Lyapunov equation

In control theory, the discrete Lyapunov equation is of the form:$A\; X\; A^H\; -\; X\; +\; Q\; =\; 0$where $Q$ is a hermitian matrix. The continuous Lyapunov equation is of form:$AX\; +\; XA^H\; +\; Q\; =\; 0$.

The Lyapunov equation occurs in many branches of control theory, such as stability analysis and optimal control. This and related equations are named after the Russian mathematician Aleksandr Lyapunov.

Application to stability

In the following theorems $A,\; P,\; Q\; in\; mathbb\{R\}^\{n\; imes\; n\}$, and $P$ and $Q$ are symmetric. The notation $P>0$ means that the matrix $P$ is positive definite

Theorem (continuous time version). If there exist $P>0$ and $Q>0$ satisfying $A^T\; P\; +\; P\; A\; +\; Q\; =\; 0$ then the linear system $dot\{x\}=A\; x$ is globally asymptotically stable. The quadratic function $V(z)=z^T\; P\; z$ is a Lyapunov function that can be used to verify stability.

Theorem (discrete time version). If there exist $P>0$ and $Q>0$ satisfying $A^T\; P\; A\; -P\; +\; Q\; =\; 0$ then the linear system $x(t+1)=A\; x(t)$ is globally asymptotically stable. As before, $z^T\; P\; z$ is a Lyapunov function.

Computational aspects of solution

The discrete Lyapunov equations can, by using Schur complements, be written as:or equivalently as:.

Specialized software is available for solving Lyapunov equations. For the discrete case, the Schur method of Kitagawa (1977) is often used. For the continuous Lyapunov equation the method of Bartels and Stewart (1972) can be used.

ee also

* Sylvester equation

References

* Kitagawa: "An Algorithm for Solving the Matrix Equation X = F X F' + S", International Journal of Control, Vol. 25, No. 5, p745–753 (1977).

* R. H. Bartels and G. W. Stewart: "Algorithm 432: Solution of the matrix equation AX + XB = C", Comm. ACM, 15 (1972), p820-826.