From my experience, these indices are used to compare time responses and doesn't concern objective functions

Thanks @Samundra for your response but now you can notice that most works are using these indices as a fitness function. I really don't know the selection of these criterias based on what?

The most appropriate answer is (as often in life): it depends!

The different performance metrics you have mentioned put different emphasizes on different control characteristics, e.g., a square error term will punish heavy control error outliers more severely than absolute error approaches while the latter focus more on accurate steady-state error free control. Hence, given some application you as a control engineer have to make up you mind to choose an appropriate metric depending on the most important needs for your control problem. This decision might be subject to a lively discussion since a ‘best choice’ for a given application will only be seldomly obvious .

Thanks @Oliver for your answer. So, you mean it depends what I'm going to do.

Yes.

It depends. e.g. if the time factor is important for the study and results, you should consider time and use the ITAE index. In the following papers, the ITAE index has been used;

Article PMU-based FOPID Controller of Large-scale Wind-PV Farms for ...

Article Optimal coordinated tuning of power system stabilizers and w...

Article Optimal fractional-order PID controller of inverter-based po...

Thanks alot @Mahdi for your answer. But I'm still confused if the time factor is important for my case, why not choose ITSE for example.

The different performance metrics you have mentioned put different emphasizes on different control characteristics, e.g., a square error term will punish heavy control error outliers more severely than absolute error approaches while the latter focus more on accurate steady-state error-free control. Hence, given some applications, you as a control engineer have to make up your mind to choose an appropriate metric depending on the most important needs for your control problem. This decision might be subject to a lively discussion since a ‘best choice’ for a given application will only be seldomly obvious.

Thanks dear @Luay for your further clarification. It is clear for me now as it depends on what i need to determine.

The best choice is to use all the mentioned criteria and comparing with each other to know the perfect performance.

Thank you dear Rihab Hamad for your suggestion, this is also a solution but I don't need every time to do this, so I asked for the first choice based on what? I think this is a global situation and depends on what I'm working with as mentioned by other researchers here.

these items are used as an objective function if you want to use the optimization method in your work, meany you can use one of them to prove the excellent your controller or approach. in the following work, I used all of them to prove that the FOPID is better than PID in my appplication.

https://www.mdpi.com/2071-1050/14/9/5095

Article Design and Optimization of Fractional Order PID Controller t...

Dear Hasan Abumeteir, I just checked your publication. It seems very good to consider. Previously, I have received some comments in this manner from reviewers but how to choose the exact one is still not defined because this depends on the application. So, as you mentioned, this is another solution to use them all if I don't know the exact criteria to use. Thanks a lot for sharing your work with us.

Dear Hussein Abubakr, I encourage you to use all of the error criteria, it is easy to define in the MatLab/Simulink

Thanks a lot dear Hasan Abumeteir, I will do.

Dear Hussein Abubakr . Maybe You should consider industry-based Control Performance Standards (CPSs) proposed by NERC (North American Electric Reliability Corporation) in 1997. They are used to assess the balance of the areas concerned and continuously improve the inspection frequency results. Such indexes are typically used for the long-term evaluation of the performance of a network relative to the frequency deviation.

see: N. Jaleeli and L.S. VanSlyck, Nerc’s new control performance standards, IEEE Transactions on Power Systems 14 (1999), pp. 1092–1099.

In fact the ISE, IAE, ITSE, and ITAE indexes depend on the control system performance requirements. Usually, they are used to tune the controller and are problem-dependent. So the controller optimally tuned for ITAE can give worse results in other indexes, and shouldn't be compared with the controller tuned for ISE (for example).

Thank you so much dear Jakub Mozaryn for this valuable information. This is exactly what I'm asking if the controller is giving good results for a specific index and not all indexes.

For control optimization. If you need a fast response and settling time, ITAE or ISE might be more suitable. If you prioritize a smooth response and lower control effort, IAE could be a better choice.