Implementation and Performance Analysis of Kalman Filters with Consistency Validation
Abstract
:1. Introduction
2. The Kalman Filters and Suboptimal Filters
2.1. Discrete Kalman Filter
2.2. Continuous Kalman Filter
2.3. Suboptimal Filters: Estimators with a General Gain
 (1)
 $\dot{P}=FP+P{F}^{T}+GQ{G}^{T}P{H}^{T}{R}^{1}HP$ with $H=0$ or ${R}^{1}=0$
 (2)
 $\dot{P}=(FKH)P+P(FKH{)}^{T}+GQ{G}^{T}+KR{K}^{T}$ with $K=0$ or $H=0$
3. Discrete Kalman Filter from Discretization of Continuous Kalman Filter
4. Illustrative Examples and Discussion
4.1. Example 1: The Scalar GaussMarkov Process
4.2. Example 2: An Additional Deterministic Control Input Is Introduced
4.3. Example 3: A Larger Gain Is Applied to the System
4.4. Example 4: The Integrated GaussMarkov Process
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
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Initialization: Initialize State Vector ${\widehat{\mathbf{x}}}_{0}$ and State Covariance Matrix ${\mathbf{P}}_{0}$ 

Time update 
(1) State propagation 
${\widehat{x}}_{k+1}^{}={\Phi}_{k}{\widehat{x}}_{k}$ 
(2) Error covariance propagation 
${P}_{k+1}^{}={\Phi}_{k}{P}_{k}{\Phi}_{k}^{\mathrm{T}}+{Q}_{k}$ or ${P}_{k+1}^{}={\Phi}_{k}{P}_{k}{\Phi}_{k}^{\mathrm{T}}+{\Gamma}_{k}{Q}_{k}{\Gamma}_{k}^{T}$ 
Measurement update 
(3) Kalman gain matrix evaluation 
${K}_{k}={P}_{k}^{}{H}_{k}^{\mathrm{T}}{[{H}_{k}{P}_{k}^{}{H}_{k}^{\mathrm{T}}+{R}_{k}]}^{1}$ 
(4) State estimate update 
${\widehat{x}}_{k}={\widehat{x}}_{k}^{}+{K}_{k}[{z}_{k}{H}_{k}{\widehat{x}}_{k}^{}]$ 
(5) Error covariance update 
${P}_{k}=\left[I{K}_{k}{H}_{k}\right]{P}_{k}^{}$ 
Initialization: Initialize State Vector $\widehat{\mathbf{x}}(0)$ and State Covariance Matrix $\mathbf{P}(0)$ 

(1) Solve the error covariance propagation by the matrix Riccati equation for P, which is symmetric positivedefinite. 
$\dot{P}=FP+P{F}^{T}+GQ{G}^{T}P{H}^{T}{R}^{1}HP$ 
(2) Calculation of Kalman gain matrix 
$K=P{H}^{T}{R}^{1}$ 
(3) State estimate update 
$\dot{\widehat{x}}=Fx+K(zH\widehat{x})$ 
Examples  System Models  Highlights of Important Issues 

1  A standard scalar GaussMarkov process 

2  Larger deterministic control input: an additional deterministic control input is introduced. 

3  Larger random input: a larger gain is applied to the scalar GaussMarkov process 

4  Integrated GaussMarkov process 

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Jwo, D.J.; Biswal, A. Implementation and Performance Analysis of Kalman Filters with Consistency Validation. Mathematics 2023, 11, 521. https://doi.org/10.3390/math11030521
Jwo DJ, Biswal A. Implementation and Performance Analysis of Kalman Filters with Consistency Validation. Mathematics. 2023; 11(3):521. https://doi.org/10.3390/math11030521
Chicago/Turabian StyleJwo, DahJing, and Amita Biswal. 2023. "Implementation and Performance Analysis of Kalman Filters with Consistency Validation" Mathematics 11, no. 3: 521. https://doi.org/10.3390/math11030521