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Approximate Methods
Published in Tasos C. Papanastasiou, Georgios C. Georgiou, Andreas N. Alexandrou, ViscousFluid Flow, 2021
Tasos C. Papanastasiou, Georgios C. Georgiou, Andreas N. Alexandrou
A typical singular perturbation problem involves two different length scales. A perturbation expansion in the original independent variable is, in general, good over a large interval corresponding to one length scale, but breaks down in a boundary layer, i.e., in a layer near a boundary where the other length scale is relevant and the dependent variable changes rapidly. This expansion is called outer approximation to the problem, and the region over which it is valid is called outer region. By properly rescaling the independent variable in the boundary layer, it is often possible to obtain an inner approximation to the solution which is valid in the boundary layer and breaks down in the outer region. A composite approximation uniformly valid over the entire domain can then be constructed by matching the inner and outer approximations. Due to the matching procedure, singular perturbation methods are also called matched asymptotic expansions. The most characteristic application of singular perturbation in fluid mechanics is the matching of potential solutions (outer approximation) to the boundary layer solutions (inner approximation) of the Navier-Stokes equation, with the inverse of the Reynolds number serving as the perturbation parameter.
Singular Perturbation and Chaos
Published in Wilfrid Perruquetti, Jean-Pierre Barbot, Chaos in Automatic Control, 2018
The notion of singular perturbation disturbance in the analysis of dynamical systems is very important and represents the theoretical base of numerous modern concepts (i.e., bifurcation, chaos, regular perturbation, singular perturbation, etc.) This last case was originally developed for the analysis of phenomena with multiple time-scales characterizing the evolution of systems belonging to the domain of the mechanics of fluids. Later, this theory was extended to the study of the other phenomena characterized by singular perturbation. For example, induction motor [35, 36], robotics [39], etc.
High-Frequency Vibration Control Using PZT Active Damping
Published in Chunling Du, Chee Khiang Pang, Multi-Stage Actuation Systems and Control, 2019
The singular perturbation design technique involves decomposing the system dynamics into slow and fast subsystems assuming that they operate on different time scales, which make them independent of each other [14,15] and allow independent controller design.
Parameter-uniform finite difference method for singularly perturbed parabolic problem with two small parameters
Published in International Journal for Computational Methods in Engineering Science and Mechanics, 2022
Tesfaye Aga Bullo, Guy Aymard Degla, Gemechis File Duressa
Singular perturbation problems emerged as a result of modeling real-life applications and their solutions exhibit boundary layer phenomena. The best example to mention is the Navier–Stokes equations with large Reynolds number in fluid dynamics, the convective heat transport problems with large Péclet number [1–3]. Based on the number of perturbation parameters, continuity or discontinuity of the coefficient; and source function or initial and/or boundary conditions throughout the considered domain, singularly perturbed parabolic problems can be categorized into various types [4–18]. Generally, singularly perturbed one-dimensional parabolic problems have boundary or interior or both boundary and interior layers depending on the defined data. Hence, in this work, we consider a class of singularly perturbed parabolic problems with two parameters whose solutions exhibit boundary layers. These types of problems arise in various areas of applications such as fluid dynamics (linear Navier–Stokes equation), chemical reactor theory, heat, and mass transfer process in composite materials with small heat conduction. Classes of singularly perturbed parabolic problems involving single perturbation parameters and sub-divided into convection-diffusion and reaction-diffusion problems are recently studied [1–7].
Event-triggered H ∞ filtering for singularly perturbed systems with external disturbance
Published in International Journal of Systems Science, 2021
Yifang Yan, Chunyu Yang, Xiaoping Ma, Linna Zhou
Because of the existence of small-time constants (Kokotovic et al., 1999; Peponides et al., 1982), many practical systems operate on different time scales. The interaction between slow and fast dynamics usually leads to high-dimensionality and ill-conditioned numerical issue in system analysis and design (Kokotovic et al., 1999; Shao, 2004; Yang & Zhang, 2009). To avert these issues, the systems with two-time-scale characteristics are described as singularly perturbed systems (SPSs), which contain a singular perturbation parameter ϵ characterising the degree of separation of fast and slow dynamics (Shao, 2004; Yang & Zhang, 2009; Yang et al., 2019). Under the existence of noisy signal, in order to estimate system states and obtain the desired performance, filtering problem has been considered for normal systems, and many valuable results are presented (Dong et al., 2013; Gao et al., 2008; Koo et al., 2017; Tanikawa, 2019). These methods cannot be applied to SPSs directly. One of the key issues of filter design for SPSs is to estimate an upper bound of ϵ (denoting as with ), such that SPSs can satisfy the performance for . Recently, the problem of filter design for SPSs with external disturbance has attracted much attention (Aliyu & Boukas, 2011; Assawinchaichote et al., 2007; Wang et al., 2016; Yang & Dong, 2008), and remains as an open area.
Robust ℋ∞ and ℒ∞–ℒ∞ sampled-data dynamic output-feedback control for nonlinear system in T–S form including singular perturbation
Published in International Journal of Systems Science, 2021
Jaejun Lee, Ji Hyun Moon, Ho Jae Lee
Singular perturbation is a phenomenon that the singularity of a parameter causes an abrupt change in the dynamic properties of a system. It is often found in modelling practical systems such as tunnel diode circuits with parasitic inductance (Assawinchaichote & Nguang, 2004), a pendulum driven by a motor with a gear train (G. H. Yang & Dong, 2008), and hypersonic viscous flow over a sphere (Martin, 1967). The stiffness that arises in such systems can be dealt with by modelling them in the form of a singularly perturbed system (SPS). An SPS typically split into subsystems according to time scales, which may result in the traditional synthesis not working well.