Position error according to motor parameter inaccuracy
Permanent Magnet Synchronous Motors (PMSM) are widely used in industrial drives due to their high power density, high torque-to- inertia ratio, small torque ripple and precise control at low speed range, possibility to torque control at zero speed, high efficiency and small size. To exploit presented advantages, a vector control should be used. Vector control allows the drive a good dynamic, effective performance especially during transients and prevents overload of the motor by controlling the torque. However, a motor shaft position sensor is required to enable the effective vector control of a PMSM. Such sensors increase the overall cost of the drive and decrease its reliability. That sensor occupies usually the end of the shaft, so elimination of the need for its installation allows new applications of such drive. This scientific problem remains an open question for and is the subject for extensive research in many scientific centers. Several approaches to this problem are reported in the literature, which are based on state observers [1,2], various versions of the Kalman filters [3,4], sliding-mode observers [5,6] or methods of applying motor saliency [7,8,9]. State observers and Kalman filters based on a motor model require complex computational operations to obtain proper accuracy, which always causes problems in real time operation. Several approaches apply motor magnetic saliency and detect the rotor position by measuring phase inductances. These methods provide a real solution at small speeds and during standstill operation, but requirements with respect to hardware and software are high. The additional scientific problem is to obtain high dynamic sensorless drive [10,11]. Assumption of the limitation the applications up to speed control (without position control) gives possibility to achieve well performance of high dynamic sensorless drive for low speed operating range.
A simple and effective structure of an observer is proposed in this paper. The presented observer structure is based on a modified concept of back EMF detection  and introduces a more complex corrector function which differs from the traditional one. The structure contains a corrector with a proportional-multi integral function (PII2) instead of the proportional correction used in the Luenberger observer . Prepared observer structure is applied to control structure presented at figure 1. The motor is fed by the PWM inverter, the control system includes vector control system of stator currents at
Presented structure gives possibility to test system in closed (observed values are used to control drive) and open (observed values are only considered) mode using manual switches as it is shown on figure 1. To achieve smooth observed values, a reference voltage can be used instead of measured ones.
For modeling the PWM converters can be used models with various complexity degrees. First approximation can be a usage of the inverter's transfer function equal to one. One can do that in the case of high carrier frequency of the inverter, since any significant delays occur in the measuring loops. At this case, the observer input signals may be smooth, and is easily obtained the proper observer performance. The other possibility to ease modeling transfer function of the inverter is zero order sample and hold block, usage the delay block or first order transfer function, which time constant or sampling interval is calculated depends on the proposed carrier frequency. These models do not require a very small simulation step size. In the case of the simulation where the effect of the inverter such as simply the impact the modulated voltage on generated state vector’s waveforms and drive performance or inverter's dead time impact is taken into account, the more accurate model is inserted in place of the simplified one. Figure 3 presents used in this model simplified structure of the PWM inverter. Structure of this model is directly derived from the method of generating the states of the switches, the change of switch state is carried out at the intersection of the reference voltage and the carrier signal. This model does not take into account the dead time phenomenon.
The cost of the exact model of inverter usage is of course much smaller calculating step, from single microseconds up to dozens of nanoseconds (for range about 10 kHz carrier frequency) in place of dozens or hundreds microseconds for simple inverter model for presented drive parameters. The calculating step value is very important due to simulation time especially during optimization procedure (thousands or hundreds of thousands of repetitions). For example, the presented drive is simulated in normal mode, at 2.10 GHz dual core Intel processor at 8.3 s, (simulation stop time 0.12 s, fixed step size – 5 μs), but start simulation without opening the model (using only the command window), will reduce the simulation time to about 6.3 s.
Presented at figure 3 inverter model generates the exemplary (for the ideal three-phase sinusoidal excitation) waveforms of the output voltage (the testing waveform obtained for carrier frequency fc=500 Hz) for bipolar voltage modulation – as presented on figure 4. Figure 5 presents the carrier frequency influence for waveforms for the excited phase current for the stationary circuit.
3. Speed controller structure
All the controllers and observer are implemented as a discrete structure. The used in this study structure of speed controller is shown in figure 6 (the currents controllers used in
4. PMSM model
Assuming ordinary simplified assumptions, the general form of the PMSM model in
and the mechanical part:
The motor model with state variables iα, iβ ω and Θ is non-linear. All state variables are measurable but in a concept without mechanical sensor ω and Θ should be estimated. The symbols iα, iβ, Ψα, Ψβ, eα, eβ are stator current, stator flux linkage and induced back EMF in
After simple conversion of the motor model, we get the model in the coordinates system rotating with the rotor -
and the mechanical part:
where ud, uq means the components of input voltage, id, iq, Ld, Lq mean currents and inductance of the motor in the
Such structure gives a possibility to ease analyze of the decoupled
Based on such general model, in simulation PMSM block expanded on nonlinearities in distribution of the flux, current measurement noise addition with control and simple model of friction are used. Model prepared in
5. Observer structure
Proposed observer system calculates position information from back EMF estimated in
According to the method presented in  it is convenient to use only first two electrical equations (1), in which the back EMF components are considered as disturbances. In such a case we can prepare an extended state formula, which can be described as:
For presented system it is possible to use ordinary Luenberger observer with correction based on error between measured and calculated currents value. Assuming that derivative of disturbances is equal zero one can write the equations describing observer (8).
or in matrix form
where K is a correction factor.
Such solution generates acceptable observer’s performance, however, a more accurate model was also taken into account .
The cost of that simplicity is required of the change of the observer’s corrector parameters settings due to operating point (
Multiplication of observer error by constant coefficient
In the case of the proportional-integral (PI) correction the observer equations take the following form:
On the basis of many simulation tests a more complex corrector structure with proportional-double integral (PII2) correction was proposed :
The advantage of introducing integral and double integral components of the observer corrector is that they ensure the astatic character of observation (parameters estimation) even during the transient speed process, in which fast changes of speed and reference currents occur – when the back EMFs waveforms are not in sinusoidal shape. The estimation of back EMF signals by the observer enables the calculation of the new values of the rotor speed and position from (10) and (11) at each step of algorithm realization. In fact, the observed signals (back EMFs) are still changing its values (sinusoidal wave) when motor isn’t at standstill and the astatic character of observer corrector is very important to follow up the position with small estimation error and small phase shift. The structure of the observer's parameters vector consists of six different elements due to symmetry of the equation (14) for current and back EMF calculation and the additional seventh ke, which is used to scaling back EMF’s estimated amplitude into speed value (11). The observer’s structure is identical in both
6. Observer parameters setting
The choice of observer parameters is an important task and should take into account dynamics in closed loop system as well as good accuracy of observation. The simulated drive is a compound object. Motor nonlinearity, discontinuity caused by the inverter structure and discrete structure of the control loop make it difficult to calculate the parameters of the observer. The solution presented in this paper is based on the special algorithm, which is used for automatic supervising the calculations, and after a certain number of cycles generates the results in form of the observer parameter set. The process of parameter synthesis was optimised by means on the base of random weight change (RWC) procedure  (Fig. 8). This algorithm is fast and insensitive to the local minimum of the optimized criterion. Depending on the value of the search step parameters, the effects of the algorithm performance that we get are like for gradient algorithm or evolutionary algorithm (Fig. 9), the possibility to quickly find a local minimum or in longer calculations the searching and finding the global minimum. The searching process is supervised by the quality index. The quality function of observer optimization is formulated as:
where eΘ is the position estimation error, ΔeΘ is the range of the error value changes of the estimated position during the transient process, t1 and t1 + τ are the time boundaries of the integral calculation. The meaning of those parameters is explained in figure 10.
The optimization procedure is performing
Because the optimization procedure of the observer parameters is based on random numbers generation, it is important to take care on proper initiation of that generator. It is very important to initiate generating values using such code:
It is used to eliminate the generation of the same values sequence, each time you start MATLAB.
7. Simulation results
The model of the PMSM control system was carried out in MATLAB Simulink ® environment. The motor was modeled with ordinary simplifying assumptions such as constant resistance and inductance in stator windings, symmetry of windings, and isotropic properties of motor (3, 4, 5). The motor model was connected with a model of a control system, which includes a vector control system of stator currents, a speed controller and a model of the analyzed observer (Figure 1). Drive model contains also a
The model of the observer was used as an element of feedback sending detected signals of rotor position and speed. The motor model was calculated with a very small step of integration, which simulates its continuous character. The step value was within the range of 0.02
Selected waveforms of speed, currents and position error are presented below. These images well illustrate the operation of observer compared with sensor mode. These waveforms were obtained as responses to the step change of speed reference, generated in the form of a step sequence starting from zero speed to 10 rad/s and to 5 rad/s at time 0.1 s. Motor load changes from zero to motor’s nominal load value at time 0.16 s. In addition, figures 11 and 12 show ±2 % range of reference value. Waveform 11 and 12 prove the well performance of sensorless mode drive at low speed – even at 5 rad/s. That drive still remains robustness on disturbance (rapid reference speed and load change). Enlarged part of figure 11 (Fig. 12) shows clearly the setpoint achieving process. Figures 13 and 14 show waveforms of currents in
|Position error [°]|
|R factor||L factor||loaded 5 rad/s||unloaded 5 rad/s||unloaded 10 rad/s|
The final test was prepared to determine the robustness to the incorrect estimate of the initial position. The question was how big may be position difference between estimated and the real one, to prevent the motor startup. Tests have shown, that the possible range of the initial position error, for which the engine will start correctly, it is 80 degrees of arc (Fig. 24). Figures 20-23 show the sine and cosine waveforms of the estimated and the real shaft position. Corresponding to sine and cosine waveforms from figures 20-23, figure 24 shows the “measured” speed waveforms. The robustness on initial position error estimation is proven.
Stable performance of the sensorless drive with PMSM at low speed was noticed in the closed loop mode. Small position error and robust on inaccuracy motor parameter estimation were proven. The resistance tolerance is about 20 %, whereas inductance error tolerance is below 9 %. A wider range of resistance change could be achieved by using resistance observation. A test for robustness on the disturbance in the measured signal was passed: even a disruption of the phase currents by injection of the random signal does not disturb the performance of the sensorless drive. Such drive can achieve high dynamics according to the definition presented in abstract. Robustness to the incorrect estimate of initial position was also tested. The used PI controller discrete implementation gives possibility to use the same speed controller settings in both operating mode – sensor and sensorless. The sensorless drive using modified Luenberger observer gives possibility of the high performance of the high dynamic drive, where the “high dynamic” definition is introduced in this chapter. An additional effect of this work is to present the outline of how to create and how to operate the complex models of electrical motor drives in the context of the selection of model components of varying complexity which affect the calculation time and quality and accuracy of the results.
stator inductance:L = 5.7 mH
stator resistance per phase:R = 0.7 Ω
total moment of inertia:J = 0.007273 kg m2
This work was supported by grant N N510 325937.Appendix
Liu J. Hao S. Ma R. Hao M. Research “. on Integer. State Observer. in Current. Control for. P. M. S. M,” 2008 228 231
Jones L. A. Lang J. H. state “. A. observer for. the permanent-magnet. synchronous motor,”. I. E. E. E. Transactions on. Industrial Electronics. vol 3 374 382Aug. 1989
T. Chan F. Borsje P. Wang W. Application “. of Unscented. Kalman filter. to sensorless. permanent-magnet synchronous. motor drive,”. presented at. the Electric. Machines Drives Conference. I. E. M. D. C. 2009 631 638
International Conference on Electronic Computer Technology, S. Gu M. F. He Y. Zhang H. Study “. on Extend. Kalman Filter. at Low. Speed in. Sensorless P. M. S. M. Drives,” presented. at the. 2009 311 316
Parasiliti F. Petrella R. Tursini M. Sensorless “. speed control. of a. P. M. synchronous motor. based on. sliding mode. observer extended Kalman. filter,” 2001 1 533 540
Vittek J. Bris P. Stulrajter M. Makys P. Comnac V. Cernat M. Chattering “. free sliding. mode control. law for. the drive. employing P. M. S. M. position control,”. 2008 115 120
Kulkarni A. B. Ehsani M. novel “. A. position sensor. elimination technique. for the. interior permanent-magnet. synchronous motor. drive,” I. E. E. E. Transactions on. Industry Applications. vol 1 144 150Feb. 1992
European Conference on, Rieder U. H. Schroedl M. simulation “. A. method for. analyzing saliencies. with respect. to enhanced. I. N. F. O. R. M-capability for. sensorless control. of P. M. motors in. the low. speed range. including standstill,”. in Power. Electronics Applications 2005p. 8pp.
Ying Y. Jianguo Z. Youguang G. Jianxun J. Numerical “. simulation of. a. P. M. S. M. model considering. saturation saliency. for initial. rotor position. estimation,” in. Control Conference. 2008CCC 2008. 27th Chinese, 2008, 114 118
Genduso F. Miceli R. Rando C. Galluzzo G. R. Back “. Sensorless-Control E. M. F. Algorithm for. High-Dynamic Performance. P. M. S. M,” I. E. E. E. Transactions on. Industrial Electronics. vol 6 2092 2100Jun. 2010
, Urbanski K. Sensorless “. control of. P. M. S. M. high dynamic. drive at. low speed. range,” in. 2011 728 732
Luenberger D. An “. introduction to. observers,” I. E. E. E. Transactions on. Automatic Control. vol 6 596 602Dec. 1971
of a PM Synchronous Motor Drive by Kalman Filter,” in , Germano F. Parasiliti Tursini M. Sensorless “. Speed Control. of a. P. M. Synchronous Motor. Drive by. Kalman Filter,”. in Proc. 1994 2 540 544
in Electrical and Electronic Eng., Urbanski K. Zawirski K. Adaptive “. observer of. rotor speed. position for. P. M. S. M. sensorless control. system,” C. O. M. P. E. L. Int J. for Computation. Maths 23 4 1129 1145 2004
- 15. F. Parasiliti, R. Petrella, and M. Tursini, “Sensorless speed control of a PM synchronous motor by sliding mode observer,” in Industrial Electronics, 1997. ISIE’97., Proceedings of the IEEE International Symposium on, 1997, pp. 1106-1111.
Conference and Exhibition on Power Electronics and Motion Control, Kosice, Urbanski K. Zawirski K. Rotor “. speed position detection. for P. M. S. M. control system,”. in Proceedings. of 9th. Int 2000 6 239 243
, Urbanski K. Zawirski K. Sensorless “. control of. S. R. M. using position. observer,” in. 2007 1 6
Burton B. Kamran F. Harley R. G. Habetler T. G. Brooke M. A. Poddar R. Identification “. control of. induction motor. stator currents. using fast. on-line random. training of. a. neural network,”. I. E. E. E. Transactions on. Industry Applications. vol 3 697 704Jun. 1997