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On Differential Controllability and Observability Functions

Yu Kawano, Bart Besselink, Jacquelien M. A. Scherpen

TL;DR

The paper investigates differential controllability and observability functions central to differential balancing for nonlinear model reduction. By assuming suitable optimal state feedback, it derives upper bounds linking differential and incremental energy functions and shows the differential observability function can be constructed from its incremental counterpart. It then establishes positive definiteness conditions for the differential energy functions through zero-state detectability and contraction-based convergence, including a duality framework between controllability and observability via differential Lyapunov and Riccati equations. These results generalize linear Gramian concepts to nonlinear settings, providing tools for energy-based analysis and design of reduced-order nonlinear models.

Abstract

Differential balancing theory for nonlinear model reduction relies on differential controllability and observability functions. In this paper, we further investigate them from two different perspectives. First, we establish novel connections between these differential energy functions and their incremental counterparts by assuming the existence of the corresponding optimal state feedback for each controllability function. Specifically, an upper bound on the incremental controllability/observability function is provided by the corresponding differential energy function. Conversely, an upper bound on the differential controllability function can be estimated from the incremental controllability function. Furthermore, the differential observability function can be constructed from the incremental observability function. Second, we explore the positive definiteness of the differential controllability/observability function in the context of controllability/observability and stability.

On Differential Controllability and Observability Functions

TL;DR

The paper investigates differential controllability and observability functions central to differential balancing for nonlinear model reduction. By assuming suitable optimal state feedback, it derives upper bounds linking differential and incremental energy functions and shows the differential observability function can be constructed from its incremental counterpart. It then establishes positive definiteness conditions for the differential energy functions through zero-state detectability and contraction-based convergence, including a duality framework between controllability and observability via differential Lyapunov and Riccati equations. These results generalize linear Gramian concepts to nonlinear settings, providing tools for energy-based analysis and design of reduced-order nonlinear models.

Abstract

Differential balancing theory for nonlinear model reduction relies on differential controllability and observability functions. In this paper, we further investigate them from two different perspectives. First, we establish novel connections between these differential energy functions and their incremental counterparts by assuming the existence of the corresponding optimal state feedback for each controllability function. Specifically, an upper bound on the incremental controllability/observability function is provided by the corresponding differential energy function. Conversely, an upper bound on the differential controllability function can be estimated from the incremental controllability function. Furthermore, the differential observability function can be constructed from the incremental observability function. Second, we explore the positive definiteness of the differential controllability/observability function in the context of controllability/observability and stability.

Paper Structure

This paper contains 19 sections, 7 theorems, 49 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Relations between Differential and Incremental Energy Functions
  4. Incremental Controllability and Observability Functions
  5. Relations between Controllability Functions
  6. Relations between Observability Functions
  7. Positive Definiteness of Differential Energy Functions
  8. Differential Observability Functions
  9. Differential Controllability Functions
  10. Example
  11. Conclusion
  12. Proofs of Theorems in Section \ref{['sec:energy']}
  13. Proof of Theorem \ref{['thm:di_con']}
  14. Proof of Theorem \ref{['thm:id_con']}
  15. Proof of Theorem \ref{['thm:di_ob']}
  16. ...and 4 more sections

Key Result

Theorem III.1

For a nonlinear system eq:sys, suppose that there exist an open convex subset $W \subset {\mathbb R}^n$ and $k: W \to {\mathbb R}^m$ of class $C^2$ such that Then, the incremental controllability function eq:icon_func is upper bounded as for any $(x_0, x'_0) \in W \times W$.

Figures (3)

  • Figure 1: Time response of \ref{['eq:sys_z']} with \ref{['eq:ex']}
  • Figure 2: Time response of \ref{['eq:dsys']} with \ref{['eq:ex']} when $(u(t), \delta u(t))\equiv (0,0)$
  • Figure 3: $|Q(x)|$

Theorems & Definitions (18)

  • Theorem III.1
  • proof
  • Theorem III.2
  • proof
  • Theorem III.3
  • proof
  • Theorem III.4
  • proof
  • Theorem IV.1
  • proof
  • ...and 8 more