margin
Gain and phase margin of a system.
If no output arguments are given, both gain and phase margin are plotted on a bode diagram.
Otherwise, the margins and their corresponding frequencies are computed and returned.
A more robust criterion to assess the stability of a feedback system is the sensitivity Ms
computed by function sensitivity.
Inputs
LTI model. Must be a single-input and single-output (SISO) system.
Imaginary parts below tol are assumed to be zero.
If not specified, default value sqrt (eps) is taken.
Outputs
Gain margin (as gain, not dBs).
Phase margin (in degrees).
Frequency for the gain margin (in rad/s).
Frequency for the phase margin (in rad/s).
Algorithm
Uses function roots to calculate the frequencies
w_gamma, w_phi from special polynomials created
from the transfer function of sys as listed below
in section «Equations».
Equations
CONTINUOUS-TIME SYSTEMS
Gain Margin
_ _
L(jw) = L(jw) BTW: L(jw) = L(-jw) = conj (L(jw))
num(jw) num(-jw)
------- = --------
den(jw) den(-jw)
num(jw) den(-jw) = num(-jw) den(jw)
imag (num(jw) den(-jw)) = 0
imag (num(-jw) den(jw)) = 0
|
Phase Margin
|num(jw)|
|L(jw)| = |-------| = 1
|den(jw)|
_ 2 2
z z = Re z + Im z
num(jw) num(-jw)
------- * -------- = 1
den(jw) den(-jw)
num(jw) num(-jw) - den(jw) den(-jw) = 0
real (num(jw) num(-jw) - den(jw) den(-jw)) = 0
|
DISCRETE-TIME SYSTEMS
Gain Margin
jwT log z
L(z) = L(1/z) BTW: z = e --> w = -----
j T
num(z) num(1/z)
------ = --------
den(z) den(1/z)
num(z) den(1/z) - num(1/z) den(z) = 0
|
Phase Margin
|num(z)|
|L(z)| = |------| = 1
|den(z)|
|
L(z) L(1/z) = 1 num(z) num(1/z) ------ * -------- = 1 den(z) den(1/z) num(z) num(1/z) - den(z) den(1/z) = 0 |
PS: How to get L(1/z)
4 3 2
p(z) = a z + b z + c z + d z + e
-4 -3 -2 -1
p(1/z) = a z + b z + c z + d z + e
-4 2 3 4
= z ( a + b z + c z + d z + e z )
4 3 2 4
= ( e z + d z + c z + b z + a ) / ( z )
|
See also: sensitivity, roots
Source Code: margin