What are the requirements to call a controller real-time?
What are the entire standards for that?
I do not think that just rendering the output quickly before the sampling interval passes is sufficient for the real-time certification.
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I don't think your question calls for pointing to any of the various official standard definitions of "real-time" (e.g., ANSI/IEEE POSIX 1003.1b-1993 and POSIX 1003.1i-1995). And the real-time computing research and practitioner communities have little consensus on a "definition," instead having a great many ad hoc interpretations. The following actually captures the essence of "real-time" and its relationship to control systems (among others):
Many people implicitly have an informal mental model that considers information or an action as being "real-time"
• if, or to the extent that, it is manifest to them with some expected delay that can be related to its perceived currency
• i.e., in an expected time frame that the information or action has satisfactory usefulness (or at least interest) to them.
That mental model is natural despite its informality. However, it must be formalized to create one for theoretical and any specific application purposes --such as for maintaining a control loop's dynamics.
That requires reasoning about “real-time” from a set of first principles, including "timeliness," "predictability of timeliness," and "uncertainty," to formulate a precise model which is satisfactorily useful for designing and measuring the behavior of the (say) control system.