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I'm looking for good software to design control systems, and more specifically to neatly design "block diagrams" for control systems.

Is there something like that?

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You can use Matlab Simulink.

I think Simulink is the most satisfactory and easy way to build control systems with traditional block diagrams.

https://www.mathworks.com/help/simulink/gs/simulink-block-diagrams.html

Some introductory block diagram models in Simulink:

http://ctms.engin.umich.edu/CTMS/index.php?example=Introduction&section=SimulinkControl

Example PID control block diagram with Matlab Simulink:

PID control block diagram with Matlab Simulink