Covariance Intersection Fusion Robust Steady-State Kalman Filter for Two-Sensor Systems with Time-Delayed Measurements

Abstract

For two-sensor systems with time-delayed measurements and uncertain noise variances, this paper presents a measurements transformation approach which transforms the systems with time-delayed measurements into the equivalent systems without measurement delays. Further the local robust steady-state Kalman filter with conservative upper bounds of unknown noise variances is presented, and then the covariance intersection (CI) fusion robust steady-state Kalman filter is also presented. The robustness of these filters is proved based on the Lyapunov equation. It is proved that the robust accuracy of the CI fuser is higher than that of each local robust Kalman filter. A Monte-Carlo simulation example shows its correctness and effectiveness.

1 Introduction

The multi-sensor information fusion has received great attentions and has been widely applied in many high-technology fields, such as tracking, signal proceeding, GPS position, robotics and so on. Usually, the standard systems without time-delayed observations are considered, but in many fields, such as the communication and control engineering, the systems with observations delays exist [1, 2].

The optimal Kalman filtering needs to exactly know system model and noise variances, the robust Kalman filters are designed to solve the filtering problems for uncertain systems. In recent years, several results have been derived for any admissible uncertainty of model parameters [3, 4] based on the Riccati equations.

Recently, the covariance intersection fusion (CI) method has been presented by Julier and Uhlman [5], which can handle the systems with unknown variances and cross-covariances.

In this paper, the two-sensor systems with uncertain noise variances and time-delayed measurements are considered. The local steady-state robust Kalman filter is presented and the covariance intersection (CI) fusion robust Kalman filter is proposed by the convex combination of the local robust Kalman filters. The robustness of the filters is proved based on the Lyapunov equation.

2 Measurement Transformation

Consider the two-sensor uncertain system with time-delayed measurements

$$ x\left( {t + 1} \right) = \Phi x\left( t \right) + \Gamma w\left( t \right) $$

(24.1)

$$ z_{i} \left( t \right) = H_{i} x\left( {t - k_{i} } \right) + e_{i} \left( t \right),i = 1,2 , $$

(24.2)

where $t$ is the discrete time, $x\left( t \right) \in R^{n}$ is the state, $z_{i} \left( t \right) \in R^{{m_{i} }}$ is the measurement of the $i{\text{th}}$ subsystem, $k_{i} \ge 0$ is the time-delay, $w\left( t \right) \in R^{r} ,e_{i} \left( t \right) \in R^{{m_{i} }}$ are uncorrelated white noises with zeros mean and unknown actual variances $\bar{Q}$ and $\bar{R}_{i}$, respectively. $\Phi$, $\Gamma$ and $H_{i}$ are known constant matrices. Assume that $Q$ and $R_{i}$ are conservative upper bounds of $\bar{Q}$ and $\bar{R}_{i}$, respectively, i.e.

$$ \bar{Q} \le Q,\bar{R}_{i} \le R_{i} ,i = 1,2 $$

(24.3)

where $A \le B$ means that $B - A \ge 0$ is a semi-positive definite matrix. Assume that each subsystem is completely observable and completely controllable.

Introducing the new measurements $y_{i} \left( t \right)$ and the measurement noises $v_{i} \left( t \right)$

$$ y_{i} \left( t \right) = z_{i} \left( {t + k_{i} } \right),v_{i} \left( t \right) = e_{i} \left( {t + k_{i} } \right),i = 1,2 $$

(24.4)

From (24.2), we have the observation equations without time-delayed

$$ y_{i} \left( t \right) = H_{i} x\left( t \right) + v_{i} \left( t \right),i = 1,2 $$

(24.5)

where $v_{i} \left( t \right)$ also has the variances $\bar{R}_{i}$. From (24.4), we have

$$ \hat{x}_{i}^{z} \left( {t|t} \right) = \hat{x}_{i} \left( {t|t - k_{i} } \right),i = 1,2 $$

(24.6)

where $\hat{x}_{i}^{z} \left( {t|t} \right)$ are the estimates of $x\left( t \right)$ based on $\left( {z_{i} \left( t \right),z_{i} \left( {t - 1} \right), \cdots } \right)\,\hat{x}_{i}^{{}} \left( {t|t - k_{i} } \right)$ are the estimates of $x\left( t \right)$ based on $\left( {y_{i} \left( {t - k_{i} } \right),y_{i} \left( {t - k_{i} - 1} \right), \cdots } \right)$.

Define the local steady-state cross-covariance as

$$ P_{ij}^{z} = E\left[ {\tilde{x}_{i}^{z} \left( {t|t} \right)\tilde{x}_{j}^{zT} \left( {t|t} \right)} \right],P_{ij}^{{}} \left( {k_{i} ,k_{j} } \right) = E\left[ {\tilde{x}_{i}^{{}} \left( {t|t - k_{i} } \right)\tilde{x}_{j}^{T} \left( {t|t - k_{j} } \right)} \right] $$

(24.7)

where $\tilde{x}_{i}^{z} \left( {t|t} \right) = x\left( t \right) - \hat{x}_{i}^{z} \left( {t|t} \right)$, $\tilde{x}_{i}^{{}} \left( {t|t - k_{i} } \right) = x\left( t \right) - \hat{x}_{i}^{{}} \left( {t|t - k_{i} } \right)$, from (24.6), we can get $P_{ij}^{z} = P_{ij} \left( {k_{i} ,k_{j} } \right)$. When $i = j$, defining $P_{i}^{z} = P_{ii}^{z}$, $P_{i} \left( {k_{i} } \right) = P_{ii} \left( {k_{i} ,k_{i} } \right)$, we have $P_{i}^{z} = P_{i} \left( {k_{i} } \right)$.

The problem is to find the local robust steady-state Kalman filter $x_{i}^{z} \left( {t|t} \right)$ and the CI fused robust steady-state Kalman filter $x_{CI}^{z} \left( {t|t} \right)$.

3 Local Robust Steady-State Kalman Filter

For two-sensor system (24.1) and (24.5), the local conservative steady-state Kalman one-step predictor with conservative variances $Q$ and $R_{i}$ are given by Sun and Deng [6], Kailath et al. [7]

$$ \hat{x}_{i} \left( {t + 1|t} \right) = \psi_{pi} \hat{x}_{i} \left( {t|t - 1} \right) + K_{pi} y_{i} \left( t \right) $$

(24.8)

$$ \psi_{pi} = \Phi - K_{pi} H_{i} ,K_{pi} = \Phi \Sigma_{i} H_{i}^{T} \left( {H_{i} \Sigma_{i} H_{i}^{T} + R_{i} } \right)^{ - 1} $$

(24.9)

where $\Psi_{pi}$ is a stable matrix and conservative one-step predictor error variance $\Sigma_{i}$ satisfies the steady-state Riccati equation

$$ \Sigma_{i} = \Phi \left[ {\Sigma_{i} - \Sigma_{i} H_{i}^{T} \left( {H_{i} \Sigma_{i} H_{i}^{T} + R_{i} } \right)^{ - 1} H_{i} \Sigma_{i} } \right]\Phi^{T} + \Gamma Q\Gamma^{T} $$

(24.10)

From (24.8), it can be rewritten as the Layapunov equation

$$ \Sigma_{i} = \psi_{pi} \Sigma_{i} \psi_{pi}^{T} + \Gamma Q\Gamma^{T} + K_{pi} R_{i} K_{pi}^{T} $$

(24.11)

Defining the actual steady-state one-step predictor error variance as

$$ \bar{\Sigma }_{i} = {\rm E}\left[ {\tilde{x}_{i} \left( {t + 1|t} \right)\tilde{x}_{i}^{\rm T} \left( {t + 1|t} \right)} \right],\,\tilde{x}_{i} \left( {t + 1|t} \right) = x\left( {t + 1} \right) - \hat{x}_{i} \left( {t + 1|t} \right) $$

(24.12)

Theorem 1

The Kalman one-step predictor (24.8)–(24.11) is robust for all admissible actual variances $\bar{Q}$ and $\bar{R}_{i}$ satisfying $\bar{Q} \le Q,\bar{R}_{i} \le R_{i}$, in the sense that

$$ \bar{\Sigma }_{i} \le \Sigma_{i} $$

(24.13)

Proof

From (24.1), we have $\hat{x}_{i} \left( {t + 1|t} \right) = \Phi \hat{x}_{i} \left( {t|t} \right)$, applying (24.12) yields $\tilde{x}_{i} \left( {t + 1|t} \right) = \Phi \tilde{x}_{i} \left( {t|t} \right) + \Gamma w\left( t \right)$, where $\tilde{x}_{i} \left( {t|t} \right) = \left[ {I_{n} - K_{fi} H} \right]\tilde{x}_{i} \left( {t|t - 1} \right) - K_{fi} v_{i} \left( t \right)$ and $K_{fi} = \Sigma_{i} H_{i}^{T} \left( {H_{i} \Sigma_{i} H_{i}^{T} + R_{i} } \right)^{ - 1}$, we have the actual prediction error formula

$$ \tilde{x}_{i} \left( {t + 1|t} \right) = \psi_{pi} \tilde{x}_{i} \left( {t|t - 1} \right) + \Gamma w\left( t \right) - K_{pi} v_{i} \left( t \right) $$

(24.14)

According to (24.12), applying (24.14) yields the actual steady-state one-step predictor error variance as

$$ \bar{\Sigma }_{i} = \psi_{pi} \bar{\Sigma }_{i} \psi_{pi}^{T} + \Gamma \bar{Q}\Gamma^{T} + K_{pi} \bar{R}_{i} K_{pi}^{T} $$

(24.15)

Defining $\Delta \Sigma_{i} = \Sigma_{i} - \bar{\Sigma }_{i}$, subtracting (24.15) from (24.11) yields the Lyapunov equation

$$ \Delta \Sigma_{i} = \psi_{pi} \Delta \Sigma_{i} \psi_{pi}^{T} + \Gamma \left( {Q - \bar{Q}} \right)\Gamma^{T} + K_{pi} \left( {R_{i} - \bar{R}_{i} } \right)K_{pi}^{T} $$

(24.16)

Applying (24.3), noting that $\psi_{pi}$ is a stable matrix, and applying the property of the Lyapunov equation [1] yield that $\Delta \Sigma_{i} \ge 0$, i.e. $\bar{\Sigma }_{i} \le \Sigma_{i} .\;\square$

For (24.1) and (24.5), the steady-state multi-step Kalman predictors are given by [6, 7]

$$ \hat{x}_{i} \left( {t + k_{i} |t} \right) = \Phi^{{k_{i} - 1}} \hat{x}_{i} \left( {t + 1|t} \right),k_{i} \ge 2 $$

(24.17)

The local steady-state multi-step predictor error variances are given as

$$ P_{i} \left( {k_{i} } \right) = \Phi^{{k_{i} - 1}} \Sigma_{i} \left( {\Phi^{{k_{i} - 1}} } \right)^{T} + \sum\limits_{j = 0}^{{k_{i} - 2}} {\Phi^{j} \Gamma Q\Gamma^{T} \left( {\Phi^{j} } \right)^{T} } ,k_{i} \ge 2 $$

(24.18)

Defining the actual steady-state multi-step predictor error variance as

$$ \bar{P}_{i} \left( {k_{i} } \right) = {\rm E}\left[ {\tilde{x}_{i} \left( {t + k_{i} |t} \right)\tilde{x}_{i}^{\rm T} \left( {t + k_{i} |t} \right)} \right],\,\tilde{x}_{i} \left( {t + k_{i} |t} \right) = x\left( {t + k_{i} } \right) - \hat{x}_{i} \left( {t + k_{i} |t} \right) $$

(24.19)

Theorem 2

The conservative Kalman multi-step predictor (24.17)–(24.18) is robust for all admissible actual variances $\bar{Q}$ and $\bar{R}_{i}$ satisfying $\bar{Q} \le Q,\bar{R}_{i} \le R_{i}$. i.e.

$$ \bar{P}_{i} \left( {k_{i} } \right) \le P_{i} \left( {k_{i} } \right) $$

(24.20)

Proof

Iterating $N - 1$ steps for (24.1), we obtain the non-recursive formula as

$$ x\left( {t + k_{i} } \right) = \Phi^{{k_{i} - 1}} x\left( {t + 1} \right) + \sum\limits_{j = 0}^{{k_{i} - 2}} {\Phi^{j} } \Gamma w\left( {t - 1} \right) $$

(24.21)

Substituting (24.17) and (24.21) into $\tilde{x}_{i} \left( {t + k_{i} |t} \right) = x\left( {t + k_{i} } \right) - \hat{x}_{i} \left( {t + k_{i} |t} \right)$, we have

$$ \tilde{x}_{i} \left( {t + k_{i} |t} \right) = \Phi^{{k_{i} - 1}} \tilde{x}\left( {t + 1|t} \right) + \sum\limits_{j = 0}^{{k_{i} - 2}} {\Phi^{j} \Gamma w\left( {t - 1} \right)} $$

(24.22)

Substituting (24.22) into (24.19) yields the actual steady-state filtering error variance as

$$ \bar{P}_{i} \left( {k_{i} } \right) = \Phi^{{k_{i} - 1}} \bar{\Sigma }_{i} \left( {\Phi^{{k_{i} - 1}} } \right)^{T} + \sum\limits_{j = 0}^{{k_{i} - 2}} {\Phi^{j} \Gamma \bar{Q}\Gamma^{T} \left( {\Phi^{j} } \right)^{T} } ,\,k_{i} \ge 2 $$

(24.23)

Defining $\Delta P_{i} \left( {k_{i} } \right) = P_{i} \left( {k_{i} } \right) - \bar{P}_{i} \left( {k_{i} } \right)$, subtracting (24.23) from (24.18) yields

$$ \Delta P_{i} \left( {k_{i} } \right) = \Phi^{{k_{i} - 1}} \left( {\Sigma_{i} - \bar{\Sigma }_{i} } \right)\left( {\Phi^{{k_{i} - 1}} } \right)^{T} + \sum\limits_{j = 0}^{{k_{i} - 2}} {\Phi^{j} \Gamma \left( {Q - \bar{Q}} \right)\Gamma^{T} \left( {\Phi^{j} } \right)^{T} } $$

(24.24)

Applying (24.3) and (24.13) yields $\Delta P_{i} \left( {k_{i} } \right) \ge 0$, (24.20) holds. $\square$

4 CI Fusion Robust Steady-State Kalman Filter

For two-sensor system (24.1) and (24.2), applying the CI fused algorithm [5], the CI fusion robust steady-state Kalman filters are given as

$$ \hat{x}_{CI}^{z} \left( {t|t} \right) = P_{CI}^{z} \left( {\omega \left( {P_{1}^{z} } \right)^{ - 1} \hat{x}_{1}^{z} \left( {t|t} \right) + \left( {1 - \omega } \right)\left( {P_{2}^{z} } \right)^{ - 1} \hat{x}_{2}^{z} \left( {t|t} \right)} \right) $$

(24.25)

$$ P_{CI}^{z} = \left[ {\omega \left( {P_{1}^{z} } \right)^{ - 1} + \left( {1 - \omega } \right)\left( {P_{1}^{z} } \right)^{ - 1} } \right]^{ - 1} $$

(24.26)

Applying (24.25), (24.26), (24.6) and (24.7) yields

$$ \hat{x}_{CI}^{z} \left( {t|t} \right) = P_{CI}^{z} \left( {\omega P_{1}^{ - 1} \left( {k_{1} } \right)\hat{x}_{1} \left( {t|t - k_{1} } \right) + \left( {1 - \omega } \right)P_{2}^{ - 1} \left( {k_{2} } \right)\hat{x}_{2} \left( {t|t - k_{2} } \right)} \right) $$

(24.27)

$$ P_{CI}^{z} = \left[ {\omega P_{1}^{ - 1} \left( {k_{1} } \right) + \left( {1 - \omega } \right)P_{2}^{ - 1} \left( {k_{2} } \right)} \right]^{ - 1} $$

(24.28)

with the constraint $\omega \ge 0$, when $k_{i} = 1$, we have $P_{1} \left( 1 \right) = \Sigma_{1} ,P_{2} \left( 1 \right) = \Sigma_{2}$.

The weighting coefficient $\omega$ is obtained by minimizing the performance index

$$ \mathop {\hbox{min} }\limits_{\omega } trP_{CI}^{z} = \mathop {\hbox{min} }\limits_{{\omega \in \left[ {0,1} \right]}} tr\left\{ {\left[ {\omega P_{1}^{ - 1} \left( {k_{1} } \right) + \left( {1 - \omega } \right)P_{2}^{ - 1} \left( {k_{2} } \right)} \right]^{ - 1} } \right\} $$

(24.29)

where the symbol ${\text{tr}}$ denotes the trace of matrix. The optimal weights $\omega$ can be quickly obtained by the 0.618 method or the Fibinacci method.

Theorem 3

The covariance intersection fused filter (24.27) and (24.28) has the actual error variance $\bar{P}_{CI}$ as

$$ \begin{aligned} \bar{P}_{CI}^{z} =\, & E\left[ {\tilde{x}_{CI}^{z} \left( {t|t} \right)\tilde{x}_{CI}^{zT} \left( {t|t} \right)} \right] \\ =\, & P_{CI}^{z} \left[ {\omega^{2} P_{1}^{ - 1} \left( {k_{1} } \right)\bar{P}_{1} \left( {k_{1} } \right)P_{1}^{ - 1} \left( {k_{1} } \right) + \omega \left( {1 - \omega } \right)P_{1}^{ - 1} \left( {k_{1} } \right)\bar{P}_{12} \left( {k_{1} ,k_{2} } \right)P_{2}^{ - 1} \left( {k_{2} } \right)} \right. \\ &\left. { +\omega \left( {1 - \omega } \right)P_{2}^{ - 1} \left( {k_{2} } \right)\bar{P}_{21} \left( {k_{2} ,k_{1} } \right)P_{1}^{ - 1} \left( {k_{1} } \right) + \left( {1 - \omega } \right)^{2} P_{2}^{ - 1} \left( {k_{2} } \right)\bar{P}_{2} \left( {k_{2} } \right)P_{2}^{ - 1} \left( {k_{2} } \right)} \right]P_{CI}^{z} \\ \end{aligned} $$

(24.30)

where $\bar{P}_{12}^{{}} \left( {k_{1} ,k_{2} } \right) = E\left[ {\tilde{x}_{1} \left( {t|t - k_{1} } \right)\tilde{x}_{2}^{T} \left( {t|t - k_{2} } \right)} \right]$ and

$$ \begin{aligned} \bar{P}_{12} \left( {k_{1} ,k_{2} } \right) = & \Phi^{{k_{1} - 1}} \psi_{p1}^{{k_{2} - k_{1} }} \bar{\Sigma }_{12} \left( {\Phi^{{k_{2} - 1}} } \right)^{T} \\ & + \sum\limits_{{r = k_{1} - 1}}^{{k_{2} - 2}} {\Phi^{{k_{1} - 1}} \psi_{pi}^{{k_{1} - r - 1}} \Gamma \bar{Q}\Gamma^{T} \left( {\Phi^{r} } \right)^{T} } + \sum\limits_{r = 0}^{{k_{1} - 2}} {\Phi^{r} \Gamma \bar{Q}\Gamma \left( {\Phi^{r} } \right)^{T} }^{{,k_{2} \ge k_{1} \ge 2}} \\ \end{aligned} $$

(24.31)

$$ \begin{aligned}\bar{P}_{12} \left( {k_{1} ,k_{2} } \right) &= \Phi^{{k_{1} - 1}} \bar{\Sigma }_{12} \psi_{p2}^{{k_{1} - k_{2} }} \left( {\Phi^{{k_{2} - 1}} } \right)^{T} \\ &\;\,+ \sum\limits_{{r = k_{2} - 1}}^{{k_{1} - 2}} {\Phi^{r} \Gamma \bar{Q}\Gamma^{T} \psi_{p2}^{{\left( {k_{2} - r - 1} \right)T}} \Psi^{{\left( {k_{2} - 1} \right)T}} } + \sum\limits_{r = 0}^{{k_{2} - 2}} {\Phi^{r} \Gamma \bar{Q}\Gamma \left( {\Phi^{r} } \right)^{T} }^{{,k_{1} \ge k_{2} \ge 2}} \\ \end{aligned} $$

(24.32)

Especially

$$ \bar{P}_{12} \left( {1,1} \right) = \bar{\Sigma }_{12} ,\bar{\Sigma }_{12} = \psi_{p1} \bar{\Sigma }_{12} \psi_{p2}^{T} + \Gamma \bar{Q}\Gamma^{T} $$

(24.33)

$$ \bar{P}_{12} \left( {k_{1} ,1} \right) = \Phi^{{k_{1} - 1}} \bar{\Sigma }_{12} \psi_{p2}^{{k_{1} - 1}} + \sum\limits_{r = 0}^{{k_{1} - 2}} {\Phi^{r} \Gamma \bar{Q}\Gamma^{T} \Psi_{p2}^{rT} } $$

(24.34)

$$ \bar{P}_{12} \left( {1,k_{2} } \right) = \psi_{p1}^{{k_{2} - 1}} \bar{\Sigma }_{12} \left( {\Phi^{{k_{2} - 1}} } \right)^{T} + \sum\limits_{r = 0}^{{k_{2} - 2}} {\psi_{pi}^{r} \Gamma \bar{Q}\Gamma^{T} \left( {\Phi^{r} } \right)^{T} } $$

(24.35)

Proof

From (24.28), we have $x\left( t \right) = P_{CI}^{z} \left[ {\omega P_{1}^{ - 1} \left( {k_{1} } \right) + \left( {1 - \omega } \right)P_{2}^{ - 1} \left( {k_{2} } \right)} \right]x\left( t \right)$. Using (24.27), we easily obtain the CI actual fused filtering error

$$ \tilde{x}_{CI}^{z} \left( {t|t} \right) = P_{CI}^{z} \left[ {\omega P_{1}^{ - 1} \left( {k_{1} } \right)\tilde{x}_{1} \left( {t|t - k_{1} } \right) + \left( {1 - \omega } \right)P_{2}^{ - 1} \left( {k_{2} } \right)\tilde{x}_{2} \left( {t|t - k_{2} } \right)} \right] $$

(24.36)

which yields (24.30). Equations (24.31)–(24.35) have been proved in Ref. [6]. $\square$

Remark 1

Applying (24.20), Ref. [5] proved that the two-sensor CI fuser is robust for all admissible $\bar{Q}$ and $\bar{R}_{i}$ satisfying (24.3), i.e.

$$ \bar{P}_{CI}^{z} \le P_{CI}^{z} $$

(24.37)

5 Accuracy Analysis

Theorem 4

For the two-sensor system (24.1)–(24.2) with time-delayed measurements, the local steady-state robust Kalman filter and CI fuser have the accuracy relations

$$ \bar{P}_{i}^{z} = \bar{P}_{i} \left( {k_{i} } \right),P_{i}^{z} = P_{i} \left( {k_{i} } \right) $$

(24.38)

$$ tr\bar{P}_{i}^{z} \le trP_{i}^{z} ,i = 1,2 $$

(24.39)

$$ tr\bar{P}_{CI}^{z} \le trP_{CI}^{z} \le trP_{i}^{z} ,i = 1,2 $$

(24.40)

Proof

From the robustness (24.20) the accuracy relation (24.39) holds. From (24.37), the first inequality of (24.40) holds. Applying (24.29), taking $\omega = 1$ yields $trP_{CI}^{z} = trP_{1}^{z}$ and $\omega = 0$ yields $trP_{CI}^{z} = trP_{2}^{z}$, Hence when $\omega \in \left[ {0,1} \right]$, we have the accuracy relation $trP_{CI}^{z} \le trP_{i}^{z} ,i = 1,2\;\square$.

Remark 2

Inequalities (24.39) and (24.40) show that the robust accuracy of the CI fuser is higher than that of each local robust filter.

6 Simulation Example

Consider the two-sensor tracking system (24.1)–(24.2)with time-delayed measurements, where $\Phi = \left[ {\begin{array}{*{20}c} 1 & {T_{0} } \\ 0 & 1 \\ \end{array} } \right],\Gamma = \left[ {\begin{array}{*{20}c} {0.5{\rm T}_{0}^{2} } \\ {T_{0} } \\ \end{array} } \right],\,H_{1} = \left[ {\begin{array}{*{20}c} 1 & 0 \\ \end{array} } \right],\,H_{2} = I_{2} ,\,T_{0} = 0.25$ is the sampled period, $x\left( t \right) = \left[ {x_{1} \left( t \right),x_{2} \left( t \right)} \right]^{T}$ is the state, $x_{1} \left( t \right)$ and $x_{2} \left( t \right)$ are the position and velocity of target at time $tT_{0}$. $w\left( t \right)$ and $v_{i} \left( t \right)$ are independent Gaussion white noises with zero mean and unknown variances $Q$ and $R_{i}$ respectively. In the simulation, we take $Q = 0.5,\;R_{1} = 0.58,\;R_{2} = diag\left( {4,0.25} \right),\;\bar{Q} = 0.45,\;\bar{R}_{1} = 0.5,\;\bar{R}_{2} = diag\left( {3,\;0.16} \right),\;k_{1} = 1,\;k_{2} = 2$

In order to give a geometric interpretation of the accuracy relations, the covariance ellipse is defined as the locus of points $\left\{ {x:x^{T} P^{ - 1} x = c} \right\}$, where $P$ is the variance matrix and $c$ is a constant. Generally, we select $c = 1$. It has been proved in [8] that $P_{1} \le P_{2}$ is equivalent to that the covariance ellipse of $P_{1}$ is enclosed in that of $P_{2}$. The accuracy comparison of the covariance ellipses is shown in Fig. 24.1. From Fig. 24.1, we see that the ellipse of the actual variances $\bar{\Sigma }_{1} or\bar{P}_{2} \left( 2 \right)$ is enclosed in that of $\Sigma_{1} orP_{2} \left( 2 \right)$, respectively, which verify the consistent (24.13) and (24.20). The ellipse of actual CI fused variance $\bar{P}_{CI}$ is enclosed in that of $P_{CI}$, which verifies the robustness of (24.37).

Fig. 24.1

The covariance ellipses of robust Kalman filters

In order to verify the above theoretical accuracy relations, taking $\rho = 200$ runs, the curves of the mean square errors (MSE) of local and fused Kalman filters are shown in Fig. 24.2, which verifies the accuracy relations (24.39), (24.40) and the accuracy relations in Table 24.1.

Fig. 24.2

The MSE curves of local and fused filters

Table 24.1 The accuracy comparison of local and fused filters

7 Conclusion

For two-sensor systems with uncertain noise variances and time-delayed measurements, the local and CI robust fused robust steady-state Kalman filters have been presented, and their robustness was proved based on the Lyapunov equation. The robust accuracy of CI fuser is higher than that the robust accuracy of each local filter