Indirect Field-oriented Control Of Im Based On Synergetic Control Theory

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Indirect Field-Oriented Control of Induction Machines Based on Synergetic Control Theory Yu Zhang, Zhenhua Jiang, Member, IEEE, Xunwei Yu

Abstract—Field-oriented control is one of the most significant control methods for high performance AC electric machines and drives. In particular, for induction machines, indirect field oriented control is a simple and highly reliable scheme which has essentially become an industry standard. This paper synthesizes and develops an indirect field-oriented speed control for induction motors based on synergetic control theory. Compared with the conventional PI control approach, our results show the speed controller based on synergetic control is more stable, robust and insensitive to system parameter variations. Effects of controller parameter variations on the system performance have also been studied. Index Terms—Indirect field-oriented control, electric drives, synergetic control, induction machines, controller parameters.

ωmech

NOMENCLATURE

the rotor speed , in actual (mechanical) radians per second. ωs supply frequency. ωmechref reference rotor speed. Te electromagnetic torque. TL load torque. P number of poles. Jeq inertial constant. Id, Iq direct- and quadrature-axis components of the induction motor armature current. Vd, Vq direct- and quadrature-axis components of the induction motor voltage. Rs stator resistance. Rr rotor resistance. Ls stator inductance. Lr rotor inductance. Lm mutual inductance.

T

I. INTRODUCTION

HE control of induction machines and drives has been highly developed in recent years. Induction generators have been widely used for wind power generation. Induction motors have been the workhorses in industry for variablespeed applications in a wide power range. There are a number of significant control methods available for induction motors including scalar control, vector or field-oriented control, direct This work was partially supported by the U.S. National Science Foundation under grant ECCS-0652300. Y. Zhang, Z. Jiang, and X. Yu are with the Department of Electrical and Computer Engineering, University of Miami, Coral Gables, FL 33146 USA Email: [email protected], [email protected], [email protected]

©2008 IEEE.

torque and flux control, and adaptive control [1]. Scalar control is aimed at controlling the induction machine to operate at the steady state, by varying the amplitude and frequency of the fundamental supply voltage [2]. A method to use of an improved V/f control for high voltage induction motors was proposed in [3]. The scalar controlled drive, in contrast to vector or field-oriented controlled one, is easy to implement, but provides somewhat inferior performance. This control method provides limited speed accuracy especially in the low speed range and poor dynamic torque response. Vector or field-oriented control has been one of the most significant developments in this area. The invention of vector control in the beginning of 1970s demonstrated that an induction motor could be controlled like a separately excited dc motor [4]. The direct method of field-oriented control requires flux-position information, which can be directly measured or deduced from other motor quantities. Direct measurement requires that the motor be modified to install flux sensors, so this method is not appropriate for generalpurpose industrial motors [5]. In [6], the universal field oriented controller was applicable in all existing field oriented controller schemes due to the generality of its reference fame. Reference [7] develops a decoupling mechanism and a speed control method based on sliding-mode control theory for a direct rotor field-oriented induction motor. In the indirect method of orientation, the flux is estimated from motorinverse dynamics, and one of the three basic implementation schemes based on stator-, airgap-, or rotor-flux orientations can be used. Indirect field oriented control of induction motors is robust and globally stable [8]. In order to regulate the motor state, the partial state feedback linearization together with a proportional-integral controller was used. The most critical disadvantage is that it is very sensitive to parameter variation [9]. To improve the field-oriented control, a full linearizing state feedback control based on differential geometric theory was proposed [10], [11]. These methods required relatively complicated and nonlinear calculations in the control algorithm. A neural network based field oriented control scheme for induction motors was proposed in [12]. Its results showed that ANN could be used as an intelligent alternative to the conventional field-orient control. The detuning correction and efficiency optimization of an indirect field-oriented induction motor drive for achieving higher characteristic were presented in [13]. Measured results showed that both the efficiency and rotor speed dynamic responses were improved by the proposed fuzzy detuning correction control and efficiency optimization control approaches. A new control approach, the synergetic approach to control

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theory, was introduced in [14]-[15]. Recent work has been reported on the application of the synergetic approach to switching power converter control, speed control for PMSM [16]-[18], in which the high performance level, design simplicity and flexibility of synergetic controllers have been demonstrated through both simulation and experiments. Furthermore, the feasibility of designing adaptive synergetic controllers by selecting robust target manifolds was discussed in [19]. Synergetic control theory has several advantages: it’s well suited to digital control; it operates at constant switching frequency which lessens the burden of filtering design. This paper describes an approach for indirect field-oriented control of induction machines based on the synergetic control method, taking speed control of an induction motor as an example. The proposed method can also be applied in control of induction generators. Section II presents a model of the induction motor in state-space form is presented. In section III, a synergetic speed controller for an induction motor is designed. The simulation results are presented and discussed in section IV. Effects of controller parameter variations on the system performance are also studied.

The inherent coupling effect in scalar control of induction motors gives sluggish response and the system is easily prone to instability because of a high-order system effect. This problem can be solved by vector or field-oriented control. It can make an induction motor be controlled like a separately excited DC motor. The control of AC drives can have highperformance. Because of DC machine-like performance, the vector control is also known as decoupling, orthogonal, or transvector control. Before the implementation of any control mode, it was necessary to define the function equations. Because of the explanation above, an induction motor model was established using a rotating (d, q) field reference (without saturation). ωdAλrq Lls Llr Rr ωd λsq R s

d λsd dt

Lm

d λrd dt

Vrd

Vsq

ωd λsd d λsq dt

Lls

Llr

Lm

ωdA λrd d λrq dt

Rr

0

Lm

0

Lr

⎥ ⎦

Hence, the electrical part of an induction motor can be described by a fourth-order model, which is given in (4), by combining equations (1) - (3): ⎡i&sd ⎤ ⎡isd ⎤ ⎡ − Lr 0 Lm 0 ⎤ ⎡Vsd ⎤ ⎢& ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ i 0 Lm ⎥⎥ ⎢Vsq ⎥ − Lr 1 ⎢ isq ⎥ ⎢ sq ⎥ ⎢ 0 ( A⎢ ⎥ + ) (4) = ⎢& ⎥ 2 ⎢L − Ls 0 0 ⎥ ⎢Vrd ⎥ i i ⎢ rd ⎥ Lm − Lr Ls ⎢ rd ⎥ ⎢ m ⎢ ⎥ ⎥ Lm − Ls ⎦ ⎢⎣Vrq ⎥⎦ 0 ⎢ i&rq ⎥ ⎢ irq ⎥ ⎣ 0 ⎣ ⎦ ⎣ ⎦

A=

⎡ Lr Rs ⎢ ⎢ 2 ⎢ − (ωdA Lm −ω s Lr Ls ) ⎢ − Rs Lm ⎢ ⎢ ⎣⎢ − Ls Lm (ω s −ωdA )

ωdA L2m −ωs Lr Ls

− Lm Rr

− Lr Lm (ωs −ωdA ) ⎥

Lr Rs

Lr Lm (ωs −ωdA )

Ls Lm (ωs −ωdA )

Ls Rr − (ωs L2m −ωdA Lr Ls )

− Lm Rr ωs L2m − Lr LsωdA

− Rs Lm

Vrq

(1)

⎤

Ls Rr

⎥ ⎥ ⎥ ⎥ ⎥ ⎦⎥

(5) By superposition, adding the torques acting on the d-axis and the q-axis of the rotor windings, the instantaneous torque produced in the electromechanical interaction is: p Tem = (λrq ird − λrd irq ) (6) 2 Substituting for flux linkages, the electromagnetic torque can be expressed in terms of inductances as: p Tem = Lm (isq ird − isd irq ) (7) 2 Finally the mechanical part of the motor is modeled by: p Tem − TL 2 Lm (isq ird − isd irq ) − TL d = ωmech = (8) dt J eq J eq

Ls = Lsl + Lm

(b) q-axis Fig. 1. Equivalent circuits of induction motor in d-q reference frame

As for the stator, the equation system is: d Vsd = Rs isd + λsd − ωd λsq dt d Vsq = Rs isq + λsq + ωd λsd dt As for the rotor, the equation system is:

⎢ ⎣

⎢ ⎥ ⎢ irq ⎦ ⎥ ⎣

where ωdA = ωslip = ωs − ωm

(a) d-axis

Rs

⎢ ⎥ ⎢ λrq ⎦ ⎥ ⎣

where:

II. MODEL OF INDUCTION MOTORS

Vsd

d λrd − ωdA λrq dt (2) d Vrq = Rr irq + λrq + ωdA λrd dt In this study, we consider a squirrel-cage induction motor, so the d and q-axis components of the rotor voltage are zero. We can relate fluxes to currents as follows: ⎡ λsd ⎤ ⎡isd ⎤ 0 Lm 0 ⎤ ⎡ Ls ⎢ ⎥ ⎢ ⎥ ⎢ λ i 0 L 0 Lm ⎥⎥ s ⎢ sq ⎥ = M ⎢ sq ⎥ given: M = ⎢ (3) ⎢ Lm ⎢ λrd ⎥ ⎢ ird ⎥ 0 Lr 0 ⎥ Vrd = Rr ird +

ωm =

p ωmech 2

ωd = ωs

Lr = Lrl + Lm

III. SYNERGETIC CONTROL DESIGN A. Synergetic Control Synthesis Procedure The general synergetic synthesis procedure is reviewed in this section. We consider an n-dimension nonlinear dynamic system that can be described by the following equation. dx(t ) = f (x , u , t ) (9) dt where x is the system state variable vector, u is the control vector, and t is the time. A controller, which produces the control vector u, is used to force the system to operate in a

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desired manner. The synergetic synthesis of the controller begins by defining a macro-variable given in (10). ψ = ψ ( x, t ) (10) where ψ is the macro-variable and ψ(x,t) is a user-defined function of system state variables and independent time. The objective of the synergetic controller is to direct the system to operate on the manifold ψ =0 (11) The characteristics of the macro-variable can be chosen by the designer according to the control specifications such as the control objective, the settling time, limitations in the control output, and so on. In the trivial case, the macro-variable can be a simple linear combination of the state variables. The same process can be repeated, defining as many macro-variables as control channels. The macro-variable is evolved in a desired manner by introducing a constraint that is expressed in the following equation. Tψ& f ( x, d , t ) + ψ = 0, T > 0 (12) where T is a controller parameter that indicates the converging speed of the closed-loop system to the manifold specified by that the macro-variable equals to zero. Taking into account the chain rule of differentiation that is given by dψ ( x, t ) ∂ψ ( x, t ) dx(t ) = ⋅ (13) dt ∂x dt substitution of (1) and (2) into (4) yields ∂ψ ( x, t ) T f ( x, u , t ) + ψ ( x , t ) = 0 (14) ∂x Upon solving (6) for u, the control law can be found as u = g ( x, t ,ψ ( x, t ), T ) (15) From (15), it can be seen that the control output depends not only on the system state variables, but also on the selected macro-variable and time constant T. In other words, the designer can choose the characteristics of the controller by selecting a suitable macro-variable and a time constant T. B. Synergetic Control Synthesis Procedure The method described in the previous section requires that we define the same number of macro-variables as control channels in the system. Thus, it requires the definition of two macro-variables, which are functions of the state variables as shown in (10). We chose these two terms: ⎧⎪ ψ 1 = isd (16) ⎨ ⎪⎩ψ 2 = K1 (ωmech − ωmechref ) + K 2 isq + K 3 ∫ (ωmech − ωmechref )dt where K1, K2, K3 are controller parameters. This selection is not arbitrary, since the physical reason can be deduced by inspection of the macro-variables.

integral-like action that can be interpreted as negative of the qaxis current reference - isq* in a decoupled speed-torque control scheme. If we choose K2 as 1, then ψ 2 = K 2 isq − isq* = isq − isq* is the negative of the q-axis current error. The objective is then to force this error to zero. Upon substitution of the macro-variables into the evolving equation as shown in (12), we get 0 = T1 (i&sd ) + isd 0 = T2 ( K1ω& mech + K 2 i&sq + K 3 (ωmech − ωmechref )) + {K1 (ωmech − ωmechref ) + K 2 isq

(17)

+ K 3 ∫ (ωmech − ωmechref )dt}

Solving for Vsd, and Vsq, we have 1 Vsd = ( Lr Rs isd + (ωdA L2m − ωs Lr Ls )isq − Lm Rr ird Lr L2 − Lr Ls − Lr Lm (ωs − ωdA )irq + m isd ) T1

(18)

p L2m − Lr Ls K1 2 Lm (isq ird − isd irq ) − TL Vsq = Lr K2 J eq +

1 (−(ωdA L2m − ωs Lr Ls )isd + Lr Rs isq Lr

+ Lr Lm (ωs − ωdA )ird − Lm Rr irq ) L2 − Lr Ls K 3 + m (ωmech − ωmechref ) Lr K2 +

(19)

L2m − Lr Ls K1 (ωmech − ωmechref ) Lr T2 K 2

L2m − Lr Ls 1 L2 − Lr Ls K 3 isq + m (ωmech − ωmechref )dt Lr T2 Lr T2 K 2 ∫ Because the rotor currents can not be measured directly, we use a current model estimator to estimate the rotor current. +

IV. SIMULATION RESULT Fig.2 shows the block diagram of the induction motor control. In this paper we chose a squirrel-cage induction motor. The rotor voltages Vrd and Vrq are zero. The studied synergetic controller has inputs: d-q reference frame stator currents isd, isq, rotor angular speed ωs , rotor mechanical speed ωmech , speed reference ωref , and load torque TL . The synergetic controller computes the control laws and outputs two voltage levels in the d-q reference fame Vsd and Vsq.

The macro-variable ψ 1 in (16), which defines a manifold: ψ 1 = isd = 0 , ensures the convergence of the d-axis current isd to zero on this invariant manifold. Then, we can achieve the rotor speed control by controlling only the q-axis current isq. The macro-variable ψ 2 consists of three terms: isq,

ωmech − ωmechref , ∫ (ωmech − ωmechref )dt . With the last terms with regards to the rotor speed, we can achieve a proportional-

Fig. 2. Induction motor model with a synergetic controller and a rotor current estimator.

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The system is modeled and simulations are performed in MATLAB/Simulink. The load torque is increased from 5 N.m to 10 N.m when t = 6s. All initial conditions of the induction motor and all initial time derivatives are set to zero.

60

40 30

TABLE I SPECIFICATIONS OF INDUCTION MOTOR MODEL

2.77Ω

Rr

2.34Ω

Ls

0.3826 H

Lr

0.3808H

Lm

0.3687 H

ωref

150 rad/s

J

0.025Kg Nm

ωs

120 π

Te

Rs

10 0 -10 4

6

8

10

20

Synergetic

Macro-variable2

0

-20

-40

-60

-80 0

0.5

1

t

(c) Behavior of Macro-variable

80

1.5 -3 x 10

ψ2 PI Synergetic

60 40 20 0 -20 0

2

4

t

6

8

10

(d) Behavior of isd 100

PI synergetic

PI Snergetic

165

t

(b) Behavior of Te

Fig. 5 shows the behavior of the system variables when the synergetic controller parameter K3 changes from 1.5 through 2.0, 2.5 to 3.0. It is shown that increasing K3 gives a bigger overshoot in the rotor speed (at the instant of 6s). This is consistent with that parameter K3 corresponds to the integral term in the macro-variableψ 2 . It is noted that increasing K3 results in a shorter settling time in Te control (at the instant of 6s). It is also seen from Figs. 5-e and 5-f that the change of K3 has a negligible effect on the evolving of the macro-variables.

50 isq

160 155 Wmech

2

isd

The effects of controller parameter variations on the system performance are also studied. In this study, the synergetic controller parameter K1 varies from 0.45 through 0.5, 0.55 to 0.6. Fig. 4 shows that increasing K1 gives a smaller overshoot in the rotor speed. In the meantime, by increasing K1, it takes less time to for the macro-variables to reach the manifolds. But the increase of K1 causes larger oscillations of the electromagnetic torque Te. It can be concluded that increasing K1 helps to achieve better speed control but results in worse torque characteristic. So a proper parameter K1 should be used in practical applications to a trade off different factors.

170

20

-20 0

The simulation results of an induction motor control with the synergetic controller and a traditional PI controller are presented in Fig. 3. Compared with the PI controller, the synergetic controller has better dynamic characteristics: the overshoot under the synergetic control is smaller; the settling time is shorter; it tracks the speed reference faster than the PI controller. Notice, when the load torque TL increases its value, there is a small transitory oscillation on the tracking of the actual speed to the reference, which is stabilized very quickly. It can be seen that the synergetic mode offers a robust control compared to the PI control particularly when applying a load torque change to the system (at the instant of 6s).

PI Synergetic

50

150

0

145 140

-50 0

135 130 0

2

4

t

6

(a) Behavior of wmech

8

10

2

4

6 8 10 t (e) Behavior of isq Fig. 3. Behavior of system variables under synergetic control and PI control

180

0.01

160

0.005 isd

Wmech

5

140

0 -0.005

120 100 0

0.2

0.4

t

0.6

0.8

-0.01 0

1

2

4

6

t

8

10

0.8

1

(e) Behavior of isd (Marco-variable1)

(a) Behavior of Wmech, from 0 to 1s 100

155

80

150 isq

Wmech

60 40

145

140

Increasing K1

20 0 0

6

6.2

6.4

6.6

t

6.8

0.2

0.4

7

0.6

t

(f) Behavior of isq, t is from 0 to 1s (b) Behavior of Wmech, from around 6s to 7s

0

10 -20 Macro-variable2

8

Te

6 4

-40 -60 -80

2 -100 0

0 0.5

1

1.5

t

2

2.5

0.2

0.4

t

3

0.6

0.8

1

-3

x 10

(g) Behavior of macro-variable2 , from 0s to 0.0001s (c) Behavior of Te, from 0.5s to 3s

0.01

12

0.005 Macro-variable2

11 10 Te

9 8 7

0 -0.005 -0.01 K1 increasing

-0.015

6

-0.02

5 6

6.2

6.4

t

6.6

(d) Behavior of Te, from around 6s to 7s

6.8

7

6

6.1

6.2

t

6.3

6.4

6.5

(h) Behavior of macro-variable2, from around 6s to 6.5s Fig. 4. Behavior of system variables under synergetic control with different values of K1

180

0.01

160

0.005

isd

Wmech

6

140

0

-0.005

120

100 0

0.2

0.4

t

0.6

0.8

1

-0.01 0

(a) Behavior of Wmech, from 0s to 1s

2

4

t

6

8

10

(e) Behavior of isd (Marco-variable1)

155

20

macro-variable2

0

Wmech

150

145

-20 -40 -60 -80

140

6

6.5

7 t (b) Behavior of Wmech, from around 6s to 7.5s

7.5

-100 0

0.5

1 t

(f) Behavior of macro-variable

120

Te

ψ2

Fig. 6 shows the system performance when the synergetic controller parameter T varies from 0.001 to 0.0001 and 0.00001, It is shown in Figs. 6-a and 6-b that the change of parameter T does not have a strong impact on the rotor speed. It is seen from Figs. 6-c and 6-d that the change of T has a notable impact on the regulating speed of the synergetic control. This is verified by the fact that the evolving of macrovariables strongly depends on the time constant of the firstorder evolving equation.

100 80 60 40 20 0 0

0.2

0.4

t

0.6

0.8

1

(c) Behavior of Te, from 0s to 1s

12

10

Te

2

-3

x 10

Fig. 5. Behavior of system variables under synergetic control with different values of K3

140

8

6

4

1.5

6

6.2

6.4

t

6.6

6.8

(d) Behavior of Te, from around 6s to 7s

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V. CONCLUSION A synergetic control approach has been presented to design indirect field-oriented control of induction motor drives in this paper. The operating characteristics of the induction motor with the proposed synergetic controller are compared with those using a conventional PI controller. Simulation results show that the induction motor under the synergetic control approach is more stable, robust and insensitive to parameter variations than under the PI control scheme. The effects of the parameter variations of the synergetic control on the induction motor’s transient and steady-state performances are studied. The results show that the controller with bigger K1 has better dynamic characteristics of speed control but worse stable characteristics of Te; increasing K3 results in worse dynamic characteristics of speed control but better characteristics of Te; increasing Te gives a more gentle approaching of the macrovariable to the manifold. Therefore, in order to achieve good

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control performances, it is necessary to consider all effecting factors to make an optimal decision on the choice of controller parameters.

REFERENCES [1] [2]

161 160.5

[3]

Wmech

160 159.5

[4]

159

[5]

158.5

T=0.00001 T=0.001 T=0.0001 157.5 0.1 0.12

[6]

158

0.14

0.16 t

0.18

0.2

[7]

(a) Behavior of Wmech, around 0.16s

[8]

142.8

[9]

Wmech

142.78 142.76

[10]

142.74 T=0.00001 T=0.001 T=0.0001 6.105 6.11

142.72 142.7 6.085

6.09

6.095

t

6.1

[11] [12]

(b) Behavior of Wmech, around 6.1s [13]

0.01

isd

0.005

[14]

0

[15]

-0.005

-0.01 0

[16] 2

4

t

6

8

10

[17]

(c) Behavior of isd (Marco-variable1) 10

[18]

macro-variable2

0 -10

[19]

-20 -30 -40 -50 0

0.01

0.02

0.03 t

(d) Behavior of macro-variable

0.04

T=0.0001 T=0.001 T=0.00001 0.05 0.06

ψ 2 , from 0 to 0.06s

Fig. 6. The behavior of system variables under synergetic control for different values of T

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