Improved Gradient Estimation for Fast Extremum Seeking: A Parametric Proportional-Integral Observer-Based Approach

In this article, a complete parametric proportional-integral observer (PPIO)-based approach is proposed to improve the performance of gradient estimation for the fast extremum seeking (ES) scheme acting on a Hammerstein plant. Unlike the prevailing gradient estimation approach of the fast ES which uses a Luenberger observer without an explicit way to obtain the observer gains, a systemic complete PPIO is established based on a complete parametric solution to a type of generalized Sylvester matrix equations. The proposed PPIO presents complete parameterization of all the gain matrices as well as the left eigenvectors in terms of some sets of design parameters that represent the degrees of design freedom. Then, the gradient estimator is constructed by multiplying the states of PPIO and the demodulation signal. Moreover, a synthetic objective function, which includes weighted performance indices of the transient error and the steady-state accuracy, is formulated. The performance of the gradient estimator is improved by minimizing the synthetic objective function through adjusting the degrees of freedom of the PPIO, and all explicit values of the parametric gain matrices are derived with the adjusted degrees of freedom. In turn, a faster and more accurate gradient estimation scheme can be obtained and significantly improve the convergence of the closed-loop system. Besides, the proposed PPIO-based estimator has excellent performance under the noise condition. Simulation examples and an application to the lean-burn combustion system are used to illustrate the effectiveness of the proposed gradient estimation scheme.