# Performance Analysis and Design Strategy for a Second-Order, Fixed-Gain, Position-Velocity-Measured (α-β-η-θ) Tracking Filter

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## Abstract

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## Featured Application

**Design and evaluation of monitoring systems in intelligent vehicles, robots, and so on.**

## Abstract

## 1. Introduction

- Simple implementation and low computational overhead: Optimal gain calculation is not required in the fixed-gain filters. Thus, the number of matrix operations is small compared with the Kalman filter and its variants [11].
- Applicability to the analytical evaluation of the Kalman filter: Fixed-gain filters are also useful for analytical evaluations of the Kalman filter because they can be characterized as steady state Kalman filters [12].

## 2. Definitions of Problem and Symbols

## 3. The $\alpha $-$\beta $-$\eta $-$\theta $ Filter

#### 3.1. Definition

#### 3.2. The $\alpha $-$\beta $ Filter

#### 3.3. Relationship to Kalman Filters

**z**is a measurement vector,

**F**is the transition matrix, ${\mathit{P}}_{k}$ is the error covariance matrix at time $kT$,

**Q**is the covariance matrix for process noise, ${\mathit{K}}_{k}$ is the optimal gain (Kalman gain) at time $kT$ and

**B**is the covariance matrix for the measurement noise.

#### 3.4. Optimal Filter for a Random-Acceleration Model and Its Problems

**Q**expressed as in [10]:

**Q**, which does not include correlations in process noise [1,2]. Other process noise can be incorporated using arbitrary process noise; see [4]. The performance of this $\alpha $-$\beta $-$\eta $-$\theta $ filter was evaluated in [31] only in terms of several simple numerical calculations, and strategies for designing tracking indices were not discussed. These problems must be solved to establish a design strategy and to properly evaluate the filter’s performance.

## 4. Derivation of Performance Indices and Stability Conditions

#### 4.1. Smoothing Performance Index

#### 4.2. Tracking Performance Index

#### 4.3. RMS Index

#### 4.4. Stability Condition

## 5. Optimal Gain Design Strategy

#### 5.1. Optimal Gain Design Using the RMS Index

- The selection of an appropriate model (e.g., RA, random-velocity) is not considered. Thus, this selection is conducted empirically [4].

#### 5.2. Procedure and Notes of the Proposed Strategy

- Set ${R}_{\mathrm{xv}}$ from the sensor performance.
- Design ${a}_{\mathrm{D}}$ based on the approximate target acceleration.
- Determine $\alpha $, $\beta $ and $\theta $ by solving Equation (32).
- Determine $\eta $ with Equation (17).

- Equation (32) can be solved by simple gradient descent with several initial values [33]. This is because the range of parameter searching is not so wide due to the stability conditions.
- This design process is conducted only once before using the filter. Although the computational costs of the above optimization process are not small, this does not affect the simple tracking process of the $\alpha $-$\beta $-$\eta $-$\theta $ filters.

#### 5.3. Relationship with Steady State PVM Kalman Filters

**Q**, which is also a well-used setting in real applications [1,2].

## 6. Steady State Performance Analysis

- Proposed filter: the $\alpha $-$\beta $-$\eta $-$\theta $ filter with the proposed strategy.
- RA filter: the $\alpha $-$\beta $-$\eta $-$\theta $ filter with the RA model using optimal q (from Equation (16)) with respect to the RMS index.
- Best $\alpha $-$\beta $ filter: the conventional $\alpha $-$\beta $ filter obtained with the proposed strategy, assuming $\eta =\theta =0$.

#### 6.1. Relationship between Performance and ${a}_{\mathrm{D}}$

#### 6.2. Relationship between Performance and ${R}_{\mathrm{xv}}$

## 7. Application to UWB Doppler Radar Simulation

- Medium maneuvering target assuming simple near-field sensing.
- High maneuvering target assuming the target executes an abrupt motion.

#### 7.1. Tracking of Medium Maneuvering Target

#### 7.1.1. Simulation Setup

#### 7.1.2. Filter Design

#### 7.1.3. Evaluation Results

#### 7.2. Tracking of High-Maneuvering Target

## 8. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

UWB | Ultra-wideband |

PVM | Position-velocity-measured |

RA | Random-acceleration |

RMS | Root-mean-square |

## Appendix A. Derivation of Equation (20)

## Appendix B. Derivation of Equation (26)

## Appendix C. Derivation of Equations (34)–(36)

**P**are denoted as ${P}^{i,j}$. With Equations (11) and (33), $\tilde{\mathit{P}}$ is calculated as:

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**Figure 1.**Relationship between ${a}_{\mathrm{D}}$ and ${\u03f5}_{\mathrm{rms}}$ for (

**a**) ${R}_{\mathrm{xv}}=1$ and (

**b**) ${R}_{\mathrm{xv}}=10$; RA: random-acceleration.

**Figure 2.**Relationship between ${R}_{\mathrm{xv}}$ and ${\u03f5}_{\mathrm{rms}}$ for (

**a**) ${a}_{\mathrm{D}}^{2}=0.01$ and (

**b**) ${a}_{\mathrm{D}}^{2}=0.1$.

**Figure 3.**UWB Doppler radar simulation scenario: (

**a**) radar positions and true orbit; (

**b**) true acceleration; UWB: ultra-wideband.

**Figure 4.**Simulation results for (

**a**) ${R}_{\mathrm{xv}}=9$ and (

**b**) ${R}_{\mathrm{xv}}=1$; PVM: position-velocity-measured.

Variables | Description | Unit |
---|---|---|

T | Sampling interval | (s) |

k | Discrete sampling index | Dimensionless |

${\left(\right)}_{,k}$ | Parameter at index k | |

${x}_{\mathrm{p}}$ | Predicted position | (m) |

${v}_{\mathrm{p}}$ | Predicted velocity | (m/s) |

${x}_{\mathrm{s}}$ | Smoothed (estimated) position | (m) |

${v}_{\mathrm{s}}$ | Smoothed (estimated) velocity | (m/s) |

${x}_{\mathrm{o}}$ | Observed (measured) position | (m) |

${v}_{\mathrm{o}}$ | Observed (measured) velocity | (m/s) |

$\alpha $ | Filter gain for ${x}_{\mathrm{s}}$ with respect to ${x}_{\mathrm{o}}$ | Dimensionless |

$\beta $ | Filter gain for ${v}_{\mathrm{s}}$ with respect to ${x}_{\mathrm{o}}$ | Dimensionless |

$\eta $ | Filter gain for ${x}_{\mathrm{s}}$ with respect to ${v}_{\mathrm{o}}$ | Dimensionless |

$\theta $ | Filter gain for ${v}_{\mathrm{s}}$ with respect to ${v}_{\mathrm{o}}$ | Dimensionless |

$\tilde{\left(\right)}$ | Forecasts | |

$\widehat{\left(\right)}$ | Estimates | |

${\left(\right)}^{\mathrm{T}}$ | Transpose of matrix | |

${\left(\right)}^{-1}$ | Inversion of matrix | |

$\mathit{x}$ | State vector of target composed of position and velocity | |

$\mathit{z}$ | Measurement vector | |

$\mathit{F}$ | Transition matrix from k to $k+1$ | |

$\mathit{P}$ | Error covariance matrix with respect to $x$ | |

$\mathit{Q}$ | Covariance matrix of process noise | |

$\mathit{K}$ | Kalman gain matrix | |

$\mathit{B}$ | Covariance matrix of measurement noise | |

${B}_{\mathrm{x}}$ | Error variance of ${x}_{\mathrm{o}}$ | (m${}^{2}$) |

${B}_{\mathrm{v}}$ | Error variance of ${v}_{\mathrm{o}}$ | (m${}^{2}$/s${}^{2}$) |

${\mathit{Q}}_{\mathrm{ra}}$ | Process noise matrix in random-acceleration (RA) model | |

q | Variance of random-acceleration (RA) process noise | (m${}^{2}$/s${}^{4}$) |

${R}_{\mathrm{xv}}$ | Ratio of ${B}_{\mathrm{x}}$ to ${T}^{2}{B}_{\mathrm{v}}$ | Dimensionless |

E$\left(\right)$ | Mean with respect to k | |

${\sigma}_{\mathrm{p}}^{2}$ | Smoothing performance index | (m${}^{2}$) |

${e}_{\mathrm{fin}}$ | Tracking performance index | (m) |

${a}_{\mathrm{c}}$ | Acceleration assumed in the derivation of ${e}_{\mathrm{fin}}$ | (m/s${}^{2}$) |

${\u03f5}_{\mathrm{rms}}$ | Root-mean-square (RMS) index | (m) |

$\mu $ | Evaluating function in the proposed gain design strategy | Dimensionless |

${a}_{\mathrm{D}}$ | Design parameter for the proposed strategy | Dimensionless |

${\mathit{Q}}_{\mathrm{gen}}$ | Arbitrary process noise matrix | |

a | (1,1) element of ${\mathit{Q}}_{\mathrm{gen}}$ | (m${}^{2}$) |

b | (1,2) (or (2,1)) element of ${\mathit{Q}}_{\mathrm{gen}}$ | (m${}^{2}$/s) |

c | (2,2) element of ${\mathit{Q}}_{\mathrm{gen}}$ | (m${}^{2}$/s${}^{2}$) |

$\u03f5$ | RMS prediction error of Monte Carlo simulations | (m) |

Tracking Filter | Input | Design Strategy | Preset Parameter | $\mathit{k}\to \mathit{\infty}$ |
---|---|---|---|---|

$\alpha $-$\beta $ filter | Position | Based on RA model [12] | q or tracking index [10] | Kalman filter |

Proposed strategy | ${a}_{\mathrm{D}}$ of Equation (31) | |||

$\alpha $-$\beta $-$\eta $-$\theta $ filter | Position | Based on RA model [27] | q or tracking index [10] | Position-velocity-measured |

and velocity | Proposed strategy | ${a}_{\mathrm{D}}$ of Equation (31) | (PVM) Kalman filter |

**Table 3.**Steady state RMS prediction error of the proposed filter for various ${a}_{\mathrm{c}}$ (${R}_{\mathrm{xv}}=9$).

${\mathit{a}}_{\mathbf{c}}$ (m/s${}^{2}$) | ${\mathit{a}}_{\mathbf{d}}^{2}$ | Mean Steady State RMS Error (cm) |
---|---|---|

0.1 | 0.00111 | 2.82 |

0.4 | 0.0178 | 2.32 |

0.6 | 0.040 | 2.33 |

1.0 | 0.111 | 2.57 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Saho, K.; Masugi, M. Performance Analysis and Design Strategy for a Second-Order, Fixed-Gain, Position-Velocity-Measured (*α*-*β*-*η*-*θ*) Tracking Filter. *Appl. Sci.* **2017**, *7*, 758.
https://doi.org/10.3390/app7080758

**AMA Style**

Saho K, Masugi M. Performance Analysis and Design Strategy for a Second-Order, Fixed-Gain, Position-Velocity-Measured (*α*-*β*-*η*-*θ*) Tracking Filter. *Applied Sciences*. 2017; 7(8):758.
https://doi.org/10.3390/app7080758

**Chicago/Turabian Style**

Saho, Kenshi, and Masao Masugi. 2017. "Performance Analysis and Design Strategy for a Second-Order, Fixed-Gain, Position-Velocity-Measured (*α*-*β*-*η*-*θ*) Tracking Filter" *Applied Sciences* 7, no. 8: 758.
https://doi.org/10.3390/app7080758