Search Results (4)

The design and calculation of nonlinear observers for parameter estimation in multi-time-scale nonlinear systems present significant challenges due to the inherent complexity and stiffness of such systems. This study proposes a framework for designing observers for two-time-scale nonlinear systems, with the objective of overcoming the aforementioned challenges. The design procedure involves reducing the original two-time-scale nonlinear system to a lower-dimensional model that captures only the slow dynamics while approximating the fast states through the use of an algebraic slow motion invariant manifold function. Subsequently, an exponential observer can be devised for this reduced system, which is valid for both state and parameter estimation. By employing the output from the original system, this observer can be adapted for online state and parameter estimation for the detailed two-time-scale system. The challenges associated with estimating parameters in two-time-scale nonlinear systems, the complexities of designing observers for such systems, and the computational burden associated with computing observers for ill-conditioned systems can be effectively addressed through the application of this design framework. A rigorous error analysis validates the convergence of the proposed observer towards the states and parameters of the original system. The viability and necessity of this observer design framework are demonstrated through a numerical example and an anaerobic digestion process. This study presents a practical approach for state and parameter estimation with observers for two-time-scale nonlinear systems. Full article

The global quasi-linearization (GQL) is used as a method to study and to reduce the complexity of mathematical models of mechanisms of chemical kinetics. Similar to standard methodologies, such as the quasi-steady-state assumption (QSSA), the GQL method defines the fast and slow invariant subspaces and uses slow manifolds to gain a reduced representation. It does not require empirical inputs and is based on the eigenvalue and eigenvector decomposition of a linear map approximating the nonlinear vector field of the original system. In the present work, the GQL-based slow/fast decomposition is applied for different combustion systems. The results are compared with the standard QSSA approach. For this, an implicit implementation strategy described by differential algebraic equations (DAEs) systems is suggested and used, which allows for treating both approaches within the same computational framework. Hydrogen–air (with 9 species) and ethanol–air (with 57 species) combustion systems are considered representative examples to illustrate and verify the GQL. The results show that 4D GQL for hydrogen–air and 14D GQL ethanol–air slow manifolds outperform the standard QSSA approach based on a DAE-based reduced computation model. Full article

Previous studies have demonstrated, experimentally and theoretically, the existence of slow–fast evolutions, i.e., slow chaotic spiking sequences in the dynamics of a semiconductor laser with AC-coupled optoelectronic feedback. In this work, the so-called Flow Curvature Method was used, which provides the slow invariant manifold analytical equation of such a laser model and also highlights its symmetries if any exist. This equation and its graphical representation in the phase space enable, on the one hand, discriminating the slow evolution of the trajectory curves from the fast one and, on the other hand, improving our understanding of this slow–fast regime. Full article

Chemical kinetic systems are modeled by dissipative ordinary differential equations involving multiple time scales. These lead to a phase flow generating anisotropic volume contraction. Kinetic model reduction methods generally exploit time scale separation into fast and slow modes, which leads to the occurrence of low-dimensional slow invariant manifolds. The aim of this paper is to review and discuss a computational optimization approach for the numerical approximation of slow attracting manifolds based on entropy-related and geometric extremum principles for reaction trajectories. Full article