Robust optimization using a gradient index: MEMS applications

This paper discusses a simple and effective robust optimization formulation and illustrates its application to MicroElectroMechanical Systems (MEMS) devices. The proposed formulation improves robustness of the objective function by minimizing a gradient index (GI), defined as a function of gradients of performance functions with respect to uncertain variables. The level of constraint feasibility is also enhanced by adding a term determined by a constraint value and the gradient index. In the robust optimal design procedure, a deterministic optimization for performance improvement is followed by a sensitivity analysis with respect to uncertainties such as MEMS fabrication errors and changes of material properties. During the process of the deterministic optimization and sensitivity analysis, dominant performances and critical uncertain variables are identified to define the GI. Our approach for robust design requires no statistical information on the uncertainties and yet achieves robustness effectively. Two MEMS application examples including a micro accelerometer and a resonant-type micro probe are presented.     

Component mode synthesis-based design optimization method for local structural modification

This research integrates component mode synthesis with a multilevel design optimization strategy to develop a design methodology for making the dynamical modification of complex structures simpler and tractable. The component mode synthesis formulates the eigenvalue equation of the entire structure in terms of vibration characteristics of local structure components. With this particular feature, the component mode synthesis helps to facilitate a two-stage procedure for the dynamic modification of a complex structure; the lower-level design optimization modifies the local structure components whose performances are prescribed by the optimal solution of the upper-level design optimization. The paper first discusses the formulation and the derivation of the Kuhn-Tucker necessary conditions for the proposed design optimization procedure and then presents numerical examples to demonstrate its numerical implementation and applicability.     

Multidisciplinary optimization of a transport aircraft wing using particle swarm optimization

The purpose of this paper is to demonstrate the application of particle swarm optimization to a realistic multidisciplinary optimization test problem. The papers new contributions to multidisciplinary optimization are the application of a new algorithm for dealing with the unique challenges associated with multidisciplinary optimization problems, and recommendations for the utilization of the algorithm in future multidisciplinary optimization applications. The selected example is a bi-level optimization problem that demonstrates severe numerical noise and has a combination of continuous and discrete design variables. The use of traditional gradient-based optimization algorithms is thus not practical. The numerical results presented indicate that the particle swarm optimization algorithm is able to reliably find the optimum design for the problem presented. The algorithm is capable of dealing with the unique challenges posed by multidisciplinary optimization, as well as the numerical noise and discrete variables present in the current example problem.     

Numerical solution of a conspicuous consumption model with constant control delay

We derive optimal pricing strategies for conspicuous consumption products in periods of recession. To that end, we formulate and investigate a two-stage economic optimal control problem that takes uncertainty of the recession period length and delay effects of the pricing strategy into account.This non-standard optimal control problem is difficult to solve analytically, and solutions depend on the variable model parameters. Therefore, we use a numerical result-driven approach. We propose a structure-exploiting direct method for optimal control to solve this challenging optimization problem. In particular, we discretize the uncertainties in the model formulation by using scenario trees and target the control delays by introduction of slack control functions.Numerical results illustrate the validity of our approach and show the impact of uncertainties and delay effects on optimal economic strategies. During the recession, delayed optimal prices are higher than the non-delayed ones. In the normal economic period, however, this effect is reversed and optimal prices with a delayed impact are smaller compared to the non-delayed case.     

Energy supply planning and supply chain optimization under uncertainty

In energy supply planning and supply chain design, the coupling between long-term planning decisions like capital investment and short-term operation decisions like dispatching present a challenge, waiting to be tackled by systems and control engineers. The coupling is further complicated by uncertainties, which may arise from several sources including the market, politics, and technology. This paper addresses the coupling in the context of energy supply planning and supply chain design. We first discuss a simple two-stage stochastic program formulation that addresses optimization of an energy supply chain in the presence of uncertainties. The two-stage formulation can handle problems in which all design decisions are made up front and operating parameters act as ‘recourse’ decisions that can be varied from one time period to next based on realized values of uncertain parameters. The design of a biodiesel production network in the Southeastern region of the United States is used as an illustrative example. The discussion then moves on to a more complex multi-stage, multi-scale stochastic decision problem in which periodic investment/policy decisions are made on a time scale orders of magnitude slower than that of operating decisions. The problem of energy capacity planning is introduced as an example. In the particular problem we examine, annual acquisition of energy generation capacities of various types are coupled with hourly energy production and dispatch decisions. The increasing role of renewable sources like wind and solar necessitates the use of a fine-grained time scale for accurate assessment of their values. Use of storage intended to overcome the limitations of intermittent sources puts further demand on the modeling and optimization. Numerical challenges that arise from the multi-scale nature and uncertainties are reviewed and some possible modeling and numerical solution approaches are discussed.     

Process optimal design in non-isothermal backward extrusion of a titanium alloy by the finite element method

A new approach to process optimal design in non-isothermal, non-steady metal forming is presented. In this approach, an optimal design problem is formulated on the basis of an integrated thermo-mechanical finite element process model so as to treat diverse process parameters, either thermal or mechanical, as the design variables to be optimized, and a derivative based approach is adopted for conducting optimization. Described in detail are the integrated process model, a formulation for process optimal design, and the schemes for the evaluation of design sensitivity, in particular, a scheme for reflecting the effect of remeshing on design sensitivity. The validity of the schemes for the evaluation of design sensitivity is examined by performing a numerical test. Also examined is the integrated process model regarding its capability of predicting defect formation, through comparison with experimental observations. Then, the proposed optimal design technique is applied to process optimization in non-isothermal backward extrusion of a titanium alloy, with emphasis on preventing defect formation.     

区间不确定性需求下的D-LFLP模型及算法

考虑物流网络需求的不确定性,运用区间分析理念以区间数度量不确定性变量与参数,建立区间需求模式下的物流网络设计的混合整数规划模型,定义风险系数与最大约束偏差,对模型进行目标函数与约束条件的确定性转化,设计问题求解的区间递阶优化遗传算法,对不同情景状态下目标函数的区间最优解与节点决策方案进行运算。算例测试表明该算法可操作性更强,求解结果具有区间最优解与情景决策的优越性。     

PolyTop: a Matlab implementation of a general topology optimization framework using unstructured polygonal finite element meshes

We present an efficient Matlab code for structural topology optimization that includes a general finite element routine based on isoparametric polygonal elements which can be viewed as the extension of linear triangles and bilinear quads. The code also features a modular structure in which the analysis routine and the optimization algorithm are separated from the specific choice of topology optimization formulation. Within this framework, the finite element and sensitivity analysis routines contain no information related to the formulation and thus can be extended, developed and modified independently. We address issues pertaining to the use of unstructured meshes and arbitrary design domains in topology optimization that have received little attention in the literature. Also, as part of our examination of the topology optimization problem, we review the various steps taken in casting the optimal shape problem as a sizing optimization problem. This endeavor allows us to isolate the finite element and geometric analysis parameters and how they are related to the design variables of the discrete optimization problem. The Matlab code is explained in detail and numerical examples are presented to illustrate the capabilities of the code.     

Robust optimal decisions with imprecise forecasts

A robust minimax approach for optimal investment decisions with imprecise return forecasts and risk estimations in financial portfolio management is considered. Single-period and multi-period mean-variance optimization models are extended to worst-case design with multiple rival risk estimations and return forecasts. In multi-period stochastic formulation of classical mean-variance portfolio optimization problem, an investor makes an investment decision based on expectations and/or scenarios up to some intermediate times prior to the horizon and, consequently, rebalances or restructures the portfolio. Multi-period portfolio optimization entails the construction of a scenario tree representing a discretized estimate of uncertainties and associated probabilities in future stages. It is well known that return forecasts and risk estimations are inherently inaccurate and there are different rival estimates, or scenario trees. Robust optimization models are presented and imprecise nature of moment forecasts to reduce the risk of making a decision based on the wrong scenario is addressed. The worst-case performance is guaranteed in view of all rival risk and return scenarios and will only improve when any scenario other than the worst-case is realized. The ex-ante performance of minimax models is tested using historical data and backtesting results are presented.     

Integrated design optimization of base-isolated concrete buildings under spectrum loading

Base isolation has become a practical control strategy for protecting structures against seismic hazards. Most previous studies on the optimum design of base-isolated structures have been focused on the design optimization of either the base isolation or the superstructure. It is necessary to simultaneously optimize both the base isolation and the superstructure as a whole to seek the most cost-efficient design for such structures. This paper presents an effective numerical optimization technique for the seismic design of base-isolated concrete building structures under spectrum loading. Attempts have been made to automate the integrated spectrum analysis and design optimization procedure and to minimize the total cost of the base-isolated building subject to design performance criteria in terms of the interstory drifts of the superstructure and the lateral displacement of the isolation system. In the optimal design problem formulation, the cost of the superstructure can be expressed in terms of concrete member sizes while assuming all these members to be linearly elastic under earthquake actions. However, the isolation system is assumed to behave nonlinearly, and its cost can be related to the effective horizontal stiffness of each isolator. Using the principle of virtual work, the lateral drift responses of concrete base-isolated buildings can be explicitly formulated and the integrated optimization problem can be solved by the optimality criteria method. The technique is capable of achieving the optimal balance between the costs of the superstructure and the isolation system while the design performance criteria can be simultaneously satisfied. One practical building example with and without base isolation is used to illustrate the effectiveness of the optimal design technique.     

基于BMI的一类不确定分段线性系统的最优控制设计

将不确定分段线性系统的最优控制问题转化成最优控制性能界的优化问题．其中性能上界的优化是以反馈增益为寻优参数的一组双线性矩阵不等式(BMI)问题，对此将遗传算法和内点法结合， 设计了一种混合算法进行求解．最后的算例表明控制律的设计及其求解算法的有效性．     

A variable-accuracy metamodel-based architecture for global MDO under uncertainty

A method for simulation-based multidisciplinary robust design optimization (MRDO) of problems affected by uncertainty is presented. The challenging aspects of simulation-based MRDO are both algorithmic and computational, since the solution of a MRDO problem typically requires simulation-based multidisciplinary analyses (MDA), uncertainty quantification (UQ) and optimization. Herein, the identification of the optimal design is achieved by a variable-accuracy, metamodel-based optimization, following a multidisciplinary feasible (MDF) architecture. The approach encompasses a variable (i) density of the design of experiments for the metamodel training, (ii) sample size for the UQ analysis by quasi Monte Carlo simulation and (iii) tolerance for the multidisciplinary consistency in MDA. The focus is on two-way steady fluid-structure interaction problem, assessed by partitioned solvers for the hydrodynamic and the structural analysis. Two analytical test problems are shown, along with the design of a racing-sailboat keel fin subject to the stochastic variation of the yaw angle. The method is validated versus a standard MDF approach to MRDO, taken as a benchmark and solved by fully coupled MDA, fully converged UQ, without metamodels. The method is evaluated in terms of optimal design performances and number of simulations required to achieve the optimal solution. For the current application, the optimal configuration shows performances very close to the benchmark solution. The convergence analysis to the optimum shows a promising reduction of the computational cost.     

Optimization of a complex flexible multibody systems with composite materials

The paper presents a general optimization methodology for flexible multibody systems which is demonstrated to find optimal layouts of fiber composite structures components. The goal of the optimization process is to minimize the structural deformation and, simultaneously, to fulfill a set of multidisciplinary constraints, by finding the optimal values for the fiber orientation of composite structures. In this work, a general formulation for the computation of the first order analytical sensitivities based on the use of automatic differentiation tools is applied. A critical overview on the use of the sensitivities obtained by automatic differentiation against analytical sensitivities derived and implemented by hand is made with the purpose of identifying shortcomings and proposing solutions. The equations of motion and sensitivities of the flexible multibody system are solved simultaneously being the accelerations and velocities of the system and the sensitivities of the accelerations and of the velocities integrated in time using a multi-step multi-order integration algorithm. Then, the optimal design of the flexible multibody system is formulated to minimize the deformation energy of the system subjected to a set of technological and functional constraints. The methodologies proposed are first discussed for a simple demonstrative example and applied after to the optimization of a complex flexible multibody system, represented by a satellite antenna that is unfolded from its launching configuration to its functional state.     

Two-level optimization of airframe structures using response surface approximation

In this paper, a two-level optimization approach is developed for the preliminary and conceptual design of airframe structures. The preliminary design, involving a single objective multidisciplinary optimization, constitutes the lower level where ASTROS (Automated STRuctural Optimization System) is employed for multidisciplinary optimization. The conceptual design, which is carried out at the upper level, aims mainly at configuration design. The multiple objectives are incorporated as a single objective function by using the K-S function formulation. The objective function and constraints at the upper level are modelled through response surface approximation. During the upper level optimization process, the branch and bound method is applied for solving the problem with discrete design variables. The proposed strategy is demonstrated by the optimization of an Intermediate Complexity Wing (ICW) model. Received June 23, 1999     

An LMI approach to design robust fault detection filter for uncertain LTI systems

In this paper, the robust fault detection filter design problem for uncertain linear time-invariant (LTI) systems with both unknown inputs and modelling errors is studied. The basic idea of our study is to use an optimal residual generator (assuming no modelling errors) as the reference residual model of the robust fault detection filter design for uncertain LTI systems with modelling errors and, based on it, to formulate the robust fault detection filter design as an H_∞ model-matching problem. By using some recent results of H_∞ optimization, a solution of the optimization problem is then presented via a linear matrix inequality (LMI) formulation. The main results include the development of an optimal reference residual model, the formulation of robust fault detection filter design problem, the derivation of a sufficient condition for the existence of a robust fault detection filter and a construction of it based on the LMI solution parameters, the determination of adaptive threshold for fault detection. An illustrative design example is employed to demonstrate the effectiveness of the proposed approach.     

Structural anti-optimization with interval design parameters

The anti-optimization problem of structures with uncertain design variables is studied by combing the conventional optimization and interval analysis. The uncertain design parameters, which usually exist in the object function and constraint conditions, are modeled as interval sets. The proposed method can endure the variation of structural performance resulting from the variation of uncertain design parameters. According to the variation range of them, the range or interval of the optimal objective function and the optimal solution can be determined. In this sense, the optimal solution is one domain rather than a point. Numerical examples are used to illustrate the feasibility and superiority of the non-probabilistic optimization method in comparison with the conventional and probabilistic optimization methods.     

基于鲁棒可靠性的不确定时滞系统最优H∞控制器设计

用区间变量描述控制系统参数的不确定性,提出了不确定时滞系统鲁棒H_∞控制的鲁棒可靠性方法,基于鲁棒可靠性的不确定时滞系统最优状态反馈H_∞控制器设计方法,将系统的最优控制器设计归结为基于线性矩阵不等式(LMI)的优化问题．所设计的控制器可以在满足对所有不确定性鲁棒可靠的前提条件下,具有最优的H_∞鲁棒性能,并能在控制系统的设计中综合考虑控制性能、控制代价和鲁棒可靠性．数值算例说明了所提方法的有效性和可行性．     

不确定随机时滞系统的鲁棒H∞保性能控制

讨论了一类具有时变时滞的不确定It类型随机系统的鲁棒H∞保性能控制问题. 运用Lyapunov-Krasovskii泛函方法, 设计使得闭环系统鲁棒随机指数均方稳定, 且具有给定H∞干扰抑制度 γ 的状态反馈保性能控制器. 控制器存在的充分条件以线性矩阵不等式(LMIs)的形式表示. 进一步, 通过求解具有LMIs约束的凸优化问题, 给出了不确定随机时滞系统的最优保性能控制器的设计方法. 最后通过一个数值例子验证了所提方法的有效性.     

H ∞ Control for uncertain system with time-delay and nonlinear external disturbance via adaptive control method

This paper considers the H _∞ control problem for the uncertain time delay system with nonlinear external disturbance. Given a system containing nonlinear external disturbance, our purpose is to design a robust controller such that the uncertain system is asymptotically stable with a generalized H _∞ disturbance attenuation level ρ. In this paper we propose a new approach to design the controller for the uncertain system which is composed of a linear controller and an adaptive controller. The problem is solved by introducing a switching function and using the idea of formulation. Based on Lyapunov stability theory, new sufficient conditions are obtained in terms of linear matrix inequalities. The effectiveness and advantages of the proposed controller design are shown via a numerical example.     

Continuum topology optimization with non-probabilistic reliability constraints based on multi-ellipsoid convex model

Using a quantified measure for non-probab ilistic reliability based on the multi-ellipsoid convex model, the topology optimization of continuum structures in presence of uncertain-but-bounded parameters is investigated. The problem is formulated as a double-loop optimization one. The inner loop handles evaluation of the non-probabilistic reliability index, and the outer loop treats the optimum material distribution using the results from the inner loop for checking feasibility of the reliability constraints. For circumventing the numerical difficulties arising from its nested nature, the topology optimization problem with reliability constraints is reformulated into an equivalent one with constraints on the concerned performance. In this context, the adjoint variable schemes for sensitivity analysis with respect to uncertain variables as well as design variables are discussed. The structural optimization problem is then solved by a gradient-based algorithm using the obtained sensitivity. In the present formulation, the uncertain-but bounded uncertain variations of material properties, geometrical dimensions and loading conditions can be realistically accounted for. Numerical investigations illustrate the applicability and the validity of the present problem statement as well as the proposed numerical techniques. The computational results also reveal that non-probabilistic reliability-based topology optimization may yield more reasonable material layouts than conventional deterministic approaches. The proposed method can be regarded as an attractive supplement to the stochastic reliability-based topology optimization.