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Velocity-Free Horizontal Position Control of Quadrotor Aircraft via Nonlinear Negative Imaginary Systems Theory

Ahmed G. Ghallab, Ian R. Petersen

Abstract

This paper presents a velocity-free position control strategy for quadrotor unmanned aerial vehicles based on nonlinear negative imaginary (NNI) systems theory. Unlike conventional position control schemes that require velocity measurements or estimation, the proposed approach achieves asymptotic stability using only position feedback. We establish that the quadrotor horizontal position subsystem, when augmented with proportional feedback, exhibits the NNI property with respect to appropriately defined horizontal thrust inputs. A strictly negative imaginary integral resonant controller is then designed for the outer loop, and robust asymptotic stability is guaranteed through satisfaction of explicit sector-bound conditions relating controller and plant parameters. The theoretical framework accommodates model uncertainties and external disturbances while eliminating the need for velocity sensors. Simulation results validate the theoretical predictions and demonstrate effective position tracking performance.

Velocity-Free Horizontal Position Control of Quadrotor Aircraft via Nonlinear Negative Imaginary Systems Theory

Abstract

This paper presents a velocity-free position control strategy for quadrotor unmanned aerial vehicles based on nonlinear negative imaginary (NNI) systems theory. Unlike conventional position control schemes that require velocity measurements or estimation, the proposed approach achieves asymptotic stability using only position feedback. We establish that the quadrotor horizontal position subsystem, when augmented with proportional feedback, exhibits the NNI property with respect to appropriately defined horizontal thrust inputs. A strictly negative imaginary integral resonant controller is then designed for the outer loop, and robust asymptotic stability is guaranteed through satisfaction of explicit sector-bound conditions relating controller and plant parameters. The theoretical framework accommodates model uncertainties and external disturbances while eliminating the need for velocity sensors. Simulation results validate the theoretical predictions and demonstrate effective position tracking performance.

Paper Structure

This paper contains 15 sections, 5 theorems, 49 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries on Negative Imaginary Systems
  3. Linear Negative Imaginary Systems
  4. Nonlinear Negative Imaginary Systems
  5. Robust Stability of Positive Feedback Interconnections
  6. Quadrotor System
  7. Coordinate Frames and Kinematics
  8. Dynamics Model
  9. Control Design and Stability Analysis
  10. Problem Formulation
  11. Inner-Loop Controller: Proportional Feedback
  12. Outer-Loop Controller: Strictly Negative Imaginary Design
  13. Main Stability Result
  14. Simulation Results
  15. Conclusions

Key Result

Lemma 1

Suppose that the system eq:LTI_system with $D=0$ is controllable and observable. Then $G(s)$ is negative imaginary if and only if there exists a positive definite matrix $P=P^T>0$ satisfying such that the storage function $V(x)=\frac{1}{2}x^TPx$ satisfies along system trajectories.

Figures (6)

  • Figure 1: Open-loop interconnection of systems $H_1$ and $H_2$ at steady state.
  • Figure 2: Positive feedback interconnection of an NNI system $H_1$ and a WS-NNI system $H_2$.
  • Figure 3: Quadrotor configuration showing body-fixed frame $\{B\}$ and inertial frame $\{E\}$.
  • Figure 4: Open-loop interconnection of reshaped position subsystem \ref{['eq:horizontal_subsystem']} and controller \ref{['eq:SNI_controller']} at steady state.
  • Figure 5: Block diagram of the proposed velocity-free horizontal position control system. The inner loop feeds back position $\bm{\xi}_h$ through the proportional gain $\boldsymbol{K_p}$; the outer loop feeds back through the SNI controller $C_v(s)$.
  • ...and 1 more figures

Theorems & Definitions (19)

  • Definition 1: lanzon2008
  • Lemma 1: ghallab2018extending
  • Definition 2: lanzon2008
  • Definition 3: ghallab2018extending
  • Remark 1
  • Definition 4
  • Definition 5
  • Remark 2
  • Theorem 1: ghallab2018extending
  • Remark 3
  • ...and 9 more