1. Introduction

The use of electric motors as an energy source is expanding. Electric vehicles are a representative example. PID (Proportional Integral Differential) control, which is generally used to control motors, has good position or speed control performance ^[1-14]. Overshoot in PI control can be reduced by improving the performance of the transient state of the controller is improved by integrating the error and IP control that determines the control value in proportion to the state to be controlled. On the other hand, it does not have a certain responsiveness to load torque or parameter changes, which may cause the position or speed control to become unstable. The mathematical error of the model or the disturbance applied needs to be compensated for to overcome this point ^[1-5].

The motor rotates at a constant speed, and the speed decreases significantly due to the addition of a load, and it takes considerable time to recover. Such load fluctuations have a great influence on control performance. There was a method of responding to load fluctuations by designing an observer to estimate the load and compensate for the estimated value. It is also important to control the current of the three-phase inverter that supplies power to the motor. Therefore, it compensates for the current controller by designing an observer that estimates the nonlinear element as a disturbance in the electric model of the motor.

Modern control is performed using state variables. If the state is unmeasurable or unrealistic, the state observer is designed to estimate the desired state and used for system control. The Luenberger state observer is a method for estimating each state by adjusting the gain of the observer so that the error between the output state value of the observer and the output state value of the system converges to zero. This method includes nonlinear terms, model errors, and disturbances as the state variables of a system to define a state and includes it in the state observer to estimate it. It can estimate a state that cannot be observed directly or measured using a state observer ^[4].

In this study, a state observer was designed to estimate the state variables required to control the speed, position, and current of the rotor shaft of a synchronous motor system with a permanent magnet rotor and controls the states. The motor of this study is a motor driven by the principle that the rotor rotates synchronously by making a rotating magnetic field by flowing a current through the three-phase wired stator coil to move the rotor attached to the surface of the permanent magnet. Compared to DC (Direct Current) motors with brushes, it does not require brushes because it has a rotor as a permanent magnet. On the other hand, this needs to be controlled by supplying current to the stator coil with a three-phase inverter according to the position of the rotor permanent magnet in the three-phase stator coil to produce a rotating magnetic field.

IPMSM (Interior Permanent Magnet Synchronous Motor) is a synchronous motor whose rotor is a permanent magnet. This is a structure in which a mechanical system and an electrical system are combined. The mechanical system constitutes a time constant with parameters, such as the moment of inertia of the rotor and the damping coefficient, which are based on the angular velocity or position of the rotation axis of the motor. The electrical system constitutes an electrical time constant with the inductance and resistance components of the stator winding connected in three phases by y or $\Delta $. The motor is a structure in which a rotating magnetic field is generated inside the motor with the current supplied to the stator windings from a three-phase inverter according to the location of the permanent magnet of the motor, generating rotational force in the permanent magnet of the rotor. Controlling the IPMSM requires knowledge of the position state of the permanent magnet of the motor rotor, and it uses a position sensor, such as an encoder or a resolver coupled to the rotating shaft. On the other hand, these sensors generate noise depending on the state of the motor.

In some cases, it is difficult to construct a sensor for measuring states, such as speed and torque, and it is very difficult to sense a state, such as disturbance. Therefore, it uses a method for estimating the state variables necessary to construct a controller by designing a mathematical state observer. The state vector was set to state variables using the speed, the position of the rotor shaft of the motor, and the current for torque control. The dynamic equation of the motor was converted into the state equation of the system. The output equation was constructed with the measurable state in the state variables as the output state. The state vector was estimated by designing the observer using the state equation and output equation, which are mathematical models of the motor system, and by designing the Luenberger state observer. A PI controller was configured using the desired reference input and the estimated state to follow as feedback.

For the control of the IPMSM, each control of the three-phase current and the method of controlling the position or the speed of the rotor by inputting the torque generated by the three-phase current were used. The electric motor was a MIMO (Multiple Input Multiple Output) system divided into a mechanical system and an electrical system to easily design and apply an observer to compensate for possible disturbance. The electric system is also separated into the magnetic flux axis and the torque axis, and divided into three SISO (Single Input Single Output) systems to proceed with the design of the compensate controller.

This paper is organized as follows. Section 2 describes the design of the Luenberger observer. Section 3 explains the speed controller design. In section 4, the experimental results were evaluated. Finally, section 5 presents the conclusion.

2. Design of the Luenberger Observer

The IPMSM model synchronized to the DQ axis as in Eqs. (1) to (4). Eq. (1) is the mechanical dynamics equation related to the rotating shaft of the motor. Eqs. (2) and (3) are the electric circuit equations expressed on the DQ axis. The current of Eq. (3) in Eq. (2) is the method of generating the torque of Eq. (4) as follows:

$\mathrm{T}_{\mathrm{e}}$ is the torque caused by electromagnetic. $\mathrm{J}_{\mathrm{m}}$ is the moment of inertia of the motor rotor. $\omega _{\mathrm{r}}$ is the mechanical velocity of the rotor. $\mathrm{B}_{\mathrm{m}}$ is the viscous friction coefficient of the motor rotor. $\mathrm{T}_{\mathrm{l}}$ is the load torque. $\mathrm{V}_{\mathrm{d}}$ and $\mathrm{V}_{\mathrm{q}}$ are input voltages of the d-axis and the q-axis, respectively. $\mathrm{L}_{\mathrm{d}}$ and $\mathrm{L}_{\mathrm{q}}$ are the inductances of the d-axis and q-axis, respectively. $\mathrm{i}_{\mathrm{d}}$, $\mathrm{i}_{\mathrm{q}}$ are currents of the d-axis and q-axis, respectively. $\mathrm{R}_{\mathrm{s}}$ is the phase resistance of the stator. $\varphi _{\mathrm{f}}$ is the permanent magnet flux of the rotor. $\mathrm{p}$ is the pole pair of the permanent magnet.

In the mechanical system of Eq. (1), if $T_{e}$ is $u_{m}$, and the equation of state is derived, the following can be obtained.

Here, the definition of the state variable for deriving the state equation of Eq. (5) is as follows:

$\Delta \left(1/J_{m}\right)$ in Eq. (6) represents a parameter error. If the state observer is designed using the state equation of Eq. (5), the following equation is obtained.

The estimation accuracy can be increased by appropriately adjusting $m_{1}$ and $m_{2}$ with the estimation gain of the observer in Eq. (7).

Fig. 1 shows the circuit equation of the motor. The figure on the left is an equivalent circuit for the d-axis, which is the same axis as the magnetic flux, made by the structure in which the magnet is embedded in the rotor. The figure on the right is a q-axis equivalent circuit that mainly generates torque.

Fig. 1. DQ axis equivalent circuit of IPMS.

The left side of Fig. 1 is an equivalent circuit of Eq. (2), and the right side is an equivalent circuit of Eq. (3). If $V_{d}$ for Eq. (2) is $u_{d}$, the equation of state can be expressed as

Here, the definition of the state variable for deriving the state equation of Eq. (8) is as follows:

$\Delta \left(1/L_{d}\right)$ in Eq. (8) represents the parameter error. The derivative becomes 0 because the terms of the state variable $x_{d,2}$ do not change in the steady state. If the state observer is designed using the state equation of Eq. (9), the following equation is obtained.

The estimation accuracy can be increased by appropriately adjusting $l_{d,1}$ and $l_{d,2}$ with the estimation gain of the observer in Eq. (7).

When $V_{q}$ in Eq. (3) is $u_{q}$, the equation of state can be expressed as

Here, the definition of the state variable for deriving the state equation of Eq. (11) is as follows:

$\Delta \left(1/L_{q}\right)$ in Eq. (12) represents a parameter error. The derivative becomes 0 because the terms of the state variable $x_{q,2}$ do not change in the steady state. If the state observer is designed using the state equation of Eq. (11), the following Eq. (13) can be obtained.

The estimation accuracy can be increased by appropriately adjusting $l_{q,1}$ and $l_{q,2}$ with the estimation gain of the observer in Eq. (7).

3. Speed Controller Design

A general PI controller was first constructed to reflect the state estimated in the previous chapter. The proportional controller improves the steady-state error, and the transient state can achieve a sufficiently good response performance with only a proportional controller. An IP controller compensates for the overshoot characteristics in the transient state. Accordingly, the control input is configured in a form in which the disturbance estimated by the disturbance observer is added to the IP controller.

The IP controller for each system from Eqs. (1) to (3) is as follows:

In Eq. (14), $\omega _{ref}$ is the reference speed. $i_{d,ref}$ in Eq. (15) can be set to control the magnetic flux and is generally controlled to 0. $i_{q,ref}$ in Eq. (16) can be set as follows using the input $T_{e}$ of the speed control in Eqs. (14) and (4).

4. Simulation

PSIM was used for a simulation program. The model of IPMSM used the model provided in PSIM with the motor parameters in Table 1. The motor parameters were the nominal values.

The IPMSM was started at a reference angular velocity of 125.6 [rad/s] and operated at a speed of ${-}$125.6 [rad/s] for two seconds to judge the performance of the designed disturbance observer and controller. A constant load of 0.5 [Nm] was applied to the rotating shaft in one second. For the disturbance state estimated by the angular velocity control, Fig. 2 compares the results of compensating the PI controller and not compensating.

The upper part of Fig. 2 is the result of only the pure controller without compensating the estimated disturbance state to the controller. An overshoot of approximately 9.95 [%] occurred, and the rise time to reach 90 [%] took approximately 0.087 seconds. The lower part of Fig. 2 is the controller that compensates for the estimated disturbance. At the start of 0 seconds, an overshoot of approximately 15 [%] occurred, and the rise time to reach 90 [%] took approximately 0.084 seconds.

An undershoot of approximately 17.9 [%] occurred when a constant load of 0.5 [Nm] was applied at one second and a disturbance was applied when the angular velocity was approximately 103.1 [rad/s]. After approximately 0.65 seconds, the angular velocity was restored to within 99 [%] of the reference angular velocity. In the lower figure, an undershoot of 9.95 [%] occurred when a disturbance was applied when the angular velocity was 113.1 [rad/s]. After approximately 0.6 seconds, the angular velocity was restored to within 99 [%] of the reference angular velocity.

Fig. 3 shows each state after compensating for the estimated disturbance to the controller. The first figure estimates load fluctuations and disturbance $\hat{f}_{\omega }$ occurring in the mechanical system. Eq. (6) shows that the estimation error was within 0.1% in the steady state. The second figure was the d-axis current and the estimated d-axis current, and the fourth figure was the q-axis current and the q-axis current. The state estimation error was less than 0.1 [%]. The third figure shows $f_{d}$ and $\hat{f}_{d}$ as a result of estimating $f_{d}$ in Eq. (9). There is some error in the transient state of the steady state, but the estimation error in the steady state is within 0.1 [%].

The fifth figure shows $f_{q}$ and $\hat{f}_{q}$ as the result of estimating $f_{q}$ in Eq. (12). The result of estimating the disturbance in Eq. (12) also indicates that the estimation error was within 0.1 [%] in the steady state.

Fig. 2. States at an angle velocity of 125.6 rad/s.

Fig. 3. IPMSM driving results at an angle velocity of 125.6 rad/s.

Table 1. IPMSM parameters.

5. Conclusion

Estimating and compensating for disturbances is necessary for precise speed and tracking control of the IPMSM. In general, the PID controller is set to a fixed gain for a predetermined state, so responding to disturbances occurring in real time is inappropriate and causes poor control performance. In order to achieve the desired control performance of position or velocity, estimation, and compensation for disturbance are essential. The state estimator using the Luenberger observer was separated into a mechanical system and an electric system, and the electric system was designed by dividing the state observer related to the magnetic flux axis and the state observer related to the torque axis again.

As a result, the error between the estimated state and the disturbance in the steady state was within 0.1 [%]. As a result of the speed tracking control, a steady state error of less than 0.1 [%] was obtained. In addition, an undershoot of 17.9 [%] occurred due to speed control without compensating for disturbance. An undershoot of 9.95 [%] resulted from compensating for the estimated disturbance and speed control. This study found that the adaptation to disturbances obtained improved performance.

The estimation performance of the designed observer can be improved if the observed state output value of the system is accurate or if the error of the system model is small. In real systems, however, noise is included during measurements, and the performance of the state observer is degraded because of parameter errors caused by uncertainty in the system model, so estimation errors for disturbances may increase. Therefore, research to improve the performance of a disturbance estimation by designing a robust state observer is required.