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Helium Atom, Variational Method and Perturbation Theory
Published in Caio Lima Firme, Quantum Mechanics, 2022
The perturbation theory is a mathematical method to find approximate solution for a problem without exact solution which has as a starting point the exact solution of the related problem. The perturbation theory is only applicable when it is possible to add a small term to the formulated problem.
Approaches to Quantum Transport
Published in David K. Ferry, An Introduction to Quantum Transport in Semiconductors, 2017
Now these two Green’s functions do not exhaust the possibilities, and these are normally described as the equilibrium Green’s functions. There are certainly other Green’s functions with which we will deal in later chapters. We will see in the next chapter that the kernel in Eq. (2.80) is, in fact, quite close to the spectral density that we introduced in Eq. (1.8), and we will discuss this in the next chapter. Finding Green’s functions from the Schrödinger equation, or from the Liouville equation, is not difficult for simple Hamiltonians, as for any quantum mechanical problem. For complicated Hamiltonians, such as in the case of many-body interactions or electron-phonon interactions, it is not so simple, and one normally resorts to perturbation theory. However, perturbation theory is not without its own problems, in that the terms may be difficult to evaluate or the series may not converge. We will deal with these problems in later chapters.
Band Structure Details and Photoconductivity
Published in N.V. Joshi, Photoconductivity, 2017
Equations (4.22) are obtained with the first-order perturbation theory; however, appreciable changes have not been reported with the second-order perturbation. These transitions are frequently observed in compensated materials, and hence absorption and photo-luminescence spectroscopy [20] provide information about the distance between donor and acceptor. In compound semiconductors it is very useful to know at which site (cation or anion) a particular impurity atom is located.
Inter-relationship between approximate dynamic inversion and MRAC augmented with proportional-integral controller
Published in Journal of Control and Decision, 2023
Santanu Mallick, Ujjwal Mondal
On the other hand, ADI converts the nonlinear system towards an equivalent linear form. This technique is basically the approximation of exact dynamic inversion or feedback linearisation. ADI drives a minimum phase nonaffine-in-control system to a given stable reference model. This control law is a solution to fast dynamics and Tikhonov’s Theorem in singular perturbation theory. It is found that the control signal moves towards exact dynamic inversion and the state of the system approaches towards the state of the given reference value provided that controller dynamics are very fast (Teo & How, 2009). For a nonaffine-in-control system, ADI using time scale separation can be implemented both for single-input and multi-input systems. In this method, fast dynamics are used for the inversion of the system as this depends on the time-scale property between system dynamics and dynamics of inverting controller (Hovakimyan et al., 2005). The adaptive DI using time-scale separation utilises online approximation. The control signal provides a fast dynamical equation, it inverts a series-parallel model, the state of which tracks the state of a given nonaffine-in-control system (Lavretsky & Hovakimyan, 2005). In the nonlinear system which is nonaffine with high order mismatched disturbance and actuator saturation, an augmented intermediate subsystem is created for compensation of the difference between the input with and without saturation (Yang & Pei, 2020).
An extended observer-based robust nonlinear speed sensorless controller for a PMSM
Published in International Journal of Control, 2019
Syed Ali Asad Rizvi, Attaullah Y. Memon
Following the singular perturbation theory (Khalil, 2002) and defining the estimation errors as where δ( · ) is the lumped uncertainty as a result of parametric uncertainties (B, J) and unknown load torque Tl. The fast time scaled error dynamics are For the case when the uncertainty is absent (δ( · ) = 0, the observer gains can be chosen such that is Hurwitz stable. However, when the uncertainty is present, the idea is to suppress it by dominating it with a high gain. By decreasing γ, the robustness and the convergence rate of observer can be improved. However, for too small values of γ, the observer is prone to measurement noise (Khalil, 2002). This noise issue is taken care of by the continuous SMO. The proposed EHGO does not require knowledge of load torque due to the extended lumped disturbance state which also allows it to tolerate uncertainty in the knowledge of mechanical parameters. Moreover, the EHGO-based speed estimate remains fairly insensitive to boundary layer of SMO.
Miracles, misconceptions and scotomas in the theory of solitary waves
Published in Geophysical & Astrophysical Fluid Dynamics, 2019
The inverse scattering method applies only a very special set of PDEs which are exactly integrable in the sense that there exists an infinite number of conservation laws. Chaos, in the sense of Poincaré and Lorenz, is impossible. Like the KdV equation, most (but not all!) of the exactly integrable wave equations are only one-space-dimensional. However, these wave equations can be justified by singular perturbation theory. When applied to dimension reduction, the usual schemes such as matched asymptotic expansions, multiple scales series, renormalisation group theory and the method of strained coordinates, etc., are collectively labelled “reductive perturbation theory”.