I have a problem about stability region of the system that is discrete.
I would appreciate if you could answer my question.
I have a circuit that including several capacitance, inductance, and
resistance. I get the A (state space matrix) of the system and then
discrete this matrix by Backward Euler method.
Also The eigenvalues of the matrix(A) have been calculated.
My question is:
For stability testing of the system that is discrete by BEM, Do
eigenvalues of matrix must be
1)Inside the unit circle centered at zero?
or
2)Outside the unit circle centered at one?
If your answer is 1, please see this PDF file(Fig 7.1 b)
https://www.siam.org/books/ot98/sample/OT98Chapter7.pdf
why said this in PDF?
From an ODE, its discrete approximation via the forward Euler method is not the same as that obtained from the backward Euler method, which is an implicit one. Indeed this last one needs to solve an algebraic equation for the unknown variable.
This is simply explained in the wikipedia article
https://en.wikipedia.org/wiki/Backward_Euler_method
justifying the unit circle centered at one.
It is worth noting that the local stability of an equilibrium point (i.e. that given from the linear approximation of an autonomous ODE ) may be different from that of the related fixed point obtained after discretization. This depends on the discretization step and on the approximation method. For example, see Figs 7.2b, 7.3d of your Pdf file. It is the same for Runge-Kutta of order 4 and 5.
Also note that, if the ODE is nonlinear, near bifurcation solutions (boundary between 2 different qualitative behaviors in a parameter space), or near a basin boundary (phase space), the discrete approximation becomes very sensitive to small variations of the ODE parameters, the approximation method, and its step.
hi
Christian Mira · has nicely explained it. Just a small trivial example
1. Consider dy/dt = ay with y=y0 at t=0
2. Continuous solution is y = (e^at )*y0
3. Discretize this with t=T*n we get y=( e^aT)^n *y0 = say p^n*y0
4. In continuous case stability will mean a
For further discussion, once again I ask my questions
Please see these Book
I have a circuit that including several capacitance, inductance, and
resistance. I get the A (state space matrix) of the system and then
discrete this matrix by Backward Euler method.
Also The eigenvalues of the matrix(A) have been calculated.
My question is:
To check system stability region, which is discrete by BEM, what is the
stability region of the system?
Actually as result the system becomes stable, in which region of z-plane
the eigenvalues of matrix(A) should be there ?
1)Inside the unit circle centered at zero?
or
2)Outside the unit circle centered at one?
or
3)Inside the circle centered at 0.5 with radius 0.5?
In this PDF file(Fig 7.1 b),which I insert its link, says "outside the
unit circle centered at one"
https://www.siam.org/books/ot98/sample/OT98Chapter7.pdf
but in the Table 87.1 in this book "The Circuits and Filters Handbook,
Second Edition, edited by Wai-Kai Chen"
the answer is 3.
It seems this is a q of mapping s plane to z plane by some modeling. (the best one is bi-linear transform) here using BEM see for example (Proakis DSP)
1. derivative in s plane is s
2. in discrete time BEM it is (yn-yn-1)/T
3 in z trf this is (1-z^-1)/T
4 equating s= (1-z^-1)/T is the transformation
5.The above mapping takes lhp of s into inside circle of center (+1/2 +0.j) in Z plane.This lies inside the unit circle and is always stable. In general poles in lhp in s plane assure stability. In Z plane stability requires poles within unit circle using definition z=e^sT (from Laplace of delayed functions , as then s= si+jw ) gives with si=0 the unit circle in the z plane.
6. It follows with BEM transformation that what is unstable in s domain could be still stable in Z domain as long as mod( zpole ) is within unit circle.As an example with p=(1-aT)and T very large ( a positive) p^n will die down to zero but for ensuring stability in s domain zpole must be within the circle in 5.Perhaps this might explain the answer to the q.
Hope it helps
Cheers
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