International Journal of Systems Science Vol. 00, No. 00, xxx 200x, 1–19
A Sliding Mode Observer for Sensorless Induction Motor Speed Regulation Claudio Aurora and Antonella Ferrara (Received 00 Month 200x; In final form 00 Month 200x) A novel sliding mode observer for current–based sensorless speed control of induction motors is presented in this paper. The control objective is to guarantee asymptotic tracking of pre–specified references for speed and rotor flux magnitude, without sensors measuring the mechanical speed and the flux, and assuming to have some kind of uncertainties on the value of the rotor resistance. To this end, the proposed observer is designed by coupling a second-order sliding mode observer of stator current with a non-linear flux and speed observer, adaptive with respect to the rotor resistance. Estimation of unknown inputs is based on a different and original approach with respect to the widely used equivalent control–based techniques. As for the control aspects, in the present proposal, the problem of chattering, typical of sliding mode controllers, is made less critical since the derivative of the stator currents are used as discontinuous forcing actions, while the actual control signals are continuous, thus limiting the mechanical stress. Keywords: sliding mode observers; induction motors; sensorless control; adaptation.
1
Introduction
Sensorless control of motor drives has become more and more frequently used in industrial and practical application during recent years. This approach couples the advantages given by the use of induction motors with respect to the other kind of electric machines (see, for instance, (1)) with the possibility of reducing the realization costs of the control system, thanks to the elimination of the sensors relevant to the mechanical variables (2). Yet, the problem of controlling speed (or torque) and flux in induction motor drives is quite hard because of the system nonlinearities and the strict coupling between the state variables. Generally, no flux sensors are provided, and even when a speed measurement is available, flux observers convergence risks to be compromised by significant parameters values variations (the most critical,
Contact author: Antonella Ferrara C. Aurora is with Danieli Automation via B. Stringher, 4 33042 Buttrio (UD), Italy, e-mail:
[email protected] A. Ferrara is with the Department of Computer Engineering and Systems Science, University of Pavia, via Ferrata 1, 27100 Pavia, Italy, fax: +39 0382 985373, e-mail:
[email protected]
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the rotor resistance, may change up to 200% of the nominal value), while the measurements of stator currents turn out to be affected by noise, due to electromagnetic disturbances or to harmonics. As a consequence, high performances and high robustness properties are required to the control and the observer algorithms. A great number of valid proposals for sensorless control of induction motors have appeared in the literature recently (see, for instance (3) and the references therein cited). A lot of efforts have also been dedicated to the problem of estimation of the unknown motor parameters (4). In this context, the sliding mode control methodology (5), (6) capable of guaranteeing high levels of robustness against matched disturbances and parameter variations seems to be well applicable (7), (8). In (9) the speed estimation is obtained by filtering a discontinuous signal, relaying on the concept of “equivalent control” (6). A different scheme, based on the same theoretical concept, is proposed in (10). Sliding mode observers for sensorless induction motor control are also proposed in (11). In (12), and (13), the sliding mode approach is adopted for parameters estimation and noise filtering. During the last years, the increasing interest in higher order sliding modes (14), (15), (16), introduced for differentiation (17), (18), and widely applied to the field of mechanical systems (see, among others, (19), (20), (21), and (22)), has led to many interesting proposals of application of these advanced variable structure control schemes to induction motor control (see, for instance, (13), (23), and (24)). In this paper, a sliding mode control algorithm for current-fed induction motors is presented. In classical current–fed control schemes the stator currents are assumed as control signals. The proposed control scheme allows to maintain the conventional control loops structure of current–fed control schemes, simply replacing the control algorithm, and leaving the current-loop and the inverter PWM logics unaltered. To circumvent the problem of chattering, the time derivatives of the currents have been regarded as auxiliary control signals, while the actual control signals are continuous. The novelty of the proposal here discussed is the coupling of the proposed control algorithm with an original speed and flux observer. Estimation of unknown inputs and uncertain parameters is achieved by driving an adaptive nonlinear observer of rotor flux with auxiliary signals, provided by a second order super–twisting–based sliding mode observer of stator current. Convergence of flux, speed and rotor time constant estimates to the real values is guaranteed. Note that a similar observer/control scheme has been presented by the authors in the preliminary work (12), where, in contrast to the present work, only first order sliding mode observers were adopted. The speed and flux adaptive observer proposed in the paper requires the knowledge of the load torque (while the control law would require only the
3
knowledge of an upperbound of it). Unfortunately, except for a limited field of applications, this information is usually unavailable. For this reason, a load torque estimator has been included in the proposed induction motor control scheme. It is designed relying again on the super-twisting method. Simulations, even at zero speed, show that the observer-based control algorithm provides high regulation accuracy and robustness. This appreciable features are confirmed by experimental results, also reported in the present paper.
2
Model of the Induction Motor and problem formulation
In a fixed reference frame a−b, the fifth order induction motor model is defined by the following equations dψa dt dψb dt dia dt di b dt dω dt
= −αψa − ωψb + M αia = −αψb + ωψa + M αib dψa 1 + (ua − Rs ia ) dt σLs dψb 1 = −β + (ub − Rs ib ) dt σLs Kf 1M Γl = (ψa ib − ψb ia ) − ω− J Lr J J = −β
(1)
where the state variables are the rotor speed ω, the rotor fluxes (ψa , ψb ) and the stator currents (ia , ib ). Stator voltages (ua , ub ) are the control signals, Γl is the load torque; J is the moment of inertia and Kf the friction coefficient; (Rr , Rs ) and (Lr , Ls ) are the rotor and stator windings resistances and inductances, respectively, and M is the mutual inductance. An induction motor with one pole pair is considered. To simplify notations, the following parameters have been introduced
α=
Rr M2 1 M ,σ =1− ,β= , Lr Ls Lr σ Ls Lr
(2)
For current-fed induction motors with high-gain current loops the motor control algorithm can be constructed on the basis of the following reduced
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order motor model dψa = −αψa − ωψb + M αia dt dψ b = −αψb + ωψa + M αib dt Kf dω 1M Γl = (ψa ib − ψb ia ) − ω− dt J Lr J J
(3)
by considering the stator currents (ia , ib ) as control inputs, the rotor fluxes (ψa , ψb ) as the state variables, Γl as an external input, and α as an unknown parameter (depending on the rotor resistance value). The quantities ωr (t) and Ψ2r (t) are the reference signals for the rotor speed and the square modulus of the rotor flux Ψ2 = ψa2 + ψb2 , respectively1 . Then, ˜ can be defined as the tracking errors ω ˜ and Ψ (
ω ˜ = ω − ωr ˜ = Ψ2 − Ψ2r = ψa2 + ψ 2 − Ψ2r Ψ b
(4)
such that their time derivatives are 1M Γl ˜˙ = (ψa ib − ψb ia ) − − ω˙ r ω J Lr J ¡ ¢ ˜˙ = −2α ψ 2 + ψ 2 + 2M α (ψa ia + ψb ib ) − 2Ψr Ψ ˙r Ψ a b
(5)
The problems addressed in the paper are the following: • to design a control algorithm that guarantees that speed and flux tracking errors are driven to zero with exponential law; • to design a speed and flux observer, to be included in the control scheme, adaptive with respect to the unknown rotor time constant (Sensorless Control ).
1 To
allow for correct operation of the control algorithm, the first and second time derivatives of the speed and flux references are assumed to be bounded.
5
3
Sliding Mode Speed and Flux Control
Hereafter, the induction motor basic sliding mode control methodology is first briefly recalled for the reader’s convenience, the new current-based sliding mode control algorithm is designed and, finally, the proposed speed and flux observer is discussed.
3.1
Preliminary Issues: Voltage-based Sliding Mode Control
Among the various sliding mode control solutions for induction motors proposed in the literature, the one presented in (6) can be regarded as the reference one. Its purpose is to directly control the inverter switching by use of three switching reference signals for the stator voltages (u1 , u2 , u3 ): to consider them in place of the transformed ones (ua , ub ), it is necessary to transform them according to the simple law ·
u1 = T u2 u3
(6)
1 1 − 1 − 2 2 2 √ √ 3 3 3 0+ − 2 2
(7)
ua ub
¸
with
r T =
The control design is based on the definition of three sliding functions which identify the manifold in the system state space such that, when the system trajectory lies on it, the system exhibits the desired dynamics. More precisely, to drive speed and flux tracking errors to zero with exponential law, and to guarantee the symmetry condition to the stator voltages system, the following sliding functions s1 = kω ω ˜ +ω ˜˙ ˜ +Ψ ˜˙ s2 = kΨ Ψ Rt s3 = 0 (u1 + u2 + u3 ) dτ
(8)
6
are selected. To determine the control law that is expected to steer the sliding functions (8) to zero in finite time, one has to consider the dynamics of s = (s1 , s2 , s3 )T , described by ds = F + Du dt
(9)
where uT = (u1 , u2 , u3 ), while vector F T = (f1 , f2 , 0) and matrix D can be explicitly found by differentiating s1 and s2 . The components of vector F may be regarded as bounded disturbances, which are in turn continuous functions of motor parameters, speed, rotor fluxes, reference signals and of their first and second time derivatives. Matrix D can be written as ¸ k1 0 0 · D 1 D = 0 k2 0 d 0 0 k3
(10)
1M 1 " # 0 −ψ ψ a b T D1 = J Lr σLs 2αM ψa ψb 0 σLs
(11)
with D1 defined as
£ ¤ and d = 1 1 1 ; k1 , k2 , and k3 are positive constant design parameters, introduced to virtually increase the control amplitude. Defining the transformed sliding functions s∗ = Ωs, where matrix Ω = D−1 exists everywhere in the system state space, except for kΨk = 0, where det D 6= 0, their time derivative, describing the state motion on the subspace s∗ = 0, results in ds∗ dΩ ∗ = ΩF + Ds + u dt dt
(12)
Then, the following switching control law u = −u0 sign (s∗ )
(13)
where u0 is a sufficient high control amplitude, can be chosen to ensure the finite time reaching of s∗ = 0, (6).
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3.2
The proposed State Feedback Sliding Mode Control
To design a sliding mode control algorithm by assuming the stator currents time derivatives as control inputs, it is first necessary to derive the sliding functions to impose the desired behavior of speed and flux errors. To this end, let (
s1 = kω ω ˜ +ω ˜˙ ˜ +Ψ ˜˙ s =k Ψ 2
(14)
Ψ
with the dynamics of sT = (s1, s2 ) described by ds = F + Di˙ dt
(15)
¡ ¢ where i˙ T = i˙ a , i˙ b is the two dimensional control. Vector F T = (f1 , f2 ) can be found in the same way as indicated in the previous section, while · D=
k1 0 0 k2
¸
" # 1M −ψb ψa 0 J Lr ψa ψb 0 2αM
(16)
with k1 and k2 positive constants. By transforming the sliding functions through the use of the matrix Ω = D−1 , and remembering that, in this case too, matrix Ω exists when kΨk 6= 0, the state motion on the subspace s∗ = 0 turns out to be characterized by the equation dΩ ∗ ˙ ds∗ = ΩF + Ds + i dt dt
(17)
By choosing the switching control law i˙ = −i˙ 0 sign (s∗ )
(18)
for sufficiently high values of the design parameter i˙ 0 the so–called reaching condition (6) is satisfied, and the objective of reaching the manifold s∗ = 0 in finite time is attained.
8
The proposed sliding mode control algorithm does not require the knowledge of the load torque, but only the knowledge of an upperbound of it.
4
Sliding Mode Speed and Flux Observation
As already mentioned, flux measurements are never available in induction machines. On the other hand, robustness and cheapness, two of the main advantages offered by induction motors, can be compromised by the need of providing speed sensors. In the sequel, a second order sliding mode speed and flux observer is described. Relying only on the stator current measurement, correct estimation of rotor flux and mechanical speed is guaranteed, even in presence of uncertain rotor resistance. 4.1
The Adaptive Second Order Sliding Mode Speed and Flux Observer
An original adaptive second order sliding mode speed and flux observer is proposed. Differently from (10), and (11), it does not rely on the equivalent control method (6), according to which unknown quantities are obtained by filtering a discontinuous signal. More specifically, it is based on the well–known super–twisting approach (14). First, let us consider a generic rotor flux observer in the form ˆ dψa = gψa dt ˆ dψb = g ψb dt
(19)
The aim of this section is to suitably design functions gψa and gψb , as well as to couple (19) with a suitable speed observer to come up with the flux and speed observer which is the main contribution of the present work. To this end, the following super–twisting–based sliding mode observer of stator current is designed 1 dˆıa 1 2 sign (˜ = −βg + (u − R i ) + w − K |˜ ı | ıa ) a s a a a ψ λ a σLs dt 1 1 dˆıb 2 sign (˜ = −βg + (u − R i ) + w − K |˜ ı | ıb ) s ψ b b b λ b b dt σLs
(20)
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where the dynamics of the auxiliary state variables wa and wb is defined as dwa = −Kα sign (˜ıa ) dt dw b = −Kα sign (˜ıb ) dt
(21)
According to (14), the finite time vanishing of the estimate errors ˜ıa1 = ˆıa − ia and ˜ıb = ˆıb − ib is ensured, independently from the convergence of the flux observer, by proper selection of the positive gains Kλ and Kα of the discontinuous injections introduced to enforce the sliding mode behavior, under the key assumption that the unknown quantities appearing in (20) are bounded. To proceed in the design of the flux and speed observer, the observation errors dynamics is considered, i.e., 1 d˜ıa dψ˜a = −β + wa − Kλ |˜ıa | 2 sign (˜ıa ) dt dt 1 d˜ıb dψ˜b dt = −β dt + wb − Kλ |˜ıb | 2 sign (˜ıb )
(22)
Then, analogously to (2), the following auxiliary quantities are introduced (
za = ˜ıa + β ψ˜a zb = ˜ıb + β ψ˜b
(23)
which exhibit the dynamics 1 dza = wa − Kλ |˜ıa | 2 sign (˜ıa ) dt 1 dz b = wb − Kλ |˜ıb | 2 sign (˜ıb ) dt
(24)
Relying on the auxiliary variables za , zb , and on the components of the current observation error ˜ıa1 , ˜ıb1 , reconstruction of the fluxes observation errors ψ˜a = ψˆa − ψa and ψ˜b = ψˆb − ψb turns out to be feasible, i.e.
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1 ˜ ψa = β (za − ˜ıa1 ) 1 ψ˜b = (zb − ˜ıb1 ) β
(25)
The flux observer (19) is now rewritten as ˆ dψa = gψa = −ˆ αψˆa − ω ˆ ψˆb + M α ˆ ia + fψa dt ˆ dψb = g = −ˆ αψˆb + ω ˆ ψˆa + M α ˆ ib + fψb ψb dt
(26)
where ω ˆ and α ˆ are estimated by ´ K dˆ ω 1 M ³ˆ Γl f ˆ = ω ˆ− + fω ψ i − ψ i a b b a − dt J Lr J J α dˆ = fα dt
(27)
under the assumption that the load torque Γl is known, as assumed in (3) (but this assumption will be removed in the sequel). Functions fψa fψb fω fα are additional terms, to be defined, introduced to impose the desired behavior to the observation errors ψ˜a , ψ˜b , ω ˜o = ω ˆ − ω, and to the estimation error α ˜=α ˆ − α. Relying on (26), the flux observation errors dynamics turn out to be ˜ dψa = −˜ αψˆa − ω ˜ o ψˆb − αψ˜a − ω ψ˜b + M α ˜ ia + fψa dt dψ˜b = −˜ αψˆb + ω ˜ o ψˆa − αψ˜b + ω ψ˜a + M α ˜ ib + fψb dt while from (27) it results
(28)
11
´ K d˜ ωo 1 M ³˜ f = ψa ib − ψ˜b ia − ω ˜ o + fω dt J Lr J α d˜ = fα dt
(29)
As in (2), the unknown quantity α has been considered as an approximately constant parameter, since its dynamics, influenced by temperature variation due to ohmic heating, is reasonably slow if compared with the dynamics of the other electro-magnetic and mechanical variables. The Lyapunov approach can now be used to properly define functions fψa fψb fω fα in order to achieve the desired convergence of the nonlinear observer defined by equations (26) and (27). To this aim, the following Lyapunov function is introduced
1 V = 2
½ ¾ 1 2 1 2 2 2 ˜ ˜ ψa + ψb + ω ˜ + α ˜ γω o γα
(30)
in which γω > 0 and γα > 0. By differentiating V , it results
V˙ =
½ ¾ 1 1 ˙ ˙ ˙ ˜ ˜ ˜ ˜ ˙ ˜o + ψa ψa + ψb ψb + ω ˜oω α ˜α ˜ γω γα
(31)
Substitution of (28) and (29) into (31), and proper grouping leads to ³ ´ Kf 1 2 V˙ = −α ψ˜a2 + ψ˜b2 + fψa ψ˜a + fψb ψ˜b − ω ˜ J γω o µ ³ ´ ³ ´¶ 1 ˜ ˆ ˜ ˆ +˜ α fα + ψa M ia − ψa + ψb M ib − ψb µγα ¶ ´ 1 1 1 M ³˜ ˜ ˜ ˆ ˜ ˆ +˜ ωo fω + ψa ib − ψb ia + ψb ψa − ψa ψb γω γω J Lr
(32)
Relying on knowledge of variables ψ˜a and ψ˜b (25), and remembering that α > 0, Kf > 0 and J > 0, V˙ ≤ 0 can be guaranteed by imposing
12
fψa = −KΨ ψ˜a ˜ fψb = −KΨ ψb ³ ´ 1M³ ´ ˜ ˆ ˜ ˆ ˜ ˜ f = γ ψ ψ − ψ ψ − ψ i − ψ i ω ω a b a b b a b a J Lr ³ ³ ´ ³ ´´ f = γ ψ˜ ψˆ − M i + ψ˜ ψˆ − M i α α a a a b b b
(33)
with constant KΨ > 0. With the choice (33), the design of the flux and speed observer is completed. The overall sensorless control scheme for induction motors is shown in Fig. 1 in full details. Before discussing the stability and convergence issues, it is useful to make some comments on the sensorless control proposal just described. First, it is worth recalling that, for some machines, the big ones, the electrical part is not always faster than the mechanical part (see, for instance, (25)), and so, in that case, the performances of our scheme (as well as of many sensorless schemes appeared in the literature) could degradate unless the tuning of the observer gains is suitably done so that the current observation error is steered to zero first. Yet, it is rather unusual to apply sensorless schemes to control big machines, since it is on small machines that the cost saving due to the elimination of mechanical sensors is significant with respect to the cost of the machine itself. In other terms, for big machines the cost benefit of the elimination of the mechanical sensors is not so relevant to justify the unavoidable reduction of reliability. 4.2
Stability and Convergence Considerations
The convergence of the proposed flux and speed observer can be analyzed as follows. First, one can observe that the current observer (20) converges independently from the the flux observer, provided that the flux observation error and its time derivatives are bounded, and this is proved, considering again the Lyapunov function in (30), and its time derivative in (31). By substituting functions fψa fψb fω fα indicated in (33), one has ³ ´ K 1 f V˙ = −(α + KΨ ) ψ˜a2 + ψ˜b2 − ω ˜2 J γω o
(34)
i.e., V˙ ≤ 0, so that it results that the flux and speed observation errors, as well as α ˜ are bounded. From equations (28)–(29), it is apparent that the same
13
holds for the time derivatives of the flux and speed observation errors, and of the estimation error of α. Moreover, on the basis of (30), and (34), and relying on the same arguments adopted in the proof of Lemma 1 in (26), persistency of excitation conditions are satisfied, i.e., the second equation in (27) can be put in the form · ¸ h³ ´³ ´i· ˜ ¸ dˆ α ψa ψ˜ ˆ ˆ = γα Γ (t) ˜a = γα ψa − M ia ψb − M ib ˜ ψb ψb dt
(35)
Persistency of excitation is then guaranteed provided that Z
t+T
Γ (τ ) ΓT (τ ) dτ
(36)
t
is positive definite for T > 0 and for any t ≥ 0 (27). This condition is satisfied, since, ∀ t ≥ 0, it results Z
t+T
µ³
t
ψˆa − M ia
´2
´2 ¶ ³ ˆ dτ ≥ 0 + ψb − M ib
(37)
And a similar analysis can be performed with reference to ω ˆ . Then, it follows that the equilibrium point (ψ˜a , ψ˜b , ω ˜o, α ˜ ) = (0, 0, 0, 0) of system (28)–(29) is globally exponentially stable. Now it is necessary to consider again the control law indicated in Section 3. When the flux and speed observer is included in the loop, the sliding variable s∗ in (18) depends on the modified tracking errors (
ω ˜ =ω ˆ − ωr ˜ =Ψ ˆ 2 − Ψ2r = ψˆa2 + ψˆ2 − Ψ2r Ψ b
(38)
that is, the observed variables replace the actual ones which cannot be measured. Yet, on the basis of the previous Lyapunov analysis, it results that the flux and speed observation errors, as well as α ˜ are bounded with their time derivatives. Clearly, the same property holds for ω ˆ , ψˆa , ψˆb , α ˆ , and their time derivatives. Then, a sufficiently high value of the design parameter i˙ 0 in (18) can be found to satisfy the reaching condition so that the objective of reaching the manifold s∗ = 0 in finite time is attained. As a consequence, the tracking errors (38) are steered to zero asymptotically. Since, for t → ∞, the observed quantities ω ˆ , ψˆa , ψˆb tend to ω, ψa , ψb , respectively, it follows that the control problem in question is solved.
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5
Load Torque Estimation
The speed and flux adaptive observer described in the previous section requires the knowledge of the load torque Γl (27). Unfortunately, except for a limited field of applications, this information is usually unavailable. In this case, a load torque estimator needs to be included in the proposed induction motor control scheme. In the sequel, a second–order super–twisting sliding mode load torque esˆ l replaces the corresponding timator is proposed. The load torque estimate Γ true value Γl in (27), yielding ´ K ˆl dˆ ω 1 M ³ˆ Γ f ψa ib − ψˆb ia − = ω ˆ− + fω dt J Lr J J
(39)
During transients, convergence of speed estimation is guaranteed by the additional term fω (33). Relying on the fact that the load torque can be regarded as an unknown bounded input, the same quantity fω is here used to drive a load torque estimator having the following structure 1 ˆ 2 Γl = Γ1 − Gλ |fω | sign (fω ) dΓ1 = −Gα sign (fω ) dt
(40)
where Gλ > 0, Gα > 0. The first equation in (29), describing the dynamics of speed estimation error, turns out to be modified as follows ´ K ˜l d˜ ω 1 M ³˜ Γ f = ψa ib − ψ˜b ia − ω ˜− + fω dt J Lr J J
(41)
As a consequence, the time derivative of the Lyapunov function (30) becomes ³ ´ K 1 ˜l 1 Γ f V˙ = − (α + KΨ ) ψ˜a2 + ψ˜b2 − ω ˜2 − ω ˜ J γω γω J
(42)
on the basis of which it a standard input-to-state-stability analysis (28) of ˜ l regarded as an input can be performed. system (28)–(29) with respect to Γ
15
The load torque estimator (40), combined with the sliding mode speed and flux observer previously described, provides satisfactory performances, even at zero speed working conditions, as confirmed by simulations and experimental results reported in the following sections.
6 6.1
Simulation Examples Simulation setup
To validate the proposed control algorithm (18) and the speed and flux adaptive observer, simulations have been carried out by means of Matlab and Simulink, adopting the same parameters of the experimental setup shown in (2), in which a 600 W one pole pair induction motor with a rated speed of 1000 rpm is used. The main purposes were to inspect both performances and robustness properties as far as reference tracking and observation accuracy are concerned. As for the control algorithm, another task is to verify that the limit imposed to the maximum value of the stator current time derivatives does not compromise the dynamical performances during transients. The speed and flux modulus references and the load torque profile are shown in Fig. 2: both the first and the second time derivatives of speed and flux reference signals are bounded. The simulation here shown has been carried out with a value of the rotor resistance equal to 150% of the nominal one. Additional simulative tests have been performed to check the performances of the load torque estimator (40), including a zero speed test, typically representing a severe benchmark for all sensorless control algorithms.
6.2
Simulation results
The control algorithm performances are illustrated in Fig. 3: current tracking error due to the current regulator is also shown. Fig. 4 reports, in the restricted time interval [0.95, 1.05] s, both the discontinuous waveform (phase a) of the control signal, and the stator current ia of the motor, thus letting us appreciate the filtering action of the integrators, of the current loop, and of the motor itself. Moreover, the limit imposed on the time derivative of the stator current (2000 A/s) seems not to affect negatively speed regulation during load torque transients. When the load torque is known, the adaptive sliding mode speed and flux observer proves to be fast and accurate: the previous analysis demonstrates that speed and flux regulation seems not to be affected by the presence of the observer in the control scheme. Fig. 5 shows that performances in speed and
16
flux modulus observation are satisfactory, even before that the convergence of the estimate of the parameter α takes place, demonstrating good robustness of the speed and flux observer. Both current and flux observation errors are quickly steered to zero, as shown in Fig. 6. When the load torque is unknown, the load torque estimator(40) is introduced in the control scheme. In the first graph (Fig. 7), speed and flux references are the ones shown in Fig. 2. Torque estimation is accurate, and does not affect the performances of speed and flux regulation. The load torque estimator has been evaluated also during a zero speed test. The satisfactory results obtained during such a simulation test are shown in Fig. 8.
7
Experimental Setup and Results
The proposed sliding mode observer was also validated in the laboratory. A 0.75 kW induction motor (Sever Italia s.r.l. 1ZK80A2) is supplied by a NFO Sinus frequency inverter (NFO Drives AB). Unlike traditional PWM, the NFO device provides sinusoidal voltage waveforms at variable–frequency, thus allowing noise–free acquisition of the electric variables. The stator phase currents and voltages are measured by Hall-type sensors (LEM LTS 15-NP and LEM CV 3-500, respectively). An optical incremental encoder provides rotor speed measurement. Acquisition and 12-bit A/D conversion of all the electric signals is performed by a Humosoft interface card, with sampling frequency of 100 kHz. Real-Time Workshop (Simulink) is used to implement the adaptive sliding mode observer previously discussed. Matlab is running on a Pentium IV 2.66 GHz personal computer, where the data acquisition board is installed. The built-in flux regulation of the NFO inverter is maintained, while tracking of the speed reference is realized by means of an external control loop. The error between the speed set–point and the speed estimate coming from the observer is sent to a PI regulator, which generates, as a control signal, a stator frequency reference for the NFO device. The required time–varying load torque is obtained by coupling the induction machine with a dc motor connected to an adjustable resistive load. As shown in Fig. 9, the experimental test was carried out by imposing a double speed ramp from 0 to 100 rad/s and from 100 to 150 rad/s. The resistive load was not changed, thus making the load torque directly proportional to the rotor speed. Flux and rotor resistance estimates provided by the adaptive observer are presented in Fig. 10. In the adopted experimental setup, these variables are not directly involved in the control loop: nevertheless, correct estimation of
REFERENCES
17
their value is required to guarantee proper operation of the whole regulation scheme. In Fig. 11, estimation errors of speed and load torque are reported. The proposed second order sliding mode observer appears capable of guaranteeing good performances and excellent robustness also in the laboratory benchmark. After the brief transient occurring during the start–up phase, convergence of each observed variable to the corresponding real value is quickly achieved: particularly satisfactory experimental results in speed and load torque estimation are shown during the second speed ramp.
8
Conclusions
In this paper a new sensorless sliding mode control scheme for induction motors is proposed. The control strategy sets a limit to the maximum value of the time derivatives of the stator currents, assumed as discontinuous control inputs, thus preventing excessive mechanical stress of the machine. A novel speed and flux observer, based on a second order sliding mode stator current observer, allows a fast and precise evaluation of both the variables, also in presence of a slowly time-varying rotor resistance. A second order super–twisting load torque estimator is also included in the scheme. Appreciable performances and robustness are assessed by simulation and experimental results.
REFERENCES
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REFERENCES
INDUCTION MOTOR
PWM INVERTER u*a
u*b load torque
CURRENT-BASED SLIDING MODE SPEED AND FLUX CONTROL
CURRENT REGULATOR
a-b d-q u*d
u*q
PI
1-2-3
PI
a-b ia
d-q
ib
a-b
ia*
ib*
ia*
ib*
i0
SPEED, FLUX, LOAD TORQUE AND ROTOR RESISTANCE SLIDING MODE OBSERVER
a ya yb w
i0 J=atan(yb/ya) s1*
s2*
rotating reference frame angle computation
-1
W=D
sliding functions transformation
s1
s2
sliding functions computation
Y*
2
Y= ya +yb
2
flux modulus computation
w*
flux modulus and speed references
Figure 1. The proposed induction motor control scheme.
REFERENCES
Speed reference
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Flux modulus reference 1.5
100
1 (Wb)
(rad/s)
80 60 40
0.5
20 0
0
1
2
0
3
0
1
Time (s)
2
3
Time (s)
Applied load torque 6 5
(Nm)
4 3 2 1 0
0
1
2
3
Time (s)
Figure 2. Speed and rotor flux reference signals and load torque profile in simulations.
Speed tracking error
Flux tracking error
2
0.5
(Wb)
(rad/s)
1
0
0
−1
−2
0
1
2
3
−0.5
0
1
Time (s)
3
Current tracking error (phase b) 4
2
2 (A)
(A)
Current tracking error (phase a) 4
0
−2
−4
2 Time (s)
0
−2
0
1
2 Time (s)
3
−4
0
1
2
3
Time (s)
Figure 3. Speed and rotor flux modulus tracking errors; tracking error of the stator current.
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REFERENCES
Current time derivative discontinuous reference (phase a) 4000
(A)
2000
0
−2000
−4000 1.15
1.16
1.17
1.18
1.19
1.2 Time (s)
1.21
1.22
1.23
1.24
1.25
1.22
1.23
1.24
1.25
Stator current (phase a) 10
(A)
5
0
−5
−10 1.15
1.16
1.17
1.18
1.19
1.2 Time (s)
1.21
Figure 4. The switching control input, i.e. the time derivative of the stator current (phase a) and the measured current ia , showing the high harmonics filtering.
Speed estim. error
Flux modulus estim. error
2
0.5
(Wb)
(rad/s)
1
0
0
−1
−2
0
1
2
3
−0.5
0
1
Time (s)
2
3
Time (s) Rotor resistance estimate
8
(1/s)
7 6 5 4 3
0
0.5
1
1.5 Time (s)
2
2.5
3
Figure 5. Speed and flux modulus tracking performances, and rotor resistance estimation, with the new adaptive sensorless sliding mode observer.
REFERENCES
Current estim. error (phase a)
Current estim. error (phase b) 0.5
(A)
(A)
0.5
0
−0.5
23
0
1
2
0
−0.5
3
0
1
Time (s) Flux estim. error (phase a)
0.1
0.1 (Wb)
0.2
(Wb)
3
Flux estim. error (phase b)
0.2
0
−0.1
−0.2
2 Time (s)
0
−0.1
0
1
2
−0.2
3
0
1
Time (s)
2
3
Time (s)
Figure 6. Stator currents and rotor fluxes components estimate errors.
Speed tracking error
Flux tracking error
0.5
0.1 (Wb)
0.2
(rad/s)
1
0
−0.5
−1
0
−0.1
0
1
2
3
−0.2
0
1
Time (s)
2
3
Time (s)
Load Torque estimation
Load Torque estimation error
6
2
5 1 (Nm)
(Nm)
4 3
0
2 −1
1 0 0
1
2
3
−2
0
1
Time (s)
Figure 7. Load torque estimation.
2 Time (s)
3
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REFERENCES
Speed tracking error
Flux tracking error
0.5
0.1 (Wb)
0.2
(rad/s)
1
0
−0.5
−1
0
−0.1
0
1
2
3
−0.2
0
1
Time (s)
2
3
Time (s)
Load Torque estimation
Load Torque estimation error
6
2
5 1 (Nm)
(Nm)
4 3
0
2 −1
1 0 0
1
2
−2
3
0
1
Time (s)
2
3
Time (s)
Figure 8. Load torque estimation during a zero speed test.
Speed reference and real speed 200
(rad/s)
150 100 50 0 −50
0
5
10
15
20
25
15
20
25
Time(s)
Load Torque
(Nm)
1
0.5
0
−0.5
0
5
10 Time(s)
Figure 9. Speed reference, real speed and load torque during experimental test.
REFERENCES
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Flux modulus estimate 1.4 1.2
(Wb)
1 0.8 0.6 0.4 0.2 0
0
5
10
15
20
25
20
25
Time(s)
Rotor resistance estimate 25
(Ohm)
20 15 10 5 0
0
5
10
15 Time(s)
Figure 10. Flux modulus and rotor resistance estimation during experimental test.
Speed estimation error 10 5
(rad/s)
0 −5 −10 −15 −20 −25
0
5
10
15
20
25
20
25
Time (s) Load Torque estimation error 1
(Nm)
0.5
0
−0.5
−1
0
5
10
15 Time(s)
Figure 11. Speed and load torque estimation errors during experimental test.
