LQR-Based Optimal Control of Inverted Pendulum System with State Estimation and Stability Analysis

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Year : 2026 | Volume : 04 | 01 | Page :
    By

    Amit Pandhare,

  • Aditya Kumbhar,

  • Vaibhav Godase,

  1. UG Students, Department of Electronics and Telecommunication Engineering, SKN Sinhgad College of Engineering, Pandharpur, Maharashtra, India
  2. UG Students, Department of Electronics and Telecommunication Engineering, SKN Sinhgad College of Engineering, Pandharpur, Maharashtra, India
  3. Assistant Professor, Department of Electronics and Telecommunication Engineering, SKN Sinhgad College of Engineering, Pandharpur, Maharashtra, India

Abstract

The inverted pendulum on a cart is a canonical benchmark problem in control systems engineering, capturing the essential challenges of stabilizing an inherently unstable, underactuated, and nonlinear plant. Classical Proportional-Integral-Derivative (PID) controllers, while widely employed in industrial practice, exhibit fundamental performance limitations when applied to such systems, primarily due to their inability to account for multivariable coupling, process noise, and the absence of a systematic optimization framework. This paper presents the complete design, theoretical analysis, and simulation validation of a Linear Quadratic Regulator (LQR) combined with a Kalman filter-based state estimator for optimal stabilization and state reconstruction of a cart-pole inverted pendulum. The system dynamics are derived using the Lagrangian formulation and linearized about the unstable upright equilibrium to enable the application of linear optimal control theory. The LQR controller is synthesized by minimizing a quadratic cost functional that balances state regulation against control effort, while the Kalman filter reconstructs unmeasured state variables from noisy sensor measurements in an optimal minimum-variance sense. Closed-loop stability is rigorously verified through eigenvalue analysis of the controlled system and confirmed via Lyapunov’s direct method. Numerical simulations conducted in MATLAB/Simulink with realistic physical parameters demonstrate that the proposed approach achieves a settling time of 1.82 seconds, a peak overshoot of 3.7 percent, and a control effort of 18.43 N·s—representing improvements of 48.6 percent, 83.3 percent, and 47.1 percent, respectively, over a well-tuned PID baseline. Results confirm superior transient performance, robust behavior under parameter perturbations of plus or minus 20 percent, and accurate state estimation convergence, collectively validating the efficacy of the LQR-Kalman architecture for practical implementation on unstable mechanical systems.

Keywords: LQR control; State estimation; Kalman filter; Optimal control; Inverted pendulum; Stability analysis.

How to cite this article:
Amit Pandhare, Aditya Kumbhar, Vaibhav Godase. LQR-Based Optimal Control of Inverted Pendulum System with State Estimation and Stability Analysis. International Journal of Advanced Control and System Engineering. 2026; 04(01):-.
How to cite this URL:
Amit Pandhare, Aditya Kumbhar, Vaibhav Godase. LQR-Based Optimal Control of Inverted Pendulum System with State Estimation and Stability Analysis. International Journal of Advanced Control and System Engineering. 2026; 04(01):-. Available from: https://journals.stmjournals.com/ijacse/article=2026/view=239789


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Ahead of Print Subscription Review Article
Volume 04
01
Received 26/02/2026
Accepted 28/02/2026
Published 07/04/2026
Publication Time 40 Days
18

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