Singular perturbation theory: techniques with applications to engineering
Solutions of many singular problems can be found by use of appropriate rescalings. We must now decide which terms constitute the leading-order dominant balance i. The second case is referred to as the singular distinguished limit. Singular perturbation problems for differential equations can arise in a number of ways and are typically more complicated than their algebraic counterparts. In what follows, we describe three methodologies for investigating solutions to singularly perturbed differential equations. The choice of technique to be applied depends on the form of the problem and also on the desired properties to be studied.
Singularly perturbed differential equations arise in many applications, such as wave propagation and quantum mechanics. Progress can be made by studying Stokes phenomena and exponential asymptotics for the solutions in the complex plane Chapman et al Singularly perturbed differential equations can yield solutions containing regions of rapid variation rapid compared to the regular length scale for the problem.
Constructing a solution to a differential equation or system involves several steps: identifying the locations of layers boundary or internal , deriving asymptotic approximations to the solution in the different regions corresponding to different distinguished limits in the equations , and ultimately, forming a uniformly valid solution over the entire domain.
Solutions obtained for the layers singular distinguished limits are usually termed inner solutions while the slowly varying solutions for the regular distinguished limits are referred to as outer solutions. The uniformly valid solution can be constructed through asymptotic matching of the inner and outer solutions, which relies on the fundamental assumption that the different solution forms overlap at on some identifiable region see Figure 1.
Procedures for matching asymptotic expansions have been examined by Kaplun, Van Dyke and others Lagerstrom , Van Dyke , Kevorkian and Cole , Eckhaus , but there are still some fundamental theoretical issues to be resolved. Unlike WKB theory, this approach can also be applied to nonlinear equations, and this versatility allows a wide range of problems to be tackled:.
Equation 9 is a weakly-nonlinear oscillator , reducing to the linear oscillator equation at leading order. The growing cumulative error in phase and amplitude apparent in the regular expansion is a consequence of the limitations of the ansatz, particularly in deviations from the unperturbed natural frequency of the system. Various perturbation methods have been developed for dealing with such problems.
These include:. Similar ideas also arise in homogenization theory , considering the averaging of spatially periodic structures of materials Bensoussan et al , Holmes We have only scratched the surface of this research area, but it is hoped that the above illustrates the power and usefulness of the methods grouped under singular perturbation theory. As discussed above, singular perturbation theory tackles difficult problems by investigating various reduced problems and then assembling the results together in an appropriate form.
These reductions could, for example, simply be to a lower order polynomial in an algebraic problem, or could be more significant, such as in the reduction of a PDE to an ODE, or a functional equation to one of algebraic form. The reduced problems can still be mathematically challenging, with the construction of a uniformly valid solution requiring an involved analysis. While some singular perturbation methods are based on rigorous analysis, the vast range of applications and available techniques typically restrict against such results.
Consequently, the methods are often classed as formal techniques. However, this is not considered to be a significant problem: any a priori assumptions can be checked for consistency once suitable expansions have been derived; furthermore, formal results obtained by these techniques have been known to provide direction for additional rigorous theory Smith , Eckhaus In fact, some authors have seen the generality of the methods described above as representative of some more fundamental notion.
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For example, Kruskal went as far as to introduce the term asymptotology in referring to the art of dealing with applied mathematical systems in limiting cases Kruskal and considered singular perturbation theory and asymptotic methods in general as a component of asymptotology. Averaging , Multiple Scale Analysis , Normal forms. Journal of Mathematical Analysis and Applications :1, Journal of Optimization Theory and Applications :2, International Journal of Systems Science 34 :3, Applicable Analysis 82 :2, Journal of Robotic Systems 19 :8, Journal of Guidance, Control, and Dynamics 24 :6, Maintenance, Modeling and Optimization, Neural Nets for Feedback Control.
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On the Singular Perturbations for Fractional Differential Equation
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Improved reduced-order model for control of flexible-link robots. Sliding-mode observers design for singular perturbation systems. Design of hierarchically distributed expert controllers for large-scale systems. Balanced realizations of singularly perturbed systems. Perturbation methods in control of flexible link manipulators. Singularly perturbed control systems: recent progress. Asymptotic behaviour of the norm of input-output operators corresponding to singularly perturbed systems with multiplicative white noise. Discrete simplex sliding mode control and its application to singular perturbation systems.
Mathematical and Analytical Techniques with Applications to Engineering
Asymptotic optimization of a linear differential system controlled by a Markov decision process. A neural network controller for flexible-link robots. Control of two-link flexible manipulators via generalized canonical transformation. Control of a hummingbird minipositioner with a multi-transputer MARC controller. Control of singularly perturbed hybrid stochastic systems. Dynamic model of a one-link robot manipulator with both structural and joint flexibility.
Decentralized and distributed control approaches and algorithms. Invariant manifolds and their application to robot manipulators with flexible joints.
Suboptimal control of singularly perturbed systems and periodic optimization. Multiscale singularly perturbed control systems: limit occupational measures sets and averaging. Perturbation theory for semi-Markov control problems. Decomposition of near-optimum regulators with nonstandard multiparameter singular perturbations.
Singularly perturbed Markov control problem: limiting average cost.
- Singular Perturbation Theory.
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A singular perturbation approach to stabilization of the internal dynamics of multilink flexible robots. Modeling and control of a two-link flexible robot manipulator. Higher-order optimal control design via singular perturbation. Proceedings of the , American Control Conference, Adaptive neural network control of flexible link robots based on singular perturbation. Fuzzy servocontrollers: the hierarchical approach. PID composite controller and its tuning for flexible link robots.
Adaptive aggregation of modular control. Proceedings of the Asian Fuzzy Systems Symposium ,