Optimization Kuta Software

[Sharon Harris]

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Abstract: In this study, we investigated the modelling and optimization of drinking water supply system reliability in the village of Zaben, Czech Republic. An in depth overview of the water supply network in the municipality, passport processing and accident and malfunction recording is provided based on data provided by the owner and operator of the water mains as well as the data collected by our own field survey. Using the data processed from accident and failure reports in addition to water main documentation, the water supply network in Zaben was evaluated according to the failure modes and effects analysis methods. Subsequently, individual water supply lines were classified based on their structural condition. In addition, a proposed plan for financing the reconstruction of the water supply mains in Zaben was created. As such, this study provides an overall assessment of the water supply network in Zaben alongside a proposed plan for the structural restoration of the water supply system, which accounts for the theoretical service life of the system and the financial resources of the owner. Keywords: water pipes; structural integrity; failure modes

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Trajectory optimization, an optimal control problem (OCP) in essence, is an important issue in many engineering applications including space missions, such as orbit insertion of launchers, orbit rescue, formation flying, etc. There exist two kinds of solving methods for OCP, i.e., indirect and direct methods. For some simple OCPs, using the indirect methods can result in analytic solutions, which are not easy to be obtained for complicated systems. Direct methods transcribe an OCPs into a finite-dimensional nonlinear programming (NLP) problem via discretizing the states and the controls at a set of mesh points, which should be carefully designed via compromising the computational burden and the solution accuracy. In general, the larger number of mesh points, the more accurate solution as well as the larger computational cost including CPU time and memory [1]. There are many numerical methods have been developed for the transcription of OCPs, and the most common method is by using Pseudospectral (PS) collocation scheme [2], which is an optimal choice of mesh points in the reason of well-established rules of approximation theory [3]. Actually, there have several mature optimal control toolkits based PS methods, such as DIDO [4], GPOPS [5]. The resulting NLP problem can be solved by the well-known algorithm packages, such as IPOPT [6] or SNOPT [7]. However, these algorithms cannot obtain a solution in polynomial-time, and the resulting solution is locally optimal. Moreover, a good initial guess solution should be provided for complicated problems.

It is to note that only the cases of mesh points are considered in [15], while in some certain applications, especially aerospace applications, to achieve more accuracy, the more mesh points are required [16]. However, purely increasing will lead to ill-conditioned NLP problem [17], which is hard to solve or cannot to be solved. In order to overcome such ill-conditioned phenomenon, some well-conditioned PS methods for ordinary differential equations (ODEs) have been proposed in [17, 18], in which, Birkhoff PS method stemmed from Birkhoff interpolation [19] is introduced to solve higher-order ODEs. In [16, 19], the first-order Chebyshev Birkhoff PS (CBPS) method is proposed to transcribe general OCP into NLP, and the advantages of CBPS over other PS methods, especially for large mesh grids, is demonstrated by its application in solving an orbital transfer problem.

In this chapter, we apply PS methods using Bikrhoff polynomials to the convex optimization framework for OCPs, particularly, the first order and second-order Birkhoff PS (BPS) methods with LGL and CGL collocation schemes are applied to transcribe a class of cascaded second-order systems, which may be convex or not. The first-order BPS method in convex optimization framework is similar to that in [16], hence we focus on the PS transcription for convexified OCP by using second-order Birkhoff polynomials. The main contributions of this paper lie in two aspects: (1) the unified matrix formulation of OCP by using BPS method within the convex optimization framework is proposed and validated; and (2) the computational performances and solution accuracies resulted from various PS schemes are extensively exploited thereby a useful conclusion that, using BPS method renders the remarkable drops in the condition number for the generated programming problem therefore lowering the computational cost.

The remainder of this chapter is organized as follows. The preliminaries of convex programming and PS method for general convexified optimal control problems are presented in Sect. 4.2, then the proposed well-conditioned PS method via Birkhoff polynomials within the convex optimization framework is detailed in Sect. 4.3. The demonstrated examples of a simple cart problem as well as a rescue orbit searching problem for validating the effective-ness and efficiency of the proposed method are presented in Sect. 4.4 followed by the conclusions in Sect. 4.5.

In general, the free-final-time OCP can be converted into a fixed-final-time problem by normalizing the time into the domain of [0, 1], consequently, only the fixed-final-time problem is considered in this chapter. A typical OCP is given as the following problem G.

To solve the general problem G by convex programming method, the performance index (4.1) and the constraints (4.2)\(\sim \)(4.4) should be the form of linear or second-order cone (SOC), thus the convexification techniques are required for converting the nonlinear or concave constraints to the corresponding convex formulation. The most important issue lies in the conversion process is to guarantee the solution of the converted problem is still that of the original problem.

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