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Robust Energy Shaping Control of an Underactuated Inverted Pendulum

M. Reza J. Harandi, Mehrzad Namvar

TL;DR

The paper tackles the challenge of stabilizing underactuated systems by applying interconnection and damping assignment passivity-based control (IDA-PBC) to the rotary inverted pendulum (RIP). It delivers a precise, analytical solution to the kinetic- and potential-energy matching PDEs, enabling exact total energy shaping with a designed $M_d(q_2)$ and $V_d(q)$ and a region of attraction that can be enlarged via controller parameters. A novel robust term is incorporated to reject a class of nonintegrable matched disturbances, with a Lyapunov-based proof guaranteeing closed-loop stability. Numerical simulations demonstrate upright stabilization and robust disturbance rejection, highlighting practical viability and providing a foundation for future hardware implementation and port-Hamiltonian system extensions.

Abstract

Although the stabilization of underactuated systems remains a challenging problem, the total energy shaping approach provides a general framework for addressing this objective. However, the practical implementation of this method is hindered by the need to analytically solve a set of partial differential equations (PDEs), which constitutes a major obstacle. In this paper, a rotary inverted pendulum system is considered, and an interconnection and damping assignment passivity-based control (IDA-PBC) scheme is developed by deriving concise analytical solutions to the kinetic and potential energy PDEs. Furthermore, a novel robust term is incorporated into the control law to compensate for a specific class of disturbances that has not been addressed within the existing IDA-PBC literature. The effectiveness of the proposed method is validated through numerical simulations, demonstrating satisfactory control performance.

Robust Energy Shaping Control of an Underactuated Inverted Pendulum

TL;DR

The paper tackles the challenge of stabilizing underactuated systems by applying interconnection and damping assignment passivity-based control (IDA-PBC) to the rotary inverted pendulum (RIP). It delivers a precise, analytical solution to the kinetic- and potential-energy matching PDEs, enabling exact total energy shaping with a designed and and a region of attraction that can be enlarged via controller parameters. A novel robust term is incorporated to reject a class of nonintegrable matched disturbances, with a Lyapunov-based proof guaranteeing closed-loop stability. Numerical simulations demonstrate upright stabilization and robust disturbance rejection, highlighting practical viability and providing a foundation for future hardware implementation and port-Hamiltonian system extensions.

Abstract

Although the stabilization of underactuated systems remains a challenging problem, the total energy shaping approach provides a general framework for addressing this objective. However, the practical implementation of this method is hindered by the need to analytically solve a set of partial differential equations (PDEs), which constitutes a major obstacle. In this paper, a rotary inverted pendulum system is considered, and an interconnection and damping assignment passivity-based control (IDA-PBC) scheme is developed by deriving concise analytical solutions to the kinetic and potential energy PDEs. Furthermore, a novel robust term is incorporated into the control law to compensate for a specific class of disturbances that has not been addressed within the existing IDA-PBC literature. The effectiveness of the proposed method is validated through numerical simulations, demonstrating satisfactory control performance.
Paper Structure (6 sections, 2 theorems, 43 equations, 4 figures)

This paper contains 6 sections, 2 theorems, 43 equations, 4 figures.

Table of Contents

  1. INTRODUCTION
  2. IDA-PBC framework for UaRs
  3. Total Energy Shaping Controller Design for the RIP
  4. Disturbance Rejection
  5. Simulation Results
  6. Conclusion

Key Result

Theorem 1

Consider the rotary inverted pendulum whose dynamics are described by 1 and dyn. To stabilize the equilibrium point $q^\ast$, a total energy shaping controller of the form 4 is applied. The controller parameters are chosen as follows where The desired potential energy function is given by Moreover, the components of $\alpha(q_2)$ are defined as Here, $\psi_{4_0}$, $k_1$, $k_2$, and $\kappa$ ar

Figures (4)

  • Figure 1: Schematic of the rotary inverted pendulum.
  • Figure 2: Simulation results of Theorem \ref{['th1']}. The configuration variables converge to $q^\ast$ that means the upright position is stabilized.
  • Figure 3: Simulation results of Theorem \ref{['th1']} in the presence of external disturbance. A steady-state error remains in the configuration variables.
  • Figure 4: Simulation results of Lemma \ref{['l1']} in the presence of external disturbance. Configuration variables converge to zero using the proposed robust controller.

Theorems & Definitions (5)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Lemma 1
  • Remark 3