Electronic – How to infer instability from the frequency response of control system

control systemfrequency responsestability

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?

Best Answer

Some comments to your question:

(1) A frequency response analysis can describe the systems behaviour only if the system is LINEAR (which is not the case for the thermostat system).

(2) For linear systems, it is best to open the feedback loop (however, for finding the conditions and the correct place for the opening requires some deeper knowledge). Then, you can analyze the loop gain response and apply one of the stability crireria (Nyquist, Bode). This method allows to determine the degree (quality) of stability (phase and/or gain margin)

(3) There is a rough method to evaluate the closed-system response (stable yes/no): The magnitude will exhibit a peaking in the "critical" frequency region where the system could oscillate. In this case, the SLOPE of the phase function at the peaking frequency must be NEGATIVE for a stable system (and will be positive in case of instability).

(4) A circuit involving thermostat devices should be analyzed in the time domain only.