# Angle Tracking Observer with Improved Accuracy for Resolver-to-Digital Conversion

^{*}

## Abstract

**:**

## 1. Introduction

_{P}+ k

_{I}/s), which makes the conventional ATO a type-II system.

_{P}+ k

_{I}/s + k

_{D}/s

^{2}) [20]. Caruso et al. [21] focused on the compensation of position error by making the tracking system have a time-dependent error, which would finally be cancelled by the PI controller. Zhang et al. [22] proposed a composite observer for fast-varying speed using the arctangent method to compensate the estimated speed from ATO. Drawing on advantages from both arctangent and ATO methods, this composite observer enhances accuracy in estimating acceleration-related position. Neural networks have also been applied independently [23] and combined with third-order ATO [18] for reducing position errors, but the implementation is complicated and time-consuming.

## 2. Principles of Resolver-to-Digital Conversion

#### 2.1. Principle of Resolver

_{ref}is the angular frequency.

_{1}(θ,t) and u

_{2}(θ,t) have the form of sinusoidal and cosinusoidal envelopes. Then, envelope detection is necessary to obtain the needed signal. In practice, there are always disturbances from imperfect characteristics, namely the amplitude imbalances, DC offsets and imperfect quadrature, which arise from eccentric rotor, unequal winding and non-orthogonal symmetry of the windings. Hence, the signal after envelope detection can be expressed in the form:

_{sin}and u

_{cos}are two ortho-symmetric signals, whose trigonometric features aid in subsequent demodulation.

#### 2.2. Classical Resolver-to-Digital Conversion

#### 2.2.1. Arctangent Method

_{sin}and u

_{cos}, given by:

#### 2.2.2. Conventional PLL-Based ATO

_{I}and k

_{P}are the coefficients for the integral and proportional terms, respectively, and Figure 2b shows the linearized structure.

#### 2.2.3. Compensated Type-III ATO

^{2}) with a time-saving and simple technique of adding a compensation module, which achieves a higher precision in tracking the position of an accelerating motor. However, this method fails to eliminate steady-state error resulting from a first- or higher-order acceleration signal.

## 3. Improved ATO with Dynamic Compensation

#### 3.1. Proposed Compensated Scheme of ATO

^{3}). The estimated speed signal is employed to construct an internal feedback loop. The basic idea of the proposed ATO is to improve the system order of the LF and therefore make the ATO become a type-IV system. By carefully designing the parameters and form of the transfer function, low-order items in the numerator of the system transfer function can be cancelled, leaving only the highest-order item, as is illustrated in the following derivation.

_{I}and k

_{P}.

^{n}. To be specific, this ATO can eliminate position errors when n ≤ 3 and reduce error when n ≥ 4.

#### 3.2. Parameter Tuning Guidance

_{I}and k

_{P}, the coefficients for the integral and proportional terms, respectively, and γ from the compensation module.

#### 3.2.1. Establishing Relationship Between k_{I} and k_{P}

_{I}and k

_{P}in conventional ATO can be employed [5]. A typical characteristic equation for a second-order system is ${s}^{2}+2\zeta {\omega}_{n}s+{\omega}_{n}^{2}=0$, where ζ equals 0.707 when the overshoot and responsiveness reach a satisfying trade-off. In our case where the second-order system in Equation (7) is denoted by ${k}_{I}={\omega}_{n}^{2}$ and k

_{P}= 2ζω

_{n}, the relationship of ${k}_{I}={k}_{P}^{2}/(4\times {0.707}^{2})$ can be established.

#### 3.2.2. Determining k_{P} and γ

_{P}and γ is to maintain γ > k

_{P}. Figure 5a shows the relationship between the two parameters and bandwidth. The bandwidth shows a diagonal-tilt increasement, which indicates that a larger k

_{P}and smaller γ will reach a higher bandwidth.

_{n}) of 1300, 780 and 520 rad/s (via varying parameter γ) in Figure 5b, the error transfer function (Equation (10)) of the proposed ATO is investigated under a specific condtion of k

_{P}= 141.4 and k

_{I}= 100

^{2}. From inspection of these three bandwidths, it can be concluded that a larger bandwidth (smaller γ) suppresses the position estimation error more effeciently, which corresponds to the analysis of Equations (12) and (13).

_{P}+ 23.6; considering a smaller value of γ, the condition results in γ = k

_{P}+ 23.6. Along with k

_{P}= γ − 23.6 and ${k}_{I}={k}_{P}^{2}/(4\times {0.707}^{2})$, Figure 5c evaluates the relationship between γ and the system’s bandwidth, which is well-fitted linearly, ω

_{n}= 10.7γ − 560. Hence, with a desired bandwidth ω

_{n}, γ can be chosen as γ = 0.0935ω

_{n}+ 53.

#### 3.2.3. Tuning Guideline

- (1)
- Choosing γ according to the desired bandwidth by referring to γ = 0.0935ω
_{n}+ 53; - (2)
- Determining k
_{P}via k_{P}= γ − 23.6; - (3)
- Determining k
_{I}via ${k}_{I}={k}_{P}^{2}/(4\times {0.707}^{2})$.

## 4. Simulation and Experimental Results

#### 4.1. Simulation Results

^{n}) to imitate the demodulated signals coming from RDC.

^{2}. Results from the conventional ATO, type-III ATO and the proposed ATO are labelled separately with e

_{1}, e

_{2}and e

_{3}, respectively.

_{1}demonstrated a steady-state error of 0.145 degree and e

_{2}, e

_{3}had zero estimation error after 0.2 s. The type-III and proposed ATO experienced an almost similar trend in the observed time span but estimation error from the proposed one had a relatively larger overshoot (8.2 × 10

^{−3}degree) than that of type-III ATO (6.1 × 10

^{−3}degree). However, the settling time was almost the same.

^{3}. e

_{1}of conventional ATO shows a large output error range, which presents a theoretically expected linearly increasing trend (at 5 s, e

_{1}= 2.18 degrees). e

_{2}and e

_{3}of Figure 7 present that although there is relatively large overshoot and longer settling time, when the systems have enough time to attain steady state, the proposed ATO will present zero position estimation error and type-III ATO will keep a position estimation error of around 7 × 10

^{−4}degree.

^{4}. As shown in Figure 8, compared with accumulative errors of 5.3 degrees and 0.0036 degree at 5 s from the conventional and type-III ATO, the proposed one only demonstrates a steady-state error of 7.1 × 10

^{−4}degree.

#### 4.2. Experimental Results

_{sin}and u

_{cos}). The signal acquisition board works as an analog-to-digital converter (ADC), which can sample the analog output signals and convert them into digital ones. The digital signals will be uploaded to the upper computer and applied with the same three ATO methods used in simulation results. It is noteworthy that, using the HIL method, the actual value of angular position can be known to evaluate error suppression performance of the RDC, whereas a practical resolver fails to provide this information [24].

_{1}, e

_{2}and e

_{3}are the position errors of the conventional ATO, type-III ATO and the proposed ATO, respectively. With time increasing, the conventional ATO has a linearly growing position error and the type-III ATO has a steady state position error of 4 × 10

^{−5}degree. And the position error of the proposed ATO is fluctuating around 0 degree, which indicates it has the ability to estimate angular position without error.

^{−4}degree to 6 × 10

^{−4}degree, respectively, and nearly zero error for the proposed ATO.

^{−5}degree. This result is better than conventional one with two orders of magnitude and the error is smaller than the type-III ATO, which ranges from 2.4 × 10

^{−5}degree to 2.8 × 10

^{−5}degree.

#### 4.3. Discussion

^{n}). Hence, we consider the compensation module, parameter design, accuracy improvement and time-saving implementation as points in our novelty.

## 5. Conclusions

^{n}). This ATO can eliminate position errors when n ≤ 3 and is effective at error reduction when n ≥ 4, which presents a solution for application in precisely estimating high-order acceleration signals. Theoretical analysis, simulation and experiments are carried out to verify and demonstrate the advantages of the proposed ATO over previous ones. We also evaluate the relationship between the parameters and bandwidth, noise level and error suppression of the ATO for succinct tuning guidance. Further research will be focused on improving the dynamic response of the proposed ATO.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Block diagram of a conventional phase-locked loop (PLL)-based angle tracking observer (ATO): (

**a**) actual structure; (

**b**) linearized structure.

**Figure 3.**Block diagram of a type-III ATO presented in [5].

**Figure 5.**(

**a**) Demonstration of relationship between the parameters in the proposed ATO and the system’s bandwidth. γ and k

_{P}are variable from 160 to 210 and from 120 to 155, respectively, with k

_{I}determined by ${k}_{I}={k}_{P}^{2}/(4\times {0.707}^{2})$. (

**b**) Amplitude–frequency response of the proposed ATO’s error transfer function with different bandwidths (ω

_{n}= 1,300,780,520 rad/s), corresponding to γ equal to 165,185,205 and k

_{P}= 141.4, k

_{I}= 100

^{2}. (

**c**) Relationship between the ATO’s bandwidth and parameter γ, with the other parameters determined by k

_{P}= γ − 23.6 and ${k}_{I}={k}_{P}^{2}/(4\times {0.707}^{2})$. The linearly fitted result corresponds well with this relationship.

**Figure 6.**Position errors of different ATOs in the simulation with a constant acceleration input (θ = 4πt

^{2}). e

_{1}, e

_{2}and e

_{3}are the position estimation errors of conventional ATO, type-III ATO in [5] and the proposed ATO, respectively.

**Figure 7.**Position errors of different ATOs in the simulation with a first-order acceleration input (θ = 4πt

^{3}). e

_{1}, e

_{2}and e

_{3}are the position estimation errors of conventional ATO, type-III ATO in [5] and the proposed ATO, respectively.

**Figure 8.**Position errors of different ATOs in the simulation with a second-order acceleration input (θ = πt

^{4}). e

_{1}, e

_{2}and e

_{3}are the position estimation errors of conventional ATO, type-III ATO in [5] and the proposed ATO, respectively.

**Figure 9.**Position errors of different ATOs in the simulation with a sine-wave input, θ = 2π + πsin(2πt). e

_{1}, e

_{2}and e

_{3}are the position estimation errors of conventional ATO, type-III ATO in [5] and the proposed ATO, respectively. (

**a**) Evolution of e

_{1}, e

_{2}and e

_{3}with a time span of 5 seconds; (

**b**) zoomed-in view of e

_{1}, e

_{2}and e

_{3}.

**Figure 11.**Demonstration of typical sine and cosine signals after envelope detection for $\theta =\frac{\pi}{15}{t}^{3}$, recorded from the signal acquisition board.

**Figure 12.**Experimental results of position estimation errors for the three kinds of ATOs at an acceleration signal of $\theta =\frac{\pi}{15}{t}^{3}$. e

_{1}, e

_{2}and e

_{3}are the estimation errors of the conventional ATO, type-III ATO and the proposed ATO, respectively. (

**a**) Errors’ evolution with a time span of 5 s. (

**b**) Zoomed-in view of e

_{1}, e

_{2}and e

_{3}.

**Figure 13.**Experimental results of position estimation errors for the three kinds of ATOs at an acceleration signal of θ=πt

^{3}. e

_{1}, e

_{2}and e

_{3}are the estimation errors of the conventional ATO, type-III ATO and the proposed ATO, respectively. (

**a**) Errors evolution with a time span of 5s. (

**b**) Zoomed-in view of e

_{1}, e

_{2}and e

_{3}.

**Figure 14.**Experimental results of position estimation errors for the three kinds of ATOs at an acceleration signal of $\theta =\frac{\pi}{60}{t}^{4}$. e

_{1}, e

_{2}and e

_{3}are the estimation errors of the conventional ATO, type-III ATO and the proposed ATO, respectively. (

**a**) Errors evolution with a time span of 5s. (

**b**) Zoomed-in view of e

_{1}, e

_{2}and e

_{3}.

**Figure 15.**Experimental results of position estimation errors for the three kinds of ATOs at an acceleration signal of θ=sin t. e

_{1}, e

_{2}and e

_{3}are the estimation errors of the conventional ATO, type-III ATO and the proposed ATO, respectively. (

**a**) Errors evolution with a time span of 5s. (

**b**) Zoomed-in view of e

_{1}, e

_{2}and e

_{3}.

Method | k_{I} | k_{P} | T | γ |
---|---|---|---|---|

Conventional ATO | 100^{2} | 141.4 | N/A | N/A |

Type-III ATO in [5] | 100^{2} | 141.4 | 0.0158 | N/A |

The proposed ATO | 100^{2} | 141.4 | N/A | 165 |

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**MDPI and ACS Style**

Qin, H.; Wu, Z.
Angle Tracking Observer with Improved Accuracy for Resolver-to-Digital Conversion. *Symmetry* **2019**, *11*, 1347.
https://doi.org/10.3390/sym11111347

**AMA Style**

Qin H, Wu Z.
Angle Tracking Observer with Improved Accuracy for Resolver-to-Digital Conversion. *Symmetry*. 2019; 11(11):1347.
https://doi.org/10.3390/sym11111347

**Chicago/Turabian Style**

Qin, Haoye, and Zhong Wu.
2019. "Angle Tracking Observer with Improved Accuracy for Resolver-to-Digital Conversion" *Symmetry* 11, no. 11: 1347.
https://doi.org/10.3390/sym11111347