Realization theory of discrete-time linear switched systems

The paper presents realization theory of discrete-time linear switched systems. A discrete-time linear switched system is a hybrid system, such that the continuous sub-system associated with each discrete state is linear. In this paper we present necessary and sufficient conditions for an input-output map to admit a discrete-time linear switched state-space realization. The conditions are formulated as finite rank conditions of a generalized Hankel-matrix. In addition, we present a characterization of minimality of discrete-time linear switched systems in terms of reachability and observable.Further, we prove that minimal realizations are unique up to isomorphism. We also discuss procedures for converting a linear switched system to a minimal one and we present an algorithm for constructing a state-space representation from input-output data.The paper uses the theory rational formal power series in non-commutative variables. The latter theory was successfully applied to bilinear and state-affine systems in the past.

💡 Research Summary

The paper develops a comprehensive realization theory for discrete‑time linear switched systems (DT‑LSS), a class of hybrid systems in which each discrete mode is associated with a linear continuous‑time sub‑system. The authors model the input‑output behavior of a DT‑LSS as a map f from the free monoid Σ* (all finite strings of mode symbols) to an output space ℝ^p. By interpreting f as a rational formal power series (FPS) over non‑commutative variables, they bring to bear the algebraic machinery of free algebras and rational series, which has previously been successful for bilinear and state‑affine systems.

The central technical construct is a generalized Hankel matrix H_f. Its rows and columns are indexed by prefixes and suffixes of mode strings, and the (u, v) entry equals f(uv). Because the variables do not commute, H_f has a block‑structured, non‑Toeplitz form. The first major result shows that a map f admits a DT‑LSS state‑space realization if and only if H_f has finite rank, and that this rank coincides with the minimal state dimension n. In other words, finiteness of the Hankel rank is both necessary and sufficient for realizability.

Minimality is then characterized in the familiar way: a realization is minimal precisely when it is both reachable (every state can be generated by some finite input‑mode sequence) and observable (different states produce distinct output sequences). Reachability and observability are captured by matrices R and O constructed directly from H_f. If rank(R)=rank(O)=n, the system is minimal, and any two minimal realizations are isomorphic—there exists a bijective linear transformation that simultaneously maps all mode‑dependent state‑transition matrices, input matrices, and the common output matrix onto each other. This mirrors the classical Kalman‑Ho theory but requires careful handling of the non‑commutative mode algebra.

From a computational standpoint, the authors propose a data‑driven algorithm to construct a minimal DT‑LSS from finite input‑output samples. The procedure consists of:

1. Hankel estimation – collect output data for a set of input‑mode strings and assemble a finite sub‑matrix of H_f.
2. Rank‑revealing factorization – apply singular value decomposition (SVD) to the estimated Hankel matrix, retain the dominant singular values to obtain a low‑rank approximation, and thereby estimate the minimal order n.
3. Construction of R and O – use the low‑rank factors to form reachable and observable matrices.
4. Extraction of system matrices – compute each mode‑dependent state‑transition matrix A_q by projecting the shifted Hankel blocks onto the reachable subspace, and recover the input matrices B_q and the common output matrix C from the factorization.
5. Minimality check and reduction – if the obtained model is not yet minimal, perform a further reduction by eliminating the common nullspace of R and O, analogous to the standard controllability‑observability decomposition.

The algorithm is essentially a non‑commutative analogue of the Ho‑Kalman realization algorithm, and it guarantees that the resulting model reproduces the observed input‑output behavior up to the chosen truncation order.

The theoretical development rests heavily on the algebra of rational FPS. The authors recall that a series is rational iff it can be expressed using a finite number of linear operations, concatenations, and the Kleene‑star (formal inverse of I − Σ A_q x_q). This equivalence provides the bridge between the abstract series f and a concrete state‑space representation. By extending the rational series framework to the switched‑system setting, the paper unifies the treatment of linear, bilinear, and state‑affine models under a single algebraic umbrella.

Beyond the core results, the paper discusses several practical implications. Minimal DT‑LSS models are valuable for controller synthesis, observer design, and model‑based fault detection, especially in applications where mode transitions are frequent (e.g., power electronics, automotive transmission systems, networked control). The ability to derive such models directly from measured data reduces the reliance on detailed physical modeling and enables rapid prototyping. Moreover, the authors hint at extensions to more general hybrid systems that include resets, guards, or continuous‑time dynamics, suggesting that the rational FPS approach could serve as a foundational tool for a broader class of non‑commutative hybrid models.

In summary, the paper delivers a rigorous, algebraically grounded realization theory for discrete‑time linear switched systems. It establishes finite‑rank Hankel conditions for realizability, characterizes minimality via reachability and observability, proves uniqueness of minimal realizations up to isomorphism, and supplies a constructive algorithm for model identification from input‑output data. By leveraging the theory of rational formal power series in non‑commutative variables, the work bridges gaps between classical linear system theory, bilinear/state‑affine analysis, and modern hybrid system design, offering both deep theoretical insight and practical tools for engineers and researchers.