---
title: Discrete-Time Switched Linear Systems
url: https://www.emergentmind.com/topics/discrete-time-switched-linear-systems
type: topic
---

# Discrete-Time Switched Linear Systems

A discrete-time switched linear system (DTSLS) is a dynamical system where the state evolves according to a sequence of linear transformations selected from a finite family of matrices, according to a switching signal. This class encapsulates a significant category of hybrid systems, modeling situations where the system structure is subject to abrupt changes. DTSLS appear extensively in control theory, abstraction-based system synthesis, verification, and identification, particularly for complex systems with both stable and unstable components. Their stability, controllability, identifiability, and model reduction have been addressed through Lyapunov methods, graph-theoretic analysis, data-driven approaches, and computable algebraic criteria.

## 1. Fundamental Model and Stability Criteria

A discrete-time switched linear system is defined by 
\[
x(t+1) = A_{\sigma(t)} x(t), \quad x(0) \in \mathbb{R}^d
\]
where $\{A_i\}_{i \in \mathcal{P}}$ is a finite set of full-rank matrices $A_i \in \mathbb{R}^{d \times d}$, and $\sigma : \mathbb{N}_0 \rightarrow \mathcal{P}$ is a switching signal (possibly restricted by additional logic, dwell-time, or admissibility constraints) [1310.3595]. Stability analysis primarily targets global asymptotic stability (GAS), defined as Lyapunov stability plus attractivity to the origin for all initial conditions.

The main sufficient condition for GAS under a switching signal $\sigma$ is the existence of quadratic Lyapunov-like functions $V_i(x) = x^\top P_i x$ and scalars $\lambda_i > 0$ for each mode $i$ such that
\[
V_i(x(t+1)) \le \lambda_i V_i(x(t)), \qquad V_j(x) \le \mu_{ij} V_i(x)
\]
with $\lambda_i < 1$ for Schur-stable modes, $\lambda_i > 1$ for unstable ones, and $\mu_{ij} \ge 1$ quantifying inter-mode jumps [1310.3595].

The switching signal must satisfy, asymptotically,
\[
\limsup_{t \rightarrow \infty} \frac{ 
\sum_{(k,\ell)\in E}(\ln \mu_{k\ell}) \rho_{k\ell}(t) + 
\sum_{j \in P_U} (\ln \lambda_j) \kappa_j(t) }
{ 
\sum_{j \in P_{AS}} (-\ln \lambda_j) \kappa_j(t) }
< 1
\]
where $\rho_{k\ell}(t)$ counts transitions, and $\kappa_j(t)$ tracks total dwell-time on mode $j$; $P_{AS}$ and $P_U$ index stable and unstable modes, respectively.

## 2. Graph-Theoretic Synthesis of Stabilizing Signals

A key innovation is the graph-theoretic representation $G = (V, E)$, with vertices corresponding to modes and edges encoding admissible switches between subsystems. Infinite walks on this graph correspond bijectively to switching signals. The existence of a stabilizing signal becomes equivalent to the existence of a finite closed walk $W$ (circuit) such that
\[
\frac{
\sum_{(k, \ell) \in E} (\ln \mu_{k\ell}) \rho_{k\ell}(W) + 
\sum_{j \in P_U} (\ln \lambda_j) \kappa_j(W)
}{
\sum_{j \in P_{AS}} (-\ln \lambda_j) \kappa_j(W)
}
< 1
\]
[1310.3595]. This recasts the stabilization task as a search for such "contractive" walks.

The design algorithm follows:
- Compute Lyapunov matrices and switching costs,
- Formulate and solve an LP encoding the ratio condition,
- Extract a circuit $W$ using Hierholzer’s algorithm,
- Construct $\sigma$ by infinite repetition of $W$, guaranteeing GAS.

This unifies stabilization for DTSLS with arbitrary or restricted switching, including cases with unstable modes, and yields explicit, polynomial-time computable stabilizing signals.

## 3. Extension: Data-Driven and Restricted Switching

Recent work generalizes these graph-based criteria to data-driven scenarios and additional switching restrictions.
- Data-driven approaches estimate Lyapunov-like parameters $(P_i, \lambda_i, \mu_{ij})$ using Hankel-type matrices constructed from time-series data, dispensing with explicit subsystem models [2003.05774, 2002.02087].
- The stabilization algorithm remains graph-based: negative-cycle detection (Bellman–Ford) yields contractive cycles compatible with given dwell-time and admissibility constraints.
- Minimum dwell-time guarantees can be computed directly from data traces, with the dwell-time bound
\[
\tau > \frac{\ln \mu}{-\ln \lambda_s}
\]
where $\lambda_s$ is a uniform contraction bound and $\mu$ is the maximal cross-mode jump [2002.02087].

Extensions to ranged dwell-time are achieved by introduction of $L$-switching-cycles and equivalent LMIs, generalizing multiple and clock-dependent Lyapunov conditions; the method attains nonconservativeness as $L$ increases [2105.14113].

## 4. Algorithms for Stabilizability under Unstable and Restricted Modes

For families comprising solely unstable subsystems, stabilizability requires identification of a Schur-stable composite block (i.e., a sequence of mode activations whose product is Schur). The sufficiency condition is then expressed as norm and commutator bounds:
\[
\|A_{N+1}^m\| \le \rho < 1; \quad \|A_\ell A_{N+1} - A_{N+1}A_\ell\| \le \varepsilon
\]
and a scalar contraction inequality:
\[
\rho e^{\lambda m(p+q)} + C \varepsilon e^{\lambda(m(p+q) + mN)} \le 1
\]
where $C$ tracks path-dependent combinatorics [2011.01530, 2005.10493]. Explicit switching signals are constructed via walks on augmented graphs enforcing prescribed dwell-time and switch admissibility, and robustness to perturbations is achieved via commutator bounds [1903.09440].

## 5. Deterministic and Randomized Synthesis, Probabilistic Genericity

Deterministic synthesis algorithms solve LPs (edge-indicator) to identify circuits with minimal contraction ratios, or apply negative-cycle detection to weighted digraphs [1405.1857]. For very large graphs or systems, probabilistic cycle-search algorithms (randomized online walk generation under connectivity and weight statistics) almost surely produce stabilizing signals, with exponentially decaying failure probabilities for large mode sets [1405.1857].

Genericity results show that under mild statistical weight/connectivity conditions, almost all large DTSLS admit a stabilizing contractive cycle; the computation scales with the cycle length rather than the system dimension.

## 6. Connection to Realization and Model Reduction

Realization theory for DTSLS provides necessary and sufficient conditions for a behavior (input-output map) to admit a state-space realization in the switched linear class, via finite-rank generalized Hankel matrices and rational formal power series [1103.1343]. Minimal realizations are unique up to isomorphism, and algorithms exist for constructing minimal models from input-output data. Model reduction based on narrowing the class of admissible switching sequences (e.g., regular languages, NDFA) enables projection-based reduction of state dimension, preserving input-output equivalence on specified switching patterns [1409.1444]. This abstraction substantially improves control synthesis and verification scalability.

## 7. Summary Table: Core Stabilizability and Synthesis Conditions

| Criterion                                      | Formula / Principle                                                       | Reference      |
|------------------------------------------------|--------------------------------------------------------------------------|---------------|
| GAS under arbitrary switching                  | Ratio condition on Lyapunov costs                                         | [1310.3595]   |
| Synthesis of stabilizing signal                | Infinite repetition of finite contractive graph walk                      | [1310.3595], [1405.1857] |
| Data-driven stability                          | LMIs on Hankel matrices; negative cycles in weighted graph                | [2003.05774], [2002.02087] |
| Unstable subsystems with Schur-stable block    | Contraction and commutator norm inequalities                              | [2011.01530], [2005.10493], [1903.09440] |
| Minimum dwell-time (data-based)                | $\tau > \frac{\ln \mu}{-\ln \lambda_s}$                                  | [2002.02087] |
| Ranged dwell-time (general)                    | $L$-switching-cycle LMIs (multiple and clock-dependent Lyapunov)          | [2105.14113] |

These algorithmic, algebraic, and data-driven methodologies together form the principal toolbox for analysis and synthesis of discrete-time switched linear systems, supporting robust control, stabilization, and system identification in broad settings.

Source: https://www.emergentmind.com/topics/discrete-time-switched-linear-systems