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Realization of multi-input/multi-output switched linear systems from Markov parameters

Fethi Bencherki, Semiha Türkay, Hüseyin Akçay

TL;DR

The paper addresses the problem of realizing a MIMO-SLS from doubly indexed Markov parameters by developing a four-stage algorithm that first yields a topologically equivalent LTV realization, then extracts discrete submodels, estimates the switching sequence with complementary schemes, and finally aligns submodels to a common basis for output prediction. The key methodology combines time-varying Hankel-parameter realization, unimodality- and dwell-time-based discrete-state clustering, and forward/backward correction plus Markov-parameter matching to robustly recover the active discrete state sequence. Theoretical results establish uniform realizability under mild assumptions and provide perturbation bounds showing robustness to noise, while the numerical examples validate discrete-state recovery, switching estimation, and basis alignment on noiseless and noisy data and compare favorably with existing SARX-based methods. The work offers a practical, modular pipeline for obtaining interpretable MIMO-SLS models from Markov-parameter data, with potential integration into single-trajectory Markov-parameter estimation workflows and broader applications in hybrid-system identification.

Abstract

This paper presents a four-stage algorithm for the realization of multi-input/multi-output (MIMO) switched linear systems (SLSs) from Markov parameters. In the first stage, a linear time-varying (LTV) realization that is topologically equivalent to the true SLS is derived from the Markov parameters assuming that the submodels have a common MacMillan degree and a mild condition on their dwell times holds. In the second stage, zero sets of LTV Hankel matrices where the realized system has a linear time-invariant (LTI) pulse response matching that of the original SLS are exploited to extract the submodels, up to arbitrary similarity transformations, by a clustering algorithm using a statistics that is invariant to similarity transformations. Recovery is shown to be complete if the dwell times are sufficiently long and some mild identifiability conditions are met. In the third stage, the switching sequence is estimated by three schemes. The first scheme is based on forward/backward corrections and works on the short segments. The second scheme matches Markov parameter estimates to the true parameters for LTV systems and works on the medium-to-long segments. The third scheme also matches Markov parameters, but for LTI systems only and works on the very short segments. In the fourth stage, the submodels estimated in Stage 2 are brought to a common basis by applying a novel basis transformation method which is necessary before performing output predictions to given inputs. A numerical example illustrates the properties of the realization algorithm. A key role in this algorithm is played by time-dependent switching sequences that partition the state-space according to time, unlike many other works in the literature in which partitioning is state and/or input dependent.

Realization of multi-input/multi-output switched linear systems from Markov parameters

TL;DR

The paper addresses the problem of realizing a MIMO-SLS from doubly indexed Markov parameters by developing a four-stage algorithm that first yields a topologically equivalent LTV realization, then extracts discrete submodels, estimates the switching sequence with complementary schemes, and finally aligns submodels to a common basis for output prediction. The key methodology combines time-varying Hankel-parameter realization, unimodality- and dwell-time-based discrete-state clustering, and forward/backward correction plus Markov-parameter matching to robustly recover the active discrete state sequence. Theoretical results establish uniform realizability under mild assumptions and provide perturbation bounds showing robustness to noise, while the numerical examples validate discrete-state recovery, switching estimation, and basis alignment on noiseless and noisy data and compare favorably with existing SARX-based methods. The work offers a practical, modular pipeline for obtaining interpretable MIMO-SLS models from Markov-parameter data, with potential integration into single-trajectory Markov-parameter estimation workflows and broader applications in hybrid-system identification.

Abstract

This paper presents a four-stage algorithm for the realization of multi-input/multi-output (MIMO) switched linear systems (SLSs) from Markov parameters. In the first stage, a linear time-varying (LTV) realization that is topologically equivalent to the true SLS is derived from the Markov parameters assuming that the submodels have a common MacMillan degree and a mild condition on their dwell times holds. In the second stage, zero sets of LTV Hankel matrices where the realized system has a linear time-invariant (LTI) pulse response matching that of the original SLS are exploited to extract the submodels, up to arbitrary similarity transformations, by a clustering algorithm using a statistics that is invariant to similarity transformations. Recovery is shown to be complete if the dwell times are sufficiently long and some mild identifiability conditions are met. In the third stage, the switching sequence is estimated by three schemes. The first scheme is based on forward/backward corrections and works on the short segments. The second scheme matches Markov parameter estimates to the true parameters for LTV systems and works on the medium-to-long segments. The third scheme also matches Markov parameters, but for LTI systems only and works on the very short segments. In the fourth stage, the submodels estimated in Stage 2 are brought to a common basis by applying a novel basis transformation method which is necessary before performing output predictions to given inputs. A numerical example illustrates the properties of the realization algorithm. A key role in this algorithm is played by time-dependent switching sequences that partition the state-space according to time, unlike many other works in the literature in which partitioning is state and/or input dependent.

Paper Structure

This paper contains 31 sections, 14 theorems, 128 equations, 13 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related work
  3. Motivation for the SLS realization
  4. Organization of the paper
  5. Problem formulation
  6. LTV realization from Markov parameters
  7. Time complexity of Algorithm 1
  8. Robustness of Algorithm 1
  9. Discrete state estimation
  10. Time complexity of Algorithm 2
  11. Robustness of Algorithm 2
  12. Switch detection from a modified LTV model
  13. Time complexity of Algorithm 3
  14. Robustness of Algorithm 3
  15. Switch detection from Markov parameters
  16. ...and 16 more sections

Key Result

Lemma 3.1

Suppose Assumptions sysasmp--varphiassmp hold and $q,r\geq 2n$. Then, the MIMO-SLS model (ssx)--(varphit) is uniform on $[2n+1 \;\;N-2n+1]$. $\blacksquare$

Figures (13)

  • Figure 1: A meta-algorithm for the SLS realization from Markov parameters sequence.
  • Figure 2: The switching sequence in the example.
  • Figure 3: The discrete state estimation via Algorithm 2 (see Appendix).
  • Figure 4: The switching sequence generated in Section \ref{['912']}.
  • Figure 5: The switching sequence estimation via Algorithms $(3',3,3")$ in order.
  • ...and 8 more figures

Theorems & Definitions (18)

  • Definition 3.1
  • Lemma 3.1
  • Theorem 3.1
  • Theorem 3.2
  • Lemma 4.1
  • Theorem 4.1
  • Theorem 4.2
  • Example 5.1
  • Example 5.2
  • Example 5.3
  • ...and 8 more