Discrete-Time Switched Linear Systems

• Discrete-Time Switched Linear Systems are dynamic models that evolve through a sequence of linear transformations selected from a finite set, capturing abrupt system changes.
• They utilize Lyapunov methods, graph-theoretic synthesis, and data-driven approaches to ensure robust stability and control, even with unstable modes.
• These systems are widely applied in control theory, system verification, and model reduction, enabling efficient computation of stabilizing signals.

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) (Kundu et al., 2013). 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 (Kundu et al., 2013).

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

$$\frac{ \sum_{(k, \ell) \in E} (\ln \mu_{k\ell}) \rho_{k\ell}(W) + \sum_{j \in P_U} (\ln \lambda_j) \kappa_j(W) }{ \sum_{j \in P_{AS}} (-\ln \lambda_j) \kappa_j(W) } < 1$$

(Kundu et al., 2013). 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 (Kundu, 2020, Kundu, 2020).
• 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 (Kundu, 2020).

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 (Xiang, 2021).

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 (Kundu, 2020, Kundu, 2020). 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 (Kundu, 2019).

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 (Kundu et al., 2014). 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 (Kundu et al., 2014).

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 (Petreczy et al., 2011). 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 (Bastug et al., 2014). 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	(Kundu et al., 2013)
Synthesis of stabilizing signal	Infinite repetition of finite contractive graph walk	(Kundu et al., 2013, Kundu et al., 2014)
Data-driven stability	LMIs on Hankel matrices; negative cycles in weighted graph	(Kundu, 2020, Kundu, 2020)
Unstable subsystems with Schur-stable block	Contraction and commutator norm inequalities	(Kundu, 2020, Kundu, 2020, Kundu, 2019)
Minimum dwell-time (data-based)	$\tau > \frac{\ln \mu}{-\ln \lambda_s}$	(Kundu, 2020)
Ranged dwell-time (general)	$L$-switching-cycle LMIs (multiple and clock-dependent Lyapunov)	(Xiang, 2021)

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.