technique for induction motors. The method ... There are many approaches to induction motor control .... Finally ud, us is related to the actual control U,, ub in (1).
Robust Adaptive Control of Induction Motors Without Flux Measurements C. M. Kwan, F. L. Lewis, and K. S. Yeung Automation and Robotics Research Institute The University of Texas at Arlington 7300 Jack Newel1 Blvd. S. Fort Worth, TX 76118. Research supported by NSF grant IRI-9216545 most induction motor control schemes. Load torque and rotor resistance can be unknown but bounded.
Nomenclature stator resistance rotor resistance stator current stator flux linkage rotor current rotor flux linkage magnitude of rotor flux linkage voltage input angular speed angle between the rotor flux linkages number of pole pairs angle of rotation stator inductance rotor inductance mutual inductance 1- M~/(L,L,) rotor inertia load torque electric motor torque (.) in the (d,q) frame (.) in the (a,b) frame reference trajectory
1. I n t r o d u c t i o n The induction motor is quite popular for fixed-speed applications. Since rotor currents are induced, no brushes and slip rings are needed. It is maintenance free, simple in operation, rugged and generally less expensive than either DC or synchronous motors [3]. On the other hand its model is much more complicated than other machines and because of this, it is considered as "the benchmark problem in nonlinear system" by the Editorial Board of IEEE Transactions on Automatic Control in comments on a recent paper written by Marino et al. [6]. There are many approaches to induction motor control [2] [6] [8] [ l l ] and references therein. The most recent ones are the adaptive input-output linearization method by [6] and the sliding mode control method of [l 11. The field oriented model of an induction motor was developed by Blaschke [2] in the seventies. By using a nonlinear state transformation, a special structure emerges. Nonlinear control, which later was known as feedback linearization, together with PI control is used to control the motor. This technique is very useful except that it is not robust to parameter variations. Uncertainties such as load torque, rotor resistance are usually unknown and may have a large degree of variations (up to 50% change in rotor resistance) [6]. The paper is organized as follows. We shall first describe a fifth-order model in [6] and then an equivalent field oriented model in Section 2. In Section 3 we shall develop two robust control schemes: one is nonadaptive and the other one is adaptive. The adaptive one is dynamical in nature and can explicitly counteract mismatched parametric uncertainties. Load torque and rotor resistance can be unknown but bounded. Full state is not needed for implementation. A method to avoid rotor flux measurement will be presented in Section 3.3. Theory as well as simulations in Section 3.4 show that robust adaptive control scheme is better than nonadaptive robust method.
estimate of (.) sliding variables nominal values of (.) R A W(0JJ.T)
M 2 R I / ( G b 2 ) +RJ(/(aLs) npWJLJ unknown motor parameters Abstract In this paper, we present a new robust adaptive control technique for induction motors. The method is robust to parameter variations and the stability analysis is much simpler than other approaches such as Marino's adaptive input-output linearization method. Another main advantage of our method is that we only require a reduced-order system to be linearly parametrizable (LP); the rest of the system dynamics can be highly nonlinear with no LP requirement. Full state feedback is not needed for implementation. In particular, the rotor flux measurement is not needed which consequently eliminates one of the major disadvantages in
2. Model of Induction Motor
2.1 Induction Motor Model A fifth-order model, which includes rotor dynamics, under the assumptions of equal mutual inductances and linear magnetic circuit, is given by [61
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A state transformation was suggested by Blaschke [2] as
w=w,
Substituting (3), (4) and (5) into (2) yields the field oriented model
do = Il-yldiq - %E+ 61(-1/J),
i, y,U, denote current, flux linkage and stator voltage input to the machine. Subscripts a, b denote the components of a vector in a fixed stator reference frame (a,b). Subscripts r and s denote components of rotor and stator respectively. The meaning of other symbols in (1) are listed in the Nomenclature. We shall use speed a,rotor fluxes yrs. v r b and stator currents is,, is, as state variables. Following the same notations of [6], we drop the subscripts r and s. We shall resume these notations when we discuss the method of avoiding the rotor flux measurements in Section 3.3. Let x = [Wy,y~bi,ib]~be the state vector and let
dt
J
(64
6 = [er = [TL-TLNR,-Rr~]' be the unknown parameter ~ load deviations from the nominal values TLN and R r of torque TL and rotor resistance R,. Also let U = [U, UblT be the control vector, and
3. Robust Adaptive Control of Induction Motor
S = M~(((JL~W,
a =R r N k ,
y = M2RrN/(oLsc)+RJ(crL,) I .J = npM/(JL). Note that a,P, y, CL are known parameters. Then (1) can be compactly written as
x = f(X)
-b U&
-b
Ubgb + 61fi + 62f2
Sliding mode control method was applied to induction motor control in [ l l ] and [8]. However, their methods are only robust to certain types of parameter uncertainties, i.e. matched uncertainties. For mismatched uncertainties such as 61, 62 in (6a) and (6b), which are not in the range space of ud and uq, there is no explicit way to handle them by previous sliding methods. The main reason for their inabilities is that their sliding variables were defined as a linear combination of states which is a usual practice in sliding theory [4] [IO]. As we shall see in a moment, our new sliding variable formulation is dynamical in stead of static. The dynamics are generated from an adaptive reference signal which, when system is in sliding mode, can estimate the unknown parameters on-line and use direct adaptive control to counteract these mismatched uncertainties. In Section 3.1 and 3.2, we shall assume the rotor fluxes are measurable. Then in Section 3.3, we shall eliminate this stringent requirement. Two general assumptions are needed: Assumption I : The reference trajectories 0' and vdr are differentiable and bounded. Assumption 2: The load torque TL and rotor resistance R, are unknown but bounded.
(2)
where
The model involves a trahsformation from the stator fixed frame (a,b) to a frame (d,q), which rotates along the flux vector (yfa,yb). The transformations between currents and flux magnitudes in different frames are given by
3.1 Nonadaptive Control Scheme The technique here exploits a special structure of the induction motor model. We first treat id, i, as some fictitious control signals to a subsystem. Then we use sliding mode control to realize these fictitious signals. We emphasize that the sliding variables are nonlinear combinations of states.
(3)
[ W J =[ -sinp COS P where
~1~ly.l
cos sin p yb
Step I : Selection of desired id and i, to control (6a) and (6b)
(4)
Our control objective is to regulate the rotor speed and the magnetic flux magnitude. Denote 0' and v d r as the desired reference levels of w and v d respectively. First, we rewrite (6a-6b) as
p = tan-'pJ
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reach sliding mode is controlled by the magnitude of vi. The larger the qi, the short the time will be to reach sliding [lo].
Remark I: When system is in sliding mode, the dynamics are completely determined by the behavior of the reducedwhere el
=o
- or,
order system (7a), (7b) and p is always bounded because of its definition in (3) and (4),and it has no effects on other equations in (7).
e2 = v d - fd.
If we choose
Remark 2 : Since the sliding controller is usually implemented by replacing the signum functions in (10) by saturation functions of boundary layer widths of ai, i = 1, 2, in order to smooth out chattering, it is necessary to study the system behavior with saturation functions. Note that control (IO) will cause CY^ to go inside the boundary layer I ~ iO
-
-
The following parameter update law is used
e = -e = -rWTPe. Using (20) and
(27)
Boundedness of o, e, e , 8 implies ud and uq are bounded. Hence 61 and 62 in (15) are also bounded, which further implies oi's are uniformly continuous. This together with (27) show that o 1 tends to zero (Lemma 2.12 of 171). Similarly we can show that 02 also tends to zero.
V=-bTPe+b r e + b T O (18) 2 2 2 with P = PT> 0 and r = rT > 0. Differentiating (18) along the trajectory (17) gives V =-bT(PA 2
IolIdt t 0.
Remark 3: Note that there is one singularity in (22b). If
(22a)
h
[-f,z - ) ~ w ~ ( P I I ~ I-+h%sgn(csdl, P~I~~)
[l+Bz/[aLr)]=O, then Ud will be infinite. To avoid this, a technique known as parameter projection [9] can be utilized,
er) + q,i = 1, 2,
which can restrict 82 to be within the known bounds of 0 2 (The bounds of 8 2 are known by Assumption 2). The case of using saturation functions can be similarly treated as in Remark 2 and hence is omitted here. The resulting system can be easily shown to be stable, i.e. bounded-input bounded-output (BIBO).
h
hg, =(fun,with
Bi
being replaced by
i = 1, 2, pij: elements of matrix P, qi > 0, then ( 19) becomes V = L T Q e -q1101-1qztozl I O . (23) 2
3.3 Elimination of Rotor Flux Measurements In the above two sections, the rotor fluxes yr.,,v r b have to be available. However it is not easy to measure the rotor
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iitliiplivc I I I ~ I Io~i l t p t l t Iccrlhi~ckIllclhod. ' l h \ phctioiricrtori I S ;I ch;iriictcristic o f irduction motor dynamics. (h) Using the motor data in 161, simulation results for the robust adaptive controller in Section 3.2 are summarized in Figs. 2a and 2b. We set Q = I, r = diag{ 10,lO). And all other parameters are the same as those in Section 3.1. The robust adaptive controller is definitely better than the performance of nonadaptive one which is shown in Fig. 1.
fluxes. There are many techniques to cstiiiiate the roto1 fluxes such as observer theory and Kalman filters ([I21 and references therein). The main problem of these approaches is that there is no closed-loop stability assurance when the observed states are used in the controller [5]. Here we use the method suggested by Soto and Yeung [ l l ] to eliminate this restriction. The stator fluxes vsa,v S b are available and can be related to the stator voltages through [6] dv= = -Rsisa+ U=. dvsb __ Rsisb +U& (28) dt dt where U,,, u,b are available and defined in (1). The subscripts r, s are refering to the quantities of rotor and stator, respectively. The stator fluxes vSa,v s b are related to the rotor and stator currents through
just
A
Now all regulation errors converge to zero. 8 , converges to A
the true 81.62, however, does not converge to the actual 82. These phenomena verify our theoretical predictions in Step 1 of Section 3.2. The applied voltage U, has the same magnitude as that of [6] and is well within inverter limits.
4.
= Lisa + Mi,, v s b = Lsisb+ Mi&. (29) Rotor fluxes are related to the stator and rotor currents through vsa
v r a = Lira+ Mi,,, Integrating (28), we get vSd
= Vsa(0) +
6
(U,
vrb
= L&
+ Misb.
There are five important contributions in this paper. The first one is that we have derived two novel robust control schemes, one adaptive and one nonadaptive, to a "nonlinear benchmark problem" known as induction motors. The adaptive one estimates unknown rotor resistance and load torque on-line and achieves significant better results than that of the non-adaptive one. The second one is that we have combined sliding and adaptive techniques in a novel way so that mismatched uncertainties can be explicitly handled. The third contribution is that our method only requires a reduced-order system to be linear parametrizable (LP). The design is modular and systematic and hence can be easily applied to nonlinear systems with similar structures. The fourth one is that we do not need motor accelerations and the derivative of flux magnitude, as compared with conventional sliding control schemes such as the works of [ll] and [8]. The fifth one is that, unlike other approaches [2] [6] [8], our method does not need rotor flux measurement in the controller. Two on-line integrators are used to generate the rotor fluxes.
(30)
- RSis3dt, (31)
Substituting (31) into (29) yields i, = (vsa - Lsisa)M, ib = (vsb - Lsisb)M. Using (32) in (30) gives the rotor fluxes Vm = L&a
Conclusion
(32)
- LSisa)M + Mi,,,
v r b = L(vsb - Lsisb)m + Misb. (33) Therefore the rotor fluxes can be constructed from known quantities, that is, integrate (28) and then use (33). To implement the above scheme, we only require two additional integrators in the controller. This should be considered as an important contribution to induction motor control.
References [l] E. Bailey and A. Arapostathis, "Simple Sliding Mode Control Scheme Applied to Robot Manipulators," Int. J. Control, 45, No. 6, pp.1197-1209, 1987. [2] F. Blaschke, "The principle of Field Orientation Applied to the New Transvector Closed-loop control system for rotating field machines," Siemens-Rev, 39, 217-220, 1972. [3] D. Brown and E. P. Hamilton, Electromechanical Energy Conversion, Macmillan Publishing Company, 1984. [4] B. Drazenovic, "The invariance conditions in variable structure systems," Automatica, 5 , 357-367, 1969. [SI J. M. Maciejowski, Multivariable Feedback Design, Addison-Wesley Publishing Company, 1989. [6] R. Marino, S. Peresada, and P. Valigi, "Adaptive inputoutput linearization control of Induction motor," IEEE. Trans. on Automatic Control, 38, 208-221, 1993. [7]K. S. Narendra and A. M. Annaswamy, Stable Adaptive Systems, Prentice-Hall, 1989. [8] A. Sabanovic and D. B. Izozimov, "Application of sliding modes to Induction motor control," I E E E Transactions Industry Application, 17, 4 1-49, 1981. [9] S. Sastry and M. Bodson, Adaptive Control: Stability, Convergence and Robustness, Prentice-Hall, 1989. [lo] J. E. Slotine and W. Li, Applied Nonlinear Control. Prentice-Hall, 1991. [ 113 R. Soto and K. S. Yeung, "Sliding-Mode control of an Induction Motor without flux measurement," Proc. 27th
3.4 Simulation Results Using the controllers in Section 3.1 and 3.2 and the results of Section 3.3, we performed some simulation studies. In these simulations, we only assume W, i,, ib are available. The rotor fluxes are reconstructed by using (33). A summary is given below. (a) Using the data in [6], we simulate the non-adaptive robust controller (10) in Section 3.1. The results are shown in Figs. l a and lb. The reference trajectories are the same as those in [63. o r i s zero from 0 to 0.3 s., 220 r/s from 0.3 to 5 s., and 350 r/s from 5 s. onwards. vdris 1.3 Wb from 0 to 5 s. and 0.8 Wb after 5 s. The discontinuities are smoothed by linear interpolations. A load disturbance of 40 Nm is added at t = 2 sec. We set R r =~ 0.075, TLN=O,ql=q2=103, kl = k2 = 25, and boundary layer width equals to 0.1. As can be seen from Figs. l a and lb, the control is not robust to the mismatched uncertainties in (6a) and (6b) because they are not in the range space of Ud and uq. Hence our nonadaptive sliding controller needs some improvements. It should be noted that the waveforms of U,, i,, ib are not due to switching in sliding mode control as it appears to be. Similar waveforms have also been observed in Marino's
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Annual Meeting and Conference of the IEEE Industry Application Society., 1992.
[I21 G. C. Verghese and S. R. Sanders, "Observers for Flux Estimation in Induction Machines," IEEE Transactions on Industrial Electronics, 35, 85-94, 1988.
. . s
Fig. la Performance of the nonadaptive robust controller.
-1
Fig. 2a Performance of the robust-adaptive controller.
I
Fig. 1 b Performance of the nonadaptive robust controller.
Fig. 2b Performance of the robust-adaptive controller.
