Brief paperState estimation for linear systems with state equality constraints☆
Introduction
Control with constraints is increasingly applied in industry (Maciejowski, 2002), and using explicit constraints, in place of their implicit inclusion using penalty and barrier methods, simplifies the design specification to focus on the performance objective. The problem which we consider here is the concomitant state estimation problem in constrained systems.
When some relations between state variables are known exactly, this type of constraint is called a hard or strong constraint. Noisy measurements are available from the observable system with the constraint and we desire to use these optimally to reconstruct a state estimate, which also is known to satisfy linear equality constraints. For systems with linear state equality constraints, it is always possible to reduce the system model parametrization, and use the reduced state equation and the conventional Kalman filter. On the other hand, there are several good reasons for not using a reduced state space for treating the constrained system. Firstly, the dimension of the reduced state space may vary between systems of different dimensionality, such as is the case for locomotion systems (Hemami & Wyman, 1979). Secondly, the reduction of the state equations makes their interpretation less natural and more difficult (Simon & Chia, 2002).
In conventional linear stochastic models with additive white process noise, for a state vector to satisfy a hard equality constraint, the process noise must have a singular covariance consistent with a linear constraint on the state. Hence, in order to design the “correct” Kalman filter for such a constrained system, the exact (singular) covariance matrix of the constrained noise must be used, which is called the constrained Kalman filter in this paper.
This paper focuses on performance comparison of the correct constrained Kalman filter with two other Kalman filters, in terms of the magnitude of estimation error covariance matrix through the monotonicity property of Riccati difference equation. One is the unconstrained Kalman filter obtained by using a bigger process noise covariance matrix than the true singular covariance and the other is the projected Kalman filter which is the projected version of the unconstrained Kalman filter onto the constraint subspace. Chia (1985) and Simon and Chia (2002) used this projection method with Kalman prediction and filtering, and proved that the state estimation error covariance of the projected estimate is smaller than that of the unconstrained estimate. Wen and Durrant-Whyte (1992) independently developed a similar method by firstly including the constraint as a perfect observation and then showing that their method is theoretically exactly the same as projecting the filtered estimate of the unconstrained Kalman filter onto the surface of the hard constraint. Recently, Mahata and Söderstom (2004) used a projection approach in estimating deterministic parameters of viscoelastic materials. Here, the additional linear constraints are imposed in the form of a partially known boundary condition to obtain better estimates.
The major contributions of this paper are two-fold:
- (1)
formulating linear stochastic systems with linear state equality constraints in a projected system representation in Section 2,
- (2)
clearly proving, without using the notion of optimality, that the constrained Kalman filter outperforms the projected and unconstrained Kalman filters in terms of the magnitude of estimation error covariance matrix in Section 3.
Notations
In this paper, matrices will be denoted by uppercase boldface (e.g., ), linear spaces are denoted by calligraphic uppercase (e.g., ), column matrices (vectors) will be denoted by lowercase boldface (e.g., ), and scalars will be denoted by lowercase (e.g., y) or uppercase (e.g., Y). For a matrix , denotes its transpose and represents the Moore–Penrose inverse of , and denotes the null space of . For a symmetric matrix or denotes the fact that is positive definite or positive semi-definite, respectively. For a random vector , represents the mathematical expectation of .
Section snippets
Linear stochastic systems with equality constraints
We investigate a method of estimating the state of systems modeled by a linear stochastic difference equation of the formwhere the state is known to be constrained in the null space of :and represents the measurement. The noise sequences and are uncorrelated with each other and have zero-mean white Gaussian distributions with variances and , respectively. The matrix (with ) is assumed to have a full row rank. If
Kalman predictors for constrained systems
In this section, after briefly describing the unconstrained Kalman predictor, we consider the projected and constrained predictors for the constrained systems described in Section 2. They are, respectively, obtained by projecting the unconstrained Kalman estimate onto the constraint subspace and by using the correct process noise covariance matrix. Performance comparison between the estimators is made by comparing the relative magnitude of covariance matrices for the Kalman predictors.
Numerical example
To show numerically the performance differences of the three estimators, consider the following linear system and measurement equation describing movement of a land-based vehicle:This example was used in Simon and Chia (2002) to compare the performance between the unconstrained and the projected Kalman predictor, where a nonlinear measurement equation was used. The first two state components of are the northerly and easterly
Concluding remarks
In this paper, we have analyzed the three estimators that can be used for estimating linear systems with known state equality constraints. Among them, it was proved that the truly constrained estimator is optimal and thus outperforms the unconstrained and the projected estimators. This fact was demonstrated through a numerical example. We can show that the discrete and continuous-time filter versions of Theorem 2 also hold, which can be found in Ko (2005).
Acknowledgment
The research was supported by USA National Science Foundation Grants ECS-0200449 and ECS-0225530.
Sangho Ko was born in Seoul, Korea, in 1967. He received the B.Sc. degree in aerospace and mechanical engineering from Korea Aerospace University, Goyang, Korea, the M.Sc. degree in aerospace engineering from Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, and the Ph.D. degree in mechanical engineering in University of California at San Diego (UCSD), USA, in 1989, 1992, and 2005, respectively. From 2005 to 2006, he was a Post-Doctoral Scholar in UCSD to research
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Sangho Ko was born in Seoul, Korea, in 1967. He received the B.Sc. degree in aerospace and mechanical engineering from Korea Aerospace University, Goyang, Korea, the M.Sc. degree in aerospace engineering from Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, and the Ph.D. degree in mechanical engineering in University of California at San Diego (UCSD), USA, in 1989, 1992, and 2005, respectively. From 2005 to 2006, he was a Post-Doctoral Scholar in UCSD to research system identification of combustion instability problem of gas turbine engines. Now, he is a Research Fellow in the Department of Electrical and Electronic Engineering of the University of Melbourne, Australia, since March 2006, where he is working on finite sample quality assessment of system identification. From 1992 to 1999, he was with Samsung Aerospace Industries, Ltd., Kyungnam, Korea, where he was involved in designing digital flight control system of the advanced jet trainer T-50 for the Republic of Korea Air Force.
Robert R. Bitmead holds the Cymer Corporation Chair in the Department of Mechanical and Aerospace Engineering, University of California, San Diego, where he is also Associate Vice-Chancellor for Academic Personnel. He received his Ph.D. from the University of Newcastle in 1980. He was with the Department of Systems Engineering, the Australian National University, from 1982 to1999. He is an IFAC Fellow, a Fellow of the Australian Academy of Technological Sciences and Engineering, and a Fellow of the IEEE. He was a member of the IFAC Council from 1996 to 2002 and a member of IFAC Technical Board from 1990 to 1996. He hails originally from Sydney and was General Chair of the IEEE Conference on Decision and Control there in 2000. He was Editor of Automatica for Adaptive and Intelligent Control from 2002 to 2005.
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This paper was presented at IFAC World Congress 2005, Prague, Czech Republic. This paper was recommended for publication in revised form by Associate Editor Marco Campi under the direction of Editor Ian Petersen.