Dual methods and approximation concepts in structural synthesis

Approximation concepts and dual method algorithms are combined to create a method for minimum weight design of structural systems. Approximation concepts convert the basic mathematical programming statement of the structural synthesis problem into a sequence of explicit primal problems of separable form. These problems are solved by constructing explicit dual functions, which are maximized subject to nonnegativity constraints on the dual variables. It is shown that the joining together of approximation concepts and dual methods can be viewed as a generalized optimality criteria approach. The dual method is successfully extended to deal with pure discrete and mixed continuous-discrete design variable problems. The power of the method presented is illustrated with numerical results for example problems, including a metallic swept wing and a thin delta wing with fiber composite skins

Rethinking solar photovoltaic parameter estimation: global optimality analysis and a simple efficient differential evolution method

Accurate, fast, and reliable parameter estimation is crucial for modeling, control, and optimization of solar photovoltaic (PV) systems. In this paper, we focus on the two most widely used benchmark datasets and try to answer (i) whether the global minimum in terms of root mean square error (RMSE) has already been reached; and (ii) whether a significantly simpler metaheuristic, in contrast to currently sophisticated ones, is capable of identifying PV parameters with comparable performance, e.g., attaining the same RMSE. We address the former using an interval analysis based branch and bound algorithm and certify the global minimum rigorously for the single diode model (SDM) as well as locating a fairly tight upper bound for the double diode model (DDM) on both datasets. These obtained values will serve as useful references for metaheuristic methods, since none of them can guarantee or recognize the global minimum even if they have literally discovered it. However, this algorithm is excessively slow and unsuitable for time-sensitive applications (despite the great insights on RMSE that it yields). Regarding the second question, extensive examination and comparison reveal that, perhaps surprisingly, a classic and remarkably simple differential evolution (DE) algorithm can consistently achieve the certified global minimum for the SDM and obtain the best known result for the DDM on both datasets. Thanks to its extreme simplicity, the DE algorithm takes only a fraction of the running time required by other contemporary metaheuristics and is thus preferable in real-time scenarios. This unusual (and certainly notable) finding also indicates that the employment of increasingly complicated metaheuristics might possibly be somewhat overkill for regular PV parameter estimation. Finally, we discuss the implications of these results and suggest promising directions for future development.Comment: v2, see source code at https://github.com/ShuhuaGao/rePVes

Control Augmented Structural Synthesis

A methodology for control augmented structural synthesis is proposed for a class of structures which can be modeled as an assemblage of frame and/or truss elements. It is assumed that both the plant (structure) and the active control system dynamics can be adequately represented with a linear model. The structural sizing variables, active control system feedback gains and nonstructural lumped masses are treated simultaneously as independent design variables. Design constraints are imposed on static and dynamic displacements, static stresses, actuator forces and natural frequencies to ensure acceptable system behavior. Multiple static and dynamic loading conditions are considered. Side constraints imposed on the design variables protect against the generation of unrealizable designs. While the proposed approach is fundamentally more general, here the methodology is developed and demonstrated for the case where: (1) the dynamic loading is harmonic and thus the steady state response is of primary interest; (2) direct output feedback is used for the control system model; and (3) the actuators and sensors are collocated

Development and demonstration of an on-board mission planner for helicopters

Mission management tasks can be distributed within a planning hierarchy, where each level of the hierarchy addresses a scope of action, and associated time scale or planning horizon, and requirements for plan generation response time. The current work is focused on the far-field planning subproblem, with a scope and planning horizon encompassing the entire mission and with a response time required to be about two minutes. The far-feld planning problem is posed as a constrained optimization problem and algorithms and structural organizations are proposed for the solution. Algorithms are implemented in a developmental environment, and performance is assessed with respect to optimality and feasibility for the intended application and in comparison with alternative algorithms. This is done for the three major components of far-field planning: goal planning, waypoint path planning, and timeline management. It appears feasible to meet performance requirements on a 10 Mips flyable processor (dedicated to far-field planning) using a heuristically-guided simulated annealing technique for the goal planner, a modified A* search for the waypoint path planner, and a speed scheduling technique developed for this project

Verified global optimization for estimating the parameters of nonlinear models

Nonlinear parameter estimation is usually achieved via the minimization of some possibly non-convex cost function. Interval analysis allows one to derive algorithms for the guaranteed characterization of the set of all global minimizers of such a cost function when an explicit expression for the output of the model is available or when this output is obtained via the numerical solution of a set of ordinary differential equations. However, cost functions involved in parameter estimation are usually challenging for interval techniques, if only because of multi-occurrences of the parameters in the formal expression of the cost. This paper addresses parameter estimation via the verified global optimization of quadratic cost functions. It introduces tools for the minimization of generic cost functions. When an explicit expression of the output of the parametric model is available, significant improvements may be obtained by a new box exclusion test and by careful manipulations of the quadratic cost function. When the model is described by ODEs, some of the techniques available in the previous case may still be employed, provided that sensitivity functions of the model output with respect to the parameters are available

Global Optimisation for Dynamic Systems using Interval Analysis

Engineers seek optimal solutions when designing dynamic systems but a crucial element is to ensure bounded performance over time. Finding a globally optimal bounded trajectory requires the solution of the ordinary differential equation (ODE) systems in a verified way. To date these methods are only able to address low dimensional problems and for larger systems are unable to prevent gross overestimation of the bounds. In this paper we show how interval contractors can be used to obtain tightly bounded optima. A verified solver constructs tight upper and lower bounds on the dynamic variables using contractors for initial value problems (IVP) for ODEs within a global optimisation method. The solver provides guaranteed bound on the objective function and on the first order sensitivity equations in a branch and bound framework. The method is compared with three previously published methods on three examples from process engineering

A Branch and Bound Algorithm for Numerical MaxCSP

Abstract. The Constraint Satisfaction Problem (CSP) framework allows users to define problems in a declarative way, quite independently from the solving process. However, when the problem is over-constrained, the answer "no solution" is generally unsatisfactory. A Max-CSP Pm = V, D, C is a triple defining a constraint problem whose solutions maximise constraint satisfaction. In this paper, we focus on numerical CSPs, which are defined on real variables represented as floating point intervals and which constraints are numerical relations defined in extension. Solving such a problem (i.e., exactly characterizing its solution set) is generally undecidable and thus consists in providing approximations. We propose a branch and bound algorithm that computes under and over approximations of its solution set and determines the maximum number mP of satisfied constraints. The technique is applied on three numeric applications and provides promising results

Kodiak: An Implementation Framework for Branch and Bound Algorithms

Recursive branch and bound algorithms are often used to refine and isolate solutions to several classes of global optimization problems. A rigorous computation framework for the solution of systems of equations and inequalities involving nonlinear real arithmetic over hyper-rectangular variable and parameter domains is presented. It is derived from a generic branch and bound algorithm that has been formally verified, and utilizes self-validating enclosure methods, namely interval arithmetic and, for polynomials and rational functions, Bernstein expansion. Since bounds computed by these enclosure methods are sound, this approach may be used reliably in software verification tools. Advantage is taken of the partial derivatives of the constraint functions involved in the system, firstly to reduce the branching factor by the use of bisection heuristics and secondly to permit the computation of bifurcation sets for systems of ordinary differential equations. The associated software development, Kodiak, is presented, along with examples of three different branch and bound problem types it implements

Global optimisation for dynamic systems using novel overestimation reduction techniques

The optimisation of dynamic systems is of high relevance in chemical engineering as many practical systems can be described by ordinary differential equations (ODEs) or differential algebraic equations (DAEs). The current techniques for solving these problems rigorously to global optimality rely mainly on sequential approaches in which a branch and bound framework is used for solving the global optimisation part of the problem and a verified simulator (in which rounding errors are accounted for in the computations) is used for solving the dynamic constraints. The verified simulation part is the main bottleneck since tight bounds are difficult to obtain for high dimensional dynamic systems. Additionally, uncertainty in the form of, for example, intervals is introduced in the parameters of the dynamic constraints which are also the decision variables of the optimisation problem. Nevertheless, in the verified simulation the accumulation of trajectories that do not belong to the exact solution (overestimation) makes the state bounds overconservative and in the worst case they blow up and tend towards ±∞. In this thesis, methods for verified simulation in global optimisation for dynamic systems were investigated. A novel algorithm that uses an interval Taylor series (ITS) method with enhanced overestimation reduction capabilities was developed. These enhancements for the reduction of the overestimation rely on interval contractors (Krawczyk, Newton, ForwardBackward) and model reformulation based on pattern substitution and input scaling. The method with interval contractors was also extended to Taylor Models (TM) for comparison purposes. The two algorithms were tested on several case studies to demonstrate the effectiveness of the methods. The case studies have a different number of state variables and system parameters and they use uncertain amounts in some of the system parameters and initial conditions. Both of the methods were also used in a sequential approach to address the global optimisation for dynamic systems problem subject to uncertainty. The simulation results demonstrated that the ITS method with overestimation reduction techniques provided tighter state bounds with less computational expense than the traditional method. In the case of the forward-backward contractor additional constraints can be introduced that can potentially contribute significantly to the reduction of the overestimation. Similarly, the novel TM method with enhanced overestimation reduction capabilities provided tighter bounds than the TM method alone. On the other hand, the optimisation results showed that the global optimisation algorithm with the novel ITS method with overestimation reduction techniques converged faster to a rigorous solution due to the improved state bounds