I. Riccati Equation Solution Initialization

The solutions for the Riccati matrices X and Y will be derived next. First, the following intermediate matrices are calculated:58

(6.11)

(6.12)

(6.13)

(6.14)

(6.15)

(6.16)

II. Hamiltonian Matrix H _? Solution

Next, when solving for the Hamiltonian matrix H _? additional intermediate matrices are generated:58

(6.17)

(6.18)

(6.19)

(6.20)

The first term in the H _? equation is

(6.21)

Similarly, the second term in the H _? equation is

(6.22)

or

(6.23)

Subtracting Eq. (6.23) from Eq. (6.21) gives the Hamiltonian matrix H _? of Eq. (6.25):

(6.24)

(6.25)

III. Solution to H _? Riccati Equation

The corresponding Riccati equation is

(6.26)

or

(6.27)

Equation (6.27) is equivalent to the following four equations:

(6.28)

(6.29)

(6.30)

(6.31)

Subtracting Eq. (6.30) from (6.29) gives X ₁₂ X ₂₁ = 0, or X ₁₂ = X ₂₁. Equations (6.28 6.31) simplify to become

(6.32)

(6.33)

(6.34)

Solving for X ₁₂ in Eq. (6.34) gives

(6.35)

or

(6.36)

To generate the solution for the Riccati matrix X, first solve for X ₁₁ = f( X ₁₂) using Eq. (6.32), then solve for X ₂₂