Descriptor Wiener state estimators

doi:10.1016/S0005-1098(00)00079-0

Automatica

Volume 36, Issue 11, November 2000, Pages 1761-1766

https://doi.org/10.1016/S0005-1098(00)00079-0 Get rights and content

Abstract

Based on the modern time-series analysis method, a new time-domain Wiener filtering approach is presented. Asymptotically stable Wiener state estimators are presented for discrete linear stochastic descriptor systems. They can be implemented via the autoregressive moving average (ARMA) recursive filters. They can handle the optimal state filtering, smoothing, and prediction problems in a unified framework, and can simply be obtained based on the ARMA innovation model. The solution of the Diophantine equations and Riccati equations is avoided, so that the computational burden is reduced. A simulation example shows the effectiveness of the new approach.

Introduction

Recently, the state estimation problems for descriptor (singular) systems have received great attention due to extensive application backgrounds to circuits, economics, and robotics, etc. So far, the optimal recursive state estimators for descriptor systems are limited to the Kalman filtering framework (Day, 1987; Nikoukhah, Willsky & Bernard, 1992; Boutayeb & Darouach, 1992; Darouach & Zasadzinski, 1992; Darouach, Zasadzinski & Mehdi, 1993). These descriptor Kalman estimators have the following disadvantages: Firstly, they cannot handle the optimal state filtering, smoothing and prediction problems in a unified form. Secondly, the solution of the Riccati equation is required, which can be a large computational burden. The Zhang, Chai and Liu (1998) and Deng and Liu (1999)'s descriptor Kalman estimators overcome the above disadvantages.

In the frequency domain, the polynomial approach to Wiener filtering (Ahlén & Sternad, 1991), is an important tool for solving the optimal signal estimation problems. Recently, it can also be applied to solve the state estimation problems for non-singular systems (Chisci & Mosca, 1992; Grimble, 1994). However, in order to obtain explicit and realizable Wiener filter of signal or state, the solution of the Diophantine equations is required, which can yield a larger computational burden.

In this paper, a new time-domain approach to Wiener filtering is presented based on the modern time-series analysis method (Deng & Guo, 1989; Deng, Zhang, Liu & Zhou, 1996), and is applied to solve the state estimation problems for descriptor systems. Asymptotically, stable descriptor Wiener state estimators are presented for the first time, which have a transfer function matrix form with the measurement signal as input, and can be implemented via the ARMA recursive filters. Their transfer function matrices can be obtained based on the ARMA innovation model. The solution of the Diophantine equations and Riccati equations is avoided, so that the computational burden is reduced. They have a unified form for the optimal recursive state filter, smoother and predictor. The disadvantages of the Kalman filtering method and polynomial approach are overcome. The key ideas of the new approach are (a) finding the non-recursive optimal state estimators with the white noise estimators (Deng et al., 1996) and measurement predictors; (b) based on the ARMA innovation model, give the white noise estimators in a Wiener filter form; (c) the recursive version of non-recursive state estimators yields the descriptor Wiener state estimators. Essentially, the state estimation problem is transformed into the white noise estimation problem.

Section snippets

Problem formulation and lemmas

Consider a linear discrete stochastic descriptor system $Mx(t+1)= Φ x(t)+ Γ w(t),$ $y(t)=Hx(t)+v(t),$ where state x(t)∈Rⁿ, measurement $y(t)∈R^{m},M, Φ, Γ$ and H are the constant matrices.

Assumption 1

w(t)∈R^r and v(t)∈R^m are the correlated white noises with zero mean and $E w(t) v(t) [w′(j) v′(j)] = Q_{w} S S′ Q_{v} δ_{tj}, Q_{w} S S′ Q_{v} >0,$ where E is the expectation, a prime denotes the transpose, $δ_{tt} =1, δ_{tj} =0 (t≠j)$ .

Assumption 2

M is a singular square matrix, i.e. $det M=0$ .

Assumption 3

The system is regular, i.e. for $∀z∈C, det(zM− Φ)≢0$ .

Assumption 4

The system is completely observable, i.e. for ∀z∈

Descriptor Wiener state estimators

Theorem 1

For descriptor system (1)–(2) with Assumptions 1–4, the asymptotically stable descriptor Wiener state estimators are given by $x ̂ (t | t+N)= D ̄_{N}^{−1} K ̄_{N} y(t+N)$ with the transfer function matrices $D ̄_{N}^{−1} K ̄_{N}$ and inputy(t+N), which have the ARMA recursive filter form $D ̄_{N} x ̂ (t | t+N)= K ̄_{N} y(t+N),$ where $D ̄_{N}$ and $K ̄_{N}$ are determined by the following left-coprime factorization: $D ̄_{N}^{−1} K ̄_{N} =K_{N} D ̃^{−1},$ where $D ̄_{N0} =I_{n}, D ̃$ and $A ̃$ are defined in (17). K_Nare defined as $K_{N} = ∑ i=0 β−2 Ω_{i}^{(1)} Γ L_{N−i} A ̃ + ∑ i=0 β−1 Ω_{i}^{(2)} [J_{i−N} −M_{N−i} A ̃],$ where $L_{i}, M_{i}$ andJ_iare defined in

Simulation example

Consider the completely observable descriptor system (1)–(2) with $M= 1000, Φ = 1 −0.2 −1 1, Γ = 11,$ $H=I_{2}, v(t)=α Γ w(t)+ξ(t),$ where $x(t)=[x_{1} (t), x_{2} (t)]′, w(t)$ and ξ(t) are independent Gaussian white noises with zero mean and variances σ_w²=1 and Q_ξ=diag(1,1), respectively, and α=0.5.

The problem is to find the descriptor Wiener state predictor $x ̂ (t | t−1)$ and filter $x ̂ (t | t)$ .

We easily obtain the ARMA innovation model $(1−0.8q^{−1})y(t)=Dε(t),$ where D=I₂+D₁q⁻¹, and we have the relation $Dε(t)=[1.2q^{−1}, 2q^{−1} −1]′w(t)+(1−0.8q^{−1}$

Conclusions

A new time–domain Wiener filtering approach has been presented based on the modern time-series analysis method. It avoids the solution of the Riccati and Diophantine equations.

Compared with the descriptor Kalman filtering approach, the Riccati equation is replaced by the ARMA innovation model. The iteration algorithm of the Riccati equation given by Nikoukhah, Willsky and Bernard (1992) requires to compute the pseudo-inverse of the (2n+m)×(2n+m) symmetric matrix in each iteration period, but we

Acknowledgements

This work was supported by Natural Science Foundation of China under Grant NSFC-69774019.

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^☆: This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor R. Boel under the direction of Editor T. Basar.

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Technical CommuniqueDescriptor Wiener state estimators☆

Abstract

Introduction

Section snippets

Problem formulation and lemmas

Descriptor Wiener state estimators

Simulation example

Conclusions

Acknowledgements

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Descriptor Wiener state estimators☆