56 192043; e-mail:
[email protected]
Košice Slovak Republic 2000
Abstract. This paper develops a generalised approach on flux observer structures in wide speed range for permanent magnet synchronous motors (PMSM) drives using combined topologies of the voltage and current models. Typical applications are in the direct torque and flux control (DTFC) and in the field oriented vector control (FOVC). A critical analysis on the flux estimators structures without corrective terms concludes to combine the advantages of the current models at low speed and the voltage models at medium-high speed. To realise a smooth and monotone transition depending on speed between these models, Luenberger observer structures having the dynamic compensator proportional integral type are used. A critical comparison on eight flux observer structures gives a top classification and useful recommendations. Simulation and experimental results using a DTFC of PMSM drive, including representative flux observers, evaluate the dynamic performances of the observers with real parameter deviations and demonstrate a good global robustness in wide speed range. An improvement will be obtained by on line parameters identifications. Keywords: Observers, Estimation techniques, Flux control, PM motor drives, Robust control, DTC (Direct Torque Control), Variable speed drives.
1. INTRODUCTION
2. FLUX ESTIMATOR STRUCTURES
To obtain high-dynamic performances of PMSM drives, modern control methods are used, e.g.., the field oriented vector control, the direct torque and flux control (DTFC) [1], [2]. These methods require estimations of the flux vector and the electromagnetic torque which can be performed by using the terminal variables: stator currents and voltages and eventually the rotor position. If the flux vector is correctly estimated the electromagnetic torque is evaluated by a simply computation. There are two structure categories to perform the flux estimation. The first is the estimators that use only the electromagnetic models of the PMSM without corrective terms. The second is the observers or the asymptotically estimators that use more a predictive correction to ensure a fast convergence and also to reduce the sensitivity to parameter variations [3].
The space vector models of the electromagnetic subsystem (EM) of the salient PMSM is a nonlinear coupled multiinputs multi-outputs (MIMO) system. It is assumed that the airgap magnetic flux is sinusoidal distributed, there are no damper windings, and the iron losses are neglected [1]. The EM models use voltage equations (Eu) or current equations (Ei) written in αβ stator reference (s) or in dq rotor reference (r). According to the relations (1) - (4) [1], [13] and Fig. 1a. -d, there are four flux estimators in vector format: a) Eus - estimator with voltage model in stator reference; b) Eur - estimator with voltage model in rotor reference; c) Eis - estimator with current model in stator reference; d) Eir - estimator with current model in rotor reference.
In recent years, flux observers using combined voltage and current models are presented in [4-7] -for induction motors, in [8], [9] - for synchronous reluctance motors and in [1013] for PMSM. In this paper, a generalised approach for the flux observers of PMSM is developed based on combined flux models (estimators): the current model at low speed, and the voltage model at medium-high speed. The smooth transition between these two models is determined via the observer bandwidth. Simulation and experimental results with proposed representative flux observers used in DTFC of PMSM drives show robustness to parameter variations and good dynamic performances in wide speed range applications. They constitute a real support for industrial implementations.
Eur: λ r = − jωλ r − Ro i r + u r ,
Eus: λ s = − Ro i s + u s ,
λ s (0) = λ os λ r (0) = λ ro
(1) (2)
Eis: λ s = L i s + 15 . L2 i s* e j2θ + λ 0o e jθ = Lso (θ) i s + λ o (θ)
(3)
Eir: λ r = ( Ldo id + λ 0o ) + jLqo iq = Lro i r + λ 0o
(4)
where: λ, i, u - flux, stator current and stator voltage vectors, Ro - stator phase resistance, λ0o - PM flux, Ldo, Lqo d, q axis inductance, L=Lsσ+3/2L0, L2 - mutual inductance and is* - complex conjugate stator current. A critical evaluation of these estimators regarding the sensitivity to parameter variations concludes as follows.
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is
θ
Ro -
us
λ
s
λo (θ)
is
a. Eus estimator
λs
s
L o (θ) c. Eis estimator
ω ir ur
Ro
X -
λ0o
j
-
λr
ir
b. Eur estimator
Lro
λr
d. Eir estimator
Fig. 1. Flux vector estimators of PMSM The Ei estimators without dynamic strongly depend on the estimated magnetic parameters (noted with “o” subscript), i.e., PM flux λ0o, and d, q axis inductance Ldo , Lqo. They are recommended at low speed, even zero speed. The Eu estimators with dynamics depend only on stator phase resistance Ro , but there are problems with offset errors in the integrator elements. They are recommended at mediumhigh speed. In practice, the recommended estimator pair is Eir estimator for low speed, even zero speed, and Eus estimator for medium-high speed, where Eus is practically better than Eir. Both Eus and Eir estimators are invariant on ω speed, while Eur and Eis estimators are nonlinear, depending on ω speed and θ position.
3. FLUX OBSERVER STRUCTURES Based on the flux estimator features, our objective is to realise flux observers using combinations between the flux estimators Eu and Ei. In essence, the flux observers use the Luenberger observer topology, but the usual linear K compensator applied to the correction error generated from the Eu and Ei estimators is replaced by a dynamic compensator depending on ω speed. The goal is to select the Ei estimator at low speed, including zero speed, respectively the Eu estimator at medium-high speed, with a smooth transition between them depending on ω speed. The correction is placed at the Eu estimator level that contains an integrator element in order to have a good stability and to reduce the possible input dc offset. The measured state vector is the stator current vector is. The Eu dynamic models estimate the flux vector λ^(e) depending on the induced voltage vector e = u - Ro i. Moreover, Eur requests the ω speed. The Ei models without dynamic estimate the flux vector λ^(i) depending on the stator current vector i, or they realise the inverse operation Ei -1, i.e., they estimate the stator current i^ based on flux estimation vector λ^. In conclusion, these remarks lead to a generalised approach of the PMSM flux observer structures using the following classification criteria, - implicitly, the generation criteria:
I) correction error nature: • flux error correction: the Eu and Ei estimators are in λλ. parallel connection; these observers are noted Oλ • current error correction: the Eu and Ei estimators are in λi. serial connection; these observers are noted Oλ II) correction error reference frame • αβ stator reference, noted with “s” superscript, • dq rotor reference, noted with “r” superscript. III) flux estimator pair (Eux, Eiy), where x,y ∈{s, r}, i.e., the reference frame of each pair estimator. λλs-sr notation means: flux observer For example, the Oλ λλ), ii) - correction error in with i) - flux error correction (Oλ stator reference (s), iii) - flux estimator pair is (Eus, Eir). Taking into account the i) -iii) criteria, in the Tab. 1. there are generated eight flux observer using combined voltage and current models. Fig. 2. presents the flux observer λλ and Oλ λi class, using Luenberger observer structures: Oλ topology with dynamic compensator K. Eu / Ei Eis Eir s s Eu λλ -ss λλs-sr Oλ Oλ r s Eu λλ -rs λλr-rr Oλ Oλ λλ parallel flux observers - ∆λ correction Oλ Eu / Ei Eir Eis Eus λis-ss λis-sr Oλ Oλ r s Eu λi -rs λir-rr Oλ Oλ λi serial flux observers - ∆i correction Oλ Tab. 1. Flux observers using combined voltage and current models The Tab. 2. shows a critical comparison between these flux observer structure for salient PMSM, to the end being a top classification. The following nonlinear functions contained by the observers are considered as disadvantages: Eis(θ) depending on θ; Eur(ω) - depending on ω; e±jθ rotator operators. On the other hand, there are considered as advantages at the observer level the fact that the variables ir or er are available in the rotor reference, out of the λs^ and/or λr^ flux estimations.
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9th International Conference on Power Electronics and Motion Control - EPE-PEMC 2000 Košice
i
θ
θ
s
λ cs
Eis
s
e
λ s
e
Ei
es
e
λλ -rs Oλλ
jθ
e
es
e e-jθ
-
λλ -rr Oλλ
s Eis -1 i
es
λ r
-
K
-
is
e-jθ
Eus
K
s ejθ i
Eir -1
-
K λ is-sr Oλ λ s
ω
θ
λis-rs Oλ
λ rc
Eir
λ s
θ
is
λ is-ss Oλ
es
-
ω Eur
r
θ
Eus
e-jθ
s
K
λ s
λ s
K
is
λ s
θ s
λ cs
θ
λ cs
s
ω Eur
-jθ
Eus
λλs-sr Oλλ
θ
ejθ
Eir
-
K
is
e-jθ
s
Eus
λλ s-ss Oλλ
is
-jθ
e
r
Eu
θ
is
jθ
e
Ei
s -1
es e-jθ
i s
-
K
ω Eu
e-jθ
λ r r
λ ir-rr Oλ
Ei
r -1
i r
ir
-
K
λλ and Oλ λi class Fig. 2. Flux observer structures: Oλ
λ/ Oλ λλs-ss Oλ λλs-sr Oλ λλs-rs Oλ λλr-rr Oλ λis-ss Oλ λis-sr Oλ λis-rs Oλ λir-rr Oλ
Disadvantages | Advantages r s^ jθ ir λ λr^ (θ) (ω) e Eu Ei x x x x x x x x x x x x x x x x x x x x x x x x x x x x s
| er
x
x
Top 3 1 4 2 3 1 4 2
Tab. 2. Critical comparison between flux observers for salient PMSM The K compensator is a filter having different behaviours for the two Eu inputs in order to be selected the advantages of the two flux estimators: Ei for low speed - even zero speed, and Eu for medium-high speed. The dynamic smooth transition between Ei and Eu estimators depending on ω speed is fixed by the observer frequency bandwidth [ω1, ω2]. The K compensator can have different topologies [4], [7], [10].
The design conditions of the K compensator structure are: 1. if ω < ω1 then Ei flux estimator is selected; 2. if ω > ω2 then Eu flux estimator is selected; 3. if ω∈(ω1, ω2) then both Ei and Eu flux estimators are selected in this bandwidth. The transition between these two estimators requires a smooth and monotone nature. Following these conditions, the simplest K compensator is a proportional integral (PI) type: K = kp + ki/s [10].
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9th International Conference on Power Electronics and Motion Control - EPE-PEMC 2000 Košice
Fig. 5. DTFC structure for PMSM This method is a strait-forward engineering one, ensures fast torque response, good perturbation rejections - all in a wide speed range, and easier practical implementations. A DTFC for PMSM (Fig. 5.) with a sliding mode speed controller (Rω) [11] including the representative flux λλs-sr and Oλ λis-sr is used to evaluate the observer Oλ dynamic performances of these observers. A bi-positional hysteresis flux controller (2H) and a three-position hysteresis torque controller (3H) are implemented.
λλ -sr - detailed structure Fig. 3. Flux observer Oλ s
λis-sr - detailed structure Fig. 4. Flux observer Oλ The K compensator parameter design are given by (5). We recommend for the ω1 , ω2 range to be: ω1 = 2...10 rad/s and ω2 = (3...10) ω1. kp = ω1 + ω2,
ki = ω1 ω2
(5)
This structure improves the stability of the pure integrator voltage model. The negative effect of the dc offset which could be present in the measurement circuits of the stator voltage and current is avoided due to the integral term of K. The best flux observer for salient PMSM is seem to be λλs-sr presented in detail in Fig. 3. In the worst conditions Oλ when the drive is no loaded at low speed, the flux estimation λs^* from the Ei estimator is given mainly by the PM flux λ0o. The contribution of the flux given by noised stator currents is reduced and thus the flux estimation has relatively clean waveforms even at very low speed. On the λis-sr observer, presented in detail in Fig. 4., other hand, Oλ has the following advantages: i.) both Eus, Eir-1 estimators are in closed loop, thus robustness in large speed range is expected; ii.) flux estimation λ^ is available in both reference frames: λs^ and λr^. The electromagnetic torque estimation Te^ is computed from the measured stator current and the estimated flux: Te^ = 3/2 p ( λα^ iβ - λβ^ iα )
(6)
4. SIMULATION AND EXPERIMENTAL RESULTS The direct torque and flux control method (DTFC) directly controls the electromagnetic torque and the stator flux vector using a table of optimal switching for the voltage source inverter (VSI) without to use rotational operators.
Simulation results The Matlab-Simulink package with Runge-Kutta 3 method, h=100µs sampling rate is used. The system parameters are: • PMSM rated parameters: p = 4 pole pair, λ0o = 0.1 Wb, Lqo = 0.02 H, Ldo = 0.012 H, Ro = 1.8 Ohm, Jo = 0.004 kgm2, Bo = 0.001 Nms/rad, ωo = 100 rad/s, Teo = 2 Nm, Iao = 3 A, Vdco = 100 V. • Sliding mode speed controller: Tsm=0.01s, Telim=2.3 Nm • Torque and flux hysteresis: 0.01Telim, respective 0.01λ0o • Flux observers: ω1 = 7, ω2 = 30, thus kp = 37, ki = 210. The case studies analyse the system performances and the robustness of stator flux and torque estimators to large real parameter variations and load torque disturbances at low speed (zero speed) and high speed (100 rad/s). For a stator temperature variation with 80 C, the stator resistance linear increases with 30% while PM flux linear decreases with 15%, i.e., R = 1.3 Ro and λo = 0.85 λ0o. This is the detuned case. For dynamic evaluations, the step inputs are applied as following: t = 0 s, ω* = 0 rad/s; t = 0.1 s, TL* = 1.5 Nm; t = 0.3 s, TL* = 0 Nm; t = 0.4 s, ω* = 100 rad/s; t = 0.7 s, TL* =1.5 Nm; t = 0.9 s, TL* =0 Nm. In the detuned case, transient responses for the speed ω, electromagnetic torque Te and estimated torque Te^; electromagnetic flux λ and estimated flux λ^ are presented λλs-sr (Fig. 7.). At λis-sr observer (Fig. 6.) or Oλ using Oλ s λλ -sr observer has an oscillation trend high speed, the Oλ and the sensitivity to estimated parameters corresponds to that of Eir estimator at the speed below the observer bandwidth, and to that of Eus estimator at the speed above λisthis bandwidth. For the same observer bandwidth, the Oλ s λλ -sr. The sr flux observer is more robust then Oλ simulation results prove the observers convergence and the global system robustness to parameter deviations and load torque in wide speed range. The fast torque and speed responses are not practically affected by the observer parameter errors, establishing the robustness and highperformances of the DTFC for PMSM drives with the proposed flux observers. The speed response robustness is significant realised by the sliding mode speed controller.
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9th International Conference on Power Electronics and Motion Control - EPE-PEMC 2000 Košice
λλs-sr flux observer Fig. 7. Transient responses using Oλ
λis-sr flux observer Fig. 6. Transient responses using Oλ Experimental results Hardware set-up of the experimental control system for PMSM drive (Fig. 8.) consists on: PC Pentium-166 MHz, ADA-1100 acquisition board 24 µsec/ADC channel, adapter interface, power inverter and PMSM. Acquisition signals are: ia , ib phase currents; Vdc inverter dc-link voltage; and sinθ, cosθ from a resolver position transducer to estimate θ and ω with a PLL observer. Output logic signals are Sa, Sb, Sc . Real time programs are divided in three parts: 1) ADA-1100 resources administration - assembly lang.; ii.) digital control and observer algorithms - C language; iii.) off-line graphical display interface - C language. The control system from Fig. 5. is real time implemented, h=200µsec A ring buffer list memorises three desired variables used at the end of real time running by graphical display interface.
Experimental results. For technical reasons regarding the VSI there are: Vdc = 60 V, Ilim =1.5 A, thus Temax = 0.8 Nm. The Fig. 9. shows transient responses of the speed ω and the estimated torque Te^ for a symmetrical triangular speed reference ω*, no load. Good performances of the DTFC λλs-sr flux observer in wide speed range system with the Oλ including zero speed are proved. The speed delay is of 20 ms. The Fig. 10. shows the estimated flux vector λ, with λ0o = 0.1 Wb, that proves a good estimation convergence with acceptable flux ripple in transient regime.
Te^ [Nm]
Vdc PC Pentium 166MHz
Process coupler ADA 1100
I/O / 3 ADC / 5
CONTROL SYSTEM
3 /
Adapter interface
Sabc
us VSI VAMSm
2 /
is
/ 2
sinθ, cosθ
PMSM
PT θ ω Resolver
ω*, ω [rad/s]
PROCESS
Fig. 8. Experimental control system for PMSM drive
Fig. 9. Transient responses to a triangular speed reference
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9th International Conference on Power Electronics and Motion Control - EPE-PEMC 2000 Košice
[4]
[5]
[6]
[7] Fig. 10. Flux vector λ^(λα^, λβ^), λ0o = 0.1 Wb. [8] 5. CONCLUSIONS The main conclusions and remarks are the following: • The proposed PMSM flux observer approach combines advantages of the current model Ei estimators at low speed with the voltage model Eu estimators at mediumhigh speed using a dynamic compensator speed depending. • The compensator dynamics design is based on the choosing of the observer frequency bandwidth in order to make a smooth and monotone transition between the estimators Ei and Eu, according to the errors from the parameter identifications. • These observer solve the problem of the dc-offset from the voltage and current measurement circuits. • A critical comparison on the eight generated structures of flux observers gives a top classification and practical useful recommendations. • In the simulation case studies using Oλ λλs-sr flux observer with parameter uncertainties, the theoretical assumptions regarding transient estimations are confirmed: the flux observer sensitivity corresponds to Eir estimator for speed below the observer bandwidth, and to Eus estimator for speed above this bandwidth. • The best robust flux observer is Oλ λis-sr. • The simulation and experimental results show that the global performances of the speed and torque responses in wide speed range are not practically affected by the observer errors, proving the robustness of the DTFC of PMSM drive system with these proposed observers. • An improvement in the flux observer performances will be obtained by on line parameters identification. 6. REFERENCES [1] Boldea, I., Nasar, S.A.: Electric Drives. CRC Press, Florida, 1999. [2] Vas P.: Sensorless Vector and Direct Torque Control. Oxford University Press, 1998. [3] Verghese G.C., Sanders S.R..: Observers for flux estimation in induction machines. IEEE Transactions
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[10]
[11]
[12]
[13]
on Industrial Electronics, Vol. 35, No. 1, Feb. 1988, pp. 85-94. Jansen, P.L., Lorenz, R.D.: A physically insightful approach to the design and accuracy assessment of flux observers for field oriented induction machine drives. IEEE Transactions on Industry Applications, Vol. 30, No. 1, Jan./Feb. 1994, pp. 101-110. Jansen, P.L., Lorenz, R.D., Novotny, D.W.: Observerbased direct field orientation: analysis and comparison of alternative methods. IEEE Transactions on Industry Applications, Vol.30, No.4, July/Aug. 1994, pp. 945-953. Lorenz R.D., Lipo T.A., Novotny D.W.: Motion control with induction motors. Proceedings of the IEEE, Vol. 82, No. 8, Aug. 1994, pp. 1215-1240. Elbuluk M. Langovsky N., Kankam M.D.: Design and implementation of a closed-loop observer and adaptive controller for induction motor drives. IEEE Transactions on Industry Applications, Vol. 34, No. 3, May/June 1998, pp. 435-443. Fratta A., Vagati A.: A reluctance motor drive for high dynamic performance applications. IEEE Transactions on Industry Applications, Vol. 28, No. 4, July/Aug. 1992, pp. 873-879. Vagati A., Pastorelli M., Franceschini G., Drogoreanu V.: Flux-observer-based high-performance control of synchronous reluctance motors by including cross saturation. IEEE Transactions on Industry Applications, Vol. 35, No. 3, May/June 1999, pp. 597-604. Bilewski, M., Fratta, A., Giordano, L., Vagati, A., Villata, F.: Control of high-performance interior permanent magnet synchronous drives. IEEE Transactions on Industry Applications, Vol. 29, No. 2, March/April 1993, pp. 328-337. Andreescu, G.D.: Robust direct torque vector control system with stator flux observer for PMSM drives. Proceedings of the 5th International Conference on Optimization of Electric and Electronic Equipment OPTIM'96, Brasov, Vol. 5, May 1996, pp. 1441-1454. Andreescu, G.D.: Position and speed sensorless control of PMSM drives based on adaptive observer. Proceedings of the 8th European Conference on Power Electronics and Applications EPE’99, Lausanne, Sept. 1999, CD-ROM, paper 436, pp. 1-10. Andreescu G.D.: Estimators in Control of Electrical Drives. Applications to PMSM (book in Romanian). Orizonturi Universitare Publisher, Timisoara, 1999.
THE AUTHOR Gheorghe Daniel Andreescu: received the M.S. degree in applied electronics in 1977, Ph.D. degree in automatic systems in 1999, both from the “Politehnica” University of Timisoara (PUT), Romania. Since 1984 he is with the Automation & Industrial Informatics Department PUT, where he is currently an Associate Professor. Also he works in automatic testing for avionics, e.g., to British Airways Avionics Engineering. His research interests include: sensorless control, direct torque and flux control for PMSM, observers, adaptive control, robots control, real time systems.
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