I am not sure, if this answer is what you expect. Nevertheless:

• Whether a system with feedback is stable (or not) can best be checked using the loop gain plot (BODE diagram). Now the well-known stability criterion can be applied.

• Simulation programs are able to calculate magnitude and phase functions also for closed-loop systems which are instable. Now - the question arises if we can derive from these BODE plots (for an instable system) some information about stability properties. And the answer is Yes.

• For stable closed-loop systems the phase function has - in general - a falling characteristic (in particular, in the frequency region where the pole frequency of the dominant pole pair is located.) In contrast, for unstable sysytems this phase function will exhibit a rising characteristic (posive slope). In case of an oscillator (pole on the imag. axis) this function will have a vertical form (phase jump).

• More than that, the finite (positive) slope of this rising phase function is a kind of "measure" for the degree of instability - that means: This slope is also an indication how far the system is away from the stability limit - hence an indication for the amount of the negative phase margin. Of course, for stable systems, the negative slope of the phase function is a measure for the positive phase margin.

I am not sure, if this answer is what you expect. Nevertheless:

• Whether a system with feedback is stable (or not) can best be checked using the loop gain plot (BODE diagram). Now the well-known stability criterion can be applied.

• Simulation programs are able to calculate magnitude and phase functions also for closed-loop systems which are unstable. Now - the question arises if we can derive from these BODE plots (for an unstable system) some information about stability properties. And the answer is Yes.

• For stable closed-loop systems the phase function has - in general - a falling characteristic (in particular, in the frequency region where the pole frequency of the dominant pole pair is located.) In contrast, for unstable sysytems this phase function will exhibit a rising characteristic (posive slope). In case of an oscillator (pole on the imag. axis) this function will have a vertical form (phase jump).

• More than that, the finite (positive) slope of this rising phase function is a kind of "measure" for the degree of instability - that means: This slope is also an indication how far the system is away from the stability limit - hence an indication for the amount of the negative phase margin. Of course, for stable systems, the negative slope of the phase function is a measure for the positive phase margin.

• It should be noted that the max. phase slope can conveniently be expressed with the peak of the corresponding group delay figure. Therefore, we have quasi-linear relationship between the phase margin and the inverse group delay peak.

I am not sure, if this answer is what you expect. Nevertheless:

• Whether a system with feedback is stable (or not) can best be checked using the loop gain plot (BODE diagram). Now the well-known stability criterion can be applied.

• Simulation programs are able to calculate magnitude and phase functions also for closed-loop systems which are instable. Now - the question is if we can derive from these BODE plots (for an instable system) some information about stability properties. And the answer is Yes.

• For stable closed-loop systems the phase function has - in general - a falling characteristic (in particular in the frequency region where the pole frequency of the dominant pole pair is located. In contrast, for unstable sysytems this phase function will exhibit a rising characteristic. In case of an oscillator (pole on the imag. axis) this function will have a vertical form (phase jump).

• More than that, the finite (positive) slope of this rising phase function is a kind of "measure" for the degree of instability - that means: This slope is also an indication how far the system is from the stability limit - hence an indication for the amount of the negative phase margin. Of course, for stable systems, the negative slope of the phase function is a measure for the positive phase margin.

I am not sure, if this answer is what you expect. Nevertheless:

• Whether a system with feedback is stable (or not) can best be checked using the loop gain plot (BODE diagram). Now the well-known stability criterion can be applied.

• Simulation programs are able to calculate magnitude and phase functions also for closed-loop systems which are instable. Now - the question arises if we can derive from these BODE plots (for an instable system) some information about stability properties. And the answer is Yes.

• For stable closed-loop systems the phase function has - in general - a falling characteristic (in particular, in the frequency region where the pole frequency of the dominant pole pair is located.) In contrast, for unstable sysytems this phase function will exhibit a rising characteristic (posive slope). In case of an oscillator (pole on the imag. axis) this function will have a vertical form (phase jump).

• More than that, the finite (positive) slope of this rising phase function is a kind of "measure" for the degree of instability - that means: This slope is also an indication how far the system is away from the stability limit - hence an indication for the amount of the negative phase margin. Of course, for stable systems, the negative slope of the phase function is a measure for the positive phase margin.