[Paper Review] Basis transform in switched linear system state-space models from input-output data

Fethi Bencherki, Semiha Türkay+1 more|arXiv (Cornell University)|Jun 21, 2021

Control Systems and IdentificationEngineering42 references1 citations

TL;DR

This paper proposes a basis transformation method to align locally identified submodels in switched linear systems (LSS) from input-output data, ensuring coherent output prediction across switching sequences. By enforcing a persistence of excitation (PE) condition on hybrid inputs, the method computes similarity transformations that preserve the system's input-output map, enabling accurate identification and prediction despite submodels being estimated in different state bases. The key contribution is a practical, algorithmic framework for basis correction that guarantees input-output equivalence under mild PE conditions.

ABSTRACT

This paper addresses the problem of basis correction in the context of LSS identification from input-output data. It is often the case that identification algorithms for the LSSs from input-output data operate locally. The individually identified local submodel estimates reside in distinct state bases, which mandates performing a basis correction that facilitates their coherent patching for the ultimate goal of performing output predictions for arbitrary inputs and switching sequences. We formulate a persistence of excitation condition for the inputs and the switching sequences that guarantee the presented approach's success. These conditions are mild in nature, which proves the practicality of the devised algorithm. We supplement the theoretical findings with an elaborating numerical simulation example.

Motivation & Objective

• To address the challenge of inconsistent state bases in locally identified submodels of switched linear systems (LSS).
• To enable coherent patching of submodels for accurate output prediction under arbitrary switching sequences.
• To establish a persistence of excitation (PE) condition on hybrid inputs (input and switching signals) that ensures the success of basis correction.
• To provide a practical algorithm for computing similarity transformations that preserve the LSS input-output map.
• To demonstrate the feasibility and accuracy of the approach through a numerical simulation with synthetic data.

Proposed method

• The method begins by identifying discrete states (submodels) via a subspace identification algorithm, which estimates local models up to similarity transformations.
• It applies the DBSCAN clustering algorithm to detect clusters in the state transition matrix estimates, identifying the number of discrete states σ.
• It formulates a persistence of excitation (PE) condition on the hybrid input (input and switching signal pair) to ensure identifiability and basis correction feasibility.
• It computes transformation matrices ϒ(j,1) for j = 2,3,...,σ using inverse and chain rules on cross-identification data from overlapping switching intervals.
• It applies the computed transformations ϒ(j,1) to align the state bases of submodels 2 through σ to that of submodel 1, ensuring input-output map invariance.
• It validates the method by comparing predicted outputs with true outputs under new inputs, showing negligible mismatch error.

Experimental results

Research questions

• RQ1 How can locally identified submodels of a switched linear system, estimated in different state bases, be aligned to form a consistent overall model?
• RQ2 What conditions on the hybrid input (input and switching sequence) ensure that basis correction is possible and effective?
• RQ3 Can a transformation be computed that preserves the input-output behavior of the LSS while reconciling submodels estimated in different bases?
• RQ4 How does the proposed method relate to observability, identifiability, and minimal realization in LSS?
• RQ5 What is the role of persistence of excitation in enabling basis correction and accurate output prediction?

Key findings

• The proposed basis correction algorithm successfully aligns submodels estimated in different state bases, resulting in a consistent LSS representation that preserves the input-output map.
• The numerical simulation shows perfect matching between true and predicted outputs, with estimation errors on the order of 10−13, indicating high accuracy.
• The persistence of excitation condition on the hybrid input is both necessary and sufficient for successful basis correction, and it is mild enough to be practically feasible.
• The method correctly estimates the number of discrete states σ = 3 using clustering on the Markov parameter estimates, as confirmed by visual and histogram analysis.
• The transformation matrices ϒ(2,1) and ϒ(3,1) are successfully computed and applied, aligning submodels 2 and 3 to the basis of submodel 1, enabling accurate output prediction.
• The link between basis transformation and graph theory is established, suggesting deeper structural connections in LSS realization and identification.