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Inertial-Based LQG Control: A New Look at Inverted Pendulum Stabilization

Daniel Engelsman, Itzik Klein

TL;DR

This work introduces an inertial-aided LQG framework (A-IPoC) for stabilizing an inverted pendulum on a cart under sensor-limited conditions by leveraging differential flatness to incorporate higher-order dynamics. By augmenting the state with jerk-related terms and deriving an acceleration-predictive model, the method improves state extrapolation and enhances observer-controller coupling, yielding a 27%–39% expansion of the stability region and 10%–15% reduction in crash rates relative to a conventional IPoC. Stability boundaries are estimated through a Monte Carlo approach that maps the basin of attraction under varying update frequencies, saturation, and disturbance scenarios. The results indicate meaningful gains in transient performance and robustness, with an open-source implementation available for reproducibility and wider adoption in dynamically stable platforms like Segways and bipedal systems.

Abstract

Linear quadratic Gaussian (LQG) control is a well-established method for optimal control through state estimation, particularly in stabilizing an inverted pendulum on a cart. In standard laboratory setups, sensor redundancy enables direct measurement of configuration variables using displacement sensors and rotary encoders. However, in outdoor environments, dynamically stable mobile platforms-such as Segways, hoverboards, and bipedal robots-often have limited sensor availability, restricting state estimation primarily to attitude stabilization. Since the tilt angle cannot be directly measured, it is typically estimated through sensor fusion, increasing reliance on inertial sensors and necessitating a lightweight, self-contained perception module. Prior research has not incorporated accelerometer data into the LQG framework for stabilizing pendulum-like systems, as jerk states are not explicitly modeled in the Newton-Euler formalism. In this paper, we address this gap by leveraging local differential flatness to incorporate higher-order dynamics into the system model. This refinement enhances state estimation, enabling a more robust LQG controller that predicts accelerations for dynamically stable mobile platforms.

Inertial-Based LQG Control: A New Look at Inverted Pendulum Stabilization

TL;DR

This work introduces an inertial-aided LQG framework (A-IPoC) for stabilizing an inverted pendulum on a cart under sensor-limited conditions by leveraging differential flatness to incorporate higher-order dynamics. By augmenting the state with jerk-related terms and deriving an acceleration-predictive model, the method improves state extrapolation and enhances observer-controller coupling, yielding a 27%–39% expansion of the stability region and 10%–15% reduction in crash rates relative to a conventional IPoC. Stability boundaries are estimated through a Monte Carlo approach that maps the basin of attraction under varying update frequencies, saturation, and disturbance scenarios. The results indicate meaningful gains in transient performance and robustness, with an open-source implementation available for reproducibility and wider adoption in dynamically stable platforms like Segways and bipedal systems.

Abstract

Linear quadratic Gaussian (LQG) control is a well-established method for optimal control through state estimation, particularly in stabilizing an inverted pendulum on a cart. In standard laboratory setups, sensor redundancy enables direct measurement of configuration variables using displacement sensors and rotary encoders. However, in outdoor environments, dynamically stable mobile platforms-such as Segways, hoverboards, and bipedal robots-often have limited sensor availability, restricting state estimation primarily to attitude stabilization. Since the tilt angle cannot be directly measured, it is typically estimated through sensor fusion, increasing reliance on inertial sensors and necessitating a lightweight, self-contained perception module. Prior research has not incorporated accelerometer data into the LQG framework for stabilizing pendulum-like systems, as jerk states are not explicitly modeled in the Newton-Euler formalism. In this paper, we address this gap by leveraging local differential flatness to incorporate higher-order dynamics into the system model. This refinement enhances state estimation, enabling a more robust LQG controller that predicts accelerations for dynamically stable mobile platforms.

Paper Structure

This paper contains 14 sections, 1 theorem, 53 equations, 10 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Theoretical Background
  3. Observer-based controller
  4. Mathematical model
  5. Proposed Methodology
  6. Differential flatness
  7. Linearized Dynamics
  8. Stability region
  9. Results and Analysis
  10. Time-domain performance metrics
  11. Performance analysis
  12. Performance comparison
  13. Stability comparison
  14. Conclusion

Key Result

Proposition 1

The controlled IPoC system defined in eq:sys_1, $\dot{\boldsymbol{x}}(t) = \boldsymbol{f}(\boldsymbol{x}(t), u(t))$, demonstrates differential flatness.

Figures (10)

  • Figure 1: LQG closed-loop dynamics diagram with a color-coded representation: true state (blue), measured state (orange), estimated state (green), and stochastic disturbances (brown).
  • Figure 2: A simplified mobile robot with an inverted pendulum system. Free body diagrams are commonly cited in wang2010designeide2011lqgrazmjooy2014comparisonbrunton2016koopman.
  • Figure 3: Stabilization task: From initial state to target position.
  • Figure 4: LQG response for $\rho =1$. Left: Linear states; Right: Angular states. Blue: True states; Green: Estimated states.
  • Figure 5: LQG response for $\rho = 0.2$. Left: Linear states; Right: Angular states. Blue: True states; Green: Estimated states.
  • ...and 5 more figures

Theorems & Definitions (4)

  • Definition 1
  • Proposition 1
  • proof
  • Remark 1