Observer-based Method

The last method listed above determines rotor position and speed using observers. The rough estimations of the angular rotor position by measuring the phase voltages and currents can be refined utilizing observers. The inputs of the observers are the inputs and outputs of the actual system such as phase voltages and currents;

Fig. 3.6 Third-Harmonics Voltage technique ea

eb

ec

v3

λr3

C

A BC BA CA CB AB

0o 60^o 120^o 180^o 240^o 300^o 360^o

θe

the outputs of the observers are the immeasurable states of the actual system such as flux linkage or back-EMF voltage. The states produced by the observer are fed back into the system, as would be the actually measured variables as used in a closed loop system control with specific rules for the various types of observers. For example, consider the discrete-time state-space form of a single output system,

[ ]

k Ax

[ ]

k Bu

[ ]

k

x +1 = + (3-22)

[ ]

k Cx

[ ]

k Du

[ ]

k

y = + (3-23)

which must be completely observable. Then, determine a modal matrix P for the system. Using the modal matrix P, the system is transformed into an observable form.

[ ]

k Px

[ ]

k Q x

[ ]

k

Next, the full-order is designed for immeasurable states. The eigenvalues of the observer are usually chosen to be slightly faster than the eigenvalues of actual system such that the state estimated error approaches zero as time approaches infinity. If the eigenvalues of the observer are large enough, the estimated values will converge to the actual values within a sufficiently short time. The observer takes the form

[ ]

k Fζ

[ ]

k gy

[ ]

k Hu

[ ]

k

with the observer output equation

[ ]

k Pζ

[ ]

k Q ζ

[ ]

k

w = ′ = ⁻¹ ′ (3-29)

which allow the transformations

[ ]

k Pζ

[ ]

k Q ζ

[ ]

k

Thus, the full-order observer in x[k] is derived, meaning that the immeasurable states can be obtained form ζ

[ ]

k , which converges to x[k]. However, the full-order observer consists of more-order equations, the pole assignment is not easy and the manual tuning is time consuming. The reduced-order observers are developed for substitution. The reduced-order observers depend on that some of the system outputs are linear transformations of the system states. In motor drive application, the state, current, is readily available for measurement such that only the states, back-EMF voltages or flux linkages, need to be estimated by the lower-order observer. Assume that the sample rate of the system is fast enough; the sliding mode observers are available. The sliding mode observers, offspring of sliding mode control, utilize the varying switching function in order to confine the state estimation errors to approach toward zero on a phase-plane, sliding surface. For details, consider the state-space system

In BLCD systems, usually considering the PMSM or the fundamental of the trapezoidal BLDC, the state-space X consists of the d-q transformed stator variables.

The state observer takes the same form with the addition of a switching input.

( )

S Bu

X A

Xˆ& = ˆ + −K⋅sgn (3-35)

where

the hat symbol denoted estimated values K is the switching gain and

( )

⎪⎩

⎪⎨

⎧

<

−

= ≥

0 1

0 1

S S S

sgn

S is the vector of sliding surface

Defining S = ˆX −X =e, where e is the state estimation errors, the error equations can be derived as

( )

e Ae

e&= −Ksgn (3-36)

Consider the approach and sliding condition, let K be a large positive value to make sure the following equation hold.

Te<0

e & (3-37)

Then, the observer states converge to the actual system states and the rotor position and the speed are estimated. Comparing to other methods, the achievement of rotor position estimation in wide speed and the real time estimation of the rotor speed are the critical grace. But the sampling rate, computational burden and the parameter variations need be considered.

Chapter 4

Sensorless Drive Modeling

4.1 Position Detection and Start-up Algorithm

The sensorless methods, which substitute for the Hall sensors or encoders to obtain the position information, have been introduced in the previous chapter. Each of them has specific advantages and drawbacks, but a common fateful fault. All of them can not detect the rotor position or speed at standstill or near zero speed with insufficient information required for estimating, such that are unable start-up. In order to overcome this fault, an extra start-up procedure, so-called “align and go”, has been proposed. As implied by the name, the algorithm is divided into two steps. First, excites any two phases winding of the stator to align the rotor to the specific position, and then accelerates the rotor according to the given firing sequences with decreasing time intervals. It should be noted that the initial position is not detected and the starting form unknown rotor position may be accompanied by a temporary reverse rotation or may cause a starting failure. These eventualities are not tolerable in many applications. Thus, a complete start-up procedure, which includes the detection of the initial rotor position and avoids the temporary reverse rotation or starting failure, should be developed [5]. The most proper method is based on the effect of the magnet position on the flux saturation, which cause the variations of the inductance. The details will be discussed below.

4.1.1 Initial Rotor Position Detection

Motors convert electrical energy to mechanical energy by magnetic field to generate desired torque, which is generated via the interactive force between windings

with electric current and magnet. The path of the magnetic flux produced by the permanent magnet and current is a loop crossing the airgap to the stator from the rotor.

Ideally, the B-H curve of the steel is linear and the inductance is constant due to the reluctance of the flux path is constant. But in practically, the B-H curve of the steel is a non-linear curve with saturation. Thus, the estimation of the rotor position can be detected by using the inductance variation due to the magnets position and the stator current. The relationship between inductance and flux linkage is shown as

+Li

= _PM

Phase λ

λ (4-1)

where λ_Phase is the summation of the flux from the permanent magnet, λ_PM, and the flux from the current i. L is the inductance of the excited phase. Supply the current with positive or negative direction to the phase, as shown in Fig. 4.1. The variations of the inductance are derived as

+ phase voltage and current to the variation of the inductance. The phase voltage equation is expressed as

a

The back-EMF can be neglect when a motor is at standstill. Then, solve the differential equation; the phase current can be derived as

⎟⎟⎠

According to (4-5), the phase current has a different transient dependent on the

inductance variation, which is determined by the relative position of the magnet and the direction of the current. It should be noted that i has a faster response than ⁺ i ⁻ due to the time constant R/L is larger than ⁺ R/L , as shown in Fig. 4.2. Therefore, ⁻ the position information can be obtained by monitoring the phase current i and ⁺ i ⁻ in an appropriate time interval.

As mention above, the inductance variation with respect to the electrical angle should be known for detecting the rotor position. This problem can be overcome by using the FEM software, Flux 2D, to drawn out the relationship between inductance

Fig. 4.1 The flux linkage with positive or negative current

Fig. 4.2 The responses of current with positive or negative direction λphase

i⁻ i⁺ i

λPM

∆λ−

∆λ+

t T

i⁺

| i⁻ | i

∆i R

v_an

and electrical angle, as shown in Fig. 4.3. Then, employing the information of inductance and (4-5) to gain the current responses with respect to the electrical angle in a time interval of 20 µs , as shown in Fig. 4.4 and Table 4.1 [5]. In Fig. 4.4, the current responses of the well-known six segments are calculated with the determined inductance in an appropriate interval of electrical degrees. Observing Fig. 4.4, it should be noted that the polarity of the difference between the positive current and negative current, ∆i , changes every 180 electrical degrees. Furthermore, the _n inherent differences with 120 electrical degrees exist in any two phases such that the polarity of one of three changes every 60 electrical degrees, which can provide the position information, as shown in Fig. 4.5 [5]. However, the position information from ∆i is not proper to drive a motor. The polarity of the difference between _n

m

n ∆i

∆i − , ∆∆i , can be effectively used to identify the rotor position due to it _k includes the information of the polarity of the difference between two back-EMF used in six-step drive. The polarity of one of three ∆∆i changes every 60 electrical _k degrees with 30 electrical degrees shift compared with the variation of ∆i , as shown _n in Fig. 4.6 and Table 4.2 [5]. Thus, the initial rotor position can be detected by monitoring the polarity of ∆∆i , after each of six segments has been supplied a _k voltage pulse with a 20 µs period.

Fig. 4.3 The variation of inductance due to the change of the current and the rotor position

Fig. 4.4 The current responses of the well-known six segments

Table 4.1 Six segments of an electrical cycle

Segment Symbol of current

B

A i1⁺ (i)

A

B i1⁻ (ii)

A

C i2⁺ (iii)

C

A i2⁻ (iv)

C

B i3⁺ (v)

B

C i3⁻ (vi)

Fig. 4.5 The differences between the current i and _n⁺ i _n⁻

4.1.2 Start-up Algorithm

In previous section, the initial position has been identified when a motor is at standstill. The correct phases winding on the stator are excited and the maximum electromagnetic torque is produced, so that the motor starts to rotation. Necessarily, the next commutation position should be detected for next excitation when the rotor rotates with 60 electrical degrees. However, the method proposed in previous section

Fig. 4.6 The differences between the current ∆i _n

Table 4.2 Polarity of ∆∆i on rotor position _k

Electrical position ∆∆i₁ ∆∆i₂ ∆∆i ₃

30^ο ~ 90^ο + + −

90^ο ~ 150^ο + − +

150^ο ~ 210^ο + − +

210^ο ~ 270^ο − − +

270^ο ~ 330^ο − + −

330^ο ~ 30^ο − + −

is not suitable while the rotor is rotating due to the time delay caused by the period of six voltage pulses and the negative direction torque produced by exciting incorrect segments. Thus, a different start-up algorithm needs been developed. The start-up algorithm is also based on the principle of the previous section. Consider the torques produced by the six segments with respect to the electrical angle, as shown in Fig. 4.7.

Except the exciting segment, which produces the maximum torque, there are still two segments to produce the torque with same direction in every commutation period. The torque of one is increasing and the other is decreasing. It should be noted that the increasing one is the next excitation. In other words, the increasing segment should replace the original exciting segment to be the new exciting segment while the torque produced by the increasing segment is larger than the torque produced by the original exciting segment. As to how to determine the magnitude of the torques, the test voltage pulse with a short time will be supplied into the increasing segment. Then, the switching point can be determined by comparing the current of the increasing segment with the current of the exciting segment due to the current is proportional to the torque. For example, the segment A is excited when the initial position is detected C as 30~60 electrical degrees. It means that the exciting segment is A and increasing C segment is BC. Then, cut off the segment A and excite the segment C BC with a shot time T_s in this commutation period, as shown in Fig. 4.8. The next excitation can be determined while the current of BC is larger than the current of A . It means C that BC replaces CA to be the exciting segment. The composition of the exciting segment and increasing segment with respect to electrical angle is shown in Table 4.3.

The start-up algorithm can speed up a motor from standstill to a low speed, but is not suitable for high-speed operation due to the test voltage pulse.

Fig. 4.7 The torque generation of the six segments

Table 4.3 The exciting and increasing segment at electrical angle Fig. 4.8 The current response of current the exciting and increasing segment

Electrical position Exciting Increasing

30^ο ~ 90^ο A C BC

90^ο ~ 150^ο BC BA

150^ο ~ 210^ο BA CA

210^ο ~ 270^ο CA CB

270^ο ~ 330^ο C B AB

330^ο ~ 30^ο A B AC

Ts 2*Ts

Ts 2*Ts

Ts 2*Ts

ON OF

current at 30ο ~ 90ο

current at 90ο ~ 150ο

4.2 Sliding Mode Observer Design and the Estimations of Position and Velocity

After a motor is successfully speeded up from standstill successfully, the rough start-up procedure should be replaced for better performance. In previous chapter, several well-known methods have been introduced and interpreted the characteristic and suitable application. Comparing these methods, the method based on the observer seems to have less constraint and more expansibility. Thus, the following section will discuss a sliding mode observer applied in sensorless BLDC system, which is considering the PMSM or the fundamental of the trapezoidal BLDC motors [4].

4.2.1 Sliding Mode Observer Design for Position Estimation

The main purpose of the sensorless methods is to replace Hall sensors or encoders, which provide the rotor information. In order to achieve the purpose, it should be considered that what is relative to the rotor position. According to chapter 2, the back-EMF voltages or the flux linkages are most proper. Considering the stability, the flux linkages are chosen to observe due to the back-EMF voltage is the differentiation of the flux linkage. The equivalent circuit equation in the stationary reference frame, D-Q coordinate, is expressed as

λ i

v _{s s}

dt L d

dt

R d ⎟ +

⎠

⎜ ⎞

⎝⎛ +

= (4-6)

where

]T

[ _{D Q} v = v v

]T

[ _{D Q} i = i i

T

T [ ]

]

[λ λ = K ⋅cosθ_e K ⋅sinθ_e

= _{D Q E E}

λ

θe rotor position at electrical angle KE EMF constant

It should be noted that (4-6) has a nonlinear termλ, which includes triangular function. Assume that KE is time independent such that the differentiation of λ can be rewritten as

[ ]

^T

where ω_re is the rotational speed at electrical angle. Then, let stator current i and the flux λ be the state variables; the input is the stator voltage v and the output is the stator current i. Assuming that ω_re is a constant parameter, the linear state equations can be expressed as

D

The observability of the system can be verified by checking (4-8). Then, the sliding mode observer can be constructed as

( )

^{i i}

∧ estimation state variable

Subtracting (4-8) from (4-9), the error equation can be derived as

( )

^{i i D} estimation error and D is disturbances. Considering the state equations that the outputs of the system only include the currents in D-Q coordinate, the sliding mode is only applied to the current estimation. The sliding surface is selected as

i i e s= =ˆ−

1 (4-11)

Then, employing the principle of Lyapunov, the approaching and sliding condition should be satisfied, as follow

0

Thus, the switching gain K can be derived as

⎥⎥

It should be noted that the magnitude of K has no influence on the response of the system. Usually, choosingn=2 is enough for the sliding mode exciting. Under the sliding mode, the current estimation error is confined into the sliding surface as

1 0

1 =e =s→

e & (4-16)

Furthermore, according to the principle of the equivalent control, the switching input can be treated as the equivalent input. Form (4-10) and (4-16), the equivalent inputs can be expressed as

D e A

u_eq =− _{12 2} − (4-17)

Substituting the equivalent inputs into the second row of (4-10), the error equation of the flux estimation can be derived as

D e A

e&₂ = ′ ₂ + ′ (4-18)

where

12

22 GA

A A′= −

[

G I

]

D D′= −

Form (4-18) and neglecting the disturbance, if all the eigenvalues of A′ are in the left-half plane of a complex plane, the estimated flux linkage will converge to the actual value. Calculating the eigenvalues of A′ , the relationship between the eigenvalues and feedback gain is given as

( )

^α j^β

eig A′ =− ± (4-19)

where

g ωL^re α = ₂

g L^re

re

ω ω β = + ₁

However, it should be considered that where to assign the poles and the influence of the disturbance.

Considering the closed-loop system form (4-18), the flux linkage estimation error can be derived as

( ) (

s sI A

)

D

( )

s

e₂ = − ′ ⁻¹ ′ (4-20)

Form (4-20), if the denominator order of the disturbance Laplace transforms is lower than one-order, the flux linkage estimation error will converge to zero due to the final value theorem. Form (4-7), the flux linkage is only relative to the velocity such that the velocity estimation error may lead to the flux estimation error. Therefore, the flux linkage estimation must be robust against the velocity estimation error. According to (4-9), the disturbance caused by velocity estimation error is express as

λ re

Substituting (4-21) into (4-20), the closed-loop transfer function T(s) can be derived as

Considering the block diagram, as shown as Fig. 4.9. In order to reduce the effect of the disturbance, the norm of T(s) needs been suppressed [1]. The frequency of flux linkage estimation changes widely with velocity. Considering the worst case, the capability of disturbance suppression can be quantitatively evaluated by the H_∞ norm. The H_∞ norm of the transfer function is calculated as

[]

⋅

λmax maximum eigenvalue

( )

^* complex conjugate transposition

2 induced norm

According (4-23), the induced norm of F can be calculated and expressed by using_ω αand β, as

2 2 2

1 α β

ω = ω +

re

F (4-25)

Substituting (4-25) into (4-24) and letting β be equal to zero, the maximum capability of disturbance suppression is given as follow

re

T ω

= 1

min ∞ (4-26)

Furthermore, considering the transient response, α is assigned with respect to velocity, as

ωre

α = ν (4-27)

where v is a factor to maintain the disturbance suppression [1].

From above, the current and the flux linkage have been estimated. The rotor position information can be obtained by using following equation.

⎟⎟⎠

⎜⎜ ⎞

⎝

= ⁻ ⎛

D Q

e λ

θ tan¹ λ _(4-28)

Then, the rotor position has been estimated without the influence of the back-EMF constant variation and robust to the parameter variation and velocity estimation error.

As to how to determine the velocity, the details will be discussed in the next section.

The estimation of velocity obtained from next section will be set as a parameter in above observer model.

4.2.2 Adaptive Velocity Estimation

There are two conventional methods to estimate the velocity. Using the differentiation of the rotor position is the most common method. However, the noise may be included due to the differential operation and the PWM or switching signal.

The other method employs the fact that the velocity is proportional to the magnitude of the back-EMF voltage, which is the differentiation of the flux linkage. However, the accuracy of the estimation is sensitive to the back-EMF constant variation and the direction of the velocity is unknown. Thus, the adaptive velocity estimation employing the estimated flux linkage λ^ˆ is developed, as shown in Fig. 4.10.

According to the relationship between the flux linkage and the velocity, the reference model is expressed as

λ J λ _re ˆ

ˆ&=ω (4-29)

The estimation model is defined as

Fig. 4.9 The block diagram of error equation

12

22

GA

A A ′ = −

J GJ F_ω =− +

L 1

ˆ ⋅ ∆ω λ

S

1

^e2

T(s)

+ +

λ J λ ˆ_re ~

~& =ω (4-30)

where λ^~ is the output of the estimation model and ωˆ is the estimated velocity. _re Considering the convergence of the estimation error, the estimation model is adopted with a, as

Taking the estimation error of the velocity into account, the estimation model with the closed-loop feedback is rewritten as

( )

^{Jλ G}

( )

^{λ λ}

λ J

λ _re ~ _re ˆ_re ~ ˆ ~

~^& =ω − ω −ω − ′ − (4-32)

From (4-29) and (4-32), the error equation can be derived as

( ) ( )

and the adaptive scheme is chosen as λ

where g′ is the adaptive integral gain. Considering the convergence of the error equation, the poles of the closed-loop should be assigned in left-half plane as previous section [2]. Then, the flux linkage estimation ~λ

converges to the actual value and the velocity estimation can be obtained by integrating (4-34). The velocity estimation consists of the direction and the magnitude and is robust to the variation of the back-EMF constant.

Fig. 4.10 The block diagram of the adaptive velocity estimation

Chapter 5

Simulation Result and Analysis

In this chapter, the observer designed in chapter 4 is simulated by using Matlab^®−Simulink^®.

5.1 System Descriptionsand Block Diagram

Generally, a PMAC motor drive system with speed feedback control consists of control algorithm, PWM generator and driver circuit, as shown as Fig. 5.1. According to (3-1), the motor model with the parameters shown in Table 5.1 is built as a block in Simulink^®. Besides, the blocks of PWM driver and PI controller have also been built up. Connecting each block, a PMAC motor drive system, as Fig 5.1, is set up, as

Generally, a PMAC motor drive system with speed feedback control consists of control algorithm, PWM generator and driver circuit, as shown as Fig. 5.1. According to (3-1), the motor model with the parameters shown in Table 5.1 is built as a block in Simulink^®. Besides, the blocks of PWM driver and PI controller have also been built up. Connecting each block, a PMAC motor drive system, as Fig 5.1, is set up, as

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