Passivity-based torque and flux tracking for induction motors with magnetic saturation

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Automatica 39 (2003) 2123 – 2130

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Brief Paper

Passivity-based torque and #ux tracking for induction motors with magnetic saturation
Hossam A. Abdel Fattaha , Kenneth A. Loparob;∗;1 a Electrical

b Electrical

Power and Machines Department, Faculty of Engineering, Cairo University, Giza, Egypt Engineering and Computer Science Department, Case Western Reserve University, Cleveland, OH, USA Received 18 May 1999; received in revised form 3 April 2003; accepted 4 July 2003

Abstract An observer-based, globally asymptotically stable torque and rotor #ux magnitude tracking controller for induction motors under magnetic saturation is proposed. The controller is synthesized using the passivity-based techniques. The paper considers a magnetically saturated -model of the motor without any simplifying assumptions. Motor #uxes are reconstructed by a closed-loop observer. Closed-loop stability of the overall scheme including the observer is demonstrated. Simulation results are given to illustrate the proposed scheme. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Induction motors; Magnetic saturation; Passivity-based control

1. Introduction A tremendous amount of e9ort has been invested by control theoreticians to propose solutions to the challenging induction motor control problems. The recent theoretical developments in induction motor control can be roughly classi:ed into feedback linearizing schemes (Marino, Peresada, & Tomei, 1999; Chiasson, 1998), Lyapunov-based designs (Kanellakopoulos, Krein, & Disilvestro, 1992), variable structure control (Utkin, 1993; Zheng, Abdel Fattah, & Loparo, 2000) and more recently the passivitybased approach, surveyed in Ortega, Loria, Nicklasson, and Sira-Ramirez (1998) and used in this paper. Passivity-based control methods, already successful in robotics and in the global stabilization of aAne and nonaAne systems, have been proposed to solve torque control problem for induction motors (see among others Espinosa-Perez,
This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Carlos Caudas de Wit under the direction of Editor Hassan Khalil. ∗ Corresponding author. Tel.: +1-216-368-4115; fax: +1-216-368-3128. E-mail address: [email protected] (K. A. Loparo). 1 Author work has been supported in part by the National Science Foundation Grant ECS-9906218.

0005-1098/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0005-1098(03)00251-6

Ortega, & Nicklasson, 1997), which is a subset of the general class of smooth air gap electromechanical machines, generally known as Blondel–Park transformable (Nicklasson, Ortega, & Espinosa-Perez, 1997). The method has surprisingly led to the :rst nonobserver-based scheme that does not require the measurement nor the estimation of rotor states (Espinosa & Ortega, 1994). Even though the method applies primarily to torque control, it is shown in Ortega, Nicklasson, and Espinosa-Perez (1996), how it can be extended to speed control provided that the load torque is known. In our recent paper (Abdel Fattah & Loparo, 2001), we remove the assumption of known load torque and provide a solution to the problem of speed control given any passivity-based torque tracking controller. The common assumption made in the development of these control laws is the linearity of the magnetic circuit of the machine, usually justi:ed by including the #ux magnitude in the outputs to be regulated by the controller and keeping this magnitude regulated at a value far from the saturation region (Marino et al., 1999). However there are no guarantees that the #ux magnitude remains in the linear magnetic region during machine transients. Moreover in many variable torque applications, it is desirable to operate the machine in the magnetic saturation region to allow the machine to develop higher torque (Sullivan & Sanders, 1995). Saturation e9ects are also known to be pronounced in

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H. A. Abdel Fattah, K. A. Loparo / Automatica 39 (2003) 2123 – 2130

drives operating in the :eld weakening region, or in drives that operate with varying #ux levels to achieve optimality in a speci:ed sense (Kirshnen, Novotny, & Lipo, 1985). Recently, research attention has been devoted to advanced induction motor control under magnetic saturation. Feedback input–output linearization schemes for induction motors with magnetic saturation were proposed in a :xed stator frame (Tarbouchi & LeHuy, 1998) and in a synchronously rotating frame (Novotnak, Chiasson, & Bodson, 1999). While in Tarbouchi and LeHuy (1998) the control signal is the stator voltage, in Novotnak et al. (1999) it is the stator current. Both papers treat the classical T -model of an induction motor. Unfortunately, due to the complicated nature of the T -model, drastic simpli:cations are required to facilitate the use of this model in nonlinear control synthesis. The major drawback in Tarbouchi and LeHuy (1998) (also present in the optimal #ux reference selection of Novotnak et al. (1999)) is the assumption that the stator and rotor leakage parameters s and r , as de:ned in Leonard (1985), are equal and constant. This assumption has the indirect e9ect of neglecting any cross-saturation e9ect that might appear in the dynamics of the motor. On the other hand, the model in Novotnak et al. (1999) is obtained by :rst simplifying the motor equations assuming a linear magnetic circuit and then including a mutual inductance that varies with mutual current. This approach does not include derivatives of the saturation function that should appear in a complete model (Levi, 1995). A similar modeling approach can also be found in Gokdere (1996) for incorporating magnetic saturation in a passivity-based control design methodology. It is worth pointing out that, in Gokdere (1996) similar to Novotnak et al. (1999), stator currents are used as the control signal. In our work (Abdel Fattah & Loparo, 1999), we showed that considering magnetic saturation explicitly in nonlinear control synthesis is of foremost importance especially when the machine is voltage actuated. Because the -model was experimentally found in Sullivan and Sanders (1995) to be better suited to capture the nonlinear magnetic e9ects, we propose in this paper a passivity-based method using voltage for torque and #ux tracking of an induction motor -model with magnetic saturation. To the best of our knowledge, this work is the :rst of its kind in this direction given the importance of this problem and the informal treatment that it has been given in the past. This work di9ers from previous work in magnetically saturating induction motor nonlinear control in several directions. First, no simplifying assumptions are used in the development of the model. Second, contrary to Tarbouchi and LeHuy (1998) and Novotnak et al. (1999), the closed-loop stability of the scheme is proven even with the use of an observer for #ux reconstruction. And third, we consider a voltage-based control law as opposed to a current-based control law, which is shown to be inherently robust to magnetic saturation in Abdel Fattah and Loparo (1999).

2. Preliminaries The two phase electrical equations for an induction motor in an arbitrary frame rotating at a speed !a (Leonard, 1985) are given by the vectorial equations Vs = Rs Is + ˙ s + !a J2 s ; 0 = Rr Ir + ˙ r + (!a − p!)J2 r :

(1)

Vs is the stator phase voltage vector, Is is the stator phase current vector, Ir is the rotor phase current vector, p is the number of pole pairs, ! is the rotor speed, Rs is the stator phase resistance, Rr is the rotor phase resistance, s and r are the stator and rotor #ux linkage vectors, respectively, and J2 is the skew-symmetric matrix given by
 0 −1 : (2) J2 = 1 0 Eq. (1) holds whether the induction motor magnetic circuit is considered linear or saturated. The mechanical equation can be expressed as J !˙ + b! = T − TL ;

(3)

where J ¿ 0 is the inertia, b ¿ 0 is the viscous damping , TL is the load torque and T is the generated torque. The induction motor -model was proposed in Sullivan and Sanders (1995). The model derivation is based on the simpli:ed tooth structure for a single pair of teeth in the motor. The basic di9erence between the -model and the classical T -model is in the relationship between the di9erent #uxes and currents. In the -model, Eq. (4) relates the stator and rotor currents to the stator and rotor #uxes.


 
Is Gs ( s )

s −gl I2 g l I2 = ; (4) + Ir Gr ( r ) −gl I2 gl I2

r where I2 is the identity matrix of dimension 2 and gl is de:ned as the reciprocal of the leakage inductance (Ll ), i.e. gl = 1=Ll . Here, Gs and Gr are the stator and rotor vector-valued nonlinear functions de:ned as Gx : R2 → R2 such that 

 imxa xa Gx ( x ) = Gx = = Imx ; (5) imxb xb where imxa =

gx1 ( x )  x 

xa

, gx ( x )

xa ;

(6)

imxb =

gx1 ( x )  x 

xa

, gx ( x )

xb ;

(7)

and the subscript x can be either s for stator or r for rotor, while the subscripts a and b denote the respective phase components for each variable. The scalar saturation

H. A. Abdel Fattah, K. A. Loparo / Automatica 39 (2003) 2123 – 2130

functions gs and gr only a9ect the magnitude, while keeping the directions of the #uxes and currents the same. Finally, the generated torque T is given by T = pgl sT J2 r :

(8)

Remark 1. It is worth pointing out that when the magnetic circuit of the machine is assumed to be linear, the T -model and the -model are identical. In this case, the inductance values are not only constant in both models, but they are also related by the standard –T transformation. But as it appears from the derivations above, a substantial di9erence appears when the magnetic circuit is nonlinear to include saturation. In such a case, a direct correspondence between the two models is not possible. The scalar nonlinear functions g1 and g used for the -model, introduced in Eqs. (6) and (7) represent the magnetic saturation. Because these functions relate the currents to the #uxes, they have the characteristics of an inverse saturation function. Because the function g can be considered as the inverse of a nonlinear inductance, the following properties are easily veri:ed: 1

g : R¿0 → R¿0 g1 (0) = 0

and

and

g : R¿0 → R¿0 ;

g(0) = lim+

x ¿ y ⇒ g1 (x) ¿ g1 (y)

x→0

and

g1 (x) ¿ 0; x g(x) ¿ g(y):

(9) (10) (11)

The following technical lemma, whose simple proof is omitted, will be used latter in the proof of the main results. Lemma 2. If X = [x1 x2 ]T ∈ R2 and Y = [y1 y2 ]T ∈ R2 are real vectors; and  : R¿0 → R¿0 is a monotone increasing function, then [(X )X − (Y )Y ]T [X − Y ] ¿ 0;

(12)

∀(X; Y ) ∈ W , where W = {(X; Y ) ∈ R2 × R2 |X = Y }: Moreover, [(X )X − (Y )Y ]T [X − Y ] = 0 ⇔ X = Y:

(13)

3. Model properties Based on the discussions of the previous section and dropping the arguments of the scalar functions without risk of confusion, the saturated -model of the induction motor model can be compactly written as given in Eq. (14)

˙ + Ce (!) + Re ( ) = Qe ;

(14a)

J !˙ + b! = T − TL ;

(14b)

2125

where = [ s r ]T , Qe = [Vs 0]T and 
−Rs gl I2 Rs (gs + gl )I2 ; Re ( ) = −Rr gl I2 Rr (gr + gl )I2 
0 0 Ce (!) = p! : 0 −J2

(15) (16)

The following properties can be easily established: • Ce (!) is skew-symmetric ∀! ∈ R. • Re ( ) is positive de:nite ∀ ∈ R4 . This can be easily shown by calculating the minimum eigenvalue !min (Re ( )) of the matrix Re ( ) which is always greater than zero. Moreover it can be easily shown that the electrical subsystem is passive by considering the following ‘energylike’ function: H ≡ 12 T . Di9erentiating with respect to time along the trajectory of (14), gives H˙ = T Qe − T Re ( )

that is to say H (t) − H (0) =

 0

t

(17)

T Qe dt −

 0

t

T Re ( ) dt

(18)

because H (t) ¿ 0, the passivity of the electrical subsystem from Qe to , ∀t ¿ 0 is established through the inequality  t  t

T Qe dt ¿ !min (Re ( ))

T dt − H (0): (19) 0

0

Mechanical subsystem (3) is strictly positive real for J ¿ 0 and b ¿ 0 and thus is passive. Having established these properties, the design of the controller can thus proceed according to the standard passivity-based approach. 4. Main result This main result of this work is obtained in two steps. First is the synthesis of a state feedback controller for the saturated -model of the induction motor that achieves both torque and rotor #ux magnitude tracking. Second is the design of a simple closed-loop observer for the estimation of the states of the -model (stator and rotor #uxes) from the available measurements, e.g. the stator currents and the rotor speed. The methodology used to design the controller is the so-called passivity-based approach introduced in the induction motor context in Ortega, Canuda, and Seleme (1993), explained in Ortega and Espinosa (1993) and in the recent book (Ortega et al., 1998). The main idea is to :rst express the system in a form that shows the workless nonlinear forces that do not require cancellation. A target energy function is then chosen for the closed loop system that is consistent with the structure of the system. Finally a control law and desired state trajectories are synthesized so that the

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H. A. Abdel Fattah, K. A. Loparo / Automatica 39 (2003) 2123 – 2130

energy function equals this target, while ensuring that the output tracks the desired signals. In this treatment, the passivity design is applied to the electrical equations, leaving the load torque as a passive disturbance as in Espinosa and Ortega (1994) and Ortega et al. (1996). The observer design is also quite simple by taking advantage of the Euler–Lagrange form of the -model of the induction motor. Both the controller and the observer are included in the result given below. Before the statement of the main result, the following assumptions are made regarding the desired torque and the desired rotor #ux magnitude trajectories: (A1). The desired torque Td (t) is a C 1 bounded function of time with bounded :rst derivative. (A2). The desired rotor #ux magnitude ’(t) is a C 2 bounded function of time with bounded :rst and second derivatives, that satis:es ’(t) ¿ 0

∀t ∈ R:

(20)

Theorem 3. For the magnetically saturated -model of the induction motor given by (14), for a smooth and bounded load torque TL , the following output feedback voltage-driven control law provides global tracking with all internal signals bounded for any desired torque Td (t) and rotor 7ux magnitude signals ’(t) that satisfy assumptions (A1) and (A2) Vs = L(Iˆs − Is ) ˙ sd + Rs (gs ( ˆ s ) + gl ) sd − Rs gl rd + K11 ( sd − ˆ s ) + K12 ( rd − ˆ r );

(21)

sd =

’(t) ˙ + R r gl 

+ +



1 K22 ( rd − ˆ r ) Rr g l

(22)

and


rd = ’(t)

cos 'd



sin 'd

, ’(t)&;

(23)

where 'd is the solution of the di9erential equation '˙d = p! +


+ g l Rr

1 Rr Td (t) +

T J2 K22 ( rd − ˆ r ) p’(t) ’(t)2 rd

with 'd (0) ∈ R

(24)

0

0




ˆ s




+ Rr gr ( ˆ r )

−I2 I2

ˆ r

 Vs L(Is − Iˆs ) = ; + 0 0

0



ˆ r (26)

where L ∈ R+ is a positive constant observer gain and Iˆs is the estimated stator current, which is given by Iˆs = gs ( ˆ s ) ˆ s + gl ( ˆ s − ˆ r ):

(27)

Proof. The :rst step is to show that if converges to

d then necessarily the rotor #ux magnitude  r  and the generated torque T converge to ’(t) and Td , respectively. From the choice of rd in Eq. (23), the :rst requirement is obviously achieved. In order to demonstrate the second requirement consider the following equation:


’(t) ˙ = ’(t) + Rr g l

  gr ( ˆ r ) + 1 ’(t) & gl

 Td (t) 1 T

rd J2 K22 ( rd − ˆ r ) J2 & + pgl ’(t) Rr gl ’(t)

(25)

The estimated states are given by the following observer:   



ˆ˙ s Is

ˆ s   + Ce (!) + Rs 0

ˆ r

ˆ˙ r

T J2 rd

sd

with


and K is the positive semi de:nite matrix 
K11 K12 : K= 0 K22

=



  gr ( ˆ r ) + 1 ’(t) &T J2 & gl

+ ’(t)

Td (t) ’(t) T &T J2T J2 & + ( rd − ˆ r )T K22 & pgl ’(t) Rr g l

+ ’(t)

1

T J2 K22 ( rd − ˆ r )&T J2T J2 & Rr gl ’(t) rd

1 Td (t) +

T J2 K22 ( rd − ˆ r ) pgl Rr gl rd +

’(t) T & K22 ( rd − ˆ r ): Rr g l

From Eq. (8), if → ˆ and ˆ → d then necessarily T → Td . The dynamics of the closed-loop system can be divided into three parts. The :rst part is the observer error dynamics, the second part is the dynamics of the error between the observer states and the desired states and the last part is the mechanical equation. These dynamics are explicitly given below after de:ning the following error vectors: 1. The tracking error e is de:ned as the di9erence between the desired #ux trajectory d , and the #ux vector for the actual machine, i.e. e = − d .

H. A. Abdel Fattah, K. A. Loparo / Automatica 39 (2003) 2123 – 2130

2. The observer error ˜ is de:ned as the di9erence between the #ux vector , and the observer #ux vector ˆ as ˆ

˜ = − . 3. The error between the observer states and the desired states is de:ned by + = e − ˜ = ˆ − d . The observer error dynamics are given by

˜˙ = −L[g ( ) − g ( ˆ ) ˆ ] s

s

s

s

s

s

s

− L[gl ˜ s − gl ˜ r ];

(28)

˜˙ r = p!J2 ˜ r − Rr gl ˜ r + Rr gl ˜ s − Rr [gr ( r ) r − gr ( ˆ r ) ˆ r ]:

s

(29)

sd

= −Rs (gs ( ˆ s ) + gl )+ s − Rs gl + r + K11 + s + K12 + r ;

(30)

+˙ r = ˆ˙ r − ˙ rd

1 T J2 & rd J2 [K21 + s + K22 + r ]: ’(t)

Rr Td J2 rd − [K21 + s + K22 + r ]: p’(t)

(31)

(32)

− p!J2 rd + Rr [gl + gr ( ˆ r )] rd − Rr gl sd (33)

which, after rearranging the terms, :nally leads to +˙ r = p!J2 + r − Rr [gl + gr ( ˆ r )]+ r (34)

The dynamics of the error between the observer states and the desired states can thus be compactly written as ˆ + K)+ = 0: +˙ + Ce (!)+ + (Re ( )

T T V1 = 12 (+

+ + +

+ ): s s r r

(37)

If ! is assumed not to have a :nite escape time behavior, the derivative of V1 can be computed as T V˙ 1 = −+

(Re ( ) + K)+ :

V2 =

Rr ˜ T ˜ L

s s + ˜ Tr ˜ r : 2 2

(38)

(39)

The derivative of V2 along the solution of the observer error dynamics is given by V˙ 2 = −LRr [gs ( s ) s − gs ( ˆ s ) ˆ s ]T ˜ s − LRr gl ˜ Ts ˜ s + LRr gl ˜ Tr ˜ s − LRr gl ˜ Tr ˜ r + LRr gl ˜ Ts ˜ r − LRr [gr ( r ) r − gr ( ˆ r ) ˆ r ] ˜ r :

− LRr [gr ( r ) r − gr ( ˆ r ) ˆ r ] ˜ r :

(40)

(35)

The last part is the mechanical equation given by Eq. (43).

(41)

Hence the derivative of V along the solution of the closed-loop system is V˙ = −LRr [gs ( s ) s − gs ( ˆ s ) ˆ s ] ˜ s T − LRr gl ( ˜ s − ˜ r )T ( ˜ s − ˜ r ) − +

(Re ( ) + K)+

− LRr [gr ( r ) r − gr ( ˆ r ) ˆ r ] ˜ r :

+˙ r = p!J2 ˆ r − Rr [gl + gr ( ˆ r )] ˆ r + Rr gl ˆ s

+ Rr gl + s − [K21 + s + K22 + r ]:

where V1 is given by

− LRr gl ( ˜ s − ˜ r )T ( ˜ s − ˜ r )

Substituting for Td and then adding and subtracting Rr gl sd results in

− [K21 + s + K22 + r ];

(36)

V˙ 2 = −LRr [gs ( s ) s − gs ( ˆ s ) ˆ s ] ˜ s

Substituting for & from Eq. (23) and using the expression of rd gives ’(t) ˙ +˙ r = p!J2 ˆ r − Rr [gl + gr ( ˆ r )] ˆ r + Rr gl ˆ s −

rd ’(t) − p!J2 rd −

V = V 1 + V2 ;

Rearranging the terms, one obtains

= p!J2 ˆ r − Rr [gl + gr ( ˆ r )] ˆ r

R r Td ˆ + Rr gl s − ’(t)& ˙ − ’(t)J2 & p! + p’(t) −

Consider the composite quadratic Lyapunov function

Moreover, V2 is given by

The dynamics of the error between the observer states and the desired states are given by +˙ = ˆ˙ − ˙

s

2127

(42)

Notice that in (42) [gs ( s ) s − gs ( ˆ s ) ˆ s ] ˜ s and [gr ( r ) r − gr ( ˆ r ) ˆ r ] ˜ r are positive de:nite by Lemma 2, that ( ˜ s − ˜ r )T ( ˜ s − ˜ r ) is nonnegative and that T +

(Re ( ) + K)+ is positive de:nite. Hence V˙ is negative de:nite. Because d is bounded as a direct consequence of assumptions (A1) and (A2), the choice of the reference

d and the control voltage Vs gives a closed loop system characterized by the dynamics (35), (28), (29) and the mechanical dynamics J !˙ + b! = pgl sT J2 r − TL :

(43)

Considering the Lyapunov function given in (36), whose derivative is given in (42), global asymptotic tracking is guaranteed if it can be shown that ! does not have a :nite escape time. Because of the assumptions on the load torque and the form of the desired trajectory d , the closed loop system Eqs. (35), (28) and (29) are locally Lipschitz.

2128

H. A. Abdel Fattah, K. A. Loparo / Automatica 39 (2003) 2123 – 2130 18

1.4 Desired Actual

16

Desired Actual

1.2 1

Generated Torque

Rotor Flux Magnitude

14

0.8

0.6

12 10 8 6 4

0.4

2 0.2 0

0 -2

0

1

2

3

4

5 6 Time

7

8

9

0

10

1

2

3

4

5 6 Time

7

8

9

10

8

9

10

Fig. 2. Generated torque.

Fig. 1. Rotor #ux magnitude.

250 Phase A Phase B

200 150 Applied Stator Voltage

Hence, there exists a time interval [t0 ; t1 ] where the solution exists and is unique without :nite escape time behavior. On any :nite interval [t0 ; t1 ], the Lyapunov argument, the boundedness of d (boundedness of 'd is not required since it enters through bounded sin and cos functions in the expression of rd ) and the boundness of TL guarantees that the stable linear di9erential equation with time varying input given by (43) has bounded right-hand side. Recalling standard results (Khalil, 1996), we know that ! does not have a :nite escape time. The arguments above are independent of the initial time t0 and the solution can be extended to the entire positive real time axis, therefore ! has no :nite escape time. Boundedness of ! follows from the fact that it is the solution of the passive system (3) with bounded input.

100 50 0 -50 -100 -150 -200 -250

0

1

2

3

4

5 Time

6

7

Fig. 3. Stator applied voltage.

0.2 Phase A Phase B

5. Simulation results 0.1 0 Stator Flux Error

To evaluate the performance of the proposed controller/observer developed for the induction motor -model with nonlinear magnetic circuit simulations are used. All the simulations are done for an induction motor whose saturation functions are given by Is  = 0:5495 sinh(2 s ) and Ir  = 0:5495 sinh(2 r ) with parameters given as follows: Rs = 8,, Rr = 6,, Ll = 0:062 H, J = 0:06 kg m2 , b = 0 N s m−1 and p = 2. The gain matrix K is given by K = diag(25). The load torque is not assumed to be constant but is modeled by TL = [2 0:2 0:01][1 ! !2 sign(!)]T . The desired trajectories for the #ux and the torque are ’ = 1 + 0:2 sin(t=5) and Td = 10 + 7 sin(t), respectively. The observer gain is L = 100 and the initial conditions for the observer states are 0 = [0:2 0:5 − 0:1 0:3]. Figs. 1–5

-0.1

-0.2 -0.3

-0.4 -0.5

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time

Fig. 4. Zoom of the observer error in stator #ux.

1

H. A. Abdel Fattah, K. A. Loparo / Automatica 39 (2003) 2123 – 2130

7. Conclusion

0.15 Phase A Phase B

0.1 0.05 0 Rotor Flux Error

2129

-0.05 -0.1 -0.15 -0.2 -0.25 -0.3 -0.35 0

0.1

0.2 0.3 0.4 0.5 0.6 0.7 0.8 Time

0.9

1

Fig. 5. Zoom of the observer error in rotor #ux.

successively depict the trajectories of rotor #ux magnitude, the electric torque generated, the applied stator voltage, the observer error in stator #ux and the observer error in rotor #ux.

6. Linear magnetic circuit case An important question that might arise is: How does the proposed passivity-based control algorithm, if modi:ed for a linear magnetic model, compare to existing passivity-based controllers that are also based on linear magnetic circuit models of the motor? The answer to this question can be divided into several points: 1. The proposed algorithm is simpler. This can be seen by the requirement that the controller gain matrix K is constant and not speed-dependent as is typical in the existing passivity-based control schemes. 2. As a consequence of K being constant, the convergence rate is speed independent as opposed to other passivity-based schemes where the damping that is added is proportional to the square of the rotor speed. This is a good advantage for tracking at very low speeds. Moreover, the matrix K is only required to be positive semi-de:nite to achieve asymptotic tracking. This gives more freedom in the choice of K as opposed to other schemes where typically a nonzero lower bound exists on the matrix gain. Thus, K can be more easily chosen to achieve the desired transient performance. 3. The proposed algorithm is more robust as a consequence of two facts. The newly added term proportional to the error in rotor #ux tracking ( rd − r ), which appears in the equations of the controller state 'd and the desired stator #ux trajectory sd , introduces a ‘closed-loop e9ect’ in the dynamics of both.

Because the need for explicitly including magnetic saturation e9ects into the synthesis of nonlinear control laws has recently been established, this paper addresses the problem of :nding a suitable output feedback law that takes this e9ect into consideration. Due to the attractiveness of passivity-based design, this paper solves the problem using this technique. A torque and #ux magnitude tracking output feedback controller, using speed and stator current measurements, is designed. Global asymptotic stability of the scheme is proven. An important characteristic of the proposed control scheme is that when it is applied to a linear magnetic model, it results in a simpler and more robust control algorithm than existing passivity-based controllers given in the literature. Simulation results are provided to illustrate the feasibility of the scheme. The scheme can be used for speed control by applying our recent contribution (Abdel Fattah & Loparo, 2001). An analytical study of the robustness with respect to rotor resistance variations and the possibly of including actuator saturation e9ects (i.e. voltage saturation) in the design is currently underway. References Abdel Fattah, H. A., & Loparo, K. A. (1999). Induction motor control system performance under magnetic saturation. In Proceedings of the American control conference, San Diego, CA (pp. 1668–1672). Abdel Fattah, H. A., & Loparo, K. A. (2001). Speed control of electrical machines: Unknown load torque case. IEEE Transactions on Automatic Control, 46(12), 1979–1983. Chiasson, J. (1998). A new approach to dynamic feedback linearization control of an induction motor. IEEE Transactions on Automatic Control, 43(3), 391–397. Espinosa, G., & Ortega, R. (1994). State observers are unnecessary for induction motor control. Systems Control Letters, 23, 315–323. Espinosa-Perez, G., Ortega, R., & Nicklasson, PerJ. (1997). Torque and #ux tracking of induction motors. International Journal of Robust and Nonlinear Control, 7, 1–9. Gokdere, L. U. (1996). Passivity-based methods for control of induction motors. Ph.D. thesis, University of Pittsburgh. Kanellakopoulos, I., Krein, P. T., & Disilvestro, F. (1992). Nonlinear #ux observer-based control of induction motors. In Proceedings of the American control conference (pp. 1700 –1704), Chicago, IL, USA. Khalil, H. K. (1996). Nonlinear systems (2nd ed.). Upper Saddle River. NJ: Prentice-Hall. Kirshnen, D. S., Novotny, D. W., & Lipo, T. A. (1985). On-line eAciency optimization of a variable frequency induction motor drive. IEEE Transactions on Industrial Applications, IA-20(4), 610–616. Leonard, W. (1985). Control of electric drives. Berlin: Springer. Levi, E. (1995). A uni:ed approach to main #ux saturation modeling in d–q axis models of induction machines. IEEE Transactions on Energy Conversion, 10(10), 455–461. Marino, R., Peresada, S., & Tomei, P. (1999). Global adaptive output feedback control of induction motors with uncertain rotor resistance. IEEE Transactions on Automatic Control, 44(5), 967–983. Nicklasson, P. J., Ortega, R., & Espinosa-Perez, G. (1997). Passivity-based control of a class of Blondel–Park transformable electric machines. IEEE Transactions on Automatic Control, 42(5), 629–647. Novotnak, R. T., Chiasson, J., & Bodson, M. (1999). High performance motion control of an induction motor with magnetic saturation. IEEE Transactions on Controlled System Techniques, 7(3), 315–327.

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Ortega, R., Canuda, C., & Seleme, S. (1993). Nonlinear control of induction motors: Torque tracking with unknown load disturbance. IEEE Transactions on Automatic Control, 38(11), 1675–1680. Ortega, R., & Espinosa, G. (1993). Torque regulation of induction motors. Automatica, 29(3), 621–633. Ortega, R., Loria, A., Nicklasson, P. J., & Sira-Ramirez, H. (1998). Passivity-based control of Euler–Lagrange systems: Mechanical, electrical and electromechanical applications. Berlin: Springer. Ortega, R., Nicklasson, PerJ., & Espinosa-Perez, G. (1996). On speed control of induction motors. Automatica, 32(3), 455–460. Sullivan, C. R., & Sanders, S. R. (1995). Models for induction machines with magnetic saturation of the main #ux path. IEEE Transactions on Industrial Applications, 31(4), 907–917. Tarbouchi, M., & LeHuy, H. (1998). Control by exact linearization of an induction motor in :eld weakening regime. In IECON Proceedings, Aachen, Germany (pp. 1597–1602). Utkin, V. (1993). Sliding mode control design principles and applications to electric drives. IEEE Transactions on Industrial Electronics, 40(1), 23–36. Zheng, Y., Abdel Fattah, H. A., & Loparo, K. A. (2000). Non-linear adaptive sliding mode observer-controller scheme for induction motors. International Journal of Adaptive Control and Signal Processing, 14, 245–273. Hossam A. Abdel Fattah was born in Giza, Egypt, in 1971. He received the B.Sc. and M.Sc. both with honors in Electrical Engineering from Cairo University, Egypt, in 1993 and 1995, respectively. In 1999, he was awarded the Ph.D. degree in Electrical Engineering and Computer Science (Systems and Control Division) from Case Western Reserve University, Cleveland, OH. Since 1999, he is a tenured assistant professor in

the Electrical Power and Machines Department in the Faculty of Engineering, Cairo University, Egypt. From 2000 to 2002, He was executive manager of ConTech, and since 2003 he is general manager of SysTech; both companies specialize in providing engineering services in automation. Current research interest include applied nonlinear control, process control and control of time delay systems.

Kenneth A. Loparo received the Ph.D. degree in Systems and Control Engineering from Case Western Reserve University, Cleveland, Ohio, in 1977. He was an Assistant Professor in the Mechanical Engineering Department at Cleveland State University from 1977 to 1979 and he has been on the faculty of The Case School of Engineering, Case Western Reserve University since 1979. He is Professor of Electrical Engineering and Computer Science and holds academic appointments in the Department of Mechanical and Aerospace Engineering and the Department of Mathematics. He is a Fellow of the IEEE and has received numerous awards including the Sigma Xi Research Award for contributions to stochastic control, the John S. Dieko9 Award for Distinguished Graduate Teaching, the Tau Beta Pi Outstanding Engineering and Science Professor Award, the Undergraduate Teaching Excellence Award and the Carl F. Wittke Award for Distinguished Undergraduate Teaching. He was Associate Dean of Engineering from 1994 –1997 and Chair of the Department of Systems Engineering from 1990 –1994. His research interests include stability and control of nonlinear and stochastic systems with applications to large-scale electric power systems; nonlinear :ltering with applications to monitoring, fault detection, diagnosis and recon:gurable control; information theory aspects of stochastic and quantized systems with applications to adaptive and dual control and the design of digital control systems; signal processing of physiological signals with applications to clinical monitoring and diagnosis.