A Robust Recursive State Estimation Method for Uncertain Linear Discrete-Time Systems

Abstract

This study presents a robust estimation approach for linear discrete-time systems subject to parametric uncertainties. To address model mismatch, the proposed method enhances the MHE framework, thereby improving estimation accuracy. Based on this framework, the estimator is derived by minimizing the expected estimation error. A detailed derivation is provided, along with a novel recursive formulation for the pseudo-covariance of the estimation error. The resulting estimator maintains structural similarity to the Kalman filter and supports recursive implementation. Theoretical analysis establishes convergence to a stable system, with guaranteed boundedness and asymptotic unbiasedness of the estimation error. Simulation results demonstrate that the proposed strategy maintains high effectiveness and robustness under different uncertain conditions.

1. Introduction

State estimation plays a crucial role in control systems and signal processing, and remains a fundamental topic in industrial automation research. Classical approaches, such as the Extended Kalman filter, Bayesian methods, and particle-based algorithms [1,2,3], have achieved remarkable success under ideal modeling assumptions. However, in practical engineering systems—such as industrial process control, autonomous vehicles, and sensor networks—the model parameters often vary over time due to environmental fluctuations, component aging, or external disturbances. These time-varying parameter uncertainties, coupled with sensor noise and incomplete modeling, make it difficult to obtain an accurate mathematical description of the system. As a result, the performance of conventional Kalman filter-based methods may degrade significantly, leading to inaccurate or unstable state estimation [4,5].

To address the degradation of estimation performance caused by modeling errors and parameter uncertainties, various robust filtering techniques have been developed. Robust filters are designed to ensure reliable estimation under model and parameter inaccuracies. Among them, the $H_{\infty }$ filter is constructed to guarantee robust performance under worst-case disturbances by designing a stable estimator that limits the energy gain from external disturbances to estimation errors [6,7,8]. Another category of robust estimation methods is set-valued approaches, which assume that the measurement disturbances are norm-bounded and construct an observation-consistent ellipsoid around the estimated state [9,10]. In addition, guaranteed-performance estimation frameworks have been proposed to restrict the steady-state error variance within a predefined upper bound under all admissible uncertainty scenarios [11,12]. However, these methods often encounter implementation challenges in online recursive filtering settings.

From the perspective of uncertain model descriptions, several robust state estimation schemes have been proposed. In [13], the Kalman filter was reinterpreted within a regularized least-squares framework, which facilitates recursive computation but still exhibits limitations under certain structured modeling errors. In a separate line of research, an iterative design framework was introduced to balance nominal performance and robustness against parameter uncertainties in filter design [14]. Furthermore, several filtering techniques have been developed for handling parameter uncertainties in discrete-time systems [4,15]. Despite their theoretical appeal, these approaches often suffer from conservatism and involve nontrivial parameter tuning. More recently, a robust state estimation method was proposed in [16] by considering the expected estimation performance under stochastic modeling errors, allowing modeling errors to arbitrarily affect the system parameters and eliminating the need for tuning design parameters.

To improve estimation accuracy in the presence of model mismatch, MHE has been applied in a wide range of systems, including constrained linear systems, nonlinear models, packet-dropout scenarios, and time-delay systems [17]. Building on this framework, a robust Kalman filter based on moving window estimation was introduced in [18], and a cost function within the MHE framework was formulated in [19] to reconstruct the state of linear systems with time-varying parameters. These works demonstrate the potential of combining robust filtering ideas with MHE, but they do not explicitly provide a recursive robust estimator tailored for parametric uncertainties in linear discrete-time systems.

In light of the above, this paper investigates the problem of robust state estimation in the presence of stochastic parameter uncertainties for linear discrete-time systems. Within the MHE framework, the cost function associated with the estimation error is reformulated to suit these systems. This formulation provides the optimal state estimate by minimizing the expected estimation error. A recursive formulation of the robust state estimator is derived. This approach removes the requirement for tuning design parameters. Furthermore, the asymptotic stability conditions of the robust state estimator are established by transforming the discrete Riccati equation.

The main contributions of this article are as follows:

(1)

A moving-horizon recursive robust state estimator is developed for linear discrete-time systems with bounded parametric uncertainties. By reformulating the MHE cost function using an RLS-based interpretation, the proposed estimator admits a Kalman-like recursive implementation and exploits measurements over a fixed window to improve accuracy. The uncertainty is restricted to a known bounded set (e.g., $|{\epsilon }_{k,j}| \le \rho$), which enables theoretical guarantees. When the model error affects the parameter matrix, the estimator can still maintain good estimation performance

(2)

The recursive formulation of the proposed robust state estimator is presented, and its asymptotic stability is demonstrated.

(3)

Vehicle–trailer simulations confirm the effectiveness of the proposed algorithm, with comparative results demonstrating its superiority over the classical Kalman filter.

This paper is structured in the following manner. Section 2 presents a state-space representation incorporating parameter uncertainties. It also details the recursive construction of the proposed estimator. Section 3 explores key properties of the Riccati recursion as foundational results and analyzes the asymptotic characteristics of the estimator. Section 4 presents simulation results on a vehicle-trailer system. Section 5 concludes the study, with detailed proofs given in the Appendix A.

Notations: The symbol $∥x∥$ indicates the Euclidean norm, and ${∥x∥}_W^2$ refers to the weighted 2-norm computed as $x^{T}Wx$, where W is a symmetric positive definite matrix. The Euclidean space of dimension m is denoted by ${\mathbb{R}}^m$. The notation $diag\{·\}$ denotes a block-diagonal matrix, while $col\{·\}$ denotes the vertical stacking of vectors. The operator $\mathbb{E}\{·\}$ represents the expected value. $X^{\left(n\right)}$ denotes the result of state estimation obtained using a moving horizon of length n.

2. Dynamic System Representation and Design of Robust State Estimator

2.1. State Space Model

A linear discrete-time system $\Delta$ subject to parametric modeling errors due to imperfect knowledge of the system dynamics is considered. The system evolution is governed by the following discrete-time model:

$$\Delta :\left\{\begin{array}{c}{x}_{k+1}=A_k\left({\epsilon }_k\right)x_k+B_k\left({\epsilon }_k\right)w_k,\hfill \\ y_k=Z_k\left({\epsilon }_k\right)x_k+v_k,\hfill \end{array}\right.k\ge 0.$$

(1)

where the vectors $x_k\in {\mathbb{R}}^n$ and $y_k\in {\mathbb{R}}^m$ denote the system’s internal state and the corresponding measurement output, respectively. The process disturbance $w_k$ and observation error $v_k$ are assumed to be mutually independent zero-mean Gaussian white noise processes. In addition, $x_0$, along with $w_k\in {\mathbb{R}}^n$ and $v_k\in {\mathbb{R}}^m$, are assumed to be mutually uncorrelated, satisfying the following covariance structure: $\mathbb{E}\left\{\mathrm{col}\left\{x_0-\mathbb{E}\left\{x_0\right\},w_k,v_k\right\}{(\ast )}^T\right\}=\mathrm{diag}\left\{{\Pi }_0,Q_k{\delta }_{kj},R_k{\delta }_{kj}\right\}$, where ${\Pi }_0$, $Q_k$, and $R_k$ denote the covariance matrices of $x_0$, $w_k$, and $v_k$, respectively, and ${\delta }_{kj}$ is the Kronecker delta. The model uncertainty at each time k is characterized by the vector ${\epsilon }_k$, composed of J mutually independent real-valued scalar parameters ${\epsilon }_{k,j}$, for $j=1,\dots ,J$. Specifically, ${\epsilon }_k$ is assumed to satisfy $|{\epsilon }_k| \le \rho ,\forall k\ge 0$, where $\rho >0$ is a known bound. The system matrices are assumed to depend affinely on ${\epsilon }_k$. In particular, the state transition matrix takes the form $A_k\left({\epsilon }_k\right)=A_k\left(0\right)+{\epsilon }_k$, where $A_k\left(0\right)$ denotes the nominal system matrix. These uncertainties are embedded within the system matrices $A_k\left({\epsilon }_k\right)$, $B_k\left({\epsilon }_k\right)$, and $Z_k\left({\epsilon }_k\right)$, each having appropriate dimensions and incorporating the time-varying modeling perturbations.

Remark 1.

In the system model (1), the modeling error ${\epsilon }_k$ is allowed to affect the parameter matrices in a general manner, while being restricted to a known bounded set. By comparison, Ref. [13] confines the uncertainty to a linear dependence on a norm-bounded uncertainty matrix, whereas Ref. [20] assumes the system matrices are differentiable with respect to ${\epsilon }_k$. Hence, model (1) is less restrictive and offers greater modeling generality, making it better suited to represent practical time-varying dynamics than the formulations in [13,20].

2.2. Robust State Estimator Design

For discrete-time linear systems with well-defined state-space representations, the Extended Kalman filter is commonly utilized to perform dynamic state inference [1]. Model inaccuracies can degrade estimation accuracy. To address this, various robust filtering methods have been proposed to handle uncertainties [21,22].

As shown in Ref. [13], a general framework is developed for state-space estimation when the parameters of the underlying linear model are subject to uncertainties, in which each time-update and measurement-update step of the Kalman filter admits a deterministic interpretation as the solution to a regularized least-squares problem. Therefore, the Kalman filter represents a particular instance of the RLS framework, which is illustrated by Equations (2) and (3).

$${\widehat{x}}_{k+1|k+1}=A_k\left(0\right){\widehat{x}}_{k|k+1}+B_k\left(0\right){\widehat{w}}_{k|k+1},$$

(2)

$$\left[\begin{array}{c}{\widehat{x}}_{k|k+1}\\ {\widehat{w}}_{k|k+1}\end{array}\right]=arg\underset{x_k,w_k}{min}\left[\begin{array}{c}{∥x_k-{\widehat{x}}_{k|k}∥}_{P_{k|k}^{-1}}^2+{∥w_k∥}_{Q_k^{-1}}^2+{∥y_{k+1}-Z_{k+1}\left(0\right)x_{k+1}∥}_{R_{k+1}^{-1}}^2\hfill \end{array}\right].$$

(3)

Equation (3) illustrates the following process: starting from an initial estimate ${\widehat{x}}_{k|k}$ for the state $x_k$, $P_{k|k}$ represents the corresponding estimation error covariance matrix. The estimate is subsequently updated based on the new observation $y_{k+1}$.

Moving horizon strategies are developed for constrained linear state estimation, where additional information for estimating state variables from output measurements is incorporated in the form of inequality constraints on states, noise, and other variables [17]; this formulation is adopted in the present work.

Provided that the aforementioned problem is solvable at time k, then the corresponding state estimates for the time indices $i\in \{k-n,\dots ,k\}$ can be subsequently determined. The new system cost function is derived:

$$\left[\begin{array}{c}{\widehat{x}}_{k-n|i+1}\\ {\widehat{w}}_{i|i+1}\end{array}\right]=\underset{\begin{array}{c}{x}_{k-n},w_k\hfill \\ i=k-n,\dots ,k\hfill \\ k>n\hfill \end{array}}{ \arg \min }\left[{∥x_{k-n}-{\widehat{x}}_{k-n}∥}_{{\left(P_{k-n}^{\left(n\right)}\right)}^{-1}}^2\right.\left.+\sum _{i=k-n}^k\left({∥w_i∥}_{Q_i^{-1}}^2+{∥y_{i+1}-Z_{i+1}\left(0\right)x_{i+1}∥}_{R_{i+1}^{-1}}^2\right)\right].$$

(4)

For cases where the modeling error is nonzero (${\epsilon }_0\ne 0$), Equation (4) is employed for state estimation, and the cost function is refined to incorporate the expected estimation performance under random modeling errors. In the cost function, ${\alpha }_k$ is defined as ${\alpha }_k=col\left\{x_{k-n}-{\widehat{x}}_{k-n},{\widehat{w}}_{i|i+1}\right\}$.

$$\begin{array}{cc}J({\alpha }_k)=\mathbb{E}\left[{∥x_{k-n}-{\widehat{x}}_{k-n}∥}_{{\left(P_{k-n}^{\left(n\right)}\right)}^{-1}}^2\right.& \hfill \left.+\sum _{i=k-n}^k\left({∥w_i∥}_{Q_i^{-1}}^2+{∥y_{i+1}-Z_{i+1}\left({\epsilon }_{i+1}\right)x_{i+1}∥}_{R_{i+1}^{-1}}^2\right)\right].\end{array}$$

(5)

The resulting cost function is formulated as shown in Equation (5). To simplify the representation of Equation (5), several auxiliary matrices are introduced as follows.

$$\begin{array}{c}\begin{array}{c}{A}_{h,k}\left({\epsilon }_{h,k}\right)=\left[\begin{array}{c}{A}_{k-n}\left({\epsilon }_{k-n}\right)\\ A_{k-n+1}\left({\epsilon }_{k-n+1}\right)A_{k-n}\left({\epsilon }_{k-n}\right)\\ ⋮\\ A_k\left({\epsilon }_k\right)\dots A_{k-n}\left({\epsilon }_{k-n}\right)\end{array}\right],R_{h,k+1}^{-1}=diag\left\{R_{k-n+1}^{-1},\dots ,R_{k+1}^{-1}\right\},\hfill \\ Z_{h,k+1}\left({\epsilon }_{h,k+1}\right)=diag\left\{Z_{k-n+1}\left({\epsilon }_{k-n+1}\right),\dots ,Z_{k+1}\left({\epsilon }_{k+1}\right)\right\},Q_{h,k}^{-1}=diag\left\{Q_{k-n}^{-1},\dots ,Q_k^{-1}\right\},\hfill \\ {\Phi }_{h,k}=diag\left\{{\left(P_{k-n|k}^{\left(n\right)}\right)}^{-1},Q_{k-n}^{-1},\dots ,Q_k^{-1}\right\},{\alpha }_{h,k}=col\left\{x_{k-n}-{\widehat{x}}_{k-n|k+1},{\widehat{w}}_{k-n|k-n+1},\dots ,{\widehat{w}}_{k|k+1}\right\},\hfill \\ {\beta }_{h,k+1}\left({\epsilon }_{h,k+1}\right)=\left[\begin{array}{c}{y}_{k-n+1}\\ ⋮\\ y_{k-n+i}\\ ⋮\\ y_{k+1}\end{array}\right]-\left[\begin{array}{c}{Z}_{k-n+1}\left({\epsilon }_{k-n+1}\right)A_{k-n}\left({\epsilon }_{k-n}\right)\\ ⋮\\ Z_{k-n+i}\left({\epsilon }_{k-n+i}\right)A_i\left({\epsilon }_i\right)\dots A_{k-n}\left({\epsilon }_{k-n}\right)\\ ⋮\\ Z_{k+1}\left({\epsilon }_{k+1}\right)A_k\left({\epsilon }_k\right)\dots A_{k-n}\left({\epsilon }_{k-n}\right)\end{array}\right]{\widehat{x}}_{k-n|k},\hfill \\ B_{h,k}\left({\epsilon }_{h,k}\right)=\left[\begin{array}{ccccc}{B}_{k-n}\left({\epsilon }_{k-n}\right)& & & & \\ A_{k-n+1}\left({\epsilon }_{k-n+1}\right)B_{k-n}\left({\epsilon }_{k-n}\right)& \ddots & & & \\ A_i\left({\epsilon }_i\right)\dots A_{k-n+1}\left({\epsilon }_{k-n+1}\right)B_{k-n}\left({\epsilon }_{k-n}\right)& B_{\left(i,2\right)}& B_i& & \\ ⋮& ⋮& ⋮& \ddots & \\ A_k\left({\epsilon }_k\right)\dots A_{k-n+1}\left({\epsilon }_{k-n+1}\right)B_{k-n}\left({\epsilon }_{k-n}\right)& B_{\left(k,2\right)}& B_{\left(k,3\right)}& \dots & B_k\left({\epsilon }_k\right)\end{array}\right]\hfill \\ B_{\left(i,2\right)}=A_i\left({\epsilon }_i\right)\dots A_{k-n+2}\left({\epsilon }_{k-n+2}\right)B_{k-n+1}\left({\epsilon }_{k-n+1}\right)\hfill \\ B_{\left(k,2\right)}=A_k\left({\epsilon }_k\right)\dots A_{k-n+2}\left({\epsilon }_{k-n+2}\right)B_{k-n+1}\left({\epsilon }_{k-n+1}\right)\hfill \\ B_{\left(k,3\right)}=A_k\left({\epsilon }_k\right)\dots A_{k-n+3}\left({\epsilon }_{k-n+3}\right)B_{k-n+2}\left({\epsilon }_{k-n+2}\right)\hfill \end{array}\end{array}$$

From these definitions, the cost function is subsequently derived.

$$J\left({\alpha }_{h,k}\right)=\left[\begin{array}{c}{∥{\alpha }_{h,k}∥}_{{\Phi }_{h,k}}^2+\mathbb{E}\left[{∥\begin{array}{c}{\Lambda }_{h,k+1}\left({\epsilon }_{h,k+1}\right){\alpha }_{h,k}-{\beta }_{h,k+1}\left({\epsilon }_{h,k+1}\right)\hfill \end{array}∥}_{R_{h,k+1}^{-1}}^2\right]\hfill \end{array}\right].$$

(6)

Examination of the cost function indicates that, without modeling errors, the proposed method reduces to the standard MHE. In particular, when the horizon length is set to $N=0$, the estimator collapses to the classical Kalman filter.

Given the convexity of the cost function in Equation (6), the global optimum ${\alpha }_{h,k}$ can be obtained by $\partial J\left({\alpha }_{h,k}\right)/\partial {\alpha }_{h,k}=0$, defining it as ${\alpha }_{kopt}$. This is given by its first-order derivative condition.

$$\begin{array}{c}\left\{{\Phi }_{h,k}+\mathbb{E}\left\{{\Lambda }_{h,k+1}^T\left({\epsilon }_{h,k+1}\right)R_{h,k+1}^{-1}{\Lambda }_{h,k+1}\left({\epsilon }_{h,k+1}\right)\right\}\right\}{\alpha }_{kopt}\hfill \\ =\mathbb{E}\left\{\begin{array}{c}{\Lambda }_{h,k+1}^T\left({\epsilon }_{h,k+1}\right)R_{h,k+1}^{-1}\left(\begin{array}{c}{y}_{h,k+1}-Z_{h,k+1}\left({\epsilon }_{h,k+1}\right)A_{h,k}\left({\epsilon }_{h,k}\right){\widehat{x}}_{k-n|k}\hfill \end{array}\right)\hfill \end{array}\right\}.\hfill \end{array}$$

(7)

where $y_{h,k+1}=col\left(y_{k-n+1},y_{k-n+2},\dots ,y_{k+1}\right)$.

For subsequent derivations, the following auxiliary matrices are introduced.

$$\begin{array}{c}\mathbb{E}\left\{{\Lambda }_{h,k+1}^T\left({\epsilon }_{h,k+1}\right)R_{h,k+1}^{-1}{\Lambda }_{h,k+1}\left({\epsilon }_{h,k+1}\right)\right\}={\Omega }_{h,k1},\mathbb{E}\left\{{\Lambda }_{h,k+1}^T\left({\epsilon }_{h,k+1}\right)\right\}={\Omega }_{h,k2},\hfill \\ \mathbb{E}\left\{\begin{array}{c}{\Lambda }_{h,k+1}^T\left({\epsilon }_{h,k+1}\right)R_{h,k+1}^{-1}{Z}_{h,k+1}\left({\epsilon }_{h,k+1}\right)A_{h,k}\left({\epsilon }_{h,k}\right)\hfill \end{array}\right\}={\Omega }_{h,k3}.\hfill \end{array}$$

(8)

The estimation of the system state under stochastic parameter uncertainty can be calculated using the following recursive algorithm.

(1)

Initialization. Set ${\widehat{x}}_{0|0}$ and $P_{0|0}$ as follow:

$$\begin{array}{c}{\widehat{x}}_{0|0}=P_{0|0}\mathbb{E}\left\{Z_0^T\left({\epsilon }_0\right)\right\}{R}_0^{-1}{y}_0,P_{0|0}={\left({\Pi }_0^{-1}+\mathbb{E}\left\{Z_0^T\left({\epsilon }_0\right)R_0^{-1}{Z}_0\left({\epsilon }_0\right)\right\}\right)}^{-1}.\hfill \end{array}$$

(9)

(2)

Parameter modification. Define the matrices $H_k$ as follows. Replace matrices $A_k\left(0\right)$, $B_k\left(0\right)$, $P_{k|k}$ and $Q_k$ by:

$$\begin{array}{c}{H}_k=\left[\begin{array}{cc}{H}_{k11}^{\left(n\right)}& H_{k12}^{\left(n\right)}\\ H_{k21}^{\left(n\right)}& H_{k22}^{\left(n\right)}\end{array}\right]={\Omega }_{h,k1}-{\left(Z_{k+1}\left(0\right)\left(\begin{array}{cc}{A}_{h,k}\left(0\right)& B_{h,k}\left(0\right)\end{array}\right)\right)}^T\hfill \\ \times R_{k+1}^{-1}{C}_{k+1}\left(0\right)\left(\begin{array}{cc}{A}_{h,k}\left(0\right)& B_{h,k}\left(0\right)\end{array}\right),\hfill \\ {\widehat{P}}_{k-n|k}^{\left(n\right)}={\left({\left(P_{k-n|k-n}^{\left(n\right)}\right)}^{-1}+H_{k11}^{\left(n\right)}\right)}^{-1},{\widehat{Q}}_k={\left(Q_k^{-1}+H_{k22}^{\left(n\right)}-H_{k21}^{\left(n\right)}{\widehat{P}}_{k-n|k}^{\left(n\right)}{H}_{k12}^{\left(n\right)}\right)}^{-1},\hfill \\ {\widehat{A}}_{h,k}\left(0\right)=\left(A_{h,k}\left(0\right)-{\widehat{B}}_{h,k}\left(0\right){\widehat{Q}}_k{H}_{k21}^{\left(n\right)}\right)\left(I-{\widehat{P}}_{k-n|k}^{\left(n\right)}{H}_{k11}^{\left(n\right)}\right),{\widehat{B}}_{h,k}\left(0\right)=B_{h,k}\left(0\right)-A_{h,k}\left(0\right){\widehat{P}}_{k-n|k}^{\left(n\right)}{H}_{k12}^{\left(n\right)}.\hfill \end{array}$$

(10)

(3)

State estimation updating.

$$\begin{array}{c}{P}_{k+1|k}^{\left(n\right)}=\left[\begin{array}{c}{A}_{h,k}\left(0\right){\widehat{P}}_{k-n|k}^{\left(n\right)}{A}_{h,k}^T\left(0\right)+{\widehat{B}}_{h,k}\left(0\right){\widehat{Q}}_k{\widehat{B}}_{h,k}^T\left(0\right)\hfill \end{array}\right],R_{eh,k+1}=R_{h,k+1}^{-1}+Z_{h,k+1}\left(0\right)P_{k+1|k}^{\left(n\right)}{Z}_{h,k+1}^T\left(0\right),\hfill \\ P_{k+1|k+1}^{\left(n\right)}=P_{k+1|k}^{\left(n\right)}-\left(\begin{array}{c}{P}_{k+1|k}^{\left(n\right)}{Z}_{h,k+1}^T\left(0\right)R_{eh,k+1}^{-1}{Z}_{h,k+1}\left(0\right)P_{k+1|k}^{\left(n\right)}\hfill \end{array}\right),\hfill \\ {\widehat{x}}_{k+1|k+1}={\widehat{A}}_{h,k}\left(0\right){\widehat{x}}_{k-n|k}+P_{k+1|k+1}^{\left(n\right)}\left(\begin{array}{c}{\left(P_{k+1|k}^{\left(n\right)}\right)}^{-1}\left(\begin{array}{c}{A}_{h,k}\left(0\right){\widehat{P}}_{k-n|k}^{\left(n\right)}\left[\begin{array}{cc}I& 0\end{array}\right]\hfill \\ +{\widehat{B}}_{h,k}\left(0\right){\widehat{Q}}_k\left[\begin{array}{cc}-H_{k21}^{\left(n\right)}{\widehat{P}}_{k-n|k}^{\left(n\right)}& I\end{array}\right]\hfill \end{array}\right)\hfill \\ \times {\Omega }_{h,k2}{R}_{k+1}^{-1}{y}_{k+1}-Z_{k+1}^T\left(0\right)R_{k+1}^{-1}{Z}_{k+1}\left(0\right){\widehat{A}}_{h,k}\left(0\right){\widehat{x}}_{k-n|k}\hfill \end{array}\right).\end{array}$$

(11)

Remark 2.

For $k<n$, the system state is estimated based on the available limited data. In ${\widehat{x}}_{k+1|k+1}$, the corresponding state estimate at time $k+1$ is obtained, and the final set of estimates represents that for $k+1$. Similarly, for $P_{k+1|k}^{\left(n\right)}$, the last set of estimates is taken as the value corresponding to $k+1$.

3. Stability and Convergence of the Robust Estimator

This section conducts a rigorous investigation into the convergence characteristics of the proposed estimator and demonstrates its asymptotic performance under the framework of robust state estimation applied to uncertain linear discrete-time systems.

3.1. Preliminary Results for Convergence Analysis

Assume modeling errors ${\epsilon }_{k,j}$ are normalized in magnitude and form set $\mathcal{E}$, and $\mathcal{E}=\left\{\epsilon \left|\left|{\epsilon }_{k,j}\right|\le 1,j=1,\dots ,L\right.\right\}$. Two assumptions are introduced to ensure the estimator’s asymptotic behavior.

Assumption 1.

The matrices $A_k\left(0\right)$, $B_k\left(0\right)$, $Z_k\left(0\right)$, $R_k$, $Q_k$, ${\Omega }_{k1}$, and ${\Omega }_{k2}$ are assumed to be time-invariant.

Assumption 2.

The system described in (1) exhibits exponential stability in the Lyapunov sense. Furthermore, associated matrices $A_k\left(0\right)$, $B_k\left(0\right)$, $Z_k\left(0\right)$, $R_k$, $Q_k$, and ${\Pi }_0$ remain uniformly bounded for all $k>0$ and ${\epsilon }_{k,j}\in \mathcal{E}$. This assumption is imposed to establish the limiting and steady-state properties.

Lemma 1.

According to linear algebra principles and the matrix inversion lemma [23], assuming all required matrices are invertible, the identity below holds for $T,U,V,W$.

$$\begin{array}{c}\left[\begin{array}{cc}T& U\\ V& W\end{array}\right]=\left[\begin{array}{cc}I& 0\\ V{T}^{-1}& I\end{array}\right]\left[\begin{array}{cc}T& 0\\ 0& W-V{T}^{-1}U\end{array}\right]\left[\begin{array}{cc}I& T^{-1}U\\ 0& I\end{array}\right],\hfill \\ T{(I+UT)}^{-1}={(I+TU)}^{-1}T,{(T+UVW)}^{-1}=T^{-1}-T^{-1}U{(W{T}^{-1}U+V^{-1})}^{-1}W{T}^{-1}.\hfill \end{array}$$

(12)

The notations $A_k\left(0\right)$, $B_k\left(0\right)$, and $Z_k\left(0\right)$ are hereinafter simplified as $A_k$, $B_k$, and $Z_k$, respectively. Accordingly, the following matrix is introduced:

$$\left\{\begin{array}{c}{U}_k={\left(Q_{h,k}^{-1}+H_{k22}^{\left(n\right)}-H_{k21}^{\left(n\right)}{\left(H_{k11}^{\left(n\right)}\right)}^{-1}{H}_{k12}^{\left(n\right)}\right)}^{-1},J_k=\left[\begin{array}{cc}0& U_k^{{\displaystyle \frac{1}{2}}}{H}_{k21}^{\left(n\right)}{\left(H_{k11}^{\left(n\right)}\right)}^{-{\displaystyle \frac{1}{2}}}\end{array}\right],\\ \begin{array}{c}{F}_k=\left[\begin{array}{c}{R}_{k-n}^{-{\displaystyle \frac{1}{2}}}{Z}_{k-n}\\ {\left(H_{k11}^{\left(n\right)}\right)}^{{\displaystyle \frac{1}{2}}}\end{array}\right],W_k=\left[\begin{array}{cc}I& 0\\ 0& I+{\left(H_{k11}^{\left(n\right)}\right)}^{-{\displaystyle \frac{1}{2}}}{H}_{k12}{U}_k{H}_{k21}^{\left(n\right)}{\left(H_{k11}^{\left(n\right)}\right)}^{-{\displaystyle \frac{1}{2}}}\end{array}\right],\hfill \\ T_{k1}=A_{h,k}-B_{h,k}{U}_k{H}_{k21}^{\left(n\right)}{\left[I+{\left(H_{k11}^{\left(n\right)}\right)}^{-1}{H}_{k12}^{\left(n\right)}{U}_k{H}_{k21}^{\left(n\right)}\right]}^{-1},D_k=B_{h,k}{U}_k^{{\displaystyle \frac{1}{2}}},\hfill \\ T_{k2}=D_k{\left(I+U_k^{{\displaystyle \frac{1}{2}}}{H}_{k21}^{\left(n\right)}{\left(H_{k11}^{\left(n\right)}\right)}^{-1}{H}_{k12}^{\left(n\right)}{U}_k^{{\displaystyle \frac{1}{2}}}\right)}^{-{\displaystyle \frac{1}{2}}}\hfill \end{array}\end{array}\right.$$

(13)

Then, based on Equation (13) and the parameter modification in Equation(10), the following expression is obtained:

$$\begin{array}{c}{P}_{k-n|k-n}^{\left(n\right)}={\left({\left(P_{k-n|k-n-1}^{\left(n\right)}\right)}^{-1}+Z_{k-n}^T{R}_{k-n}^{-1}{Z}_{k-n}\right)}^{-1}\hfill \\ =P_{k-n|k-n-1}^{\left(n\right)}-P_{k-n|k-n-1}^{\left(n\right)}{Z}_{k-n}^T{\left(R_{k-n}^{-1}+Z_{k-n}{P}_{k-n|k-n-1}^{\left(n\right)}{Z}_{k-n}^T\right)}^{-1}{Z}_{k-n}{P}_{k-n|k-n-1}^{\left(n\right)},\hfill \\ {\widehat{P}}_{k-n|k}^{\left(n\right)}={\left({\left(P_{k-n|k-n}^{\left(n\right)}\right)}^{-1}+H_{k11}^{\left(n\right)}\right)}^{-1}\hfill \\ =P_{k-n|k-n}^{\left(n\right)}-P_{k-n|k-n}^{\left(n\right)}{\left(H_{k11}^{\left(n\right)}\right)}^{{\displaystyle \frac{1}{2}}}{\left(I+{\left(H_{k11}^{\left(n\right)}\right)}^{{\displaystyle \frac{1}{2}}}{P}_{k-n|k-n}^{\left(n\right)}{\left(H_{k11}^{\left(n\right)}\right)}^{{\displaystyle \frac{1}{2}}}\right)}^{-1}{\left(H_{k11}^{\left(n\right)}\right)}^{{\displaystyle \frac{1}{2}}}{P}_{k-n|k-n}^{\left(n\right)},\hfill \\ {\widehat{Q}}_k={\left(Q_k^{-1}+H_{k22}^{\left(n\right)}-H_{k21}^{\left(n\right)}{\left({\left(P_{k-n|k-n}^{\left(n\right)}\right)}^{-1}+H_{k11}^{\left(n\right)}\right)}^{-1}{H}_{k12}^{\left(n\right)}\right)}^{-1}\hfill \\ ={\left(\begin{array}{c}{Q}_k^{-1}+H_{k22}^{\left(n\right)}-H_{k21}^{\left(n\right)}{\left(H_{k11}^{\left(n\right)}\right)}^{-1}{H}_{k12}^{\left(n\right)}\hfill \\ +H_{k21}^{\left(n\right)}{\left(H_{k11}^{\left(n\right)}\right)}^{-1}{\left(I+{\left(H_{k11}^{\left(n\right)}\right)}^{{\displaystyle \frac{1}{2}}}{\left(P_{k-n|k-n}^{\left(n\right)}\right)}^{-1}{\left(H_{k11}^{\left(n\right)}\right)}^{{\displaystyle \frac{1}{2}}}\right)}^{-1}{\left(H_{k11}^{\left(n\right)}\right)}^{-{\displaystyle \frac{1}{2}}}{H}_{k12}^{\left(n\right)}\hfill \end{array}\right)}^{-1}.\hfill \end{array}$$

(14)

3.2. Convergence Analysis

Theorem 1.

Suppose the aforementioned conditions hold, where $A_k$ is of full rank, the pair $(A_k,Z_k)$ is detectable, and $(T_{k1},T_{k2})$ is stabilizable. Then, for any ${\Pi }_0>0$, the sequence $P_{k-n|k-n-1}^{\left(n\right)}$ converges to a distinct positive semidefinite matrix P. Similarly, $P_{k-n|k-n}^{\left(n\right)}$ reaches a similar limit. Consequently, the proposed estimator asymptotically approaches a time-invariant stable system.

Proof. 

A detailed derivation of Theorem 1 is presented in the following. □

The derivation steps are shown in Appendix A. Use the results obtained in Appendix A to continue the derivation as follows.

By applying Equations (10), (11), (13) along with Lemma 1, $P_{k+1|k}^{\left(n\right)}$ is reformulated as follows.

$$\begin{array}{c}{P}_{k+1|k}^{\left(n\right)}=A_{h,k}{P}_{k-n|k-n-1}^{\left(n\right)}{A}_{h,k}^T+B_{h,k}{U}_k{B}_{h,k}^T\hfill \\ -\left[\left(A_{h,k}{P}_{k-n|k-n-1}^{\left(n\right)}{F}_k^T+D_k{J}_k\right){\left(W_k+F_k{P}_{k-n|k-n-1}^{\left(n\right)}{F}_k^T\right)}^{-1}{\left(A_{h,k}{P}_{k-n|k-n-1}^{\left(n\right)}{F}_k^T+D_k{J}_k\right)}^T\right]\hfill \end{array}$$

(15)

Since the last term in Equation (15) exhibits the form of a Riccati recursion, and based on the same line of reasoning used in the asymptotic analysis of Kalman filters [23], it follows that under the given assumptions-namely, that $A_k$ are full-rank matrices, the pair $(A_k,Z_k)$ is detectable, and the pair $(T_{k1},T_{k2})$ is stabilizable-the sequence $P_{k+1|k}$ converges.

As indicated by Equation (11), $P_{k+1|k}^{\left(n\right)}$ and $P_{k|k}^{\left(n\right)}$ exhibit identical convergence properties. Consequently, $P_{k|k}^{\left(n\right)}$ tends toward a distinct positive semidefinite matrix.

Let the matrix be given by:

$$L_k=A_{h,k}-\left[\left(A_{h,k}{P}_{k-n|k-n-1}^{\left(n\right)}{F}_k^T+D_k{J}_k\right){\left(W_k+F_k{P}_{k-n|k-n-1}^{\left(n\right)}{F}_k^T\right)}^{-1}{F}_k\right]$$

(16)

Given that $P_{k+1|k}^{\left(n\right)}$ converges to a unique matrix P, it follows that $L_k$ also converges to a stable constant matrix L, which is given by $L=A_{h,k}-\left(A_{h,k}P{F}_k^T+D_k{J}_k\right){\left(W_k+F_{k}P{F}_k^T\right)}^{-1}{F}_k$.

The following conclusion can be drawn.

$$\begin{array}{c}{L}_k=\left[\left(A_{h,k}-B_{h,k}{\widehat{Q}}_{h,k}{G}_{k21}^{\left(n\right)}\right)\left(I-{\widehat{P}}_{k-n|k}^{\left(n\right)}{G}_{k11}^{\left(n\right)}\right){\left(I+P_{k-n|k-n-1}^{\left(n\right)}{Z}_{k-n}^T{R}_{k-n}^{-1}{Z}_{k-n}\right)}^{-1}\right]\hfill \\ ={\widehat{A}}_{h,k}{\left(I+P_{k-n|k-n-1}^{\left(n\right)}{Z}_{k-n}^T{R}_{k-n}^{-1}{Z}_{k-n}\right)}^{-1}\hfill \end{array}$$

(17)

Define $S_k$ and $V_k$:

$$\begin{array}{c}{S}_k=\left[I-P_{k+1|k+1}^{\left(n\right)}{Z}_{k+1}^T{R}_{k+1}^{-1}{Z}_{k+1}\right]{\widehat{A}}_{h,k}=col\left(A_{f,k-n},\dots ,A_{f,k}\right)\hfill \\ V_k=\left(A_{h,k}{\widehat{P}}_{k-n|k}^{\left(n\right)}\left[\begin{array}{cc}I& 0\end{array}\right]+B_{h,k}{\widehat{Q}}_k\left[\begin{array}{cc}-G_{k21}^{\left(n\right)}{\widehat{P}}_{k-n|k}^{\left(n\right)}& I\end{array}\right]\right){\Omega }_{h,k2}{R}_{k+1}^{-1}=col\left(B_{f,k-n},\dots ,B_{f,k}\right)\hfill \end{array}$$

(18)

Then the state update in Equation (11) takes the form

$${\widehat{x}}_{k+1|k+1}=A_{f,k}{\widehat{x}}_{k-n|k}+P_{k+1|k+1}{P}_{k+1|k}^{-1}{B}_{f,k}{y}_{k+1}.$$

(19)

So according to the definition of $L_k$, we can get

$$\begin{array}{c}{A}_{f,k}={\left(I+P_{k+1|k}^{\left(n\right)}{Z}_{k+1}^T{R}_{k+1}^{-1}{Z}_{k+1}\right)}^{-1}{L}_{k,k}{\left(I+P_{k-n|k-n-1}^{\left(n\right)}{Z}_{k-n}^T{R}_{k-n}^{-1}{Z}_{k-n}\right)}^{-1}\hfill \end{array}$$

(20)

Given the prior result that $P_{k+1|k}\to P$, where P is a unique positive semidefinite matrix, this convergence remains valid under the same assumptions, so that

$$\begin{array}{c}\underset{k\to \infty }{lim}\left(I+P_{k+1|k}^{\left(n\right)}{Z}_{k+1}^T{R}_{k+1}^{-1}{Z}_{k+1}\right)=\underset{k\to \infty }{lim}\left(I+P_{k-n|k-n-1}^{\left(n\right)}{Z}_{k-n}^T{R}_{k-n}^{-1}{Z}_{k-n}\right)\hfill \end{array}$$

(21)

As indicated by Equation (21), as $k\to \infty$ the eigenvalues of $A_{f,k}$ progressively approach those of $L_{k,k}$. Consequently, the gain matrix $L_k$ converges to a limiting matrix L. Under the standing assumptions, the prediction covariance $P_{k+1|k}$ converges to a unique positive definite matrix $P>0$ and, moreover, satisfies

$$\underset{k\ge 0}{sup}tr\left(P_{k+1|k}\right)\le M,M>0.$$

Since the limiting matrix L is exponentially stable, the estimation error dynamics

$$e_{k+1}=A_{f,k}{e}_k+{\xi }_k$$

form a stable linear stochastic system with bounded process noise, where ${\xi }_k$ denotes the composite disturbance term arising from process noise, measurement noise, and modeling errors. It then follows that the estimation error covariance is uniformly bounded,

$$\underset{k\ge 0}{sup}\mathbb{E}\parallel x_k-{\widehat{x}}_{k|k}{\parallel }^2<\infty .$$

(22)

This establishes that the estimation error is mean-square bounded.

Remark 3.

Assumption 1 requires that the normalized uncertainty variables ${\epsilon }_{k,j}$ satisfy $\mathcal{E}=\left\{\epsilon \left|\left|{\epsilon }_{k,j}\right|\le 1,j=1,\dots ,L\right.\right\}$. In practice, this bound is obtained from prior identification or engineering tolerance, e.g., sensor calibration errors, modelling errors in physical parameters, or known ranges of operating conditions. Consequently, the prescribed uncertainty range directly affects the weighting of the robustness term in the cost function and, therefore, the achievable estimation accuracy.

Theorem 2.

Under the same premises, the designed robust estimator ensures asymptotic unbiasedness, and its error covariance remains bounded at each time step k.

Proof. 

A detailed derivation of Theorem 2 is presented in the following. □

The variables below are introduced to support later analysis:

$$\begin{array}{c}{\mathrm{{\rm Y}}}_{k-n}\left({\epsilon }_{k-n}\right)=\left[\begin{array}{c}{P}_{k-n|k-n-1}^{\left(n\right)}{Z}_{h,k-n}^T\left(0\right)R_{h,k-n}^{-1}{Z}_{h,k-n}\left({\epsilon }_{h,k-n}\right)\hfill \end{array}\right]=diag\left({\Gamma }_{k-2n},\dots ,{\Gamma }_{k-n}\right),\hfill \\ {\mathrm{X}}_{k-n|k-n}=\left[I+{\Gamma }_{k-n}\left(0\right)\right]x_{k-n},{\widehat{\mathrm{X}}}_{k-n|k-n}=\left[I+{\Gamma }_{k-n}\left(0\right)\right]{\widehat{x}}_{k-n|k-n},{\tilde{\mathrm{X}}}_{k-n|k-n}\hfill \\ ={\mathrm{X}}_{k-n|k-n}-{\widehat{\mathrm{X}}}_{k-n|k-n}.\hfill \end{array}$$

(23)

Define ${\stackrel{⌢}{A}}_k({\epsilon }_k)$ as follow:

$${\stackrel{⌢}{A}}_{h,k}\left({\epsilon }_{h,k}\right)=\left[I+{\mathrm{{\rm Y}}}_{k+1}\left(0\right)\right]A_{h,k}\left({\epsilon }_{h,k}\right){\left[I+{\Gamma }_{k-n}\left(0\right)\right]}^{-1},$$

(24)

then, by referring to Theorem 1 and the result in Equation (25), it can be inferred that

$$\left[\begin{array}{c}{\tilde{\mathrm{X}}}_{k+1|k+1}\\ {\widehat{\mathrm{X}}}_{k+1|k+1}\end{array}\right]={\tilde{A}}_k\left({\epsilon }_{h,k}\right)\left[\begin{array}{c}{\tilde{\mathrm{X}}}_{k-n|k-n}\\ {\widehat{\mathrm{X}}}_{k-n|k-n}\end{array}\right]+{\tilde{B}}_k\left({\epsilon }_{h,k}\right)\left[\begin{array}{c}{w}_i\\ v_{i+1}\end{array}\right].$$

(25)

In which

$$\begin{array}{c}{\tilde{A}}_k\left({\epsilon }_{h,k}\right)=\left[\begin{array}{cc}{\tilde{A}}_{k1}\left({\epsilon }_{h,k}\right)& {\tilde{A}}_{k1}\left({\epsilon }_{h,k}\right)-L_k\\ {\tilde{A}}_{k2}\left({\epsilon }_{h,k}\right)& {\tilde{A}}_{k2}\left({\epsilon }_{h,k}\right)+L_k\end{array}\right],\hfill \\ {\tilde{B}}_k\left({\epsilon }_{h,k}\right)=\left[\begin{array}{cc}\left(\begin{array}{c}I+{\mathrm{{\rm Y}}}_{k+1}\left(0\right)-V_k{Z}_{h,k+1}\left({\epsilon }_{h,k+1}\right)\hfill \end{array}\right)B_{h,k}\left({\epsilon }_{h,k}\right)& -V_k\\ V_k{Z}_{h,k+1}\left({\epsilon }_{h,k+1}\right)B_{h,k}\left({\epsilon }_{h,k}\right)& V_k\end{array}\right].\hfill \end{array}$$

(26)

where

$$\begin{array}{c}{\tilde{A}}_{k1}\left({\epsilon }_{h,k}\right)=\left(I+{\mathrm{{\rm Y}}}_{k+1}\left(0\right)-V_k{Z}_{h,k+1}\left({\epsilon }_{h,k+1}\right)\right){\left(I+{\mathrm{{\rm Y}}}_{k+1}\left(0\right)\right)}^{-1}{\stackrel{⌢}{A}}_{h,k}\left({\epsilon }_{h,k}\right),\hfill \\ {\tilde{A}}_{k2}\left({\epsilon }_{h,k}\right)=\left(\begin{array}{c}{V}_k{Z}_{h,k+1}\left({\epsilon }_{h,k+1}\right){\left(I+{\mathrm{{\rm Y}}}_{k+1}\left(0\right)\right)}^{-1}{\stackrel{⌢}{A}}_{h,k}\end{array}\left({\epsilon }_{h,k}\right)\right..\hfill \end{array}$$

(27)

Moreover, Equation (25) resembles Equation (16) in Ref. [20], and thus can be established through an analogous proof, which is omitted here for brevity. Provided that the matrix $A_{h,k}$ is stable, the condition guaranteeing bounded estimation error in the robust filtering scheme can be derived. This concludes the demonstration of Theorem 2.

4. Simulation Results

4.1. Model Introduction

A vehicle-trailer system is employed to evaluate the performance of the proposed robust state estimator [24,25]. The corresponding linearized model is formulated as follows:

$$\left\{\begin{array}{c}{x}_{k+1}^1=\left(1.0000-{\displaystyle \frac{VT}{L}}\right)x_k^1+{\displaystyle \frac{VT}{L}}{x}_k^2+\left(0.9804+{\displaystyle \frac{VT}{L}}\right)w_k^1,\hfill \\ x_{k+1}^2=\left(1.0000-{\displaystyle \frac{VT}{L}}\right)x_k^2+\left(0.9804+{\displaystyle \frac{VT}{L}}\right)w_k^2.\hfill \end{array}\right.$$

where $x_k^1$ and $x_k^2$ represent the steering angle of the tractor and the heading of the car, respectively. The length of the tractor is denoted as L, V represents constant velocity, and T represents the sampling period. Taking into account the influence of ${\epsilon }_k$, it is incorporated into the system dynamics, leading to

$$\begin{array}{c}{A}_k\left({\epsilon }_k\right)=\left[\begin{array}{cc}1.0000-{\displaystyle \frac{VT}{L}}& {\displaystyle \frac{VT}{L}}+{\epsilon }_k\\ 0.0000& 1.0000-{\displaystyle \frac{VT}{L}}\end{array}\right],B_k\left({\epsilon }_k\right)=\left[\begin{array}{cc}0.9804+{\displaystyle \frac{VT}{L}}& 0.0000\\ 0.0000& 0.9804+{\displaystyle \frac{VT}{L}}\end{array}\right].\hfill \end{array}$$

The corresponding mathematical representation is the same as the uncertain linear system $\Delta$.

Give $L=500$ cm, $T=0.1$ s, and $V=98$ cm/s, the corresponding matrix parameters are defined as:

$$\begin{array}{c}{A}_k\left({\epsilon }_k\right)=\left[\begin{array}{cc}0.9804& 0.0196+{\epsilon }_k\\ 0.0000& 0.9804\end{array}\right],B_k\left({\epsilon }_k\right)=\left[\begin{array}{cc}1.0000& 0.0000\\ 0.0000& 1.0000\end{array}\right],R_k=1.0000,\hfill \\ Q_k=\left[\begin{array}{cc}1.9608& 0.0195\\ 0.0195& 1.9605\end{array}\right],{\Pi }_0=\left[\begin{array}{cc}1.0000& 0.0000\\ 0.0000& 1.0000\end{array}\right],Z_k\left({\epsilon }_k\right)=\left[\begin{array}{cc}1.0000& -1.0000\end{array}\right].\hfill \end{array}$$

Since $A_k\left({\epsilon }_k\right)$ is upper triangular, its eigenvalues are given by its diagonal entries, i.e., ${\lambda }_1={\lambda }_2=0.9804$, which are independent of ${\epsilon }_k$. Hence, for all admissible uncertainties $|{\epsilon }_k| \le \rho$, the spectral radius satisfies $\rho \left(A_k\left({\epsilon }_k\right)\right)=0.9804<1,$ and therefore the discrete-time system is exponentially stable in the Lyapunov sense (uniformly in k). Furthermore, to enhance the experimental accuracy, each simulation is repeated $4\times 10^2$ times, and each experiment is set for $4\times 10^2$ moments. The mean squared Euclidean distance between the actual and estimated states serves as an approximation of the instantaneous estimation error variance, expressed as follows:

$$\mathbb{E}{∥x_k-{\widehat{x}}_{k|k}∥}^2\approx {\displaystyle \frac{1}{400}}{\mathrm{\Sigma }}_{j=1}^{400}{∥x_k-{\widehat{x}}_{k|k}^{\left(j\right)}∥}^2.$$

4.2. Performance Comparison of Different Algorithms

Figure 1 presents the scenario where the modeling error ${\epsilon }_k$ is held constant at −0.8508. The temporal evolution of the estimation error variance is depicted in Figure 1a. As shown, the proposed robust estimator yields an error variance approximately 1.14 dB higher than that of the Kalman filter when the true system parameters are known, which aligns with expectations since the Kalman filter achieves optimal performance under the MMSE framework. Compared with single-run state trajectories, the Estimation Error Covariance Means offers a more objective and reliable performance measure by averaging over multiple runs, effectively highlighting estimation accuracy and robustness when trajectory differences are subtle. Nevertheless, in comparison with the method [26], the proposed approach exhibits superior performance, reducing the estimation error variance by 3.76 dB. Moreover, the proposed method also outperforms the $H_{\infty }$ filter, achieving a lower estimation error variance over the entire horizon. Furthermore, Figure 1b demonstrates that as the moving horizon length increases, the estimation error variance decreases when ${\epsilon }_k$ is fixed at $-0.8508$. In particular, the variance drops from 13.76 dB at $n=0$ to 12.94 dB at $n=8$.

In Figure 2, the values of ${\epsilon }_k$ vary across different experiments and are uniformly distributed in the interval $\left[-1.0,1.0\right]$. The estimator’s performance shows an improvement of approximately 0.74 dB compared to the previous method [26], as illustrated in Figure 2a. Moreover, the proposed estimator also outperforms the $H_{\infty }$ filter by about 0.34 dB. In Figure 2b, the estimation error variance decreases from 20.54 dB to 20.15 dB with increasing moving horizon size n.

To assess the robustness of the estimator under uncertainty, modeling errors ${\epsilon }_k$, which vary at each sampling instant and follow a uniform distribution within the range $\left[-1.0,1.0\right]$, are introduced. As illustrated in Figure 3, the proposed robust state estimator consistently exhibits superior performance under these conditions.

Figure 1, Figure 2 and Figure 3 illustrate that the proposed robust state estimator achieves consistently reliable estimation performance across different levels of parameter uncertainties. Moreover, the estimation error variance exhibits a decreasing trend as n increases, highlighting the method’s effectiveness. Furthermore,

Table 1. Estimation Error Covariance Means (dB).

	Modeling Errors	Case 1	Case 2	Case 3
Algorithm
Kalman estimator: using nominal parameters		17.73	21.28	20.47
Kalman estimator: using actual parameters		12.53	17.59	16.09
H-infinity filter		16.62	20.88	20.31
Robust state estimator (n = 0)		13.67	20.54	19.7
Robust state estimator (n = 3)		13.25	20.43	19.6
Robust state estimator (n = 6)		13.09	20.22	19.44
Robust state estimator (n = 8)		12.94	20.15	19.4

summarizes the average estimation error covariance of different estimation algorithms over 400 simulation runs. Both the figures and the table validate the effectiveness of the designed robust estimator through performance demonstration.

Table 1 summarizes the average estimation error covariance of different estimation algorithms over 400 simulation runs.

Moreover, the estimation error variance exhibits a decreasing trend as n increases, highlighting the method’s effectiveness. The figures and table confirm the effectiveness of the robust estimator.

It is worth emphasizing that, in this numerical example, the modeling uncertainty ${\epsilon }_k$ is intentionally introduced only in the state transition matrix $A_k$, while the remaining system matrices are kept at their nominal values. This simplified setting is adopted to isolate the influence of parametric uncertainty and to clearly demonstrate the effectiveness of the proposed estimation strategy. It should be noted that the proposed framework itself is not restricted to second-order systems nor to uncertainties acting only on $A_k$; the theoretical formulation allows uncertainties to enter other system matrices as well, and the same derivation can be extended accordingly.

To quantify the impact of the uncertainty range on estimation accuracy, a scalar scaling factor $\rho \ge 0$ is introduced such that ${\epsilon }_{k,j}=\rho {\tilde{\epsilon }}_{k,j}$, where ${\tilde{\epsilon }}_{k,j}\sim \mathcal{U}[-1,1]$. In this setting, ${\tilde{\epsilon }}_{k,j}$ represents a normalized, dimensionless parameter deviation, while $\rho$ controls the admissible magnitude of uncertainty around the nominal model, i.e., $|{\epsilon }_{k,j}| \le \rho$ for all time instants k. This setting corresponds to practical time-varying systems in which parameters fluctuate within known bounded ranges.

The proposed estimator is evaluated for $\rho \in \{0,0.5,1.0,1.5\}$, while all other settings, including noise covariances, weighting matrices, and the horizon length fixed at $n=3$, are kept unchanged. For each value of $\rho$, the steady-state MMSE of the state estimate is computed, and the average results are reported in

Table 2. MMSE of the state estimate for different uncertainty scaling factors $\rho$.

$\mathit{\rho }$	Estimation Error Covariance Means (dB)
0	18.12
0.5	19.17
1.0	20.19
1.5	21.43

. As expected, the estimation error increases monotonically with $\rho$. When $\rho =0$, the estimator reduces to the nominal Kalman filter and achieves the minimum MMSE. As $\rho$ increases, the estimator remains stable but exhibits a gradual loss of accuracy due to the enlarged uncertainty set.

5. Conclusions

This work has established a robust state estimation framework for linear discrete-time systems with parametric uncertainties by embedding the RLS interpretation into the moving horizon estimation paradigm. Beyond algorithmic development, the main contribution lies in the theoretical characterization of the estimator’s long-term behavior. Specifically, the proposed formulation admits a recursive implementation with a Kalman filter-like structure, while guaranteeing the existence and uniqueness of a steady-state error covariance. It is further shown that the estimation error remains uniformly bounded in the mean-square sense and converges asymptotically to an unbiased estimate under bounded parametric perturbations.

In addition, the influence of the uncertainty range on estimation accuracy is explicitly investigated, revealing a monotonic relationship between the admissible parameter variation and the steady-state estimation error. This analysis provides a quantitative interpretation of robustness and clarifies the trade-off between uncertainty tolerance and estimation precision.

Future research will focus on extending the proposed theoretical framework to adaptive horizon selection mechanisms that preserve stability guarantees, as well as deriving tighter analytical bounds linking uncertainty magnitude to steady-state performance. Moreover, the applicability of the proposed estimator to non-autonomous (time-varying) systems will be investigated, with particular emphasis on establishing long-term stability and performance guarantees under time-dependent dynamics.