“Time-varying” and “nonautonomous” dynamical systems and their Lyapunov analysis

Question

It is possible to distinguish the properties "time-varying" and "nonautonomous" in dynamical systems regarding Lyapunov stability analysis?

Please see the following link (original post by the original author) for the full question. Thanks.

https://robotics.stackexchange.com/q/6642/15233

Answer 1

These are really properties of (deterministic) dynamical systems and not of Lyapunov analysis per se.

A system is time invariant if the system parameters does not depend on time

These systems are represented by:

$$\dot x = f(x,u), \dot x = f(x)$$ or when the system is linear $$\dot x = Ax + Bu, \dot x = Ax$$

A system is time varying if the system parameters does depend on time

These systems are represented by:

$$\dot x = f(x,u,t)$$ or $$\dot x = A(t) x + B(t)u$$

For a RLC circuit, $A(t)$ could represent the matrix containing time varying capacitance, inductance or resistance. Similarly, for a mass-spring-damper system, $A(t)$ could represent time varying damping, friction, and mass. Of course, all real system are time varying albeit on the scale of hours, years, or even millennia.

A (time invariant) system is autonomous if the input $u$ is a function of the state:

These systems are represented by:

$$\dot x = f(x,u(x)) = f(x)$$ or $$\dot x = Ax + Bu(x) = (A-BK)x$$ Supposing that we are using feedback $u = -Kx$. Any state feedback systems are autonomous, because your input $u$ is a function of your state.

And you might have guessed it, a (time invariant) non-autonomous system is when your input is not a function of the state

These systems are represented by:

$$\dot x = f(x,u)$$ or $$\dot x = Ax + Bu$$

For example, $u$ could be the irradiation of the sun hitting a solar panel, where as $x$ encapsulates the states of the solar panel. The solar panel is not going to affect sunshine or the sun for that matters, or the cloud passing the sun.