Method and Apparatus for Estimating Rotor Position in a Sensorless Synchronous Motor

[0038]Referring next to FIGS. 3-5, a first embodiment of the position observer 40 according to the present invention is illustrated. The current feedback (iα and iβ) and voltage signals (vα and vβ), each in the stationary reference frame, are inputs to the position observer 40. The position observer 40 uses a model 44 of a synchronous motor to calculate equivalent EMF values in the stationary reference frame (e′α and e′β) as developed according to equations (1)-(8).

[0039]The model 44 is based on a voltage model of a permanent magnet salient pole synchronous machine in the stationary reference frame, equation (1). Current feedback signals (iα and iβ) and voltage signals (vα and vβ) in the stationary reference frame are input to the position observer 40. These current and voltage signals are used by the EMF model 44, equation (6), to determine the equivalent EMF values (e′α and e′β). The equivalent EMF values are then inputs to block 46 and used to estimate the rotor position, {circumflex over (θ)}.

[ v α v β ] = [ R + t ( L avg + Δ L cos 2 θ ) t Δ L sin 2 θ t Δ L sin 2 θ R + t ( L avg - Δ L cos 2 θ ) ] · [ i α i β ] + ωλ pm [ - sin θ cos θ ] ( 1 )

where, R is the motor resistance in ohms, ω is the electrical rotational frequency of the rotor in radians/sec, λpm is the flux linkage established due to the permanent magnets in the rotor in Webers, θ is the rotor angle in radians, and the Lavg and ΔL inductances are defined in equations (2) and (3), respectively.

L avg = L q + L d 2 ( 2 ) Δ L = L d - L q 2 ( 3 )

where, Lq is the q axis inductance and Ld is the d axis inductance of the synchronous motor, each measured in Henries.

[0040]While the voltage model described in Equation (1) is based on a permanent magnet motor, the model may be adapted for other types of synchronous machines by replacing the λpm term, which is the flux linkage established due to the permanent magnets in the rotor, with a term which generally identifies the flux linkage established in the rotor of a synchronous machine.

[0041]Equation (1) may be rearranged, resulting in equations (4) and (5).

v α = R · i α + t ( L avg - Δ L + Δ L cos 2 θ ) · i α + t Δ L · i α + t Δ L sin 2 θ · i β - ωλ pm sin θ ( 4 ) v β = R · i β + t ( L avg - Δ L - Δ L cos 2 θ ) · i β + t Δ L · i β + t Δ L sin 2 θ · i α + ωλ pm cos θ ( 5 )

[0042]It is desirable to express the voltage model in a form that removes any speed dependency from the model. Equations (4) and (5) are further rearranged to isolate the position dependent terms as the equivalent EMF values (e′α and e′β). The resulting equivalent EMF model, is shown in equation (6).

[ v α v β ] = [ R + L q t 0 0 R + L q t ] · [ i α i β ] + [ e α ′ e β ′ ] ( 6 )

[0043]In order to obtain an estimate of the rotor position from equation (6), the relationship between e′α and e′β must be further analyzed. An expression for e′α and e′β may be developed from equations (4)-(6).

e α ′ = t ( Δ L cos 2 θ ) · i α + t Δ L · i α + t Δ L sin 2 θ · i β - ωλ pm sin θ ( 7 ) e β ′ = t ( ( - Δ L ) cos 2 θ ) · i β + t Δ L · i β + t Δ L sin 2 θ · i α + ωλ pm cos θ ( 8 )

[0044]Properties of the reference frame transformation may be used to further simplify equations (7) and (8). For example, when a balanced three-phase current or voltage is transformed into the stationary reference frame, the magnitude of the alpha and beta components are substantially equal for the corresponding stationary reference frame current or voltage. Further, the reference frame may be established such that the alpha component leads the beta component by ninety degrees, as illustrated in FIG. 9, such that iα equals Ia cos(θ+δ) and iβ equals Ia sin(θ+δ), where Ia is the magnitude of the motor current and δ is an arbitrary constant. Using these properties of the stationary reference frame and integrating both sides of equations (7) and (8) with respect to time, the following equations result.

∫e′α=[Ia(Ld−Lq)cos δ+λpm] cos θ  (9)

∫e′β=[Ia(Ld−Lq)cos δ+λpm] sin θ  (10)

[0045]Equations (9) and (10) are utilized by block 46 to extract the estimated rotor position, {circumflex over (θ)}, from the equivalent EMF values (e′α and e′β). Each of the equivalent EMF values from equation (6) in the stationary reference frame is integrated, blocks 48 and 50. From equations (9) and (10), it may be observed that dividing the beta component by the alpha component results in the tangent of theta. Consequently, an estimated rotor position is obtained by determining the inverse tangent of the beta component over the alpha component, as performed in block 52 and shown in equation (11).

θ ^ = tan - 1 [ ∫ e β ′ ∫ e α ′ ] ( 11 )

[0046]Referring now to FIGS. 6-8, a second embodiment of the position observer 40 according to the present invention is illustrated. The motor current and voltage, each being transformed to the stationary reference frame, are input signals to the position observer 40. The position observer 40 uses a sliding mode model 54 and an equivalent control block 56 to obtain an estimated rotor position, {circumflex over (θ)}.

[0047]Sliding mode control is a non-linear control strategy which attempts to force a dynamic system to operate at a desired operating point using a “bang-bang” type controller. A “bang-bang” type controller compares an input signal against a pre-determined operating point, for example a particular value or plane of operation, and outputs one of two values, for example a zero or a one, based on which side of the operating point the input signal is located. A set of sliding mode equations is developed according to principles of sliding mode control, and the equations are used to force desirable operating conditions for a system. The present sliding mode equations were developed using techniques described in the text by Vadim Utkin et al., “Sliding Mode Control in Electromechanical Systems,” 1st Ed., Taylor & Francis, 1999. The resultant sliding mode equation, equation (12), describes the sliding mode model 54 of the position observer 40.

[ i α ^ . i β ^ . ] = 1 L q [ v α v β ] - R L q [ i α ^ i β ^ ] - K SM L q [ sign ( i α ^ - i α ) sign ( i β ^ - i β ) ] ( 12 )

where, R is the motor resistance, Lq is the q axis inductance of the motor, and KSM is a constant observer gain.

[0048]The error dynamics of the sliding mode observer are obtained by subtracting the sliding mode equations from the equivalent EMF model of equation (6). The resulting expression for these error dynamics is given in equation (13).

[ i α _ . i β _ . ] = - R L q [ i α i β _ ] + 1 L q [ e α ′ e β ′ ] - K SM L q [ sign ( i α _ ) sign ( i β _ ) ] ( 13 )

where, iα and iβ are the error values between the observed and actual current values in the stationary reference frame.

[0049]In order to ensure that the sliding mode observer is stable, a set of Lyapunov stability functions are developed. Lyapunov stability functions are a set of equations that can be used to determine the stability of a dynamic system. The set of equations are designed such that a model of a dynamic system satisfies the criteria defined in equations (14) and (15).

f(x)≧0  (14)

where equality occurs only if ‘x’ is equal to zero.

{dot over (f)}(x(t))<0  (15)

[0050]Consequently, equations (16) and (17) were developed according to the Lyapunov stability criteria in order to ensure stability of the position observer.

V = 1 2 ( i α 2 _ + i β 2 _ ) ( 16 ) V . = ( i α _ · i α _ . + i β _ · i β _ . ) = - R L q ( i α 2 _ + i β 2 _ ) + 1 L q ( e α ′ · i α _ + e β ′ · i β _ ) - K SM L q ( i α _ + i β _ ) ( 17 )

[0051]The constant observer gain, KSM, is selected such that the value is large enough to force the observer to quickly converge on the desired operating point. The equivalent EMF components, e′α and e′β as described by equations (7) and (8), have an upper limit as determined by the machine parameters and operating characteristics of the synchronous motor. In order to force the observer to converge, the constant observer gain, KSM, is preferably selected such that KSM is greater than the maximum value of either of the equivalent EMF components, e′α and e′β. By selecting a constant observer gain greater than the maximum value of either of the equivalent EMF components, the observed current will converge to the actual current.

[0052]FIG. 6 depicts an embodiment of the position observer 40 implementing sliding mode control. The current feedback signals (iα and iβ) and voltage signals (vα and vβ) in the stationary reference frame are input to the position observer 40. These signals are used by the sliding mode model 54 as defined by equation (12) to obtain observed current values in the stationary reference frame.

[0053]The sliding mode controller forces convergence of the observed current values to the measured current values. In order to force convergence of the observed current values with the measured current values, the desired error values between the observed and actual current values in the stationary reference frame, iα and iβ, are set to zero and an equivalent control method, as shown in FIG. 8, is applied. By setting the error values, iα and iβ, to zero, an expression for the equivalent EMF value of the synchronous motor is obtained, according to equations (18) and (19).

e′α=[KSMsign( iα)]eq (18)

e′β=[KSMsign( iβ)]eq (19)

[0054]An estimate of the rotor position can be obtained as illustrated in block 56 of FIG. 8. Each component of the error values, iα and iβ, are passed through a sign function, 58 and 62, to determine whether the observed current is greater than or less than the actual current. The sign function, 58 and 62, generates a fixed, positive or negative, value according to the sign of the error value input to the function. A low pass filter, 60 and 64, is then applied to the output of the sign function, 58 and 62, to obtain the equivalent EMF values, e′α and e′β, in the stationary reference frame. The rotor position is finally estimated by determining the negative of the arctangent of the alpha component of the equivalent EMF value over the beta component 66, as shown in equation (20).

θ ^ = - tan - 1 [ e α ′ e β ′ ] ( 20 )

[0055]In operation, the position observer 40 is used to estimate the angular position of the rotor in a synchronous motor 15 controlled by a motor drive 10. The observer 40 receives current and voltage signals, preferably in the stationary reference frame, as input signals. These current and voltage signals are then used to determine the estimated angular position based on a model of the synchronous motor.

[0056]The equivalent EMF model, as given in equation (6), is an improved model for describing synchronous motors. The model is independent of rotor speed and has reduced sensitivity to motor parameters. The primary motor parameters typically involved in any EMF model are the stator resistance and the d axis and q axis stator inductances. The stator resistance is most easily and accurately obtained by direct measuring, as is known in the art. The stator inductances are typically more difficult to obtain. Reduced sensitivity to motor parameters has been accomplished, at least in part, by eliminating dependence on the d axis inductance value from the model. Further, identification of the motor's q axis inductance by a method such as that described in a co-pending application by the same inventors, Ser. No. U.S. Ser. No. 12/208,046, which is hereby incorporated by reference, may provide a range of q axis inductance values for varying operating currents. The equivalent EMF model preferably reads one of the q axis inductance values from a table of values, which are dependent on the operating current, to provide an improved estimate of rotor position.

[0057]It should be understood that the invention is not limited in its application to the details of construction and arrangements of the components set forth herein. The invention is capable of other embodiments and of being practiced or carried out in various ways. Variations and modifications of the foregoing are within the scope of the present invention. It also being understood that the invention disclosed and defined herein extends to all alternative combinations of two or more of the individual features mentioned or evident from the text and/or drawings. All of these different combinations constitute various alternative aspects of the present invention. The embodiments described herein explain the best modes known for practicing the invention and will enable others skilled in the art to utilize the invention.