Some time I found that some robot controller uses a critically damped reference control signal such as in Autonomous Surface Vehicle (ASV) heading controller. My question is how do we implement this reference signal in the actual robot implementation?
The transfer function in the simulink sketch serves as a pre-filter. In general, pre-filters are used to smooth the input since step reference signals are’nt logical for tracking (tracking step command will require infinite accelerations). Most of the time these filter are chosen as the close loop desired transfer function. Working this way creates a reference model for tracking and prevents integrator windup and derivative induced saturations in PID controllers. If you want to implement one in Matlab/Simulink you can design a continues reference model and then use the c2d command in Matlab which transforms continues systems into discrete systems given some sampling frequency.
I think you misunderstood the Simulink. The transfer function is the system and the reference signal in the picture you've posted is the step function. The input for the system in the Simulink is a constant value of 20 magnitude. You can see that from the figure. The curve goes to 20 in 18 unit time and stays for the future which indicates the system is stable.