Multiple reference consistency check for LAAS: a novel position domain approach

Abstract

Since the traditional Maximum Likelihood-based range domain multiple reference consistency check (MRCC) has limitations in satisfying the integrity requirement of CAT II/III for civil aviation, a Kalman filter-based position domain method has been developed for fault detection and exclusion in the Local Area Augmentation System MRCC process. The position domain method developed in this paper seeks to address the limitations of range domain-based MRCC by focusing not only on improving the performance of the fault detection but also on the integrity risk requirement for MRCC. In addition, the issue of the stability of the Kalman filter in relation to the position domain approach is considered. GPS range corrections from multiple reference receivers are fused by the adaptive Kalman filter at the master station for detecting and excluding the single reference receiver’ failure. The performance of the developed Kalman filter-based MRCC has been compared with the traditional method using experimental data. The results reveal that the vertical protection level is slightly better in the traditional method compared with the developed Kalman filter-based approach under the fault-free case. However, the availability can be improved to over 97% in the proposed method relative to the traditional method under the single-fault case. Furthermore, the fault-tolerant positioning result with an accuracy improvement of more than 32% can be achieved even if different fault types are considered under the single-fault case. In particular, the algorithm can be a candidate option as an augmentable complement for the traditional MRCC and can be implemented in a master station element of the LAAS integrity monitoring architecture.

Appendices

Appendix 1

The mathematical induction method is used to prove $\user2{P}_{{{{\rm v}}_{0,i} ,k}} = \user2{P}_{{{{\rm v}}_{0} ,k}}$. For k = 0, we have $\user2{P}_{{{{\rm v}}_{0,i} ,0}} = \user2{P}_{{{{\rm v}}_{0} ,0}} = {\bar{\varvec P}}_{0}$, which is satisfied by (12). Assume $\user2{P}_{{{{\rm v}}_{0,i} ,l - 1}} = \user2{P}_{{{{\rm v}}_{0} ,l - 1}}$ is true for some positive integer l. Then for k = 1, we have

$$ \begin{aligned} {\varvec{\Uppsi}}_{{{{\rm v}}_{0} ,l\left| {l - 1} \right.}} \user2{P}_{{{{\rm v}}_{0,i} ,l - 1}} {\varvec{\Uppsi}}_{{{{\rm v}}_{i} ,l\left| {l - 1} \right.}}^{\text{T} } + \user2{W}_{{{{\rm v}}_{0} ,l}} \user2{Q}_{l} \user2{W}_{{{{\rm v}}_{i} ,l}}^{\text{T} } \\ = & \left( {\user2{I} - \user2{K}_{{{{\rm v}}_{0} ,l}} \user2{H}_{{{{\rm v}}_{0} ,l}} } \right){\varvec{\Upphi}}_{{l\left| {l - 1} \right.}} \user2{P}_{{{{\rm v}}_{0,i} ,l - 1}} {\varvec{\Upphi}}_{{l\left| {l - 1} \right.}}^{\text{T} } \left( {\user2{I} - \user2{K}_{{{{\rm v}}_{i} ,l}} \user2{H}_{{{{\rm v}}_{i} ,l}} } \right)^{\text{T} } + \left( {\user2{I} - \user2{K}_{{{{\rm v}}_{0} ,l}} \user2{H}_{{{{\rm v}}_{0} ,l}} } \right)\user2{Q}_{l} \left( {\user2{I} - \user2{K}_{{{{\rm v}}_{i} ,l}} \user2{H}_{{{{\rm v}}_{i} ,l}} } \right)^{\text{T} } \\ = & \left( {\user2{I} - \user2{K}_{{{{\rm v}}_{0} ,l}} \user2{H}_{{{{\rm v}}_{0} ,l}} } \right)\left( {{\varvec{\Upphi}}_{{l\left| {l - 1} \right.}} \user2{P}_{{{{\rm v}}_{0,i} ,l - 1}} {\varvec{\Upphi}}_{{l\left| {l - 1} \right.}}^{\text{T} } + \user2{Q}_{l} } \right)\left( {\user2{I} - \user2{K}_{{{{\rm v}}_{i} ,l}} \user2{H}_{{{{\rm v}}_{i} ,l}} } \right)^{\text{T} } \\ \end{aligned} $$

(25)

Substitute (14) and the predetermined assumption $\user2{P}_{{{{\rm v}}_{0,i} ,l - 1}} = \user2{P}_{{{{\rm v}}_{0} ,l - 1}}$ into (25), we have:

$$ {\varvec{\Uppsi}}_{{{{\rm v}}_{0} ,l\left| {l - 1} \right.}} \user2{P}_{{{{\rm v}}_{0,i} ,l - 1}} {\varvec{\Uppsi}}_{{{{\rm v}}_{i} ,l\left| {l - 1} \right.}}^{\text{T} } + \user2{W}_{{{{\rm v}}_{0} ,l}} \user2{Q}_{l} \user2{W}_{{{{\rm v}}_{i} ,l}}^{\text{T} } = \user2{P}_{{{{\rm v}}_{0} ,l}} \left( {\user2{I} - \user2{K}_{{{{\rm v}}_{i} ,l}} \user2{H}_{{{{\rm v}}_{i} ,l}} } \right)^{\text{T} } $$

(26)

According to the Kalman filter theory, the following can be obtained:

$$ \user2{K}_{{{{\rm v}}_{0} ,l}} = \user2{P}_{{{{\rm v}}_{0} ,l}} \user2{H}_{{{{\rm v}}_{0} ,l}}^{\text{T} } \user2{R}_{{{{\rm v}}_{0} ,l}}^{ - 1} $$

(27)

and

$$ \user2{K}_{{{{\rm v}}_{0} ,l}} \user2{R}_{{{{\rm v}}_{0,i} l}} \user2{K}_{{{{\rm v}}_{i} ,l}}^{\text{T} } = \user2{P}_{{{{\rm v}}_{0} ,l}} \user2{H}_{{{{\rm v}}_{0} ,l}}^{\text{T} } \user2{R}_{{{{\rm v}}_{0} ,l}}^{ - 1} \user2{R}_{{{{\rm v}}_{0,i} l}} \user2{K}_{{{{\rm v}}_{i} ,l}}^{\text{T} } = \user2{P}_{{{{\rm v}}_{0} ,l}} \user2{H}_{{{{\rm v}}_{0} ,l}}^{\text{T} } \user2{E}_{i} \user2{K}_{{{{\rm v}}_{i} ,l}}^{\text{T} } = \user2{P}_{{{{\rm v}}_{0} ,l}} \user2{H}_{{{{\rm v}}_{i} ,l}}^{\text{T} } \user2{K}_{{{v}_{i} ,l}}^{\text{T} } $$

(28)

Substitute (28) and (26) into (15), we have

$$ \user2{P}_{{{{\rm v}}_{0,i} ,l}} = \user2{P}_{{{{\rm v}}_{0} ,l}} \left( {\user2{I} - \user2{K}_{{{{\rm v}}_{i} ,l}} \user2{H}_{{{{\rm v}}_{i} ,l}} } \right)^{\text{T} } +\,\user2{P}_{{{{\rm v}}_{0} ,l}} \user2{H}_{{{{\rm v}}_{i} ,l}}^{\text{T} } \user2{K}_{{{{\rm v}}_{i} ,l}}^{\text{T} } = \user2{P}_{{{{\rm v}}_{0} ,l}} $$

Hence $\user2{P}_{{{{\rm v}}_{0,i} ,l}} = \user2{P}_{{{{\rm v}}_{0} ,l}}$ is satisfied for k = 1.

Appendix 2

Begin with (23),

$$ \begin{aligned} \frac{{{\text{d}}I_{{\rm v}} }}{{{\text{d}}\alpha_{{{{\rm v}}_{i} }} }} = & \frac{\text{d}}{{{\text{d}}\alpha_{{{{\rm v}}_{i} }} }}\left\{ {\sum\limits_{i = 0}^{3} {P({{\rm v}}_{i} )\left[ {\int\limits_{{{\hat{\varvec x}}_{{{\rm v}}} + {\text{VAL}}}}^{\infty } {N_{x} \left( {{\hat{\varvec x}}_{{{{\rm v}}_{i} }} ,\frac{1}{{\alpha_{{{{{\rm v}}}_{i} }}^{2} }}\gamma_{i}^{2} \user2{P}_{{{{{\rm v}}}_{i} }} } \right){\text{d}}{\tilde{\varvec x}}_{{{\rm v}}} } + \int\limits_{ - \infty }^{{{\hat{\varvec x}}_{{{\rm v}}} - {\text{VAL}}}} {N_{x} \left( {{\hat{\varvec x}}_{{{{{\rm v}}}_{i} }} ,\frac{1}{{\alpha_{{{{{\rm v}}}_{i} }}^{2} }}\gamma_{i}^{2} \user2{P}_{{{{{\rm v}}}_{i} }} } \right){\text{d}}{\tilde{\varvec x}}_{{\rm v}} } } \right]} } \right\} \\ = & \frac{\text{d}}{{{\text{d}}\alpha_{{{{{\rm v}}}_{i} }} }}\left\{ {\sum\limits_{i = 0}^{3} {P({{\rm v}}_{i} )\left. {\left[ {\int\limits_{\text{VAL}}^{\infty } {N_{s} \left( {{\hat{\varvec x}}_{{{{{\rm v}}}_{i} }} - {\hat{\varvec x}}_{{{\rm v}}} ,\frac{1}{{\alpha_{{{{{\rm v}}}_{i} }}^{2} }}\gamma_{i}^{2} \user2{P}_{{{{{\rm v}}}_{i} }} } \right){\text{d}}s} + \int\limits_{ - \infty }^{{ - {\text{VAL}}}} {N_{s} \left( {{\hat{\varvec x}}_{{{{v}}_{i} }} - {\hat{\varvec x}}_{{{\rm v}}} ,\frac{1}{{\alpha_{{{{{\rm v}}}_{i} }}^{2} }}\gamma_{i}^{2} \user2{P}_{{{{{\rm v}}}_{i} }} } \right){\text{d}}s} } \right]} \right|_{{s = {\tilde{\varvec x}}_{{{\rm v}}} - {\hat{\varvec x}}_{{{\rm v}}} }} } } \right\} \\ = & \sum\limits_{i = 0}^{3} {P({{\rm v}}_{i} )\frac{\text{d}}{{{\text{d}}\alpha_{{{{{\rm v}}}_{i} }} }}\left\{ {\frac{1}{{\alpha_{{{{{\rm v}}}_{i} }} }}\left. {\left[ {\int\limits_{{\alpha_{{{{{\rm v}}}_{i} }} {\text{VAL}}}}^{\infty } {N_{u} \left( { - \alpha_{{{{{\rm v}}}_{i} }} \beta_{{{{{\rm v}}}_{i} }} ,\gamma_{i}^{2} \user2{P}_{{{{{\rm v}}}_{i} }} } \right){\text{d}}u} + \int\limits_{{\alpha_{{{{\rm v}}_{i} }} {\text{VAL}}}}^{\infty } {N_{u} \left( {\alpha_{{{{{\rm v}}}_{i} }} \beta_{{{{{\rm v}}}_{i} }} ,\gamma_{i}^{2} \user2{P}_{{{{{\rm v}}}_{i} }} } \right){\text{d}}u} } \right]} \right|_{{u = s \cdot \alpha_{{{{\rm v}}_{i} }} }} } \right\}} \\ = & \sum\limits_{i = 0}^{3} {P({{\rm v}}_{i} )\left\{ {\frac{1}{{\alpha_{{{{\rm v}}_{i} }} }}\left[ {0 - \left. {{\text{VAL}} \cdot N_{u} \left( { - \alpha_{{{{{\rm v}}}_{i} }} \beta_{{{{{\rm v}}}_{i} }} ,\gamma_{i}^{2} \user2{P}_{{{{{\rm v}}}_{i} }} } \right)} \right|_{{u = \alpha_{{{{\rm v}}_{i} }} \cdot {\text{VAL}}}} + 0 - \left. {{\text{VAL}} \cdot N_{u} \left( {\alpha_{{{{{\rm v}}}_{i} }} \beta_{{{{{\rm v}}}_{i} }} ,\gamma_{i}^{2} \user2{P}_{{{{{\rm v}}}_{i} }} } \right)} \right|_{{u = \alpha_{{{{{\rm v}}}_{i} }} \cdot {\text{VAL}}}} } \right]} \right.} - \left. {\frac{1}{{\alpha_{{{{{\rm v}}}_{i} }}^{2} }}\left[ {\int\limits_{{\alpha_{{{{{\rm v}}}_{i} }} \cdot {\text{VAL}}}}^{\infty } {N_{u} \left( { - \alpha_{{{{{\rm v}}}_{i} }} \beta_{{{{{\rm v}}}_{i} }} ,\gamma_{i}^{2} \user2{P}_{{{{{\rm v}}}_{i} }} } \right){\text{d}}u} + \int\limits_{{\alpha_{{{{\rm v}}_{i} }} \cdot {\text{VAL}}}}^{\infty } {N_{u} \left( {\alpha_{{{{{\rm v}}}_{i} }} \beta_{{{{{\rm v}}}_{i} }} ,\gamma_{i}^{2} \user2{P}_{{{{{\rm v}}}_{i} }} } \right){\text{d}}u} } \right]} \right\} \\ = & \sum\limits_{i = 0}^{3} {P({{\rm v}}_{i} )\left\{ {\frac{\text{VAL}}{{\alpha_{{{{{\rm v}}}_{i} }} }} \cdot \left[ {\left. {N_{u} \left( { - \alpha_{{{{{\rm v}}}_{i} }} \beta_{{{{{\rm v}}}_{i} }} ,\gamma_{i}^{2} \user2{P}_{{{{{\rm v}}}_{i} }} } \right)} \right|_{{u = \alpha_{{{{{\rm v}}}_{i} }} \cdot {\text{VAL}}}} + \left. {N_{u} \left( {\alpha_{{{{{\rm v}}}_{i} }} \beta_{{{{{\rm v}}}_{i} }} ,\gamma_{i}^{2} \user2{P}_{{{{{\rm v}}}_{i} }} } \right)} \right|_{{u = \alpha_{{{{{\rm v}}}_{i} }} \cdot {\text{VAL}}}} } \right]} \right.} \left. { + \frac{1}{{2\alpha_{{{{{\rm v}}}_{i} }}^{2} }}\left[ {{\text{erfc}}\left( {\frac{{\alpha_{{{{{\rm v}}}_{i} }} {\text{VAL}} + \alpha_{{{{{\rm v}}}_{i} }} \beta_{{{{{\rm v}}}_{i} }} }}{{\gamma_{i} \sqrt {2\user2{P}_{{{{{\rm v}}}_{i} }} } }}} \right) + {\text{erfc}}\left( {\frac{{\alpha_{{{{{\rm v}}}_{i} }} {\text{VAL}} - \alpha_{{{{{\rm v}}}_{i} }} \beta_{{{{{\rm v}}}_{i} }} }}{{\gamma_{i} \sqrt {2\user2{P}_{{{{\rm v}}_{i} }} } }}} \right)} \right]} \right\} \\ = & 0 \\ \end{aligned} $$

which is equal to (24). In which $\text{erfc} (x) = \frac{1}{\sqrt \pi }\int_{x}^{\infty } {\exp \left( { - t^{2} } \right)\text{d} t}$ is a built-in function in Matlab.