High performance induction motor drives based on `Field. Orientation' have been ...... Speed Detection of Inverter Fed Induction Motors us- ing Slot Harmonics ...
A Robust Torque Controller for Induction Motors Without Rotor Position Sensor: Analysis and Experimental Results E.G.Strangas, H.K.Khalil, B. Al Oliwi, L. Laubinger Department of Electrical Engineering Michigan State University East Lansing, MI 48823, USA
J.M.Miller Ford Motor Co. Vehicle Electronics Systems Department Dearborn, MI 48121, USA
of the internal magnetic elds. High performance also requires fast control of torque, which is achieved through a transformation of the state variables to a moving frame of reference. These transformations allow decoupling of the developed torque from other variables, but increase the sensitivity of the controller to machine parameter variations. Xu and Novotny 2] presented a transformation to the estimate of the stator ux. In this frame of reference the I. Introduction controller becomes independent of the leakage inductance of the stator, but the components of the stator current High performance induction motor drives based on `Field that control the stator ux and torque are coupled. The Orientation' have been commercially available for almost authors show by simulation and experiment the robusttwo decades. Their performance is such that they can ness of the system performance and its superiority over easily replace DC drives without loss of accuracy, stabil- the same method based on the rotor ux estimate. ity or speed of response. The most signicant barrier to the other hand, there has been extensive work based further utilization is at present the rotor position sensor, onOn transformation to the rotor ux. The magnitude and which, being external to the controller-power electronics position of this ux can be calculated from the stator assembly, is the cause of low reliability, high maintenance equations involving the estimation of back emf, or from and high installation cost. the rotor equations. These two methods are often referred Since the end of 1980s the estimation of the rotor posi- to as Direct and Indirect Field In general, tion, or of the position of the rotor ux, has been the sub- the rst method produces errorsOrientation. dependent on the inteject of intensive research. The methods utilized include a gration method and the estimates of stator parameters, variety of control theory approaches, and the state of the which may become unacceptable at low speeds, while the art is detailed in a recent book 1]. produces errors related to variation of rotor paramThe operation of an induction motor as part of a `high other eters. performance drive' without rotor position sensors requires While many drives switch between `indirect' and `dithe estimation of speed or position of the rotor or of one rect' eld orientation, at an appropriate low speed, Bose
3] proposed a hybrid controller that utilized both techniques with dierent weight lters, and showed good experimental results. In most of the stator based methods, the integration of the stator induced voltages poses signicant computational problems stemming from the DC oset and other errors. Many solutions have been utilized, including low pass lters and adaptive observers. Ohtani, Takada and Tanaka 5] presented a method that utilized stator based estimation of rotor speed but included a compensator, which in experiment permitted operation down to 1=100th of rated motor speed. Abstract|A robust controller, which tracks ux and torque commands is designed for an induction motor. It is based on a transformation to the estimate of the rotor ux. The speed of the rotor is estimated from the stator currents using a high-gain observer. The controller is robust with respect to variations of the rotor resistance. Analysis of the controller shows that the system is asymptotically stable in the motoring and brakeing regions and in the generating region for large enough torque reference. Simulations and experimental results show agreement with the analysis.
Yoo and Ha 6] presented recently two complementary ux and speed estimators that do not require integrators but are based on the stator equations. Their main estimator requires the knowledge of the sign of the rotor speed, and derivatives of stator currents, and operates in the motoring and generating region. Their complementary estimator does not require the sign of the rotor speed, but does not operate in the generating mode. Rigorous open loop analysis is presented for slow speed variations and shows convergence in their respective regions. An elaborate scheme allows for switching between estimators as the operating point enters the motoring, brakeing or generating region. The analysis does not take into consideration variation of the rotor resistance. The inability to operate an induction motor at zero frequency, or at least estimate how close to it the induction motor can operate, utilizing only rst order phenomena has also led to a collection of alternative approaches. They are based on the exploitation of secondary phenomena, e.g. saturation, higher current harmonics, rotor asymmetry or imbalance etc. as for example in 4]. Two issues of operation remain of concern: One is the dependence of the performance of the controller on some of the parameters of the motor, and the second is the operation of the controller around the zero frequency point. In this work we present a controller that requires no rotor position or speed measurements, or measurement of the terminal voltages. It is robust with respect to the rotor resistance. The use of an accurate value of the stator resistance in the controller is crucial fortunately stator temperature was available for other reasons, and was utilized to provide an accurate estimate. The approach combines traditional rotor ux estimators with high-gain observers and robust control techniques to achieve the desired torque tracking. The rotor ux is estimated using the rotor equation and an estimate of the rotor speed. The state equation is then transformed to the frame of reference dened by the estimate of the rotor ux, which translates the ux and torque commands to commands for the estimates of the direct and quadrature axes currents. A high-gain observer is utilized for the derivative of the q-axis current estimate, and from it the rotor speed is estimated. A sliding mode controller is used for the regulation of the rotor ux estimate and a saturated PI controller is used for torque tracking. The proposed controller is able to operate in the motoring and brakeing regions, and at these operating points the steady state error for torque and ux tracking estimates is zero. It can pass through the generating region without losing local stability. For this controller we present closed loop analysis, taking into consideration the variation of rotor resistance and errors in the rotor ux and speed estimates. Conditions are described, under which asymptotic tracking is achieved and lost, and these
conditions are veried by simulation and experiment. II. The Induction Motor Model
We start from the established equations 7] for the stator and rotor voltages in their natural frames: ; (1) Rr ir + Lr ddtir + M dtd is e;jp = 0 ; (2) Rs is + Ls ddtis + M dtd ir ejp = vs where is the rotor position and p is the number of the
pole pairs of the motor. Dening the rotor ux in the stator reference frame as:
r = M is + Lr irejp
(3)
and eliminating all references to rotor currents we obtain: r _r = ; RLr r + jp!r + MR Lr is r Rs is + Ls ddtis + LM dtd (r ; M is) = vs r
(4) (5)
In addition we consider the equation for the motor developed torque:
Td = (2pM=3Lr )= isr ]
(6)
Limiting ourselves further on to discussing only stator currents and voltages and rotor ux, we drop the subscripts s and r from these quantities in the equations (4) and (5). Furthermore we dene as r = Rr =Lr , s = Rs =Ls, = (1 ; )=, = 1=(Ls ), = 1=. III. Controller Derivation
To estimate the rotor ux, we start with (4). Using matrix rather than space vector notation, and with 1 0 0 ; 1 I = J = 0 1 1 0 we obtain:
^_ = (;roI + p!^ J)^ + roM i
(7)
where !^ is the rotor speed estimate, to be determined, and ro is the nominal value of r . Let e = ^ ; be the ux estimation error. We perform a change of variables to bring the machine equations into the eld orientation coordinates, but, unlike the traditional machines literature, we use the ux estimate ^ rather than itself. This idea, which was rst used in 11], allows us to keep track of the eect of the ux estimation error and perform rigorous analysis of the closed-loop system. The new variables
Ji=^d , are ^d = j^ j, ^ = 6 ^ , ^id = ^T ii=^d , ^iq = ;^T Ji T T T ^ ^ ^ ^ ^ ^ v^d = vv=d , v^q = ; Jv Jv=d, ed = ee=d , and eq = T ^ ^ Je ; Je=d . The variables ^id and ^iq can be calculated on line from the measured current i and the ux estimate ^ . The control variable v can be calculated on line from v^d and v^q using ^ . The new variables satisfy the equations
Singular perturbation theory 13] shows that, for bounded , z1 and z2 are O( ). Consequently, y^2 ; ^i_q = O( ). The speed estimate !^ is given by the equation
^_ d = ;ro^d + ro M^id (8) _ ^ = p!^ + ro M^iq =^d (9) _^id = r ^d ; (s + r M )^id + p!^^iq + roM^i2q =^d + v^d ; r ed ; p!eq (10) ^i_q = ;p!^d ; p!^^id ; (s + r M )^iq ; roM^id^iq =^d + v^q + p!ed ; r eq (11) e_d = ;r ed + (p!^ ; p! + roM^iq =^d )eq + (ro ; r )(M^id ; ^d ) (12) ^ ^ e_q = ;(p!^ ; p! + roM iq =d )ed ; r eq + (ro ; r )M^iq + p(^! ; !)^d (13)
Subtracting (20) from (11), it can be veried that (21) !^ ; ! = ; 1 +^O( ) pd
and the motor torque is given by T = (2pM=3Lr )(^iq ^d + ^ideq ; ^iq ed) (14) The key observation here is that the actual speed ! appears on the right-hand side of (11). Neglecting the ux estimation errors ed, and eq and using the nominal value ro for r , we can use (11) to estimate !. This requires calculation of the derivative of ^iq . We use a high-gain observer to approximate this derivative and generate the estimate of !. Setting y1 = ^iq y2 = ^i_q (15) it can be veried that y1 and y2 satisfy the state equations y_1 = y2 (16) 2 ^ y_2 = def = ddti2q (17)
A high-gain observer can be taken as 8] y^_ 1 = y^2 + (a1 = )(y1 ; y^1 ) (18) _y^2 = (a2 = 2 )(y1 ; y^1 ) (19) where a1 , a2 , and are positive parameters, with chosen su ciently small to assign the observer poles at order O(1= ) locations that are fast enough relative to the dynamics of the system. Taking z1 = (^y1 ; y1 )= and z2 = y^2 ; y2 yields z_1 = ;a1 z1 + z2 z_2 = ;a2 z1 +
y^2 = ;p!^ ^d ; p!^^id ; (s + ro M )^iq ; roM^id^iq =^d + v^q (20)
where
1 = p!ed ; r eq + (ro ; r )M^iq The speed estimate !^ is passed through a limiter to en-
sure its boundedness and overcome the transient peaking associated with high-gain observers 8]. The design of feedback control so that the torque T asymptotically tracks a reference torque Tref (t) is achieved using robust control to solve two separate control problems. In the rst problem, v^d is designed so that ^d asymptotically tracks a reference ux ref . In the second problem, v^q is designed so that ^iq asymptotically tracks a reference current iref . The reference values ref and iref are chosen to satisfy the relationship
Tref = (2pM=3Lr )iref ref (22) which follows from (14) upon setting ed = eq = 0. For convenience, we take ref to be constant. Flux regulation is achieved using a sliding mode controller 14, 10] while current tracking uses a PI controller.1 Flux Regulation:
Taking
s1 = k1 (^d ; ref ) + (M^id ; ^d ) k1 > 0
(23)
it can be seen that
s_1 = 1 () + v^d where 1 is a continuous function of the state and reference variables. The sliding mode controller is taken as
v^d = ;V sgn(s1 )
(24)
where sgn() is the signum nonlinearity and V is chosen large enough that j 1 j < V over the domain of interest. With (24), s1 reaches zero in nite time. In implementation, the signum nonlinearity sgn(s1 ) is replaced by the 1 In an earlier version 12], a sliding mode controller was used in the current tracking loop, but it was later replaced by a PI controller due to chattering in implementation.
saturation nonlinearity sat(s1 = 1), with small 1 > 0, to reduce chattering 10]. q-axis Current Tracking:
We use the PI controller
v^q = ;k3 s2 k3 > 0 where
(25)
Z
s2 = k2 e1+e2 k2 > 0 e1 = (^iq ;iref ) dt e2 = ^iq ;iref (26) The control v^q is passed through a limiter to ensure it boundedness. IV. Closed-loop Analysis
A. Boundedness
We start by showing that boundedness of v, !, and !^ guarantee boundedness of all state variables. The boundedness of v, and !^ is ensured by design, while the boundedness of ! is assumed to be the result of an outer feedback loop that deals with the speed control problem. Recall that for the torque control problem, ! plays the role of an exogenous signal. Hence, its boundedness must be guaranteed by some factors beyond our controller. Using
V1 = T + (Mr =s )(i + )T (i + ) as a Lyapunov function for (10)-(11), it can be veried that
TQ i + i + T + 2 (Mr =s )(i + ) v
V_1 = ;2r where
(27)
(1 + M )I ;M I Q = ;M I (M= )I is positive denite. Therefore, boundedness of v ensures boundedness of and i. Using V2 = ^T ^ as a Lyapunov function for (13), it can be seen that V_2 = ;2ro^T ^ + 2ro M ^T i Hence, boundedness of i ensures boundedness of ^ , Consequently, ^d is bounded. From the denitions of ^id , ^iq , v^d , and v^q , we see that as long as ^d is bounded away from zero,2 these four variables will be bounded. Finally, 2 The singularity condition ^ = 0 is avoided by initialization of d the controller so that ^ d tracks its reference value before the torque command is applied.
using V3 = e2d + e2q as a Lyapunov function for (12)){((13), we obtain V_3 = ;2r (e2d + e2q ) + 2(ro ; r ) (M^id ; ^d )ed + M^iq eq ] + 2p(^! ; !)^d eq (28) Since ^id , ^d , ^iq , !^ ; !, and ^d are bounded, we conclude that ed and eq are bounded. B. Asymptotic Behavior
We now analyze the asymptotic behavior of the closedloop system as t tends to 1. The closed-loop system is described by a sixth-order model, formed of equations (8) { (13), the integrator of the PI controller, and the feedback control laws. We will show that the asymptotic behavior of the system is dominated by a second-order model that can be deduced from the sixth-order one. First, we note that s1 reaches the boundary-layer fjs1 j 1 g in nite time. Inside the boundary-layer, the variable s1 will be much faster than the other variables due to smallness of 1 . Using singular perturbation theory, we can reduce the model order by setting 1 = 0 which results in s1 = 0. Second, it can be seen that, for typical values of the PI controller gains k2 and k3 , the variables s2 will be much faster than the other variables. Again, by singular perturbation theory, we can reduce the order of the model by setting s2 = 0. On the surface s1 = s2 = 0, the closedloop system is described by the fourth-order model ~_ d = ;rok1 ~d (29) e_1 = ;k2 e1 (30) e_ d = ;(ro ; r )k1 ~d ; r ed + (p!^ ; p! + ro M^iq =^d)eq (31) e_ q = ;(p!^ ; p! + ro M^iq =^d )ed ; r eq + (r0 ; ar ) M i^q + p(^! ; !)^d (32) where ~d = ^d ; ref . From (29) and (30), we see that ~d and e1 approach zero as t tends to 1. Hence, for the asymptotic behavior of the system, it is su cient to consider the second-order model e_ d = ;r ed + (p!^ ; p! + ro M^iq =^d)eq e_ q = ;(p!^ ; p! + ro M^iq =^d )ed ; r eq + (r0 ; ar ) M i^q + p(^! ; !)^d Substituting for (^! ; !) using (21), with O( ) neglected, we obatin e_ d = ;r ed + (r Miref =ref )eq ; (p!=ref )ed eq + (r =ref )e2q (33) e_ q = ;(p! + r Miref =ref )ed + (p!=ref )e2d ; (r =ref )ed eq (34)
The system (33){(34) has an equilibrium point at (ed eq ) = (0 0). For torque tracking we need this equilibrium point to be asymptotically stable, for the convergence of (ed eq ) to (0 0) ensures, from (14), that T tracks Tref at steady state. By linearization, it can be veried that the equilibrium point (0 0) is asymptotically stable if the 2 2 matrix 2
;r A1 (t) = 4 (t) ;p! ; 3R2rpTref 2 ref
3Rr Tref (t) 2p2ref
0
3 5
T ref
Coordinate
A2 (t) = 4
(t) ; 3R2rpTref 2 ref
3Rr Tref (t) 2p2ref
0
5
^ V
Transformation
S
Rotor to stator λ
^ V
ref Sliding Mode Controller
Induction
Inverter
Motor
d
i ^ ρ ^ λ d
Rotor Flux
^ iq
^ i d
s
Coordinate Transformation Stator to rotor
Estimator
^ ω
is exponentially stable that is, the origin of the linear system z_ = A1 (t)z is exponentially stable. To investigate exponential stability of A1 , we view it as a perturbation of 3 2
;r
^ V q
^ i q ref
High Gain Observer
Fig. 1. Block Diagram of the Controller
points at P 1 = (0 0), P 2 = (0 ;Miref ), and
p!ref ) ;r (r Miref + p!ref ) with a perturbation term proportional to !. The matrix P 3 = p!(r Mi2 ref + 2 r + p !2 2r + p2 !2 A2 is a familiar matrix in adaptive control theory, c.f. 9], 2 and is known to be exponentially stable if 3Rr Tref (t)=2pref Linearization at the equilibrium points shows that P 1 is is persistently exciting that is, asymptotically stable if !2 ! Z t+T0 1 3Rr Tref ( ) d 8 t 0 3 R r Tref 0 Tref p! + 2p2 >0 (35) T0 t 2p2ref ref for some positive constants T0 and 0 . The constant 0 , called the level of persistence of excitation, determines the degree of exponential stability of A2 . From stability theory, e.g. 10], it is well known that the perturbed matrix A1 will be exponentially stable if ! is small enough. We can tolerate a larger ! as the level of persistence of excitation 0 increases. The connection between ! and Tref can be easily seen in the special case when both are constant, for then A1 will a constant matrix. In fact, in this case we can easily perform nonlocal analysis to study the behavior of the system far from origin (0 0). Let us note rst the special case when Tref = ! = 0. It can be easily seen from (33){(34) that in this case the system has a unique equilibrium point at the origin. Using the Lyapunov function V3 = e2d + e2q and LaSalle's invariance principle 10], it can be shown that the origin is globally asymptotically stable. Hence, all trajectories asymptotically approach at the origin. This special case arises in the initialization phase when the motor is at rest, no torque command is applied, and the constant ux command ref is applied. It shows that during this initialization phase the ux estimation error reaches zero steady-state. This zero ux estimation error becomes the initial state when a torque command or speed is applied. When the constant values of Tref and ! are dierent from zero, the system (33){(34) has three equilibrium
P 2 is asymptotically stable if !Tref < 0 and P 3 is asymptotically stable if
(36) !
! p! + 32Rpr T2ref > 0 ref
(37)
In each case, the equilibrium point is unstable if the inequality is reversed. From the stability conditions (35){ (37), we can identify three modes of operation: Motoring mode: sign(!) = sign(Tref ). In this case, P 1 and P 3 are asymptotically stable while P 2 is unstable. Braking mode: sign(!) = ;sign(Tref ), and jp!j < j3Rr Tref =2p2ref j. In this case, P 1 and P 2 are asymptotically stable, while P 3 is unstable. Generation mode: sign(!) = ;sign(Tref ), and jp!j > j3Rr Tref =2p2ref j. In this case, P 2 and P 3 are asymptotically stable, while P 1 is unstable. Thus, the origin P 1 = (0 0) is asymptotically stable in the motoring and braking modes. Since at the end of the initialization phase, the initial state will be very close to the origin, the trajectory will indeed converge to the
4
3
[Nm]
origin when it is asymptotically stable. This shows that perfect tracking of Tref is achieved in the motoring and braking modes. Moreover, tracking is robust with respect to variations in the rotor resistance Rr since the controller uses only the nominal value ro of r = Rr =Lr . This conclusion is conrmed by simulation and experimental results presented in the forthcoming sections. Let us conclude by noting that when r jTref j jp!j < 3R2p (38) 2 ref
1
the origin will be asymptotically stable irrespective of the signs of Tref and !. This agrees with our earlier conclusion for the general case of time-varying Tref and !.
0
To verify the theoretical results, to test the eects of parameter errors and system delays, and to calculate the controller parameters, a series of simulations were run. In the simulation presented here the model rotor resistance was higher than the nominal by 80%. The torque and ux commands were held constant, while the load speed varied. In Fig. 2 and Fig. 4 it is clear that the controller generally succeeds in tracking the two commands. If the load speed becomes negative enough, and it persists, the inital speed estimate is initially correct, but quickly degrades. It became clear that, as theory had predicted ( equation (38)), the system would be locally stable at zero torque and ux errors at low speeds (point P1) as long as the torque command was adequate. In addition, the system would remain at the original stable point, P1, during short torque transients. The same excursions of torque command or load speed, if prolonged, would lead to one of the other two, undesirable stability points. This is shown clearly during the rst time that the speed becomes negative, and the speed estimate remains correct for a short period of time. Also, once the speed has returned to positive, the system recovers, showing that the instability is local. It is also noted that although there can be a signicant error in the speed estimate (Fig. 3), the torque linearity remains high (Fig. 2), as predicted by the theory.
1
2
3
4
5
Fig. 2. Simulation: Reference torque (solid line) and developed torque (dashed line).
60
[rad/s]
40 20 0 −20 −40
0
1
2
3
4
5
[s]
Fig. 3. Simulation: Load speed (solid line) and estimated speed (dashed line)
0.03
0.025
[Wb]
VI. Simulations
0
[s]
V. Machine and Controller Parameters
The motor used has been developed for automotive use it has four poles and a rating of about 1000W. It has nominal parameters were: Rs = 13:5m", Rr = 28m", M = 0:53mH ,Lr = 0:608mH , and Ls = 0:595mH . The gains for the system were: For the iq PI controller, k3 = 5, k2 = 14 for the ux controller, the gain k1 = 130, and 1 = 1=130 for the high gain observer, = 0:143.
2
0.02
0.015
0.01
0
1
2
3
4
[s]
Fig. 4. Simulation: Reference rotor ux (solid line
and actual rotor ux (dashed line)
5
VII. Experiment
Induction Motor
DC Generator In-line torque sensor
Rotor position sensor
To test the theory we developed an experimental setup, a general view of which is shown in Fig. 5. A MOSFET inverter was used with turn-on and turno times close to 2s and was supplied from a 12V source. The drive controller was implemented in an ATT DSP32C, with analog inputs and digital outputs. The program cycle was 100s, the same for all operations forward dierences were used for discretizing the dierential equations. The DSP was housed in an IBM compatible PC, which was used also for program cross-compilation and debugging. Because of its capacity, the DSP board was used to control the complete experiment, not only the induction motor. Its functions included:
MOSFET
Controlled Current sensors
Inverter
Rectifier
Space Vector PWM
Data
I.M. Controller
Speed/Torque
Acquisition
DSP board
Controller
ATT 32C
PC
Fig. 5. Experimental Setup
Measurement of currents, observers and estimators, controller, output of voltage commands,
to the induction motor controller, while the speed command was given to DC regenerative drive, as in experi monitoring of the stator temperature, and adjusting ment D. In some of the experiments, an outside loop was the value of the stator resistance in the model, used to generate these two commands. A torque sensor was to used measure the shaft torque. control of the DC regenerative drive, Because of the eects of inertia and friction, its readings closer to developed torque at high load torques and data acquisition and processing for the monitoring were constant speeds. of the experiment (currents, rotor speed, shaft torque To start close at the equilibrium point P 1, and to avoid measurements). singularities due to low ux, we initially supplied a zero command and a rated ux command to the drive The generation of the gate pulses to the inverter was ac- torque at zero After 1 ; 3s we supplied torque commands complished through a dedicated microcontroller, Intel960, and loadspeed. speed. which used a space vector Pulse Width Modulation scheme, operating at 10kHz . The PWM scheme accounted for VIII. Experiment Results dead time and for very short pulses, but its operation is not the subject of the present paper. Four temperature A series of experiments were conducted to verify the sensors were placed on the stator surface or imbedded in behavior of the controller as discussed earlier. the stator windings. They were used to monitor safe operation of the induction motor, as well as to adjust the value of the stator resistance in the model. Given the value of the stator resistance Rs0 at temperature t0 , its value at temperature t1 , for this motor, is: 70
60
+ 235 Rs1 = Rs0 tt1 + 235 0
The value of the stator resistance is critical in this application, and has to be estimated accurately. Since this was a high current, low voltage application, the stator conductors were quite thick. Because of this, the stator resistance used was dierent than its DC value. The DC regenerative drive, used as an active load of the Induction motor, was supplied from the line through a computer-controlled four-quadrant thyristor converter. The DC motor was controlled by the DSP through a PI controller operating on the delay angle of the controlled rectier. The torque command was given independently
[ rad / s ]
50
40
30
20
10
0 0
5
10
15
20 [s]
25
30
35
40
Fig. 6. Experiment A: Motor Speed Estimate(Solid Line) and Measured Speed ( Dashed Line)
The rst experiment, A, was to explore the operation of the motor in decreasing speed. The torque and ux commands were held constant, while the load speed varied. In
0.025
50 40
0.02
30 20
0.015
Wb
Amp.
10
0.01
0 −10 −20
0.005
−30 −40
0 0
5
10
15
20 [s]
25
30
35
−50 0
40
5
10
15 [s]
20
25
30
Fig. 10. Experiment B: Estimated torque component of the current, i^q
Fig. 7. Experiment A: Estimated rotor ux linkages 60 60 50 40 40
20
[ rad / s ]
Amp.
30
20
10
0
−20
0 −40 −10 0
5
10
15
20 [s]
25
30
35
40 −60 0
Fig. 8. Experiment A: Estimated torque component of the current, i^q
80
60
[ rad / s ]
40
20
0
−20
−40
5
10
15 [s]
20
25
Fig. 9. Experiment B: Motor Speed (Solid line) and Measured Speed (Dashed Line)
10
15 [s]
20
25
30
Fig. 11. Experiment C: Motor Speed (Solid Line) and Measured Speed (Dashed Line)
Fig. 6 the actual and estimated speeds are shown, while in Fig. 8 is shown the estimated value if iq . In Fig. 7 the value of the estimated magnetizing ux linkages is given. In experiment B the motor reverses speed while it is operated without load torque except for the combined inertia of itself and the DC generator. In Fig. 9 the actual and estimated speed are given, while in Fig. 10 is given the estimate of the torque component of the current, i^q .
−60 0
5
30
In experiment C, the direction of the motor is again reversed,but this time with simultaneous reversal of the torque command. In Fig. 11 the actual and estimated speed are given, while in Fig. 12 is given the estimate of the torque component of the current, i^q . Although the estimated speed can be quite inaccurate as shown in experiment C, the torque tracking is appreciably better. It is also clear in experiments B and C that during a speed reversal the operating point can pass through a locally unstable region without loss of stability. In experiment D it is attempted to simulate the operation of the drive under real torque and speed proles. The times where the torque response deviated from the commanded one, and the speed estimate is incorrect are either at starting, i.e. before the ux has been established, and the operation of the drive has settled at the stable point P1, and when the machine is pushed to the generating region for long times, as predicted by theory. It is also to be noted that after loss of local stability, the controller recovers once an operating point is commanded in the stable region.
100 80 40
60 30
40
20
[rad/s]
20
Amp.
10
0 −20
0
−40 −10
−60 −20
−80
−30
−100 0
−40 0
5
10
15 [s]
20
25
30
Fig. 12. Experiment C: Estimated torque component of the current, i^q
2
4
6
8
10 [s]
12
14
16
18
20
Fig. 15. Experiment D: Actual (imposed) Motor Speed (Solid Line) and Estimated Speed (Dashed Line)
IX. Conclusions
2.5 2 1.5
[ Nm ]
1 0.5 0 −0.5 −1 −1.5 −2 0
5
10
15 [s]
20
25
30
Fig. 13. Experiment C: Measured shaft torque (Dashed Line) and estimated electromagnetic torque (Solid Line)
1.5
1
0.5
[Nm]
0
−0.5
−1
−1.5
−2
−2.5 0
2
4
6
8
10 [s]
12
14
16
18
Fig. 14. Experiment D: Motor Torque Command(Solid Line) and Measured Shaft Torque (Dashed Line)
20
A controller was developed for induction motors, without a rotor position or voltage sensor. The controller is robust with respect to variations in the rotor resistance, while variations in the stator resistance are compensated for through the measurement of stator temperature. The controller is based on the transformation of the motor variables to the system of coordinates dened by the estimated rotor ux. It estimates the rotor speed from the derivative of the q-axis current, which it calculates with the use of a high gain observer, thus avoiding taking derivatives of measured quantities. A sliding mode controller is used to track the ux command, and a PI controller to track the torque command. Closed loop analysis is presented, taking into account the feedback loop and the asymptotic behavior of the system. The system has good asymptotic stability, when it is energized and the ux established before a nonzero torque command is applied. It operates well in the motoring and brakeing mode, while in the generating mode its performance is decreased. Its asymptotic stability in the generating region depends on the persistence of the torque command with respect to the value of the negative speed. Conditions for zero error in torque and speed tracking have been developed and are used to explain the performance of the motor in simulation and experiment. An experimental setup has been developed and was used to verify the analysis. The simulation and experimental results show good tracking in the brakeing and motoring mode, while the drive performance in the generating region agrees with the theory, allowing short excursions into the generating region, maintaining operation close to zero errors in ux and torque.
X. References
1] K. Rajashekara, A. Kawamura , and K. Matsouse, editors. Sensorless Control of AC Motor Drives. IEEE Press, 1996.
2] X. Xu and D.W. Novotny. Implementation of Direst Stator Field Orientation Control on a Versatile DSP Based System. IEEE Trans. Ind. Appl. 27:694-700, 1991.
3] B.K. Bose, M.G.Simoes et al. Speed Sensorless Hybrid Vector Controlled Induction Motor Drive Transducers, IEEE Tran. Contr. Sys. Tech.
4] A. Ferrah, K.J. Bradley and G.M. Asher. Sensorless Speed Detection of Inverter Fed Induction Motors using Slot Harmonics and Fast Fourier Transform, in IEEE PESC pp. 279-286, 1992.
5] T. Ohtani, N. Takada, and K. Tanaka. Vector Control of Induction Motor Without Shaft Encoder, IEEE Tran. Ind. Appl. 28:157-164, 1992
6] Ho-Sun Yoo and In Joong Ha. A Polar CoordinateOriented Method of Identifying Rotor Flux and Speed of Induction Motors without Rotational Transducers In IEEE IAS95 pp.137-143, 1995.
7] W. Leonhard, Control of Electrical Drives, Springer Verlag, 1984
8] F. Esfandiari and H.K. Khalil. Output feedback stabilization of fully linearizable systems. Int. J. Contr., 56:1007{1037, 1992.
9] P.A. Ioannou and J. Sun. Robust Adaptive Control. Prentice-Hall, Upper Saddle River, New Jersey, 1995.
10] H.K. Khalil. Nonlinear Systems. Prentice Hall, Upper Saddle River, New Jersey, second edition, 1996.
Elias G. Strangas received the Dipl. Eng. Degree in Electri-
cal Engineering from the National Technical University of Greece, and the Ph.D. from the University of Pittburgh in 1975 and 1980 respectively. He was with the University of Missouri-Rolla from 1983-1985. He has been with Michigan State University since 1986, where he is an Associate Professor. His research interests include numerical methods as applied to the computation of the Electromagnetic Field in Electrical Machines, the design of Power Electronics Systems and Control of Drives.
Hassan K. Khalil received the B.S. and M.S. degrees from Cairo
University, Cairo, Egypt, and the Ph.D. degree from the University of Illinois, Urbana-Champaign, in 1973, 1975, and 1978, respectively, all in Electrical Engineering. Since 1978, he has been with Michigan State University, East Lansing, where he is currently Professor of Electrical Engineering. He has consulted for General Motors and Delco Products. He has published several papers on singular perturbation methods, decentralized control, robustness, nonlinear control, and adaptive control. He is author of the book Nonlinear Systems (second edition New Jersey: Prentice Hall, 1996), coauthor, with P. Kokotovic and J. O'Reilly, of the book Singular Perturbation Methods in Control: Analysis and Design (New York: Academic, 1986), and coeditor, with P. Kokotovic, of the book Singular Perturbation in Systems and Control (New York: IEEE Press, 1986). He was the recipient of the 1983 Michigan State University Teacher Scholar Award, the 1989 George S. Axelby Outstanding Paper Award of the IEEE Transactions on Automatic Control, the 1994 Michigan State University Withrow Distinguished Scholar Award, and the 1995 Michigan State University Distinguished Faculty Award. He is an IEEE Fellow since 1989. Dr. Khalil served as Associate Editor of IEEE Transactions on Automatic Control, 1984 - 1985 Registration Chairman of the IEEE-CDC Conference, 1984 CSS Board of Governors, 1985 Program Committee member, IEEE-CDC Conference, 1986 Finance Chairman of the 1987 American Control Conference (ACC) Program Chairman of the 1988 ACC Program Committee member, 1989 ACC and General Chair of the 1994 ACC. He is now serving as Associate Editor of Automatica and action editor of Neural Networks.
Bader Al Oliwi was born in Zul, Saudi Arabia in 1965. He
received his BSc degree in Electrical Engineering from King Saud University, Saudi Arabia in 1989 and his MS degree from Michigan State University in 1993. He is currently a Ph.D. student at MSU. His research interests include robust adaptive nonlinear control and induction motors.
11] H.K. Khalil and E.G. Strangas. Robust speed control of induction motors using position and current measurement. IEEE Trans. Automat. Contr., 41:1216{ 1220, 1996. Lorenz Laubinger studied electrical engineering at the UniverHannover, Germany, from 1988 to 1994 and obtained degree
12] H.K. Khalil, E.G. Strangas, and J.M. Miller. A ofsityDipl.-Ing. Worked at Michigan State University and Ford Motor torque controller for induction motors without rotor Company on control of electric machines. Since 1997 active in the position sensors. In ICEM 96, Vigo, Spain, September eld of machine vision and pattern recognition at Perceptron Inc., Michigan. 1996.
13] P.V. Kokotovic, H.K. Khalil, and J. O'Reilly. Singular Perturbations Methods in Control: Analysis and Design. Academic Press, New York, 1986.
14] V.I. Utkin. Sliding Modes in Optimization and Control. Springer-Verlag, New York, 1992.
John M. Miller (M83, SM94) received the BSEE in 1976, MSEE
1979 and PhD 1983. Member of the technical sta, Texas Instruments, Dallas Texas from 1976 to 1980. He joined Ford Motor Company research laboratory in 1983 to work on electric vehicle programs and vehicle electrical system components. His current work assignments include hybrid electric vehicles, particularly, the
development of ac drives systems and ancillary components. His interests range from the development of novel electromechanical components to the advancement of 42V electrical distribution system for vehicles. He is a member of the MIT/Industry Consortium on Advanced Automotive Electrical Systems and Components, a member of the steering committee and a member of the Ford-MIT Alliance
on 42V architecture. He is a member of the IEEE Southeast Michigan Section executive committee where he currently holds the o ce of treasurer. He is co-organizer of the IEEE Workshop on Power Electronics in Transportation and the nancial chair. Dr. Miller holds numerous US patents and publications on automotive electrical and electronic systems.
