It is shown that the optimal damping ratio for linear second-order systems that results in minimum-time no-overshoot response to step inputs is of the bang-bang type: the optimal damping ratio is zero at the outset and is switched to some maximum value at an appropriate instant of time. The switching time is shown to be a function of the maximum damping ratio and system natural frequency. Furthermore, it is shown that the larger the maximum damping ratio is, the less time it takes for the system to reach the desired set point. Finally, it is shown that, if the optimal damping ratio is switched as a function of the system state, then the minimum-time no-overshoot criterion is satisfied, irrespective of the magnitude of uncertainty in the value of the system natural frequency.
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