I have a question about marginal stability of a system: \begin{equation} \mathbf{x}[k] = \mathbf{A}\mathbf{x}[k-1] \end{equation} I would adapt the definition of marginal stability from this question to the above discrete system. The system is marginally stable if the signal $\mathbf{x}[k]$ is bounded, i.e.: \begin{equation} \lim_{k\rightarrow\infty} \mathbf{x}[k] < M < \infty \end{equation} I have trouble finding the correct book reference.

Most of the references I have found talk about asymptotic stability, and state that spectral radius of matrix should be $\rho(\mathbf{A}) < 1$. If at least one eigenvalue of matrix $\mathbf{A}$ is outside unit circle, the above system is unstable.

I have read in few references that multiple same eigenvalues result in the unstable matrix. However, I don't think this is the case for the unit matrix: \begin{equation} \mathbf{A} = \begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix} \end{equation} In this case the matrix has two same eigenvalues with value $1$, and matrix is marginally stable. What confuses me is that you can have the following matrix: \begin{equation} \mathbf{A} = \begin{bmatrix} 2 & -1\\ 1 & 0 \end{bmatrix} \end{equation} with same eigenvalues, but this system is unstable.

Can marginal stability be characterized by the location of eigenvalues? How can I determine whether the system above is stable by analyzing the matrix $\mathbf{A}$. If possible, could you provide a reference?


Actually, this is a question that you should concern the matrix $A$'s geometric multiplicity (see the reference How do you calculate the geometric multiplicities?). For simplicity, the geometric multiplicity of matrix $A$ is equal to the null space dimension of characteristic $|A-\lambda I |$. Firstly, it should be no doubt that if any eigenvalue is outside the unit cycle, the discrete time system is unstable (since this will boost the system). Secondly, if all eigenvalues are inside the unit cycle, the system is asymptotic stable. When eigenvalues occur at the unit boundary, things will become a little bit tedious. It states that the system is stable if and only if: 1. for the eigenvalues with geometric multiplicity equal to 1, they can inside or on the unit cycle (that is to say not outside the unit cycle). 2. for the eigenvalues with larger than 1 geometric multiplicity, they must have modulos smaller than 1. I hope this can somehow solve your concerns. For the details, these statements can be verified by the expansion of the exponential matrix in discrete-time.

Ref: Panos J. Antsaklis and Anthony N. Michel, A Linear Systems Primer. Chapter 4.4, Lyapunov Stability of Linear Systems.


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