[Paper Review] Observer-based switched-linear system identification

Fethi Bencherki, Semiha Türkay+1 more|arXiv (Cornell University)|Jul 30, 2021

Control Systems and Identification42 references4 citations

TL;DR

This paper proposes a novel observer-based method for identifying discrete-time switched-linear systems from input-output data without requiring direct measurement of the continuous state. By transforming the state-space model into a switched ARX (SARX) form via a deadbeat observer, the method reduces the identification problem to sparse optimization and clustering, enabling accurate estimation of switching sequences, submodel parameters, and discrete states under mild structural assumptions and persistence of excitation conditions.

ABSTRACT

In this paper, we present a methodology to identify discrete-time state-space switched linear systems (SLSs) from input-output measurements. Continuous-state is not assumed to be measured. The key step is a deadbeat observer based transformation to a switched auto-regressive with exogenous input (SARX) model. This transformation reduces the state-space identification problem to a SARX model estimation problem. Overfitting issues are tackled. The switch and parameter identifiability and the persistence of excitation conditions on the inputs are discussed in detail. The discrete-states are identified in the observer domain by solving a non-convex sparse optimization problem. A clustering algorithm reveals the discrete-states under mild assumptions on the system structure and the dwell times. The switching sequence is estimated from the input-output data by the multi-variable output error state space (MOESP) algorithm and a variant modified from it. A convex relaxation of the sparse optimization problem yields the block basis pursuit denoising (BBPDN) algorithm. Theoretical findings are supported by means of a detailed numerical example. In this example, the proposed methodology is also compared to another identification scheme in hybrid systems literature.

Motivation & Objective

• Address the challenge of identifying switched-linear systems when the continuous state is unmeasured, a common limitation in practical system identification.
• Overcome overfitting and structural complexity in hybrid system identification by transforming the state-space model into a switched ARX (SARX) representation.
• Enable robust identification of switching sequences, submodel parameters, and discrete states through sparse optimization and clustering techniques.
• Ensure identifiability and stability by analyzing persistence of excitation conditions and structural assumptions on dwell times and system structure.
• Provide a computationally efficient and theoretically grounded framework for MIMO switched systems using convex relaxation and subspace-based algorithms.

Proposed method

• Design a deadbeat observer to transform the switched-linear state-space model into an equivalent switched ARX (SARX) model, eliminating the need for direct state measurements.
• Formulate the identification problem as a non-convex sparse optimization problem in the observer domain to detect discrete states and switching events.
• Apply a block basis pursuit denoising (BBPDN) convex relaxation to the sparse optimization problem for numerical tractability and stability.
• Use the MOESP algorithm and a modified variant to estimate the switching sequence from input-output data, leveraging subspace system identification principles.
• Employ a clustering algorithm to identify discrete states under mild assumptions on system structure and minimum dwell times.
• Utilize Hankel matrix factorization and rank conditions on Markov parameters to establish observability and structural identifiability.

Experimental results

Research questions

• RQ1 How can switched-linear systems be identified from input-output data when the continuous state is not directly measurable?
• RQ2 What transformation enables the reduction of a complex state-space identification problem to a more tractable SARX model estimation task?
• RQ3 Under what conditions is the switching sequence, submodel parameters, and discrete states identifiable from input-output data?
• RQ4 How can overfitting be mitigated in the identification of hybrid systems with switching dynamics?
• RQ5 What role do persistence of excitation and structural assumptions (e.g., dwell time, rank conditions) play in ensuring identifiability and convergence?

Key findings

• The proposed deadbeat observer transformation successfully converts the switched-linear state-space model into an equivalent SARX model, enabling identification without direct state measurement.
• The non-convex sparse optimization problem for discrete state detection is shown to be identifiable under mild structural assumptions and persistence of excitation conditions.
• The block basis pursuit denoising (BBPDN) convex relaxation provides a numerically stable and efficient solution to the sparse optimization problem.
• The clustering algorithm reliably identifies discrete states when the system satisfies minimum dwell time and structural regularity conditions.
• The MOESP-based switching sequence estimation achieves accurate reconstruction of switching instants, validated through numerical experiments.
• Theoretical analysis confirms that identifiability is preserved under the persistence of excitation condition, with a uniform lower bound on the excitation energy across all modes.