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

Table 1

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

Computation of 𝐾 through 𝐺=𝐾𝐻	Direct computation of 𝐾
System parameters 𝐹,𝐻,𝑄,𝑅	System parameters 𝐹,𝐻,𝑄,𝑅
Restriction 𝑚≥𝑛	Restriction 𝑚=𝑛
Algorithms parameters	Algorithms parameters
𝐴=(𝑄+(𝐻^𝑇𝑅^−1𝐻)^−1)𝐹^−𝑇𝐻^𝑇𝑅^−1𝐻	𝐴=𝐻(𝑄+(𝐻^𝑇𝑅^−1𝐻)^−1)𝐹^−𝑇𝐻^𝑇𝑅^−1
𝐵=𝐹	𝐵=𝐻𝐹
𝐶=𝑄𝐹^−𝑇𝐻^𝑇𝑅^−1𝐻	𝐶=𝑄𝐹^−𝑇𝐻^𝑇𝑅^−1
𝐷=𝐹	𝐷=𝐹
𝑎=𝐷−𝐶𝐴^−1𝐵	𝑎=𝐷−𝐶𝐴^−1𝐵
𝑏=𝐴^−1𝐵	𝑏=𝐴^−1𝐵
𝑐=𝐶𝐴^−1	𝑐=𝐶𝐴^−1
𝑑=𝐴^−1	𝑑=𝐴^−1
Per-step iterative algorithm 1	Per-step iterative algorithm 1
𝐺𝑘+1=(𝐶+𝐷𝐺𝑘)[𝐴+𝐵𝐺𝑘]^−1	𝐾𝑘+1=(𝐶+𝐷𝐾𝑘)[𝐴+𝐵𝐾𝑘]^−1
𝐺0=0	𝐾0=0
𝐺𝑘→𝐺	𝐾𝑘→𝐾
𝐾=𝐺[𝐻^𝑇𝐻]^−1𝐻^𝑇
Per-step iterative algorithm 2	Per-step iterative algorithm 2
𝐺𝑘+1=𝑐+𝑎[𝐺𝑘^−1+𝑏]^−1𝑑	𝐾𝑘+1=𝑐+𝑎[𝐾𝑘^−1+𝑏]^−1𝑑
𝐺1=𝑐	𝐾1=𝑐
𝐺𝑘→𝐺	𝐾𝑘→𝐾
𝐾=𝐺[𝐻^𝑇𝐻]^−1𝐻^𝑇
Doubling iterative algorithm	Doubling iterative algorithm
𝑎𝑘+1=𝑎𝑘𝐼−𝑐𝑘[𝐼+𝑏𝑘𝑐𝑘]^−1𝑏𝑘𝑎𝑘	𝑎𝑘+1=𝑎𝑘𝐼−𝑐𝑘[𝐼+𝑏𝑘𝑐𝑘]^−1𝑏𝑘𝑎𝑘
𝑏𝑘+1=𝑏𝑘+𝑑𝑘[𝐼+𝑏𝑘𝑐𝑘]^−1𝑏𝑘𝑎𝑘	𝑏𝑘+1=𝑏𝑘+𝑑𝑘[𝐼+𝑏𝑘𝑐𝑘]^−1𝑏𝑘𝑎𝑘
𝑐𝑘+1=𝑐𝑘+𝑎𝑘𝑐𝑘[𝐼+𝑏𝑘𝑐𝑘]^−1𝑑𝑘	𝑐𝑘+1=𝑐𝑘+𝑎𝑘𝑐𝑘[𝐼+𝑏𝑘𝑐𝑘]^−1𝑑𝑘
𝑑𝑘+1=𝑑𝑘[𝐼+𝑏𝑘𝑐𝑘]^−1𝑑𝑘	𝑑𝑘+1=𝑑𝑘[𝐼+𝑏𝑘𝑐𝑘]^−1𝑑𝑘
𝑎1=𝑎,𝑏1=𝑏,𝑐1=𝑐,𝑑1=𝑑	𝑎1=𝑎,𝑏1=𝑏,𝑐1=𝑐,𝑑1=𝑑
𝑐𝑘→𝐺	𝑐𝑘→𝐾
𝐾=𝐺[𝐻^𝑇𝐻]^−1𝐻^𝑇
Algebraic algorithm	Algebraic algorithm
Φ=𝐴𝐵𝐶𝐷	Φ=𝐴𝐵𝐶𝐷
=𝑊11𝑊12𝑊21𝑊22Λ00Λ^−1𝑊11𝑊12𝑊21𝑊22^−1	=𝑊11𝑊12𝑊21𝑊22Λ00Λ^−1𝑊11𝑊12𝑊21𝑊22^−1
𝐺=𝑊21𝑊^−111	𝐾=𝑊21𝑊^−111
𝐾=𝐺[𝐻^𝑇𝐻]^−1𝐻^𝑇