Proceedings of the 18th World Congress The International Federation of Automatic Control Milano (Italy) August 28 - September 2, 2011
Adaptive Interconnected Observer-Based Backstepping Control Design For Sensorless PMSM Marwa Ezzat1 , Jesus de Leon2 , Alain Glumineau1 1
LUNAM Université, Ecole Centrale de Nantes, IRCCyN UMR CNRS 6597, Nantes, France, 2 FIME, Universidad Autonoma de Nuevo Leon, Mexico. Avenida Universidad s/n, San Nicolas de los Garza, Nuevo Leon, Mexico, C.P. 66451.
[email protected],
[email protected],
[email protected].
Abstract: In this paper, a robust sensorless speed observer-controller scheme for a surface permanent magnet synchronous motor (SPMSM) is proposed. First-of-all, an adaptive high gain interconnected observer is designed. This observer estimates the rotor speed, the stator resistance and the load torque as well. A non linear backstepping controller is developed. The above observer is associated to this controller and the proof of the complete scheme is given. The overall system is tested by simulation in the framework of an industrial benchmark. Keywords: Permanent magnet synchronous motor, sensorless control, adaptive observer. dθ dt dΩ dt did dt diq dt
1. INTRODUCTION The sensorless techniques become essential and attract the attention of the industrial applications and the researches as well. Therefore, many approaches for speed/position estimation have been investigated in the literature. Different methods as electromotrive force (Chen et al. (2003)), adaptive observers (Cascella et al. (2003)), and extended Kalman filter (Boussak (2005)) have been used to estimate both the speed and the position of the PMSM. When analyzing all these results, it is clear that previous results have rarely evaluated robustness of the closed-loop system with respect to parameters variations while this is a key-point to get robust performances for sensorless control. Moreover these studies never investigate the robustness of the observer-controller scheme when the observability is lost. Nearly all the published observers have been tested with classical vector control using a proportional-integral controller. From this point-of-view, adapted nonlinear robust controls as high order sliding mode and backstepping (Ouassaid et al. (2004); Plestan et al. (2007)) could be more efficient. But all of them use the speed measurement. In (Ke and Lin (2005)) an angular velocity observer is proposed for estimating the speed and measures the position.
= Ω p Tl fv ψ f iq − Ω − J J J Rs 1 = − id + pΩiq + vd Ls Ls ψf Rs 1 = −p Ω − pΩid − iq + vq . Ls Ls Ls
=
(1)
Note that, id,q and vd,q are the measured states. On the other hand, Ω, θ and Tl are the unmeasured states that are estimated in the sequel; where p Pole pairs Load torque Tl ψf Magnet flux Ω Rotor mechanical speed θ Rotor angular position id,q Stator currents in (d − q) reference frame vd,q Stator voltages in (d − q) reference frame Rs Stator-winding resistance Ls Stator-winding inductance fv Viscous damping coefficient J Rotor moment of inertia. 3. ADAPTIVE INTERCONNECTED OBSERVERS DESIGN
2. MATHEMATICAL MODEL The SPMSM model in the synchronous (d − q)-reference frame reads as 978-3-902661-93-7/11/$20.00 © 2011 IFAC
In the sequel an adaptive interconnected observer will be designed for the sensorless PMSM. It is assumed that load torque and stator resistance are slowly varying with respect to electric and mechanic variables. Then the dynamic behavior of these two variables can be read as
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
T˙l = 0
R˙ s = 0.
(2)
Remark 1. Equation (2) means that the load torque and stator resistance values are assumed to be approximate by piecewise constant function. Only the bound of the load torque is assumed to be known. Furthermore, it is clear that the stator resistance slowly changes with the temperature. However, using step constant functions this variation can be approximated and the proposed approach works. Other approaches can be used, for instance singular perturbation methodology, however the dynamics of the PMSM is fast with respect to the variations of the stator resistance that it could be considered constant. Thus, the extended SPMSM model (1)-(2) may be seen as the interconnection between two subsystems X˙ 1 = A1 (y)X1 + F1 (X2 ) + Φ1 (u) (3) Σ1 : y 1 = C 1 X1 Σ2 :
X˙ 2 = A2 (y)X2 + F2 (X1 , X2 ) + Φ2 (u) + ΦTl (4) y 2 = C 2 X2
with 0 piq fv , A1 (·) = A2 (·) = 0− 0 J ⎤ ⎡ R s ψf i − d −p Ω − pΩid F1 (·) = , F2 (·) = ⎣ p Ls ⎦ , Ls ψ f iq 0 J
0 −
iq Ls , 0
1 1 0 vq vd 1 , Φ1 = Ls , Φ2 = Ls , Φ = − 0 0 J C1 = C2 = [ 1 0 ] . T
and X1 do not satisfy the regularly persistence condition. Then, asymptotic stability of the observer is not guaranteed. This problem is solved, by using the practical stability introduced in Section 4. Remark 3. >From (3) and (4), it is clear that A1 (y) is globally Lipschitz w.r.t. X2 , A2 (y) is globally Lipschitz w.r.t. X1 . F1 (X2 ) is globally Lipschitz w.r.t. X2 and uniformly w.r.t. (u, y) and that F2 (X2 , X1 ) is globally Lipschitz w.r.t. X2 , X1 and uniformly w.r.t. (u, y). Then, adaptive interconnected observers for subsystems (3) and (4) are given by ⎧ ⎨ Z˙ 1 = A1 (y)Z1 + F1 (Z2 ) + Φ1 (u) + S1−1 C1T (y1 − yˆ1 ) (5) O1 : S˙1 = −ρ1 S1 − AT1 (y)S1 − S1 A1 (y) + C1T C1 ⎩ yˆ1 = C1 Z1 ⎧ ˙ Z2 = A2 (y)Z2 + F2 (Z1 , Z2 ) + Φ2 (u) + ΦTˆl ⎪ ⎪ ⎪ ⎪ +(ΛSθ−1 ΛT C2T + ΓSx−1 C2T )(y2 − yˆ2 ) ⎪ ⎪ ⎪ ⎪ +KC1T (y1 − yˆ1 ) ⎪ ⎪ ⎨ ˙ −1 T T ˆ (6) O2 : T l = Sθ Λ C2T(y2 − yˆ2 ) + B(y1 − yˆ1 )T ˙ x = −ρx Sx − A2 (y)Sx − Sx A2 (y) + C2 C2 ⎪ S ⎪ ⎪ ⎪ ⎪ S˙ θ = −ρθ Sθ + ΛT C2T C2 Λ ⎪ ⎪ ⎪ −1 T ⎪ ˙ ⎪ ⎩ Λ = (A2 (y) − ΓSx C2 C2 )Λ + Φ yˆ2 = C2 Z2 T
T
ˆ with Z1 = ˆiq Rˆs and Z2 = ˆid Ω are the estimated state variables respectively of X1 and X2 . ρ1 , ρx , ρθ are positive constants, S1 and Sx are symmetric positive definite matrices Sθ (0) > 0, B(Z1 ) = k Jp ψf ˆiq , K=
−kc1 −kc2
,
Γ=
1 0 0α
with k, kc1 , kc2 , α and are positive constants.
T
X1 = [iq Rs ] and X2 = [id Ω ] are respectively the state T vectors of systems (3) and (4), u = [ud uq ] is the input, T and y = [id iq ] is the output. Furthermore, the SPMSM physical operation domain D is defined by the set of values D = {X ∈ R5 | |id | ≤ Id max , |iq | ≤ Iq max , |Ω| ≤ Ωmax , |Tl | ≤ Tl max , |Rs | ≤ Rs max } with X = [id iq Ω Tl Rs ]T and Id max , Iq max , Ωmax , Tl max , Rs max the actual maximum values for currents, speed, load torque and stator resistance, respectively. The adaptive interconnected observer, developed in the sequel for the sensorless SPMSM, is based on the interconnection between several observers satisfying some required properties, in particular the property of input persistence. The input persistence is related to the observability properties of system (3)-(4). In order to design an observer for system (3)-(4), a separate synthesis of the observer for each subsystem is required. Remark 2. When the PMSM remains in the unobservable area, X2
The second observer (6) is composed of two parts: the first part to estimate the state (id Ω) and the second part to estimate the load torque (Tl ), by using the stator currents id and iq . Furthermore, S1−1 C1T is the gain of observer (6) and (ΛSθ−1 ΛT C2T + ΓSx−1 C2T ) and KC1T are the gains of observer (6). We can see that the gain of the observer (6) is spited in two terms. The first one, (ΓSx−1 C2T ), is associated to the state estimation and depends on the solution of a Ricatti equation. The second one (ΛSθ−1 ΛT C2T ) is related to the identification parameter and depends on the solution of a differential equation. The solutions of these equations are dependent of the regularly persistence (richness of the signal) with respect to state and the parameter. Remark 4. In equation (6) the term (B(Z1 )(y1 − yˆ1 )) can be expressed as follows p B(Z1 )(y1 − yˆ1 ) ≡ k[ ψf (iq − ˆiq )] ≡ k(Te − T˜e ) J where Te and T˜e are respectively the "measured" and "estimated" electromagnetic torques.
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Lemma 1. Assume that v is a regularly persistent input for state affine system (3)-(4), and consider the following Lyapunov differential equation
Σ1
˙ S(t) = −θS(t) − AT (v(t))S(t) − S(t)A(v(t)) + C T C with S(0) > 0, then
Σ2
∃θ0 > 0, ∀θ ≥ θ0 , ∃¯ α > 0, β¯ > 0 , t0 > 0 : ¯ ¯ I ≤ S(t) ≤ βI, ∀t ≥ t0 , α where I is the identity matrix. It is clear that v = (u, X2 ) and S(t) = S1 for subsystem (3), and for subsystem (4) one has v = (u, X1 ) and S(t) = S2 . It is worth mentioning that the conditions of observability loss have been stated in (Ezzat et al. (2010)), where the PMSM is unobservable for some input value (the rotor speed equal to zero). In the PMSM observability area, the inputs v = (u, X2 ) and v = (u, X1 ), for subsystem (3) and for subsystem (4) respectively, are regularly persistent and the convergence of the observer can be assured. However, in the unobservable region PMSM (under the conditions of zero speed), such inputs are "bad input" and the observer convergence is not guaranteed. The use of practical stability properties can solve this problem. 4. STABILITY ANALYSIS OF OBSERVER UNDER UNCERTAIN PARAMETERS Under indistinguishable trajectories (unobservable area) the asymptotic convergence of any observer can not always be guaranteed because the observability properties are lost on these trajectories. Then, in such cases, it is necessary to analyze the stability of the observer and the closed loop system. The practical stability property, if satisfied, (Laskhmikanthan et al. (1990)) allows to establish that dynamics of the estimation error converge in a ball Br of radius r (x ∈ Br ⇒ x ≤ r). If r → 0 at t → ∞, then the classical asymptotic stability is obtained. Theorem 1. (Laskhmikanthan et al. (1990)). Assume that i) ¯h1 , ¯ h2 are given such that 0 < ¯ h1 < ¯ h2 ; ii) V ∈ C[IR+ × IRn , IR+ ] and V (t, e) is locally Lipschitz in e; iii) for (t, e) ∈ IR+ × Bh¯ 2 , d1 ( e ) ≤ V (t, e) ≤ d2 ( e ) and V˙ (t, e) ≤ ℘(t, V (t, e))
(7)
C[IR2+ , IR];
where d1 , d2 ∈ W and ℘ ∈ iv) d2 (¯ h1 ) < d1 (¯h2 ) holds. Then, the practical stability properties of:
Consider that the SPMSM parameters are uncertain bounded with well-known nominal values. Then, equations (3-4) can be rewritten as
(9)
⎧ ⎨ X˙ 2 = A2 (y)X2 + F2 (X1 , X2 ) + Φ2 (u) + ΦTl : (10) +ΔA (y) + ΔF2 (X1 , X2 ) ⎩y = C X 2 2 2 2
with ΔA1 (y), ΔA2 (y), ΔF1 (X2 ) and ΔF2 (X1 , X2 ) are the uncertain terms of A1 (y), A2 (y), F1 (X2 ), F2 (X1 , X2 ), respectively. It follows that the uncertain terms are represented as iq Δψf 0 − −p Ω − pΩid ΔA1 (·) = ΔLs , ΔF1 (·) = ΔLs 0 0 0 ΔA2 (y), ΔF2 (X2 , X1 ) can be written following a similar way ⎡
⎤ ΔRs − id ⎦. ΔA2 (y) = , ΔF2 (·) = ⎣ p ΔLs Δψf iq ΔJ Considering the SMPSM physical operation domain D, then there exist positive constants ρi > 0, for i = 1, ..., 4; such that ΔA1 (y) ≤ ρ1 , ΔA2 (y) ≤ ρ2 , ΔF1 (X2 ) ≤ ρ3 , ΔF2 (X1 , X2 ) ≤ ρ4 . The parameters ρi , i = 1, ..., 4 are positive constants determined from the maximal values of ΔA1 (y), ΔA2 (y), ΔF1 (X2 ) and ΔF2 (·) in the physical domain D.
0 piq fv 0− J
Let define the estimation errors as 1 = X1 − Z 1 ,
2 = X2 − Z2 ,
3 = Tl − Tˆl . (11)
>From equations (5)-(6) and (9)-(10), one gets ˙1 = [A1 (y) − S1−1 C1T C1 ] 1 + ΔA1 (y)X2 +F1 (X2 ) + ΔF1 (X2 ) − F1 (Z2 )
(12)
˙2 = [A2 (y) − ΛSθ−1 ΛT C2T C2 − ΓSx−1 C2T C2 ] 2 (13) +Φ 3 − KC1T C1 1 + ΔA2 (y)X2 +F2 (X2 , X1 ) + ΔF2 (X2 , X1 ) − F2 (Z2 , Z1 ) ˙3 = −Sθ−1 ΛT C2T C2 2 − B(Z1 )C1 1 .
(14)
2
− Λ 3 , it yields Applying the transformation 2 = ˙ 3 . Then the estimation error dynamics ˙2 = ˙2 − Λ ˙3 − Λ are given by ˙1 = [A1 (y) − S1−1 C1T C1 ] 1 + ΔA1 (y)X2 +F1 (X2 ) + ΔF1 (X2 ) − F1 (Z2 ) ˙2 = [A2 (y) − ΓSx−1 C2T C2 ] 2 + (B − K ) 1 (15) +ΔA2 (y)X2 + F2 (X2 , X1 ) +ΔF2 (X2 , X1 ) − F2 (Z2 , Z1 ) ˙3 = −Sθ−1 ΛT C2T C2 Λ 2 − Sθ−1 ΛT C2T C2 3 − B 1
(8) l˙ = ℘(t, l), l(t0 ) = l0 ≥ 0, imply the corresponding practical stability properties of e˙ = f (t, e), e(t0 ) = e0 , t0 ≥ 0. 4.1 Stability analysis
⎧ ⎨ X˙ 1 = A1 (y)X1 + F1 (X2 ) + Φ1 (u) : +ΔA (y) + ΔF1 (X2 ) ⎩y = C X 1 1 1 1
with B = B(Z1 )C1 , K = KC1T C1 . Since (u, X2 ) and (u, X1 ) are regular persistent inputs for subsystems (9)(10), respectively; and from Lemma 1, then there exist t0 ≥ 0 and real numbers ηSmax > 0, ηSmin > 0 which are i i independent of θi such that V (t, i ) = Ti Si i (1 ≤ i ≤ 3)
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
2
2
∀t ≥ t0 ηSmin i ≤ V (t, i ) ≤ ηSmax i . i i
(16)
Theorem 2. Consider the extended SPMSM dynamic model represented by (3)-(4). System (5)-(6) is an adaptive observer for system (3)-(4). Furthermore, the strongly uniformly practically stability of estimation error dynamics (15) is established. Sketch of Theorem 2 Proof. A Lyapunov function candidate is considered as Vo = V1 + V2 +V3 , where V1 = T1 S1 1 , V2 = T2 Sx 2 and V3 = T3 Sθ 3 . Taking the time derivative of Vo and using (5), (6) and (15), we have μ ˜3 V˙ o ≤ −(ρ11 − 2k1 k6 − μ ˜3 ϕ1 )V1 − (ρx − μ ˜ 4 ϕ3 − )V2 ϕ1 (17) μ ˜4 −(ρθ − )V3 + μ ˜11 1 + μ ˜22 2 , ϕ3 with μ ˜11 = 2˜ μ1 , μ ˜21 = 2˜ μ2 , and consequently, we have V˙ o ≤ −δ(V1 + V2 + V3 ) + μ( V1 + V2 ) (18) ≤ −δVo + μψ Vo , where δ = min(δ1 , δ2 , δ3 ), μ = max(˜ μ11 , μ ˜22 ), where δ1 = ρ11 − 2k1 k6 − μ ˜3 ϕ1 > 0, δ2 = ρx − μ ˜ 4 ϕ3 − μ ˜3 μ ˜4 > 0, δ = ρ − > 0, and ψ > 0, such that 3 θ ϕ1√ ϕ3√ √ ψ V1 + V2 + V3 > V1 + V2 . So that ˜ 3 ϕ1 , ρ11 > 2k1 k6 + μ
ρx > μ ˜ 4 ϕ3 +
μ ˜3 , ϕ1
ρθ >
μ ˜4 . (19) ϕ3
5. CONTROLLER Given the measurement of the currents and the voltages, to design a control law in order to the speed of the motor tracks a desired reference i.e. (Ω → Ωref ). Step 1 Let be z1 = Ω − Ωref = ξ1 − α0 the tracking error, where Ωref is the speed reference. Then, taking the time derivative, we have z˙1 = Ω˙ − Ω˙ ref = ξ˙1 − α˙ 0 pψ z˙1 = J f iq − fJv Ω − TJl − Ω˙ ref z˙1 = ξ2 + β1 pψ where ξ2 = J f iq and β1 = − fJv Ω − TJl − α˙ 0 . Defining the following candidate Lyapunov function, V1 = 1 2 2 z1 . Taking α1 = −w1 z1 − β1 , and z2 = ξ2 − α1 . Then, taking the time derivative, it follows z˙1 = ξ2 + β1 = z2 − w 1 z1 0 Then, we obtain V˙ 1 = z2 z1 − w1 z12 . Step 2 From z2 = ξ2 − α1 , it follows that pψ z˙2 = J f i˙ q − α˙ 1 pψ pψ pψ s z˙2 = J f {− Lsf Ω − pΩid − R ˙ 1 + JLfs vq Ls iq } − α z˙2 = β2 + Kvq pψ pψ pψ s where β2 = J f {− Lsf Ω−pΩid − R ˙ 1 and K = JLfs . Ls iq }− α Similarly, defining the following candidate Lyapunov function, V2 = V1 + 12 z22 , whose time derivative is given by
V˙ 2 = V˙ 1 + z˙2 z2 = z2 z1 − w1 z12 + z˙2 z2 V˙ 2 = −w1 z12 + z2 (z1 + β2 + Kvq ) = −w1 z12 − w2 z22 < 0. Then, to force the stability, the control law can be com1 puted by: vq = K (−w2 z2 − z1 + β2 ). Step 3 As mentioned above, to eliminate the reluctance torque, the current reference is fixed to zero: idref = 0, then z3 = id − idref = id . Thus its time derivative is, z˙3 = 1 s −R Ls id + pΩiq + Ls vd . Defining the following Lyapunov function, V3 = V2 + 12 z32 , it follows that 1 s V˙ 3 = −w1 z12 − w2 z22 + z3 {− R Ls id + pΩiq + Ls vd }, V˙ 3 = −w1 z12 − w2 z22 − w3 z32 < 0. To obtain, the control law for the current id is given by vd = −Ls w3 id + Rs id − pLs Ωiq . 6. STABILITY ANALYSIS OF THE OBSERVER-CONTROLLER SCHEME In order to prove the stability of the whole system (observer+controller+machine), the speed, the current id and the load torque are replaced by their estimated values in the control law. Considering the following candidate Lyapunov function: Voc = Vo + Vc = T1 S1 1 + T2 Sx 2 + T3 Sθ 3 (20) 1 1 1 + z12 + z22 + z32 2 2 2 Vo = T1 S1 1 + T2 Sx 2 + T3 Sθ 3 and Vc = 12 z12 + 12 z22 + 1 2 2 z3 are the associated Lyapunov function of the adaptive interconnected observer and of the backstepping controller ˙ respectively. √ From inequality(18), one knows that Vo ≤ −δVo + μψ Vo . the time derivative of Voc (20) reads V˙ oc ≤ −δVo + μψ Vo −w1 z12 − w2 z22 − w3 z32 (21) pψf fv z1 z1 C 1 1 + z1 B 1 2 + 3 . − J J J with B1 = [0 1] Considering the following inequalities z1 1 Sρ
11
z1 2 Sρ
x
z1 3 Sρ
θ
ξ1 1 2 2 1 Sρ + z1 11 2 2ξ1 ξ2 1 2 2 2 Sρ + ≤ z1 x 2 2ξ2 ξ3 1 2 2 2 Sρ + ≤ z1 . θ 2 2ξ3 ≤
(22)
Substituting equation (22) into (21) and by regrouping with respect to ( 1 , 2 , 3 z1 , z2 ) and z3 ) V˙ oc ≤ −δVo + μψ Vo 2 2 2 −ϑ1 1 Sρ − ϑ2 2 Sρ − ϑ3 3 Sρ (23) 2
11
2
x
2
−ϑ4 ( z1 ) − ϑ5 ( z2 ) − ϑ6 ( z3 )
θ
v ξ2 3 where ϑ1 = ( J f ) ξ21 , ϑ2 = ( −f ϑ3 = −ξ J ) 2 , 2J , pψf −fv 1 ϑ4 = w1 + 2Jξ1 − 2Jξ2 − 2Jξ3 , ϑ5 = (w2 ), ϑ 6 = w3 . Taking ϑ = min(ϑ1 , ϑ2 , ϑ3 ), and ϑ = min(ϑ4 , ϑ5 , ϑ6 ), then the inequality (23) reads
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
V˙ oc ≤ −(δ + ϑ)Vo + μψ Vo 2 2 2 −ϑ ( zΩ + zφ + zφ )
(24)
a
where
b
300
V˙ oc ≤ −ηVoc + μψ Voc ; (25) ). Consider the following change of with η = min(δ + ϑ, ϑ √ variable voc = 2 Voc . The time derivative of voc satisfies the following inequality v˙ oc ≤ −ηvoc + ψμ
9
8 250
(26)
Load Torque (Nm)
Speed (rad/sec)
7 200
150
100
6
5
4
3
2
and the solution of (26) is
50 1
ψμ (1 − e−η(t−t0 ) ). (27) η By using the same procedure such as the strongly uniformly practical stability proof of the observer, the strongly uniformly practical stability properties of (26) can be proved, that implies the corresponding strongly uniformly practical stability of the system. Hence, the estimation and the tracking errors of the closed-loop system converge towards a ball Bh¯ oc of radius ¯ hoc with ¯ hoc = ψμ η . voc = voc (t0 )e−η(t−t0 ) +
0
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Fig. 1. Reference trajectories of sensorless industrial benchmark. a. Speed Reference b. Load Torque
b
a 350
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300 0.02
7. SIMULATION RESULTS Speed Error (rad/s)
been simulated. The parameters of the SPMSM are given in (WebSetUp (2010)) where the industrial benchmark is described. See Figure (1−a) for the speed reference and Figure (1−b) for the load torque. The results shown in Figure (2-a) represent the measured speed with its reference for the nominal case and fully loaded motor. The tracking is very efficient as displayed by Figure (2-b) which shows the speed error (reference speed- measured speed). Only the estimated speed is supplied to the controller. It is clear that the observer has a good performance. Figure (4) and Figure (5) demonstrate the resistance and load torque estimation respectively for the nominal case. For the robustness test, a resistance variation of +50% and full load torque application is carried out. This case is represented by Figures (6), (7) and (8) for the speed, resistance and load torque estimation respectively. The results reveal the efficiency of the proposed observercontroller scheme.
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Fig. 2. Control. a. Reference Speed & measured speed Speed error
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REFERENCES Speed error (rad/s)
Boussak, M. (2005). Implementation and experimental investigation of sensorless speed control with initial rotor position estimation for interior permanent magnet synchronous motor drive. IEEE Trans. Power Electronics, 20(6), 1413–1421. Cascella, G.L., Salavatore, N., and Salvatore, L. (2003). Adaptive sliding-mode observer for field oriented sensorless control of spmsm. IEEE Int. Symp. Indus. Electronics, Rio de Janeiro, Brasil, 1137–1143. Chen, Z., Tomita, M., Doki, S., and Okuma, S. (2003). An extended electromotive force model for sensorless control of interior permanent-magnet synchronous motor. IEEE Trans. Indus. Electronics, 50(2). Ezzat, M., de Leon, J., Gonzalez, N., and Glumineau, A. (2010). Sensorless speed control of permanent magnet synchronous motor by using sliding mode observer. 11 th
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Fig. 3. Nominal Case. a. Estimated speed (rad/sec) versus time (sec). b. Speed estimation error (rad/sec) versus time (sec).
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a
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Fig. 4. Resistance estimation. a. Estimated resistance (Ohm) versus time (sec). b. Resistance estimation error (Ohm) versus time (sec).
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12
0
−0.02
−0.04
Tlmes Tlobs 0
0.02
14
−0.06
0
2
4
6
8
10
12
14
Time (S)
14
b
−3
1.5
6
Time (S)
Time (S)
350
0
0.04
−2
Fig. 5. Load torque estimation. a. Estimated load torque (N m) versus time (sec). b. Load torque estimation error (N m) versus time (sec).
−50
8
0
Time (S)
0
0.06
2
−0.04
Tlmes Tlobs 4
14
b
10
Torque error (Nm)
0.04 Torque (Nm)
8
2
12
b 0.06
Torque error (Nm)
Torque (Nm)
a 10
0
10
Fig. 7. +50%Rs . a. Estimated resistance (Ohm) versus time (sec). b. Resistance estimation error (Ohm) versus time (sec). a
−2
8
Time (S)
Time (S)
14
Time (S)
Fig. 6. At +50%Rs . a. Estimated speed (rad/sec) versus time (sec). b. Speed estimation error (rad/sec) versus time (sec).
Fig. 8. Load torque estimation at +50%Rs . a. Estimated load torque (Nm) versus time (sec). b. Load torque estimation error (Nm) versus time (sec). International Workshop on Variable Structure Systems, Mexico city, Mexico, 26-28 June. Ke, S.S. and Lin, J.S. (2005). Sensorless speed tracking control with backstepping design scheme for permanent magnet synchronous motors. Proceedings of IEEE Conference on Control Applications, Toronto, Canada, August 28-31. Laskhmikanthan, V., Leela, S., and Martynyuk., A.A. (1990). Practical stability of nonlinear systems. Word Scientific. Ouassaid, M., Cherkaoui, M., and Zidani, Y. (2004). A nonlinear speed control for a pm synchronous motor using an adaptive backstepping control approach. IEEE International Conference on Industrial Technology (ICIT), Hammamet, Tunisia. Plestan, F., Glumineau, A., and Bazani, G.J. (2007). New robust position control of a synchronous motor by high order sliding mode. IEEE Conference on Decision and Control, New Orleans, Louisiana, USA. WebSetUp (2010). http://www2.irccyn.ecnantes.fr/bancessai/.
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