Unbiased minimum-variance input and state estimation for linear discrete-time systems with direct feedthrough

doi:10.1016/j.automatica.2006.11.016

Automatica

Volume 43, Issue 5, May 2007, Pages 934-937

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Abstract

This paper extends previous work on joint input and state estimation to systems with direct feedthrough of the unknown input to the output. Using linear minimum-variance unbiased estimation, a recursive filter is derived where the estimation of the state and the input are interconnected. The derivation is based on the assumption that no prior knowledge about the dynamical evolution of the unknown input is available. The resulting filter has the structure of the Kalman filter, except that the true value of the input is replaced by an optimal estimate.

Introduction

Systematic measurement errors and model uncertainties such as unknown disturbances or unmodeled dynamics can be represented as unknown inputs. The problem of optimal filtering in the presence of unknown inputs has therefore received a lot of attention.

Friedland (1969) and Park, Kim, Kwon, and Kwon (2000) solved the unknown input filtering problem by augmenting the state vector with an unknown input vector. However, this method is limited to the case where a model for the dynamical evolution of the unknown input is available.

A rigorous and straightforward state estimation method in the presence of unknown inputs is developed by Hou and Müller (1994) and Hou and Patton (1998). The approach consists in first building an equivalent system which is decoupled from the unknown inputs, and then designing a minimum-variance unbiased (MVU) estimator for this equivalent system.

Another approach consists in parameterizing the filter equations and then calculating the optimal parameters by minimizing the trace of the state covariance matrix under an unbiasedness condition. An optimal filter of this type was first developed by Kitanidis (1987). The derivation of Kitanidis (1987) is limited to linear systems without direct feedthrough of the unknown input to the output and yields no estimate of the input. An extension to state estimation for systems with direct feedthrough was developed by Darouach, Zasadzinski, and Boutayeb (2003). Extensions to joint input and state estimation for systems without direct feedthrough are addressed by Hsieh (2000) and Gillijns and De Moor (2007).

In this paper, we combine both extensions of Kitanidis (1987) by addressing the problem of joint input and state estimation for linear discrete-time systems with direct feedthrough of the unknown input to the output. Using linear minimum-variance unbiased estimation, we develop a recursive filter where the estimation of the state and the input are interconnected. The estimation of the input is based on the least-squares (LS) approach developed by Gillijns and De Moor (2007), while the state estimation problem is solved using the method developed by Kitanidis (1987).

This paper is outlined as follows. In Section 2, we formulate the filtering problem and present the recursive three-step structure of the filter. Next, in 3 Time update, 4 Input estimation, 5 Measurement update, we consider each of the three steps separately and derive equations for the optimal input and state estimators. Finally, in Section 6, we summarize the filter equations.

Section snippets

Problem formulation

Consider the linear discrete-time system $x_{k + 1} = A_{k} x_{k} + G_{k} d_{k} + w_{k},$ $y_{k} = C_{k} x_{k} + H_{k} d_{k} + v_{k},$ where $x_{k} \in R^{n}$ is the state vector, $d_{k} \in R^{m}$ is an unknown input vector, and $y_{k} \in R^{p}$ is the measurement. The process noise $w_{k} \in R^{n}$ and the measurement noise $v_{k} \in R^{p}$ are assumed to be mutually uncorrelated, zero-mean, white random signals with known covariance matrices, $Q_{k} = E [w_{k} w_{k}^{T}] ⩾ 0$ and $R_{k} = E [v_{k} v_{k}^{T}] > 0$ , respectively. Results are easily generalized to the case where $w_{k}$ and $v_{k}$ are correlated by applying a preliminary transformation to

Time update

First, we consider the time update. Let ${\hat{x}}_{k - 1 | k - 1}$ and ${\hat{d}}_{k - 1}$ denote the optimal unbiased estimates of $x_{k - 1}$ and $d_{k - 1}$ given measurements up to time $k - 1$ , then the time update is given by ${\hat{x}}_{k | k - 1} = A_{k - 1} {\hat{x}}_{k - 1 | k - 1} + G_{k - 1} {\hat{d}}_{k - 1} .$ The error in the estimate ${\hat{x}}_{k | k - 1}$ is given by ${\tilde{x}}_{k | k - 1} ≔ x_{k} - {\hat{x}}_{k | k - 1}, = A_{k - 1} {\tilde{x}}_{k - 1 | k - 1} + G_{k - 1} {\tilde{d}}_{k - 1} + w_{k - 1},$ with ${\tilde{x}}_{k | k} ≔ x_{k} - {\hat{x}}_{k | k}$ and ${\tilde{d}}_{k} ≔ d_{k} - {\hat{d}}_{k}$ . Consequently, the covariance matrix of ${\hat{x}}_{k | k - 1}$ is given by $P_{k | k - 1}^{x} ≔ E [{\tilde{x}}_{k | k - 1} \tilde{x}_{k | k - 1}^{T}], = [\begin{matrix} A_{k - 1} & G_{k - 1} \end{matrix}] [\begin{matrix} P_{k - 1 | k - 1}^{x} & P_{k - 1}^{xd} \\ P_{k - 1}^{dx} & P_{k - 1}^{d} \end{matrix}] [\begin{matrix} A_{k - 1}^{T} \\ G_{k - 1}^{T} \end{matrix}] + Q_{k - 1},$ with $P_{k | k}^{x} ≔ E [{\tilde{x}}_{k |}$

Input estimation

In this section, we consider the estimation of the unknown input. In Section 4.1, we determine the matrix $M_{k}$ such that (4) yields an unbiased estimate of $d_{k}$ . In Section 4.2, we extend to MVU input estimation.

Measurement update

Finally, we consider the update of ${\hat{x}}_{k | k - 1}$ with the measurement $y_{k}$ . We calculate the gain matrix $L_{k}$ which yields the MVU estimator of the form (5). Using (5), (6), we find that ${\tilde{x}}_{k | k} = (I - L_{k} C_{k}) {\tilde{x}}_{k | k - 1} - L_{k} H_{k} d_{k} - L_{k} v_{k} .$ Consequently, (5) is unbiased for all possible $d_{k}$ if and only if $L_{k}$ satisfies $L_{k} H_{k} = 0 .$ Let $L_{k}$ satisfy (10), then it follows from (9) that $P_{k | k}^{x}$ is given by $P_{k | k}^{x} = (I - L_{k} C_{k}) P_{k | k - 1}^{x} (I - L_{k} C_{k})^{T} + L_{k} R_{k} L_{k}^{T} .$ An MVU state estimator is then obtained by calculating the gain matrix $L_{k}$ which minimizes the

Summary of filter equations

In this section, we summarize the filter equations. We assume that ${\hat{x}}_{0}$ , the estimate of the initial state, is unbiased and has known variance $P_{0}^{x}$ . The initialization step of the filter is then given by:

Initialization: ${\hat{x}}_{0} = E [x_{0}], P_{0}^{x} = E [(x_{0} - {\hat{x}}_{0}) (x_{0} - {\hat{x}}_{0})^{T}] .$ The recursive part of the filter consists of three steps: the estimation of the unknown input, the measurement update and the time update. These three steps are given by

Estimation of unknown input: ${\tilde{R}}_{k} = C_{k} P_{k | k - 1}^{x} C_{k}^{T} + R_{k}, M_{k} = (H_{k}^{T} \tilde{R}_{k}^{- 1} H_{k})^{- 1} H_{k}^{T} \tilde{R}_{k}^{- 1}, {\hat{d}}_{k} = M$

Conclusion

This paper has studied the problem of joint input and state estimation for linear discrete-time systems with direct feedthrough of the unknown input to the output. A recursive filter was developed where the update of the state estimate has the structure of the Kalman filter, except that the true value of the input is replaced by an optimal estimate. This input estimate is obtained from the innovation by weighted LS estimation, where the optimal weighting matrix is computed from the covariance

Acknowledgments

Our research is supported by Research Council KULeuven: GOA AMBioRICS, several PhD/postdoc & fellow Grants; Flemish Government: FWO: PhD/postdoc Grants, projects, G.0407.02 (support vector machines), G.0197.02 (power islands), G.0141.03 (Identification and cryptography), G.0491.03 (control for intensive care glycemia), G.0120.03 (QIT), G.0452.04 (new quantum algorithms), G.0499.04 (Statistics), G.0211.05 (Nonlinear), research communities (ICCoS, ANMMM, MLDM); IWT: Ph.D. Grants, GBOU (McKnow);

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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 Karl Henrik Johansson under the direction of Editor André Tits.

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Technical communiqueUnbiased minimum-variance input and state estimation for linear discrete-time systems with direct feedthrough☆

Abstract

Introduction

Section snippets

Problem formulation

Time update

Input estimation

Measurement update

Summary of filter equations

Conclusion

Acknowledgments

Automatica

Automatica

Automatica

Automatica

Optimal filtering

Technical communique
Unbiased minimum-variance input and state estimation for linear discrete-time systems with direct feedthrough☆