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Unifying Controller Design for Stabilizing Nonlinear Systems with Norm-Bounded Control Inputs

Ming Li, Zhiyong Sun, Siep Weiland

Abstract

This paper revisits a classical challenge in the design of stabilizing controllers for nonlinear systems with a norm-bounded input constraint. By extending Lin-Sontag's universal formula and introducing a generic (state-dependent) scaling term, a unifying controller design method is proposed. The incorporation of this generic scaling term gives a unified controller and enables the derivation of alternative universal formulas with various favorable properties, which makes it suitable for tailored control designs to meet specific requirements and provides versatility across different control scenarios. Additionally, we present a constructive approach to determine the optimal scaling term, leading to an explicit solution to an optimization problem, named optimization-based universal formula. The resulting controller ensures asymptotic stability, satisfies a norm-bounded input constraint, and optimizes a predefined cost function. Finally, the essential properties of the unified controllers are analyzed, including smoothness, continuity at the origin, stability margin, and inverse optimality. Simulations validate the approach, showcasing its effectiveness in addressing a challenging stabilizing control problem of a nonlinear system.

Unifying Controller Design for Stabilizing Nonlinear Systems with Norm-Bounded Control Inputs

Abstract

This paper revisits a classical challenge in the design of stabilizing controllers for nonlinear systems with a norm-bounded input constraint. By extending Lin-Sontag's universal formula and introducing a generic (state-dependent) scaling term, a unifying controller design method is proposed. The incorporation of this generic scaling term gives a unified controller and enables the derivation of alternative universal formulas with various favorable properties, which makes it suitable for tailored control designs to meet specific requirements and provides versatility across different control scenarios. Additionally, we present a constructive approach to determine the optimal scaling term, leading to an explicit solution to an optimization problem, named optimization-based universal formula. The resulting controller ensures asymptotic stability, satisfies a norm-bounded input constraint, and optimizes a predefined cost function. Finally, the essential properties of the unified controllers are analyzed, including smoothness, continuity at the origin, stability margin, and inverse optimality. Simulations validate the approach, showcasing its effectiveness in addressing a challenging stabilizing control problem of a nonlinear system.
Paper Structure (15 sections, 12 theorems, 49 equations, 3 figures)

This paper contains 15 sections, 12 theorems, 49 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Universal Formulas for Stabilizing Nonlinear Systems
  3. PMN Controller and Sontag's Universal Formula
  4. Lin-Sontag's Universal Formula
  5. A Unifying Controller Design Method
  6. A Unified Controller
  7. A Constructive Approach
  8. Property Analysis of the Unified Controller
  9. Smoothness and Continuity at the Origin
  10. The Unified Controller
  11. The Optimization-Based Universal Formula
  12. Stability Margin
  13. Inverse Optimality
  14. Simulations
  15. Conclusions and Future Research Directions

Key Result

Lemma 2.2

The solution of QP_clf can be explicitly expressed as where the superscript "$^{\star}$" indicates that $\mathbf{u}_{\mathrm{PMN}}^{\star}(\mathbf{x})$ is the optimal solution of QP_clf, and $\mathbf{m}_{\mathrm{PMN}}^{\star}(\mathbf{x})=-\frac{a(\mathbf{x})+\sigma(\mathbf{x})}{\|\mathbf{b}(\mathbf{x})\|^{2}} \mathbf{b}(\mathbf{x})^{\top}$.

Figures (3)

  • Figure 1: A graphical exhibition illustrating the construction of our proposed universal formula in an $\mathbf{x}$-compatible scenario: the whole area indicates a 2-D control space $\mathbf{u}\in\mathbb{R}^{2}$; the admissible control input set $\mathcal{S}_{V}(\mathbf{x})$ and $\mathcal{B}$ are represented by the green and lavender areas, respectively; the magenta dashed segment (relating to $\max\left(-\frac{a(\mathbf{x})}{\sigma(\mathbf{x})},0\right)\leq\kappa\leq\frac{\|\mathbf{b}(\mathbf{x})\|-a(\mathbf{x})}{\sigma(\mathbf{x})}$) represents solutions of PMN controllers that simultaneously satisfy the constraint \ref{['Scaling_Condition']} and the input constraint \ref{['Control_Input_Limit']}. In contrast, the blue ray (corresponding to $\kappa>\frac{\|\mathbf{b}(\mathbf{x})\|-a(\mathbf{x})}{\sigma(\mathbf{x})}$) represents solutions that satisfy the constraint \ref{['Scaling_Condition']} but not satisfying the input constraint \ref{['Control_Input_Limit']}; Sontag's universal formula is the blue point on the blue ray; Lin-Sontag's universal formula is the magenta point on the magenta dashed segment; the green dashed line represents a hyperplane $\mathbf{H}_{\mathrm{CLF}}$.
  • Figure 2: Stabilizing performances and control input behaviors for different solutions.
  • Figure 3: A comparison of the cost of different solutions.

Theorems & Definitions (31)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.8
  • proof
  • ...and 21 more