In control theory, the discrete Lyapunov equation is of the form
Contents
- Application to stability
- Computational aspects of solution
- Analytic solution
- Discrete time
- Continuous time
- References
where
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
Theorem (continuous time version). Given any
Theorem (discrete time version). Given any
Computational aspects of solution
Specialized software is available for solving Lyapunov equations. For the discrete case, the Schur method of Kitagawa is often used. For the continuous Lyapunov equation the method of Bartels and Stewart can be used.
Analytic solution
Defining the
Discrete time
Using the result that
where
Moreover, if
Continuous time
Using again the Kronecker product notation and the vectorization operator, one has the matrix equation
where
Similar to the discrete-time case, if