The Hybrid RFNN Control for a PMSM Drive System Using Rotor Flux Estimator

(1)

The Hybrid RFNN Control for a PMSM Drive System Using Rotor Flux Estimator

Chih-Hong Lin Chih-Peng Lin

Department of Electrical Engineering Department of Engineering National United University Su-Mo Enterprise Co. LTD

Miao Li, R.O.C. Taichung, R.O.C.

E-mail: [email protected] E-mail:[email protected]

Abstract—The hybrid recurrent fuzzy neural network (HRFNN) control permanent magnet synchronous motor (PMSM) drive system using rotor flux estimator is developed to control electric motorcycle in this paper. First, the dynamic models of a PMSM drive system and electric motorcycle are builted though experimental tests and parameters measurements. Then, a HRFNN control control system using rotor flux estimator is developed to control PMSM drive system in order to drive electric motorcycle. The rotor flux estimator consists of the estimation algorithm of rotor flux position and speed based on the back electromagnetic force (EMF). Moreover, the HRFNN controller consists of the supervisor control, RFNN and variable structure control (VSC) is applied to PMSM drive system using rotor flux estimator.

The parameters of RFNN are trained to control command current in order to achieve different output torque of various speeds. The electric motorcycle is operated to provide constant disturbance torque. Finally, the effectiveness of the proposed control schemes is demonstrated by experimental results.

Keywords: permanent magnet synchronous motor, recurrent fuzzy neural network, variable structure control

I. INTRODUCTION

In recent years, for the purpose of reducing air pollution and enhancing environmental protection, quite a few countries require their automotive industries to develop electric vehicles in place of gasoline-powered automobiles gradually. Here we put our attention on the development and research of electric motorcycles since motorcycles are much more widespread than automobiles for individual transportation in Asia. The AC servo drives, including PMSMs (both trapezoidal and sinusoidal types), switched reluctance motors and induction motors, have been widely used in robotics, computer-numerically-controlled (CNC) machine tools, elevators and many other applications in the area of mechatronics [1, 2] for decades.

To achieve fast four-quadrant operation and smooth starting and acceleration, the field-oriented control [1-3], or vector control, is used in the design of AC servo drives.

Compared with induction servo motors, a PM synchronous motor has such advantages as higher efficiency owing to the absence of rotor losses and lower no-load current below the rated speed; moreover, its decouple control performance is much less sensitive to the parameter variations of the motor [1, 2].

Recently much research has been done on the

applications of fuzzy neural network (FNN) systems, which have the advantages of both fuzzy systems and neural networks, in the control fields to deal with nonlinearities and uncertainties of the control systems [4-5]. Moreover, the FNN’s are universal approximators [6-7], which can approximate any dynamics to a prespecified accuracy by the learning process [6]. The fuzzy-neural approximator presented in Leu, Lee and Wang [8] tuned on line to approximate the unknown nonlinear dynamic systems for adaptive control. The supervisory FNN controller proposed in Lin, Hwang and Wai [9] comprised a supervisory controller which is designed to stabilize the system states around a defined bound region, and an FNN sliding-mode controller which combines the advantages of the sliding-mode control with robust characteristics and the FNN with on-line learning ability. The adaptive control schemes of nonlinear systems that incorporate the techniques of FNNs have also grown rapidly [10-12]. The major drawback of the existing FNN is that their application domain is limited to static problem due to their feedforward network structure, and the disadvantages of the feedforward network structure have been mentioned in the previous paragraph. On the other hand, the RFNN [13-14], which naturally involves dynamic elements in the form of feedback connections used as internal memories, has the same dynamic and robust advantages as the RNN. Therefore, this paper presents HRFNN control PMSM drive system using rotor flux estimator is developed to control electric motorcycle in this paper. The method is not dependent upon the predetermined characteristics of the motor and can adapt to any change in the motor characteristics. Finally, some experimental results are provided to demonstrate the effectiveness of the proposed control schemes.

II.CONFIGURATION OF PMSMD^RIVES^YSTEM In order to convenient analysis, the electric torque of the motor can be described as [1-3]:

] ) (

2 [ 3

qs ds q d qs fd md

e P L I i L L i i

T = + − (1) The equation of the motor dynamics is

r r L

e T B J

T = + ω + ω (2) In (1) and (2), ω is the rotor speed, _r P is the number of poles, T_L stands for the load torque (external load disturbance), B represents the viscous frictional coefficient

(2)

and J is the moment of inertia. The basic principle in controlling a PM synchronous motor drive is based on field orientation. In equation (1) and equation (2), if ids =0, the d-axis flux linkage λ_d is fixed, since L_md and I_fd are constant for a PM synchronous motor; the electromagnetic torque T_e is then proportional to i_qs, which is determined by closed-loop control. The rotor flux is produced in the d-axis only, while the current vector is generated in the q-axis for the field-oriented control. Since the generated motor torque is linearly proportional to the q-axis current as the d-axis rotor flux is constant in equation (2), the maximum torque per ampere can be achieved.

With the implementation of field-oriented control, the PM synchronous servo motor drive can be simplified as

* qs t

e Ki

T = (3) 2

/ 3 _md _fd

t PL I

K = (4) The block diagram of a PMSM motor drive system for electric motorcycle is shown in Fig. 1. The system is constituted by the following parts: a PMSM (loaded wheel of electric motorcycle), a interlocked and delay time circuits, a field-oriented mechanism including the coordinate translation, ^sinθ_s^/^cosθsand lookup table generation, a hysteresis-band comparison current-controlled PWM, a voltage source inverter (VSI), a speed control loop and a position control loop were implemented by TMS320C32 DSP control board and interface card. The field-oriented mechanism drive system was implemented by TMS320C32 DSP control system. A host PC downloads the program running on the DSP. For the speed control system, the electric motorcycle is operated to provide constant disturbance torque.

Fig. 1. Configuration of the proposed HRFNN control system using rotor flux estimator.

III.T^HEH^YBRIDRFNNCÔNTROLDÊSIGNU^SINGRÔTOR FLUX ESTIMATOR

The configuration of the proposed HRFNN control system using rotor flux estimator is depicted in Fig. 1. The HRFNN control control system using rotor flux estimator is developed to control PMSM drive system in order to drive electric motorcycle. The rotor flux estimator consists of the estimation algorithm of rotor flux position and speed based on the back electromagnetic force (EMF). The machine model of a PM synchronous motor can be rewritten in the stationary reference frame as follows [1-3]:

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎥+

⎥⎦

⎤

⎢⎢

⎣

⎡

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

−

⎥=

⎥⎦

⎤

⎢⎢

⎣

⎡

qss dss

s s s

qs dss

s s s s

qss dss

v v

L L i i

L R L R

i i dt

d

0 1 1 0 0

0

⎥⎦

⎢ ⎤

⎣ + ⎡−

r e

Ke

θ ω

cos

sin (5)

where v and _ds^s v_qs^s are d and q axis stator voltage, i_ds^s and

qss

i are d and q axis stator current, Ls is stator inductance, K_e stands back electromagnetic force constant,

Rs represents stator resistance, ω_r is rotor flux speed,

θr is rotor flux position. Then back electromagnetic force voltages can be defined as

⎥⎦

⎢ ⎤

⎣

⎡−

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

r e

r e q

d

v v

θ ω

θ

Δ ω_cos^sin (6)

Then switching surface of sliding mode can be defined as follows:

ˆ 0

ˆ Δ =

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

−

= − s R

s qs qs

dss

dss E

i i

i

S i (7)

The dynamic mode of current observers of sliding mode can be represented as follows:

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎥+

⎥⎦

⎤

⎢⎢

⎣

⎡

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

−

⎥=

⎥⎦

⎤

⎢⎢

⎣

⎡

qss dss

s s s

qs dss

s s s s

qss dss

v v

L L i i

L R L R

i i dt d

0 1 1 0 ˆ

ˆ 0

0 ˆ

ˆ

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

− + −

⎥⎥

⎦

⎤

⎢⎢

⎣ + ⎡

ˆ ) sgn(

ˆ ˆ

qss qss

dss dss

q

e d i i

i i v

K v α

(8)

whereiˆ_ds^s、iˆ_qs^s and vˆ_d、vˆ represent observational values q

at i_ds^s 、iqs^s _and v_d 、 vq , respectively, α represents switching gain, sgn(⋅) is sign function. Substituting equation (8) into equation (5), then the dynamic equations can be acquired as follows:

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

− + −

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

− + −

= ^ˆ_ˆ _sgn(^sgn(_ˆ^ˆ _s ₎⁾

s qs qs

dss dss

q q d d e R

R i i

i i v

v v K v

AE

E α ₍₉₎

where

+ _

HRFNN Speed Controller

0

*= id

*

iq

*

ia i^*b ic^*

iai_b

e Limiter

ic vab

C PMSM IGBT Power Module Inverter +

−

vbc

3/2 Phase Coordinate Translation

Sliding Mode Current Observer

Actual System Translation

Voltage Estimator

Switch Gain - +

- +

Sine PWM Circuit

Rotor Flux and speed Estimator Coordinate

Translation and Sin/Cos Generation Interlock and Isolated Circuit Current

Sensors Circuit and A/D Converter

ω∗

θˆr ωˆr

Σ

Σ Σ

s qs s dsv v ,

q dv v ,

q dv v ˆˆ,

s qs s dsi i ˆˆ,

α

α α

s qs s dsi i ,

q dv v ˆˆ, Current Loop Controller

Voltage Sensors Circuit and A/D Converter DC Power

Vdc= 48V+ -

TMS 320C32 DSP Control Board

Continuous Variable Transmission

System (CVT)

Wheels

+ _

HRFNN Speed Controller

0

*= id^*_d=0 i

*

iq_q^*

i

*

ia^*a

i ii^*b^*b iicc^*^*

iaa

i ii_b_b

e Limiter

icc

i vvabab

C PMSM IGBT Power Module Inverter +

−

vbcbc

v

3/2 Phase Coordinate Translation

Sliding Mode Current Observer

Actual System Translation

Voltage Estimator

Switch Gain - +

- +

Sine PWM Circuit

Rotor Flux and speed Estimator Coordinate

Translation and Sin/Cos Generation Interlock and Isolated Circuit Current

Sensors Circuit and A/D Converter

ω∗

θˆr ωˆr

Σ

Σ Σ

s qs s dsv v ,

q dv v ,

q dv v ˆˆ,

s qs s dsi i ˆˆ,

α

α α

s qs s dsi i ,

q dv v ˆˆ, Current Loop Controller

Voltage Sensors Circuit and A/D Converter DC Power

Vdc= 48V+ -

TMS 320C32 DSP Control Board

Continuous Variable Transmission

System (CVT)

Wheels

(3)

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

−

=

s s s s

L R L R A

0 0

(10) and switching gain α can be designed as:

ˆ ), sgn(

ˆ ) ( / ˆ ) (

min{R i_ds^s −i_ds^s L−K_e v_d−v_d i_ds^s −i_ds^s

α=

δ

−

− )/ (ˆ )sgn(ˆ )}

(iˆ_qs^s i_qs^s L K_e v_q v_q i_qs^s i_qs^s

R (11)

where σ is very small vale.

Theorem 1: If switching gain α satisfy equation (11), and equation (10) will be guaranteed globally asymptotic stability.

Proof: The Lyapunov function of positive definite can be selected as follows:

S S V 2 ^TΓ

= 1

(12) where Γ is unit positive-definite matrix. Then we differentiate equation (12)，

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

− + −

=

q q d d e R T R

R T R

v v

v K v

AE E

S S

V ˆ

Γ ˆ Γ

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

− + sgn(ˆ − )

ˆ ) sgn(

qss qss

dss dss T R

R i i

i E Γ i

α

[ ]

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

− +

−

− +

−

=

ˆ ) sgn(

ˆ ) ( /

ˆ ) (

ˆ ) sgn(

ˆ ) ( /

ˆ ) (

ˆ ) ( ˆ )

(

qss qss q

q e s s

s qs qs s

dss dss d d

e s s

s ds ds s

s R s qs s qs s ds ds

i i v

v K L i i R

i i v

v K L i i R

i i i i

α α Γ

[ ]

[

⁽^ˆ ⁾⁽^ˆ ⁾ ⁽^ˆ ⁾ ⁾⁽^ˆ ⁾

]

(ˆ ˆ )

( ² ²

qss qss q s q

s ds ds d d

e

qss qss dss

dss s s

i i v v i i v v K

i i i

L i R

−

− +

−

− +

−

=

[

ⁱ^ds^s ⁻ⁱ^ds^s ⁺ⁱ^qs^s ⁻ⁱ^qs^s

]

+α ˆ ˆ

2

1 λ

λ

Δ + (13) where

dss dss

dss dss d d

s e s ds ds s s

i i

i i v v K i

L i R

− +

−

− +

−

= ˆ

ˆ ) ˆ )(

( ˆ )

( ²

1

α λ

qss qss

qss qss q q s e

s qs s qs s

i i

i i v v K i

L i R

− +

−

− +

−

= ˆ

ˆ ) ˆ )(

( ˆ )

( ²

2

α

λ (14)

If the differentiation of Lyapunov function is negative definite，the system can be guaranteed globally asymptotic stability. By switching gain in equation (10), λ1 and λ2

in equation (14) should be less zero, then ^V<0^satisfy switching condition of sliding mode.

According to Theorem 1, the switching gain of switching surface in equal control of sliding mode is enough small, so that switching condition can be satisfied. When switching condition can be satisfied, the system in equation

(9) will achieve asymptotic stability. Therefore,

=0

= R

R E

E (15) It represent that system dynamic fall onto switching surface.

We use equations (9) and (15), and relationship can be acquired as

ˆ ) ˆ sgn(_ds^s _ds^s

e d

d i i

v K

v = − α −

(16) ˆ )

ˆ sgn(_qs^s _qs^s

q e

q i i

v K

v = − α − (17) We substitute equations (16) and (17) into equation (6), then rotor flux position of motor can be estimated and rotor angular speed of motor can be derived as follows:

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

−

− +

= −

−

=

−

ˆ ) sgn(

ˆ ) tan sgn(

ˆ ) ( ˆ ˆ tan

1 1

s qs s qs q

e

s ds s ds d

e q d r

i i v

K

i i v

K v v

α α θ

(18)

ˆ ) ˆ sin cosˆ (ˆ

ˆ_r 2 v_q _r v_d _r

P θ θ

ω = − (19)

Where P is pole number, θˆ is estimation of rotor flux _r position, ωˆ is estimation of rotor angular speed. _r

In order to avoid charting phenomenon 0f sliding mode, the sign function sgn( tf()) can be replaced by the following equation as

σ + ) (

) ( t f

t

f (20)

where

ε σ ε

σ _≥^<

⎩⎨

=⎧ ()

) ( , 0

0, t f

t

f (21)

where σ₀^andε are all positive constants.

In order to control the rotor speed of the PMSM effectively, a hybrid control system is proposed in this section. The configuration of the proposed HRFNN control system, which combines a supervisory control system, RFNN and compensated control system, is depicted in Fig. 2. The control law is assumed to take the following form [6]:

C R S

t U U U

U = + + ⁽²²⁾

where U_S is the supervisory control, U_R is the RFNN control and U_C is a compensated control. The supervisory control U_S is designed so that the states of the controlled system are stabilized around a redetermined bound region.

Due to the excessive and chattering control effort induced by the supervisory control law, the RFNN control and compensated control are introduced to reduce and smooth the control effort when the system states are inside the predefined bound region. The RFNN control U_R is the main tracking controller used to mimic an idea control law, and the compensated control U_C is designed to compensate the difference between the idea control law and the RFNN control. The supervisory control law fires only when the RFNN approximation properties can not be guaranteed.

(4)

A four-layer RFNN comprises the input (the i layer), membership (the j layer), rule (the k layer) and output layer (the o layer), is adopted to implement the RFNN controller in this study. The signal propagation and the basic function in each layer of the RFNN are introduced as follows:

Layer 1: Input Layer

For every node i in this layer, the net input and the net output are represented as

( )

^, ¹^,²

, 1

1 1

4 1

1

=

∏ −

=

i ) N ( net ) N ( net f ) N ( y

) N ( y w ) N ( x ) N ( net

i i

o o i oi

i

(23) where x₁¹=ω_m−ωˆ_r =e is the command error between the

desired command, ω , and the estimation of rotor angular _m speed, ωˆ ; _r x¹₂ =e(1-z^-1)≡Δe is the command error change, in which z⁻¹ represents a time delay; N denotes the number of iterations; yo⁴ is the output of the RFNN.

Layer 2: Membership Layer

In this layer, each node performs a membership function.

The Gaussian function, is adopted as the membership function. For the jth node

( )

(

^net

( )

^N

) (

^net ^N

)

^j ⁿ

f N y

m N x

net

j j

ij ij j i

, , 1 , ) ( exp ) ( )

(

, )

(

2 2

2 2 2 2

"

=

− −

= σ

(24) where mij and σij are, respectively, the mean and the standard deviation of the Gaussian function in the jth term of the ith input linguistic variable xi² to the node of layer 2, and n is the total number of the linguistic variables with respect to the input nodes.

Layer 3: Rule Layer

Each node k in this layer is denoted by _∏, which multiplies the input signals and outputs the result of product. For the kth rule node

(

^net ⁽^N⁾

)

^net ⁽^N⁾ ^k ^l

f ) N ( y

) N ( x w ) N ( net

k k

j jk j k

, , 1 , ,

3 3

3 3 3

"

=

=∏

(25) where x³j represents the jth input to the node of layer 3;

3

wjk, the weights between the membership layer and the rule layer, are assumed to be unity; ^l=( )ⁿ^/ⁱ ⁱ is the number of rules with complete rule connection if each input node has the same linguistic variables.

Layer 4: Output Layer

The single node o in this layer is labeled with ∑, which computes the overall output as the summation of all input signals

=∑

k ko k

o N w x N ,

net⁴( ) ⁴ ⁴( )

(

( )

)

^{( )}^, ¹

)

( ⁴ ⁴ ⁴

4 N = f net N =net N o=

y_o _o _o _o (26)

where the connecting weight w⁴ko is the output action strength of the oth output associated with the kth rule; xk⁴

represents the kth input to the node of layer 4, yo⁴=UR. Moreover, UR can be rewritten as follows:

Γ Ο

= Ο

= R ^T

R U

U (E ) (27)

where the error signal Ε is the input of the RFNN;

[

^w¹¹⁴ ^w²¹⁴ ^"^"^w^l⁴¹

]

^T

=

Ο is the adjustable parameter vector of the FNN; Γ=[x₁⁴ x₂⁴""x_l⁴]^T , in which x is k⁴

determined by the selected membership function and 1

0≤xk⁴ ≤ .

To develop the compensated control law U_C , first, a minimum approximation error ρ is defined as follows:

( )

^*

*− Ο

=U UF ^E

ρ ⁽²⁸⁾

where Ο is an optimal weight vector achieves the ^* minimum approximation error, and the absolute value of ρ is assumed to be less than a small positive constant, η (i.e.,

η

ρ < ). The error equation can be rewritten as follows:

[ ( ) ]

_⎭⎬^⎫

⎩⎨⎧ − Ο − −

+ Λ

= E Bm U UF E UC UA

E ^*

( )

_⎥⎦^⎤

⎢⎣⎡ + Ο −Ο Γ− − +

Λ

= C A

T

m ρ ^* U U

B

E (29)

Then, the Lyapunov function is defined as

( )^t ⁼ ₂¹ ⁺₂¹

(

^Ο^*⁻^Ο

) (

^T ^Ο^*⁻^Ο

)

V ^E ^PE γ

T (30)

Take the derivative of the Lyapunov function and use (29), then

( )

(

^Ο ^Ο

)

^Ο

t ^T ^T ^T

V = E PE+ E PE− ^* −

γ 1 2

1 2

1

[ ^C ^A]

T

T + −U −U

−

= m ρ

2

1E QE E PB

(

^Ο ^Ο

)

^T^Γ

(

^Ο ^Ο

)

^T^Ο

T m − − −

+E PB ^* ^*

γ

1 (31)

To satisfy V(t)≤0, the adaptation laws Ο and the compensated controller U_C are designed as follows [9]:

Γ

Ο =γETPBm ₍₃₂₎

(

^T ^m

)

UC =ηsgnE PB ⁽³³⁾

where γ >0 is denoted as adaptation gains. Substitute (32) into (31), then

( )^t ^T ^T ^m ^T ^mÛ^C ^T ^mÛÂ

V =− E QE+E PB ρ−E PB −E PB 2

1

C T m

T m

TQE E PB E PB U

E + −

−

≤ ρ

2

1 ₍₃₄₎

use (34), thus

( )^t ^≤⁻¹₂^E^T^QE^≤⁰

V ⁽³⁵⁾

Since 0V(t)≤ , )V(t is negative semidefinite (i.e., )

0 ( ) (t V

V ≤ ), which implies E and (Ο −^* Ο) are bounded using (30). Since all the variables in the right side of (30) are bounded, it implies E is also bounded. Then

)

Ξ is uniformly continuous [15-16]. By using Barbalat’s (t