A systems identification-based-method for linear, parametric approximation of the eddy currents brake's model.

Danila, Adrian ; Moraru, Sorin Aurel ; Perniu, Liviu 等

Abstract: The paper presents a study for the determination of the optimum polynomial estimate for the eddy currents brake's model. The general approach is based on the direct- and quadrature-axis theory that leads to a nonlinear set of equations. The canonic form of this dynamic set of equations can be obtained by using the field oriented principle. However, even in the canonical form, this set of equations cannot be used to design an appropriate linear, continuous or digital, control system. In this paper, the systems identification techniques are proposed to determine the optimum polynomial model of the eddy-currents brake.

Key words: eddy-currents brakes, system identification, polynomial model.

1. INTRODUCTION

The eddy current brakes are often used as energy spreading-out elements into the automotive industry. The eddy current brakes' braking torque may be simply controlled by modifying the value of the excitation current. The electromagnetic torque of the brake is produced due to the interaction between the eddy currents produced into brake's induced and the resulting magnetic field.

Additionally, a control system is used to maintain the braking torque near the predicted/desired value. The control system controls the braking torque or the angular speed of the brake. The control system has two cascade loops: an inner loop that compensates the excitation current and an outer loop that compensates the electromagnetic torque. The outer loop consists of a PI compensator whose elements are subject of optimization.

In the first chapter, the dynamic models of the eddy-currents brake are presented. In the second chapter the authors develop a general, direct- and quadrature-axis dynamic model for the eddy-currents brake. Subsequently a digitization of the continuous dynamic set of equations using the Euler-Cauchy method leads to a nonlinear numerical set of equations. In purpose to obtain the optimal estimate polynomial model of the brake, the least mean square method and the instrumental variables method are used into the third part of the paper.

2. DYNAMIC MODELS OF THE EDDY-CURRENTS BRAKES

2.1 Models available in the literature

Various dynamic models for the eddy-currents brake are available in the literature, (Pozdeev and Rozman, 1965). In Smythe's approach the problem is treated as a disc of finite radius rotating within a perpendicular to the disc surface magnetic field. The solution of field equations and the torque calculation are obtained by means of the reflection procedure especially suited to the geometry of the model. The main result is as follows:

T = [omega] x [gamma] x [R.sup.2] x [[PSI].sup.2.sub.0] x D/ [(R + [[beta].sup.2] x []gamma].sup.2] x [[omega].sup.2]).sup.2] (1)

where: T--brake's torque; [omega]--angular velocity; [[PHI].sub.0]--the magnetic flux; D--constant coefficient depending on pole arrangement; R--reluctance of the electromagnet; [gamma] = 10-9/[rho] - volume rezistivity of the disk. Based on the Smythe's model, Schieber and J. H. Wouterse, tried to obtaine a general model for the eddy-current brake availabe for both low- and high-speed regions. Wouterse model, matching to the high-speed region is as follows:

[F.sub.e] (v) = [[??].sub.e] x 2/[v.sub.k]/v + v/[v.sub.k] (2)

where: [[??].sub.e] 1/[[mu].sub.0] x [square root of c/[xi] x [pi]/4 x [D.sup.2] x [B.sup.2.sub.0]] x [square root of x/D]= is the maximum electromagnetic force, [v.sub.k] = 2/[[mu].sub.0] x [square root of 1/c x [xi] x [rho]/D] x [square root of x/D] = is the peak value of the speed, [rho]-specific resistance of the induced material; d--disk thickness; D--diameter of the soft iron pole; [xi]--ratio between the zone width and the air gap, in asympthotic distribution around poles; c--proportionality factor; v--tangential speed of the rotating disk measured at center of the pole; x--air gap between pole faces including disk thickness; [B.sub.0]--air gap induction at zero speed.

The model based on a given set of experimental data provided by Omega Technologies takes into consideration the dependency of the electromagnet/inductor reluctance on the angular velocity. The braking torque is given by:

R = [k.sub.1] x [omega]/[(1 + [k.sub.2] x [[omega].sup.2] + [k.sub.3] x [[omega].sup.4]/ 1 + [k.sub.4] x [omega] + [k.sub.5] x [[omega].sup.3]).sup.2] (3)

parameters [k.sub.1] - [k.sub.5] are to be evaluated for specific types of eddy-currents brakes using experimental data and the least mean squares method as a maximum likelihood estimator.

2.2. The direct- and quadrature-axis based model for the eddy-currents brake

Taking into consideration the assumptions presented in (Danila, 2006), the general direct- and quadrature-axis theory referred to the stator coordinate system leads to the following set of equations:

* the voltage equations and the torque equation:

-[u.sub.d] = d[[PSI].sub.d]/dt - [omega] x [[PSI].sub.q]

-[u.sub.q] = d[[PSI].sub.q]/dt + [omega] x [[PSI].sub.d] (4. a,b)

[u.sub.F'] = [R'.sub.F] x [I'.sub.F] + d[[PSI]'.sub.F]/dt (5)

[FIGURE 1 OMITTED]

[M.sub.el] = p x ([[PSI].sub.q] x [i.sub.d] - [[PSI].sub.d] x [i.sub.q]) = -[M.sub.A] + J x d[OMEGA]/dt (6)

* the flux-linkage equations:

[[PSI].sub.d] = [L.sub.d] x [i.sub.d] + [L.sub.hd] [I'.sub.F] [[PSI].sub.q] = [L.subq] x [i.sub.q] [[PSI]'.sub.F] = [L'.sub.F] x [i'.sub.F] + [L.sub.hd] x [i.sub.d] (7.a, b, c)

where: [[PSI].sub.d], [[PSI].sub.q] are the flux-linkage components on the d- and qaxis, respectively, [[PSI][??].sub.'F] is the excitation flux (referred to the induced), ud, uq, id, iq are the induced voltage and currents components on the two axis, i'F is the excitation current, Mel is the electromagnetic torque and J is the axial inertia moment referred to the brake's axis.

3. FIELD ORIENTATION OF THE COORDINATE SYSTEM AND THE DYNAMIC EQUATIONS SET DISCRETIZATION

Field orientation principle consists of referring both stator and rotor set of equations to the same coordinate system such as the two components that produce the electromagnetic torque i.e. the magnetic flux component and the current component are clarity disconnected, (Henneberger, 2004).

For the eddy-currents brake the coordinate system has to be referred to the magnetic flux-linkage of the induced. In this case the dynamic set of equations reduces to only three equations as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

where: [T.sub.d] = [L.sub.OL] + [L.sub.d]/[R.sub.OL] and [T.sub.F] = [L'.sub.F]/[R'.sub.F] are the time constants of the induced and the excitation winding respectively. In purpose to solve this set of equations, terms containing derivatives have to be discretizated using Euler-Cauchy method, (Henneberger, 2004). The main results are, (Topa, 2006):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

Based on the results above, the step response of the brake may be calculated on computers. In purpose to obtain an optimal approximate polynomial model for the brake, the least mean square method is to be used. The polynomial models have the following operational equations, (Soderstrom and Stoica, 1989):

* the ARX model:

y[t] = B([q.sup.-1])/A([q.sup.-1] x u[t] + e[t] (10)

* the ARMA model:

y[t] = B([q.sup.-1])/A([q.sup.-1] x u[t] + 1/A([q.sup.-1]) x e[t] (11)

where y[t], u[t] and e[t] are the output, the input and the white noise, respectively. Brake's parameters used for computations are presented in Table 1. In Fig. 2 is shown the step response computed in MatLAB environment for the physical and the polynomial estimates of the brake. The parameters of the polynomial model are given in Table 2.

4. CONCLUSIONS

The determination of the optimum polynomial model for the eddy-currents brake may be solved by means of system identification techniques. Further investigations have to be made to determine the appropriate validation criterion for the resulting estimate.

5. REFERENCES

Danila, A., (2006). A direct- and quadrature-axis theory based model for the eddy-currents brakes. Buletinul Institutului Politehnic din Iasi, Tomul LII (LVI), Fasc. 5A Oct. 2006, pp.273-281 ISSN 1223-8139.

Henneberger, G., (2004). Electrical Machines II. Dynamic Behavior Convertor Supply and Control, RWTH Aachen.

Soderstrom, T. and Stoica, P., (1989) System Identification, Prentice Hall International, Hemel Hempstead.

Pozdeev, A.D. and Rozman, Ia. B., (1965). Electromagnetic clutches and brakes, Editura Tehnica, Bucuresti.

Topa, I., (2006). The eddy-currents brakes in field oriented coordinate system. Buletinul Institutului Politehnic din Iasi, Tomul LII (LVI), Fasc. 5A Oct. 2006, pp.273-281 ISSN 1223-8139. Table 1. Physical parameters of the eddy-currents brake. Induced resistance [R.sub.Ol] [OMEGA] 0.0826 Induced self-inductance [R.sub.Ol] mH [2.465.10.sup.-2] Induced mutual inductance [L.sub.h] mH [1.74.10.sup.-2] Excitation resistance [R'.sub.F] [OMEGA] 600 Angular frequency [omega] 1/s 3351 Table 2. Parameters of polynomials A([q.sup.-1]) and B([q.sup.-1]). 1 2 3 4 [a.sub.est] 1 -1.8571 1.2127 -0.2905 [b.sub.est] 0 0.0316 0.0332 --