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Basis transform in switched linear system state-space models from input-output data

Fethi Bencherki, Semiha Türkay, Hüseyin Akçay

TL;DR

This work tackles the problem of aligning locally identified submodels of a linear switched system (LSS) that reside in different state bases, to yield a coherent input–output map under arbitrary inputs and switching sequences. It introduces interpolation conditions that recover products of similarity transforms from input–output data under rank conditions, and proves that persistently exciting (PE) hybrid inputs exist with length $N=O((\sigma-1)n)$. A graph-theoretic interpretation shows that a spanning tree with $\sigma-1$ edges suffices to determine all necessary transformation pairs, and Algorithm 1 is proposed to compute the corrected submodels $\check{\mathcal{P}}_\nu$ from clusters of local estimates. The numerical example confirms that the proposed basis correction yields exact output predictions after alignment, linking identifiability, PE input design, and basis transformation in LSS identification.

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.

Basis transform in switched linear system state-space models from input-output data

TL;DR

This work tackles the problem of aligning locally identified submodels of a linear switched system (LSS) that reside in different state bases, to yield a coherent input–output map under arbitrary inputs and switching sequences. It introduces interpolation conditions that recover products of similarity transforms from input–output data under rank conditions, and proves that persistently exciting (PE) hybrid inputs exist with length . A graph-theoretic interpretation shows that a spanning tree with edges suffices to determine all necessary transformation pairs, and Algorithm 1 is proposed to compute the corrected submodels from clusters of local estimates. The numerical example confirms that the proposed basis correction yields exact output predictions after alignment, linking identifiability, PE input design, and basis transformation in LSS identification.

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.

Paper Structure

This paper contains 12 sections, 5 theorems, 45 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Problem formulation
  4. Motivation for a basis transformation
  5. Related work
  6. Contributions
  7. Interpolation conditions for basis transformation
  8. PE hybrid inputs
  9. Relations with the graph theory
  10. Basis transformation algorithm
  11. Numerical example
  12. Conclusions

Key Result

Lemma 3.1

Suppose that $n \geq 2$, $i\geq 1$, and Assumptions stabmin--mindwell hold. Let $\kappa_{i-1}(t_{i-1})$ be as in (kapd). Then, $\kappa_{i-1}(t_{i-1})=\kappa_{i-1}(k_i-n)$ for all $t_{i-1} \in [k_{i-1}\;k_i-n]$.

Figures (7)

  • Figure 1: A graph representation of $\mathcal{G}$.
  • Figure 2: Two spanning trees for $\mathcal{G}$ given in Figure \ref{['fig100']}
  • Figure 3: A graph representation for $\Upsilon^{(2)}$ and the vertex set.
  • Figure 4: The switching sequence.
  • Figure 5: Retrieval of the clusters via the DBSCAN algorithm Esteretal:96
  • ...and 2 more figures

Theorems & Definitions (6)

  • Lemma 3.1
  • Theorem 3.1
  • Definition 4.1
  • Lemma 4.1
  • Theorem 4.1
  • Theorem 6.1