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

Mohammad Khajenejad

TL;DR

Our paper tackles robust stabilization and disturbance attenuation for bounded Jacobian nonlinear discrete-time systems with nonlinear observations under state and measurement noise. We propose to stabilize a higher-dimensional interval observer and design a dynamic controller to tighten the closed-loop state bounds, exploiting a separation principle between observer and controller. The approach yields tractable LMIs/SDPs for gain synthesis and demonstrates superior performance over a static strategy in simulations. This framework enables reliable operation of nonlinear DT systems with uncertainties in engineering domains and robotics.

Abstract

This paper presents an optimal dynamic control framework for bounded Jacobian nonlinear discrete-time (DT) systems with nonlinear observations affected by both state and process noise. Rather than directly stabilizing the uncertain system, we focus on stabilizing an interval observer in a higher dimensional space, whose states bound the true system states. Our nonlinear dynamic control method introduces added flexibility over traditional static and linear approaches, effectively compensating for system nonlinearities and enabling potentially tighter closed-loop intervals. Additionally, we establish a separation principle that allows for the design of observer and control gains. We further derive tractable matrix inequalities to ensure system stability in the closed-loop configuration. The simulation results show that the proposed dynamic control approach significantly outperforms a static counterpart method.

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

TL;DR

Our paper tackles robust stabilization and disturbance attenuation for bounded Jacobian nonlinear discrete-time systems with nonlinear observations under state and measurement noise. We propose to stabilize a higher-dimensional interval observer and design a dynamic controller to tighten the closed-loop state bounds, exploiting a separation principle between observer and controller. The approach yields tractable LMIs/SDPs for gain synthesis and demonstrates superior performance over a static strategy in simulations. This framework enables reliable operation of nonlinear DT systems with uncertainties in engineering domains and robotics.

Abstract

This paper presents an optimal dynamic control framework for bounded Jacobian nonlinear discrete-time (DT) systems with nonlinear observations affected by both state and process noise. Rather than directly stabilizing the uncertain system, we focus on stabilizing an interval observer in a higher dimensional space, whose states bound the true system states. Our nonlinear dynamic control method introduces added flexibility over traditional static and linear approaches, effectively compensating for system nonlinearities and enabling potentially tighter closed-loop intervals. Additionally, we establish a separation principle that allows for the design of observer and control gains. We further derive tractable matrix inequalities to ensure system stability in the closed-loop configuration. The simulation results show that the proposed dynamic control approach significantly outperforms a static counterpart method.
Paper Structure (9 sections, 5 theorems, 35 equations, 2 figures)

This paper contains 9 sections, 5 theorems, 35 equations, 2 figures.

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

9867741 Let $f :\mathcal{X} \subset \mathbb{R}^{n} \to \mathbb{R}^{p}$ and suppose $\forall x \in \mathcal{X}, J_f(x) \in [\underline{J}_f,\overline{J}_f]$, where $\underline{J}_f,\overline{J}_f$ are known matrices in $\mathbb{R}^{p \times n}$. Then, $f$ can be decomposed into a (remainder) affine where $H$ is a matrix in $\mathbb{R}^{p \times n}$, that satisfies the following

Figures (2)

  • Figure 1: Parameters and computed gains for the example system in Section \ref{['sec:example']}.
  • Figure 2: Open-loop states (first plot) and the closed-loop upper and lower framers returned by our proposed dynamic control design, i.e., $\overline{x}^{dy},\underline{x}^{dy}$, as well as the framers returned by the static feedback control approach in khajenejad2024optimalcontrolstatic, i.e., $\overline{x}^{st},\underline{x}^{st}$ (second to sixth plots).

Theorems & Definitions (16)

  • Definition 1: Interval
  • Definition 2: Jacobian Sign-Stability
  • Proposition 1: Mixed-Monotone Decomposition
  • Proposition 2: Tight Decomposition Functions for JSS Mappings 9867741
  • Definition 3: Correct Interval Framers
  • Definition 4: Framer Error
  • Definition 5: Stability and Interval Observer
  • Definition 6: $\mathcal{H}_{\infty}$-Robust & Optimal Interval Observer
  • Proposition 3
  • proof
  • ...and 6 more