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Observability for Nonlinear Systems: Connecting Variational Dynamics, Lyapunov Exponents, and Empirical Gramians

Mohamad H. Kazma, Ahmad F. Taha

TL;DR

This work advances nonlinear observability by introducing the Variational Gramian (Var-Gram), a discrete-time, variational-dynamics-based alternative to the Empirical Gramian (Empr-Gram) that remains equivalent for local observability but is computationally more efficient. It establishes a direct link between the Var-Gram and Lyapunov exponents, showing that the log-determinant of the Var-Gram relates to the Lyapunov spectrum, and derives a local observability condition via the spectral radius. The Var-Gram also supports a scalable sensor node selection framework, proving modularity of the Gramian with respect to sensor placement and submodularity of the log-determinant objective, enabling greedy algorithms with performance guarantees. Numerical case studies on nonlinear combustion networks validate the equivalence to Empr-Gram, demonstrate observability conditions via Lyapunov exponents, and illustrate the practicality and scalability of the SNS approach for nonlinear systems. The work lays a foundation for data-driven observability assessment and sensor design in nonlinear networks, with potential extensions to stochastic settings and systems with inputs.

Abstract

Observability quantification is a key problem in dynamic network sciences. While it has been thoroughly studied for linear systems, observability quantification for nonlinear networks is less intuitive and more cumbersome. One common approach to quantify observability for nonlinear systems is via the Empirical Gramian (Empr-Gram) -- a generalized form of the Gramian of linear systems. In this paper, we produce three new results. First, we establish that a variational form of discrete-time autonomous nonlinear systems yields a so-called Variational Gramian (Var-Gram) that is equivalent to the classic Empr-Gram; the former being easier to compute than the latter. Via Lyapunov exponents derived from Lyapunov's direct method, the paper's second result derives connections between existing observability measures and Var-Gram. The third result demonstrates the applicability of these new notions for sensor selection/placement in nonlinear systems. Numerical case studies demonstrate these three developments and their merits.

Observability for Nonlinear Systems: Connecting Variational Dynamics, Lyapunov Exponents, and Empirical Gramians

TL;DR

This work advances nonlinear observability by introducing the Variational Gramian (Var-Gram), a discrete-time, variational-dynamics-based alternative to the Empirical Gramian (Empr-Gram) that remains equivalent for local observability but is computationally more efficient. It establishes a direct link between the Var-Gram and Lyapunov exponents, showing that the log-determinant of the Var-Gram relates to the Lyapunov spectrum, and derives a local observability condition via the spectral radius. The Var-Gram also supports a scalable sensor node selection framework, proving modularity of the Gramian with respect to sensor placement and submodularity of the log-determinant objective, enabling greedy algorithms with performance guarantees. Numerical case studies on nonlinear combustion networks validate the equivalence to Empr-Gram, demonstrate observability conditions via Lyapunov exponents, and illustrate the practicality and scalability of the SNS approach for nonlinear systems. The work lays a foundation for data-driven observability assessment and sensor design in nonlinear networks, with potential extensions to stochastic settings and systems with inputs.

Abstract

Observability quantification is a key problem in dynamic network sciences. While it has been thoroughly studied for linear systems, observability quantification for nonlinear networks is less intuitive and more cumbersome. One common approach to quantify observability for nonlinear systems is via the Empirical Gramian (Empr-Gram) -- a generalized form of the Gramian of linear systems. In this paper, we produce three new results. First, we establish that a variational form of discrete-time autonomous nonlinear systems yields a so-called Variational Gramian (Var-Gram) that is equivalent to the classic Empr-Gram; the former being easier to compute than the latter. Via Lyapunov exponents derived from Lyapunov's direct method, the paper's second result derives connections between existing observability measures and Var-Gram. The third result demonstrates the applicability of these new notions for sensor selection/placement in nonlinear systems. Numerical case studies demonstrate these three developments and their merits.
Paper Structure (17 sections, 11 theorems, 50 equations, 6 figures, 1 table)

This paper contains 17 sections, 11 theorems, 50 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction and Contributions
  2. Preliminaries and Definitions
  3. Observability of nonlinear systems
  4. Flow maps and output mapping
  5. Distinguishability and local observability
  6. Notion of uniform observability
  7. Empirical observability Gramian
  8. A Variational Approach for Quantifying Nonlinear Observability
  9. Observability Conditions & LEs
  10. Application of Observability-Based SNS in Nonlinear Networks
  11. Numerical Case Studies
  12. Observability of a combustion reaction network
  13. Sensor node selection in nonlinear networks
  14. Summary, Limitation and Future Directions
  15. Runge-Kutta Discrete-time Model
  16. ...and 2 more sections

Key Result

Theorem II.1

Consider the discrete-time nonlinear system (2) and assume it satisfies Definition def:Hermkern (distinguishability). The system is then locally observable in the set $\mathcal{X}_0$ if there exists a finite horizon $\mathrm{N}_x < \mathrm{N}$ such that the Jacobian $\tfrac{\partial\boldsymbol{\xi}(

Figures (6)

  • Figure 1: Illustration of nonlinear observability quantification and the corresponding optimal sensor selection. Empirical and variational Gramians (Sections \ref{['sec:Empr']}, \ref{['sec:ObsGram']}) are shown to be equivalent (Theorem \ref{['theo:Equivelence']}). The Var-Gram is connected to LEs and observability (Section \ref{['sec:LyapExp']}). The Var-Gram results in a submodular sensor selection problem $\mathbf{P1}$ (Section \ref{['sec:SNS1']}), illustrated through case studies (Section \ref{['sec:casestudies']}).
  • Figure 2: Mapping of the Empr-Gram (left) and Var-Gram (right). The square colors indicate the strength and direction of the relations between the variables. Color contrast represents the strength of the relation, while the color itself indicates the sign of the interrelations: red for positive and blue for negative correlation between state variables. Notice that along the diagonal, values are always positive.
  • Figure 3: LEs computed for the $\mathrm{H}_2\mathrm{O}_2$ reaction network.
  • Figure 4: Optimally selected subset of sensor nodes for the $\mathrm{H}_2\mathrm{O}_2$ combustion network under sensor ratios $r=[2,3,4,5]/9$. Sub-figures (a)-(d) depict the sequential order of sensor selections and the corresponding nodes observed. The normalized observability relation between state variables is indicated by the edge color contrast for each sensor ratio. The color intensity represents the strength of observability.
  • Figure 5: Normalized observability relation between state variables based on $(5/9\approx0.7)$ sensor node ratio (left) and full node sensor ratio (right) for the sensed nodes within the nonlinear combustion network. For left figure, sensed nodes are optimally chosen as $\{1,\;2\;,4\;,6\;,9\}$.
  • ...and 1 more figures

Theorems & Definitions (18)

  • Definition II.1: distinguishability Hermann1977
  • Definition II.2: local observability Hermann1977
  • Definition II.3: uniform observability Hanba2009
  • Theorem II.1: Hanba2009, Th. 7
  • Proposition III.1
  • Remark III.1
  • Proposition III.2
  • Corollary III.1
  • Theorem III.1
  • Remark IV.1
  • ...and 8 more