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Minimum Variance FIR Smoothers for Discrete-Time State Space Models Bo Kyu Kwon, Soohee Han, Oh Kyu Kwon, and Wook Hyun Kwon
Abstract—We propose a fixed-lag finite-impulse-response (FIR) smoother for a discrete-time state space model. The proposed FIR smoother estimates the state at the fixed-lag time using measured output samples on the recent finite time horizon so that the variance of the estimation error is minimized. The minimum variance FIR (MVFIR) smoother is unbiased and independent of any a priori information of the state on the horizon. A numerical example shows that the proposed MVFIR smoother has better performance than the fixed-lag Kalman smoother based on the infinite impulse response structure when transitory modeling uncertainties exist.
In many cases, the state variables are assumed to be unmeasurable; only the output, as a part of a state, can be measured. It is reasonable to assume that the initial state information is also unmeasurable and, therefore, unknown because the initial state is itself a state. In this paper, we consider a linear fixed-lag FIR smoother for at time without requiring estimation of the state a priori initial state information on the receding horizon. This FIR smoother can be represented as
Index Terms—Finite-impulse-response (FIR) smoother, fixed-lag smoother, minimum variance, receding horizon, unbiased.
(1)
I. INTRODUCTION
S
MOOTHING problems have been widely investigated and are well established in estimation theory. In practice, smoothers are used to improve the performance of filters. In general, the structure of estimators, including smoothers and filters, can be divided into finite impulse response (FIR) or infinite impulse response (IIR) on the basis of the duration of their impulse response. FIR estimators make use of finite measurements on the most recent time interval, called the receding horizon, whereas the IIR estimators use all of the information from initiation to the current time. Both FIR and IIR structures have been applied to filters and predictors, and their properties and disadvantages have been defined for a long time. In the case of smoothers, only IIR smoothers were introduced and there is no result for a general FIR smoother to the best of the author’s knowledge. For IIR filters, some potential problems due to the IIR structure have been reported. IIR filters may diverge for systems with modeling errors [1], [2] or numerical errors [3]. Since the IIR filter has an internal state, undesirable signals can accumulate inside the state and may explode or oscillate. To solve these problems, FIR filters have been used as an alternative to IIR filters despite their increased computational burden. The same applies for smoothers: To overcome the shortcomings of the IIR smoothers, it is desirable to adopt the FIR structure to smoother design for fixed-lag state estimation. Manuscript received August 2, 2006; revised November 21, 2006. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Konstantinos N. Plataniotis. B. K. Kwon, S. Han, and W. H. Kwon are with the School of Electrical Engineering and Computer Science, Seoul National University, Seoul 151-742, Korea (e-mail:
[email protected]). O. K. Kwon is with the School of Electrical and Electronic Engineering and Computer Science, Inha University, Inchon 402-751, Korea (e-mail:
[email protected]). Digital Object Identifier 10.1109/LSP.2007.891840
at the current time , where is the size of the receding horizon is the impulse response with finite duration which will and be obtained later in this paper independently of the output and on the horizon. If the time lag is set to the initial state 0, the FIR smoother (1) is reduced to the FIR filter to estimate the current state . In addition, the FIR smoother (1) is required to be unbiased as follows: (2) The unbiased condition (2) must be met regardless of the initial state on the horizon. Of the linear unbiased FIR smoothers, the fixed-lag optimal FIR smoother will be obtained to minimize the variance of the estimation error, and is called a minimum variance FIR (MVFIR) smoother. Although there are no results for the general FIR smoother of the form (1), there have been some results for FIR filters for state in (1). For deterministic disestimation that correspond to crete-time systems without noise, a moving horizon least-square filter was given in [4]. For special discrete stochastic systems without system noise, a linear FIR filter was introduced for maximum likelihood criterion [5]. For general discrete-time stochastic systems, FIR filters were established by modification of the Kalman filter [6], where heuristic infinite covariance of the initial state information is used and the efficiency of the filters is unclear. In [7], an optimal unbiased FIR filter was derived assuming that the system matrix is nonsingular. Without this assumption, an FIR filter was derived in [4], in which the system is assumed to be noise free. In [8], a filter version of (1) was proposed for deterministic systems assuming that the system matrix is nonsingular. The assumptions described before may prevent FIR filters from being applied to real problems. In this paper, the more general MVFIR smoother is obtained without these limiting assumptions. It is shown to reduce to the minimum variance FIR filter to estimate the current state if in (1) and (2) are set to 0.
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Therefore, the proposed MVFIR smoother is more general than any existing FIR filter. This paper is organized as follows: In Section II, the MVFIR smoother for a general discrete-time system is proposed in a standard FIR form. The performance of the proposed MVFIR smoother is compared with that of the fixed-lag Kalman IIR smoother in Section III. Finally, our conclusions are presented in Section IV.
The finite number of measurements is expressed in terms of the at the initial time as follows: state (7) and are the finite measurement vector and the where finite measurement noise vector given by
II. MVFIR SMOOTHER FOR DISCRETE-TIME SYSTEMS (8)
Consider a linear discrete-time state space model (3) (4) where and are the state vector and the measurement vector, respectively. At the initial time , the state is a random variable with a mean and a covariance . and the measurement noise The system noise vector vector are zero-mean white Gaussian and mutually unand denote the covariance correlated. and , which are assumed to be positive defimatrices of nite matrices. These noises are uncorrelated with the initial state . The pair of the system (3) and (4) is assumed to be observable so that all modes are observed at the output and stabilized observers can be constructed. In order to obtain an FIR smoother, it is necessary to relate the most recent information of the outputs to the estimated state. To to be estimated is represented in a batch begin, the state , called the form on the most recent time interval horizon, as follows:
and
and
are obtained from
.. .
(9)
.. .
.. .
.. .
.. .
.. . (10)
The variance of the noise term in (7) can be shown to be given by (11) and are the diagonal matrices of where spectively, given by
and
, re-
(5) where by
and the finite system noise vector
are given Combining (5) and (7) yields the following linear model:
(6) As the next step, systems (3) and (4) will be represented in a batch form on the most recent time interval . On , the measurements can be expressed as the horizon follows:
(12) where is defined in (6). Now, we can obtain an MVFIR smoother by using a linear model (12). An FIR smoother (1) can be expressed as a linear (8) on the transformation of the finite measurements as follows: horizon (13)
.. .
where a smoother gain matrix
is given by
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and is chosen to minimize the variance of the estimation error. Using the linear model (12), the MVFIR smoother (13) can be rewritten as
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for
. Calculating from and taking its expectation, we then obtain
(20) (14) and taking the expectation on both sides of (14) yields the following equation:
Note that the variance of the th component of the estimation of the smoother error depends only on the th row vector gain matrix . We consider each term of the summation (20) independently. The following performance criterion is then established
To satisfy the unbiased condition (i.e., ), on the horizon, the folwithout respect to the initial state lowing constraint is required: (15)
(21)
Substituting (15) into (14), we have is the Lagrange multiplier associated with the where th unbiased constraint. To minimize with respect to and , two conditions from which obtained as
is the estimation error vector at time and (16)
must be satisfied. It follows that we have:
The object now is to determine the optimal smoother gain matrix subject to the unbiased constraint (15), which can be summarized by the following criterion: (22) subject to (17) For convenience, the matrix
where the inverse of is guaranteed because is positive definite. Premultiplying (22) by , we have (23)
in (13) is partitioned as where a second equality coming from (19) and
is obtained as
(18) is the th row vector of the smoother where and are denoted by gain and the th row vectors of and , respectively. and denote the th components of the state and , respectively. Since the unbiased conthe estimation error , the th unbiased constraint can be straint is rewritten as (19)
(24) Note that the inverse of is guaranteed by the observability of the system (3)–(4). Substituting (24) into (22) yields
(25) in (13) can be obtained by combining all according to (18). These results are summarized in the following theorem.
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Theorem 1: For the state space model (3)–(4), the minimum variance FIR smoother (1) is given by
with the optimal smoother gain matrix
determined by
(26) If the fixed-lag size is set to zero, the MVFIR smoother (26) is reduced to the minimum variance FIR filter to estimate the current state . As seen in (26), the inverse of the system matrix does not appear in the optimal smoother gain matrix . III. NUMERICAL EXAMPLE In this section, we give a numerical example to demonstrate the performance of the proposed MVFIR smoother. Consider a state space model
(27)
where is an uncertain model parameter. The system noise covariance and the measurement noise covariance are 0.01 is and 0.02, respectively. The uncertain model parameter given as
otherwise
(28)
For the smoother design, the horizon size and the fixed-lag size are and , respectively. We performed a simulation for the system (27) with temporary modeling uncertainties (28). In Fig. 1, the estimation errors of the MVFIR smoother and the fixed-lag Kalman smoother are compared. It can be seen that the estimation error of the MVFIR smoother is considerably smaller than that for the fixed-lag Kalman smoother when modeling uncertainties exist. Moreover, it is shown that the estimation error of the MVFIR smoother converges more rapidly than that of the fixed-lag Kalman smoother after temporary modeling uncertainty disappears. Therefore, the proposed MVFIR smoother can be said to be more robust than other IIR smoothers when applied to systems with modeling uncertainty.
Fig. 1. Estimation errors of MVFIR smoother and fixed-lag Kalman smoother.
IV. CONCLUSION In this paper, an MVFIR smoother for discrete-time state space models is proposed. The proposed MVFIR smoother is chosen to achieve minimum error variance with unbiased constraint. It is designed to be linear, unbiased, with an FIR structure, and is independent of any state information. For a general system with system and measurement noise, the MVFIR smoother is derived without any artificial assumptions. The nonsingularity of the system matrix is not required. Heuristic approaches such as infinite covariance of the initial state are not used. The approach in this paper is logical and systematic. Due to its FIR structure, the MVFIR smoother is believed to have better estimation performance for temporary modeling uncertainty or numerical errors than other IIR smoothers, such as the Kalman IIR smoother. It was shown through simulation that the proposed MVFIR smoother has much better performance than the Kalman IIR smoother. Therefore, the MVFIR smoother may be a good substitute in cases where an optimal IIR smoother has some disadvantages due to its IIR structure. REFERENCES [1] R. J. Fitzgerald, “Divergence of the Kalman filter,” IEEE Trans. Autom. Control, vol. AC-16, no. 6, pp. 736–747, Dec. 1971. [2] S. Sangsuk-Iam and T. E. Bullock, “Analysis of discrete-time Kalman filtering under incorrect noise covariances,” IEEE Trans. Autom. Control, vol. 35, no. 12, pp. 1304–, Dec. 1990. [3] M. S. Grewal and A. P. Anderews, Kalman filtering—Theory and Practice. Englewood Cliffs, NJ: Prentice-Hall, 1993. [4] K. V. Ling and K. W. Lim, “Receding horizon recursive state estimation,” IEEE Trans. Autom. Control, vol. 44, no. 9, pp. 1750–1753, Sep. 1999. [5] A. H. Jazwinski, “Limited memory optimal filtering,” IEEE Trans. Autom. Control, vol. AC-13, no. 5, pp. 558–563, Oct. 1968. [6] W. H. Kwon, P. S. Kim, and P. Park, “A receding horizon Kalman FIR filter for discrete time-invariant systems,” IEEE Trans. Autom. Control, vol. 44, no. 9, pp. 1787–1791, Sep. 1999. [7] W. H. Kwon, P. S. Kim, and S. H. Han, “A receding horizon unbiased fir filter for discrete-time state space models,” Automatica, vol. 38, no. 3, pp. 545–551, 2002. [8] S. H. Han, W. H. Kwon, and P. S. Kim, “Quasi-deadbeat minimax filters for deterministic state space models,” IEEE Trans. Autom. Control, vol. 47, no. 11, pp. 1904–1908, Nov. 2002.
