Saliency-based position estimation in permanent magnet synchronous motors
A method and system for the sensorless estimation of the states of the mechanical subsystem of a motor are disclosed. In particular, the method and system split the estimation problem into two sub-problems: the estimation of motor parameters as a function of position, and the determination of the states of the mechanical subsystem of the motor using the estimated parameters via a state observer. Motor current and PWM voltage in a polyphase system is measured or otherwise determined and converted into current and voltage in α-β coordinates. A least square estimator is constructed that uses the voltage and current in α-β coordinates and provides an estimate of the motor parameters, and in particular, the inductances of the motor, which are a function of the rotor position. These estimated inductances are used to drive a state observer whose dynamics have been selected to provide as an output the state of the mechanical subsystem of the motor, and in particular, the position, velocity, and acceleration of this subsystem.
1. An apparatus for providing a position estimate of a polyphase motor receiving polyphase current from a voltage inverter, the voltage inverter receiving PWM commands from a controller, the apparatus comprising:
a current measuring apparatus for measuring at least two polyphase currents;
an α-β converter coupled to the current measuring apparatus, the α-β converter operative to convert the polyphase currents into the α-β frame of reference, the α-β converter providing an output of the current in α-β coordinates;
the α-β converter further coupled to the controller and receiving the PWM commands therefrom, the α-β converter operative to convert the PWM commands into α-β coordinates and to provide as an output the PWM commands in α-β coordinates;
a least square estimator, receiving the current in α-β coordinates and the PWM commands in α-β coordinates, the least square estimator operative to provide an output vector {circumflex over (q)} of estimated parameters, the estimated parameters including estimated inductances of the motor; and
a state observer coupled to the least square estimator and receiving the estimated parameters including the estimated inductances of the motor therefrom, the state observer operative to provide an output of an estimate of a mechanical subsystem of the motor that includes a position estimate of the mechanical subsystem of the motor.
2. The apparatus of claim 1 wherein the least square estimator estimates the output vector {circumflex over (q)} of estimated parameters by solving an equation
q =( W T W ) −1 W T x,
wherein W is a matrix of n−1 voltage differences between adjacent PWM subintervals of a PWM period, and x is a vector of the polyphase currents.
3. The apparatus of claim 1 wherein the state observer is of the form:
{circumflex over({dot over (α)})}=γ 3 ε,
{circumflex over({dot over (ω)})}=γ 2 ε+{circumflex over (α)}, and
{circumflex over({dot over (θ)})}=γ 1 ε+{circumflex over (ω)},
wherein {circumflex over (θ)} is the position estimate, {circumflex over (ω)} is a speed estimate, and {circumflex over (α)} is an acceleration estimate of the mechanical subsystem of the motor, γ I are design parameters for setting poles of the subsystem to determine a speed of response of the state observer, and ε is a non-linear observation error of the form
ε= L 1 sin ( 2 θ),
wherein θ is a position of the mechanical subsystem of the motor, and
L
1
=
L
d
-
L
q
2
wherein L 1 represents the estimated inductances of the motor, L d denoting inductance in a magnetic flux direction of the polyphase currents, and L q denoting inductance in a direction perpendicular to the magnetic flux direction.
4. A method for providing a position estimate of a polyphase motor receiving polyphase current from a voltage inverter, the voltage inverter receiving PWM commands from a controller, the method comprising the steps of:
forming a matrix W, wherein the matrix W is a matrix of n−1 voltage differences between adjacent PWM subintervals of a PWM period;
obtaining measurements of at least two polyphase currents;
converting the obtained current measurements into an α-β coordinate system;
forming a vector x, wherein the vector x is a vector of the polyphase currents;
computing an output vector {circumflex over (q)} of estimated parameters using an equation {circumflex over (q)}=(W T W) −1 W T x;
providing the computed output vector {circumflex over (q)} of estimated parameters to a state observer;
updating the state observer by solving
{circumflex over({dot over (α)})}=γ 3 ε,
{circumflex over({dot over (ω)})}=γ 2 ε+{circumflex over (α)}, and
{circumflex over({dot over (θ)})}=γ 1 ε+{circumflex over (ω)},
wherein {circumflex over (θ)} is the position estimate, {circumflex over (ω)} is a speed estimate, and {circumflex over (α)} is an acceleration estimate of a mechanical subsystem of the motor, γ I are design parameters for setting poles of the subsystem to determine a speed of response of the state observer, and ε is a non-linear observation error of the form
ε= L 1 sin(2θ),
wherein θ is a position of the mechanical subsystem of the motor, and
L
1
=
L
d
-
L
q
2
,
and
wherein L d denotes inductance in a magnetic flux direction of the polyphase currents, and L q denotes inductance in a direction perpendicular to the magnetic flux direction; and
providing an output of an estimate of the mechanical subsystem of the motor that includes a position estimate of the mechanical subsystem of the motor.