An Application of STWS Technique in Solving Stiff Non-linear System: 'High Irradiance Responses' (HIRES) of Photomorphogenesis

This paper illustrates an application of the Single Term Walsh Series (STWS) technique in solving stiff non-linear system: ‘High Irradiance RESponses’ (HIRES) of Photomorphogenesis from plant physiology. The chemical reaction scheme of HIRES problem has been modelled into system of stiff non-linear differential equations. This stiff system has been solved using the STWS technique. The STWS solutions are compared with the results obtained by the well-known solvers, namely, VODE and RADAU5. The applicability of the STWS technique has been tested

Comparison of Single Term Walsh Series Technique and Extended RK Methods Based on Variety of Means to Solve Stiff Non-linear Systems

This paper presents a comparison of Single Term Walsh Series (STWS) technique and the extended Runge-Kutta (RK) methods based on variety of means such as Arithmetic Mean (AM), Harmonic Mean (HaM), Centroidal Mean (CeM) and Contraharmonic Mean (CoM) to solve stiff non-linear systems of initial value problems (IVPs). Numerical solutions of some stiff non-linear systems are investigated for their stiffness. The discrete solutions obtained through STWS technique are compared with that of the RK methods based on variety of means. The applicability of the STWS technique has been demonstrated. The results show that the STWS technique is more suitable to solve stiff non-linear systems including highly stiff problems

State Analysis of Time-Varying Singular Bilinear Systems by RK-Butcher Algorithms

The Runge-Kutta (RK)-Butcher algorithm is used to study the timevarying singular bilinear systems with the exact solutions. The results (discrete solutions) obtained using the Haar wavelets, Single-Term Walsh series (STWS) and RK-Butcher algorithms are compared with the exact solutions of the time-varying singular bilinear systems. It is found that the solution obtained using the RK-Butcher algorithm is closer to the exact solutions of the time-varying singular bilinear systems. The RK-Butcher algorithm can easily be implemented using a digital computer and the solution can be obtained for any length of time, which is an added advantage of this algorithm

On a Biparameter Maximal Multilinear Operator

It is well-known that estimates for maximal operators and questions of pointwise convergence are strongly connected. In recent years, convergence properties of so-called `non-conventional ergodic averages' have been studied by a number of authors, including Assani, Austin, Host, Kra, Tao, and so on. In particular, much is known regarding convergence in $L^2$ of these averages, but little is known about pointwise convergence. In this spirit, we consider the pointwise convergence of a particular ergodic average and study the corresponding maximal trilinear operator (over $\mathbb{R}$, thanks to a transference principle). Lacey and Demeter, Tao, and Thiele have studied maximal multilinear operators previously; however, the maximal operator we develop has a novel bi-parameter structure which has not been previously encountered and cannot be estimated using their techniques. We will carve this bi-parameter maximal multilinear operator using a certain Taylor series and produce non-trivial H\"{o}lder-type estimates for one of the two "main" terms by treating it as a singular integrals whose symbol's singular set is similar to that of the Biest operator studied by Muscalu, Tao, and Thiele.Comment: 32 pages, 1 figur

Trajectory generation for the N-trailer problem using Goursat normal form

Develops the machinery of exterior differential forms, more particularly the Goursat normal form for a Pfaffian system, for solving nonholonomic motion planning problems, i.e., motion planning for systems with nonintegrable velocity constraints. The authors use this technique to solve the problem of steering a mobile robot with n trailers. The authors present an algorithm for finding a family of transformations which will convert the system of rolling constraints on the wheels of the robot with n trailers into the Goursat canonical form. Two of these transformations are studied in detail. The Goursat normal form for exterior differential systems is dual to the so-called chained-form for vector fields that has been studied previously. Consequently, the authors are able to give the state feedback law and change of coordinates to convert the N-trailer system into chained-form. Three methods for planning trajectories for chained-form systems using sinusoids, piecewise constants, and polynomials as inputs are presented. The motion planning strategy is therefore to first convert the N-trailer system into Goursat form, use this to find the chained-form coordinates, plan a path for the corresponding chained-form system, and then transform the resulting trajectory back into the original coordinates. Simulations and frames of movie animations of the N-trailer system for parallel parking and backing into a loading dock using this strategy are included

Alternatives for jet engine control

Research centered on basic topics in the modeling and feedback control of nonlinear dynamical systems is reported. Of special interest were the following topics: (1) the role of series descriptions, especially insofar as they relate to questions of scheduling, in the control of gas turbine engines; (2) the use of algebraic tensor theory as a technique for parameterizing such descriptions; (3) the relationship between tensor methodology and other parts of the nonlinear literature; (4) the improvement of interactive methods for parameter selection within a tensor viewpoint; and (5) study of feedback gain representation as a counterpart to these modeling and parameterization ideas

Analysis and control of power systems using orthogonal expansions

In recent years, considerable attention has been focused on the application of orthogonal expansions to system analysis, parameter identification, model reduction and control system design. However, little research has been done in applying their useful properties to Power System analysis and control. This research attempts to make some inroads in applying the so called " orthogonal expansion approach " to analysis and control of Power systems, especially the latter. A set of orthogonal functions commonly called Walsh functions in system science after it's discoverer J.L. Walsh [1923] have been successfully used for parameter identification in the presence of severe nonlinearities. The classical optimal control problem is applied to a synchronous machine infinite bus system via the orthogonal expansion approach and a convenient method outlined for designing PID controllers which can achieve prespecified closed loop response characteristics. The latter is then applied for designing a dynamic series capacitor controller for a single machine infinite bus system