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Blind System Identification in Linear Parameter-Varying Systems

Javad Zahedi Moghaddam, Hamidreza Momeni, Mojtaba Danesh

TL;DR

The paper tackles blind LPV-IO identification (BLPVI) where the scheduling variable $p$ is unmeasured and inputs/outputs are available. It recasts the LPV dynamics as a discrete-time HMM and introduces the quasi-static Viterbi–Baum–Welch (QSVBW) algorithm to jointly estimate the scheduling path and nonlinear coefficient mappings, using a polynomial parameterization $g_i(p)$ with a coefficient matrix $ extbf{H}$ so that $ $mu(k) = $ extbf{P}_k^d extbf{H} oldsymbol{oldsymbol{oldsymbol{oldsymbol{ o}}}} oldsymbol{oldsymbol{ o}}(k)$. The nonlinear mappings are learned by updating $ extbf{H}$ (via a closed-form rule) while the scheduling path is inferred with the Viterbi algorithm, and the HMM parameters are refined with Baum–Welch steps, aided by cross-validation to choose the polynomial degree and PSO for initialization. A numerical study on an LPV-IO model demonstrates accurate recovery of both the scheduling variable and the nonlinear mappings, achieving high best-fit rates for outputs (≈95.7%) and scheduling paths (≈90.3%), indicating the method’s viability for BLPI in static-scheduling LPV contexts. The work provides a practical pathway to blind identification in nonlinear LPV systems and discusses avenues for extending to dynamic scheduling dependencies and MIMO settings, where observation models remain Gaussian and tractable.

Abstract

Blind System Identification (BSI) is used to extract a system model whenever input data is not attainable. Therefore, the input data and system model should be estimated simultaneously. Because of nonlinearities in a large number of systems, BSI problem is usually challenging to solve. In this paper, an innovative solution is proposed to deal with the BSI problem in nonlinear systems using the properties of the Linear Parameter-Varying (LPV) systems and Hidden Markov Models (HMM). More specifically, assuming scheduling variable is not measurable, the dynamic of the LPV system is approximated. To solve the BSI problem in this context, LPV structure is modeled as an HMM network and a modified Quasi-Static combination of Viterbi and Baum-Welch algorithms (QSVBW) is proposed to estimate the nonlinear mappings and scheduling variable signal. The applicability and the performance of the suggested QSVBW algorithm has been justified by numerical studies.

Blind System Identification in Linear Parameter-Varying Systems

TL;DR

The paper tackles blind LPV-IO identification (BLPVI) where the scheduling variable is unmeasured and inputs/outputs are available. It recasts the LPV dynamics as a discrete-time HMM and introduces the quasi-static Viterbi–Baum–Welch (QSVBW) algorithm to jointly estimate the scheduling path and nonlinear coefficient mappings, using a polynomial parameterization with a coefficient matrix so that mu(k) = . The nonlinear mappings are learned by updating (via a closed-form rule) while the scheduling path is inferred with the Viterbi algorithm, and the HMM parameters are refined with Baum–Welch steps, aided by cross-validation to choose the polynomial degree and PSO for initialization. A numerical study on an LPV-IO model demonstrates accurate recovery of both the scheduling variable and the nonlinear mappings, achieving high best-fit rates for outputs (≈95.7%) and scheduling paths (≈90.3%), indicating the method’s viability for BLPI in static-scheduling LPV contexts. The work provides a practical pathway to blind identification in nonlinear LPV systems and discusses avenues for extending to dynamic scheduling dependencies and MIMO settings, where observation models remain Gaussian and tractable.

Abstract

Blind System Identification (BSI) is used to extract a system model whenever input data is not attainable. Therefore, the input data and system model should be estimated simultaneously. Because of nonlinearities in a large number of systems, BSI problem is usually challenging to solve. In this paper, an innovative solution is proposed to deal with the BSI problem in nonlinear systems using the properties of the Linear Parameter-Varying (LPV) systems and Hidden Markov Models (HMM). More specifically, assuming scheduling variable is not measurable, the dynamic of the LPV system is approximated. To solve the BSI problem in this context, LPV structure is modeled as an HMM network and a modified Quasi-Static combination of Viterbi and Baum-Welch algorithms (QSVBW) is proposed to estimate the nonlinear mappings and scheduling variable signal. The applicability and the performance of the suggested QSVBW algorithm has been justified by numerical studies.
Paper Structure (14 sections, 35 equations, 4 figures)

This paper contains 14 sections, 35 equations, 4 figures.

Table of Contents

  1. INTRODUCTION
  2. Preliminaries
  3. LPV-IO model
  4. Discrete--Time HMM
  5. Main
  6. The Proposed QSVBW Algorithm
  7. Use cross-validation to determine the order of polynomial models that approximate the coefficients of gis in an LPV static model.
  8. Determine the initial conditions:
  9. Estimate the best sequence of the scheduling variable using the Viterbi algorithm \ref{['eqn23']}.
  10. Update the parameters of the model:
  11. Stop condition
  12. Numerical Study
  13. Discussion
  14. Conclusion

Figures (4)

  • Figure 1: An HMM architecture of an LPV--IO model with static dependency on the scheduling variable
  • Figure 2: Flowchart of the QSVBW algorithm
  • Figure 3: True output signal: $y$ (solid-blue), estimated output signal: $\hat{y}$ (dashed-black)
  • Figure 4: True scheduling variable: $p$ (solid-blue), estimated scheduling variable: $\hat{p}$ (dashed-black)