Discrete Lyapunov Equation

The discrete Lyapunov equation is a foundational mathematical tool used in control theory and stability analysis for discrete-time linear systems. It is given by the matrix equation ATPA - P = -I, where A is the state transition matrix and I is the identity matrix. The goal is to find the symmetric matrix P, which acts as a quadratic energy certificate.

Experiment

We can explore how the parameters affect the stability and the existence of a unique positive-definite solution.

1. Open the Discrete Lyapunov Equation module.
2. Set the parameter a to 0.5, b to 1, and d to 0.5.
3. Observe that the solution P is calculated and the system is classified as "Stable", with "Positive definite P" being true.
4. Change the parameter a to 2 and d to 2 with b = 0.
5. Notice that the asymptotic stability changes to "Not asymptotically stable", and "Positive definite P" becomes false, indicating the system is unstable and the quadratic energy form is no longer positive definite.
6. Try setting a to 1 and d to 0. The spectral radius becomes 1, and you will see that there is "No unique solution" because the matrix operator is singular.