Iterative and Algebraic Algorithms for the Computation of the Steady State Kalman Filter Gain

Abstract

The Kalman filter gain arises in linear estimation and is associated with linear systems. The gain is a matrix through which the estimation and the prediction of the state as well as the corresponding estimation and prediction error covariance matrices are computed. For time invariant and asymptotically stable systems, there exists a steady state value of the Kalman filter gain. The steady state Kalman filter gain is usually derived via the steady state prediction error covariance by first solving the corresponding Riccati equation. In this paper, we present iterative per-step and doubling algorithms as well as an algebraic algorithm for the steady state Kalman filter gain computation. These algorithms hold under conditions concerning the system parameters. The advantage of these algorithms is the autonomous computation of the steady state Kalman filter gain.

1. Introduction

The Kalman filter gain arises in Kalman filter equations in linear estimation and is associated with linear systems. State space systems have been widely used in estimation theory to describe discrete time systems [1–5]. It is known [1] for time invariant systems that if the signal process model is asymptotically stable, then there exists a steady state value of the Kalman filter gain. Thus, the steady state gain is associated with time invariant systems described by the following state space equations: 𝑥_𝑘+1=𝐹𝑥_𝑘+𝑤_𝑘,𝑧_𝑘=𝐻𝑥_𝑘+𝑣_𝑘, (1) for 𝑘≥0, where 𝑥_𝑘 is the 𝑛-dimensional state vector at time 𝑘, 𝑧_𝑘 is the 𝑚-dimensional measurement vector at time 𝑘, 𝐹 is the 𝑛×𝑛 system transition matrix, 𝐻 is the 𝑚×𝑛 output matrix, 𝑤_𝑘 is the plant noise at time 𝑘, and 𝑣_𝑘 is the measurement noise at time 𝑘. Also, {𝑤_𝑘} and {𝑣_𝑘} are Gaussian zero-mean white random processes with covariance matrices 𝑄 and 𝑅, respectively.

The discrete time Kalman filter [1, 6] is the most well-known algorithm that solves the filtering problem. In fact, Kalman filter faces simultaneously two problems as follows.(i)Estimation: the aim is to recover at time 𝑘 information about the state vector at time 𝑘 using measurements up till time 𝑘.(ii)Prediction: the aim is to obtain at time 𝑘 information about the state vector at time 𝑘+1 using measurements up till time 𝑘; it is clear that prediction is related to the forecasting side of information processing.Kalman filter uses the measurements up till time 𝑘 in order to produce the (one step) prediction of the state vector and the corresponding prediction error covariance matrix 𝑃_𝑘+1/𝑘, as well as producing the estimation of the state vector and the corresponding estimation error covariance matrix 𝑃_𝑘/𝑘. The Kalman filter equations needed for the computation of the prediction and estimation error covariance matrices are as follows: 𝐾_𝑘=𝑃_{𝑘/𝑘−1}𝐻^𝑇𝐻𝑃_{𝑘/𝑘−1}𝐻^𝑇+𝑅⁻¹,𝑃_𝑘/𝑘=𝐼−𝐾_𝑘𝐻𝑃_{𝑘/𝑘−1},𝑃_𝑘+1/𝑘=𝑄+𝐹𝑃_𝑘/𝑘𝐹^𝑇, (2)(3)(4) for 𝑘≥0, with initial condition 𝑃_0/−1=𝑃₀ for the time instant, where there are no measurements given. Note that 𝐾_𝑘 is the Kalman filter gain.

From (2) to (4), we are able to derive the Riccati equation, which is an iterative equation with respect to the prediction error covariance: 𝑃_𝑘+1/𝑘=𝑄+𝐹𝑃_{𝑘/𝑘−1}𝐹^𝑇−𝐹𝑃_{𝑘/𝑘−1}𝐻^𝑇𝐻𝑃_{𝑘/𝑘−1}𝐻^𝑇+𝑅⁻¹𝐻𝑃_{𝑘/𝑘−1}𝐹^𝑇. (5)

In the general case, where 𝑅 and 𝑃₀ are positive definite matrices, using in (5) the matrix inversion lemma: [𝐴+𝐵𝐶𝐷]⁻¹=𝐴⁻¹−𝐴⁻¹𝐵𝐶⁻¹+𝐷𝐴⁻¹𝐵⁻¹𝐷𝐴⁻¹, (6) the Riccati equation is formulated as 𝑃_𝑘+1/𝑘=𝑄+𝐹𝑃⁻¹_{𝑘/𝑘−1}+𝐻^𝑇𝑅⁻¹𝐻⁻¹𝐹^𝑇. (7)

The Riccati equation is a nonlinear iterative equation with respect to the prediction error covariance. For time invariant systems, it is well known [1] that if the signal process model is asymptotically stable, then there exists a steady state value 𝑃_𝑝 of the prediction error covariance matrix. In fact, the prediction error covariance tends to the steady state prediction error covariance.

The steady state prediction error covariance satisfies the steady state Riccati equation𝑃_𝑝=𝑄+𝐹𝑃_𝑝⁻¹+𝐻^𝑇𝑅⁻¹𝐻⁻¹𝐹^𝑇. (8)

Then, from (2), it is clear that there also exists a steady state value 𝐾 of the Kalman filter gain [7]. The steady state gain can be calculated by 𝐾=𝑃_𝑝𝐻^𝑇𝐻𝑃_𝑝𝐻^𝑇+𝑅⁻¹. (9)

Also, from (3), it is clear that there also exists a steady state value 𝑃_𝑒 of the estimation error covariance matrix [7], which can be calculated by 𝑃_𝑒=𝑃_𝑝−𝑃_𝑝𝐻^𝑇𝐻𝑃_𝑝𝐻^𝑇+𝑅⁻¹𝐻𝑃_𝑝=𝑃_𝑝⁻¹+𝐻^𝑇𝑅⁻¹𝐻⁻¹. (10)

It is obvious from (9) that the steady state Kalman filter gain can be derived via the steady state prediction error covariance. The covariance matrix in Kalman filter plays an important role in many applications [1, 4, 6, 8–10]. The steady state prediction error covariance can be derived by solving the Riccati equation emanating from Kalman filter. The discrete time Riccati equation has attracted recent attention. In view of the importance of the Riccati equation, there exists considerable literature on its algebraic solutions; for example, in [1, 7, 11, 12], the authors have derived an eigenvector solution, while the author of [13] has included solving scalar polynomials. Other methods are based on the iterative solutions [1, 13–18] concerning per-step or doubling algorithms. The iterative algorithms that provide the steady state Kalman filter gain together with the prediction error covariance are the Chandrasekhar algorithms [1], as well as the iterative algorithm that calculate the Kalman gain only once for a period of the stationary channel, as opposed to each data sample in the conventional filter [19]. A geometric illustration of the Kalman filter gain is given in [20].

In this paper, we present algorithms for the steady state Kalman filter gain autonomous computation. These algorithms hold under conditions concerning the system parameters. The paper is organized as follows: two new per-step iterative algorithms, a new doubling iterative algorithm, and an algebraic algorithm for the computation of the steady state Kalman filter gain are presented in Section 2. In Section 3, two examples verify the results. Finally, Section 4 summarizes the conclusions.

2. New Algorithms for the Steady State Kalman Filter Computation

2.1. Assumptions

We assume the general case, where 𝑅 and 𝑃₀ are positive definite matrices.

The Kalman filter gain 𝐾_𝑘 is a matrix of dimension 𝑛×𝑚.

We define the matrix 𝐺_𝑘=𝐾_𝑘𝐻. (11) It is clear that 𝐺_𝑘 is a nonsymmetric matrix of dimension 𝑛×𝑛.

It is also clear that there exists a steady state value 𝐺=𝐾𝐻. (12)

Also, we define the matrix 𝑆=𝐻^𝑇𝑅⁻¹𝐻. (13) Note that 𝑆 is an 𝑛×𝑛 symmetric positive semidefinite matrix and 𝑆 is a positive definite if rank(𝐻)=𝑛; this means that 𝑆 is a nonsingular matrix in the case, rank(𝐻)=𝑛 with 𝑚≥𝑛, [21].

2.2. Indirect Steady State Kalman Filter Gain Computation

In this section, we present algorithms for 𝐺 computation. Then, we show how to compute the steady state Kalman filter 𝐾 through 𝐺=𝐾𝐻.

2.2.1. Iterative Algorithms for 𝐺 Computation

In this section, we present two iterative per-step algorithms and an iterative doubling algorithm for 𝐺 computation.

Per-Step Iterative Algorithm 1. Using (2) and (11), it is derived that 𝐺_𝑘=𝐾_𝑘𝐻=𝑃_{𝑘/𝑘−1}𝐻^𝑇𝐻𝑃_{𝑘/𝑘−1}𝐻^𝑇+𝑅⁻¹𝐻=𝑃_{𝑘/𝑘−1}𝐻^𝑇𝐻𝑃_{𝑘/𝑘−1}𝐻^𝑇+𝑅⁻¹𝑅𝑅⁻¹𝐻=𝑃_{𝑘/𝑘−1}𝐻^𝑇𝐻𝑃_{𝑘/𝑘−1}𝐻^𝑇+𝑅⁻¹×𝐻𝑃_{𝑘/𝑘−1}𝐻^𝑇+𝑅−𝐻𝑃_{𝑘/𝑘−1}𝐻^𝑇𝑅⁻¹𝐻=𝑃_{𝑘/𝑘−1}𝐻^𝑇𝐻𝑃_{𝑘/𝑘−1}𝐻^𝑇+𝑅⁻¹𝐻𝑃_{𝑘/𝑘−1}𝐻^𝑇−+𝑅𝐻𝑃_{𝑘/𝑘−1}𝐻^𝑇+𝑅⁻¹𝐻𝑃_{𝑘/𝑘−1}𝐻^𝑇𝑅⁻¹𝐻=𝑃_{𝑘/𝑘−1}𝐻^𝑇𝐼−𝐻𝑃_{𝑘/𝑘−1}𝐻^𝑇+𝑅⁻¹𝐻𝑃_{𝑘/𝑘−1}𝐻^𝑇𝑅⁻¹𝐻=𝑃_{𝑘/𝑘−1}𝐻^𝑇𝑅⁻¹𝐻−𝑃_{𝑘/𝑘−1}𝐻^𝑇𝐻𝑃_{𝑘/𝑘−1}𝐻^𝑇+𝑅⁻¹𝐻𝑃_{𝑘/𝑘−1}𝐻^𝑇𝑅⁻¹𝐻=𝑃_{𝑘/𝑘−1}−𝑃_{𝑘/𝑘−1}𝐻^𝑇𝐻𝑃_{𝑘/𝑘−1}𝐻^𝑇+𝑅⁻¹𝐻𝑃_{𝑘/𝑘−1}×𝐻^𝑇𝑅⁻¹𝐻=𝑃⁻¹_{𝑘/𝑘−1}+𝐻^𝑇𝑅⁻¹𝐻⁻¹𝐻^𝑇𝑅⁻¹𝐻=𝑃⁻¹_{𝑘/𝑘−1}+𝑆⁻¹𝑆. (14)

Thus, arises 𝐾_𝑘𝐻=𝑃⁻¹_{𝑘/𝑘−1}+𝑆⁻¹𝑆. (15)

Using the Riccati equation (7), (15), the nonsingularity of 𝑆, and some algebra we have 𝑃_𝑘+1/𝑘=𝑄+𝐹𝑃⁻¹_{𝑘/𝑘−1}+𝐻^𝑇𝑅⁻¹𝐻⁻¹𝐹^𝑇=𝑄+𝐹𝑃⁻¹_{𝑘/𝑘−1}+𝑆⁻¹𝐹^𝑇=𝑄+𝐹𝐾_𝑘𝐻𝑆⁻¹𝐹^𝑇⟹𝑃_𝑘+1/𝑘𝑆=𝑄𝑆+𝐹𝐾_𝑘𝐻𝑆⁻¹𝐹^𝑇𝑆=𝑄𝐹^−𝑇𝑆𝑆⁻¹𝐹^−𝑇𝑆⁻¹+𝐹𝐾_𝑘𝐻𝑆⁻¹𝐹^−𝑇𝑆⁻¹⟹𝑃_𝑘+1/𝑘𝑆𝑆⁻¹𝐹^−𝑇𝑆=𝑄𝐹^−𝑇𝑆+𝐹𝐾_𝑘𝐻⟹𝑆⁻¹𝐹^−𝑇𝑆=𝑆⁻¹𝑃⁻¹_𝑘+1/𝑘𝑄𝐹^−𝑇𝑆+𝐹𝐾_𝑘𝐻⟹𝐾_𝑘+1𝐻𝑆⁻¹𝐹^−𝑇𝑆=𝐾_𝑘+1𝐻𝑆⁻¹𝑃⁻¹_𝑘+1/𝑘𝑄𝐹^−𝑇𝑆+𝐹𝐾_𝑘𝐻. (16)

Also, from (2) and (13), we can write 𝐾_𝑘𝐻𝑃_{𝑘/𝑘−1}𝐻^𝑇+𝑅=𝑃_{𝑘/𝑘−1}𝐻^𝑇⟹𝐾_𝑘𝐻𝑃_{𝑘/𝑘−1}𝐻^𝑇+𝐾_𝑘𝑅=𝑃_{𝑘/𝑘−1}𝐻^𝑇⟹𝐾_𝑘𝑅=𝑃_{𝑘/𝑘−1}𝐻^𝑇−𝐾_𝑘𝐻𝑃_{𝑘/𝑘−1}𝐻^𝑇⟹𝐾_𝑘𝑅=𝐼−𝐾_𝑘𝐻𝑃_{𝑘/𝑘−1}𝐻^𝑇⟹𝐾_𝑘=𝐼−𝐾_𝑘𝐻𝑃_{𝑘/𝑘−1}𝐻^𝑇𝑅⁻¹⟹𝐾_𝑘𝐻=𝐼−𝐾_𝑘𝐻𝑃_{𝑘/𝑘−1}𝐻^𝑇𝑅⁻¹𝐻⟹𝐾_𝑘𝐻=𝐼−𝐾_𝑘𝐻𝑃_{𝑘/𝑘−1}𝑆. (17)

Since the matrices 𝑃_{𝑘/𝑘−1},𝑆 are nonsingular, the last equation yields 𝐾_𝑘𝐻𝑆⁻¹𝑃⁻¹_{𝑘/𝑘−1}=𝐼−𝐾_𝑘𝐻. (18)

Substituting in (16) the matrix 𝐾_𝑘+1𝐻𝑆⁻¹𝑃⁻¹_𝑘+1/𝑘 by (18), it follows 𝐾_𝑘+1𝐻𝑆⁻¹𝐹^−𝑇𝑆=𝐾_𝑘+1𝐻𝑆⁻¹𝑃⁻¹_𝑘+1/𝑘𝑄𝐹^−𝑇𝑆+𝐹𝐾_𝑘𝐻=𝐼−𝐾_𝑘+1𝐻𝑄𝐹^−𝑇𝑆+𝐹𝐾_𝑘𝐻=𝑄𝐹^−𝑇𝑆+𝐹𝐾_𝑘𝐻−𝐾_𝑘+1𝐻𝑄𝐹^−𝑇𝑆−𝐾_𝑘+1𝐻𝐹𝐾_𝑘𝐻, (19) whereby it is implied that 𝐾_𝑘+1𝐻𝑆⁻¹𝐹^−𝑇𝑆+𝐾_𝑘+1𝐻𝑄𝐹^−𝑇𝑆+𝐾_𝑘+1𝐻𝐹𝐾_𝑘𝐻=𝑄𝐹^−𝑇𝑆+𝐹𝐾_𝑘𝐻⟹𝐾_𝑘+1𝐻𝑆⁻¹𝐹^−𝑇𝑆+𝑄𝐹^−𝑇𝑆+𝐹𝐾_𝑘𝐻=𝑄𝐹^−𝑇𝑆+𝐹𝐾_𝑘𝐻⟹𝐾_𝑘+1𝐻=𝑄𝐹^−𝑇𝑆+𝐹𝐾_𝑘𝐻×𝑆⁻¹𝐹^−𝑇𝑆+𝑄𝐹^−𝑇𝑆+𝐹𝐾_𝑘𝐻⁻¹. (20)

Thus, the above equation can be written as 𝐾_𝑘+1𝐻=𝑄𝐹^−𝑇𝑆+𝐹𝐾_𝑘𝐻𝑆⁻¹𝐹^−𝑇𝑆+𝑄𝐹^−𝑇𝑆+𝐹𝐾_𝑘𝐻⁻¹. (21)

Combining (21) with (11), the following nonlinear iterative equation with respect to 𝐺_𝑘 is derived: 𝐺_𝑘+1=𝑄𝐹^−𝑇𝑆+(𝐹)𝐺_𝑘×𝑆⁻¹𝐹^−𝑇𝑆+𝑄𝐹^−𝑇𝑆+(𝐹)𝐺_𝑘⁻¹=𝐶+𝐷𝐺_𝑘𝐴+𝐵𝐺_𝑘⁻¹, (22) where 𝐴=𝑄+𝑆⁻¹𝐹^−𝑇𝑆,𝐵=𝐹,𝐶=𝑄𝐹^−𝑇𝑆,𝐷=𝐹. (23)

The algorithm uses the initial condition 𝐺₀=𝐾₀𝐻=𝑃₀𝐻^𝑇[𝐻𝑃₀𝐻^𝑇+𝑅]⁻¹𝐻. It is known [1] that the prediction error covariance tends to the steady state prediction error covariance and that the convergence is independent of the initial uncertainty, that is, independent of the value of the initial condition 𝑃₀. Thus, we are able to assume zero initial condition 𝑃₀=0 and so we are to use the initial condition 𝐺₀=0.

It is clear that 𝐺_𝑘 tends to a steady state value 𝐺 and by (22) 𝐺 satisfies 𝐺=𝐶+𝐷𝐺𝐴+𝐵𝐺⁻¹. (24)

Per-Step Iterative Algorithm 2. We rewrite (22) as 𝐺_𝑘+1=𝐶+𝐷𝐺_𝑘𝐴+𝐵𝐺_𝑘⁻¹=𝐶𝐴⁻¹𝐴+𝐷+𝐶𝐴⁻¹𝐵−𝐶𝐴⁻¹𝐵𝐺_𝑘𝐴+𝐵𝐺_𝑘⁻¹=𝐶𝐴⁻¹𝐴+𝐵𝐺_𝑘+𝐷−𝐶𝐴⁻¹𝐵𝐺_𝑘𝐴+𝐵𝐺_𝑘⁻¹=𝐶𝐴⁻¹+𝐷−𝐶𝐴⁻¹𝐵𝐺_𝑘𝐴+𝐵𝐺_𝑘⁻¹=𝐶𝐴⁻¹+𝐷−𝐶𝐴⁻¹𝐵𝐴𝐺_𝑘⁻¹+𝐵⁻¹=𝐶𝐴⁻¹+𝐷−𝐶𝐴⁻¹𝐵𝐺_𝑘⁻¹+𝐴⁻¹𝐵⁻¹𝐴⁻¹. (25) Thus, the following nonlinear iterative equation with respect to 𝐺_𝑘 is derived: 𝐺_𝑘+1=𝑐+𝑎𝐺_𝑘⁻¹+𝑏⁻¹𝑑, (26) where 𝑎=𝐷−𝐶𝐴⁻¹𝐵=𝐹−𝑄𝑄+𝑆⁻¹⁻¹𝐹,𝑏=𝐴⁻¹𝐵=𝑆⁻¹𝐹^𝑇𝑄+𝑆⁻¹⁻¹𝐹,𝑐=𝐶𝐴⁻¹=𝑄𝑄+𝑆⁻¹⁻¹,𝑑=𝐴⁻¹=𝑆⁻¹𝐹^𝑇𝑄+𝑆⁻¹⁻¹. (27) The algorithm uses the initial condition 𝐺₀=𝐾₀𝐻=𝑃₀𝐻^𝑇[𝐻𝑃₀𝐻^𝑇+𝑅]⁻¹𝐻. It is known [1] that the prediction error covariance tends to the steady state prediction error covariance and that the convergence is independent of the initial uncertainty, that is, independent of the value of the initial condition 𝑃₀. Thus, we are able to assume zero initial condition 𝑃₀=0. In this case, in order to avoid 𝐺₀⁻¹, we are to use the initial condition 𝐺₁=𝑐.

It is clear that 𝐺_𝑘 tends to a steady state value 𝐺 and by (26) 𝐺 satisfies 𝐺=𝑐+𝑎𝐺⁻¹+𝑏⁻¹𝑑. (28)

Doubling Iterative Algorithm. Ιn (22), setting 𝐺_𝑘=𝐾_𝑘𝐻=𝑌_𝑘𝑋_𝑘⁻¹, (29) we take 𝑌_𝑘+1𝑋⁻¹_𝑘+1=𝐶+𝐷𝑌_𝑘𝑋_𝑘⁻¹𝐴+𝐵𝑌_𝑘𝑋_𝑘⁻¹⁻¹=𝐶𝑋_𝑘+𝐷𝑌_𝑘𝐴𝑋_𝑘+𝐵𝑌_𝑘⁻¹ (30) or 𝑋_𝑘+1𝑌_𝑘+1𝑋=Φ^𝑘𝑌_𝑘, (31) where Φ==𝐴𝐵𝐶𝐷𝑄+𝑆⁻¹𝐹^−𝑇𝑆𝐹𝑄𝐹^−𝑇=𝑑𝑆𝐹⁻¹𝑑⁻¹𝑏𝑐𝑑⁻¹𝑐𝑑⁻¹𝑏+𝑎 (32) is a matrix of dimension 2𝑛×2𝑛 and 𝐴, 𝐵, 𝐶, 𝐷 as in (23).

We are able to use zero initial condition 𝑃₀=0, so 𝐺₀=𝐾₀𝐻=𝑌₀𝑋₀⁻¹=0; that is, 𝑋₀𝑌₀=𝐼0, (33) and hence 𝑋_𝑘𝑌_𝑘=Φ^𝑘𝐼0. (34)

We define Φ^2𝑘=𝑑_𝑘⁻¹𝑑_𝑘⁻¹𝑏_𝑘𝑐_𝑘𝑑_𝑘⁻¹𝑐_𝑘𝑑_𝑘⁻¹𝑏_𝑘+𝑎_𝑘 (35) with initial condition Φ²⁰=𝑑₀⁻¹𝑑₀⁻¹𝑏₀𝑐₀𝑑₀⁻¹𝑐₀𝑑₀⁻¹𝑏₀+𝑎₀𝑑=Φ=⁻¹𝑑⁻¹𝑏𝑐𝑑⁻¹𝑐𝑑⁻¹.𝑏+𝑎 (36)

Then, 𝑋2^𝑘𝑌2^𝑘=Φ^2𝑘𝐼0, (37) and, using the doubling principle [1] Φ^2𝑘+1=Φ^2𝑘Φ^2𝑘, we have 𝑑⁻¹_𝑘+1𝑑⁻¹_𝑘+1𝑏_𝑘+1𝑐_𝑘+1𝑑⁻¹_𝑘+1𝑐_𝑘+1𝑑⁻¹_𝑘+1𝑏_𝑘+1+𝑎_𝑘+1=𝑑_𝑘⁻¹𝑑_𝑘⁻¹𝑏_𝑘𝑐_𝑘𝑑_𝑘⁻¹𝑐_𝑘𝑑_𝑘⁻¹𝑏_𝑘+𝑎_𝑘𝑑_𝑘⁻¹𝑑_𝑘⁻¹𝑏_𝑘𝑐_𝑘𝑑_𝑘⁻¹𝑐_𝑘𝑑_𝑘⁻¹𝑏_𝑘+𝑎_𝑘. (38)

Then we are able to derive, after some algebra, the following nonlinear iterative equations: 𝑎_𝑘+1=𝑎_𝑘𝐼−𝑐_𝑘𝐼+𝑏_𝑘𝑐_𝑘⁻¹𝑏_𝑘𝑎_𝑘,𝑏_𝑘+1=𝑏_𝑘+𝑑_𝑘𝐼+𝑏_𝑘𝑐_𝑘⁻¹𝑏_𝑘𝑎_𝑘,𝑐_𝑘+1=𝑐_𝑘+𝑎_𝑘𝑐_𝑘𝐼+𝑏_𝑘𝑐_𝑘⁻¹𝑑_𝑘,𝑑_𝑘+1=𝑑_𝑘𝐼+𝑏_𝑘𝑐_𝑘⁻¹𝑑_𝑘, (39) with initial conditions 𝑎₁=𝐷−𝐶𝐴⁻¹𝐵=𝐹−𝑄𝑄+𝑆⁻¹⁻¹𝑏𝐹,₁=𝐴⁻¹𝐵=𝑆⁻¹𝐹^𝑇𝑄+𝑆⁻¹⁻¹𝑐𝐹,₁=𝐶𝐴⁻¹=𝑄𝑄+𝑆⁻¹⁻¹,𝑑₁=𝐴⁻¹=𝑆⁻¹𝐹^𝑇𝑄+𝑆⁻¹⁻¹. (40)

Then, since 𝑋2^𝑘𝑌2^𝑘=Φ^2𝑘𝐼0=𝑑_𝑘⁻¹𝑐_𝑘𝑑_𝑘⁻¹, (41) it is clear that 𝑐_𝑘=𝑌₂^𝑘𝑋₂⁻¹_𝑘=𝐺₂^𝑘 tends to a steady state value 𝐺.

2.2.2. Algebraic Algorithm for 𝐺 Computation

In this section, we present an algebraic algorithm for 𝐺 computation. As in (29), setting 𝐺_𝑘=𝐾_𝑘𝐻=𝑌_𝑘𝑋_𝑘⁻¹ (42) and using the parameters 𝐴, 𝐵, 𝐶, 𝐷 by (23), we derive Φ==𝐴𝐵𝐶𝐷𝑄+𝑆⁻¹𝐹^−𝑇𝑆𝐹𝑄𝐹^−𝑇,𝑆𝐹 (43) which is a matrix of dimension 2𝑛×2𝑛. Since detΦ=det(𝐴)det𝐷−𝐶𝐴⁻¹𝐵=det𝑄+𝑆⁻¹𝐹^−𝑇𝑆×det𝐹−𝑄𝐹^−𝑇𝑆𝑄+𝑆⁻¹𝐹^−𝑇𝑆⁻¹𝐹=det𝑄+𝑆⁻¹det𝐹⁻¹det(𝑆)×det𝐹−𝑄𝐹^−𝑇𝑆𝑆⁻¹𝐹^𝑇𝑄+𝑆⁻¹⁻¹𝐹=det(𝑆)det𝐹−𝑄𝑄+𝑆⁻¹⁻¹𝐹×det𝐹⁻¹det𝑄+𝑆⁻¹=det(𝑆)det𝐼−𝑄𝑄+𝑆⁻¹⁻¹×det(𝐹)det𝐹⁻¹det𝑄+𝑆⁻¹=det(𝑆)det𝑄+𝑆⁻¹(𝑆)𝑆−𝑄=detdet⁻¹=1, (44) it is evident that Φ is a nonsingular matrix and its eigenvalues occur in reciprocal pairs.

Thus, (43) can be written Φ==𝐴𝐵𝐶𝐷𝑄+𝑆⁻¹𝐹^−𝑇𝑆𝐹𝑄𝐹^−𝑇𝑆𝐹=𝑊𝐿𝑊⁻¹, (45) where 𝐿=Λ00Λ⁻¹ (46) is a diagonal matrix containing the eigenvalues of Φ, with Λ diagonal matrix with all the eigenvalues of Φ lying outside the unit circle, and 𝑊𝑊=¹¹𝑊₁₂𝑊₂₁𝑊₂₂ (47) is the matrix containing the corresponding eigenvectors of Φ, with 𝑉=𝑊⁻¹=𝑉₁₁𝑉₁₂𝑉₂₁𝑉₂₂. (48)

We are able to use zero initial condition 𝑃₀=0, so 𝐺₀=𝐾₀𝐻=𝑌₀𝑋₀⁻¹=0; that is, 𝑋₀𝑌₀=𝐼0, (49) and hence 𝑋_𝑘𝑌_𝑘=Φ^𝑘𝐼0. (50)

Then, from (50) and (45)–(48), we are able to write 𝑋_𝑘𝑌_𝑘=Φ^𝑘𝐼0=𝑊𝐿𝑊^−1𝑘𝐼0=𝑊₁₁𝑊₁₂𝑊₂₁𝑊₂₂Λ^𝑘00Λ^−𝑘𝑉₁₁𝑉₁₂𝑉₂₁𝑉₂₂𝐼0; (51)

that is, 𝑋_𝑘𝑌_𝑘=𝑊₁₁Λ^𝑘𝑉₁₁+𝑊₁₂Λ^−𝑘𝑉₂₁𝑊₂₁Λ^𝑘𝑉₁₁+𝑊₂₂Λ^−𝑘𝑉₂₁. (52)

Substituting in (42) the matrices 𝑋_𝑘, 𝑌_𝑘 from (52), we have that 𝐺_𝑘=𝑌_𝑘𝑋_𝑘⁻¹=𝑊₂₁Λ^𝑘𝑉₁₁+𝑊₂₂Λ^−𝑘𝑉₂₁×𝑊₁₁Λ^𝑘𝑉₁₁+𝑊₁₂Λ^−𝑘𝑉₂₁⁻¹. (53)

Furthermore, the diagonal matrix Λ⁻¹ contains all the eigenvalues of Φ lying inside the unit circle, which follows that lim_𝑘→∞Λ^−𝑘=0. Then, 𝐺_𝑘 tends to a steady state value 𝐺 with 𝐺=lim_𝑘→∞𝐺_𝑘, and from (53) arises 𝐺=𝑊₂₁𝑊⁻¹₁₁. (54)

2.2.3. Steady State Kalman Filter Gain Computation

All algorithms presented in Sections 2.2.1 and 2.2.2 compute the steady state value 𝐺. Taking into account the assumptions of Section 2.1, we are able to conclude that, under the condition rank(𝐻)=𝑛, the steady state gain is 𝐾=𝐺𝐻^𝑇𝐻⁻¹𝐻^𝑇. (55)

2.3. Direct Steady State Kalman Filter Gain Computation

In this section, we present algorithms for the direct computation of the steady state Kalman filter 𝐾. The proposed algorithms compute directly the steady state Kalman filter gain, that is, without using 𝐺=𝐾𝐻. All these algorithms hold under the assumption that 𝑛=𝑚. Note that, since rank(𝐻)=𝑛, 𝐻 and 𝑆 are nonsingular matrices.

2.3.1. Iterative Algorithms for 𝐾 Computation

In this section, we present two iterative per-step algorithms and an iterative doubling algorithm for 𝐾 computation.

Per-Step Iterative Algorithm 1. Using (11), (22), and (13), we are able to derive the following nonlinear iterative equation with respect to the Kalman filter gain 𝐾_𝑘: 𝐾_𝑘+1=𝐺_𝑘+1𝐻⁻¹=𝑄𝐹^−𝑇𝑆+𝐹𝐺_𝑘𝑆⁻¹𝐹^−𝑇𝑆+𝑄𝐹^−𝑇𝑆+𝐹𝐺_𝑘⁻¹𝐻⁻¹=𝑄𝐹^−𝑇𝐻^𝑇𝑅⁻¹𝐻+𝐹𝐺_𝑘×𝐻⁻¹𝑅𝐻^−𝑇𝐹^−𝑇𝐻^𝑇𝑅⁻¹𝐻+𝑄𝐹^−𝑇𝐻^𝑇𝑅⁻¹𝐻+𝐹𝐺_𝑘⁻¹𝐻⁻¹=𝑄𝐹^−𝑇𝐻^𝑇𝑅⁻¹𝐻+𝐹𝐾_𝑘𝐻×𝑅𝐻^−𝑇𝐹^−𝑇𝐻^𝑇𝑅⁻¹𝐻+𝐻𝑄𝐹^−𝑇𝐻^𝑇𝑅⁻¹𝐻+𝐻𝐹𝐾_𝑘𝐻⁻¹=𝑄𝐹^−𝑇𝐻^𝑇𝑅⁻¹+𝐹𝐾_𝑘×𝐻𝑅𝐻^−𝑇𝐹^−𝑇𝐻^𝑇𝑅⁻¹𝐻+𝐻𝑄𝐹^−𝑇𝐻^𝑇𝑅⁻¹𝐻+𝐻𝐹𝐾_𝑘𝐻⁻¹=𝑄𝐹^−𝑇𝐻^𝑇𝑅⁻¹+𝐹𝐾_𝑘×𝑅𝐻^−𝑇𝐹^−𝑇𝐻^𝑇𝑅⁻¹+𝐻𝑄𝐹^−𝑇𝐻^𝑇𝑅⁻¹+𝐻𝐹𝐾_𝑘⁻¹=𝑄𝐹^−𝑇𝐻^𝑇𝑅⁻¹+𝐹𝐾_𝑘×𝐻𝑄𝐹^−𝑇𝐻^𝑇𝑅⁻¹+𝑅𝐻^−𝑇𝐹^−𝑇𝐻^𝑇𝑅⁻¹+𝐻𝐹𝐾_𝑘⁻¹. (56)

The nonsingularity of 𝑆 and (13) allow us to write the equality in (56) as 𝐾_𝑘+1=𝑄𝐹^−𝑇𝐻^𝑇𝑅⁻¹+𝐹𝐾_𝑘×𝐻𝑄𝐹^−𝑇𝐻^𝑇𝑅⁻¹+𝐻𝑆⁻¹𝐹^−𝑇𝐻^𝑇𝑅⁻¹+𝐻𝐹𝐾_𝑘⁻¹=𝐶+𝐷𝐾_𝑘𝐴+𝐵𝐾_𝑘⁻¹, (57) where 𝐴=𝐻𝑄𝐹^−𝑇𝐻^𝑇𝑅⁻¹+𝐻𝑆⁻¹𝐹^−𝑇𝐻^𝑇𝑅⁻¹=𝐻𝑄+𝑆⁻¹𝐹^−𝑇𝐻^𝑇𝑅⁻¹,𝐵=𝐻𝐹,𝐶=𝑄𝐹^−𝑇𝐻^𝑇𝑅⁻¹,𝐷=𝐹. (58)

The initial condition is 𝐾₀=𝑃₀𝐻^𝑇[𝐻𝑃₀𝐻^𝑇+𝑅]⁻¹. It is known [1] that the prediction error covariance tends to the steady state prediction error covariance and that the convergence is independent of the initial uncertainty, that is, independent of the value of the initial condition 𝑃₀. Thus, we are able to assume zero initial condition 𝑃₀=0 and so we are to use the initial condition 𝐾₀=0.

It is clear that 𝐾_𝑘 tends to a steady state value 𝐾 satisfying 𝐾=𝐶+𝐷𝐾𝐴+𝐵𝐾⁻¹. (59)

Per-Step Iterative Algorithm 2. Using (57), we are able to derive the following nonlinear iterative equation with respect to the Kalman filter gain 𝐾_𝑘: 𝐾_𝑘+1=𝐶+𝐷𝐾_𝑘𝐴+𝐵𝐾_𝑘⁻¹=𝐶𝐴⁻¹𝐴+𝐷+𝐶𝐴⁻¹𝐵−𝐶𝐴⁻¹𝐵𝐾_𝑘𝐴+𝐵𝐾_𝑘⁻¹=𝐶𝐴⁻¹𝐴+𝐵𝐾_𝑘+𝐷−𝐶𝐴⁻¹𝐵𝐾_𝑘𝐴+𝐵𝐾_𝑘⁻¹=𝐶𝐴⁻¹+𝐷−𝐶𝐴⁻¹𝐵𝐾_𝑘𝐴+𝐵𝐾_𝑘⁻¹=𝐶𝐴⁻¹+𝐷−𝐶𝐴⁻¹𝐵𝐴𝐾_𝑘⁻¹+𝐵⁻¹=𝐶𝐴⁻¹+𝐷−𝐶𝐴⁻¹𝐵𝐾_𝑘⁻¹+𝐴⁻¹𝐵⁻¹𝐴⁻¹=𝑐+𝑎𝐾_𝑘⁻¹+𝑏⁻¹𝑑, (60) where 𝐴,𝐵,𝐶,𝐷 are given by (58) and 𝑎=𝐷−𝐶𝐴⁻¹𝐵,𝑏=𝐴⁻¹𝐵,𝑐=𝐶𝐴⁻¹,𝑑=𝐴⁻¹. (61)

The algorithm uses the initial condition 𝐾₀=𝑃₀𝐻^𝑇[𝐻𝑃₀𝐻^𝑇+𝑅]⁻¹. It is known [1] that the prediction error covariance tends to the steady state prediction error covariance and that the convergence is independent of the initial uncertainty, that is, independent of the value of the initial condition 𝑃₀. Thus, we are able to assume zero initial condition 𝑃₀=0. In this case, in order to avoid 𝐾₀⁻¹, we are to use the initial condition 𝐾₁=𝑐.

It is clear that 𝐾_𝑘 tends to a steady state value 𝐾 satisfying 𝐾=𝑐+𝑎𝐾⁻¹+𝑏⁻¹𝑑. (62)

Doubling Iterative Algorithm. Ιn (57), setting 𝐾_𝑘=𝑌_𝑘𝑋_𝑘⁻¹, (63) we take 𝑌_𝑘+1𝑋⁻¹_𝑘+1=𝐶+𝐷𝑌_𝑘𝑋_𝑘⁻¹𝐴+𝐵𝑌_𝑘𝑋_𝑘⁻¹⁻¹=𝐶𝑋_𝑘+𝐷𝑌_𝑘𝐴𝑋_𝑘+𝐵𝑌_𝑘⁻¹ (64) or 𝑋_𝑘+1𝑌_𝑘+1𝑋=Φ^𝑘𝑌_𝑘, (65) where =𝑑Φ=𝐴𝐵𝐶𝐷⁻¹𝑑⁻¹𝑏𝑐𝑑⁻¹𝑐𝑑⁻¹𝑏+𝑎 (66) is a matrix of dimension 2𝑛×2𝑛 and 𝐴, 𝐵, 𝐶, 𝐷 as in (58).

Working as in the doubling iterative algorithm of Section 2.2.1 and using zero initial condition 𝑃₀=0, so 𝐾₀=𝑌₀𝑋₀⁻¹=0; we are able to derive the following nonlinear iterative equations: 𝑎_𝑘+1=𝑎_𝑘𝐼−𝑐_𝑘𝐼+𝑏_𝑘𝑐_𝑘⁻¹𝑏_𝑘𝑎_𝑘,𝑏_𝑘+1=𝑏_𝑘+𝑑_𝑘𝐼+𝑏_𝑘𝑐_𝑘⁻¹𝑏_𝑘𝑎_𝑘,𝑐_𝑘+1=𝑐_𝑘+𝑎_𝑘𝑐_𝑘𝐼+𝑏_𝑘𝑐_𝑘⁻¹𝑑_𝑘,𝑑_𝑘+1=𝑑_𝑘𝐼+𝑏_𝑘𝑐_𝑘⁻¹𝑑_𝑘, (67) with initial conditions 𝑎₁𝑏=𝑎,₁𝑐=𝑏,₁𝑑=𝑐,₁=𝑑. (68) It is clear that 𝑐_𝑘=𝑌₂^𝑘𝑋₂⁻¹_𝑘=𝐾₂^𝑘 tends to a steady state value 𝐾.

2.3.2. Algebraic Algorithm for 𝐾 Computation

In this section, we present an algebraic algorithm for 𝐾 computation. Working as in the algebraic algorithm of Section 2.2.2 and using the parameters 𝐴, 𝐵, 𝐶, 𝐷 by (58), we derive =𝑊Φ=𝐴𝐵𝐶𝐷¹¹𝑊₁₂𝑊₂₁𝑊₂₂Λ00Λ⁻¹𝑊₁₁𝑊₁₂𝑊₂₁𝑊₂₂⁻¹, (69) which is a matrix of dimension 2𝑛×2𝑛.

Then, the steady state Kalman filter is 𝐾=𝑊₂₁𝑊⁻¹₁₁. (70)

2.4. Advantages of the Proposed Algorithms

All algorithms for the computation of the steady state Kalman filter gain 𝐾, presented in Section 2, are summarized in Table 1. It is clear that the direct computation of the Kalman filter gain is feasible only if the following restriction holds: 𝑛=𝑚. The advantage of the presented algorithms is the autonomous computation of the steady state Kalman filter gain. Especially, the steady state Kalman filter gain is important, when we want to compute the parameters of the steady state Kalman filter 𝑥_{𝑘+1/𝑘+1}=𝐼−𝐾𝐻𝐹𝑥_𝑘/𝑘+𝐾𝑧_𝑘+1=𝐼−𝐺𝐹𝑥_𝑘/𝑘+𝐾𝑧_𝑘+1. (71) It is obvious from (71) that the parameters of the steady state Kalman filter are related to the steady state Kalman filter gain.

Table 1 

Algorithms for the computation of the steady state Kalman filter gain 𝐾.

In particular, the steady state prediction error covariance can be computed via the steady state gain and is given by 𝑃_𝑝=𝐼−𝐾𝐻⁻¹𝐾𝑅𝐻𝐻^𝑇𝐻⁻¹. (72)

Indeed, from (2), arises 𝐾=𝑃_𝑝𝐻^𝑇[𝐻𝑃_𝑝𝐻^𝑇+𝑅]⁻¹, which leads to 𝐾𝐻𝑃_𝑝𝐻^𝑇+𝑅=𝑃_𝑝𝐻^𝑇⟹𝐾𝐻𝑃_𝑝𝐻^𝑇+𝐾𝑅=𝑃_𝑝𝐻^𝑇⟹𝑃_𝑝𝐻^𝑇−𝐾𝐻𝑃_𝑝𝐻^𝑇=⟹𝑃𝐾𝑅𝐼−𝐾𝐻_𝑝𝐻^𝑇=⟹𝑃𝐾𝑅_𝑝𝐻^𝑇=𝐼−𝐾𝐻⁻¹⟹𝑃𝐾𝑅_𝑝𝐻^𝑇𝐻=𝐼−𝐾𝐻⁻¹𝐾𝑅𝐻. (73) Since rank(𝐻)=𝑛, the matrix 𝐻^𝑇𝐻 is nonsingular [21]; thus from the last equation arises immediately the formula of the steady state prediction error covariance in (72).

Also, by (3), the steady state estimation error covariance can be computed via the steady state prediction error covariance 𝑃_𝑒=𝐼−𝐾𝐻𝑃_𝑝. (74)

3. Examples

In this section, two examples verify the results of Section 2.

Example 1. A model of dimensions 𝑛=1 and 𝑚=2 is assumed with parameters: 12,.𝐹=0.8,𝐻=𝑄=5,𝑅=0.1000.4 (75) In this example, we have rank(𝐻)=𝑛=1 with 𝑚>𝑛.
Using all algorithms presented in Section 2.2, we computed 𝐺=0.9902. (76)
Then, using (55), we computed the steady state gain 𝐾=0.19800.3961. (77)

Example 2. A model of dimensions 𝑚=𝑛=2 is assumed with parameters: ,,,.𝐹=−0.90.7−0.30.1𝐻=1322𝑄=1003𝑅=0.1000.4 (78) In this example, we have rank(𝐻)=2.
Using all algorithms presented in Section 2.2, we computed 𝐺=.0.83820.03170.06470.9841 (79)
Then, using (55), we computed the steady state gain 𝐾=.−0.40330.62070.4597−0.1975 (80)
We also computed the same steady state gain, using all algorithms presented in Section 2.3, since 𝑚=𝑛=2.

4. Conclusions

The Kalman filter gain arises in Kalman filter equations in linear estimation and is associated with linear systems. The gain is a matrix through which the estimation and the prediction of the state as well as the corresponding estimation and prediction error covariance matrices are computed. For time invariant and asymptotically stable systems, there exist steady state values of the estimation and prediction error covariance matrices. There exists also a steady state value of the Kalman filter gain.

The steady state Kalman filter gain is usually derived via the steady state prediction error covariance by first solving the corresponding Riccati equation. In view of the importance of the Riccati equation, there exists considerable literature on its algebraic or iterative solutions, including the Chandrasekhar algorithms, which are the only iterative algorithms that provide the steady state Kalman filter gain together with the prediction error covariance.

Iterative per-step and doubling algorithms as well as an algebraic algorithm for the steady state Kalman filter computation were presented. These algorithms hold under conditions concerning the system parameters. The advantage of these algorithms is the autonomous computation of the steady state Kalman filter gain. This is important if we want to compute only the steady state Kalman filter gain or to compute the parameters of the steady state Kalman filter, which are related to the steady state Kalman filter gain.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.