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Synthesis of Functional Unknown Input Observers for LTV and MIMO LTI Systems with Arbitrary Relative Degree

Alexey A. Margun, Alexey A. Bobtsov, Denis V. Efimov, Alexandr D. Panin, Mariia O. Rassolova

TL;DR

The paper advances functional unknown input observers for systems with unmeasured disturbances by proposing two complementary designs: a GPEBO-based UIO for a class of LTV plants in chain-of-integrators form, and a functional UIO for MIMO LTI systems with arbitrary relative degree. The first method yields asymptotic convergence under a stable $R(t)$, while the second provides a systematic procedure to compute a functional matrix $Q$ and guarantees exponential convergence under detectable $(A,C)$ and rank-based conditions, with explicit kernel-based computation of $Q$. The results are extended to nonlinear systems, showing how nonlinearities treated as unknown inputs can be accommodated with implementable observers through auxiliary variables and derivative-free formulations. Simulations on SISO LTV and MIMO plants illustrate effective estimation of state functionals and partial state vectors under disturbances, highlighting the practical impact for control, regulation, and robust state estimation in complex dynamical settings.

Abstract

This article focuses on the development of functional unknown input observers for systems with arbitrary relative degree. Two distinct approaches are presented to address this challenge. The first approach is tailored to a class of time-varying systems expressed in a canonical controllable form. This method leverages the Generalized Parameter Estimation-Based Observer framework. The article derives the conditions for applicability of this solution and outlines the limitations on the number of estimable state variables. The second approach targets multi-input multi-output systems. In contrast to existing methods, the proposed solution is applicable to systems with arbitrary relative degree, significantly broadening its scope of application. The theoretical results are validated through simulation studies, which demonstrate the effectiveness of the proposed observers.

Synthesis of Functional Unknown Input Observers for LTV and MIMO LTI Systems with Arbitrary Relative Degree

TL;DR

The paper advances functional unknown input observers for systems with unmeasured disturbances by proposing two complementary designs: a GPEBO-based UIO for a class of LTV plants in chain-of-integrators form, and a functional UIO for MIMO LTI systems with arbitrary relative degree. The first method yields asymptotic convergence under a stable , while the second provides a systematic procedure to compute a functional matrix and guarantees exponential convergence under detectable and rank-based conditions, with explicit kernel-based computation of . The results are extended to nonlinear systems, showing how nonlinearities treated as unknown inputs can be accommodated with implementable observers through auxiliary variables and derivative-free formulations. Simulations on SISO LTV and MIMO plants illustrate effective estimation of state functionals and partial state vectors under disturbances, highlighting the practical impact for control, regulation, and robust state estimation in complex dynamical settings.

Abstract

This article focuses on the development of functional unknown input observers for systems with arbitrary relative degree. Two distinct approaches are presented to address this challenge. The first approach is tailored to a class of time-varying systems expressed in a canonical controllable form. This method leverages the Generalized Parameter Estimation-Based Observer framework. The article derives the conditions for applicability of this solution and outlines the limitations on the number of estimable state variables. The second approach targets multi-input multi-output systems. In contrast to existing methods, the proposed solution is applicable to systems with arbitrary relative degree, significantly broadening its scope of application. The theoretical results are validated through simulation studies, which demonstrate the effectiveness of the proposed observers.

Paper Structure

This paper contains 13 sections, 61 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Functional unknown input observer synthesis for a class of LTV systems
  3. Problem statement
  4. Functional unknown input observer synthesis
  5. Functional unknown input observer synthesis for MIMO systems with arbitrary relative degree
  6. Problem statement
  7. Unknown input observer synthesis for plants with unit relative degree
  8. Functional unknown input observer synthesis for plants with arbitrary relative degree
  9. Unknown input observer synthesis for a class of nonlinear systems
  10. Simulation results
  11. Simulation of the observer for LTV system
  12. Simulation of the observer for MIMO system
  13. Conclusion

Figures (2)

  • Figure 1: Observation errors of the functional observer for SISO system
  • Figure 2: Observation errors of the functional observer for MIMO system.