2. State Key Laboratory of Fire Science, Department of Automation and Institute of Advanced Technology, University of Science and Technology of China, Hefei 230027, China;
3. Key Laboratory of Technology in GeoSpatial Information Processing and Application System, Chinese Academy of Sciences, Beijing 100190, China;
4. College of Information Engineering, Zhejiang University of Technology, Hangzhou 310023, China
Due to its conciseness which may facilitate solutions, the linear Gaussian system is a preferable model to describe many physical systems, especially in engineering field such as target tracking, robot navigation, and fault diagnosis [1]. The Kalman filter has been proved to be an optimal filter for linear Gaussian systems for the filtering problem [2]. Furthermore, based on the concept of the conventional Kalman filter, a variety of improved Kalman filters have been investigated to solve different practical problems [3][8]. This paper is devoted to solve the optimal filtering problem for a class of linear Gaussian systems where the system states are updated at a fast uniform sampling rate and the measurements are sampled at a slow uniform sampling rate. The updating rate of system states is several times the sampling rate of measurements and the multiple is constant. In other words, there is currently no available measurement methods for systems whose states update during the period between two adjacent measurements. Therefore, the conventional Kalman filter is invalid for this case due to the asynchronism between system states updating and measurements sampling.
The proposed problem in this paper can be categorized as either a filtering problem of a special class of systems suffering the measurement dropouts or delays, or a multirate information fusion problem. The distributed
The proposed filter has a typical application in wheeled robot navigation systems. The external references are often used to correct the navigation system by means of global positioning system (GPS) or the other approaches which are timeconsuming [19][21], such that they will suffer from the multirate issue. Another application of the proposed filter can be found in the air quality evaluation systems to estimate the street air pollutant concentration in real time, which is of great significance due to its dominance for government to make more effective control strategies to improve the atmospheric environment. The conventional method termed as modelbased method and the emerging technology called mobilesensing method can be combined to improve the performance based on their complementary characteristics, but in an asynchronous way [22][24]. In this paper, we consider the application of the proposed filter design for these two practical systems.
The rest of the paper is organized as follows. Section Ⅱ formulates the problem. Section Ⅲ discusses the solution in detail. Section Ⅳ is composed of two case studies, i.e., the wheeled robot navigation system and the air quality evaluation system. The paper is concluded in Section Ⅴ.
Ⅱ. PROBLEM STATEMENTConsidering a linear Gaussian system described by the following linear stochastic difference equation
$ {\pmb{ x}}_{t+1}={F}{\pmb{ x}}_t+{G}{\pmb{ u}}_t+{\Gamma}{\pmb{ w}}_t $  (1a) 
$ {\pmb{ y}}_{N t}={H}{\pmb{ x}}_{N t}+{\pmb{ v}}_{N t} $  (1b) 
where
The following assumptions are needed.
Assumption 1: The noises are independent Gaussian white processes such that
$\begin{array}{l} {E}\left\{\begin{bmatrix}{\pmb{ w}}_t\\{\pmb{ v}}_t\end{bmatrix}\begin{bmatrix}{\pmb{ w}}_\tau^\prime &{\pmb{ v}}_\tau^\prime\end{bmatrix}\right\} =\begin{bmatrix}{Q}&{\pmb{ 0}}\\{\pmb{ 0}}&{R}\end{bmatrix}\delta_{t\tau}\end{array} $  (2a) 
$ {E}\{{\pmb{ x}}_0\}={\pmb{ \chi}} $  (2b) 
$ {E}\{({\pmb{ x}}_0{\pmb{ \chi}})({\pmb{ x}}_0{\pmb{ \chi}})^\prime\}={\pmb{ \rho}} $  (2c) 
where
Assumption 2: The pairwise structure
Assumption 3: The noise variance matrices
Remark 1: According to Assumption 1, the system and measurement noises are assumed to be the independent Gaussian white processes. This is reasonable for some practical systems. It is well known that many random variables (RVs) in nature are actually the sum of many independent RVs. According to the fact that the sum of independent RVs tends to a Gaussian RV regardless of their probability distribution functions [25]; this is why the noises are assumed to have Gaussian distribution. As for the independence assumption, the coloured noise can be modelled by a linear system with white noise as its input, indicating that the coloured noise can be "whitened" by state argumentation method.
The problem is to develop a linear minimum variance filter
The proposed filter is first discussed, followed by the noise variance estimator.
A. Improved Kalman FilterThe Kalman filter has been proved to be an optimal filter for linear Gaussian systems. It estimates the system states and then obtains feedback in the form of measurement, dividing the equations into two groups: predictor equation and corrector equation. The former is responsible for projecting forward the current state estimations and error covariance to obtain the priori estimations for the next time instant. The latter is responsible for the feedback, that is, incorporating a new measurement into the priori estimations to obtain an improved posteriori estimation. The process is repeated with the previous posteriori estimation used to predict the new priori estimation. Hence, the Kalman filter performs as a timevariant recursive optimal filter.
Suppose that
$ \begin{align} {\pmb{ \widehat{x}}}_{Nt+1}={F}{\pmb{ \widehat{x}}}_{Nt}+{G}{\pmb{ u}}_{Nt} \end{align} $  (3) 
which is an unbiased estimation. Define the estimation error
$ \begin{align} {P}_{Nt+1} ={E}\{{\pmb{ \overline{x}}}_{Nt+1} {\pmb{ \overline{x}}}_{Nt+1}^\prime\} \end{align} $  (4a) 
$\begin{array}{l} &={E}\{({\pmb{ x}}_{Nt+1}{\pmb{ \widehat{x}}}_{Nt+1})({\pmb{ x}}_{Nt+1}{\pmb{ \widehat{x}}}_{Nt+1})^\prime\} \notag\\ &={E}\{ ({F}{\pmb{ \overline{x}}}_{Nt}+{\Gamma}{\pmb{ w}}_{Nt})({F}{\pmb{ \overline{x}}}_{Nt}+{\Gamma}{\pmb{ w}}_{Nt})^\prime \} \notag\\ &={E}\{ {F}{\pmb{ \overline{x}}}_{Nt}{\pmb{ \overline{x}}}_{Nt}^\prime{F}^\prime+{\Gamma}{\pmb{ w}}_{Nt}{\pmb{ w}}_{Nt}^\prime{\Gamma}^\prime \} \notag \\ &={F}{P}_{Nt}{F}^\prime+{\Gamma}{Q}{\Gamma}^\prime \end{array} $  (4b) 
which is essentially the predicted estimation error variance in the conventional Kalman filter.
During the period of
$ {\pmb{ \widehat{x}}}_{Nt+\tau} ={F}{\pmb{ \widehat{x}}}_{Nt+\tau1}+{G}{\pmb{ u}}_{Nt+\tau1} $  (5a) 
$ ={F}^\tau{\pmb{ \widehat{x}}}_{Nt}+\sum\limits_{i=0}^{\tau1}{F}^{\tau1i}{G}{\pmb{ u}}_{Nt+i} $  (5b) 
with the estimation error variance of
$ {P}_{Nt+\tau} ={F}{P}_{Nt+\tau1}{F}^\prime+{\Gamma}{Q}{\Gamma}^\prime $  (6a) 
$ ={F}^\tau{P}_{Nt}{F}^{\tau\prime}+\sum\limits_{i=0}^{\tau1}{F}^i{\Gamma}{Q}{\Gamma}^\prime{F}^{i\prime} $  (6b) 
At the time
$ {\pmb{ \widetilde{x}}}_{Nt+N}={\mathcal{F}}{\pmb{ \widehat{x}}}_{Nt}+{\pmb{ \mu}}_{Nt} $  (7a) 
$ {\widetilde{P}}_{Nt+N} ={\mathcal{F}}{P}_{Nt}{\mathcal{F}}^\prime+{\mathcal{Q}} $  (7b) 
where
The effect on filtering performance brought by
The ASYNCKF algorithm is a timevarying recursive filter implemented by calculating the optimal Kalman gain
Considering the period without measurements of
$ {\pmb{ {x}}}_{Nt+\tau} ={F}{\pmb{ {x}}}_{Nt+\tau1}+{G}{\pmb{ u}}_{Nt+\tau1} +{\Gamma}{\pmb{ w}}_{Nt+\tau1} $  (8a) 
$ ={F}^\tau{\pmb{ \widehat{x}}}_{Nt}+\sum\limits_{i=0}^{\tau1}{F}^{\tau1i}{G}{\pmb{ u}}_{Nt+i}+\sum\limits_{i=0}^{\tau1}{F}^{\tau1i}{\Gamma}{\pmb{ w}}_{Nt+i} $  (8b) 
and then setting
$ {\pmb{ x}}_{Nt+N}={\mathcal{F}}{\pmb{ x}}_{Nt}+{\pmb{ \mu}}_{Nt}+{\pmb{ \omega}}_{Nt} $  (9a) 
$ {\pmb{ y}}_{Nt}={H}{\pmb{ x}}_{N t}+{\pmb{ v}}_{N t} $  (9b) 
where
$ \begin{align} {\pmb{ \widehat{x}}}_{t+1}=({I}{K}{H})({F}{\pmb{ \widehat{x}}}_t+{G}{\pmb{ u}}_t)+{K}{\pmb{ y}}_t \end{align} $  (10) 
where the matrix
$ {E}={\mathcal{F}}{P}{\mathcal{F}}^\prime+{\mathcal{Q}} $  (11a) 
$ {K}={E}{H}^\prime({H}{E}{H}^\prime+{R})^{1} $  (11b) 
$ {P}=({I}{K}{H}){\mathcal{F}}. $  (11c) 
It has been proven that the timeinvariant filter will converge to the optimal timevariant Kalman filter with probability one, and therefore the convergence of the timeinvariant form of ASYNCKF can be easily deduced [27]. Note that we use the word "timeinvariant" here to show the invariance of
The proposed regulator can be described as a noise variance estimator
$ {\pmb{ x}}_{Nt+N}=q^{N}{\mathcal{F}}{\pmb{ x}}_{Nt+N}+{G}{\pmb{ \mu}}_{Nt}+{\Gamma}{\pmb{ \omega}}_{Nt} $  (12) 
where
$ {\pmb{ x}}_{Nt+N}=({I}q^{N}{\mathcal{F}})({G}{\pmb{ \mu}}_{Nt}+{\Gamma}{\pmb{ \omega}}_{Nt}). $  (13) 
Substituting (13) into (9b) yields
$ {\pmb{ y}}_{Nt}={H}({I}q^{N}{\mathcal{F}})({G}q^{N}{\pmb{ \mu}}_{Nt}+{\Gamma}q^{N}{\pmb{ \omega}}_{Nt})+{\pmb{ v}}_{Nt} $  (14) 
and then using
$ {I}q^{N}{\mathcal{F}}= \frac{{\rm adj}({I}q^{N}{\mathcal{F}})}{{\rm det}({I}q^{N}{\mathcal{F}})} $  (15) 
where
$ \begin{align} &{\rm det}({I}q^{N}{\mathcal{F}}){\pmb{ y}}_{Nt}= \notag\\ &\qquad {H}{\rm adj}({I}q^{N}{\mathcal{F}})({G}q^{N}{\pmb{ \mu}}_{Nt}+{\Gamma} q^{N}{\pmb{ \omega}}_{Nt}) \notag\\ &\qquad\ \ +{\rm det}({I}q^{N}{\mathcal{F}}){\pmb{ v}}_{Nt}. \end{align} $  (16) 
Conducting the leftcoprime factorization [28] on both sides, we have
$ \begin{align} &{A}(q^{N}){\pmb{ y}}_{Nt}= \notag\\ &{B}_\mu(q^{N}){\pmb{ \mu}}_{Nt}+{B}_\omega(q^{N}){\pmb{ \mu}}_{Nt}+{A}(q^{N}) {\pmb{ v}}_{Nt} \end{align} $  (17) 
where
$ {X}(q^{N})={X}_0+q^{N}{X}_1+\cdots+q^{N\ell_x}{X}_{\ell_x} $ 
with
$ \mathit{\boldsymbol{z}}_{Nt}={B}_\omega(q^{N})\mathit{\boldsymbol{\mu}}_{Nt}+{A}(q^{N})\mathit{\boldsymbol{v}}_{Nt} $  (18) 
where
Define the correlation function of
$ {\Lambda}^{(i)}=\sum^{\ell_\lambda}\limits_{k=i}({B}_{\omega, k} {\mathcal{Q}}{B}_{\omega, ki}^\prime+ {A}_{k}{R}{A}_{ki}^\prime) $  (19) 
where
$ {\Delta}{\pmb{ \theta}}={\pmb{ \vartheta}} $  (20) 
where
$ {\underline{\Delta}}{\pmb{ \theta}}={\pmb{ \underline{\vartheta}}} $  (21) 
where
$ {\theta}={\underline{\Delta}}^{1}{\pmb{ \underline{\vartheta}}}. $  (22) 
Substituting the estimations
$ {\pmb{ \widehat{\theta}}}_{Nt}={\underline{\Delta}}^{1}{\pmb{ \underline{\widehat{\vartheta}}}} _{Nt} $  (23) 
where
So the remaining work is to design the estimators for
$ {\widehat{\Lambda}}_{Nt}^{(i)}=\frac{1}{t}\sum\limits_{k=i}^{t}\gamma^{tk}{\pmb{ z}}_{Nk}{\pmb{ z}}_{N(ki)}^\prime $  (24a) 
$ \begin{align} &=\frac{1}{t}\left(\gamma\sum\limits_{k=i}^{t1}\gamma^{t1k}{\pmb{ z}}_{Nk}{\pmb{ z}}_{N(ki)}^\prime+{\pmb{ z}}_{Nt}{\pmb{ z}}_{N(ti)}^\prime\right) \notag\\ &=\frac{t1}{t}\frac{1}{t1}\left(\gamma\sum\limits_{k=i}^{t1}\gamma^{t1k}{\pmb{ z}}_{Nk}{\pmb{ z}}_{N(ki)}^\prime+{\pmb{ z}}_{Nt}{\pmb{ z}}_{N(ti)}^\prime\right) \notag\\ &=\left(1\frac{1}{t}\right)\gamma{\widehat{\Lambda}}_{NtN}^{(i)}+\frac{1}{t}{\pmb{ z}}_{Nt}{\pmb{ z}}_{N(ti)}^\prime \notag\\ &=\gamma{\widehat{\Lambda}}_{NtN}^{(i)}+\frac{1}{t}\left({\pmb{ z}}_{Nt}{\pmb{ z}}_{N(ti)}^\prime\gamma{\widehat{\Lambda}}_{Nt}^{(i)}\right)\end{align} $  (24b) 
where
The method in (24a) and (24b) is termed as the {fading estimation}. The alternative is the {finitememory estimation} with the form of
$ {\widehat{\Lambda}}_{Nt}^{(i)}=\frac{1}{T}\sum\limits_{k=tT+1}^{t}{\pmb{ z}}_{Nk}{\pmb{ z}}_{N(ki)}^\prime $  (25) 
whose recursive form does not exist. However, this method is still a promising solution. As seen in Fig. 1, the blue plots represent for the weights assignment on history data of fading estimation, it is found that all the history data are shrunken sharply except the current one if
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Fig. 1 The comparison between fading estimation and finitememory estimation on the weights assigned to the history data. 
Substituting
Two simulation tests for the wheeled robot navigation system and air quality evaluation system by applying the proposed selftuning asynchronous filter will be presented.
A. Wheeled Robot Navigation SystemA global positioning system (GPS) aided wheeled robot navigation system is composed of three parts: the space segment, the control segments and the user segment, as illustrated in Fig. 2. The wheeled robot is mounted with a GPS receiver to determine its kinematic states. However, the GPS has a low updating frequency, and its signal may weaken due to various reasons.
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Fig. 2 Illustration for wheeled robot navigation system. 
The kinematics of a wheeled robot on roads can be approximately treated as the motion on the twodimensional plane, which can be decomposed to the eastward motion and northward motion. Let
$ x_{E, t+1}=x_{E, t}+T\dot{x}_{E, t}+0.5T^2(u_{E, t}+w_{E, t}) $  (26a) 
$ \dot{x}_{E, t+1}=\dot{x}_{E, t}+T(u_{E, t}+w_{E, t}) $  (26b) 
$ x_{N, t+1}=x_{N, t}+T\dot{x}_{N, t}+0.5T^2(u_{N, t}+w_{N, t}) $  (26c) 
$ \dot{x}_{N, t+1}=\dot{x}_{N, t}+T(u_{N, t}+w_{N, t}) $  (26d) 
where
$ y_{E, Nt}=x_{E, Nt}+v_{E, Nt} $  (27a) 
$ y_{N, Nt}=x_{N, Nt}+v_{N, Nt} $  (27b) 
where
Introducing the state variables
$ \begin{array}{l} F = \left[ {\begin{array}{*{20}{c}} 1&T&0&0\\ 0&1&0&0\\ 0&0&1&T\\ 0&0&0&1 \end{array}} \right],\;\;\;{\cal F} = \left[ {\begin{array}{*{20}{c}} 1&{NT}&0&0\\ 0&1&0&0\\ 0&0&1&{NT}\\ 0&0&0&1 \end{array}} \right]\\ G = \Gamma = \left[ {\begin{array}{*{20}{c}} {0.5{T^2}}&0\\ T&0\\ 0&{0.5{T^2}}\\ 0&T \end{array}} \right],\;\;\;H = \left[ {\begin{array}{*{20}{c}} 1&0&0&0\\ 0&0&1&0 \end{array}} \right]\\ Q = \left[ {\begin{array}{*{20}{c}} {\sigma _E^2}&0\\ 0&{\sigma _N^2} \end{array}} \right],\;\;R = \left[ {\begin{array}{*{20}{c}} {\varsigma _E^2}&0\\ 0&{\varsigma _N^2} \end{array}} \right]. \end{array} $ 
It is noted that the modelling procedure is very typical by applying some basic physical principles. It is also termed as the target tracking problem which is a popular research topic worldwide [27], [28].
Before the simulation, some parameters should be determined: the sampling time interval
Case 1: The test is done using the asynchronous filter without considering the variation of noise characteristics.
Case 2: The test is done using the selftuning asynchronous filter.
Case 3: The test is done using the conventional Kalman filter. The last measurement received will be used to project forward the estimation while the measurements are missing.
The results are shown in Fig. 3 where the black curves represent the real tracks and those in red, blue, and green for the estimations under different cases. The error is defined as the linear distance between the estimations and the real locations. The comparison between Case 1 and Case 2 indicates that the noise variance estimator is of significant importance for the influence brought by variation of noise characteristics can be eliminated to the largest extent. Using the last measurement to project forward the estimation can smooth the estimations during the period without measurements, but the comparison between Case 2 and Case 3 shows the superiority of the selftuning asynchronous filter. Furthermore, the mean value of errors under the three cases are
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Fig. 3 Localization results and errors under the Cases 1, 2 and 3. 
Partially enlarging Fig. 3 (b) yields the Fig. 4, where the blue circles represent measurements, the red curves are estimations, and the black curves are the real tracks. It is found that the data derived by deadreckoning method without correction will diverge to infinity, but the GPS will correct the estimation in time while the error gets larger. Although the performance of deadreckoning method deteriorates over time, it performs quite well in a short period, which has complemented the absence of GPS. Hence the wheeled robot navigation system is usually integrated with the two methods by fusing their outputs, to achieve a relatively accurate estimation in real time.
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Fig. 4 Partially enlarged detail in the Fig. 3 (b). 
Based on the existing dispersion model shown in [30], [31], we will present a discrete linear parameterised boxtype dispersion model, which is a parameterised semiempirical model which makes use of a priori assumptions about the flow and dispersion conditions. As illustrated in Fig. 5 the streets can be treated as a closed box with its top exchanging air with the highlevel atmosphere for the very deepness of the street canyon. The inner atmosphere is relatively stable and the pollutant concentration will decrease with a fixed proportion
$ \begin{align} x_{t+1}\pi=k(x_{t}\pi)+u_{t}+w_{t} \end{align} $  (28) 
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Fig. 5 Illustration for atmospheric dispersion system. 
where
The monitoring cars equipped with atmospheric sensors can be used to correct the estimation. Rearranging (28) and modeling the mobilesensing method yields
$ x_{t+1}=kx_{t}+\chi+u_{t}+w_{t} $  (29a) 
$ y_{Nt}=x_{Nt}+v_{Nt} $  (29b) 
where
Set the parameters as dispersion coefficient
The results are shown in Fig. 6. The first test result in (a) shows the asymptotical stability, where the weight
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Fig. 6 Simulation results for 
In this paper, the optimal filtering problem for linear Gaussian systems is considered. A selftuning asynchronous filter is proposed whose noise variance estimator is able to adjust filtering coefficients adaptively. Two case studies, the wheeled robot navigation system and the air quality evaluation system, have demonstrated the effectiveness of the proposed approach.
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