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I can find a lot of information on how to calculate gain and phase margins from Bode plots, but can't find any proper justification for these rules in terms of the positions of the poles.

In principle, you are asking for a relation between the systems behaviour in the time domain (stability properties) and the frequency domain (pole locations), correct?

Well - this can be answered as follows:

  • When you are calculating the step response (time domain) you have to solve a differential equation by introducing an exponential "Ansatz" exp(st) (from the beginning without knowing the meaning of the symbol"s"; you only know its dimension: 1/time). As a result, you arrive at a equation to be solved - this is the so called "characteristic equation" P(s)=0 of the system. Solving this equation - together with a proper interpretation of this solution - gives you the following information: The unknown quantitty "s" is complex and can be interpreted as a complex frequency "s=sigma+jw". As a consequence, the step response will be of the form exp(sigma*t)*sin(wt). From this, you can derive that the real part of "s" must be negative for a decaying step response (sigma<0).

  • When you are calculating the (second-order) transfer function H(s)=N(s)/D(s) you will see that the denominator D(s) consists of a second-order polynominal of the form (1+As+Bs²). And you will further notice that this polynominal is identical to the above mentioned char. polynoiminal P(s). This identity establishes the relation between time and frequency domain because the poles of the transfer function (resulting from P(s)=0) are identical to the roots of the char. equation. As we have seen that the real part "sigma" of this root must be negative, we have the requirement in the frequency domain: The real part "sigma" of the pole must be negative.

  • As far as I know, there is no formula which describes the relation between the phase margin and the pole location. However, there is a fixed relation between (correction:) pole location and the pole quality factor [Qp=1/2*cos(phi)], phi=angle between neg. real axis and pole vector. On the other hand, the gain peaking of the transfer function (in the vicinity of the pole frequency) is related to Qp - and we also can relate the gain peaking to the phase margin. Hence, there is a - more or less - indirect relation between phase margin and pole location (real part).

  • One general comment: Finding a DIRECT relation between system poles and stability margin is "problematic" because the stability margins are defined for an open-loop system (LOOP GAIN) whereas the pole location is investigated for the closed-loop system.

Remark: However, it can be shown that - as a rather good approximation - the following expression holds for a second-order system and phase margins PM<65 deg: PM=50/Qp(in deg). In this context, it shoulsshould be taken into account that phase margins exceeding 65 deg,. are rather uncritical.

I can find a lot of information on how to calculate gain and phase margins from Bode plots, but can't find any proper justification for these rules in terms of the positions of the poles.

In principle, you are asking for a relation between the systems behaviour in the time domain (stability properties) and the frequency domain (pole locations), correct?

Well - this can be answered as follows:

  • When you are calculating the step response (time domain) you have to solve a differential equation by introducing an exponential "Ansatz" exp(st) (from the beginning without knowing the meaning of the symbol"s"; you only know its dimension: 1/time). As a result, you arrive at a equation to be solved - this is the so called "characteristic equation" P(s)=0 of the system. Solving this equation - together with a proper interpretation of this solution - gives you the following information: The unknown quantitty "s" is complex and can be interpreted as a complex frequency "s=sigma+jw". As a consequence, the step response will be of the form exp(sigma*t)*sin(wt). From this, you can derive that the real part of "s" must be negative for a decaying step response (sigma<0).

  • When you are calculating the (second-order) transfer function H(s)=N(s)/D(s) you will see that the denominator D(s) consists of a second-order polynominal of the form (1+As+Bs²). And you will further notice that this polynominal is identical to the above mentioned char. polynoiminal P(s). This identity establishes the relation between time and frequency domain because the poles of the transfer function (resulting from P(s)=0) are identical to the roots of the char. equation. As we have seen that the real part "sigma" of this root must be negative, we have the requirement in the frequency domain: The real part "sigma" of the pole must be negative.

  • As far as I know, there is no formula which describes the relation between the phase margin and the pole location. However, there is a fixed relation between (correction:) pole location and the pole quality factor [Qp=1/2*cos(phi)], phi=angle between neg. real axis and pole vector. On the other hand, the gain peaking of the transfer function (in the vicinity of the pole frequency) is related to Qp - and we also can relate the gain peaking to the phase margin. Hence, there is a - more or less - indirect relation between phase margin and pole location (real part).

  • One general comment: Finding a DIRECT relation between system poles and stability margin is "problematic" because the stability margins are defined for an open-loop system (LOOP GAIN) whereas the pole location is investigated for the closed-loop system.

Remark: However, it can be shown that - as a rather good approximation - the following expression holds for a second-order system and phase margins PM<65 deg: PM=50/Qp(in deg). In this context it shouls be taken into account that phase margins exceeding 65 deg, are rather uncritical.

I can find a lot of information on how to calculate gain and phase margins from Bode plots, but can't find any proper justification for these rules in terms of the positions of the poles.

In principle, you are asking for a relation between the systems behaviour in the time domain (stability properties) and the frequency domain (pole locations), correct?

Well - this can be answered as follows:

  • When you are calculating the step response (time domain) you have to solve a differential equation by introducing an exponential "Ansatz" exp(st) (from the beginning without knowing the meaning of the symbol"s"; you only know its dimension: 1/time). As a result, you arrive at a equation to be solved - this is the so called "characteristic equation" P(s)=0 of the system. Solving this equation - together with a proper interpretation of this solution - gives you the following information: The unknown quantitty "s" is complex and can be interpreted as a complex frequency "s=sigma+jw". As a consequence, the step response will be of the form exp(sigma*t)*sin(wt). From this, you can derive that the real part of "s" must be negative for a decaying step response (sigma<0).

  • When you are calculating the (second-order) transfer function H(s)=N(s)/D(s) you will see that the denominator D(s) consists of a second-order polynominal of the form (1+As+Bs²). And you will further notice that this polynominal is identical to the above mentioned char. polynoiminal P(s). This identity establishes the relation between time and frequency domain because the poles of the transfer function (resulting from P(s)=0) are identical to the roots of the char. equation. As we have seen that the real part "sigma" of this root must be negative, we have the requirement in the frequency domain: The real part "sigma" of the pole must be negative.

  • As far as I know, there is no formula which describes the relation between the phase margin and the pole location. However, there is a fixed relation between (correction:) pole location and the pole quality factor [Qp=1/2*cos(phi)], phi=angle between neg. real axis and pole vector. On the other hand, the gain peaking of the transfer function (in the vicinity of the pole frequency) is related to Qp - and we also can relate the gain peaking to the phase margin. Hence, there is a - more or less - indirect relation between phase margin and pole location (real part).

  • One general comment: Finding a DIRECT relation between system poles and stability margin is "problematic" because the stability margins are defined for an open-loop system (LOOP GAIN) whereas the pole location is investigated for the closed-loop system.

Remark: However, it can be shown that - as a rather good approximation - the following expression holds for a second-order system and phase margins PM<65 deg: PM=50/Qp(in deg). In this context, it should be taken into account that phase margins exceeding 65 deg. are rather uncritical.

5 added 54 characters in body
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I can find a lot of information on how to calculate gain and phase margins from Bode plots, but can't find any proper justification for these rules in terms of the positions of the poles.

In principle, you are asking for a relation between the systems behaviour in the time domain (stability properties) and the frequency domain (pole locations), correct?

Well - this can be answered as follows:

  • When you are calculating the step response (time domain) you have to solve a differential equation by introducing an exponential "Ansatz" exp(st) (from the beginning without knowing the meaning of the symbol"s"; you only know its dimension: 1/time). As a result, you arrive at a equation to be solved - this is the so called "characteristic equation" P(s)=0 of the system. Solving this equation - together with a proper interpretation of this solution - gives you the following information: The unknown quantitty "s" is complex and can be interpreted as a complex frequency "s=sigma+jw". As a consequence, the step response will be of the form exp(sigma*t)*sin(wt). From this, you can derive that the real part of "s" must be negative for a decaying step response (sigma<0).

  • When you are calculating the (second-order) transfer function H(s)=N(s)/D(s) you will see that the denominator D(s) consists of a second-order polynominal of the form (1+As+Bs²). And you will further notice that this polynominal is identical to the above mentioned char. polynoiminal P(s). This identity establishes the relation between time and frequency domain because the poles of the transfer function (resulting from P(s)=0) are identical to the roots of the char. equation. As we have seen that the real part "sigma" of this root must be negative, we have the requirement in the frequency domain: The real part "sigma" of the pole must be negative.

  • As far as I know, there is no formula which describes the relation between the phase margin and the pole location. However, there is a fixed relation between (correction:)pole pole location and the pole quality factor [Qp=1/2*cos(phi)][Qp=1/2*cos(phi)], phi=angle between neg. real axis and pole vector. On the other hand, the gain peaking of the transfer function (in the vicinity of the pole frequency) is related to Qp - and we also can relate the gain peaking to the phase margin. Hence, there is a - more or less - indirect relation between phase margin and pole location (real part).

  • One general comment: Finding a DIRECT relation between system poles and stability margin is "problematic" because the stability margins are defined for an open-loop system (LOOP GAIN) whereas the pole location is investigated for the closed-loop system.

Remark: However, it can be shown that - as a rather good approximation - the following expression holds for a second-order system and phase margins PM<65 deg: PM=50/Qp(in deg). In this context it shouls be taken into account that phase margins exceeding 65 deg, are rather uncritical.

I can find a lot of information on how to calculate gain and phase margins from Bode plots, but can't find any proper justification for these rules in terms of the positions of the poles.

In principle, you are asking for a relation between the systems behaviour in the time domain (stability properties) and the frequency domain (pole locations), correct?

Well - this can be answered as follows:

  • When you are calculating the step response (time domain) you have to solve a differential equation by introducing an exponential "Ansatz" exp(st) (from the beginning without knowing the meaning of the symbol"s"; you only know its dimension: 1/time). As a result, you arrive at a equation to be solved - this is the so called "characteristic equation" P(s)=0 of the system. Solving this equation - together with a proper interpretation of this solution - gives you the following information: The unknown quantitty "s" is complex and can be interpreted as a complex frequency "s=sigma+jw". As a consequence, the step response will be of the form exp(sigma*t)*sin(wt). From this, you can derive that the real part of "s" must be negative for a decaying step response (sigma<0).

  • When you are calculating the (second-order) transfer function H(s)=N(s)/D(s) you will see that the denominator D(s) consists of a second-order polynominal of the form (1+As+Bs²). And you will further notice that this polynominal is identical to the above mentioned char. polynoiminal P(s). This identity establishes the relation between time and frequency domain because the poles of the transfer function (resulting from P(s)=0) are identical to the roots of the char. equation. As we have seen that the real part "sigma" of this root must be negative, we have the requirement in the frequency domain: The real part "sigma" of the pole must be negative.

  • As far as I know, there is no formula which describes the relation between the phase margin and the pole location. However, there is a fixed relation between (correction:)pole location and the pole quality factor [Qp=1/2*cos(phi)], phi=angle between neg. real axis and pole vector. On the other hand, the gain peaking of the transfer function (in the vicinity of the pole frequency) is related to Qp - and we also can relate the gain peaking to the phase margin. Hence, there is a - more or less - indirect relation between phase margin and pole location (real part).

  • One general comment: Finding a DIRECT relation between system poles and stability margin is "problematic" because the stability margins are defined for an open-loop system (LOOP GAIN) whereas the pole location is investigated for the closed-loop system.

I can find a lot of information on how to calculate gain and phase margins from Bode plots, but can't find any proper justification for these rules in terms of the positions of the poles.

In principle, you are asking for a relation between the systems behaviour in the time domain (stability properties) and the frequency domain (pole locations), correct?

Well - this can be answered as follows:

  • When you are calculating the step response (time domain) you have to solve a differential equation by introducing an exponential "Ansatz" exp(st) (from the beginning without knowing the meaning of the symbol"s"; you only know its dimension: 1/time). As a result, you arrive at a equation to be solved - this is the so called "characteristic equation" P(s)=0 of the system. Solving this equation - together with a proper interpretation of this solution - gives you the following information: The unknown quantitty "s" is complex and can be interpreted as a complex frequency "s=sigma+jw". As a consequence, the step response will be of the form exp(sigma*t)*sin(wt). From this, you can derive that the real part of "s" must be negative for a decaying step response (sigma<0).

  • When you are calculating the (second-order) transfer function H(s)=N(s)/D(s) you will see that the denominator D(s) consists of a second-order polynominal of the form (1+As+Bs²). And you will further notice that this polynominal is identical to the above mentioned char. polynoiminal P(s). This identity establishes the relation between time and frequency domain because the poles of the transfer function (resulting from P(s)=0) are identical to the roots of the char. equation. As we have seen that the real part "sigma" of this root must be negative, we have the requirement in the frequency domain: The real part "sigma" of the pole must be negative.

  • As far as I know, there is no formula which describes the relation between the phase margin and the pole location. However, there is a fixed relation between (correction:) pole location and the pole quality factor [Qp=1/2*cos(phi)], phi=angle between neg. real axis and pole vector. On the other hand, the gain peaking of the transfer function (in the vicinity of the pole frequency) is related to Qp - and we also can relate the gain peaking to the phase margin. Hence, there is a - more or less - indirect relation between phase margin and pole location (real part).

  • One general comment: Finding a DIRECT relation between system poles and stability margin is "problematic" because the stability margins are defined for an open-loop system (LOOP GAIN) whereas the pole location is investigated for the closed-loop system.

Remark: However, it can be shown that - as a rather good approximation - the following expression holds for a second-order system and phase margins PM<65 deg: PM=50/Qp(in deg). In this context it shouls be taken into account that phase margins exceeding 65 deg, are rather uncritical.

4 added 54 characters in body
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I can find a lot of information on how to calculate gain and phase margins from Bode plots, but can't find any proper justification for these rules in terms of the positions of the poles.

In principle, you are asking for a relation between the systems behaviour in the time domain (stability properties) and the frequency domain (pole locations), correct?

Well - this can be answered as follows:

  • When you are calculating the step response (time domain) you have to solve a differential equation by introducing an exponential "Ansatz" exp(st) (from the beginning without knowing the meaning of the symbol"s"; you only know its dimension: 1/time). As a result, you arrive at a equation to be solved - this is the so called "characteristic equation" P(s)=0 of the system. Solving this equation - together with a proper interpretation of this solution - gives you the following information: The unknown quantitty "s" is complex and can be interpreted as a complex frequency "s=sigma+jw". As a consequence, the step response will be of the form exp(sigma*t)*sin(wt). From this, you can derive that the real part of "s" must be negative for a decaying step response (sigma<0).

  • When you are calculating the (second-order) transfer function H(s)=N(s)/D(s) you will see that the denominator D(s) consists of a second-order polynominal of the form (1+As+Bs²). And you will further notice that this polynominal is identical to the above mentioned char. polynoiminal P(s). This identity establishes the relation between time and frequency domain because the poles of the transfer function (resulting from P(s)=0) are identical to the roots of the char. equation. As we have seen that the real part "sigma" of this root must be negative, we have the requirement in the frequency domain: The real part "sigma" of the pole must be negative.

  • As far as I know, there is no formula which describes the relation between the phase margin and the pole location. However, there is a fixed relation between the real part "sigma" of pole(correction:)pole location and the pole quality factor (Qp=1[Qp=1/2*sigma2*cos(phi)], phi=angle between neg. real axis and pole vector. On the other hand, the gain peaking of the transfer function (in the vicinity of the pole frequency) is related to Qp - and we also can relate the gain peaking to the phase margin. Hence, there is a - more or less - indirect relation between phase margin and pole location (real part).

  • One general comment: Finding a DIRECT relation between system poles and stability margin is "problematic" because the stability margins are defined for an open-loop system (LOOP GAIN) whereas the pole location is investigated for the closed-loop system.

I can find a lot of information on how to calculate gain and phase margins from Bode plots, but can't find any proper justification for these rules in terms of the positions of the poles.

In principle, you are asking for a relation between the systems behaviour in the time domain (stability properties) and the frequency domain (pole locations), correct?

Well - this can be answered as follows:

  • When you are calculating the step response (time domain) you have to solve a differential equation by introducing an exponential "Ansatz" exp(st) (from the beginning without knowing the meaning of the symbol"s"; you only know its dimension: 1/time). As a result, you arrive at a equation to be solved - this is the so called "characteristic equation" P(s)=0 of the system. Solving this equation - together with a proper interpretation of this solution - gives you the following information: The unknown quantitty "s" is complex and can be interpreted as a complex frequency "s=sigma+jw". As a consequence, the step response will be of the form exp(sigma*t)*sin(wt). From this, you can derive that the real part of "s" must be negative for a decaying step response (sigma<0).

  • When you are calculating the (second-order) transfer function H(s)=N(s)/D(s) you will see that the denominator D(s) consists of a second-order polynominal of the form (1+As+Bs²). And you will further notice that this polynominal is identical to the above mentioned char. polynoiminal P(s). This identity establishes the relation between time and frequency domain because the poles of the transfer function (resulting from P(s)=0) are identical to the roots of the char. equation. As we have seen that the real part "sigma" of this root must be negative, we have the requirement in the frequency domain: The real part "sigma" of the pole must be negative.

  • As far as I know, there is no formula which describes the relation between the phase margin and the pole location. However, there is a fixed relation between the real part "sigma" of pole and the pole quality factor (Qp=1/2*sigma). On the other hand, the gain peaking of the transfer function (in the vicinity of the pole frequency) is related to Qp - and we also can relate the gain peaking to the phase margin. Hence, there is a - more or less - indirect relation between phase margin and pole location (real part).

  • One general comment: Finding a DIRECT relation between system poles and stability margin is "problematic" because the stability margins are defined for an open-loop system (LOOP GAIN) whereas the pole location is investigated for the closed-loop system.

I can find a lot of information on how to calculate gain and phase margins from Bode plots, but can't find any proper justification for these rules in terms of the positions of the poles.

In principle, you are asking for a relation between the systems behaviour in the time domain (stability properties) and the frequency domain (pole locations), correct?

Well - this can be answered as follows:

  • When you are calculating the step response (time domain) you have to solve a differential equation by introducing an exponential "Ansatz" exp(st) (from the beginning without knowing the meaning of the symbol"s"; you only know its dimension: 1/time). As a result, you arrive at a equation to be solved - this is the so called "characteristic equation" P(s)=0 of the system. Solving this equation - together with a proper interpretation of this solution - gives you the following information: The unknown quantitty "s" is complex and can be interpreted as a complex frequency "s=sigma+jw". As a consequence, the step response will be of the form exp(sigma*t)*sin(wt). From this, you can derive that the real part of "s" must be negative for a decaying step response (sigma<0).

  • When you are calculating the (second-order) transfer function H(s)=N(s)/D(s) you will see that the denominator D(s) consists of a second-order polynominal of the form (1+As+Bs²). And you will further notice that this polynominal is identical to the above mentioned char. polynoiminal P(s). This identity establishes the relation between time and frequency domain because the poles of the transfer function (resulting from P(s)=0) are identical to the roots of the char. equation. As we have seen that the real part "sigma" of this root must be negative, we have the requirement in the frequency domain: The real part "sigma" of the pole must be negative.

  • As far as I know, there is no formula which describes the relation between the phase margin and the pole location. However, there is a fixed relation between (correction:)pole location and the pole quality factor [Qp=1/2*cos(phi)], phi=angle between neg. real axis and pole vector. On the other hand, the gain peaking of the transfer function (in the vicinity of the pole frequency) is related to Qp - and we also can relate the gain peaking to the phase margin. Hence, there is a - more or less - indirect relation between phase margin and pole location (real part).

  • One general comment: Finding a DIRECT relation between system poles and stability margin is "problematic" because the stability margins are defined for an open-loop system (LOOP GAIN) whereas the pole location is investigated for the closed-loop system.

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