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Optimal Feedback Stabilizing Control of Bounded Jacobian Discrete-Time Systems via Interval Observers

Mohammad Khajenejad

Abstract

This paper addresses optimal feedback stabilizing control for bounded Jacobian nonlinear discrete-time (DT) systems with nonlinear observations, affected by state and process noise. Instead of directly stabilizing the uncertain system, we propose stabilizing a higher-dimensional interval observer whose states enclose the true system states. Our nonlinear control approach introduces additional flexibility compared to linear methods, compensating for system nonlinearities and allowing potentially tighter closed-loop intervals. We also establish a separation principle, enabling independent design of observer and control gains, and derive tractable linear matrix inequalities, resulting in a stable closed-loop system.

Optimal Feedback Stabilizing Control of Bounded Jacobian Discrete-Time Systems via Interval Observers

Abstract

This paper addresses optimal feedback stabilizing control for bounded Jacobian nonlinear discrete-time (DT) systems with nonlinear observations, affected by state and process noise. Instead of directly stabilizing the uncertain system, we propose stabilizing a higher-dimensional interval observer whose states enclose the true system states. Our nonlinear control approach introduces additional flexibility compared to linear methods, compensating for system nonlinearities and allowing potentially tighter closed-loop intervals. We also establish a separation principle, enabling independent design of observer and control gains, and derive tractable linear matrix inequalities, resulting in a stable closed-loop system.

Paper Structure

This paper contains 9 sections, 6 theorems, 33 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem Formulation
  4. Proposed Control Design
  5. Interval Framers
  6. Closed-Loop System and Separation Principle
  7. Control Design
  8. Illustrative Example
  9. Conclusion and Future Work

Key Result

Proposition 1

efimov2013interval Let $A \in \mathbb{R}^{n \times p}$ and $\underline{x} \leq x \leq \overline{x} \in \mathbb{R}^n$. Then ,$A^+\underline{x}-A^{-}\overline{x} \leq Ax \leq A^+\overline{x}-A^{-}\underline{x}$. As a corollary, if $A$ is non-negative, $A\underline{x} \leq Ax \leq A\overline{x}$.

Figures (1)

  • Figure 1: Open-loop states (first plot), as well as the closed-loop upper and lower framers and actual states (second to sixth plots), returned by our proposed control design.

Theorems & Definitions (14)

  • Definition 1: Interval, Maximal and Minimal Elements, Interval Width
  • Proposition 1
  • Definition 2: Jacobian Sign-Stability
  • Proposition 2: Mixed-Monotone Decomposition
  • Proposition 3: Tight and Tractable Decomposition Functions for JSS Mappings 9867741
  • Definition 3: Correct Interval Framers
  • Definition 4: Framer Error
  • Definition 5: Stability and Interval Observer
  • Proposition 4
  • proof
  • ...and 4 more