Application Of Singular Perturbation Method Research Articles

Editorial

The lead article for this issue of EJAM is Prof. Bernard J. Matkowsky's personalized survey-style article on the theory and application of singular perturbation methods to noisy dynamical systems in the limit of small noise. This article is based on his John von Neumann Prize lecture presented at the Society of Industrial and Applied Mathematics (SIAM) Annual Meeting in July 2017. The John von Neumann Lecture is awarded by SIAM for outstanding and distinguished contributions to the field of applied mathematical sciences and for the effective communication of these ideas to the community. From 1990–1996, Matkowsky was an inaugural editorial board member for EJAM.

Temperature control of a solar furnace for material testing

A solar furnace is a thermodynamic device that concentrates sun radiation in order to achieve high-temperatures at a focus, where a sample of the material to be tested is located. This article address the problem of designing a control architecture for solar furnaces. It is motivated by the use of a solar furnace as an instrument in material science research to perform two types of experimental tests. In the first type, samples are tested in high-temperature stress cycles. In these experiments it is important to control the temperature profile. The second test type is characterised by imposing a solar radiation flux profile. On both experiment types there are strong nonlinear effects, a significant degree of parametric uncertainty and disturbances, such as solar radiation fluctuation due to clouds and to sun's apparent movement. Although the actuator (shutter) is much faster than the thermic system, it may destabilise the overall controlled system. The contribution of this article consists in the design of a control architecture for solar furnaces and the application of singular perturbation methods in order to derive conditions in the form of bounds for the speed of actuator dynamics so that the closed-loop system is stable.

Application of singular perturbation method in analyzing traffic density waves

Car following model is one of microscopic models for describing traffic flow. Through linear stability analysis, the neutral stability lines and the critical points are obtained for the different types of car following models and two modified models. The singular perturbation method has been used to derive various nonlinear wave equations, such as the Korteweg-de-Vries (KdV) equation and the modified Korteweg-de-Vries (mKdV) equation, which could describe different density waves occurring in traffic flows under certain conditions. These density waves are mainly employed to depict the formation of traffic jams in the congested traffic flow. The general soliton solutions are given for the different types of car following models, and the results have been used to the modified models efficiently.

Adaptive control of flexible joint manipulators: Comments on two papers

We discuss two papers [Spong, M. W. (1989) Syst. Control. Lett., 13, 15–21; Chang, Y.-Z. and R. W. Daniel (1992) Automatica, 28, 969–974] on the adaptive control of flexible joint robot manipulators. Our goal is to clarify the contributions of the two papers by correcting some technical errors and to provide some additional insight into the application of singular perturbation methods for the control of flexible joint robots.

Application of singular perturbation methods for three-dimensional minimum-time interception

In this paper, a feedback control law based on the singular perturbation method is developed for three-dimensional minimum-time interception. Whereas the heading and flight-path angles are considered fast variables with the same time scale, the relative position and the specific energy are considered slow variables. A zeroth-order optimal control algorithm is developed, and an extension to higher-order analysis is discussed. With the demonstrations of several numerical examples, it is shown that this time-scale separation is physically reasonable and results in a uniformly valid control law for long-, medium-, and short-range interception.

A singular perturbation analysis of optimal aerodynamic and thrust magnitude control

This paper illustrates the application of singular perturbation methods to optimal thrust magnitude control (TMC) and optimal lift control in flight mechanics. The modeling is restricted to horizontal plane dynamics. A multiple time scale analysis results in nonlinear feedback control solutions for lift and thrust during a turn to a specified down range position. The analysis is carried out to rust order with respect to the position state variables. Numerical results for a medium range and a short range air-launched missile are given, and comparisons are made to two alternative propulsion control concepts. The multiple time scaling procedure used here is applicable to solving a wide class of optimal control problems. It avoids the problem in asymptotic methods of picking the unknown adjoints to suppress unstable modes in the boundary layer and reduces the two-point boundary value problem to a series of pointwise function extremizations. Hence, the optimal control solution is essentially analytic and algebraic.

Singular perturbations and the sounding rocket problem

In this paper, Goddard's problem of maximizing the final altitude of a sounding rocket (a singular problem of optimal control) is analyzed using singular perturbation methods. The problem is first cast in singular perturbation form and then solved to zero order by adding boundary-layer corrections to the reduced solution. For a quadratic drag law, a closed-form solution is obtained, although consideration of a numerical example indicates that this solution is not useful for practical sounding rockets. However, use of state variable transformations allows a very accurate numerical approximation to be constructed. It is concluded that application of singular perturbation methods to the well-known sounding rocket problem indicates that these methods may have utility in dealing with singular problems of optimal control.

Energy Management and Singular Perturbations in Flight Mechanics

This paper discusses the application of singular perturbation methods to problems in flight mechanics. Emphasis is given to the selection of slow and fast states and to an approach for obtaining nonlinear optimal control solutions in feedback form. Energy state transformations play a useful role in preconditioning the state dynamics for singular perturbation analysis; however, multiple boundary layer methods and new approaches to boundary layer matching are required to obtain closed-loop solutions. Weighted minimum time/fuel solutions for aircraft and missile applications are discussed.

A review of the nonlinear theory of plasma oscillations

This is a coherent review of various aspects of nonlinear longitudinal plasma oscillations. The two cases of a single mode and of a continuum of modes are treated separately. In the monochromatic case, the important role of resonant particles is discussed. It is shown that secular terms arise when the Vlasov and Poisson equations are solved by a regular perturbation method in powers of the electric field. The application of singular perturbation methods to remove secularities is treated. In the second case the quasi-linear theory is reviewed and Dupree's extension of it is also discussed. After Al'tshul and Karpman's method of summing the principal secular terms to all orders, the kinetic equation of waves, wave-particle interactions and resonant wave interactions are discussed. Finally the plasma echo phenomenon is treated briefly.

Application of Singular Perturbation Methods to Resonance Problems

Application of Singular Perturbation Methods to Resonance Problems