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. 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.
Mathematical description
Consider a linear dynamical system, where is the state vector, is the control input and and are the system matrices. The objective is to minimize the quadratic cost functionsubject to the constraint. Hhere and are positive definite, symmetric, weighting matrices, referred to as the state cost and control cost. The optimization leads to, where is a symmetric matrix and satisfies the continuous-time algebraic Riccati equation
Chandrasekhar introduced the factorization so that
The second term is regarded linear since the operation is a projection on a reduced-dimensional space.
Example
Let us illustrate the Chandrasekhar equations using a simple example, where we takethen we have and therefore
For this example, the Chandrasekhar equations become