Constrained linear state estimation—a moving horizon approach

doi:10.1016/S0005-1098(01)00115-7

Automatica

Volume 37, Issue 10, October 2001, Pages 1619-1628

https://doi.org/10.1016/S0005-1098(01)00115-7 Get rights and content

Abstract

This article considers moving horizon strategies for constrained linear state estimation. Additional information for estimating state variables from output measurements is often available in the form of inequality constraints on states, noise, and other variables. Formulating a linear state estimation problem with inequality constraints, however, prevents recursive solutions such as Kalman filtering, and, consequently, the estimation problem grows with time as more measurements become available. To bound the problem size, we explore moving horizon strategies for constrained linear state estimation. In this work we discuss some practical and theoretical properties of moving horizon estimation. We derive sufficient conditions for the stability of moving horizon state estimation with linear models subject to constraints on the estimate. We also discuss smoothing strategies for moving horizon estimation. Our framework is solely deterministic.

Introduction

The Kalman filter is the standard choice for estimating the state of a linear system when the measurements are noisy and the process disturbances are unmeasured. One reason for the popularity of the Kalman filter is that it possesses many important theoretical properties such as stability. Often additional insight about the process is available in the form of inequality constraints. With the addition of inequality constraints, however, general recursive solutions such as Kalman filtering are unavailable. One strategy for determining an optimal state estimate is to reformulate the estimation problem as a quadratic program. This formulation allows for the natural addition of inequality constraints. While there exist many strategies to solve efficiently quadratic programs with the particular structure of the linear estimation problem (cf. Biegler, 1998), the problem grows without bound as we collect more measurements.

Building on the success of receding horizon control (for recent reviews, see Mayne, 1997; Lee & Cooley, 1997; Mayne, Rawlings, Rao, & Scokaert, 2000), moving horizon estimation (MHE) has been suggested as a practical strategy to incorporate inequality constraints in estimation (cf. Muske, Rawlings, & Lee, 1993; Muske & Rawlings, 1995; Robertson, Lee, & Rawlings, 1996; Tyler, 1997; Rao & Rawlings, 2000b). The basic strategy of MHE is to reformulate the estimation problem as a quadratic program using a moving, fixed-size estimation window. The fixed-size estimation window is necessary to bound the size of the quadratic program. Because only a subset of the data is considered, stability questions arise. The contribution of this article is that we prove stability for moving horizon estimation. We also briefly discuss smoothing strategies. The central theme of our analysis is the relationship between the full information estimation problem and its moving horizon approximation. This relationship is analyzed using forward dynamic programming and allows us to derive sufficient conditions for stability. Our stability results build on some of the general results of Rao and Rawlings (2000b).

Section snippets

Problem statement

Let the system generating the data sequence {y_k} be modeled by the following linear, time-invariant, discrete-time system $x_{k+1} =Ax_{k} +Gw_{k},$ $y_{k} =Cx_{k} +v_{k},$ where it is known that the states and disturbances satisfy the following constraints: $x_{k} ∈ X, w_{k} ∈ W, v_{k} ∈ V .$ We assume $x_{k} ∈ R^{n}$ , $y_{k} ∈ R^{p}$ , and $w_{k} ∈ R^{m}$ and the sets $X$ , $W$ , and $V$ are polyhedral and convex (i.e. $X ={x : Dx⩽d}$ ) with $0∈ W$ and $0∈ V$ . Let $x(k; z,{w_{j}})$ denote the solution of model (1) at time k subject to the initial condition z and disturbance sequence {w_j}_j=0^k−1: $x(k;$

Moving horizon approximation

Efficient strategies exist for solving the quadratic program (2). However, the problem size grows with time as the estimator processes more data. As a result, the problem complexity scales at least linearly with T. To make the estimation problem tractable, we need to bound the problem size. One strategy to reduce the problem (2) to a fixed dimension quadratic program is to employ a moving horizon approximation. The basic strategy of the moving horizon approximation is to consider explicitly a

Stability analysis

When the inequality constraints (3) are not present, the solution to the quadratic program (2) may be obtained analytically, yielding the Kalman filter. The relationship between least squares and the Kalman filter is well known (cf. Bryson & Frazier, 1963; Rauch, Tung, & Striebel, 1965). Even with the addition of constraints, the estimator enjoys analogous stability properties. In particular, the constrained estimator is stable in the sense of an observer. The following discussion of observer

Smoothing update

In our development of the MHE, we use a filter update to summarize the past information. With the filter update we transfer the prior information to current estimate window by conditioning the estimates at time T using $x ̂_{T−N}$ . The conditioning is the result of the approximate arrival cost $(x_{T−N} − x ̂_{T−N})′Π_{T−N}^{−1} (x_{T−N} − x ̂_{T−N})$ achieving its minimum at $x ̂_{T−N}$ . A schematic of the filter update strategy is shown in Fig. 1.

Rather than conditioning the estimate at time T on $x ̂_{T−N}$ , we may also condition the

Conclusion

We have demonstrated that moving horizon estimation (MHE) is a practical strategy for constrained state estimation. Three separate formulations were presented. The key result of this work is that if the full information estimator is stable, then MHE is also stable provided one does not introduce extra bias with the prior information. To characterize this condition, we analyzed the estimation problem using forward dynamic programming and the notion of arrival cost.

We believe one of the strengths

Acknowledgements

The authors gratefully acknowledge the financial support of the industrial members of the Texas-Wisconsin Modeling and Control Consortium and NSF through grant #CTS-9708497. The authors express their thanks to Professor D.Q. Mayne for helpful discussions and feedback about this work. The authors also acknowledge P.K. Findeisen, who studied some of the issues presented in this paper as part of a Master's research project (Findeisen 1997) at the University of Wisconsin-Madison. We are grateful to

Christopher V. Rao received the B.S. from Carnegie Mellon University in 1994 and the Ph.D. from the University of Wisconsin in 2000, both in Chemical Engineering. He is currently a post doctoral researcher in the Department of Bioengineering at the University of California, Berkeley and Lawrence Berkeley National Laboratory. His current research interests are in the areas of computational molecular biology and signal transduction.

References (23)

R.R. Bitmead et al.
Monotonicity and stabilizability properties of solutions of the Riccati difference equation: Propositions, lemmas, theorems, fallacious conjectures and counterexamples
System Control Letters
(1985)
D.Q. Mayne et al.
Constrained model predictive control: Stability and optimality
Automatica
(2000)
C. Striebel
Sufficient statistics in the optimum control of stochastic systems
Journal of Mathematical Analysis and Application
(1965)
T. Başar et al.
H^∞—Optimal control and related minimax design problems: A dynamic game approach
(1995)
Biegler, L. T. (1998). Efficient solution of dynamic optimization and NMPC problems. Proceedings of the international...
Bryson, A. E., & Frazier, M. (1963). Smoothing for linear and nonlinear dynamic systems. Proceedings of the Optimum...
A.E. Bryson et al.
Applied optimal control
(1975)
C.E. de Souza et al.
Riccati equation in optimal filtering of nonstabilizable systems having singular state transition matrices
IEEE Transactions on Automatic Control
(1986)
Findeisen, P. (1997). Moving horizon state estimation of discrete time systems. Master's thesis, University of...
M. Frank et al.
An algorithm for quadratic programming
Naval Research Logistics Quarterly
(1956)

A.H. Jazwinski

Stochastic processes and filtering theory

(1970)

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James B. Rawlings was born in Gary, Indiana, USA in 1957. He received the B.S. from the University of Texas in 1979 and the Ph.D. from the University of Wisconsin in 1985, both in Chemical Engineering. He spent one year at the University of Stuttgart as a NATO post-doctoral fellow and then joined the faculty at the University of Texas. He moved to the University of Wisconsin in 1995 and is currently the Chair and Paul A. Elfers Professor of the Department of Chemical Engineering and the co-director of the Texas-Wisconsin Modeling and Control Consortium (TWMCC). His research interests are in the areas of chemical process modeling and control, nonlinear model predictive control, moving horizon state estimation and monitoring, particle technology and crystal engineering.

Jay H. Lee was born in Seoul, Korea, in 1965. He obtained his B.S. degree in Chemical Engineering from the University of Washington, Seattle, in 1986, and his Ph.D. degree in Chemical Engineering from California Institute of Technology, Pasadena, in 1991. From 1991 to 1998, he was with the Department of Chemical Engineering at Auburn University, AL, as an Assistant Professor and an Associate Professor. From 1998 to 2000, he was with School of Chemical Engineering at Purdue University, West Lafayette, as an Associate Professor. Currently, he is a Professor in the School of Chemical Engineering at Georgia Institute of Technology, Atlanta. He has held visiting appointments at E.I. Du Pont de Nemours, Wilmington, in 1994 and at Seoul National University, Seoul, Korea, in 1997. He was a recipient of the National Science Foundation's Young Investigator Award in 1993. His research interests are in the areas of system identification, robust control, model predictive control and nonlinear estimation.

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

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Brief PaperConstrained linear state estimation—a moving horizon approach☆

Abstract

Introduction

Section snippets

Problem statement

Moving horizon approximation

Stability analysis

Smoothing update

Conclusion

Acknowledgements

System Control Letters

Automatica

Journal of Mathematical Analysis and Application

H∞—Optimal control and related minimax design problems: A dynamic game approach

Applied optimal control

Riccati equation in optimal filtering of nonstabilizable systems having singular state transition matrices

IEEE Transactions on Automatic Control

An algorithm for quadratic programming

Naval Research Logistics Quarterly

Stochastic processes and filtering theory

Brief Paper
Constrained linear state estimation—a moving horizon approach☆

H^∞—Optimal control and related minimax design problems: A dynamic game approach