Single Inverted Pendulum LQR Control Simulation with Reduced-Order State Observer Design

This technical document explores two methodological approaches for designing reduced-order state observers within the framework of single inverted pendulum LQR control simulations. Through systematic implementation of these observer design techniques, we achieve enhanced state estimation accuracy, thereby enabling more precise control performance for the inverted pendulum system.

The first methodological approach employs a linear observer design, which provides a computationally efficient solution for state estimation. This method utilizes linear matrix equations and eigenvalue placement techniques to construct the observer gain matrix. Implementation typically involves solving the algebraic Riccati equation using MATLAB's lqr() function or equivalent numerical solvers, with observer dynamics governed by the standard state-space representation dx_hat/dt = A*x_hat + B*u + L*(y - C*x_hat).

The second approach investigates nonlinear observer design, which incorporates system nonlinearities through extended Kalman filter (EKF) formulations or sliding mode observer techniques. This method requires Jacobian matrix computations and nonlinear stability analysis, often implemented using recursive prediction-correction algorithms. While requiring more sophisticated coding structures involving Euler integration or Runge-Kutta methods, this approach delivers superior estimation accuracy under large-angle pendulum deviations.

To demonstrate practical implementation, we provide comprehensive numerical examples for each observer design methodology. The simulations include MATLAB/Simulink code snippets showcasing observer initialization parameters, state update routines, and performance comparison metrics. Through quantitative analysis of estimation error convergence rates and control stability margins, we evaluate the relative advantages and computational trade-offs of each observer design strategy in real-time control applications.

This document serves as a comprehensive technical resource for implementing reduced-order state observers in single inverted pendulum systems, providing both theoretical foundations and practical coding guidelines for optimal LQR control implementation.