When can we judge a control system to be unstable from its frequency response?.For example, thermostat computes the error signal (desired room temperature - actual room temperature) and then actuates accordingly. If the error is positive, the thermostat increases the room temperature to value of the error signal and if the error is negative, the thermostat decreases to the error value. I am aware that there is a time delay for the actuator to reach the error and may also oscillate before reaching the error value.
If hypothetical room temperature fluctuates, our error signal would also fluctuate and so would our actuation. We therefore check the system's response for sinusoidal error signal for a range of frequencies. Now, how and when do we reach the conclusion of instability from frequency response? My understanding is that quality of response of the control system depends on rise time. Large rise time and high frequency error signal is not desirable. Could someone elaborate on this or correct my understanding?