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

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

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

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

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

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

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

Example

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

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 $Z^{T}Z=z_{1}^{2}+Z^{2}$ and therefore

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

For this example, the Chandrasekhar equations become

{\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}}.

References

^ Chandrasekhar, S. (2013). Radiative transfer. Courier Corporation.
^ Ito, K., & Powers, R. K. (1987). Chandrasekhar equations for infinite dimensional systems. SIAM journal on control and optimization, 25(3), 596-611.
^ Kailath, T. (1972, December). Some Chandrasekhar-type algorithms for quadratic regulators. In Proceedings of the 1972 IEEE Conference on Decision and Control and 11th Symposium on Adaptive Processes (pp. 219–223). IEEE.
^ Lainiotis, D. (1976). Generalized Chandrasekhar algorithms: Time-varying models. IEEE Transactions on Automatic Control, 21(5), 728-732.
^ Freitas, F. D., Ishihara, J. Y., & Borges, G. A. (2006, June). Continuous-Time H/spl infin/Control Design of Large Scale Systems Using Chandrasekhar~ fs Equations. In 2006 American Control Conference (pp. 2239–2244). IEEE.

[1] Chandrasekhar, S. (2013). Radiative transfer. Courier Corporation.

[2] Ito, K., & Powers, R. K. (1987). Chandrasekhar equations for infinite dimensional systems. SIAM journal on control and optimization, 25(3), 596-611.

[3] Kailath, T. (1972, December). Some Chandrasekhar-type algorithms for quadratic regulators. In Proceedings of the 1972 IEEE Conference on Decision and Control and 11th Symposium on Adaptive Processes (pp. 219–223). IEEE.

[4] Lainiotis, D. (1976). Generalized Chandrasekhar algorithms: Time-varying models. IEEE Transactions on Automatic Control, 21(5), 728-732.

[5] Freitas, F. D., Ishihara, J. Y., & Borges, G. A. (2006, June). Continuous-Time H/spl infin/Control Design of Large Scale Systems Using Chandrasekhar~ fs Equations. In 2006 American Control Conference (pp. 2239–2244). IEEE.

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