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Prescribed-Performance-Aware Hybrid-Gain-Based Robust Controller

Amit Shivam, Kiran Kumari, Fernando A. C. C. Fontes

Abstract

This paper proposes a prescribed performance function aware hybrid gain finite time sliding mode control framework for a class of nonlinear systems subject to matched disturbances. The hybrid gain structure ensures bounded control effort while retaining finite time convergence, and the incorporation of PPFs enables explicit enforcement of transient performance requirements. Theoretical guarantees are first established for first order systems, characterizing finite time convergence, disturbance rejection, and residual bounds. The approach is then extended to second order dynamics, where a sliding manifold is designed using PPF constraints to facilitate controlled shaping of position and velocity transients. Simulation studies illustrate the proposed design under matched peak control conditions. Comparative results for second-order systems demonstrate that, while a well tuned non-PPF hybrid gain controller achieves competitive tracking performance, the PPF-aware formulation strictly enforces prescribed transient constraints and yields consistent reductions of approximately 9 to 12 percent in integral error and control energy metrics without increasing peak actuation effort.

Prescribed-Performance-Aware Hybrid-Gain-Based Robust Controller

Abstract

This paper proposes a prescribed performance function aware hybrid gain finite time sliding mode control framework for a class of nonlinear systems subject to matched disturbances. The hybrid gain structure ensures bounded control effort while retaining finite time convergence, and the incorporation of PPFs enables explicit enforcement of transient performance requirements. Theoretical guarantees are first established for first order systems, characterizing finite time convergence, disturbance rejection, and residual bounds. The approach is then extended to second order dynamics, where a sliding manifold is designed using PPF constraints to facilitate controlled shaping of position and velocity transients. Simulation studies illustrate the proposed design under matched peak control conditions. Comparative results for second-order systems demonstrate that, while a well tuned non-PPF hybrid gain controller achieves competitive tracking performance, the PPF-aware formulation strictly enforces prescribed transient constraints and yields consistent reductions of approximately 9 to 12 percent in integral error and control energy metrics without increasing peak actuation effort.
Paper Structure (18 sections, 4 theorems, 61 equations, 2 figures, 1 table)

This paper contains 18 sections, 4 theorems, 61 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Prescribed-performance and design objective
  4. PPF-Aware Hybrid-Gain Based Controller Design
  5. First-order system
  6. Region A- $W \ge \varepsilon_0$
  7. Region B- $\varepsilon < W \le \varepsilon_0$
  8. Second-order system
  9. Parameter tuning and feasibility guidelines
  10. Choose PPF parameters
  11. Outer-region feasibility
  12. Inner-region feasibility
  13. Explicit tuning for the mixed-power inner gain
  14. Second-order case
  15. Simulation Results
  16. ...and 3 more sections

Key Result

Theorem 1

Consider the first-order system eq: first order under controller eq: control law fos. Suppose $k_0 > \bar{d}_\xi$, then for any initial condition satisfying $|x(0)| < \rho(0)$, the closed loop system ensures

Figures (2)

  • Figure 1: Results for first order system
  • Figure 2: Comparative numerical simulation results

Theorems & Definitions (9)

  • Remark 1: Interpretation and use of Assumption \ref{['ass:xi_bound']}
  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Corollary 1: Gaussian inner gain
  • Theorem 2
  • proof
  • Remark 2