order Extended Luenberger type non-linear observer model. ... prove that the Observer is able to control the drive in steady state conditions as well as in.
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POSITION SENSORLESS OPERATION OF A SWITCHED RELUCTANCE DRIVE BASED ON OBSERVER
*BRIGHTON I? 1691993
C.Elmas and H. Zelaya-De La P m Birmingham Universty, U.K. Abstract. It is known that the operation of the Switched Reluctance Motor (SRM) requires the knowledge of the rotor position for current control and commutation. This information is generally provided by a shaft encoder or resolver. In some cases,the environment in which the motor operates may cause difliculties in maintaining satisfactoly performance. Therefore, the elimination of position sensor has gain wide attention. In this paper a new algorithm for estimation of rotor position and speed are described for an SRM.The method uses a Reducedorder Extended Luenberger type non-linear observer model. The algorithm uses the information obtained from the measurements of input voltages and currents. The algohirthm was implemented with a TMS32OC30 DSP. The results obtained from the on-line operation prove that the Observer is able to control the drive in steady state conditions as well as in transient. Kevwords. Switched reluctance motor, position sensorles operation, non-linear observer.
INTRODUCTION
The operational nature of a switched reluctance motor ( S F M ) requires accurate information from the rotor position. A shaft encoder or a resolver is usually utilised for this information. In most cases, however, the environment in which the motor operates may cause dficulties in maintaining satisfactory performance. It is, therefore, highly desirable that the system has a minimal number of sensors. Thus, some of the variables that define the SRM have to be estimated. During the past decade a great deal at the work has been reported on the use of some methods to determine the rotor position of many types of motors and ranges without rotor position encoders (i.e. for induction motors, permanent magnet motors, and switched reluctance motors). In the area of the SRM, many researchers have dealt with the problem of eliminating the position encoder by measuring terminal voltages and currents as an alternative indirect method of position sensing.
In the subject of indirect position sensing, there are two principal methods. The first one is so-calledwave form detection method that determines the position of the rotor by using inductance or mutual inductance effects, (Bass et al. [l], Bass et al. [2], Acarnley et al. [3], Mwngi and Stephenson [4],Panda and Amaratunga [ 5 ] ) . The second method is to use an observer theory. Lumsdaine and Lang [6]. The wave form detection methods monitor the terminal current and voltage of the motor and look for single event extrema or zeroes by using known motor models Q 1993 The European Power Electronics Association
to relate the detected events to position. Although these. methods have eliminated the need for direct position sensing, this kind of sensorless operation, however, imposes limitations on the operating range of drive (Acamley and Ertugrul [7]).
The second method, as mentioned above, is an observer application that reconstructs the state of the SRM drive system on the basis of known system inputs and system measurements. During the past decade, a considerable amount of work has been reported on the use of the observer theory in many different types of drives, e.g., Garcia-Cerrada [SI proposed an observer model on an induction motor to estimate rotor resistance and position. Tajima and Hory [9] have also recently reported an observer model to estimate speed instantaneously with flux. Furuhashy et al. [IO] and French et al. [ I l l have also recently reported an observer model for a Brushless DC motor estimating speed and position as well as some motor parameters. Due to the highly non-linear nature of the Switched Reluctance Motor, researchers have been reluctant to implement this theory to the SRM. Only one publication has been found in the literature by Lumsdaine and Lang on a variable reluctance motor to estimate both position and speed. Although, they have successfully implemented a linear observer theory, the non-linearity due to the speed perturbation was neglected by means of very large rotor inertia. As a result of this, the performance of the observer may not be fast enough for transient operation. In this paper a new algorithm for estimation of rotor position and speed is described for the SRM. The method uses a Reduced-Order Extended Luenberger (ROEL) type non-linear observer model for joint state
a3
and parameter estimation (i.e. rotor position, speed and load torque) to follow the detailed dynamics of the motor itself. With this method, a sophisticated model of the motor dynamics, driven by the same voltages applied to the actual motor and corrected by differences between estimated and measured currents, continuously estimates rotor position as well as speed. Since the objective of this paper is to estimate rotor speed and position, these states have to be included in the system model this is called "combined" or "joint" state and parameter estimation. This multiplication of states could transfer the linear time-invariant system into a time-variant non-linear system. Thus, as a solution, a non-linear observer has to be sought. A popular non-linear observer is the Extended Kalman Filter (EKF) in the presence of noise when the system is considered as stochastic. In recent years there has been a vast amount of research devoted to estimating joint state and parameter in electrical drives. For this issue some interesting contributions can be found in a series of papers, e.g., Capolino and Du [12], Bal and Grant [13], and Atkinson et al. [14], etc. But the operation may suffer from lack of robustness results to stochastic control systems.
Often noise-free measurements are available, then, the system becomes deterministic and therefore an asymptotic non-linear observer is an alternative. Such problem has been an active research area for decades. Like the Extended Kalman Filter (EKF), the design method for non-linear observer is based on the extended linearisation of the observer error system, called the Extended Luenberger Observer @LO) on a given interval of time (Zeitz [IS]). Unfortunately, in electrical drive application, the socalled Extended Luenberger Observer, has gain little attention until1 recently. In an JEE paper, OrlowskaKowalska [I61 has successfully addressed one of the first application of the EL0 in an induction motor drive. She has, however, reported only the simulation results in ideal conditions, thus, the on-line application of the EL0 is yet to be reported and an open question. MODELLING A SWITCHED RELUCTANCE MOTOR DRIVE
v=fi+dYr dt where v = [va, y,, vc, VdIT is the terminal voltage vector, i = [ia, ib, i,-,idlT is the phase current vector, and \v = pya,
w, \vc, VdlT is the flux linkage vector. R
is the diagonal matrix of phase resistance. From the assumption of magnetic linearity, the flux linkage and the phase Currents are related by yr = L(8) i
where L(B) is the inductance matrix and 0 is the rotor position. From the energy conservation law and the assumption of magnetic linearity, the mechanical equation of the machine is given by do J-=-Bo+T-T, dt
The voltages across the terminals of the machine are related to the phase Currents and the flux linkage by the matrix equation
(3)
de dt
-=a
(4)
where
or
where w is the rotor speed, B is the friction coefficient, J is the combined inertia of the rotor plus the mechanical load, T, is the load toque, and T is the electromagnetic torque. The Eqs (I), (3), and (4) together are called Augmented Motor Model. Combining Eq. ( I ) to (2), a state space representation of the machine equations can be shown as follow:
(7)
do dt
-=-__
The modelling an SRM is a straightfonvard, and provide a good description for its operation. A brief review is included here to establish the notation. To simpllfy the modelling of the SRM, magnetic nonlinearity and mutual inductance are omitted, and a model with lumped parameters is presented.
(2)
T-T,, J
de -=a dt
Bo J (9)
In Eq. (3), the load torque TL is considered as external disturbance and sometimes considered as known or zero [6]. However, this assumption could lead to have a totally invalid response since, in practice, the load torque does exist and is hardly known precisely even at steady state operation due to the speed perturbations.
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Lumsdaine and Lang had used this idea and eliminated T, term in Eq. (3) by choosing a very large inertia. For these obvious reasons, the load torque T, should be known somehow. This would also enhance the overall system performance (Milanovic et al. [17], Brdys and h [18], and Westerholt et al. [19]. One reasonable solution for this problem is estimation of the T,. By adding this quantity to the state equation as a state, then T, can be transformed into unknown and inaccessible state variable (Milanovk et al. 1989). Since the changing rate of the load for a very short sampling time (typically 100 ps in our case) is slower than the electromagnetic process, we can reasonably assume that the dTL/dt is zero. Then, we can write a new augmented system of equations for the SRM in state space form as:
0 t,T, 0
0 0 t,Tb t,T, 0 0
0 t,Td 0
0
t,
0 1-tSB J 0 0
1
Assuming the phase currents are measurable, then the output equation will be: y(t) = cNxN(t)
(11)
where the subscript N represents the electromagnetic part whereas the subscript P represents the parameter parts of the motor. Since the motor augmented model combines both the states and the parameters, it is nonlinear because of the multiplication of states [14]. Motor matrix in Eq. (10) and (1 1) now are written as:
If we want to implement the observer by the use of a digital computer, we must convert a continuous-time system to a discrete-time system. In this case, state estimation would be one function of an overall digital control scheme. A forward difference approximation yields xfi+1) = x@) + 4 f(X@), e ( K h
O(Kh
TL@))+~
where x@) represents the Present state and x@+l) represent the next state and 4 is the sampling interval. The general discretisation becomes
x ~ + 1=)
[
FN Fp(k)][XN ( k,
FN(0, 6% TL) fi)=
(19)
If the phase currents 6 and phase voltages vn are then the phase flux wn, can be directly obtained from Eq. (1) as:
In this Case, there is no need to estimate v,. It should be notice that any initial error would yield invalid result. However, the initial condition can be reset
Whenever 4 equals zero, due to the Operational nature of SRMs and taken \irn = 0 as long as the corresponding phase is not energised. As seen from Eq. (14), the augmented motor model is made up of two components: electromagnetic and parameter. Since we assume that all states of the electromagnetic part are available then the obsemer dimension a n be r e d u d to be equal to the dimention of the unmeasurable mrt of the s h e variable.
[
(k) Xp(k)l+ B0' (k)][v(~)I (14)
in which
v@)= IVa,vb, vc, VdIT,
RELUCTANCE DRIVE In order to implement the Luenberger theory to the non-linear model, first we must obtain a linear mathematical model for the SRM. To do this, it is assumed that the variables deviate only slightly from state trajectory %(t), then Eq. (13) may be expanded into a Taylor series expansion about this point as follows:
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x(k+l)=f(f(k)) + g ( K ( k ) ) [ ~ ( k -E ) @)I + higher order terms+ Bu(k)
(21)
where the derivative dvdu is evaluated at (x = E). If the variation (x E ) is small, the higher order terms in (x - X) may be neglected, then ~ q . 6 2 may ) be written
-
In the above treatment it is assumed that F is a stable matrix and constant during each sampling interval.
By combining of the Reduced-orderobserver state z and the measurements as follows
[:I=
by
A(%) =
&(Ti) dx
(22)
n-m
J-[c]x(k)+[:] 3 T
an estimate of the state x(t) can be obtained as:
% @+1) = Mz(k) + N y Q
as
Having obtained the linearised model of the SRM, the linear Luenberger Observer theory can directly be used to produce the state estimates of the non-linear model.
If the last step estimate is chosen as the operating point f is conveniently shown as for a short sampling-time, - follows:
E
(lt)
~(k) - Tx(k)
in which
; = + f,-1(6) ., where
x(k+l)=f(f(k))+A(X) [x(k)- E(k)]+Bu(k) (25)
i-! =dia i-1. i-1 i-1 i-d(0) 1. ] a@)' b(6)' c(6)' (e)
T(%)B= H ( f )
(26)
where "(E) is the SolUtiOn with respect to the reference position X :
-
T ( f ) X(a) FT(a) = K(f)C G(B)-T[A(k)- i ( a ) ]
(27) (28)
and the nxn matrix (29)
has an inverse [N MI=[:]
(30)
then the (n-m)th order observer has the form of
+a%) (31)
z(k+i) = FZ&) +KY&) + H u ~
(34)
(35)
In this section the system of the form
with measurement y = Cx(t) are considered. If there is again a unique (n-m)xn matrix T such that
(33)
In the case of no magnetic and memy modeled inductance variation, then, the error between the ad and the observed stateis defined by:
z (k)=ii &)
Assuming X is equal to the latest estimated value and piecewise constant, then Linear Luenberger Observer may be applied within every sampling period.
(32)
g[
(36)
(37)
then, the error dynamics will have following form: E(k+I) =FE(k)+T(o(k),O(k),TL(k))x(k)
(38)
If the gain matrix K is chosen in a way that all eigenvalues (Q of the F satisfies Ihil
