Chandrasekhar algorithm

Chandrasekhar algorithm refers to an efficient method to solve matrix Riccati equation, which uses symmetric factorization and was introduced by Subrahmanyan Chandrasekhar in his book, Radiative Transfer.^[1] This technique was later adapted for use in control theory, leading to the development of the Chandrasekhar equations, which refer to a set of linear differential equations that reformulates continuous-time algebraic Riccati equation (CARE).^[2][3][4][5]

Mathematical description

Consider a linear dynamical system ${\displaystyle {\dot {x}}(t)=Ax(t)+Bu(t)}$, where ${\displaystyle x(t)}$ is the state vector, ${\displaystyle u(t)}$ is the control input and ${\displaystyle A}$ and ${\displaystyle B}$ are the system matrices. The objective is to minimize the quadratic cost function

${\displaystyle J=\int _{0}^{\infty }[x^{T}Qx+u^{T}Ru)]dt}$

subject to the constraint ${\displaystyle {\dot {x}}(t)=Ax(t)+Bu(t)}$. Hhere ${\displaystyle Q}$ and ${\displaystyle R}$ are positive definite, symmetric, weighting matrices, referred to as the state cost and control cost. The optimization leads to ${\displaystyle u=-R^{-1}B^{T}Px}$, where ${\displaystyle P(t)}$ is a symmetric matrix and satisfies the continuous-time algebraic Riccati equation

${\displaystyle -{\dot {P}}=A^{T}P+PA-PBR^{-1}B^{T}P(t)+Q.}$

Chandrasekhar introduced the factorization ${\displaystyle P(t)=Z(t)Z(t)^{T}}$(${\displaystyle Z}$ need not be a square matrix) so that

${\displaystyle {\dot {Z}}=(A-BR^{-1}B^{T}ZZ^{T})Z+QZ^{-T}.}$

The second term is regarded linear since the operation ${\displaystyle Z^{T}Z}$ is a projection on a reduced-dimensional space.

Example

Let us illustrate the Chandrasekhar equations using a simple example, where we take

${\displaystyle A={\begin{bmatrix}1&2\\0&-1\end{bmatrix}},\quad B={\begin{bmatrix}0\\1\end{bmatrix}},\quad Q=I={\begin{bmatrix}1&0\\0&1\end{bmatrix}},\quad R={\begin{bmatrix}1\end{bmatrix}},\quad Z={\begin{bmatrix}z_{1}\\z_{2}\end{bmatrix}},}$

then we have ${\displaystyle Z^{T}Z=z_{1}^{2}+Z^{2}}$ and therefore

${\displaystyle ZZ^{T}Z={\begin{bmatrix}z_{1}\\z_{2}\end{bmatrix}}(z_{1}^{2}+z_{2}^{2}).}$

For this example, the Chandrasekhar equations become

${\displaystyle {\begin{bmatrix}{\dot {z}}_{1}\\{\dot {z}}_{2}\end{bmatrix}}={\begin{bmatrix}z_{1}-z_{1}^{3}+2z_{2}-z_{1}z_{2}^{2}\\-z_{2}\end{bmatrix}}+{\frac {1}{z_{1}^{2}+z_{2}^{2}}}{\begin{bmatrix}z_{1}\\z_{2}\end{bmatrix}}.}$