The use of Riccati equations in optimal control theory

[John Finn]

I know that linear control theory, in the form $\dot{x}=Ax+Bu$, $\dot{u}=Cx+Du$, can be put in the form of a matrix Riccati equation. But is there really an advantage to doing so?

 

[jim mcnamara]

Thread moved.

 

[Mark44]

I know that linear control theory, in the form $\dot{x}=Ax+Bu$, $\dot{u}=Cx+Du$, can be put in the form of a matrix Riccati equation. But is there really an advantage to doing so?

I don't know anything about linear control theory or matrix Riccati equations, but the above looks like linear algebra as it relates to systems of differential equations, which I do know something about.
Assuming A, B, C, and D are constants, the two equations above can be rewritten in this form:
$\begin{bmatrix}\dot x \\ \dot u \end{bmatrix} = \begin{bmatrix}A & B \\ C & D \end{bmatrix}\begin{bmatrix} x \\ u \end{bmatrix}$

The advantage of writing the system in this form is that this matrix differential equation can be solved for x and u by diagonalizing the 2 x 2 matrix I wrote using standard techniques.

 

[DaveE]

Sorry, it's been decades since I did any optimal control stuff, even then I wasn't great at it, academically speaking. But I recall this was a pretty normal approach. Something about accessibility to useful math.

Anyway, I thought this looked like a useful review paper:
https://dml.cz/bitstream/handle/10338.dmlcz/124651/Kybernetika_09-1973-1_4.pdf