[Paper Review] Partial-realization theory and algorithms for linear switched systems: A formal power series approach

Mihály Petreczky, Jan H. van Schuppen|arXiv (Cornell University)|Oct 25, 2010

Stability and Control of Uncertain Systems47 references3 citations

TL;DR

This paper develops partial-realization theory and algorithms for linear switched systems (LSS) using rational formal power series. It introduces two algorithms—based on Hankel matrix column extraction and matrix factorization—that compute minimal partial realizations, with convergence to a complete minimal realization when the Hankel matrix rank stabilizes.

ABSTRACT

The paper presents partial-realization theory and realization algorithms for linear switched systems. Linear switched systems are a particular subclass of hybrid systems. We formulate a notion of a partial realization and we present conditions for existence of a minimal partial realization. We propose two partial-realization algorithms and we show that under certain conditions they yield a complete realization. Our main tool is the theory of rational formal power series.

Motivation & Objective

• Develop a formal partial-realization theory for linear switched systems (LSS), extending classical realization theory to hybrid systems.
• Address the challenge of identifying minimal LSS models from limited input-output data, relevant for systems identification and model reduction.
• Demonstrate that rational formal power series theory provides a robust framework for deriving realization algorithms for LSS.
• Lay the foundation for extending partial-realization theory to broader classes of hybrid systems, including piecewise-affine systems with guards.
• Bridge theoretical formalism with practical algorithm design, particularly for subspace identification-like methods in LSS.

Proposed method

• Formulate input-output behavior of LSS as a family of rational formal power series, leveraging the correspondence between LSS and such series.
• Define the Hankel matrix of the input-output map as the Hankel matrix of the corresponding formal power series family.
• Propose Algorithm 1 to construct a partial realization via column selection from a finite sub-matrix of the Hankel matrix.
• Propose Algorithm 2 to compute a partial realization through factorization of a finite Hankel sub-matrix, ensuring isomorphism with Algorithm 1.
• Use rank stabilization of finite Hankel sub-matrices (H_{Φ,N,N}, H_{Φ,N+1,N}, H_{Φ,N,N+1}) as a stopping criterion for convergence to a complete minimal realization.
• Leverage theoretical results from formal power series to derive conditions under which partial realizations are minimal and complete.

Experimental results

Research questions

• RQ1 Under what conditions does a minimal partial realization exist for a given linear switched system?
• RQ2 Can the structure of rational formal power series be used to derive constructive algorithms for partial realization of LSS?
• RQ3 How can the rank of finite Hankel matrix sub-matrices be used to determine when a partial realization becomes a complete minimal realization?
• RQ4 To what extent can the proposed formal power series framework be extended to more general hybrid systems, such as piecewise-affine systems?
• RQ5 What is the relationship between generalized Markov-parameters and the realization of LSS, and how can they be estimated from data?

Key findings

• The proposed partial-realization algorithms (Algorithm 1 and Algorithm 2) yield isomorphic minimal realizations when applied to the same input-output data.
• A minimal complete realization of the input-output behavior is obtained if the rank of the finite Hankel sub-matrix stabilizes, i.e., when rank(H_{Φ,N,N}) = rank(H_{Φ,N+1,N}) = rank(H_{Φ,N,N+1}).
• The algorithms return a minimal LSS realization for any N such that N+1 equals the dimension of a potential minimal realization of the input-output map.
• The theory of rational formal power series provides a unifying framework that enables the derivation of partial-realization results for LSS and potentially other hybrid systems.
• The approach is extendable to model reduction via moment matching, where a lower-order LSS is constructed as a partial realization of a finite sequence of generalized Markov-parameters.
• The framework supports geometric and topological analysis of spaces of LSS, enabling distance metrics and parametric identification tools.