# Time-Invariant (non)autonomous Systems

Regarding potential distinctions between autonomous, non-autonomous, time-invariant and time-varying systems, I have found out opinions supporting that:

1. autonomous systems are time-invariant and non-autonomous are time-varying, (A treatise on stability of autonomous and non-autonomous systems, R. Adhikari - ‎2013)

2. linear systems can be time varying or invariant, nonlinear systems can be autonomous or non-autonomous Book

3. systems can be simultaneously characterised as any combination among the above, which indicates that they cannot be the same thing. (Adversary Control Tactics for Cyberphysical Systems, Kontouras, 2020 - non-autonomous linear time invariant) (Realization and identiﬁcation of autonomous linear periodically time-varying systems, Markovsky et al.-autonomous linear time varying)

The example is about LTI but it would be the same for LTV.

This system:

$$\ddot{y}-b\dot{y}-ay=u(t)$$

$$y=x_{1}$$, $$\dot{x_{1}}=x_{2}$$, $$\dot{x_{2}}=ax_{1}+bx_{2}+u(t)$$

$$\dot{\vec{x}}=\begin{bmatrix} 0 & 1\\ a&b \end{bmatrix}\vec{x}+\begin{bmatrix} 0\\ 1 \end{bmatrix}u(t)$$

, is time-invariant since $$A$$, $$B$$ matrices are not time-dependent, and since there is a time-dependent input $$u(t)$$, it should additionally be non-autonomous.

If we feed the output $$y$$ back, the input $$u(t)$$, will now become a function of state variables. In this sense, the system should be autonomous.

$$\dot{\vec{x}}=\begin{bmatrix} 0 & 1\\ a&b \end{bmatrix}\vec{x}+\begin{bmatrix} 0\\ 1 \end{bmatrix}u(t)$$

$$y=x_{1}$$

$$u(t)=-ky$$

Do the previous make sense? Is it that open-loop control systems are in principle non-autonomous, while closed-loop ones are autonomous, in the absence of additional time-dependent disturbances?