For matrix and Hermitian , the Lyapunov equation is

where is to be determined. If is Hurwitz, then

1. This equation has a unique Hermitian solution .
2. If is PSD, is PSD. Or if is PD, is PD.

Furthermore, the solution to the Lyapunov equation is

Application#

The Lyapunov equation is useful in context of 👁️ Observability and 🕹️ Controllability: specifically, it gives us an easier way to compute the observability and controllability Grammians.

Starting with observability, note that the observability Grammian is

In a LTI system, and are arbitrary, so we can set and . Then, this matrix is the solution of the Lyapunov equation

This is simply a system of equations, and since we know and , we can solve for .

Similarly, for the controllability Grammian

we let and , and with a change of variables, we have

which is the solution of the Lyapunov equation

Again, we know and , so we can solve for .

Finally, if there exists positive definite and satisfying the Lyapunov equation, is Hurwitz. This can be used with a found controllability Grammian to show stability, even for time-varying or nonlinear systems.