What is the purpose of a use case diagram ?
A use-case diagram intends to show how a subject brings useful functionality in relation with its actors. Each use case is meant to have some value for the actors or other stakeholders of the system.
According to the UML 2.5 specs:
Each UseCase specifies some behavior that a subject can perform in
collaboration with one or more Actors. UseCases define the offered
Behaviors of the subject without reference to its internal structure.
An Actor models a type of role played by an entity that interacts with
the subjects of its associated UseCases (e.g., by exchanging signals
and data). Actors may represent roles played by human users, external
hardware, or other systems.
So in principle, the system itself is not meant to be its own actor. The actor should correspond to a role that is expressed independently of the system. Please note that my wording does not exclude that the system fulfils this role itself.
What does this mean for your self-interacting system ?
Labeling the actor 'System' is therefore a very bad idea. The actor should correspond to a role that is expressed independently of the system. You could for example think of an actor: Supervision system
(which could be another system, or the system itself):

Another possibility is to dig into the details:
A subject of a UseCase could be a system or any other element that may
have behavior, such as a Component or Class.
So you could show the use-cases from the point of view of a subsystem, and for example have a subject Measurement Subsystem
and an actor Monitoring Subsystem
:

What do you really want to represent ?
Introspective use cases can be the symptom of the wrong diagram choice.
It is possible that you're less interested in how the system contributes to the goals of its users and how it relates with its external environment, but more in its internals. In this case, check if an activity, sequence or communication diagrams could not better suit your needs.