⚡ electrical engineering

Stability Criteria via Common Non-strict Lyapunov Matrix for Discrete-time Linear Switched Systems

This paper investigates the stability conditions for discrete-time linear switched systems by utilizing a common non-strict Lyapunov matrix.

Original authors: Xiongping Dai, Yu Huang, Mingqing Xiao

Published 2026-06-03

📖 5 min read🧠 Deep dive

CC BY 3.0

Original authors: Xiongping Dai, Yu Huang, Mingqing Xiao

Original paper licensed under CC BY 3.0 (http://creativecommons.org/licenses/by/3.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a machine that can switch between two different modes of operation, let's call them Mode A and Mode B. Every second, a "switching signal" decides whether the machine runs in Mode A or Mode B. The machine's state (like its position or energy) changes based on which mode is active.

The big question this paper asks is: Will this machine eventually calm down and stop moving (stabilize), or will it go wild, no matter how the switching signal behaves?

The "Magic Rule" (The Common Lyapunov Matrix)

Usually, to prove a system is safe, you need to show that every time you switch modes, the machine gets strictly closer to zero (like a ball rolling down a hill that gets steeper every time).

However, this paper deals with a trickier, more realistic scenario. Imagine the machine has a "Magic Rule" (called a Common Non-strict Lyapunov Matrix).

Strict Rule: Every step must make the machine smaller.
Non-strict Rule (This Paper): The machine never gets bigger, but sometimes it stays exactly the same size. It's like walking on a flat surface: you don't fall down, but you don't necessarily move toward the exit either. You might just walk in circles.

The authors ask: If we have this "Non-strict" rule (where things don't grow, but might stall), can we still guarantee the machine will eventually stop?

The Three Main Findings

1. The "Non-Chaotic" Switching (The Boring Switcher)

The Analogy: Imagine a switcher who is a bit indecisive but not crazy. They might switch back and forth, but they always pause for a while on one setting before switching again. They don't jump around randomly every millisecond.
The Result: If the machine is stable on its own (Modes A and B are both "safe" individually) and the switcher is "non-chaotic" (they pause long enough), the machine will eventually stop. Even if the switcher pauses on a mode that doesn't shrink the machine, the fact that they eventually switch to the other mode (which does shrink it) ensures the machine calms down.

2. The "Recurrent" Switching (The Looping Switcher)

The Analogy: Imagine a switcher who loves patterns. They might switch in a complex rhythm, but eventually, they repeat the exact same sequence of switches over and over again.
The Result: The authors found a way to split the machine's possible states into two groups:

The "Safe" Group: States that will eventually shrink to zero.
The "Stuck" Group: States that just keep spinning in a loop, never shrinking.
The paper proves that if the "Safe" and "Stuck" groups don't overlap in a specific way, the machine will almost always stabilize. It's like saying, "Unless you start in this very specific, rare spot, you will eventually stop."

3. The "Absolute" Stability (The Ultimate Test)

The Analogy: What if the switcher is a total chaos agent? They switch in any pattern imaginable, including the worst possible ones designed to keep the machine running forever. Can we tell if the machine is safe?
The Result: Yes, but only for small machines (2D or 3D).

For 2D Machines: You only need to check a tiny list of combinations: Does Mode A shrink? Does Mode B shrink? Does switching A then B shrink? If all three do, the machine is safe forever.
For 3D Machines: You need to check a slightly longer list (combinations of 1, 2, 3, 4, 5, 6, and 8 switches). If all those specific combinations shrink the machine, then no matter how crazy the switching gets, the machine will eventually stop.

The "Finiteness" Property

The paper concludes with a fascinating idea called the Spectral Finiteness Property.
Usually, to know if a system is safe, you might need to check an infinite number of switching patterns. But this paper says: No, you don't.
For these specific types of machines (2D or 3D with the "Magic Rule"), you only need to check a finite number of patterns. If those few patterns work, the infinite future is safe. It's like checking the first few pages of a book to know the ending; you don't need to read the whole library.

Summary in Plain English

The authors solved a puzzle about machines that switch between two settings. They found that even if the settings don't force the machine to shrink every single time (as long as they never make it grow), we can still predict if the machine will stop.

If the switching is somewhat orderly, it stops.
If the switching is repetitive, it stops (unless you start in a very weird spot).
If the switching is completely chaotic, we can still be 100% sure it stops, but only if we check a short, specific list of switching patterns. If those patterns work, the machine is safe forever.

This gives engineers and mathematicians a concrete "checklist" to ensure safety without having to simulate infinite possibilities.

Technical Summary: Stability Criteria via Common Non-strict Lyapunov Matrix for Discrete-time Linear Switched Systems

Problem Statement
The paper investigates the asymptotic stability of discrete-time linear switched systems governed by a finite set of matrices $S = \{S_1, \dots, S_K\} \subset \mathbb{R}^{d \times d}$ . The dynamics are defined by $x_n = S_{\sigma_n} \cdots S_{\sigma_1} x_0$ , where $\sigma: \mathbb{N} \to \{1, \dots, K\}$ is an arbitrary switching signal.

The core focus is on systems where the matrices share a common, but not necessarily strict, Lyapunov matrix $P$ . Specifically, there exists a symmetric positive-definite matrix $P$ such that:
$P - S_k^T P S_k \geq 0 \quad \text{for all } k = 1, \dots, K.$
This condition implies that the induced norm $\|x\|_P = \sqrt{x^T P x}$ is non-increasing under each subsystem ( $\|S_k\|_P \leq 1$ ), but it does not guarantee strict contraction ( $\|S_k\|_P < 1$ ). The authors address the challenge that even if individual subsystems are stable ( $\rho(S_k) < 1$ ), the switched system may not be absolutely stable (stable for all switching signals) under this weak Lyapunov condition.

Methodology
The authors employ a combination of linear algebra, spectral theory, and ergodic theory. Key methodological components include:

State Space Splitting: Utilizing a splitting theorem (based on prior work [13]) for recurrent switching signals. For a recurrent signal $\sigma$ , the state space $\mathbb{R}^d$ decomposes into a stable manifold $E_s(\sigma)$ and a central manifold $E_c(\sigma)$ . Stability is determined by whether $E_c(\sigma)$ is trivial.
$\omega$ -Limit Set Analysis: The paper analyzes the $\omega$ -limit sets of matrix products $\langle S_{\sigma_n} \cdots S_{\sigma_1} \rangle$ . Under the weak Lyapunov condition, these limit sets are compact and possess specific geometric properties regarding the norm $\|\cdot\|_P$ .
Non-chaotic and Recurrent Signals: The authors distinguish between "non-chaotic" signals (which contain arbitrarily long constant subsequences) and "recurrent" signals (which return arbitrarily close to their initial state in the shift topology).
Spectral Finiteness Property: The paper investigates whether the generalized spectral radius $\rho(S)$ is attained by a finite product of matrices. This property allows for the decidability of absolute stability.
Dimensional Reduction: The proofs for absolute stability rely on analyzing the dimensions of the subspaces $K_{\|\cdot\|_P}(S_k)$ (vectors where the norm is preserved) and their intersections.

Key Contributions and Results

1. Stability under Non-chaotic Switching (Theorem A)
The paper establishes that if $S$ satisfies the weak Lyapunov condition and every subsystem is strictly stable ( $\rho(S_k) < 1$ ), then the system is stable for any non-chaotic switching signal. This extends previous results that required generic (regular) signals.

2. Almost Sure Stability (Theorem B)
For $d=2$ or $d=3$ , if the intersection of the norm-preserving subspaces is trivial ( $K_{\|\cdot\|_P}(S_1) \cap K_{\|\cdot\|_P}(S_2) = \{0\}$ ), the system is stable for almost every switching signal with respect to any non-atomic ergodic probability measure. This highlights that instability, if it exists, occurs only on a set of measure zero (typically specific periodic signals).

3. Absolute Stability Criteria for $d=2$ and $d=3$ (Theorems C and D)
The paper provides necessary and sufficient conditions for absolute stability (stability for all switching signals) when $d=2$ or $d=3$ :

Case $d=2$ : $S$ is absolutely stable if and only if $\rho(S_1) < 1$ , $\rho(S_2) < 1$ , and $\rho(S_1 S_2) < 1$ .
Case $d=3$ : $S$ $S$ is absolutely stable if and only if $\rho(A) < 1$ $ρ (A) < 1$ for all products $A \in \{S_1, S_2\}^\ell$ $A \in {S_{1}, S_{2}}^{ℓ}$ where $\ell \in \{1, 2, 3, 4, 5, 6, 8\}$ $ℓ \in {1, 2, 3, 4, 5, 6, 8}$ .
- Notably, the authors observe that checking words of length 7 is unnecessary.

4. Spectral Finiteness Property
As a consequence of the absolute stability criteria, the paper proves that for $d=2$ and $d=3$ , if the system satisfies the weak Lyapunov condition and has a generalized spectral radius $\rho(S) = 1$ , then $S$ possesses the spectral finiteness property. This means the spectral radius is realized by a finite product of matrices. Specifically:

For $d=2$ , $\rho(S) = \max \{ \rho(S_1), \rho(S_2), \sqrt{\rho(S_1 S_2)} \}$ .
For $d=3$ , $\rho(S)$ is the maximum of the $n$ -th roots of the spectral radii of products of length $n \in \{1, \dots, 6, 8\}$ .

5. Counterexamples and Limitations
The authors provide examples (e.g., Example 6.6) demonstrating that:

Subsystem stability ( $\rho(S_k) < 1$ ) does not imply absolute stability if the weak Lyapunov condition holds but the spectral finiteness condition fails (e.g., $\rho(S_1 S_2) = 1$ ).
The "extension" of a set $S$ to a symmetric set $S \cup S^T$ does not preserve the spectral radius properties required for these stability conclusions.

Significance and Claims
The paper claims to resolve the decidability of absolute stability for discrete-time linear switched systems with a common non-strict Lyapunov matrix in dimensions $d=2$ and $d=3$ . The significance lies in:

Decidability: Providing finite, computable criteria (checking a finite number of matrix products) to determine absolute stability, which is generally undecidable for arbitrary switched systems.
Refinement of Stability Concepts: Distinguishing between stability driven by non-chaotic signals, almost sure stability, and absolute stability, showing that under the weak Lyapunov condition, instability is often confined to specific periodic switching patterns.
Spectral Finiteness: Establishing that the spectral finiteness property holds for these specific classes of systems, linking the geometric condition of the Lyapunov matrix to the algebraic property of finite spectral radius realization.

The authors conclude by posing a conjecture for $d \geq 4$ : whether periodic switched stability implies absolute stability under the same weak Lyapunov condition, noting that the current techniques do not immediately extend to higher dimensions.