A new steepest descent method with global convergence properties

In this study, we tend to propose a replacement hybrid algorithmic rule which mixes the search directions like Steepest Descent (SD) and Quasi-Newton (QN). First, we tend to develop a replacement search direction for combined conjugate gradient (CG) and QN strategies. Second, we tend to depict a replacement positive CG methodology that possesses the adequate descent property with sturdy Wolfe line search. We tend to conjointly prove a replacement theorem to make sure global convergence property is underneath some given conditions. Our numerical results show that the new algorithmic rule is powerful as compared to different standard high scale CG strategies.

Now -a-days PID (proportional-integral-derivative) controller is used in every type of small or large industries. For this the digital PID controller is so popular. But it is necessary to design and tune the PID controller in a proper method for applying the plant. The paper represents a method of tuning a digital proportional-integral-derivative (PID) controller. The name of the method is Steepest Gradient Descent Method (SDGM). It is an iterative method. The SDGM method is very simple, easy to use and each iteration is very fast. It is also a guaranteed method to find the minima through numerous times of iterations as long as it exists. By applying this optimization method PID controller is tuned both of digital and analog process. In the paper by applying the method of some plant the comparison result of analog and digital are shown. From the comparison it is found that the digital scheme is preferable than analog scheme. Index Terms—PID controller, SDGM. I. INTRODUCTION The fi rst controllers with proportional, integral, and derivative (PID) feedback control action became commercially available during the 1930s. The 1940s saw widespread acceptance in industry of pneumatic PID controllers, and their electronic counterparts entered the market in the 1950s. Digital hardware has been routinely used since the 1980s with significant impact on process control. Even several decades after three-mode controllers were introduced; the vast majority of controllers used in the chemical process industry are based on PI/PID models (1). The popularity of these controllers has led to research on tuning methods, resulting in hundreds of publications on this topic. Ziegler-Nichols tuning relations and Cohen-Coon tuning rules are among the earliest published methods. Tuning relations based on error criteria are more recent model-based tuning rules, which offer improvements over earlier tuning methods. Tuning rules also exist for unstable processes as well as for tuning in the presence of plant model mismatch (2). Despite the numerous approaches available for controller tuning, surveys indicate that poorly tuned control loops are abundant in industry. In general, the gain scheduling of the PID controllers is an optimization problem. The method of steepest descent (SDGM) is also known as The Gradient Descent, which is basically an optimization algorithm to find the local minimum of a function. By applying the SDGM to the PID controller, the controller is tuned and applied to some conventional plant then the digital and analog results are compared to the paper.

Application of gradient descent method to the sedimentary grain-size distribution fitting

Gradient descent (GD) method is used to fit the the laser grain-size distribution of the sediments with a sum of four weighted lognormal functions. The numerical results indicate that the GD method is not only easy to operate but also could effectively optimize the parameters of the fitting function with the error decreasing steadily. Meanwhile the overall fitting results are satisfactory. As a new way of data fitting, the GD method could also be used to solve other optimization problems.

The steepest descent method was designed by Cauchy (1847) and is the simplest of the gradient methods for the optimization of general continuously differential functions in n variables. Its importance is due to the fact that it gives the fundamental ideas and concepts of all unconstrained optimization methods. It introduces a pattern common to many optimization methods. In this pattern, an iteration consists of two parts: the choice of a descent search direction dk followed at once by a line search to find a suitable stepsize αk. The search direction in the steepest descent method is exactly the negative gradient.

On-Line Eigenvector Algorithms for the Identification of Dynamic Systems

Two molecular models, one representing a “small” molecule and the other a “large” molecule, both with several free rotors, are used to test, by comparison with exact count, seven approximation formulas for the calculation of density of harmonic vibration–rotation states, at energies between 1000 and 20 000 cm−1. The techniques of the classical approach, the Laplace transform, steepest descents, and symmetric functions are represented among the seven formulas. Also shown is the effect of the various approximations on the energy dependence of the computed unimolecular rate constant, and on ion breakdown curves calculated by the statistical theory of mass spectra. It is concluded that the techniques of the Laplace transform [J. Chem. Phys. 46, 3736 (1967)] and steepest descents, as formulated in an Appendix, yield the best all-around approximation, while a semiclassical-type formula [J. Chem. Phys. 41, 1883 (1964)] requires least computational effort and still gives excellent results in most cases. In an Appendix, the method of steepest descents is generalized and its connection with the Laplace transform noted.

In this paper, the optimization of a hybrid-cycle OTEC plant is carried out by the method of steepest descent. The hybrid cycle is a combination of a closed-cycle OTEC plant and a spray flash desalination plant. The total heat exchange area per net power is used as an objective function. The numerical results are shown for a 10MW hybird-cycle OTEC plant with plate-type heat exchangers. Ammonia is used as a working fluid. The results are compared with that of a closed cycle OTEC system.

We discuss an optimal control approach for a 2D Stokes flow with a free surface. The aim is to optimize the shape of a polymer film by adjusting the ambient pressure in a casting process. The resulting minimization problem is solved by the method of steepest descent. Numerical results will be presented. Furthermore we state the adjoint system for the Lagrangian formalism. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A classical approach to extending the validity of Airy's dispersion relation for surface gravity waves by Friedrichs (1948) to gentle slopes (of special inclinations) is here re-examined with extended small-slope asymptotics using the full linear harmonic function theory combined with the method of steepest descent. A new dispersion relation emerges that appears to give significantly increased accuracy over sloping beds when tested on the plane beach problem with various forms of the mild-slope equation (MSE) and global error reductions of the order 50% are noted in some ‘from deep to shore’ computations. Unlike the classical formula, the new formula predicts a discontinuous wavenumber at a place where the bottom slope is discontinuous. Preliminary tests examining the reflection coefficient with the basic (early version) MSE over ramp-type profiles indicate that this is not a major problem and numerical results using wavenumber calculated by the new dispersion relation are qualitatively similar to those of the modified MSE (MMSE) developed in Chamberlain & Porter (1995). When the new formula is applied (with mass conservation) to the MMSE on the ramp, results are almost identical to those of a full linear model for inclines having a gradient up to 8:1. It is also shown that the dominant asymptotic analysis, responsible for the new formula, is valid for all slope angles $\alpha{<}\pi/2$ and not just the special angles considered by Friedrichs.

Greenhouses should be designed to meet requirements of construction costs, durability against loads and the possibility of providing suitable environment to plant growth. To get guides to design an optimal greenhouse from the above-mentioned three viewpoints, one of the optimization techniques, which is called the steepest descent method, is applied in this study. The direct light transmission into the greenhouse in winter is regarded as one of the most important environmental factors.The steepest descent method gives the combination of the design variables which constitute the optimal (minimum or maximum) value in the objective function under some constraints. In this study the design variables were, for example, the shape of the house, the moment of inertia and crosssection features of each member and the ratch measure. The allowable stress or deflection of each member played a role of constraints in the course of the optimization. The objective function was the construction cost or the direct light transmission. In the steepest descent method, however, only one objective function is allowed. For this reason the following two steps were used. First the construction cost (Js) is chosen as the objective function and is minimized. Then the inequality (1) is added to the constraints and the light transmission is maximized, where k is the coefficient of construction cost allowance and larger than 1.0, and Jsopt is the minimum construction cost.Because in the steepest descent method all variables should be continuous and differentiable, some discontinuous variables, e.g. the number of purlins, were approximated with appropriate continuous functions. For the same reason any size and shape of steel member were assumed to exist. The crosssection features of the member are expressed with three independent variables on the basis of the empirical relations.The optimizations were carried out under the conditions of a snow load of 30cm and a wind speed of 50m/s. The span was fixed at 9m. The latitude and the coefficient of construction cost allowance in the optimization of the light transmission were 34°N and 1.10, respectively. The following points became clear by the optimizations.1) The steepest descent method can be applied to greenhouse design and the results show appreciable improvement both in the construction cost and in the light transmission in comparison with an example of commercial greenhouses.2) The results of the optimization of the construction cost (cost optimization, hereafter) and the light transmission of the N-S oriented greenhouse (N-S optimization, hereafter) show a symmetric shape, whereas the optimal E-W oriented greenhouse has a ridge nearer to the south wall, so that the south roof slope becomes steeper.3) Every optimization results in a ratch measuer of about 4m, which is wider than that of commercial greenhouses.4) The moment of inertia of the rafters and the posts ranges from 150 to 290cm4. In the cases of the cost and N-S optimizations, the rafters have a larger moment of inertia than the posts. In the E-W optimization, the north rafter has an appreciably larger moment of inertia than the three others.5) The cost optimization shows that members of the crosssection with larger depth and width are more suitable for the rafters and the posts, if the moment of inertia is the same. This holds true for the north members in the E-W optimization and for the posts in the N-S optimization. In the E-W oriented greenhouse, a member with shorter depth and larger section area and a member with shorter width are more suitable for the south rafter and for the south post, respectively.6) In this study the strength of the glass was not taken into consideration. This resulted in very wide purlin intervals, which seem to be unrealistic, but this still proves that larger purlins with wider intervals are more suitable

A steepest descent based bit allocation method with polynomial iteration complexity for minimizing the sum of frame distortions under a total bit rate constraint is presented for a predictive video coder. Previous algorithms that solve this problem have been based on Lagrangian relaxation followed by unconstrained optimization. Some of these methods have complexity exponential in the prediction depth in the worst case. Moreover, the convergence properties of the faster methods are difficult to analyze. A steepest descent method is utilized here for its comparatively low computational complexity, excellent performance and ease of analysis. Sufficient conditions for global optimality are presented without assuming independent encoding of frames. Results for dependent bit allocation through the assignment of quantizer step-sizes to frames in MPEG-2 encoded sequences suggest that these optimality conditions are to a large extent satisfied in practice. A PSNR improvement of up to 1.5 dB is obtained over the standard TM5 rate control algorithm for MPEG-2.

This paper contributes a mathematical framework for rotating electric machine optimal design. The proposed method utilizes a hybrid finite element/air-gap macro-element method, with steepest descent optimization. In particular, the air-gap element is advantageous for torque optimization, since it directly yields a smooth magnetic field function in the air-gap and an analytic expression for torque, based on the magnetic potential solution at the vertices of the finite element mesh. The sensitivity of an arbitrary objective function with respect to changes in geometric design parameters can be used in the steepest descent method. For illustration purposes, the optimization process is applied to a simple electromagnet for torque maximization.

In this paper, some properties of the steepest descent path of the integral describing transient plane wave propagation in anisotropic cold plasmas are determined. The relationship of the steepest descent method to the possible arrival of evanescent waves is illustrated by a numerical example. Also, it is shown through error estimates that the result obtained by the steepest descent method is valid for finite z and large t.

Solution of polynomial equations by method of steepest descent