A Quadratic Control Framework for Dynamic Systems
Igor Ladnik
TL;DR
The paper presents a unified quadratic optimal-control framework for linear and nonlinear discrete-time systems that targets trajectory tracking by minimizing a quadratic cost in deviations from desired trajectories. It combines exact LQR solutions for linear dynamics with an iterative iLQR scheme for nonlinear dynamics, employing backward/forward recursions and Jacobian-based linearization to update control policies. The approach is validated on canonical benchmarks (Rayleigh oscillator, inverted pendulum, two-link manipulator, and quadcopter) and shown to achieve accurate tracking under nonlinearities, with a clear path to integration into MPC for real-time constrained control. This work provides a practical, modular methodology for robust trajectory tracking and adaptive control in complex dynamical systems, suitable for robotics and aerial-vehicle applications.
Abstract
This article presents a unified approach to quadratic optimal control for both linear and nonlinear discrete-time systems, with a focus on trajectory tracking. The control strategy is based on minimizing a quadratic cost function that penalizes deviations of system states and control inputs from their desired trajectories. For linear systems, the classical Linear Quadratic Regulator (LQR) solution is derived using dynamic programming, resulting in recursive equations for feedback and feedforward terms. For nonlinear dynamics, the Iterative Linear Quadratic Regulator (iLQR) method is employed, which iteratively linearizes the system and solves a sequence of LQR problems to converge to an optimal policy. To implement this approach, a software service was developed and tested on several canonical models, including: Rayleigh oscillator, inverted pendulum on a moving cart, two-link manipulator, and quadcopter. The results confirm that iLQR enables efficient and accurate trajectory tracking in the presence of nonlinearities. To further enhance performance, it can be seamlessly integrated with Model Predictive Control (MPC), enabling online adaptation and improved robustness to constraints and system uncertainties.