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PMSM SPEED SENSORLESS DIRECT TORQUE CONTROL
BASED ON EKF
ABSTRACT
Direct torque controlled permanent magnet synchronous motor (PMSM) has rapid response and good static and dynamic performance. In the direct torque control method, the observation accuracy of stator flux linkage directly determines the performance of the entire system. System with mechanic speed sensor has lower reliability and higher system cost.
In the traditional DTC control, flux linkage is observed through pure voltage integration. The model is very simple, but in practical applications, the disadvantages impact of integrator such as the sensitivity to initial value and DC offset has influenced stator flux linkage observation accuracy. In order to solve these problems an improved integration is applied to observe stator flux linkage in induction motor, which could only get accurate phase information. In another technique low-pass filter is applied instead of integrator to observe flux linkage, which results in flux linkage phase ahead and its amplitude smaller.
Aiming at speed sensor less DTC controlled surface permanent magnet synchronous motor, Extended Kalman filter (EKF) researched to estimate both stator flux linkage and rotor speed in the paper. The disadvantage of pure integrator has been overcome, and the advantages of DTC method such as rapid torque response and strong robustness are still maintained. In the meantime, the problems resulting from mechanical speed sensor has been resolved. Therefore, speed sensor less direct torque control for surface permanent magnet synchronous motor is realized. The motor start problems are solved as EKF do not need accurate initial rotor position information to achieve stability convergence. DTC-based on EKF flux linkage has no obvious ripples due to more accurate EKF observer. Torque dynamic response time is basically the same, which indicates EKF control method do not affect the DTC dynamic performance. The torque ripples using DTC-based on EKF method is significantly reduced and steady performance has been greatly improved.
Simulation results have shown that the advantage of direct torque control method such as rapid torque response is maintained, at the same time, the system based on EKF is robust to motor parameters and load disturbance. The dynamic and static performances are dramatically improved.
I. INTRODUCTION
Direct torque controlled permanent magnet synchronous motor (PMSM) has rapid response and good static and dynamic performance; so many scholars have conducted research to this field and achieved certain results [1-4]. In the direct torque control method, the observation accuracy of stator flux linkage directly determines the performance of the entire system. System with mechanic speed sensor has lower reliability and higher system cost. So how to get stator flux linkage and speed information has become the research hotspot.
In the traditional DTC control, flux linkage is observed through pure voltage integration. The model is very simple, but in practical applications, the disadvantages impact of integrator such as the sensitivity to initial value and DC offset has influenced stator flux linkage observation accuracy. In order to solve these problems, in literature [5], an improved integration is applied to observe stator flux linkage in induction motor, which could only get accurate phase information. In literature [6], low-pass filter is applied instead of integrator to observe flux linkage, which results in flux linkage phase ahead and its amplitude smaller. Amplitude and phase compensation is researched in literature [7], which cannot fundamentally resolve the shortcomings of pure integrator.
Aiming at speed sensorless DTC controlled surface permanent magnet synchronous motor (SPMSM), Extended Kalman filter (EKF) observer is researched to estimate both stator flux linkage and rotor speed in the paper. The disadvantage of pure integrator has been overcome, and the advantages of DTC method such as rapid torque response and strong robustness are still maintained. In the meantime, the problems resulting from mechanical speed sensor has been resolved.
DIRECT TORQUE CONTROL (DTC)
Direct Torque Control (DTC) is a method that has emerged to become one possible alternative to the well-known Vector Control of Induction Motors [1–3]. This method provides a good performance with a simpler structure and control diagram. In DTC it is possible to control directly the stator flux and the torque by selecting the appropriate VSI state. The main advantages offered by DTC are:
– Decoupled control of torque and stator flux.
– Excellent torque dynamics with minimal response time.
– Inherent motion-sensor less control method since the motor speed is not required to achieve the torque control.
– Absence of coordinate transformation (required in Field Oriented Control (FOC)).
– Absence of voltage modulator, as well as other controllers such as PID and current controllers (used in FOC).
– Robustness for rotor parameters variation. Only the stator resistance is needed for the torque and stator flux estimator.
These merits are counterbalanced by some drawbacks:
– Possible problems during starting and low speed operation and during changes in torque command. Requirement of torque and flux estimators, implying the consequent parameters identification (the same as for other vector controls).
– Variable switching frequency caused by the hysteresis controllers employed.
– Inherent torque and stator flux ripples.
– Flux and current distortion caused by sector changes of the flux position.
– Higher harmonic distortion of the stator voltage and current waveforms compared to other methods such as FOC.
– Acoustical noise produced due to the variable switching frequency. This noise can be particularly high at low speed operation.
A variety of techniques have been proposed to overcome some of the drawbacks present in DTC [4]. Some solutions proposed are: DTC with Space Vector Modulation (SVM) [5]; the use of a duty--ratio controller to introduce a modulation between active vectors chosen from the look-up table and the zero vectors [6–8]; use of artificial intelligence techniques, such as Neuro-Fuzzy controllers with SVM [9]. These methods achieve some improvements such as torque ripple reduction and fixed switching frequency operation. However, the complexity of the control is considerably increased.
A different approach to improve DTC features is to employ different converter topologies from the standard two-level VSI. Some authors have presented different implementations of DTC for the three-level Neutral Point Clamped (NPC) VSI [10–15]. This work will present a new control scheme based on DTC designed to be applied to an Induction Motor fed with a three-level VSI. The major advantage of the three-level VSI topology when applied to DTC is the increase in the number of voltage vectors available. This means the number of possibilities in the vector selection process is greatly increased and may lead to a more accurate control system, which may result in a reduction in the torque and flux ripples. This is of course achieved, at the expense of an increase in the complexity of the vector selection process.
To understand the answer to this question we have to understand that the basic function of a variable speed drive (VSD) is to control the flow of energy from the mains to the process.
Energy is supplied to the process through the motor shaft.
Two physical quantities describe the state of the shaft: torque and speed. To control the flow of energy we must therefore, ultimately, control these quantities.
In practice, either one of them is controlled or we speak of “torque control” or “speed control”. When the VSD operates in torque control mode, the speed is determined by the load.
Likewise, when operated in speed control, the torque is determined by the load.
Initially, DC motors were used as VSDs because they could easily achieve the required speed and torque without the need for sophisticated electronics.
However, the evolution of AC variable speed drive technology has been driven partly by the desire to emulate the excellent performance of the DC motor, such as fast torque response and speed accuracy, while using rugged, inexpensive and maintenance free AC motors.
In this section we look at the evolution of DTC, charting the four milestones of variable speed drives, namely:
• DC Motor Drives 7
• AC Drives, frequency control, PWM 9
• AC Drives, flux vector control, PWM 10
• AC Drives, Direct Torque Control 12
We examine each in turn, leading to a total picture that identifies the key differences between each.
AC Drives
Introduction
• Small size
• Robust
• Simple in design
• Light and compact
• Low maintenance
• Low cost
The evolution of AC variable speed drive technology has been partly driven by the desire to emulate the performance of the DC drive, such as fast torque response and speed accuracy, while utilizing the advantages offered by the standard AC motor.
[pic]
Controlling variables are Voltage and Frequency
• Simulation of variable AC sine wave using modulator
• Flux provided with constant V/f ratio
• Open-loop drive
• Load dictates torque level
Unlike a DC drive, the AC drive frequency control technique uses parameters generated outside of the motor as controlling variables, namely voltage and frequency. Both voltage and frequency reference are fed into a modulator which simulates an AC sine wave and feeds this to the motor’s stator windings. This technique is called Pulse Width Modulation (PWM) and utilizes the fact that there is a diode rectifier towards the mains and the intermediate DC voltage is kept constant. The inverter controls the motor in the form of a PWM pulse train dictating both the voltage and frequency. Significantly, this method does not use a feedback device which takes speed or position measurements from the motor’s shaft and feeds these back into the control loop. Such an arrangement, without a feedback device, is called an “open-loop drive”.
Advantages
• Low cost
• No feedback device required – simple because there is no feedback device, the controlling principle offers a low cost and simple solution to controlling economical AC induction motors.
This type of drive is suitable for applications which do not require high levels of accuracy or precision, such as pumps and fans.
• Field orientation not used
• Motor status ignored
• Torque is not controlled
• Delaying modulator used
With this technique, sometimes known as Scalar Control, field orientation of the motor is not used. Instead, frequency and voltage are the main control variables and are applied to the stator windings. The status of the rotor is ignored, meaning that no speed or position signal is fed back.
Therefore, torque cannot be controlled with any degree of accuracy. Furthermore, the technique uses a modulator which basically slows down communication between the incoming voltage and frequency signals and the need for the motor to respond to this changing signal.
[pic]
Features
• Field-oriented control - simulates DC drive
• Motor electrical characteristics are simulated- “Motor Model”
• Closed-loop drive
• Torque controlled INDIRECTLY
To emulate the magnetic operating conditions of a DC motor, i.e. to perform the field orientation process, the flux-vector drive needs to know the spatial angular position of the rotor flux inside the AC induction motor. With flux vector PWM drives, field orientation is achieved by electronic means rather than the mechanical commentator/brush assembly of the DC motor.
Firstly, information about the rotor status is obtained by feeding back rotor speed and angular position relative to the stator field by means of a pulse encoder. A drive that uses speed encoders is referred to as a “closed-loop drive”. Also the motor’s electrical characteristics are mathematically modeled with microprocessors used to process the data.
The electronic controller of a flux-vector drive creates electrical quantities such as voltage, current and frequency, which are the controlling variables, and feeds these through a modulator to the AC induction motor. Torque, therefore, is controlled INDIRECTLY.
Advantages
Good torque response
• Accurate speed control
• Full torque at zero speed
• Performance approaching DC drive
Flux vector control achieves full torque at zero speed, giving it a performance very close to that of a DC drive.
Drawbacks
• Feedback is needed
• Costly
• Modulator needed
To achieve a high level of torque response and speed accuracy, a feedback device is required. This can be costly and also adds complexity to the traditional simple AC induction motor.
Also, a modulator is used, which slows down communication between the incoming voltage and frequency signals and the need for the motor to respond to this changing signal. Although the motor is mechanically simple, the drive is electrically complex.
[pic]
Controlling Variables
With the revolutionary DTC technology developed by ABB, field orientation is achieved without feedback using advanced motor theory to calculate the motor torque directly and without using modulation. The controlling variables are motor magnetizing flux and motor torque. With DTC there is no modulator and no requirement for a tachometer or position encoder to feed back the speed or position of the motor shaft. DTC uses the fastest digital signal processing hardware available and a more advanced mathematical understanding of how a motor works. The result is a drive with a torque response that is typically 10 times faster than any AC or DC drive. The dynamic speed accuracy of DTC drives will be 8 times better than any open loop AC drives and comparable to a DC drive that is using feedback. DTC produces the first “universal” drive with the capability to perform like either an AC or DC drive.
As can be seen from Table, both DC Drives and DTC drives use actual motor parameters to control torque and speed. Thus, the dynamic performance is fast and easy. Also with DTC, for most applications, no tachometer or encoder is needed to feed back a speed or position signal. Comparing DTC (Figure 4) with the two other AC drive control blocks shows up several differences, the main one being that no modulator is required with DTC. With PWM AC drives, the controlling variables are frequency and voltage which need to go through several stages before being applied to the motor. Thus, with PWM drives control is handled inside the electronic controller and not inside the motor.
[pic]
PERMANENT MAGNET SYNCHRONOUS MOTOR
A permanent magnet synchronous motor (PMSM) is a motor that uses permanent magnets to produce the air gap magnetic field rather than using electromagnets. These motors have significant advantages, attracting the interest of researchers and industry for use in many applications.
Permanent Magnet Materials
The properties of the permanent magnet material will affect directly the performance of the motor and proper knowledge is required for the selection of the materials and for understanding PM motors. The earliest manufactured magnet materials were hardened steel. Magnets made from steel were easily magnetized. However, they could hold very low energy and it was easy to demagnetize. In recent years other magnet materials such as Aluminum Nickel and Cobalt alloys (ALNICO), Strontium Ferrite or Barium Ferrite (Ferrite), Samarium Cobalt (First generation rare earth magnet) (SmCo) and Neodymium Iron-Boron (Second generation rare earth magnet) (NdFeB) have been developed and used for making permanent magnets. The rare earth magnets are categorized into two classes: Samarium Cobalt (SmCo) magnets and Neodymium Iron Boride (NdFeB) magnets. SmCo magnets have higher flux density levels but they are very expensive. NdFeB magnets are the most common rare earth magnets used in motors these days. A flux density versus magnetizing field for these magnets is illustrated in figure.
[pic]
Fig. Flux Density versus Magnetizing Field of Permanent Magnetic Materials
Classification of Permanent Magnet Motors
1. Direction of field flux
PM motors are broadly classified by the direction of the field flux. The first field flux classification is radial field motor meaning that the flux is along the radius of the motor. The second is axial field motor meaning that the flux is perpendicular to the radius of the motor. Radial field flux is most commonly used in motors and axial field flux have become a topic of interest for study and used in a few applications.
2. Flux density distribution
PM motors are classified on the basis of the flux density distribution and the shape of current excitation. They are PMSM and PM brushless motors (BLDC). The PMSM has a sinusoidal-shaped back EMF and is designed to develop sinusoidal back EMF waveforms. They have the following:
1. Sinusoidal distribution of magnet flux in the air gap
2. Sinusoidal current waveforms
3. Sinusoidal distribution of stator conductors.
BLDC has a trapezoidal-shaped back EMF and is designed to develop trapezoidal back EMF waveforms. They have the following:
1. Rectangular distribution of magnet flux in the air gap
2. Rectangular current waveform
3. Concentrated stator windings.
3. Permanent magnet radial field motors
In PM motors, the magnets can be placed in two different ways on the rotor. Depending on the placement they are called either as surface permanent magnet motor or interior permanent magnet motor.
Surface mounted PM motors have a surface mounted permanent magnet rotor. Each of the PM is mounted on the surface of the rotor, making it easy to build, and specially skewed poles are easily magnetized on this surface mounted type to minimize cogging torque. This configuration is used for low speed applications because of the limitation that the magnets will fly apart during high-speed operations. These motors are considered to have small saliency, thus having practically equal inductances in both axes. The permeability of the permanent magnet is almost that of the air, thus the magnetic material becoming an extension of the air gap. For a surface permanent magnet motor Ld = Lq.
The rotor has an iron core that may be solid or may be made of punched laminations for simplicity in manufacturing. Thin permanent magnets are mounted on the surface of this core using adhesives. Alternating magnets of the opposite magnetization direction produce radially directed flux density across the air gap. This flux density then reacts with currents in windings placed in slots on the inner surface of the stator to produce torque. Figure shows the placement of the magnet.
[pic]
Fig. Surface Permanent Magnet Motor
Interior PM motors have interior mounted permanent magnet rotor as shown in figure. Each permanent magnet is mounted inside the rotor. It is not as common as the surface mounted type but it is a good candidate for high-speed operation. There is inductance variation for this type of rotor because the permanent magnet part is equivalent to air in the magnetic circuit calculation. These motors are considered to have saliency with q axis inductance greater than the d axis inductance ( Lq > Ld ).
[pic]
Fig. Interior Permanent Magnet Motor
SCALAR CONTROL V/F OF PMSM
Constant volt per hertz control in an open loop is used more often in the squirrel cage IM applications. Using this technique for synchronous motors with permanent magnets offers a big advantage of sensorless control. Information about the angular speed can be estimated indirectly from the frequency of the supply voltage. The angular speed calculated from the supply voltage frequency can be considered as the value of the rotor angular speed if the external load torque is not higher than the breakdown torque. The mechanical synchronous angular speed is proportional to the frequency fs of the supply voltage
[pic]
where p is the number of pole pairs. The RMS value of the induced voltage of AC motors is given as
[pic]
By neglecting the stator resistive voltage drop and assuming steady state conditions, the stator voltage is identical to the induced one and the expression of magnetic flux can be written as
[pic]
To maintain the stator flux constant at its nominal value in the base speed range, the voltage-to-frequency ratio is kept constant, hence the name V/f control. If the ratio is different from the nominal one, the motor will become overexcited or under excited. The first case happens when the frequency value is lower than the nominal one and the voltage is kept constant or if the voltage is higher than that of the constant ratio V/f . This condition is called over excitation, which means that the magnetizing flux is higher than its nominal value. An increase of the magnetizing flux leads to a rise of the magnetizing current. In this case the hysteresis and eddy current losses are not negligible. The second case represents under excitation. The motor becomes underexcited because the voltage is kept constant and the value of stator frequency
is higher than the nominal one. Scalar control of the synchronous motor can also be demonstrated via the torque equation of SM, similar to that of an induction motor. The electromagnetic torque of the synchronous motor, when the stator resistance Rs is not negligible, is given as
[pic]
EXTENDED KALMAN FILTER
KALMAN FILTER:
The Kalman filter is essentially a set of mathematical equations that implement a predictor-corrector type estimator that is optimal in the sense that it minimizes the estimated error covariance—when some presumed conditions are met.
• The Discrete Kalman Filter
This section describes the filter in its original formulation (Kalman 1960) where the measurements occur and the state is estimated at discrete points in time.
1. The Process to be Estimated
The Kalman filter addresses the general problem of trying to estimate the state of [pic] a discrete-time controlled process that is governed by the linear stochastic difference equation
[pic]
with a measurement [pic] that is
[pic]
The random variables [pic] represent the process and measurement noise (respectively). They are assumed to be independent (of each other), white, and with normal probability distributions
[pic]
In practice, the process noise covariance and measurement noise covariance matrices might change with each time step or measurement, however here we assume they are constant.
The [pic] matrix A in the difference equation relates the state at the previous time step [pic] to the state at the current step k, in the absence of either a driving function or process noise. Note that in practice A might change with each time step, but here we assume it is constant. The [pic] matrix B relates the optional control input [pic]to the state x. The [pic] matrix H in the measurement equation relates the state to the measurement zk. In practice H might change with each time step or measurement, but here we assume it is constant.
2. The Computational Origins of the Filter
We define [pic] (note the “super minus”) to be our a priori state estimate at step k given knowledge of the process prior to step k, and [pic] to be our a posteriori state estimate at step k given measurement [pic] . We can then define a priori and a posteriori estimate errors as
[pic]
The a priori estimate error covariance is then
[pic]
and the a posteriori estimate error covariance is
[pic]
In deriving the equations for the Kalman filter, we begin with the goal of finding an equation that computes an a posteriori state estimate [pic] as a linear combination of an a priori estimate [pic] and a weighted difference between an actual measurement [pic] and a measurement prediction [pic]. Some justification is given in “The Probabilistic Origins of the Filter” found below.
[pic]
The difference [pic] in equation (4.7) is called the measurement innovation, or the residual. The residual reflects the discrepancy between the predicted measurement [pic] and the actual measurement [pic]. A residual of zero means that the two are in complete agreement.
The [pic] matrix K is chosen to be the gain or blending factor that minimizes the a posteriori error covariance. This minimization can be accomplished by first substituting into the above definition for [pic], performing the indicated expectations, taking the derivative of the trace of the result with respect to K, setting that result equal to zero, and then solving for K. For more details see (Maybeck 1979; Jacobs 1993; Brown and Hwang 1996). One form of the resulting K that minimizes is given by
[pic]
we see that as the measurement error covariance R approaches zero, the gain K weights the residual more heavily. Specifically,
[pic]
On the other hand, as the a priori estimate error covariance [pic] approaches zero, the gain K weights the residual less heavily. Specifically,
[pic]
Another way of thinking about the weighting by K is that as the measurement error covariance R approaches zero, the actual measurement [pic] is “trusted” more and more, while the predicted measurement [pic] is trusted less and less. On the other hand, as the a priori estimate error covariance [pic] approaches zero the actual measurement [pic] is trusted less and less, while the predicted measurement [pic] is trusted more and more.
3. The Probabilistic Origins of the Filter
The justification is rooted in the probability of the a priori estimate [pic] conditioned on all prior measurements [pic] (Bayes’ rule). For now let it suffice to point out that the Kalman filter maintains the first two moments of the state distribution,
[pic]
The a posteriori state estimate reflects the mean (the first moment) of the state distribution— it is normally distributed if the conditions are met. The a posteriori estimate error covariance reflects the variance of the state distribution (the second non-central moment). In other words,
[pic]
For more details on the probabilistic origins of the Kalman filter, see (Brown and Hwang 1996).
4. The Discrete Kalman Filter Algorithm
We will begin this section with a broad overview, covering the “high-level” operation of one form of the discrete Kalman filter (see the previous footnote). After presenting this high-level view, we will narrow the focus to the specific equations and their use in this version of the filter.
The Kalman filter estimates a process by using a form of feedback control: the filter estimates the process state at some time and then obtains feedback in the form of (noisy) measurements. As such, the equations for the Kalman filter fall into two groups: time update equations and measurement update equations. The time update equations are responsible for projecting forward (in time) the current state and error covariance estimates to obtain the a priori estimates for the next time step. The measurement update equations are responsible for the feedback—i.e. for incorporating a new measurement into the a priori estimate to obtain an improved a posteriori estimate. The time update equations can also be thought of as predictor equations, while the measurement update equations can be thought of as corrector equations. Indeed the final estimation algorithm resembles that of a predictor-corrector algorithm for solving numerical problems as shown below in Figure.
[pic]
Figure: The ongoing discrete Kalman filter cycle. The time update projects the current state estimate ahead in time. The measurement update adjusts the projected estimate by an actual measurement at that time.
The specific equations for the time and measurement updates are presented below in tables.
Again notice how the time update equations in table project the state and covariance estimates forward from time step k-1 to step k. Initial conditions for the filter are discussed in the earlier references.
Table: Discrete Kalman filter time update equations.
[pic]
Table: Discrete Kalman filter measurement update equations.
[pic]
The first task during the measurement update is to compute the Kalman gain [pic] . The next step is to actually measure the process to obtain[pic] , and then to generate an a posteriori state estimate by incorporating the measuremen. Again equation is simply repeated here for completeness. The final step is to obtain an a posteriori error covariance estimate. After each time and measurement update pair, the process is repeated with the previous a posteriori estimates used to project or predict the new a priori estimates. This recursive nature is one of the very appealing features of the Kalman filter—it makes practical implementations much more feasible than (for example) an implementation of a Wiener filter (Brown and Hwang 1996) which is designed to operate on all of the data directly for each estimate. The Kalman filter instead recursively conditions the current estimate on all of the past measurements. Figure 4.2 below offers a complete picture of the operation of the filter, combining the high-level diagram of Figure with the equations from tables.
[pic]
Figure: A complete picture of the operation of the Kalman filter, combining the high-level diagram of above figure with the equations from tables.
THE EXTENDED KALMAN FILTER (EKF)
1.The Process to be Estimated
The Kalman filter addresses the general problem of trying to estimate the state [pic]of a discrete-time controlled process that is governed by a linear stochastic difference equation. But what happens if the process to be estimated and (or) the measurement relationship to the process is non-linear? Some of the most interesting and successful applications of Kalman filtering have been such situations. A Kalman filter that linearizes about the current mean and covariance is referred to as an extended Kalman filter or EKF. In something akin to a Taylor series, we can linearize the estimation around the current estimate using the partial derivatives of the process and measurement functions to compute estimates even in the face of non-linear relationships. Let us assume that our process again has a state vector [pic], but that the process is now governed by the non-linear stochastic difference equation
[pic]
with a measurement [pic] that is
[pic]
where the random variables [pic] again represent the process and measurement. In this case the non-linear function f in the difference equation relates the state at the previous time step k-1 to the state at the current time step k. It includes as parameters any driving function [pic] and the zero-mean process noise [pic]. The non-linear function h in the measurement equation relates the state [pic] to the measurement[pic].
In practice of course one does not know the individual values of the noise [pic]at each time step. However, one can approximate the state and measurement vector without them as
[pic]
And [pic]
Where [pic] is some a posteriori estimate of the state (from a previous time step k).
It is important to note that a fundamental flaw of the EKF is that the distributions (or densities in the continuous case) of the various random variables are no longer normal after undergoing their respective nonlinear transformations. The EKF is simply an ad hoc state estimator that only approximates the optimality of Bayes’ rule by linearization. Some interesting work has been done by Julier et al. in developing a variation to the EKF, using methods that preserve the normal distributions throughout the non-linear transformations (Julier and Uhlmann 1996).
2. The Computational Origins of the Filter
To estimate a process with non-linear difference and measurement relationships, we begin by writing new governing equations that linearize an estimate.
[pic]
where
• [pic] are the actual state and measurement vectors,
• [pic] are the approximate state and measurement vectors,
• [pic] is an a posteriori estimate of the state at step k,
• The random variables [pic] represent the process and measurement noise,
• A is the Jacobian matrix of partial derivatives of f with respect to x, that is
[pic]
• W is the Jacobian matrix of partial derivatives of f with respect to w,
[pic]
• H is the Jacobian matrix of partial derivatives of h with respect to x,
[pic]
• V is the Jacobian matrix of partial derivatives of h with respect to v,
[pic]
Note that for simplicity in the notation we do not use the time step subscript k with the Jacobians A, W, H and V, even though they are in fact different at each time step.
Now we define a new notation for the prediction error,
[pic]
and the measurement residual,
[pic]
Remember that in practice one does not have access to [pic], it is the actual state vector, i.e. the quantity one is trying to estimate. On the other hand, one does have access to [pic], it is the actual measurement that one is using to estimate [pic].
[pic]
where [pic]represent new independent random variables having zero mean and covariance matrices [pic] , with [pic]. Notice that the equations are linear, and that they closely resemble the difference and measurement equations and from the discrete Kalman filter. This motivates us to use the actual measurement residual [pic] and a second (hypothetical) Kalman filter to estimate the prediction error [pic]. This estimate, call it [pic] , could then be used to obtain the a posteriori state estimates for the original non-linear process as
[pic]
The random variables have approximately the following probability distributions (see the previous footnote):
[pic]
Given these approximations and letting the predicted value of [pic]be zero, the Kalman filter equation used to estimate [pic] is
[pic]
the second (hypothetical) Kalman filter:
[pic]
Equation can now be used for the measurement update in the extended Kalman filter, with [pic], and the Kalman gain [pic] coming with the appropriate substitution for the measurement error covariance.
The complete set of EKF equations is shown below in tables. Note that we have substituted [pic] for to remain consistent with the earlier “super minus” a priori notation, and that we now attach the subscript K to the Jacobians A, W, H, and V, to reinforce the notion that they are different at (and therefore must be recomputed at) each time step.
As with the basic discrete Kalman filter, the time update equations in table project the state and covariance estimates from the previous time step k-1 to the current time step k. [pic] are the process Jacobians at step k, and [pic] is the process noise covariance at step k. As with the basic discrete Kalman filter, the measurement update equations in table correct the state and covariance estimates with the measurement[pic] . [pic] and V are the measurement Jacobians at step k, and [pic]is the measurement noise covariance at step k. (Note we now subscript R allowing it to change with each measurement.)
Table: EKF time update equations.
[pic]
Table: EKF measurement update equations.
[pic]
The basic operation of the EKF is the same as the linear discrete Kalman filter. Figure below offers a complete picture of the operation of the EKF, combining the high-level diagram of Figure with the equations from tables
[pic]
Figure: A complete picture of the operation of the extended Kalman filter, combining the high level diagram of Figure
An important feature of the EKF is that the Jacobian [pic] in the equation for the Kalman gain [pic]serves to correctly propagate or “magnify” only the relevant component of the measurement information. For example, if there is not a one-to-one mapping between the measurement [pic]and the state via h, the Jacobian [pic]affects the Kalman gain so that it only magnifies the portion of the residual [pic]that does affect the state. Of course if over all measurements there is not a one-to-one mapping between the measurement [pic]and the state via h, then as you might expect the filter will quickly diverge. In this case the process is unobservable.
II. MODEL OF SPMSM
(-( Coordinate is chosen in order to design EKF observer. The voltage equation of SPMSM is as following.
[pic]
[pic]
Where [pic] are stator voltage [pic]coordinate components, [pic] are stator current components, [pic] and[pic] are stator flux linkage components, [pic] is stator resistance, [pic] is stator inductance, [pic] is rotor speed, [pic] is rotor position angle.
III. DESIGN OF EKF OBSERVOR
The state and output equations of discrete linear system with random interference is as following [8].
[pic]
Where system noise w(k ) takes into account the system disturbances and model errors, while v(k) represents the measurement noise, which takes into account all measure noise and measure errors. Both v(k )and w(k )are zero-mean white noise with covariance of R and Q , respectively and independent from each other.
Both the optimal state and its covariance are computed in a two-step loop. The first one (prediction step) performs a prediction of both quantities based on the previous estimates.
The equations are as the following:
[pic]
The second step (innovation step) corrects the predicted state and estimation and its covariance matrix through a feedback correction scheme that makes use of the actual measured quantities, which are realized by the following equations.
[pic]
The flux linkage equation of SPMSM is as the following.
[pic]
[pic]
Compared to the dynamic process of SPMSM, sampling interval time is so small that speed can be considered unchanged, which is [pic]. Also, we have the equation [pic]
[pic]is selected as state variable,[pic]are chosen as input and output variable respectively which can easily obtained from the measurements. Thus the dynamic state model for SPMSM was as following.
[pic]
Where [pic]
[pic]
Below equation is obtained by linearization and discretization of above equation
[pic]
Where
[pic]
System noise w(k ) represents the error caused by parameters changes and linearization and discretization, while v(k ) represents errors caused by input and output measurements. Given system initial state, state estimates can be got through recursive operation.
IV. SPEED SENSORLESS DTC CONTROL
Schematic diagram of speed sensor less DTC control for SPMSM based on EKF is shown as figure. Switch voltage vector selection is shown as TABLE.
[pic]
Fig. Schematic diagram of speed sensorless DTC control for SPMSM based on EKF
[pic]
Table. Switch voltage vector selection table
V. SIMULATION RESULT
Simulation model is established using MATLAB/SIMULINK tools. EKF algorithm is achieved by S function. Parameters of SPMSM are shown is TABLE.
TABLE-MOTOR PARAMETERS
[pic]
Parameters of EKF are as following to ensure filter not divergent.
[pic]
VI. CONCLUSION
Speed sensorless DTC control based on EKF for SPMSM is proposed. For a relatively accurate SPMSM model, flux linkage and rotor speed and rotor position can be estimated more precisely by EKF algorithm. Ripples on torque and stator flux are reduced. The motor start problems are solved as EKF do not need accurate initial rotor position information to achieve observer stability convergence. DTC control for PMSM based on EKF has just begun. Many places are imperfect and need for further study.
REFERENCES
[1] L.Zhong, M.F.Rahman, W.Y. Hu , K.W.Lim, “Analysis of direct torque control in permanent magnet synchronous motor drives,” IEEE Trans. On Power Electronics, 1997, 13(5), pp:528 536.
[2] Jun Hu, Bin Wu. “New integration algorithms for estimating motor flux over a wide speed range,” IEEE Trans. on Power Electronics, 1998, 13(5),pp:969-977.
[3] Cenwei Shi, Jianqi Qiu, Mengjia Jin, “Study on the performance of different direct torque control methods for permanent magnet synchronous machines,” Proceeding of the Csee, 2005, 25(16),pp:141-146.
[4] Limei Wang, Yanping Gao. “Direct torque control for permanent magnet synchronous motor based on space voltage vector pulse width modulation,” Journal of Shenyang University of Technology, 2007,29(6), pp:613-617.
[5] Zhiwu Huang, Yi Li, Xiaohong Nian, “Simulation of direct torque control based on modified integrator,” Computer Simulation, 2007,24(02),pp:149-152.
[6] Liyong Yang, Zhengxi Li, Rentao Zhao, “Stator flux estimator based programmable cascaded digital low-pass filter,” Electric Drive,2007,37(10),pp:25-28.
[7] NRN.Klris, AHM. Yatim, “An improved stator flux estimation in steady-state operation for direct torque control of induction machine,” IEEE Trans on Industry Applications, 2002, 38(1),pp: 110-116.
[8] Yingpei Liu. Research on PMSM speed sensorless control for elevator drive [D]. Tianjin University: 2007.6[pic]
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