Optimal Estimation for Continuous-Time Nonlinear Systems Using State-Dependent Riccati Equation (SDRE)
Adnan Tahirovic, Azra Redzovic
TL;DR
By expressing nonlinear dynamics as $\dot{x}=A(x)x+B(x)u$ and optimizing a quadratic-like cost with SDRE, the paper unifies control and state estimation in a nonlinear analogue of LQG. It introduces an SDRE-KF estimator, compares it to EKF and PF on a simple pendulum and a Van der Pol oscillator, and demonstrates that SDRE-KF can achieve comparable or superior estimation accuracy depending on the system. The results underscore the practical viability and educational value of the SDRE-based approach for nonlinear state estimation and control. Future work proposes adaptive state-dependent parameterizations and policy-iteration extensions to further enhance performance.
Abstract
This paper introduces a unified approach for state estimation and control of nonlinear dynamic systems, employing the State-Dependent Riccati Equation (SDRE) framework. The proposed approach naturally extends classical linear quadratic Gaussian (LQG) methods into nonlinear scenarios, avoiding linearization by using state-dependent coefficient (SDC) matrices. An SDRE-based Kalman filter (SDRE-KF) is integrated within an SDRE-based control structure, providing a coherent and intuitive strategy for nonlinear system analysis and control design. To evaluate the effectiveness and robustness of the proposed methodology, comparative simulations are conducted on two benchmark nonlinear systems: a simple pendulum and a Van der Pol oscillator. Results demonstrate that the SDRE-KF achieves comparable or superior estimation accuracy compared to traditional methods, including the Extended Kalman Filter (EKF) and Particle Filter (PF). These findings underline the potential of the unified SDRE-based approach as a viable alternative for nonlinear state estimation and control, providing valuable insights for both educational purposes and practical engineering applications.