Application of nonlinear sliding-mode control to permanent magnet synchronous machine:.

Abstract

This article is devoted to the study of the performances of the nonlinear order by sliding-mode with the integration of an integrator regulator, applied to the synchronous permanent magnet machine. In a first stage, we presented the design of the order by this new approach followed by a structure of the optimal ordering of the adjustment with the law of commutation against reaction of state with integrator. The second stage is devoted to the development of an order to variable structure with a strategy of three surfaces, which allow the elimination of the overloads that can result in a destruction of the system to regulate.

Keywords: Synchronous permanent magnet machine, sliding-mode, slip surface, nonlinear, decoupling, simulation, robustness.

Introduction

In the case of linear systems with constant parameters, the traditional laws of ordering of type PI and PID give good results. On the other hand, the nonlinear systems or those with non-constant parameters, these traditional laws of order can be insufficient since they are non-robust, especially when the requirements on the precision and other dynamic characteristics of the system are severe. In this case one must call upon laws of order insensitive to the variations, the nonlinearities and the disturbances [1].

The order with variable structure (OVS) is of nonlinear order by nature; it constitutes a good solution with the problems quoted above. This order with variable structure has a very great advantage, since its law of order changes in a discontinuous way.

In order to create a " hyper surface " called as slip, to force the dynamics of the system to correspond to that definite by the equation of "the hyper surface", then the commutations of the order are carried out according to the variables of states [2].

The system is in sliding-mode when the state is maintained on this hyper surface [7][4]. In this condition the dynamics of the system remains insensitive to the variations of the parameters of the process, to certain parasitic disturbances and to the simplifying assumptions at the time of modelling.

The order (OVS) is well adapted, to treat the systems which generate ill-defined models. It has several advantages such as the precision desired, simplicity, stability, a very weak response time and a good robustness.

Definition

A system is known as variable structure if it admits a representation by differential equations of the type:

dx / dt = f(t, x) (1)

where : x is a vector of dimension n : 1, 2, 3, ... n

x = [([x.sub.1], [x.sub.2], [x.sub.3] ... [x.sub.n]).sup.t]

[f.sub.i][(t, [x.sub.1], ... [x.sub.n]).sup.t]

Are continuous functions per pieces and having discontinuities on a surface S.

The structure is known as variable because of the discontinuous change between two or several structures [3]. The study of such systems is very advantageous particularly in mechanics, in physics, in electricity and in chemistry ... This advantage is due to the properties of stability acquired by the total system independent of each subsystem [f.sub.i](x). The circuits of conversion of power constitute an example typical practice of a system with variable structure. Indeed, for each position of the switch, the system is controlled by a system of differential equation.

Theory of the Sliding-Modes

The theory of the sliding-modes finds an adequate application in the theory of order of the systems to relay and in the circuits of electronics of power. The systems with variable structure are characterized by the choice of a function and a suitable logic of commutation. This choice at any moment ensures switching between these structures and makes it possible to impose the behavior wished for the total system [9] [13].

The hyper surface S(x) = 0 of dimension

n-1 divides space into two parts [5] [6].

G = [G.sup.+] if S(x)>0

G = [G.sup.-] if S(x)<0

Around the hyper surface S(x) = 0, the function

f(x, t), is discontinuous. The latter takes two values on each side of surface in the vicinity of a point (x) :

f = [f.sup.+] in [G.sup.+]

f = [f.sup.-] in [G.sup.-]

Where : S(x, t), is the switching function.

The variety of commutation is of size equal to (n) deducting the number of the switching function available. In this case it is the number of exits which one must stabilize.

Trajectories associated with function f are summarized in three configurations where they are described by the temporal evolutions:

* The first configuration represents the trajectories of [f.sup.+] and [f.sup.-] that highlight the repulsion for [f.sup.-] (respectively for [f.sup.+]).

* The second represents the trajectories where the phenomena of attraction exist for

[f.sup.+] (respectively for [f.sup.-]) and of

[f.sup.-] (respectively for [f.sup.+]).

* The third represents trajectories of

[f.sup.+] and [f.sup.-] which converge towards the surface of commutation [S.sub.0] and which have the characteristic to slip on it. This phenomenon is called << sliding-mode >> [4].

[FIGURE 1 OMITTED]

Advantages and Validity

The advantages of the order by mode of slip are very significant. They are appreciable and well-known since the eighties [11]. This type of order gives good precisions, stability in a very short response time, simplicity towards the design and guaranteed a good robustness. The major advantage resides in the characteristic of adaptation as concerns the systems which have vague models. This inaccuracy can be due to the following reasons:

* Inaccuracy of the parameters;

* Variations of the parameters;

* Simplifications of the dynamic model.

This type of order can be used in an analogical way not only in the problems of continuation of the models but also in regulation. It is affirmed [5] that the simplicity of the implementation and the adaptation are applicable in the linear and nonlinear processes [7].

Design and Trajectory in the Plan of Phase

The recent design of the regulators by sliding-mode deals with the problems of stability and desired performances in a systematic way. The new technique requires mainly the three following stages [12] :

1--The choice of desired surface;

2--The determination of the suitable order;

3--The insurance of the conditions of convergence.

The technique of this order consists in bringing back the trajectory of state of a system towards slip surface then to make commutation by the intermediary of a logic of adapted commutation, up to the point of balance [5].

The trajectory consists of three distinct parts as the figure (2) shows.

** Mode of convergence (MC): where the variable to be regulated moves starting from the point of initial balance.

** Mode of slip (MS): where the variable of state reaches the slip surface.

** Mode of the permanent mode (MPM): Where the behavior of the system is around the point of balance.

Direct Sampling Function

It is the first condition of convergence. It is proposed by V.I. Utkin [1]. This function is expressed in the following form:

S(x)[??](x)<0 (2)

It is necessary to introduce, under this condition, the exact values on the left and on the right of commutation for S(x) and its derivative. The conditions of convergences ensure a dynamics of the system corresponding to the slip surfaces [8].

Function of Lyaponov

It is a matter of formulating a positive scalar function V(x)>0 for the variables of states of the system, and to choose the law of commutation which will decrease this function. That means mathematically: V(x)<0.

The function suggested is used, generally, to ensure stability of nonlinear systems [10]. It is defined, like its derivative as follows:

V(x) = 1/2 [S.sup.2](x) (3)

[??](x) = S(x)[??](x) (4)

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

So that the function of Lyaponov decreases, its derivative need be negative, i.e., to ensure the following condition:

S(x)[??](x)<0 (5)

This equation shows that the square of the distance towards the surface measured By

[S.sup.2](x) decrease all the time; whereas, the trajectory of the system is obliged to move towards the surface on the two sides as figure (1) shows. It is an ideal sliding-mode.

Calculation of the Order

Once the surface slip as well as the criterion of convergence are chosen, it remains to determine the corresponding order to bring back the variable to be controlled towards the surface and then towards its point of balance by maintaining the condition of existence of the sliding-modes. The order must commutate between the values [u.sub.max] and [u.sub.min] in a manner instantaneously. It is an essential assumption in the systems design with variable structures controlled by the sliding-modes. Alternation depends on the sign of the slip surface, as the figure (4) shows. In this case, the oscillations of very high frequencies called "chattering" appear in the mode of slip.

[FIGURE 4 OMITTED]

The structure of a controller consists of two parts, the first relates to the exact linearization, the second relates to stability. The second part is very significant in the technique of the order by sliding-mode, because it is used to eliminate the effects of inaccuracy from the model and to reject the disturbances caused by the phenomena external. This behavior is the cause of the variation of the profit (K). If this last is very small, the response time will be very long. On the other hand, if it is large, then response time will be fast, but the phenomenon of "Chattering " is likely to appear in permanent mode. The size of our order is of the form:

u(t) = [u.sub.eq](t) + [u.sub.n] (6)

[u.sub.eq](t) corresponds to the equivalent order suggested by V.U. Utkin and A.F. Filipov. This order is the simplest and the most direct. It is calculated according to the recognition of the behavior of the system during the sliding-mode where:

[??](x) (7)

The equivalent order can be interpreted as the modulated average value that the order takes during the very fast commutation between

[u.sub.max] and [u.sub.min], as shown in figure (4).

Order with Threshold

During rapid commutations of the order, at the time of the sliding-modes, the phenomenon of "chattering" appears. High frequency components, are add to the spectrum of the order. These components are undesirable because, they can damage the actuators, or deteriorate the system by causing the excitement of the high modes which one did not consider at the moment of modelling. In order to reduce "chatterings" one imposes a variation of the value of the order [u.sub.n], according to the distance between the variable of state and the slip surface [4]. Other methods introduce thresholds (dead zone) on the commutation of the sign function.

This new order is characterized by a threshold ([epsilon]) as shown in the figure (5). The continuous component ([u.sub.eq]) acts alone in the band which surrounds surface S(x) of slip. Since the discontinuous part [u.sub.n] is null, then the oscillations on the results will be attenuated in a considerable way. On the other hand, there will be a static difference on the result, at the moment of the regulation, when ([epsilon]) increase.

The expression of the discontinuous order would be:

[u.sub.n] = 0 if [absolute value of S(x)] < [epsilon] (8)

[u.sub.n] = Ksign(S(x)) if [absolute value of S(x)] > [epsilon] (9)

Oscillations can persist in permanent mode, from where a softening of the order [u.sub.n] is need.

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

Order Softened

In order to decrease the value of the order [u.sub.n], the softened order is characterized by a threshold

([[epsilon].sub.1]) or two ([[epsilon].sub.1], [[epsilon].sub.2]). As the figure(7) shows, one distinguishes three zones which depend on the distance from the point considered on the slip surface:

* Zone 1 : The distance is higher than the threshold ([[epsilon].sub.2])in this case the function sign is effective.

* Zone 2 : The distance is lower than the threshold ([[epsilon].sub.1]) and consequently [u.sub.n] is null. It is the dead zone.

* Zone 3 : the point is in the band [[[epsilon].sub.1], [[epsilon].sub.2]], [u.sub.n] is, then, a linear function of the distance from the right-hand side of the slope equals K / [[epsilon].sub.2] - [[epsilon].sub.1].

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

It is noticed that the larger the threshold is, the less the commutations are. In this case the problem of precision arises. The system evolving in the band is thus likely never to reach the origin of the plan of phase (desired point). Figure (8), illustrates the slip surface as well as the softened order in the case of a control speed with the (OVS) of the synchronous permanent magnet considered machine. The example shows us that the introduction of the threshold induces a static error since the surface S(x) = 0 is never reached.

Solution Suggested

Whatever the method of softening used to eliminate or limit the oscillations, the undesirable phenomenon always exists. This last, can be solved by a method which leads to increase slightly the order of surface S(x) of slip [6]. The suggested method needs to introduce the derivative of the error in the calculation of the area. One introduces an acceleration term into the case of the control speed. The new equation which governs surface will be:

S(x) = [[lambda].sub.x] e(x) + [??](x) (10)

With :

[[lambda].sub.x] : Positive constant which interprets the band-width of the desired control.

The use of the new surface known as "increased surfaces" thus involves an increase in the frequency of commutation of the order and consequently one will have a strong reduction in the oscillations;

e(x) = [[OMEGA].sub.ref] - [[OMEGA].sub.r] : Error between the considered speed instruction and the measured speed;

[??](x) = [[??].sub.ref] - [??](x) : Error of acceleration.

Application to the Synchronous Permanent Magnet Machine

Model of reference, based on the orientation of flow

The model of the synchronous permanent magnet machine is defined as follows:

[di.sub.d] / dt = -[a.sub.1] + [a.sub.2][i.sub.q][OMEGA] + [a.sub.3][v.sub.d]

[di.sub.q] / dt = -[b.sub.1][i.sub.q] - [b.sub.2][i.sub.d][OMEGA] - [b.sub.3][OMEGA] + [b.sub.4][v.sub.q]

d[OMEGA] / dt = -[c.sub.3] - [c.sub.4][C.sub.r] + ([c.sub.1][i.sub.d] + [c.sub.2])[i.sub.q] (11)

Where the coefficients are defined as follows:

[a.sub.1] = R / [L.sub.d] ; [a.sub.2] = p [L.sub.q] / [L.sub.d] ; [a.sub.3] = 1 / [L.sub.d];

[b.sub.1] = R / [L.sub.q] ; [b.sub.2] = p [L.sub.d] / [L.sub.q] ; [b.sub.3] = p [[phi].sub.f] / [L.sub.q] ; [b.sub.4] = 1 / [L.sub.q] ; [c.sub.1] p [L.sub.d] - [L.sub.q] / J ; [c.sub.2] = p [[phi].sub.f] / J ;

[c.sub.3] = [F.sub.] / J ; [c.sub.4] = 1 / J.

Synthesis of the Regulators

(Strategy of adjustment on three surfaces)

The technique of the sliding-modes requires a choice of surfaces which ensures the constraints of the order. The error of the adjustment is selected like a surface because, on one hand, the continuation speed is imposed by the order [i.sup.*.sub.q] and in the other hand the surface S([OMEGA]) must be of a relative degree of order 1. [4][5]. The error is defined as follows: S([OMEGA])=[[OMEGA].sub.ref] - [OMEGA]

The new technique requires that surface S([OMEGA]) should be invariant and gravitational. This solution is proposed by the following function of Lyapunov:

[v.sub.1] = 1/2 [S.sup.2]([OMEGA])

The derivative is thus:

[[??].sub.1] = S([OMEGA])[??]([OMEGA]) (12)

To meet the requirement imposed before, one imposes the following dynamics on S([OMEGA]):

[??]([OMEGA]) = -[K.sub.1]SignS([OMEGA]) (13)

With :

[K.sub.1] : Positive constant

Thus the surface S([OMEGA]) converges asymptotically towards the zero value, in the same way the speed [OMEGA] converges towards its reference [[OMEGA].sub.ref] since the temporal derivative of surface S([OMEGA]) is given by the following equation:

[??]([OMEGA]) = [[??].sub.ref] + [c.sub.3][OMEGA] + [c.sub.4][C.sub.r] - ([c.sub.1][i.sub.d] + [c.sub.2][i.sub.q]) (14)

While imposing on S([OMEGA]) the suggested dynamics, the exit of the speed regulator will be thus:

[i.sub.qref] = [[??].sub.ref] + [c.sub.3][OMEGA] + [c.sub.4][C.sub.r] + [K.sub.1]SignS([OMEGA]) / [c.sub.1][i.sub.d] + [c.sub.2] (15)

Since the current [i.sub.qref] exists the denominator need be different from zero.

[i.sub.qref] [not equal to] 0 [right arrow] [i.sub.qref] [not equal to] - [c.sub.2] / [c.sub.1] (16)

The condition (16) is always realised because the current is maintained null.

The strategy of the adjustment is illustrated by the diagram described by the figure (9). The nonlinear adjustment by sliding-mode uses, in this case, the method of adjustment in cascade. The new structure includes a loop of speed regulation. The latter generates the reference of the current [i.sup.*.sub.q] which imposes the order [v.sup.*.sub.q]. The order [v.sup.*.sub.d] is imposed by the regulation of current [i.sub.d].

[FIGURE 9 OMITTED]

Simulation

For the validation of the structure of the order suggested, one simulated the strategies of order of three surfaces. The simulation is made in the following way:

One made a loadless starting with an instruction of 200 [Rad/s] with application of the nominal load between 1 and 2 [sec] then an inversion of the rotation direction of (200 to -200) [Rad/s].

One notes:

* The rejection of disturbance is very fast;

* A very weak response time;

* A practically null static error;

* A decoupling excel for the technique suggested.

The curves prove that speed follows its reference in the two directions of rotation. The stator current remains in an acceptable interval.

Between t = 1 [s] and 4 [s], speed remains insensitive with the variations of the parameters and the disturbances due to the application of the nominal couple.

For a performance evaluation, of our machine, with the technique of order by the sliding mode, one did a robustness test of this adjustment vis-a-vis the machine parameters. One increases resistance of 200% and a reduction of 80% of inductances. As for constant mechanics, it is increased to 80%. See figure (11).

It is noted that the speed of our machine remains practically insensitive with the variations caused by the disturbances made, what confirms the robustness of the adjustment suggested.

Conclusion

We presented, in the first part, a method of analysis and synthesis of the nonlinear law of order by the continuation of a basic discontinuous model. The order uses the method of Lyapunov. The second part was devoted to the application of the various algorithms to the order of the synchronous machine to permanent magnet. The nonlinear order by sliding mode using vectorial decoupling and nonlinear decoupling shows its effectiveness even if the parameters (mechanical and electric) of the machine undergo variations. Various simulations made show that the nonlinear system of regulation gives very good performances. The obtained results, enables us to conclude that our system of regulation proposed is robust even if the disturbances are not known. One also adds that the regulation suggested can be applied in fields requiring high performances such as the field of robotics, the field of the machine tools ...

Results of Simulation

[FIGURE 10 OMITTED]

[FIGURE 11 OMITTED]

References

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[10] E. Bouhassoun, M.O. Mahmoudi, M.S. Boucherit, " Commande par Mode Glissant d'une M.S.A.P. avec pilotage vectoriel. >>, ICEL 98, USTO, Oran 1998.

[11] R. J. Wai and F. J. Lin, " Fuzzy neural network sliding-mode position controller for induction servo motor drive " Proc. Inst. Elec. Eng. Elect. Power applications, Vol. 146, No. 3, PP,297-308, May 1999.

[12] F. Barrero, A. Gonzalez, A. Torralba, E Galvan and L. G. Franquelo, " Speed Control of Induction Motors Using a Novel Fuzy Sliding-Mode Structure " IEEE Transaction on fuzzy systems, VOL.10, No. 3, June 2002.

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A. Kechich (1), B. Mazari (2) and I. K. Bousserhane (3)

(1) Institut d'electrotechnique, B.P417 C .U. Bechar, Tel/Fax : [phone]

(2) Institut d'Electrotechnique, U.S.T.O. B.P1505, Oran El M'naouer, Algerie

(3) Institut d'electrotechnique, B.P417 C .U. Bechar E-mail: [e-mail]