The implementation of nonlinear control depends on the accuracy of the system model, which, however, is often restricted by parametric and structural uncertainty in the underlying dynamics. In this paper, we propose methods of estimating parameters and states that aim at matching the identified model and the true dynamics not only in the direct output measurements, i.e., in an $L^2$-sense, but also in the higher-order time derivatives of the output signals, i.e., in a Sobolev sense. A Lie-Sobolev gradient descent-based observer-estimator and a Lie-Sobolev moving horizon estimator (MHE) are formulated, and their convergence properties and effects on input–output linearizing control and model predictive control (MPC) respectively are studied. Advantages of Lie-Sobolev state and parameter estimation in nonlinear processes are demonstrated by numerical examples and a reactor with complex dynamics.

非线性控制的实现取决于系统模型的精度，但是，该精度通常受到基础动力学中参数和结构不确定性的限制。在本文中，我们提出了一种估计参数和状态的方法，这些方法旨在不仅在直接输出测量中（即，在${\mathrm{大号}}^{\mathrm{2个}}$感，但也可以是输出信号的高阶时间导数，即Sobolev感。建立了一个基于Lie-Sobolev梯度下降的观测器估计器和一个Lie-Sobolev移动视野估计器（MHE），分别研究了它们的收敛特性以及对输入输出线性化控制和模型预测控制的影响。通过数值实例和具有复杂动力学的反应堆证明了Lie-Sobolev状态和参数估计在非线性过程中的优势。