9th IFAC Smposium on Nonlinear Control Sstems Toulouse, France, September 4-6, 3 ThA Robustness Analsis of a Position Observer for Surface Mount Permanent Magnet Snchronous Motors vis à vis Rotor Salienc Harish K Pillai a,, Romeo Ortega b, Michael Hernandez c, Thomas Devos c, Francois Malrait c a Dept of Elec Engg, IIT Bomba, Mumbai 476, India b Laboratoire des Signau et Sstèmes, CNRS SUPELEC, 99 Gif sur Yvette, France c Schneider Electric, STIE, 33, rue Andre Blanchet, 7 Pac-sur-Eure, France Abstract A gradient descent based nonlinear observer for surface mount permanent magnet snchronous motors (PMSMs) with remarkable stabilit properties was recentl proposed in [7] A ke assumption for the derivation of the observer is the absence of rotor salienc, which is the case in surface mount PMSMs A question of great practical interest is to assess the performance of the observer in the presence of salienc This is the topic of stud of the present paper It is shown that the robustness of the observer is full determined b the sinusoidal stead state values of the currents, providing some guidelines for the selection of their reference values to ensure good position estimation Introduction Position information is required for field orientation control of PMSMs, which is the industr standard for high performance applications In some cases, installing position sensors is troublesome, for instance, in some vacuum pumps, cranes and elevators Also, in some other applications, like household equipments such as refrigerators and air conditioners, cost constraints stmies the use of speed sensors The above problems motivate to develop sensorless algorithms for PMSMs, for which numerous works have been published, see [, 6] for a recent review of the literature A ver simple observer of the rotor position of PMSMs that ehibits some remarkable stabilit properties was recentl presented in [7] The analsis, done for the full nonlinear model, proves global (under some conditions, even eponential) convergence of the position estimate and ields ver simple robust tuning rules In [5] the position observer Corresponding author Email addresses: hp@eeiitbacin (Harish K Pillai), ortega@lsssupelecfr (Romeo Ortega), michaelhernandez@frschneider-electriccom (Michael Hernandez), thomasdevos@frschneider-electriccom (Thomas Devos), francoismalrait@frschneider-electriccom (Francois Malrait) was combined with an ad hoc linear speed estimator and a standard field oriented controller, which are often used in applications, ielding ver encouraging eperimental results In [8] the position observer is combined with a speed and load torque observer and a passivit based controller to obtain an asmptoticall stable controller that regulates the mechanical speed of the motor, measuring onl the electrical coordinates See also [, 9] where this observer is also used The position observer of [7] is designed for non salient PMSMs, for which it is possible to derive a gradient descent method whose robustness properties are well known In the phase of salienc, that is present in all practical machines, the gradient cannot be computed and the question of performance of the observer naturall arises In this paper, we carr out a nonlinear analsis of the effect of salienc on the error equations when the machine is operating in sinusoidal stead state The remaining of the paper is organized as follows Section presents the model of the salient PMSM The observer of [7] is briefl recalled in Section 3 and the error equations due to the presence of salienc are derived in Section 4 Some observations about the equilibria of these equations are listed in Section 5 alongwith some representative simulation diagrams Preprint submitted to Elsevier Ma 3, 3 Copright 3 IFAC 353
Mathematical Model of Salient PMSM The classical fied frame αβ model of the PMSM in sinusoidal regime establishes the relation between the voltages and currents as [3, 4] v αβ = Ri αβ + d dt λ αβ () where λ αβ = (λ α, λ β ) R(i d ), i αβ = (i α, i β ) R(i d ) and v αβ = (v α, v β ) R(i d ) are the flu linkages, the currents and the voltages in the α and β phases, respectivel, and R is the resistance of the stator coils In the unsaturated case the flu linkages are defined b λ α = L αα i α + L αβ i β + cos nθ () λ β = L βα i α + L ββ i β + sin nθ where L αα and L ββ are self inductances of α and β phases, respectivel, and L αβ and L βα are the mutual inductances between the two phases is the magnetic flu due to the permanent magnet in the rotor, nθ is the electrical angle between the α- phase and the rotor ais where θ is the mechanical angle between the α-phase and the rotor ais and n is the number of pole-pairs in the machine For the salient machine, the self and mutual inductances of the phases var with the electrical angle between the phases and the rotor ais A model that captures this variation is given b the following equations L αα = L s + L g cos nθ L ββ = L s L g cos nθ L αβ = L βα = L g sin nθ, Substituting these equations into () gives the equation λ αβ = [L s + L g Q(θ)]i αβ + c(θ), (3) where, to simplif the notation, we have defined [ ] [ ] cos nθ sin nθ cos nθ Q(θ) :=, c(θ) := sin nθ cos nθ sin nθ (4) The final model of the salient PMSM is then given b (), (3) and (4) A non salient machine is modeled with the same equations, under the assumption that L g = As is standard practice, and without loss of generalit, we assume L αβ = L βα The observer problem is to generate a convergent estimate of θ from the measurements of i αβ and v αβ, and the knowledge of the (positive constant) phsical parameters R, L s, L g and 3 The Position Estimator of [7] From (3) it is obvious that if the flu is known, and L g =, it is possible to reconstruct θ via the trigonometric identit ( ) θ = tan λβ L s i β λ α L s i α On the other hand, from () it is clear that one can estimate the flu λ αβ (up to a constant due to the unknown initial conditions) b integrating a quantit that depends on the measured quantities The ke observation of [7] is that a gradient can be computed to add to this integral a correction term that will drive the estimation in the right direction More precisel, consider the following equations ẋ = v αβ Ri αβ = L s i αβ, with the standard Euclidean norm Clearl, estimates the flu λ αβ up to a constant Now, note that in the case of no salienc, equation (3) looses the middle term involving L g and therefore = λ m when = λ αβ Therefore, it is reasonable to propose a gradient search that tries to minimize the criterion (λ m L s i αβ ) which leads to the flu observer proposed in [7] ẋ = v αβ Ri αβ + γ( L s i αβ )(λ m L s i αβ ) (5) where γ > is a scaling factor For the salient machine L g, bringing into (3) an unknown θ dependent term It is not clear how to formulate a criterion with a computable gradient in this case In this paper, we analze the performance of the flu observer designed for the non-salient machine [7], when applied to a machine with salienc 4 Error Equations In this section, the dnamics of the observation errors are derived and presented in a suitable rotating frame for the analsis of their equilibria To Copright 3 IFAC 354
streamline the presentation we introduce the rotation matri [ ] cos nθ sin nθ ep( nθj) =, sin nθ cos nθ where J := [ ] Notice that this matri defines the well known transformation from αβ to dq coordinates [4] In particular, the i d and i q currents are given b i dq = ep( nθj)i αβ As will become clear below the values of these currents will pla a fundamental role on the performance of the observer Proposition 4 Consider the salient PMSM model (), (3) and (4) together with the flu observer (5) Define the error signals e = λ αβ and it s scaled and rotated form ξ = ep( nθj)e (6) Introduce, moreover, the time scale change Then, dξ dτ dξ dτ where dt dτ = γλ m = Ωξ σ(ξ)(ξ i d ) = Ωξ σ(ξ)(ξ + i q) (7) σ(ξ) := (ξ i d ) + (ξ + i q), (8) with the scaled currents i dq [ ] i i dq := d i q and the scaled speed Proof From (3) Ω = n γλ m := L g i dq L s i αβ = e + λ αβ L s i αβ = e + L g Qi αβ + c In the sequel the arguments of the functions are omitted θ 3 Thus, and L s i αβ = e + L g Qi αβ + c = e + L g i αβ + λ m +L g e Qi αβ + e c + L g c Qi αβ λ m L s i αβ = e L g i αβ Thus, the error dnamics is ė = ẋ λ L g e Qi αβ e c L g c Qi αβ = γ[ e L g i αβ L g e Qi αβ e c L g c Qi αβ ] (e + L g Qi αβ + c) (9) Now, differentiating (6) we get ξ = n dθ dt Jξ ep( nθj)ė On simplifing, we get ξ = n θjξ γλ mσ(ξ) [ ξ i d ξ + i q ] The proof is completed introducing the time scaling Remark The equations (7) full describe the behavior of the position observer (5) when applied to a salient machine Note that the functions i d and i q pla a central role in the error equations (7) in the stead state operation of the machine the are constants, proportional to the currents i d and i q Note that the scaling factor L g is, in some sense, a measure of the salienc of the machine as it depends on L g scaled b the magnetic flu of the permanent magnet of the machine 5 Observations about the Error Equations For the sake of brevit, we now list several observations about eistence and location of equilibria for the error equations (7), when i dq is constant that is, for the machine operating in stead statewe have also added relevant figures from simulations that support these observations Formal proofs of these observations would be reported elsewhere Copright 3 IFAC 355
For a nonzero Ω, ever equilibrium point of the error equations lie on a critical circle defined b C(ξ) := ( ξ i d + ) ( ) + ξ + i q (i d + ) + (i q) 4 For fied values of Ω, i d and i q, there is at least one equilibrium point and at most three equilibrium points When more than one equilibrium point eists, then Ω 3 and ( + i d ) + (i q) 8 9 The origin is an equilibrium point if and onl if σ(ξ) = or σ(ξ) = Note that the case of a non-salient machine falls in this categor as L g = and therefore σ(ξ) = The disk {ξ R ξ + (i d + ) + (i q) } = K ( a ) (( a ) + ( + b) ) K = a = 4 = K ( + b) (( a ) + ( + b) ) b = 8 5 5 5 5 5 5 5 5 = K ( a ) (( a ) + ( + b) ) K = a = 4 = K ( + b) (( a ) + ( + b) ) b = 8 5 5 5 5 5 5 5 5 is globall asmptoticall stable (GAS) If (i d + ) + (i q) =, then a) If < Ω < /, then there are three equilibria The origin is a stable node, whereas the other two equilibrium points are a saddle and an unstable equilibrium point b) If Ω = /, then there are two equilibria c) If Ω > /, then the origin is the onl equilibrium It is a stable node for Ω and a stable focus for Ω > If (i d + ) + (i q) >, then there eists an equilibrium point ξ, such that σ = σ(ξ ) > This equilibrium point is stable If (σ + ) Ω, then ξ is a stable node, otherwise it is a stable focus If (i d + ) + (i q) /, then there eists onl one equilibrium point, which is unstable However, a stable limit ccle also eists in this case, whose basin of attraction is the whole plane minus the equilibrium point If / < (i d + ) + (i q) < 8/9, then there eists onl one equilibrium point For low speeds, the equilibrium point is unstable and there eists a limit ccle At higher speeds, the equilibrium point becomes a stable focus 4 Figure : The case of (i d + ) + (i q) = having one and three equilibrium points If > (i d + ) + (i q) 8/9, then there eists two critical speeds Ω < Ω, such that a) if Ω < Ω, then onl one equilibrium point eists For low speeds, this equilibrium point is unstable and there eists a limit ccle around this equilibrium point As the speed increases, the limit ccle shrinks until finall it opens out and the equilibrium point becomes a stable focus b) if Ω Ω Ω, then there are three equilibrium points One of them is stable, a second one is a saddle and a third one is an unstable equilibrium point c) if Ω > Ω, then there eists onl one equilibrium point, which is stable The case of (i d + ) + (i q) = and having one stable equilibrium point is displaed in Figure a Notice that the origin is a stable focus in this case The situation of three equilibrium points appearing Copright 3 IFAC 356
= K ( a ) (( a ) + ( + b) ) = K ( + b) (( a ) + ( + b) ) K = a = 5 b = = K ( a ) (( a ) + ( + b) ) = K ( + b) (( a ) + ( + b) ) K = a = 5 b = 4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 = K ( a ) (( a ) + ( + b) ) = K ( + b) (( a ) + ( + b) ) 5 K = a = 5 b = Figure 3: The case of (i d + ) + (i q ) < / having an unstable equilibrium point and a limit ccle 5 5 5 5 5 5 5 5 3 Figure : Various cases of equilibrium points when (i d + ) + (i q ) > Figure 4 shows the various cases arising when 8/9 (i d +) +(i q) < Thus Figure 4a is a case of Ω > Ω : one stable equilibrium point Figure 4b is the case of Ω < Ω < Ω : three equilibrium points with one stable, one unstable focus and the third a saddle Figure 4c is the case of Ω < Ω : one unstable equilibrium point with a limit ccle around it 6 Conclusions for the case (i d + ) + (i q) = is displaed in Figure b Again notice here that the origin is a stable equilibrium point (in fact, a stable node) Among the other two equilibrium points, the one nearer to the origin is a saddle, whereas the one further awa is an unstable focus The case of (i d + ) + (i q) > and having one stable equilibrium point is displaed in Figure a, whereas a case of three equilibrium points is displaed in Figure b Of the three equilibrium points, the one closest to the origin is a stable equilibrium point (in fact, a stable node) Among the other two equilibrium points, the one appearing towards the top of the diagram is a saddle, whereas the one further to the right is an unstable focus In Figure 3, we have a case where (i d + ) + (i q) / The equilibrium point is an unstable focus Observe that there eists a limit ccle around the equilibrium point This limit ccle is stable, as in all points other than the equilibrium point, eventuall evolves towards this limit ccle 5 We have done a theoretical analsis of the sensorless position observer given in [7] for non-salient machines Onl under the condition of (i d + ) + (i q) =, would the origin be an equilibrium point of the error dnamics Thus, it is onl under this ver specific condition that the error in estimation of flu can actuall go to zero In all the other cases, there is some error in the estimated flu and therefore the estimated position However, this error is never ver large a globall attractive disc eists for the error sstem Under specific conditions, three equilibrium points can eist When three equilibrium points do eist, the one closest to the origin is a stable equilibrium The other two equilibrium points are a saddle and an unstable node/focus In all other cases, there is onl one equilibrium point which is either a stable equilibrium point or an unstable equilibrium with a stable limit ccle around the unstable equilibrium point Copright 3 IFAC 357
= K ( a ) (( a ) + ( + b) ) References = K ( + b) (( a ) + ( + b) ) 5 5 5 5 K = a = 3 b = 7 5 5 5 5 = K ( a ) (( a ) + ( + b) ) K = 5 a = 3 = K ( + b) (( a ) + ( + b) ) b = 7 5 5 5 5 5 5 5 5 = K ( a ) (( a ) + ( + b) ) K = a = 3 = K ( + b) (( a ) + ( + b) ) b = 7 [] P P Acarnle and J F Watson, Review of positionsensorless operation of brushless permanent-magnet machines, IEEE Trans on Ind Electron, vol 53, no, pp35-36, Apr 6 [] W Dib, R Ortega and J Malaize, Sensorless control of permanent-magnet snchronous motor in automotive applications: Estimation of the angular position, IECON, November 7,, Melbourne, Australia [3] S Ichikawa, M Tomita, S Doki and S Okuma, Sensorless control of PMSM using on-line parameter identification based on sstem s identification theor, IEEE Trans Industrial Electronics, Vol 53, No, pp 363 373, April 6 [4] PC Krause, Analsis of Electric Machiner, Mc- Graw Hill, New York, 986 [5] J Lee, K Nam, R Ortega, L Pral and A Astolfi, Sensorless control incorporating a nonlinear observer for surface mount permanent magnet snchronous motors, IEEE Transactions on Power Electronics, Vol 5, No, pp 9 97, [6] K Nam, AC Motor Control and Electric Vehicle Applications, CRC Press, [7] Ortega R, L Pral, A Astolfi, J Lee and K Nam, Estimation of rotor position and speed of permanent magnet snchronous motors with guaranteed stabilit, IEEE Transaction on Control Sstems Technolog, vol 9, no 3, pp 6 64, Ma [8] D Shah, G Espinosa, R Ortega and M Hilairet, An asmptoticall stable sensorless speed controller for non salient permanent magnet snchronous motors, 8th IFAC World Congress, Aug 8 Sept,, Milano, Ital (To appear in Int J on Robust and Nonlinear Control) [9] P Tomei and C Verrelli, Observer based speed tracking control for sensorless PMSM with unknown torque, IEEE Trans Automat Contr, Vol 56, No 6, June, pp 484 488 5 5 5 5 5 5 5 5 Figure 4: Different cases arising when 8/9 (i d + ) + (i q) < 6 Copright 3 IFAC 358