Adaptive Fading-Memory Receding-Horizon Filters and Smoother for Linear Discrete Time-Varying Systems

Abstract

In this paper, an adaptive fading-memory receding-horizon (AFMRH) filter is proposed by combining the receding-horizon structure and the adaptive fading-memory method. In the recent finite horizon, state error covariance is adapted with an adaptive fading factor; then the process noise covariance matrix adaption is realized by adjusting the properties of systems. An AFMRH fixed-lag smoother is also proposed by combining the proposed AFMRH filtering algorithm and a Rauch–Tung–Striebel smoothing algorithm to improve the estimation accuracy. Because the proposed AFMRH filter and smoother are reduced to the optimal receding-horizon (RH) filter and smoother when all measurements have the same weight, the proposed adaptive RH estimators could provide a more general solution than the optimal RH filter and smoother. To reduce the complexity and improve the estimation performance of the proposed RH estimators, an adaptive horizon adjustment method and a switching filtering algorithm based on an adaptive fading factor are also proposed. In particular, the proposed adaptive horizon adjustment method is designed to be computationally efficient, which makes it suitable for online and real-time applications. Through computer simulation, the performance and adaptiveness of the proposed approaches were verified by comparing them with existing fading-memory approaches.

1. Introduction

The Kalman filter has been widely investigated and applied in various engineering fields owing to its optimality and computational efficiency [1,2,3,4,5].

However, the optimality of the Kalman filter is guaranteed only if the system model is accurate and the information of the noise statistics is exactly known. In practical applications, an accurate model is difficult to obtain and the noise characteristics of the system are generally unknown. Thus, the estimation performance of the Kalman filter may degrade or diverge in the presence of model uncertainties and numerical errors [5,6].

To address the issue of the divergence of the conventional Kalman filter, fading memory Kalman filters were investigated to improve the filtering robustness. Because heavy weighting of recent measurements increases the stability margin of the Kalman filter, the fading-memory algorithm provides robustness to the filter and an effective way of preventing the divergence of the Kalman filter. Moreover, it can facilitate a more general solution than conventional Kalman filters, because the fading-memory Kalman filter is reduced and equivalent to a conventional Kalman filter when the fading factor is taken as one.

Another solution to solve the divergence problem of the conventional Kalman filter is by using the receding horizon (RH) estimation method. Unlike Kalman-filter-based approaches, the state estimation of an RH filter uses only recent finite measurements; thus, it is insensitive to erroneous modeling of the system and numerical errors. RH estimation has good properties such as bounded-input bounded-output (BIBO) stability, dead-beat property, fast-tracking ability, and robustness against modeling uncertainties and numerical errors. Furthermore, the RH estimators in [7,8,9,10,11,12,13,14] do not require a priori knowledge of initial information. Thus, as an alternative to Kalman-filter-based methods, RH estimators have been extensively investigated recently [7,8,9,10,11,12,13,14,15,16,17,18,19].

To achieve a better estimation performance, adopting the fading-memory method for designing the RH estimators is of significance. By combining the RH estimation and fading-memory methods, the robust characteristics of both the methods can be simultaneously incorporated into the design of RH estimators. With a constant fading factor, the fading-memory RH filters were proposed for linear continuous time-invariant and discrete time-varying state-space models in [17] and [10], respectively. For the RH smoothing problem, the fading-memory algorithm was applied to the fixed-lag RH smoother with a recursive form in [11]. The results in these approaches show that the combination of RH estimation and fading-memory strategy can provide faster tracking ability and better robustness concerning temporary modeling uncertainties than the fading Kalman filter and nonfading-memory RH filter. Thus, fading-memory RH estimation is reasonable and expected to improve estimation performance.

However, previous approaches have only considered a constant fading factor, which makes it difficult to achieve the desired estimation performance for unpredictable disturbances. Moreover, although optimal RH estimators are combined with the fading-memory method, fading-memory RH estimators with a constant fading factor cannot guarantee optimal filtering [20]. Furthermore, a large fading-factor value may cause a loss of filtering accuracy. These problems can be solved by integrating the adaptive fading-memory algorithm into RH estimation and the estimation performance of an RH filter can be considerably improved compared to that of the constant one. In the case of fading-memory Kalman filtering, many adaptive fading-memory (AFM) Kalman filters have been proposed to adjust the fading factors [6,20,21,22,23,24,25,26,27]. However, the contribution to RH estimation based on the adaptive fading-memory strategy is limited.

Therefore, this study focuses on a comprehensive estimation algorithm by combining the advantages of the AFM estimation and the RH structure. Based on the concept of the AFM technique and the recursive RH filtering structure, an AFMRH filtering algorithm is proposed in this study. Additionally, an RH fixed-lag smoother can provide a more accurate estimate than an RH filter [7]. Because there is no report on an adaptive RH smoother and the filtering result can be expanded to the fixed-lag smoothing problem, an AFMRH smoother based on the Rauch–Tung–Striebel (RTS) algorithm [28,29] is also proposed for a linear discrete time-varying state-space model. If all measurements have the same weight, the proposed AFMRH filter and smoother are reduced to optimal RH filter and smoother, respectively. Therefore, it is significant that the proposed filter and smoother can provide a more general solution than the previous optimal RH filter and smoother.

Furthermore, the estimation performance of the proposed AFMRH estimators can be improved by choosing the appropriate horizon length, because the horizon length is a key design parameter of the RH filter. For example, if the model uncertainty or numerical error exists, the RH estimator with a short horizon length can provide better robustness compared to the RH filter with a long one. Moreover, it can enable fast-tracking ability and computational efficiency. Conversely, if the model accurately represents the real system, the estimate of the RH filter with a long horizon length is more accurate than that of the RH filter with a short one. Although the RH filter with a long horizon length is the proper choice for an accurate system model, the computational complexity considerably increases as the horizon length grows. Thus, the horizon length should be properly chosen to compromise between the estimation accuracy and the computational efficiency.

Several algorithms for determining an appropriate value of the horizon length have been suggested for linear time-invariant systems [30,31]. The proposed algorithms provide an effective way for linear time-invariant systems. However, they suffer from heavy computational burden. Moreover, if there are unpredictable temporary model uncertainties, the fixed and constant horizon lengths of the proposed algorithms are not satisfactory even for the linear time-invariant systems. Furthermore, these algorithms cannot guarantee the solution for time-varying systems and are difficult to apply in practice [32]. For temporarily uncertain systems, switching an RH filter that comprises two RH filters with different horizon lengths was proposed for time-invariant systems [33]. Because the temporary uncertainty is detected using the chi-square test statistic, and a suitable RH filter is selected and operated to estimate the state, the proposed switching RH filter performs well and provides good estimation results for both nominal and temporarily uncertain systems. However, the threshold value used to determine the temporary uncertainty should be precomputed, such that it is difficult to apply on time-varying systems. For time-varying systems, the switching RH filter-bank-based adaptive horizon-selection method was proposed in [34]. Because the horizon length is adjusted each time to adapt to the changes in system characteristics, the proposed algorithm exhibits better estimation performance than previous approaches with a fixed-constant horizon length. However, this method incurs a heavy computational cost because several different RH filters should always be operated simultaneously, and the calculation should be repeated until an appropriate value of the horizon length is achieved.

Therefore, a novel adaptive-horizon-length adjustment (AHLA) algorithm was also proposed in this study to improve the computational efficiency and estimation performance of the proposed AFMRH estimators. The horizon length is constantly adjusted such that the absolute residual values of the three sequential estimates with different horizon lengths is minimized and applied for the next three estimates. Because the proposed AHLA method uses results in the estimation process, it can have less computational burden than the previous approach in [34].

Although the estimation performance of the proposed AFMRH filter can be improved by applying the AHLA method, the computational cost of the proposed AFMRH estimators can increase because of a large adjusted horizon length. As the RH filter converges to the Kalman filter as the horizon length increases [13], it is better to choose a Kalman filter that provides the optimal estimate and computational efficiency than an RH filter with large horizon length. Moreover, the adaptive fading factor can also be used to determine whether a temporary model uncertainty or numerical error exists. Thus, a simple but efficient switching AFMRH (SAFMRH) filtering algorithm based on adaptive fading factor and AHLA method is also proposed to improve the computational efficiency. It is expected that the proposed AHLA method and SAFMRH filtering algorithm can improve the estimation performance and computational efficiency for both the nominal and temporarily uncertain systems.

The rest of this paper is organized as follows:

In Section 2, an AFMRH filter is proposed by combining the recursive optimal RH filter with the adaptive fading factor. Then, based on the results of AFMRH filtering, a fixed-lag AFMRH smoothing algorithm is proposed using the Rauch–Tung–Striebel smoothing technique in Section 3. Thereafter, in Section 4, a computationally efficient AHLA method and switching filtering algorithm based on the adaptive fading factor are proposed to improve the estimation accuracy and computational efficiency of the proposed filter and smoother. In Section 5, the performances of the proposed AFMRH estimators are evaluated via numerical simulations. Finally, the conclusions are presented in Section 6.

2. An Adaptive Fading-Memory Receding-Horizon Filter

2.1. An Optimal Receding-Horizon Filter

Consider the linear discrete time-varying state-space model given by

$$\begin{array}{ccc}\hfill x_{k+1}& =& A_k{x}_k+G_k{w}_k,\hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill y_k& =& C_k{x}_k+v_k,\hfill \end{array}$$

(2)

where $x_k$ is the state vector, $y_k$ is the output vector, and the pair $(A_k,C_k)$ is assumed to be observable. The process noise vector $w_k$ and the output noise vector $v_k$ are assumed to be zero-mean white Gaussian noise and uncorrelated with each other and with the initial state $x_0$. The process and measurement noise covariance matrices are denoted as $Q_k$ and $R_k$, respectively, and are assumed to be positive definite.

On the finite horizon $[k-Nk-1]$, a recursive optimal RH filter to determine the a priori state estimate ${\widehat{x}}_{k\mid k-1}$ can be expressed as [13]:

$$\begin{array}{ccc}\hfill {\widehat{x}}_{k-N+i+1\mid k-1}& =& A_{k-N+i}\left\{{\widehat{x}}_{k-N+i\mid k-1}+K_{k-N+i\mid k-1}\left(y_{k-N+i}-C_{k-N+i}{\widehat{x}}_{k-N+i\mid k-1}\right)\right\},\hfill \end{array}$$

(3)

$$\begin{array}{ccc}\hfill P_{k-N+i+1\mid k-1}& =& A_{k-N+i}\left(P_{1,k-N+i\mid k-1}-K_{k-N+i\mid k-1}{C}_{k-N+i}{P}_{k-N+i\mid k-1}\right)A_{k-N+i}^T\hfill \\ & & +G_{k-N+i}{Q}_{k-N+i}{G}_{k-N+i}^T\prime \hfill \end{array}$$

(4)

for $0\le i\le N-1$, where the gain matrix $K_{k-N+i\mid k-1}$ is defined as

$$\begin{array}{ccc}\hfill K_{k-N+i\mid k-1}& =& P_{k-N+i\mid k-1}{C}_{k-N+i}^T{\left(C_{k-N+i}{P}_{k-N+i\mid k-1}{C}_{k-N+i}^T+R_{k-N+i}\right)}^{-1}.\hfill \end{array}$$

(5)

The initial conditions ${\widehat{x}}_{k-N\mid k-1}$ and $P_{k-N\mid k-1}$ are obtained as

$$\begin{array}{ccc}\hfill {\widehat{x}}_{k-N\mid k-1}& =& P_{k-N\mid k-1}{\widehat{w}}_{k-N\mid k-1},\hfill \end{array}$$

(6)

$$\begin{array}{ccc}\hfill P_{k-N\mid k-1}& =& S_{k-N\mid k-1}^{-1},\hfill \end{array}$$

(7)

where ${\widehat{w}}_{k-N\mid k-1}$ and $S_{k-N\mid k-1}$ represent the a priori information state and its information matrix, respectively, which are obtained from the following recursive calculation:

$$\begin{array}{ccc}\hfill {\widehat{w}}_{k-l-1\mid k-1}& =& C_{k-l}^T{R}_{k-l}^{-1}{y}_{k-l}+A_{k-l-1}^T{\widehat{w}}_{k-l\mid k-1}-A_{k-l-1}^T{S}_{k-l\mid k-1}{G}_{k-l}\hfill \\ & & \times {\left(Q_{k-l}^{-1}+G_{k-l}^T{S}_{k-l\mid k-1}{G}_{k-l}\right)}^{-1}{G}_{k-l}^T{\widehat{w}}_{k-l\mid k-1},\hfill \end{array}$$

(8)

$$\begin{array}{ccc}\hfill S_{k-l-1\mid k-1}& =& C_{k-l}^T{R}_{k-l}^{-1}{C}_{k-l}+A_{k-l-1}^T{S}_{k-l\mid k-1}{A}_{k-l-1}-A_{k-l-1}^T{S}_{k-l\mid k-1}{G}_{k-l}\hfill \\ & & \times {\left(Q_{k-l}^{-1}+G_{k-l}^T{S}_{k-l\mid k-1}{G}_{k-l}\right)}^{-1}{G}_{k-l}^T{S}_{k-l\mid k-1}{A}_{k-l-1},\hfill \end{array}$$

(9)

for $1\le l\le N-1$, with the initial conditions ${\widehat{w}}_{k-1\mid k-1}=0$ and $S_{k-1\mid k-1}=0$.

2.2. An Adaptive Fading-Memory Receding-Horizon Filter

Because the recursive optimal RH filter estimates the current state using recent finite-horizon measurements, it guarantees BIBO stability, fast tracking, and robustness against modeling uncertainties. It is still based on precise mathematical models and can only deal with dynamic linear systems.

Although the recursive optimal RH filter has good properties, it is derived from a precise system model. Thus, when the system model and prior statistical information of noises are unknown or inaccurate, its estimation performance can degrade. As the mismatched system model and inaccurate noise covariances can be adjusted according to measurements, an AFMRH filter is proposed by combining recursive optimal RH filtering with an adaptive filtering algorithm based on the adaptive fading factor.

The equations of the AFMRH filter can be described with those of the recursive optimal RH filter in Equations (3)–(9), except the error covariance update equation, as

$$\begin{array}{ccc}\hfill P_{k-N+i+1\mid k-1}& =& {\lambda }_{k-N+i\mid k-1}{A}_{k-N+i}\left(P_{1,k-N+i\mid k-1}-K_{k-N+i\mid k-1}{C}_{k-N+i}{P}_{k-N+i\mid k-1}\right)\hfill \\ & & \times A_{k-N+i}^T+G_{k-N+i}{Q}_{k-N+i}{G}_{k-N+i}^T,\hfill \end{array}$$

(10)

where ${\lambda }_{k-N+i\mid k-1}\ge 1$ is the adaptive fading factor. By adjusting the adaptive fading factor, most recent measurements are adaptively overweighted so that the estimation error due to incomplete information can be reduced.

Now, the algorithm for selecting the optimal adaptive fading factor is presented. The residual, which is the difference between the measurement and predicted measurements, is represented as

$${\delta }_{k-N+i\mid k-1}=y_{k-N+i}-C_{k-N+i}{\widehat{x}}_{k-N+i\mid k-1}.$$

(11)

By substituting (11) into Equation (3) and taking the expectation on both sides, the covariance of the residual is formulated as

$$\begin{array}{ccc}\hfill {\Delta }_{k-N+i\mid k-1}& =& \mathbf{E}\left[{\delta }_{k-N+i\mid k-1}{\delta }_{k-N+i\mid k-1}^T\right]\hfill \\ & =& \mathbf{E}\left[\left(y_{k-N+i}-C_{k-N+i}{\widehat{x}}_{k-N+i\mid k-1}\right){\left(y_{k-N+i}-C_{k-N+i}{\widehat{x}}_{k-N+i\mid k-1}\right)}^T\right]\hfill \\ & =& C_{k-N+i}{P}_{k-N+i\mid k-1}{C}_{k-N+i}^T+R_{k-N+i}.\hfill \end{array}$$

(12)

and the auto-covariance of the residual is represented as

$$\begin{array}{ccc}& & \mathbf{E}\left[{\delta }_{k-N+i+j\mid k-1}{\delta }_{k-N+i\mid k-1}^T\right]\hfill \\ & =& \mathbf{E}\left[\left(y_{k-N+i+j}-C_{k-N+i+j}{\widehat{x}}_{k-N+i+j\mid k-1}\right){\left(y_{k-N+i}-C_{k-N+i}{\widehat{x}}_{k-N+i\mid k-1}\right)}^T\right]\hfill \\ & =& C_{k-N+i+j}{A}_{k-N+i+j-1}\left(I-K_{k-N+i+j-1}\right)A_{k-N+i+j-2}\left(I-K_{k-N+i+j-2}\right)\hfill \\ & & \times A_{k-N+i+j-3}\cdots \left(I-K_{k-N+1}\right)A_{k-N+i}{\Lambda }_{k-N+i},\hfill \end{array}$$

(13)

for $1\le j\le N-i-1$, where ${\Lambda }_{k-N+i}$ is definded as

$$\begin{array}{c}\hfill {\Lambda }_{k-N+i}=P_{k-N+i\mid k-1}{C}_{k-N+i}^T-K_{k-N+i}{\Delta }_{k-N+i\mid k-1}.\end{array}$$

(14)

When the gain matrix $K_{k-N+i}$ is optimal, i.e., Equation (5) holds,

$$\begin{array}{ccc}\hfill {\Lambda }_{k-N+i}& =& P_{k-N+i\mid k-1}{C}_{k-N+i}^T-K_{k-N+i}{\Delta }_{k-N+i\mid k-1}\hfill \\ & =& P_{k-N+i\mid k-1}{C}_{k-N+i}^T-P_{k-N+i\mid k-1}{C}_{k-N+i}^T(C_{k-N+i}{P}_{k-N+i\mid k-1}{C}_{k-N+i}^T\hfill \\ & & +R_{k-N+i}{)}^{-1}\left(C_{k-N+i}{P}_{k-N+i\mid k-1}{C}_{k-N+i}^T+R_{k-N+i}\right)=0,\hfill \end{array}$$

(15)

the aucto-covariance of the residual is zero for $0\le i\le N-1$ and $1\le j\le N-i-1$.

In practice, however, ${\Lambda }_{k-N+i}$ in Equation (15) may not be zero owing to the incomplete information; thus the accuracy of the recursive optimal RH filter can be evaluated by using the following scalar function:

$$\begin{array}{c}\hfill J_{k-N+i\mid k-1}=\sum _n\sum _m{\Lambda }_{n,m,k-N+i\mid k-1}^2,\end{array}$$

(16)

where ${\Lambda }_{n,m,k-N+i\mid k-1}$ represents the $(n,m)th$ elements of ${\Lambda }_{k-N+1}$. The absolute minimum of $J_{k-N+i\mid k-1}\left({\lambda }_{k-N+i}\right)$ means that the estimation result is mostly close to the optimal one. Therefore, the AFMRH filtering problem can be defined to find the adaptive fading factor selected to minimize the following cost function:

$$\begin{array}{c}\hfill \underset{{\lambda }_{k-N+i\mid k-1}}{min}{J}_{k-N+i\mid k-1}.\end{array}$$

(17)

However, the calculation to find the optimal solution of Equation (17) requires a heavy computational load and is not unsuitable for the online state estimation. Thus the following algorithm is introduced in [6].

By substituting the gain matrix (5) into the optimal condition

$$\begin{array}{ccc}& & P_{k-N+i\mid k-1}{C}_{k-N+i}^T-K_{k-N+i}{\Delta }_{k-N+i\mid k-1}\hfill \\ & & =P_{k-N+i\mid k-1}{C}_{k-N+i}^T\{I-P_{k-N+i\mid k-1}{C}_{k-N+i}^T(C_{k-N+i}\hfill \\ & & \times P_{k-N+i\mid k-1}{C}_{k-N+i}^T+R_{k-N+i}{)}^{-1}\}{\Delta }_{k-N+i\mid k-1}=0,\hfill \end{array}$$

(18)

the following relation is obtained:

$$\begin{array}{c}\hfill C_{k-N+i}{P}_{k-N+i\mid k-1}{C}_{k-N+i}^T+R_{k-N+i}={\Delta }_{k-N+i\mid k-1}.\end{array}$$

(19)

As $Q_{k-N+i}$, $R_{k-N+i}$ and the horizon initial error covariance $P_{k-N\mid k-1}$ are all positive definite, the optimal condition (18) is equivalent to Equation (12). By substituting the error covariance update Equations (10) into (19), we obtain

$$\begin{array}{c}\hfill {\lambda }_{k-N+i\mid k-1}{C}_{k-N+i}{P}_{k-N+i\mid k-1}{C}_{k-N+i}^T={\Delta }_{k-N+i\mid k-1}-R_{k-N+i}.\end{array}$$

(20)

By denoting $H_{k-N+i\mid k-1}$ and $L_{k-N+i\mid k-1}$ as

$$\begin{array}{ccc}\hfill H_{k-N+i\mid k-1}& =& {\widehat{\Delta }}_{k-N+i\mid k-1}-C_{k-N+i}{G}_{k-N+i-1}{Q}_{k-N+i-1}{G}_{k-N+i-1}^T{C}_{k-N+i}^T\hfill \\ & & -R_{k-N+i},\hfill \end{array}$$

(21)

$$\begin{array}{ccc}\hfill L_{k-N+i\mid k-1}& =& C_{k-N+i}{A}_{k-N+i-1}{P}_{k-N+i-1\mid k-1}{A}_{k-N+i-1}^T{C}_{k-N+i}^T,\hfill \end{array}$$

(22)

where ${\widehat{\Delta }}_{k-N+i\mid k-1}$ is the estimate of ${\Delta }_{k-N+i\mid k-1}$, we have

$$\begin{array}{c}\hfill {\lambda }_{k-N+i\mid k-1}{L}_{k-N+i\mid k-1}=H_{k-N+i\mid k-1}.\end{array}$$

(23)

The unbiased estimate of ${\Delta }_{k-N+i\mid k-1}$ can be obtained by using the recent measurements and estimates on the finite horizon $[k-Nk-1]$ as

$${\widehat{\Delta }}_{k-N+i\mid k-1}=\frac{1}{N}\left(\sum _{j=0}^i{\delta }_{k-N+j\mid k-1}{\delta }_{k-N+j\mid k-1}^T+\sum _{j=i+1}^{N-1}{\delta }_{k-N+j\mid k-2}{\delta }_{k-N+j\mid k-2}^T\right).$$

(24)

For computational efficiency, the predicted residual covariance ${\Delta }_{k-N+i\mid k-1}$ can also be sequentially calculated as

$$\begin{array}{ccc}\hfill {\widehat{\Delta }}_{k-N+i\mid k-1}& =& {\widehat{\Delta }}_{k-N+i-1\mid k-1}+\frac{1}{N}({\delta }_{k-N+i\mid k-1}{\delta }_{k-N+i\mid k-1}^T\hfill \\ & & -{\delta }_{k-N+i\mid k-2}{\delta }_{k-N+i\mid k-2}^T),\hfill \end{array}$$

(25)

where the initial horizon value of the predicted residual is obtained from the result of the previous horizon as ${\widehat{\Delta }}_{k-N\mid k-1}={\widehat{\Delta }}_{k-1\mid k-2}$ and

$$\begin{array}{c}\hfill {\widehat{\Delta }}_{N\mid N-1}=\frac{1}{N}\sum _{i=1}^N{\delta }_{i\mid N-1}{\delta }_{i\mid N-1}^T,\end{array}$$

(26)

for the first horizon $[1N-1]$.

Then, the optimal fading factor can be obtained by taking the trace on both sides of Equation (23) as

$$\begin{array}{ccc}\hfill {\lambda }_{k-N+i\mid k-1}& =& max\left\{1,tr\left(H_{k-N+i\mid k-1}\right)/tr\left(L_{k-N+i\mid k-1}\right)\right\}.\hfill \end{array}$$

(27)

Thus, on the horizon $[k-Nk]$, the AFMRH filtering algorithm can be summarized as following Algorithm 1:

Algorithm 1 The AFMRH filtering algorithm

Initialization: ${\hat{\Delta }}_{N\mid N-1}\leftarrow \frac{1}{N}{\sum }_{i=1}^N{\delta }_{i\mid N-1}{\delta }_{i\mid N-1}^T$ for the first horizon $[1N-1]$

Step 1: Backward information smoothing on the horizon $[k-Nk-1]$

Initialization: ${\widehat{w}}_{k-1\mid k-1}\leftarrow 0$, $S_{k-1\mid k-1}\leftarrow 0$   for $l\leftarrow 1$ to $N-1$ do     $$\begin{gathered} {\widehat{w}}_{k-l-1\mid k-1}\leftarrow C_{k-l}^T{R}_{k-l}^{-1}{y}_{k-l}+A_{k-l-1}^T{\widehat{w}}_{k-l\mid k-1}-A_{k-l-1}^T{S}_{k-l\mid k-1}{G}_{k-l} \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times {\left(Q_{k-l}^{-1}+G_{k-l}^T{S}_{k-l\mid k-1}{G}_{k-l}\right)}^{-1}{G}_{k-l}^T{\widehat{w}}_{k-l\mid k-1} \\ {S}_{k-l-1\mid k-1}\leftarrow C_{k-l}^T{R}_{k-l}^{-1}{C}_{k-l}+A_{k-l-1}^T{S}_{k-l\mid k-1}{A}_{k-l-1}-A_{k-l-1}^T{S}_{k-l\mid k-1} \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times G_{k-l}{\left(Q_{k-l}^{-1}+G_{k-l}^T{S}_{k-l\mid k-1}{G}_{k-l}\right)}^{-1}{G}_{k-l}^T{S}_{k-l\mid k-1}{A}_{k-l-1} \end{gathered}$$   end for

Step 2: Finite horizon adaptive fading filtering on the horizon $[k-Nk-1]$

Initialization: ${\widehat{x}}_{k-N\mid k-1}\leftarrow S_{k-N\mid k-1}^{-1}{\widehat{w}}_{k-N\mid k-1}$, $P_{k-N\mid k-1}\leftarrow S_{k-N\mid k-1}^{-1}$, ${\widehat{\Delta }}_{k-N\mid k-1}\leftarrow {\widehat{\Delta }}_{k-1\mid k-2}$   for $i\leftarrow 0$ to $N-1$ do       $$\begin{gathered} K_{k-N+i\mid k-1}\leftarrow P_{k-N+i\mid k-1}{C}_{k-N+i}^T(C_{k-N+i}{P}_{k-N+i\mid k-1}{C}_{k-N+i}^T \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} +R_{k-N+i}{)}^{-1} \\ {\widehat{x}}_{k-N+i+1\mid k-1}\leftarrow A_{k-N+i}{\widehat{x}}_{k-N+i\mid k-1}+A_{k-N+i}{K}_{k-N+i\mid k-1}(y_{k-N+i} \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} -C_{k-N+i}{\widehat{x}}_{k-N+i\mid k-1}) \\ {\delta }_{k-N+i\mid k-1}\leftarrow y_{k-N+i}-C_{k-N+i}{\widehat{x}}_{k-N+i\mid k-1} \\ {\widehat{\Delta }}_{k-N+i\mid k-1}\leftarrow {\widehat{\Delta }}_{k-N+i-1\mid k-1}+\frac{1}{N}({\delta }_{k-N+i\mid k-1}{\delta }_{k-N+i\mid k-1}^T \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} -{\delta }_{k-N+i\mid k-2}{\delta }_{k-N+i\mid k-2}^T) \\ {H}_{k-N+i\mid k-1}\leftarrow {\widehat{\Delta }}_{k-N+i\mid k-1}-C_{k-N+i}{G}_{k-N+i}{Q}_{k-N+i}{G}_{k-N+i}^T{C}_{k-N+i}^T \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} -R_{k-N+i} \\ {L}_{k-N+i\mid k-1}\leftarrow C_{k-N+i}{A}_{k-N+i}{P}_{k-N+i\mid k-1}{A}_{k-N+i}^T{C}_{k-N+i}^T \\ {\lambda }_{k-N+i\mid k-1}\leftarrow max\left\{1,tr\left(H_{k-N+i\mid k-1}\right)/tr\left(L_{k-N+i\mid k-1}\right)\right\} \\ {P}_{k-N+i+1\mid k-1}\leftarrow {\lambda }_{k-N+i\mid k-1}{A}_{k-N+i}(P_{1,k-N+i\mid k-1}-K_{k-N+i\mid k-1}{C}_{k-N+i} \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times P_{k-N+i\mid k-1})A_{k-N+i}^T+G_{k-N+i}{Q}_{k-N+i}{G}_{k-N+i}^T \end{gathered}$$   end for

3. An Adaptive Fading-Memory Receding-Horizon Fixed-Lag Smoother

In the case where the system can tolerate delay, smoothers can provide better accuracy estimates than filters. However, to the best of the author’s knowledge, an adaptive fixed-lag RH smoothing algorithm has not been investigated. Hence, in this section, an AFMRH fixed-lag smoothing algorithm based on the AFMRH filter and Rauch–Tung–Striebel (RTS) smoother is proposed to further improve the estimation performance of the proposed filter.

The proposed smoother comprises two passes: forward AFMRH filtering and backward RTS smoothing. In the forward pass, the AFMRH filter is used to obtain the a priori and a posteriori estimates and their error covariance matrices. In the time interval from the fixed-lag time up to the current time, the estimated states and their error covariance matrices are stored for the backward pass. Thereafter, the backward pass based on the RTS smoothing algorithm runs by utilizing the stored data to estimate the fixed-lag state.

3.1. An Adaptive Fading-Memory Receding-Horizon Filter for a Posteriori State Estimation

We first introduce the AFMRH filter for the a priori and a posteriori state estimation, which is the forward-pass filtering algorithm of the proposed smoother.

On the horizon $[k-Nk]$, a posteriori estimate ${\widehat{x}}_{k-N+i\mid k}^{+}$ and its error covariance matrix $P_{k-N+i\mid k}^{+}$ can be obtained as follows:

$$\begin{array}{ccc}\hfill {\widehat{x}}_{k-N+i\mid k}& =& A_{k-N+i}{\widehat{x}}_{k-N+i-1\mid k}^{+},\hfill \end{array}$$

(28)

$$\begin{array}{ccc}\hfill P_{k-N+i\mid k}& =& {\lambda }_{k-N+i}{A}_{k-N+i}{P}_{k-N+i-1\mid k}^{+}{A}_{k-N+i}^T+G_{k-N+i}{Q}_{k-N+i}{G}_{k-N+i}^T,\hfill \end{array}$$

(29)

$$\begin{array}{ccc}\hfill K_{k-N+i\mid k}& =& P_{k-N+i\mid k}{C}_{k-N+i}^T{\left(C_{k-N+i}{P}_{k-N+i\mid k}{C}_{k-N+i}^T+R_{k-N+i}\right)}^{-1},\hfill \end{array}$$

(30)

$$\begin{array}{ccc}\hfill {\delta }_{k-N+i\mid k}& =& y_{k-N+i}-C_{k-N+i}{\widehat{x}}_{k-N+i\mid k},\hfill \end{array}$$

(31)

$$\begin{array}{ccc}\hfill {\widehat{x}}_{k-N+i\mid k}^{+}& =& {\widehat{x}}_{k-N+i\mid k}+K_{k-N+i\mid k}{\delta }_{k-N+i\mid k},\hfill \end{array}$$

(32)

$$\begin{array}{ccc}\hfill P_{k-N+i\mid k}^{+}& =& \left(I-K_{k-N+i\mid k}{C}_{k-N+i}\right)P_{k-N+i\mid k},\hfill \end{array}$$

(33)

where ${\lambda }_{k-N+i}$ is obtained from (21), (22), and (27) with the estimated residual covariance ${\widehat{\Delta }}_{k-N+i\mid k}$ obtained by using the recent finite measurements on the horizon $[k-Nk]$ as

$${\widehat{\Delta }}_{k-N+i\mid k}=\frac{1}{N+1}\left(\sum _{j=0}^i{\delta }_{k-N+j\mid k}{\delta }_{k-N+j\mid k}^T+\sum _{j=i+1}^N{\delta }_{k-N+j\mid k-1}{\delta }_{k-N+j\mid k-1}^T\right).$$

(34)

Similar to the design of the filter, the predicted residual covariance can be sequentially obtained for reducing the computational cost as

$$\begin{array}{ccc}\hfill {\widehat{\Delta }}_{k-N+i\mid k}& =& {\widehat{\Delta }}_{k-N+i-1\mid k}+\frac{1}{N+1}\left({\delta }_{k-N+i\mid k}{\delta }_{k-N+i\mid k}^T-{\delta }_{k-N+i\mid k-1}{\delta }_{k-N+i\mid k-1}^T\right),\hfill \end{array}$$

(35)

where the initial value is obtained from the result of the previous horizon as ${\widehat{\Delta }}_{k-N\mid k}={\widehat{\Delta }}_{k-1\mid k-1}$ and

$$\begin{array}{c}\hfill {\widehat{\Delta }}_{N\mid N}=\frac{1}{N+1}\sum _{i=0}^N{\delta }_{i\mid N}{\delta }_{i\mid N}^T,\end{array}$$

(36)

for the first horizon $\left[1N\right]$.

During the time interval $[k-hk]$, the a priori and a posteriori estimates and their corresponding covariance matrices are stored in memory for the backward RTS smoothing, where h is the fixed-lag size.

The initial conditions of forward recursion are calculated from the following recursive equations [14]:

$$\begin{array}{ccc}\hfill {\widehat{x}}_{k-N-1\mid k}^{+}& =& {\left(P_{k-N-1\mid k}^{+}\right)}^{-1}{\widehat{w}}_{k-N-1\mid k}^{+}=S_{k-N-1\mid k}^{+}{\widehat{w}}_{k-N-1\mid k}^{+},\hfill \end{array}$$

(37)

$$\begin{array}{ccc}\hfill P_{k-N-1\mid k}^{+}& =& (S_{k-N-1\mid k}^{+}{)}^{-1}\hfill \end{array}$$

(38)

where ${\widehat{w}}_{k-N-1\mid k}^{+}$ and $S_{k-N-1\mid k}^{+}$ are the a posteriori estimate of the information state and the a posteriori information matrix, respectively, obtained from

$$\begin{array}{ccc}\hfill S_{k-l\mid k}& =& C_{k-l}^T{R}_{k-l}^{-1}{C}_{k-l}+S_{k-l\mid k}^{+},\hfill \\ \hfill S_{k-l-1\mid k}^{+}& =& A_{k-l-1}^T{S}_{k-l\mid k}{A}_{k-l-1}-A_{k-l-1}^T{S}_{k-l\mid k}{G}_{k-l}(Q_{k-l}^{-1}\hfill \end{array}$$

(39)

$$\begin{array}{ccc}& & +G_{k-l}^T{S}_{k-l\mid k}{G}_{k-l}{)}^{-1}{G}_{k-l}^T{S}_{k-l\mid k}{A}_{k-l-1},\hfill \end{array}$$

(40)

$$\begin{array}{ccc}\hfill {\widehat{w}}_{k-l\mid k-1}& =& C_{k-l}^T{R}_{k-l}^{-1}{y}_{k-l}+{\widehat{w}}_{k-l\mid k}^{+},\hfill \\ \hfill {\widehat{w}}_{k-l-1\mid k}^{+}& =& A_{k-l-1}^T{\widehat{w}}_{k-l\mid k}-A_{k-l-1}^T{S}_{k-l\mid k}{G}_{k-l}(Q_{k-l}^{-1}+G_{k-l}^T\hfill \end{array}$$

(41)

$$\begin{array}{ccc}& & \times S_{k-l\mid k}{G}_{k-l}{)}^{-1}{G}_{k-l}^T{\widehat{w}}_{k-l\mid k},\hfill \end{array}$$

(42)

with the initial conditions ${S_{k\mid k}}^{+}=0$ and ${\widehat{w}}_{k\mid k}^{+}=0$.

3.2. Backward RTS Smoothing

The backward recursion of the RTS smoothing algorithm runs backward from current time k down to the fixed-lag time h to estimate the fixed-lag state ${\widehat{x}}_{k-h\mid k}^{b+}$, in which the superscript b denotes backward estimation.

The backward state and its error covariance matrix conditions of backward smoothing are initialized as ${\widehat{x}}_{k\mid k}^{b+}={\widehat{x}}_{k\mid k}^{+}$ and $P_{k\mid k}^{b+}=P_{k\mid k}^{+}$, respectively, which are obtained from the forward filtering pass. Then, the smoothed estimates and their corresponding covariance matrices are calculated recursively from k down to h as [14]

$$\begin{array}{ccc}\hfill K_{k-l\mid k}^b& =& P_{k-l\mid k}^{+}{A}_{k-l+1}{P}_{k-l+1\mid k}^{-1}\hfill \end{array}$$

(43)

$$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\begin{array}{ccc}\hfill {\widehat{x}}_{k-l\mid k}^{b+}& =& {\widehat{x}}_{k-l\mid k}^{+}+K_{k-l\mid k}^b\left({\widehat{x}}_{k-l+1\mid k}^{+}-{\widehat{x}}_{k-l\mid k}\right),\hfill \end{array}$$

(44)

$$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\begin{array}{ccc}\hfill P_{k-l\mid k}^{+b}& =& P_{k-l\mid k}^{+}-K_{k-l\mid k}^b\left(P_{k-l+1\mid k}-P_{k-l+1\mid k}^{+}\right){\left(K_{k-l\mid k}^b\right)}^T,\hfill \end{array}$$

(45)

for $1\le l\le h$, where $x_{k-l\mid k}^{+b}$ and $P_{k-l\mid k}^{+b}$ are the backward state estimate and corresponding covariance matrix, respectively, and $K_{k-l\mid k}^b$ is the gain matrix of the backward smoother.

4. Adaptive Horizon Length Adjustment Method and Switching Adaptive Fading-Memory Receding-Horizon Filter

The horizon length influences the estimation performance of the RH filter. In particular, the computational cost of the RH filter can be considerably reduced by using a short horizon length.

If modeling uncertainties or numerical errors exist, the RH filter with a short horizon length could generate less estimation error and computational complexity than that with a long horizon length. Moreover, as it could provide faster tracking speed than the long-horizon RH filter when model uncertainty disappears, the RH filter with a short horizon is suitable for cases with temporary model uncertainties or numerical errors [33].

By contrast, when the model represents the real system well and numerical errors do not exist, there is a significant trade-off between estimation accuracy and computational cost. In this situation, as the horizon length of the RH filter increases, its estimation accuracy increases, but the computational cost also significantly increases. The Kalman filter could be the proper choice instead of the RH filter with a long horizon length when the system model is accurate because the estimate of the RH filter converges to that of the Kalman filter with increasing horizon length and the computational cost of the Kalman filter is significantly smaller than that of the RH filter.

Accordingly, a simple but efficient way to enhance the computational efficiency and estimation accuracy is to adopt the AHLA method and use the Kalman filter instead of the RH filter with a long horizon length when the system model is accurate. Therefore, a SAFMRH filter based on an AHLA method and the adaptive fading factor is proposed.

First, an AFMRH filtering algorithm with a novel AHLA method is introduced.

The main idea of the AHLA method is the circulative set of three sequentially obtained estimates. During the three time steps, the three estimates are sequentially obtained from the AFMRH filter with different horizon lengths; then their absolute residual values are compared to find the most suitable one. The horizon length of the estimate that has the minimum of the absolute residual value is selected for the next cycle. The concept of the AHLA method is depicted in Figure 1.

The detailed AHLA method is as follows (explained in case of estimating the state in the cycle $[k-2k]$):

• At time $k-2$, the horizon initial states ${\widehat{x}}_{k-N_{k-2,k}-2\mid k-2}$, ${\widehat{x}}_{k-N_{k-2,k}-1\mid k-2}$, and ${\widehat{x}}_{k-N_{k-2,k}\mid k-2}$ and their error covariance matrices corresponding to the estimates of the states $x_{k-2}$, $x_{k-1}$, and $x_k$, respectively, are obtained from backward information filtering (6)–(9) with the initial conditions ${\widehat{w}}_{k-2\mid k-2}=0$ and $S_{k-2\mid k-2}=0$, where $N_{k-2,k}$ is the horizon length of the AFMRH filter for estimating the state in the time interval $[k-2k]$. Note that the horizon initial conditions ${\widehat{x}}_{k-N_{k-2,k}-2\mid k-2}$ and ${\widehat{x}}_{k-N_{k-2,k}\mid k-2}$ are obtained by using measurements $[k-N_{k-2,k}-2k-2]$, not $[k-N_{k-2,k}-2k-3]$ and $[k-N_{k-2,k}-1k-1]$, respectively.
  At the same time, ${\widehat{x}}_{k-2\mid k-2}$ is obtained from the Kalman filtering recursion (3) and (4). Additionally, the predicted estimate ${\widehat{x}}_{k-1\mid k-2}$ is obtained from Equation (3) with ${\widehat{x}}_{k-2\mid k-2}$ and $P_{k-2\mid k-2}$.
  It is also noted that ${\widehat{x}}_{k-1\mid k-2}$ is obtained from the AFMRH filter with the horizon length $N_{k-2,k}+1$.
• At time $k-1$, ${\widehat{x}}_{k-1\mid k-2}$ is obtained from the Kalman filtering recursion given by Equations (3) and (4) with the initial conditions ${\widehat{x}}_{k-N_{k-2,k}-1\mid k-2}$ and $P_{k-N_{k-2,k}-1\mid k-2}$, which are obtained in the previous step.
  Note that ${\widehat{x}}_{k-1\mid k-2}$ is obtained from the AFMRH filter with the horizon length $N_{k-2,k}$.
• At time k, ${\widehat{x}}_{k\mid k-1}$ is obtained from the Kalman filtering recursion given by Equations (3) and (4) with the initial conditions ${\widehat{x}}_{k-N_{k-2,k}\mid k-2}$ and $P_{k-N_{k-2,k}\mid k-2}$, which are already obtained at $k-2$.
  At the same time, ${\widehat{x}}_{k-1\mid k-1}$ is stored in the estimation process. Moreover, the absolute values of residuals $|{\delta }_{k-1\mid k-2}^{+}|$, $|{\delta }_{k-1\mid k-2}^c|$, and $|{\delta }_{k\mid k-1}^{-}|$ corresponding to ${\widehat{x}}_{k-1\mid k-2}$, ${\widehat{x}}_{k-1\mid k-2}$, and ${\widehat{x}}_{k-1\mid k-1}$, respectively, are obtained from Equation (11).
  Note that ${\widehat{x}}_{k-1\mid k-1}$ is obtained from the AFMRH filter with the horizon length $N_{k-2,k}-1$.
• Thereafter, the horizon length of the estimate that has the minimum of the absolute residual value is selected as the horizon length $N_{k+1,k+3}$ for the next cycle $[k+1k+3]$.
  Note that the horizon length must be in the interval $[N_{min},N_{max}]$, where $N_{min}$ is the order of the system and $N_{max}$ is the maximum allowed horizon length considered as the design parameter.

Second, the SAFMRH filtering algorithm is introduced by combining the proposed AHLA method and switching filter strategy. Although the horizon length is selected properly by the AHLA method, a large horizon length may cause an unacceptable computational load for the RH filter. Because the estimate of the RH filter converges to that of the Kalman filter as the horizon increases and Kalman filter has less computational complexity than the RH filter, the computation efficiency and estimation accuracy could be improved by using the Kalman filter instead of the RH filter when the model uncertainty does not exist. Thus, the switching rule of the Kalman filter and the proposed AFMRH filter with the AHLA method is as follows:

• In cases in which $\lambda \le 1$ and $N\ge N_{max}$ indicate that the system model is reliable, the estimate and its error covariance are obtained directly from the Kalman filter given by Equations (3) and (4).
• By contrast, for $\lambda >1$ or $N<N_{max}$, the estimate and its error covariance matrix are obtained from the proposed AFMRH filter with the AHLA method. If the filter is switched from the Kalman filter to the AFMRH filter, the initial value of the horizon length is always taken as the nominal one, which is the design parameter.

5. Simulation Results and Discussion

In this section, several simulation results are presented to verify the effectiveness and performance of the proposed state-estimation algorithms.

To verify the adaptiveness and robustness of the proposed estimation algorithms, numerical simulations were performed for the discrete F-404 gas turbine aircraft engine model with a temporary modeling uncertainty [33] as

$$x_{k+1}=\left[\begin{array}{ccc}0.9305+{\delta }_k& 0& 0.1107\\ 0.0077& 0.9802+{\delta }_k& -0.0173\\ 0.0142& 0& 0.8953+{\delta }_k\end{array}\right]x_k+\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]w_k,$$

(46)

$$\begin{array}{ccc}\hfill y& =& \left[\begin{array}{ccc}1+0.1{\delta }_k& 0& 0\\ 0& 1+0.1{\delta }_k& 0\end{array}\right]x_k+v_k,\hfill \end{array}$$

(47)

where the initial state is taken as $x_0=[000{]}^T$ and the actual process and measurement noise covariance matrices are taken as $Q=0.2^2$ and $R=0.{01}^2{I}_{2\times 2}$, respectively.

To show the estimation performance of the proposed methods in the case of model mismatch, the model’s uncertain parameter ${\delta }_k$ is defined as

$$\begin{array}{c}\hfill {\delta }_k=\left\{\begin{array}{cc}0.1,& 200\le k\le 250,\\ 0,& otherwise.\end{array}\right.\end{array}$$

All simulations presented in this section were performed using MATLAB R2020b on an Intel i5-9400 CPU, 2.90-GHz, 64-bit machine with 8-GB RAM.

First, to verify the effectiveness of the proposed AFMRH filter, the proposed AFMRH filter was compared with a standard Kalman filter [3], recursive optimal RH filter [13], and AFM Kalman filter [22]. The initial state and its error covariance for both the standard Kalman filter and AFM Kalman filter were set as $x_0=[111{]}^T$ and $P_0={10}^3{I}_{3\times 3}$, respectively, and the horizon length for both the standard recursive optimal RH filter and the proposed AFMRH filter was taken as $N=20$. Moreover, the minimum and maximum allowed horizon lengths were taken as $N_{min}=3$ and $N_{max}=25$, respectively.

The second state-estimation errors of four filters are shown in Figure 2.

Figure 2 shows that the proposed AFMRH filter successfully estimates the real state and has better estimation performance compared to others in the case of model mismatch. Moreover, for the model mismatch, both the adaptive filters have small estimation errors and fast convergence property compared with the nonadaptive filter. In particular, the proposed AFMRH filter progressively reduces the estimation error compared with that of the AFM Kalman filter when model uncertainty exists. Moreover, the estimation errors of the proposed AFMRH filter rapidly decrease even though the model uncertainty exists, whereas those of the AFM Kalman filter continuously increase. This result shows that the proposed AFMRH filter could possess early adaptability to changes in the system because its adaptive fading factors and the estimates are obtained by using the finite measurements in the future (based on the estimation time).

In Figure 3, the changes in adaptive fading factors of the proposed AFMRH filter are also compared with those of the AFM Kalman filter. The results reveal that the adaptive fading factor could slowly converge to 1 because the accumulated estimation error of the AFM Kalman filter affects the determination of the adaptive fading factor due to its estimation structure. This means that the estimate of the AFM Kalman filter slowly converges to the optimal one.

However, as the proposed filter is designed with an RH estimation structure, which estimates the state using only the recent finite measurements, the estimates of the RH filter are not affected by previous estimation errors. Thus, the proposed AFMRH filter is expected to possess better adaptiveness than the AFM Kalman filter. As expected, the adaptive fading factor of the proposed filter converges to 1 much faster than that of the AFM Kalman filter. This result shows that the AFM method enhances the estimation performance of the RH structured filter and facilitates better adaptiveness than the Kalman filter with the AFM method. Moreover, the filtering structure was also shown to affect the determination of the adaptive fading factor.

Second, the effectiveness of the proposed AHLA method was verified by comparing the simulation results of the recursive optimal RH filter with those of the proposed AHLA method and a standard one.

Figure 4 and Figure 5 show the horizon lengths and estimation errors of a standard recursive optimal RH filter and the recursive optimal RH filter with the proposed AHLA method, respectively. As seen in Figure 5, the horizon length increases when the model uncertainty does not exist $(k<200)$, whereas it decreases when the model uncertainty exists $(200\le k\le 250)$ with the passage of time. This result shows that the proposed AHLA method provides an adaptively adjusted and suitable horizon length for the model mismatch. Moreover, the simulation result in Figure 4 reveals that the estimation errors of the recursive optimal RH filter with the proposed AHLA method are significantly smaller than those of a standard recursive optimal RH filter in the case of model mismatch. Thus, a suitable horizon length could be obtained using the proposed AHLA method to improve the estimation performance of the RH filter.

Third, to verify the accuracy and computational efficiency of the proposed AFMRH filter with AHLA method and SAFMRH filter, their estimation results are compared with those of the proposed AFMRH filter and a constant fading RH (CFMRH) filter [8]. The constant fading factor for the CFMRH filter is taken as $1.02$.

The second state-estimation errors and time-averaged values of the root-mean-square estimation (RMSE) errors are compared in Figure 6 and

Table 1. RMSE errors $(\times {10}^{-2}{}^{\circ }\mathrm{C})$.

Time Interval	$\left[50500\right]$	$\left[50200\right]$	$\left[201350\right]$
Standard Kalman filter	$155.011$	1714	$271.428$
AFM Kalman filter	$33.076$	1714	$57.922$
Recursive optimal RH filter	$42.600$	1809	$74.641$
AFMRH filter	5368	1809	9138
CFMRH filter	$11.880$	1930	$20.678$
AFMRH filter + AHLA	$\mathbf{3798}$	1782	$\mathbf{6266}$
SAFMRH filter	4754	$\mathbf{1772}$	7994

, respectively.

In Figure 6, it can be easily observed that the estimation errors of the proposed AFMRH filter with AHLA method and SAFMRH filter are remarkably smaller than those of CFMRH filter in the case of model mismatch. Moreover, it is also observed that the estimation errors of the proposed AFMRH filter with AHLA method and SAFMRH filter are smaller than the proposed AFMRH filter, as expected. In particular, the estimation errors of the proposed AFMRH filter with AHLA method and SAFMRH filter quickly converge to zero compared with others RH filters after the model uncertainty disappear.

A comparison of all RMSE errors reveals that RMSE errors of the proposed AFMRH filter with the AHLA method and SAFMRH filter are smaller than other fading-memory RH filters irrespective of whether model uncertainty exists. For $k<200$, i.e., in the situation where the model uncertainty does not exist, the proposed SAFMRH filter has the smallest value of RMSE error. For $200\le k\le 350$, i.e., when the model mismatch exists, the RMSE errors of the proposed AFMRH filter with the AHLA method and SAFMRH filter are much smaller than that of others.

The reasons for this result are as follows: because the switching condition could be met when the model mismatch does not exist and the Kalman filter is used instead of the RH filter with a long horizon length, the SAFMRH filter could provide a more accurate estimate than other fading-memory RH filters. Moreover, because the horizon length in the AHLA method is considered as a design parameter, the AHLA method-based filters yield better estimation performances than the AFMRH filter with constant horizon length. Furthermore, when the AHLA method is incorporated in its algorithm, the proposed SAFMRH filter could also provide better estimates concerning whether the model uncertainty exists or not. Thus, the combination of the AHLA method and switch filtering scheme with AFMRH filtering improves the robustness and estimation accuracy of the proposed AFMRH filter.

Additionally, in the SAFMRH filtering algorithm, the computational cost for backward recursion to find the initial horizon conditions is less than that of the AFMRH filter successively about three times. The computationally efficient Kalman filtering algorithm is used instead of the proposed AFMRH filter with a long horizon length when the conditions are met. Thus, the proposed SAFMRH filtering algorithm is expected to reduce the computational cost of the AFMRH filter. The time-averaged computation times of all the RH filters are compared in

Table 2. Average computation time (ms).

Standard Kalman filter	$0.037$
AFM Kalman filter	$0.056$
Recursive optimal RH filter	$0.506$
CFMRH filter	$1.856$
AFMRH filter	$0.841$
AFMRH filter + AHLA	$0.575$
SAFMRH filter	$\mathbf{0}.\mathbf{359}$

to show the improvement in computational efficiency.

As expected, the results show that the proposed AFMRH filter with AHLA method and SAFMRH filtering algorithm considerably reduces the computation time compared with the AFMRH filter. Moreover, the proposed SAFMRH filter consumes much less computation time than other fading-memory RH filters. Therefore, the proposed AFMRH filter with AHLA method and SAFMRH filter are helpful and appropriate for a system with a temporary model uncertainty.

Finally, the improved performance of the proposed AFMRH smoother is verified by comparing the simulation results of the proposed AFMRH filter, recursive optimal RH smoother [14], and CFMRH smoother [11]. The horizon length and fixed lag size of all the RH smoothers described in this section are taken as $N=20$ and $h=7$, respectively. Further, the constant fading factor for the CFMRH smoother is taken as $0.65$.

The estimation errors of estimation algorithms are shown in Figure 7.

As indicated, the proposed AFMRH smoother provides a more robust estimate than recursive optimal RH and CFMRH smoothers for the model mismatch. Moreover, a comparison of the estimation errors of the proposed AFMRH filter and smoother revealed that the proposed AFMRH smoother exhibited better estimation performance than the proposed AFMRH filter, as expected. Thus, the proposed AFMRH smoother is more efficient and appropriate than the AFMRH filter for a system with model uncertainty if some time delay is tolerable. Furthermore, the proposed AFMRH smoother provides significant improved estimation performance compared to the nonfading and constant-fading-memory RH smoother.

6. Conclusions

In this study, adaptive RH filters and a smoother were proposed for linear discrete time-varying systems. The proposed AFMRH filter was first obtained by combining receding-horizon and adaptive fading-memory methods. The adaptive fading factor was optimally obtained using the measurements and estimates in the recent finite horizon, and applied to the recursive RH filtering algorithm to adjust the state error covariance. Moreover, as an extension of the proposed AFMRH filter, an AFMRHF smoother based on the RTS smoothing algorithm was also proposed to improve the estimation accuracy of the proposed RH filter. Furthermore, an AHLA method and a switching AFMRH filtering method were proposed to improve the estimation accuracy and reduce the computational cost of the proposed AFMRH filter.

Owing to the RH estimation structure, the proposed approaches guarantee BIBO stability and may have a fast convergence speed and robustness against temporary model uncertainties. Moreover, if the fading factor is taken as one, the proposed AFMRH filter and smoother are equivalent to the existing recursive optimal RH filter and smoother. Notably, the proposed AFMRH estimation algorithms can provide a more general solution than the previous optimal recursive RH filter. Furthermore, as the proposed AHLA method uses results in the estimation process, it may have less computational burden and provide a more suitable method for online and real-time applications compared with previous approaches. The proposed algorithms can be applied to state estimation problems in linear discrete-time systems with unknown noise statistics, model uncertainties, and external disturbances. The simulation results show that the adaptive RH estimators proposed in this study can exhibit better estimation performance compared with the Kalman-filter-based AFM filter and existing RH estimators.