An interval Kalman filtering with minimal conservatism

doi:10.1016/j.amc.2012.02.050

Applied Mathematics and Computation

Volume 218, Issue 18, 15 May 2012, Pages 9563-9570

https://doi.org/10.1016/j.amc.2012.02.050 Get rights and content

Abstract

The interval Kalman filtering (IKF) can handle parametric interval uncertainties of the system matrices, and it computes the lower and upper boundaries of the estimated states. In this paper, we propose an alternative form of interval Kalman filtering to reduce the conservatism inherent to the existing interval Kalman filtering. First, we address why the existing interval Kalman filtering scheme induces conservatism in the boundary estimation. Then, to remove the conservatism, we derive noise covariance matrices taking into account of interval uncertainties as well as process and measurement noises. Following the typical derivation process of the standard Kalman filtering, a new recursive form of interval Kalman filtering is derived. Through numerical simulations, the superiority of the new algorithm over existing IKF is illustrated.

Introduction

Robust control research has a long history. In robust control, two approaches are noticeable. The first approach is the traditional robust control [2]. The main interest of the traditional robust control problem is to handle control systems in the presence of model uncertainty bounded by maximum singular value (operator 2-norm). The other approach is to design a robust controller for the plant with parametric interval uncertainty [3]. Indeed, great amount of literature is available under the name of “interval”: for instance, interval algebra [4], [5], Schur stability of interval matrices [6], Hurwitz stability of interval matrices [7], interval polynomial matrices [8], eigenvalues of interval matrices [9], [10], [11], and robust control with parameter uncertainty [12]. Both problems (norm-based uncertainty and interval-based uncertainty) have been substantially studied and thus systematically formulated in numerous publications. Robust control problems are significantly linked with estimation problems; specifically with Kalman filtering, because in control problems, states are assumed available by either measurement or estimation. There are various types and numerous purposes of Kalman filtering. Associated with model uncertainty, robust Kalman filtering has attracted considerable amount of attention from control engineers. Most of literature with regard to robust Kalman filtering, however, has been devoted to norm-based uncertainties, structure-fixed uncertainties, and uncertain observations [13], [14], [15], [16], [17]. About a decade years ago, there was a novel trial to handle interval uncertainties in nominal parameters under Kalman filtering territory, namely interval Kalman filtering (IKF) [1]. The novel feature of IKF is the ability of coping with parametric interval variations in plant model along the time axis, but still keeping an optimality of standard Kalman filtering scheme. It can be said that the IKF is a correspondence of parametric robust control problem in estimation theory. Even though there are some performance improvements over existing algorithms [1], with possible actual applications such as ballistic missile tracking [18], unfortunately however there has been no theoretical enhancement nor research activity on IKF from that time on. The principal motivation of this paper is to reduce some conservatism inherent to IKF by way of considering the parametric uncertainties as additional disturbances. The paper is structured as follows. In the following section, some basic concepts and preliminarily required definitions are addressed. In Section 3, existing interval Kalman filtering (IKF) is briefly reviewed. The main result of this paper is established in Section 4, and some numerical simulations are carried out in Section 5. Conclusions are given in Section 6.

Section snippets

Basic concepts and definitions

Throughout the paper, we use the following concepts.

Definition 2.1

A parameter x is called interval parameter if it lies between two closed extreme upper and under boundary values $\bar{x}$ and x, i.e., $x \in x^{I} = [\underset{̲}{x}, \bar{x}]$ , x ∈ R and $\bar{x} \in R$ . For all $x \in x^{I}$ , there exists a corresponding λ such that $x = λ \bar{x} + (1 - λ) \underset{̲}{x}$ with 0 ⩽ λ ⩽ 1, λ ∈ R. A n × m matrix A is written as A = [a_ij] ∈ R^n×m, i = 1, … , n, j = 1, … , m. An interval matrix $A^{I}$ is a matrix with interval elements such as: $A^{I} = [a_{ij}^{I}]$ .

An interval matrix can be written as $A^{I} = [\underset{̲}{A}, \bar{A}]$ where A = [a_ij] and $\bar{A} =$

Interval Kalman filtering (IKF)

Interval Kalman filtering (IKF) was firstly addressed in [1]. The main motivation of interval Kalman filtering is to handle uncertainties arising along the time axis. The main idea of interval Kalman filtering is straightforward and intuitive; in this section we briefly review the algorithm before proceeding to our main result. In (1), (2), matrices A_k can be rewritten as $A_{k} = A^{o} + Δ A_{k}, Δ A_{k} \in Δ_{A}^{I} = [- Δ_{A}, Δ_{A}]$ , where Δ_A is a constant matrix with positive elements. Likewise, B_k and C_k can be rewritten as $B_{k} =$

Alternative IKF (AIKF)

For the derivation of alternative interval Kalman filtering (AIKF), let us change (1), (2) such as: $x_{k + 1} = A_{k} x_{k} + B_{k} ξ_{k} = (A^{o} + Δ A_{k}) x_{k} + (B^{o} + Δ B_{k}) ξ_{k},$ $= A^{o} x_{k} + Δ A_{k} x_{k} + (B^{o} + Δ B_{k}) ξ_{k},$ $y_{k} = (C^{o} + Δ C_{k}) x_{k} + η_{k} = C^{o} x_{k} + Δ C_{k} x_{k} + η_{k} .$ The terms ΔA_kx_k + (B^o + ΔB_k)ξ_k and ΔC_kx_k + η_k are considered as process and measurement noises respectively. To continue, it is necessary to establish covariance matrices for process noise and measurement noise, which are defined as: $Q = E {[Δ A_{k} x_{k} + (B^{o} + Δ B_{k}) ξ_{k}] [Δ A_{k} x_{k} + (B^{o} + Δ B_{k}) ξ_{k}]^{T}}$ and $R = E {[Δ C_{k} x_{k} + η_{k}] [Δ C_{k} x_{k} + η_{k}]^{T}}$ .

Example 1

Let us consider the following discrete-time linear system (’A’ matrix was obtained from Example B of [1]) with 50% interval uncertainties in the nominal plant: $x_{k + 1} = [(\begin{matrix} 0.4 & 0.1 \\ - 0.1 & 0.2 \end{matrix}) + Δ A_{k}] x_{k} + [(\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}) + Δ B_{k}] ξ_{k}$ $y_{k} = [(\begin{matrix} 1 & 1 \end{matrix}) + Δ C_{k}] x_{k} + η_{k}$ where $Δ A_{k} \in Δ_{A}^{I} = (\begin{matrix} [- 0.2, 0.2] & [- 0.05, 0.05] \\ [- 0.05, 0.05] & [- 0.1, 0.1] \end{matrix}); Δ B_{k} \in Δ_{B}^{I} = (\begin{matrix} [- 0.5, 0.5] & [- 0.5, 0.5] \\ [- 0.5, 0.5] & [- 0.5, 0.5] \end{matrix});$ $Δ C_{k} \in Δ_{C}^{I} = (\begin{matrix} [- 0.5, 0.5] & [- 0.5, 0.5] \end{matrix})$ with $Q = E {ξ ξ^{T}} = (\begin{matrix} 10^{- 3} & 0 \\ 0 & 10^{- 3} \end{matrix})$ and R = {ηη^T} = 10⁻³, and the sampling step is 0.1. Plots of Fig. 1 are the test results from interval Kalman filtering ((a) and (b))

Conclusions

This paper was devoted to the alternative form of interval Kalman filtering (IKF). After addressing conservatism inherent to IKF, which makes the system diverge easily, we established a new algorithm based on interval computation. The alternative interval Kalman filtering (AIKF) does not require interval arithmetics in propagation and correction processes since interval uncertainties are contained into process and measurement noise covariances. Following the standard derivation process of

Acknowledgement

The research of this paper was supported by Basic Research Projects in High-tech Industrial Technology of Gwangju Institute of Science and Technology (GIST), and by National Research Foundation of Korea (NRF) (No. 2011–0002847 and 2011–0021474).

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An interval Kalman filtering with minimal conservatism

Abstract

Introduction

Section snippets

Basic concepts and definitions

Interval Kalman filtering (IKF)

Alternative IKF (AIKF)

Example 1

Conclusions

Acknowledgement

Appl. Math. Comput.

Syst. Control Lett.

Appl. Math. Comput.

Interval Kalman filtering

IEEE Trans. Aerosp. Electron. Syst.

Essentials of Robust Control

Robust Control: The Parameter Approach

Interval Analysis

Applied Interval Analysis

Sufficient conditions for the asymptotic stability of interval matrices

Int. J. Control