State Estimation and Control for Continuous-Time Nonlinear Systems: A Unified SDRE-Based Approach

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 the 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.

💡 Research Summary

The paper proposes a unified framework that simultaneously addresses state‑feedback control and state estimation for continuous‑time nonlinear systems by leveraging the State‑Dependent Riccati Equation (SDRE) methodology. Traditional Linear‑Quadratic‑Gaussian (LQG) theory is limited to linear dynamics; extending it to nonlinear plants usually requires linearization (as in the Extended Kalman Filter, EKF) or stochastic sampling (as in Particle Filters, PF). Both approaches have drawbacks: EKF’s linearization can cause significant performance degradation or even divergence in strongly nonlinear regimes, while PF demands substantial computational resources that hinder real‑time deployment.

The authors first recast a nonlinear system (\dot{x}=f(x,u)) into a state‑dependent linear‑like representation (\dot{x}=A(x)x+B(x)u), where the matrices (A(x)) and (B(x)) depend on the current state. This factorization enables the use of a quadratic performance index (J=\int_0^\infty