PMSM SPEED SENSORLESS DIRECT TORQUE CONTROLBASED 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

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motor parameters and load disturbance. The dynamic and static performances are dramatically

improved.

I. INTRODUCTIONDirect 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.

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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.

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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.

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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.

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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.

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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.

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

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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.

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Controlling VariablesWith 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.

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PERMANENT MAGNET SYNCHRONOUS MOTORA 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.

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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-

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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.

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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 ).

Fig. Interior Permanent Magnet Motor

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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

where p is the number of pole pairs. The RMS value of the induced voltage of AC motors is

given as

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

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 16

electromagnetic torque of the synchronous motor, when the stator resistance Rs is not negligible,

is given as

EXTENDED KALMAN FILTERKALMAN 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 a

discrete-time controlled process that is governed by the linear stochastic difference equation

with a measurement that is

The random variables represent the process and measurement noise (respectively).

They are assumed to be independent (of each other), white, and with normal probability

distributions

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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 matrix A in the difference equation relates the state at the previous time step 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

matrix B relates the optional control input to the state x. The 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 (note the “super minus”) to be our a priori state estimate at step k given

knowledge of the process prior to step k, and to be our a posteriori state estimate at

step k given measurement . We can then define a priori and a posteriori estimate errors as

The a priori estimate error covariance is then

and the a posteriori estimate error covariance is

In deriving the equations for the Kalman filter, we begin with the goal of finding an equation that

computes an a posteriori state estimate as a linear combination of an a priori estimate

and a weighted difference between an actual measurement and a measurement prediction

. Some justification is given in “The Probabilistic Origins of the Filter” found below.

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The difference in equation (4.7) is called the measurement innovation, or the

residual. The residual reflects the discrepancy between the predicted measurement and the

actual measurement . A residual of zero means that the two are in complete agreement.

The 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 , 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

we see that as the measurement error covariance R approaches zero, the gain K weights the

residual more heavily. Specifically,

On the other hand, as the a priori estimate error covariance approaches zero, the gain K

weights the residual less heavily. Specifically,

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Another way of thinking about the weighting by K is that as the measurement error covariance R

approaches zero, the actual measurement is “trusted” more and more, while the predicted

measurement is trusted less and less. On the other hand, as the a priori estimate error

covariance approaches zero the actual measurement is trusted less and less, while the

predicted measurement is trusted more and more.

3. The Probabilistic Origins of the Filter

The justification is rooted in the probability of the a priori estimate conditioned on all prior

measurements (Bayes’ rule). For now let it suffice to point out that the Kalman filter

maintains the first two moments of the state distribution,

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,

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.

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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.

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.

Table: Discrete Kalman filter measurement update equations.

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The first task during the measurement update is to compute the Kalman gain . The next step

is to actually measure the process to obtain , 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.

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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 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 , but that the process is now

governed by the non-linear stochastic difference equation

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with a measurement that is

where the random variables 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 and

the zero-mean process noise . The non-linear function h in the measurement equation relates

the state to the measurement .

In practice of course one does not know the individual values of the noise at

each time step. However, one can approximate the state and measurement vector without them as

And

Where 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.

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where

• are the actual state and measurement vectors,

• are the approximate state and measurement vectors,

• is an a posteriori estimate of the state at step k,

• The random variables represent the process and measurement noise,

• A is the Jacobian matrix of partial derivatives of f with respect to x, that is

W is the Jacobian matrix of partial derivatives of f with respect to w,

H is the Jacobian matrix of partial derivatives of h with respect to x,

V is the Jacobian matrix of partial derivatives of h with respect to v,

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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,

and the measurement residual,

Remember that in practice one does not have access to , it is the actual state vector, i.e. the

quantity one is trying to estimate. On the other hand, one does have access to , it is the actual

measurement that one is using to estimate .

where represent new independent random variables having zero mean and

covariance matrices , with . 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 and a

second (hypothetical) Kalman filter to estimate the prediction error . This estimate, call it

, could then be used to obtain the a posteriori state estimates for the original non-linear

process as

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The random variables have approximately the following probability distributions (see the

previous footnote):

Given these approximations and letting the predicted value of be zero, the Kalman filter

equation used to estimate is

the second (hypothetical) Kalman filter:

Equation can now be used for the measurement update in the extended Kalman filter, with

, and the Kalman gain 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

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.

are the process Jacobians at step k, and 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

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covariance estimates with the measurement . and V are the measurement Jacobians at step

k, and 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.

Table: EKF measurement update equations.

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

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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 in the equation for the Kalman

gain 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 and the state via h, the Jacobian affects the Kalman gain so that it only

magnifies the portion of the residual that does affect the state. Of course if over

all measurements there is not a one-to-one mapping between the measurement and the state

via h, then as you might expect the filter will quickly diverge. In this case the process is

unobservable.

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II. MODEL OF SPMSM- Coordinate is chosen in order to design EKF observer. The voltage equation of SPMSM is as

following.

Where are stator voltage coordinate components, are stator current

components, and are stator flux linkage components, is stator resistance, is stator

inductance, is rotor speed, is rotor position angle.

III. DESIGN OF EKF OBSERVORThe state and output equations of discrete linear system with random interference is as following

[8].

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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:

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.

The flux linkage equation of SPMSM is as the following.

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Compared to the dynamic process of SPMSM, sampling interval time is so small that speed can

be considered unchanged, which is . Also, we have the equation

is selected as state variable, 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.

Where

Below equation is obtained by linearization and discretization of above equation

Where

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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 CONTROLSchematic diagram of speed sensor less DTC control for SPMSM based on EKF is shown as

figure. Switch voltage vector selection is shown as TABLE.33

Fig. Schematic diagram of speed sensorless DTC control for SPMSM based on EKF

Table. Switch voltage vector selection table

V. SIMULATION RESULTSimulation model is established using MATLAB/SIMULINK tools. EKF algorithm is achieved

by S function. Parameters of SPMSM are shown is TABLE.

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TABLE-MOTOR PARAMETERS

Parameters of EKF are as following to ensure filter not divergent.

VI. CONCLUSION

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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.

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[4] Limei Wang, Yanping Gao. “Direct torque control for permanent magnet synchronous motor

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[5] Zhiwu Huang, Yi Li, Xiaohong Nian, “Simulation of direct torque control based on modified

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