International Scholarly Research Notices

Research Article

On the Convergence of the Modified Riccati Equation

Table 3

Proposed implementation of the modified Riccati equation (pmRE).
Matrix operationMatrix dimensionsCalculation burden
𝑃 − 1 𝑘 / 𝑘 − 1 ( 𝑛 × 𝑛 ) ( 1 / 6 ) ( 1 6 𝑛 3 − 3 𝑛 2 − 𝑛 )
𝑀 ≡ 𝑃 − 1 𝑘 / 𝑘 − 1 + 𝐻 𝑇 𝑅 − 1 𝐻 ( 𝑛 × 𝑛 ) + ( 𝑛 × 𝑛 ) ∗ ( 1 / 2 ) ( 𝑛 2 + 𝑛 )
𝑀 − 1 ( 𝑛 × 𝑛 ) ( 1 / 6 ) ( 1 6 𝑛 3 − 3 𝑛 2 − 𝑛 )
𝐹 ⋅ 𝑀 − 1 ( 𝑛 × 𝑛 ) ⋅ ( 𝑛 × 𝑛 ) 2 𝑛 3 − 𝑛 2
𝐹 𝑀 − 1 ⋅ 𝐹 𝑇 ( 𝑛 × 𝑛 ) ⋅ ( 𝑛 × 𝑛 ) ∗ 𝑛 3 + ( 1 / 2 ) ( 𝑛 2 − 𝑛 )
Ω ≡ 𝑄 + 𝐹 𝑀 − 1 𝐹 𝑇 ( 𝑛 × 𝑛 ) + ( 𝑛 × 𝑛 ) ∗ ( 1 / 2 ) ( 𝑛 2 + 𝑛 )
𝑝 𝑑 ⋅ Ω ( 𝑛 × 𝑛 ) ∗ ( 1 / 2 ) ( 𝑛 2 + 𝑛 )
𝐹 ⋅ 𝑃 𝑘 / 𝑘 − 1 ( 𝑛 × 𝑛 ) ⋅ ( 𝑛 × 𝑛 ) 2 𝑛 3 − 𝑛 2
𝐹 𝑃 𝑘 / 𝑘 − 1 ⋅ 𝐹 𝑇 ( 𝑛 × 𝑛 ) ⋅ ( 𝑛 × 𝑛 ) ∗ 𝑛 3 + ( 1 / 2 ) ( 𝑛 2 − 𝑛 )
Ψ ≡ 𝑄 + 𝐹 𝑃 𝑘 / 𝑘 − 1 𝐹 𝑇 ( 𝑛 × 𝑛 ) + ( 𝑛 × 𝑛 ) ∗ ( 1 / 2 ) ( 𝑛 2 + 𝑛 )
( 1 − 𝑝 𝑑 ) ⋅ Ψ ( 𝑛 × 𝑛 ) ∗ ( 1 / 2 ) ( 𝑛 2 + 𝑛 )
𝑃 𝑘 + 1 / 𝑘 = ( 1 − 𝑝 𝑑 ) Ψ + 𝑝 𝑑 Ω ( 𝑛 × 𝑛 ) + ( 𝑛 × 𝑛 ) ∗ ( 1 / 2 ) ( 𝑛 2 + 𝑛 )
𝐵 p m R E = ( 1 / 3 ) ( 3 4 𝑛 3 + 3 𝑛 2 + 5 𝑛 )
*Symmetric matrix.