On the Convergence of the Modified Riccati Equation

Table 2

Classical implementation of the modified Riccati equation (cmRE).

Matrix operation	Matrix dimensions	Calculation burden
𝐻⋅𝑃𝑘/𝑘−1	(𝑚×𝑛)⋅(𝑛×𝑛)	2𝑛^2𝑚−𝑛𝑚
𝐻𝑃𝑘/𝑘−1⋅𝐻^𝑇	(𝑚×𝑛)⋅(𝑛×𝑚)^∗	𝑛𝑚^2+𝑛𝑚−(1/2)(𝑚^2+𝑚)
𝑂𝑘≡𝐻𝑃𝑘/𝑘−1𝐻^𝑇+𝑅	(𝑚×𝑚)+(𝑚×𝑚)^∗	(1/2)(𝑚^2+𝑚)
𝑂^𝑘−1	(𝑚×𝑚)	(1/6)(16𝑚^3−3𝑚^2−𝑚)
𝐾𝑘≡𝑂^𝑘−1⋅𝐻𝑃𝑘/𝑘−1	(𝑚×𝑚)⋅(𝑚×𝑛)	2𝑛𝑚^2−𝑛𝑚
𝑊≡𝑃𝑘/𝑘−1𝐻^𝑇⋅𝐾𝑘	(𝑛×𝑚)⋅(𝑚×𝑛)^∗	𝑛^2𝑚+𝑛𝑚−(1/2)(𝑛^2+𝑛)
𝑝𝑑⋅𝑊	(𝑛×𝑛)^∗	(1/2)(𝑛^2+𝑛)
Π1≡𝑃𝑘/𝑘−1−𝑝𝑑⋅𝑊	(𝑛×𝑛)+(𝑛×𝑛)^∗	(1/2)(𝑛^2+𝑛)
𝐹⋅Π1	(𝑛×𝑛)⋅(𝑛×𝑛)	2𝑛^3−𝑛^2
Π2≡𝐹Π1⋅𝐹^𝑇	(𝑛×𝑛)⋅(𝑛×𝑛)^∗	𝑛^3+(1/2)(𝑛^2−𝑛)
𝑃𝑘+1/𝑘=𝑄+Π2	(𝑛×𝑛)+(𝑛×𝑛)^∗	(1/2)(𝑛^2+𝑛)
𝐵cmRE=(1/2)(6𝑛^3+𝑛^2+𝑛)+3𝑛^2𝑚+3𝑛𝑚^2+(1/6)(16𝑚^3−3𝑚^2−𝑚)

^*Symmetric matrix.