Hello,

If, by "the scheme is stable", you mean "the discrete-time system x(k+1) = A*x(k) is asymptotically stable," then this will be equivalent to "the spectral radius of the matrix A is less than 1" because, for the asymptotical stability, we need the largest absolute value of the eigenvalues of A to be less than 1.

Please see the following paper, for more than one matrices switching:

Article Stability of a Set of Matrices with Applications to Automatic Control

Thank you sir. Appreciate your time.

Dear Om Prakash Yadav

If the spectral radius is less than 1, it is not true that the norm of the matrix is less than 1!. The relation between spectral radius and the matrix norm is as follows ρ(A) ≤ |||A|||, and we can prove that

ρ(A) ≤ (|||Ak|||)1/k as k goes to infinity.

So if ρ(A) ≤ 1, it is not necessary that |||A||| < 1!

But the converse is true.

If |||A||| < 1, then ρ(A) ≤ 1.

Ex: An upper triangular matrix with zero entries of the diagonal.

ρ(A) = 0 < 1, but |||A|| >1. ( for many non zero uper entries >1)

Regards

Thank you Dr. Issam Kaddoura. So what can be the sufficient condition for stability of a system?

Dear Om Prakash,

What information available about your system?

(1) Since the space Mn(R) of nxn real matrices is finite dimensional all norms on this space are equivalent.

(2) If || . || is a norm on Rn then we have an associated matrix norm || . ||’ by setting ||A||’= sup{||Ax||: ||x||=1}. Not all norms on Mn(R) are associated to some norm on Rn. For example the Frobenius norm is not an associated one.

(3) If || . ||’ is an associated norm on Mn(R) then ρ(A) ≤ ||A||’ and we do not have equality in general, however for every matrix A and every 0

Dear Dr. Omran Kouba , Thank you very much for help. I would appreciate if you could add some insights for the forllowing problem,

Suppose I have a matrix A=[0.4 0.8; 0 0.7]. Then 0.7=ρ(A)1, 1.2 and 1.5 respectively, to be precise. May I conclude the iterative scheme x_k=Ax_{k-1} will be stable.

Thanks for your time.

Dear Dr. Issam Kaddoura I am having a matrix of tridiagonal type and studying the stability of an iterative scheme x_{k+1}=Ax_k.

Thanks for your time.

Om Prakash Yadav

The answer to your question is Yes, because there is a norm N(.) on R2 such that the associated norm N' satisfies N'(A)

Thanks a lot, Sir.

Dear Omran Kouba

What if one eigenvalue has a positive real part regardless of the spectral selected norm? Do you think the system preserves stability?

Regards

Dear Issam Kaddoura

For discrete time systems stability depends on the magnitude of the eigenvalues of A , not the sign of the real part. Eigenvalues inside the unit circle = stability.

Regards.

Dear Omran Kouba,

It is a confusing criteria for stability.

We are familiar with the first and second lyapunov approach and some analog methods such as the hurwitz criteria to guarantee the dynamical system stability of the equilibrium states. All based on the sign analysis of the eigenvalues and not their magnitude!

Anyway, I appreciate if you mention some references that discuss and define this type of stability.

Best regards

Dear Issam Kaddoura

Here is a tutorial on discrete time linear systems.

https://homes.esat.kuleuven.be/~maapc/static/files/SYSTHEORY/Slides/Lecture6/Lecture6-Discrete%20time%20system%20properties.pdf

You can also consult this

http://www.control.tu-berlin.de/images/0/07/DCS_ADTS_2x3.pdf

Much appreciated Prof. Kouba, Omran Kouba