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High-Resolution Permanent Magnet Drive Using Separated Observers for Acceleration Estimation and Control^{ †}

^{1}

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^{†}

## Abstract

**:**

## 1. Introduction

## 2. PM Motor System

^{−5}deg, as reported in the datasheet [29]. More importantly, the resolution is expected to improve to 1.08 × 10

^{−5}deg using the proposed motion control method. Detailed PM motor specifications are listed in Table 1.

## 3. High-Resolution Position Sensing

^{9}= 0.7 deg. As listed in Table 1, the corresponding scanner resolution is 78 μm. By using the interpolated position ${\mathsf{\theta}}_{\mathrm{intp}}$, the position resolution is increased to 360 deg/2

^{9+16}= 1.08 × 10

^{−5}deg. The scanner resolution is theoretically below 1 μm, assuming a rigid body on the scanner mirror and motor shaft. This increased position resolution is useful for acceleration estimation and acceleration control.

## 4. Dynamic Model of PM Motor

_{ML}and K

_{ML}are equivalent damper and spring caused by mirror load; and B

_{M}and K

_{M}are damper and spring considering the motor rotation dynamic. Under this effect, the corresponding torque load can be modelled by

_{ML}and θ

_{M}are the position, respectively, of the mirror and motor, and ${\mathrm{T}}_{\mathrm{dis}}$ is used to model the vibration torque induced by mirror unbalance load. Based on the above derivation, the PM motor model is formulated in S-domain to easily analyze the dynamic property. The model input is actual current I and the torque load T

_{L}, where the output is position θ

_{M}. The transfer function of θ

_{M}is developed by

_{t}is the torque constant of the PM motor. As reported in [30,31], the motor electromagnetic torque is equal to T

_{em}(s) = K

_{t}× I(s). Figure 7 explains the conventional motion control for the PM motor with mirror load. In this system, the cascaded position and speed controller are developed for the manipulation of current command. In this controller, the position controller is formulated by proportional ${\mathrm{K}}_{\mathrm{p}1}$ control, while the speed controller is based on proportional ${\mathrm{K}}_{\mathrm{p}2}$ and integral ${\mathrm{K}}_{\mathrm{i}2}$ control. By selecting controller gains ${\mathrm{K}}_{\mathrm{p}1}$/${\mathrm{K}}_{\mathrm{p}2}$/${\mathrm{K}}_{\mathrm{i}2}$, the over-damped transient dynamic without overshoot can be achieved.

## 5. Observer-Based Speed Estimation

#### 5.1. Proposed Speed Observer

_{M}and actual motor current I, where the output is estimated speed $\hat{\mathsf{\omega}}$. The corresponding transfer function for $\hat{\mathsf{\omega}}$(s) estimation can be then depicted by

_{1}, K

_{2}, and K

_{3}are observer controller gains for the estimation bandwidth determination. In addition, the estimated motor torque ${\hat{\mathrm{T}}}_{\mathrm{em}}$ is designed as torque feedforward for the speed observer. In (4), ${\hat{\mathrm{T}}}_{\mathrm{em}}$ can be obtained through the estimated torque constant ${\hat{\mathrm{K}}}_{\mathrm{t}}$ times the actual motor current I, as given by

_{M}and actual motor current I. The corresponding transfer function of $\widehat{\mathsf{\omega}}$ can be derived by

#### 5.2. Differentiation Noises Elimination

_{ns}(s) by the blue dashed line. The magnitude of |${\widehat{\mathsf{\omega}}}_{\mathrm{direct}}$(s)/θ

_{ns}(s)| increases with frequency noises. This speed estimation might not be suited for PM motors for high-bandwidth operation.

_{ns}(s) in the speed observer and ${\widehat{\mathsf{\omega}}}_{\mathrm{direct}}$(s)/θ

_{ns}(s) through Equation (8). Considering the proposed observer for speed estimation, $\widehat{\mathsf{\omega}}$(s)/θ

_{ns}(s) is equivalent to a HPF. Once high-frequency noises occur on measured position, the influence of noises ${\mathsf{\theta}}_{\mathrm{ns}}$ on $\widehat{\mathsf{\omega}}$ maintain a constant magnitude below the observer estimation frequency.

_{ns}(s) in Equation (9), only observer proportional gain K

_{2}and integral gain K

_{3}are used for ${\widehat{\mathsf{\omega}}}_{\mathrm{en}}$ estimation. Thus, the transfer function of ${\widehat{\mathsf{\omega}}}_{\mathrm{en}}$(s)/θ

_{ns}(s) is derived by

_{1}, K

_{2}

^{,}and K

_{3}to ensure the closed-loop stability. Although the stability is the same between $\widehat{\mathsf{\omega}}$(s) and ${\widehat{\mathsf{\omega}}}_{\mathrm{en}}$(s), the dynamic property is different, due to there being no differential gain K

_{1}in the numerator of Equation (10). Without the differential operator, both the high-frequency dynamic response and noises are decreased. Figure 10 illustrates the frequency response of ${\widehat{\mathsf{\omega}}}_{\mathrm{en}}$(s)/θ

_{ns}(s) as the green dashed line. Comparing to $\widehat{\mathsf{\omega}}$(s)/θ

_{ns}(s), the influence of position noises is decreased beyond the speed observer bandwidth. In this paper, the observer bandwidth is designed at 2.5 kHz to ensure the sufficient speed estimation dynamic for PM motor motion control. Considering the mirror-reflected torque harmonics beyond 3 kHz, these disturbances can be compensated based on another proposed acceleration observer in Section 6.

#### 5.3. Simulation Result

## 6. Acceleration Estimation and Control

#### 6.1. Acceleration Feedback Control

_{L}and torque load ${\mathrm{T}}_{\mathrm{L}}$. Based on the controller topology in Figure 7, the corresponding DRF is represented by

_{M}or mirror inertia J

_{ML}. It is noted that conventional PM motors result in small inertia for high dynamic response. More importantly, due to high rotation speed, the harmonics of mirror unbalanced load can be up to 3–5 kHz. It leads to various challenges under the conventional control topology in Figure 7.

#### 6.2. Acceleration Estimation

_{1}. Figure 14 shows the signal process of the separated acceleration observer. To minimize the differentiation noise balancing the high-frequency dynamic, a separated observer without the differential controller is the candidate. In this acceleration observer, the speed regulation is designed instead of the differential controller. On this basis, the measured position and estimated enhanced speed are both used as observer inputs, while the motor torque estimation in Equation (5) is designed as the feedforward control for better estimation bandwidth. In this case, the corresponding transfer function of estimated acceleration $\widehat{\mathsf{\alpha}}$ without differentiation noises is represented by

#### 6.3. Simulation Results

## 7. Experimental Results

^{2}inertia to stimulate the high-frequency vibrational load. Table 1 in Section 2 lists key characteristics of the test PM motor. A sine/cosine encoder with 512 pulses is installed for the position measurement. Figure 3 illustrates the overall control flowchart of the PM motor. The current regulation is implemented based on the amplifier hardware. All other motion control and observer-based estimation is implemented in a 32-bit digital signal process, TI-TMS320F28379. It is noted that the bilinear transformation is used to realize all control and estimation algorithms mentioned in Section 5 and Section 6.

#### 7.1. High-Resolution Position Interpolation

_{crs}and interpolated position θ

_{intp}, under a 10 Hz sinusoidal position command. In this test, the PM motor is operated without a mirror to clearly demonstrate the position resolution neglecting vibrational load. As seen from the magnified plot, the resolution of coarse position θ

_{crs}is around ±0.7 deg (78 μm) without the interpolation. By contrast, the interpolated position θ

_{intp}is close to the position command, where the resolution is smaller than 0.01 deg (1 μm). It is concluded that the position interpolation can greatly increase the measurement resolution under the same sensing hardware.

#### 7.2. Observer-Based Speed Estimation

^{2}mirror load. In Figure 18a, if the direct differentiation is applied, a visible filter-reflected phase delay is observed under a 2.5 kHz LPF. Moreover, a small amount of speed variation is resultant during peak speed commands. This variation is caused by the friction torque when the motor mirror rotates across zero position. Comparing to ${\widehat{\mathsf{\omega}}}_{\mathrm{direct}}$, two observer-based estimated speeds are illustrated in Figure 18b. By adding the torque feedforward in Figure 8, the nearly zero phase delay is achieved on both $\widehat{\mathsf{\omega}}$ and ${\widehat{\mathsf{\omega}}}_{\mathrm{en}}$. More importantly, the proposed enhanced estimation ${\widehat{\mathsf{\omega}}}_{\mathrm{en}}$ results in the lowest speed variation among the three estimated speeds. As a result, better performance is expected if proposed ${\widehat{\mathsf{\omega}}}_{\mathrm{en}}$ is used for the motion control and acceleration estimation.

#### 7.3. Observer-Based Acceleration Estimation

#### 7.4. Motion Control Response

^{2}mirror load. In Figure 20a, the conventional motion control in Figure 7 is applied where ${\widehat{\omega}}_{\mathrm{direct}}$ in Equation (8) is used for the speed regulation. Although cascaded position P control and speed PI control is designed to remove transient overshoot, the slow 2 ms transient response and visible vibration-reflected position error are resultant. By contrast, in Figure 20b, the transient setting time is improved from 2 ms to 1.27 ms once the observer-based ${\widehat{\omega}}_{\mathrm{en}}$ is used for speed regulation. Because of the low-noise speed feedback, a faster dynamic response is achieved under the same controller bandwidth.

## 8. Comparison between Simulation and Experimental Results

## 9. Conclusions

- A position interpolation is used to increase the position resolution. The high-performance speed and acceleration estimations are implemented with this interpolation process.
- The proposed speed observer reduces the differentiation noise on speed estimation. A better dynamic response of the PM motor is achieved.
- A separated acceleration observer is proposed for acceleration estimation. The lowest steady-state error is achieved through acceleration control to suppress the high-frequency mirror vibrational harmonics.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 5.**Position interpolation between encoder pulse counts: (

**a**) conceptual interpolation process and (

**b**) experimental result from the test motor (9-bit pulses + 16-bit ADC).

**Figure 8.**Speed estimation using the proposed observer with torque feedforward for phase delay compensation.

**Figure 9.**Frequency responses of different speed estimations at different frequencies (ideal parameter estimation, ${\widehat{\mathrm{K}}}_{\mathrm{t}}$ = K

_{t}and ${\widehat{\mathrm{J}}}_{\mathrm{ALL}}$ = J

_{ALL}).

**Figure 10.**Frequency response of estimation noise attenuation among different speed estimations (ideal parameter estimation, ${\widehat{\mathrm{K}}}_{\mathrm{t}}$ = K

_{t}and ${\widehat{\mathrm{J}}}_{\mathrm{ALL}}$ = J

_{ALL}).

**Figure 11.**Simulation of different speed estimations under a sinusoidal position feedback signal (0.002% white position noise).

**Figure 12.**Comparison of disturbance rejection function |θ

_{L}(s)/T

_{L}(s)| between conventional control in (11) and acceleration control in (12).

**Figure 13.**Advanced PM motor motion control with proposed acceleration control to improve the disturbance rejection at high frequency.

**Figure 14.**Proposed separated acceleration observer based on the feedbacks of measured position and estimated speed from the prior speed observer in Figure 8.

**Figure 15.**Simulation of different acceleration estimations under a sinusoidal position feedback signal (0.002% white noise).

**Figure 17.**Comparison of different position signals under a sinusoidal position command: position command θ*, coarse position θ

_{crs}in Figure 4, and proposed interpolated position θ

_{intp}.

**Figure 18.**Comparison of different speed estimations under a 50 Hz speed command with 0.64 g·cm

^{2}mirror load: (

**a**) ω* and ${\widehat{\mathsf{\omega}}}_{\mathrm{direct}}$ and (

**b**) $\widehat{\mathsf{\omega}}$ and ${\widehat{\mathsf{\omega}}}_{\mathrm{en}}$ (2.5 kHz speed observer bandwidth).

**Figure 19.**Comparison of different acceleration estimations under a 50 Hz acceleration command with 0.64 g·cm

^{2}mirror load): (

**a**) α* and ${\widehat{\mathsf{\alpha}}}_{\mathrm{direct}}$ (

**b**) α* and $\widehat{\mathsf{\alpha}}$ (8.6 kHz acceleration observer bandwidth).

**Figure 20.**Comparison of position step control performance: (

**a**) conventional motion control with direct speed estimation; (

**b**) motion control with proposed speed estimation; and (

**c**) speed estimation plus proposed acceleration control (0.64 g·cm

^{2}mirror load).

Characteristics | Values |
---|---|

Rotor poles | 4-pole |

Rated torque | 0.0127 Nm |

Rated current | 1 A |

Position rotation | 40 deg (maximum) |

Resistance | 1.7 Ω |

Inductance | 0.22 mH |

Rated voltage | ±15 V |

Inertia | 0.82 g·cm^{2} |

Sample frequency | 100 kHz |

Resolution per degree | 40 mm/360 deg = 111 μm/deg |

Control accuracy (existing drive) | 4.41 × 10^{−5} deg |

Control accuracy (proposed drive) | 1.08 × 10^{−5} deg |

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**MDPI and ACS Style**

Lin, Y.-J.; Chou, P.-H.; Yang, S.-C.
High-Resolution Permanent Magnet Drive Using Separated Observers for Acceleration Estimation and Control. *Sensors* **2022**, *22*, 725.
https://doi.org/10.3390/s22030725

**AMA Style**

Lin Y-J, Chou P-H, Yang S-C.
High-Resolution Permanent Magnet Drive Using Separated Observers for Acceleration Estimation and Control. *Sensors*. 2022; 22(3):725.
https://doi.org/10.3390/s22030725

**Chicago/Turabian Style**

Lin, Yi-Jen, Po-Huan Chou, and Shih-Chin Yang.
2022. "High-Resolution Permanent Magnet Drive Using Separated Observers for Acceleration Estimation and Control" *Sensors* 22, no. 3: 725.
https://doi.org/10.3390/s22030725