Are there any concrete application of the Lyapunov theorem for LTI systems?

Question

Consider a LTI system $\dot x = Ax$.

This system is globally asymptotical stable iff given any $Q \succ 0$, there exists a unique $P \succ 0$ such that $A^{T}P+PA+Q=0$ holds.

https://en.wikipedia.org/wiki/Lyapunov_equation

This is a neat theorem, however, it seems that it is much fast just to call $\text{eigs}(A)$ in MATLAB and check the sign of the eigenvalues rather than finding two PD matrices. So I don't get why I need to know something like this.

Is there any concrete application arising from Lyapunov equation for LTI systems?

If not, where else does it show up in dynamical system theory or control theory?

Answer 1

Yes, there are many applications. For instance,

• it allows you to derive tests to establish the stability of a system for which the matrix $A$ is uncertain (robust analysis). This can be used to design controllers for uncertain linear systems (robust control).
• it allows you to design controllers that can optimize certain properties of the closed-loop system using optimization-based methods.
• it allows you to prove the stability of certain important complex control methods such as Model Predictive Control, for instance.
• it allows you to establish the stability of more complex linear systems such as time-varying, parameter-varying, hybrid, and stochastic systems.