A modified Benders decomposition method for efficient robust optimization under interval uncertainty

The goal of robust optimization problems is to find an optimal solution that is minimally sensitive to uncertain factors. Uncertain factors can include inputs to the problem such as parameters, decision variables, or both. Given any combination of possible uncertain factors, a solution is said to be robust if it is feasible and the variation in its objective function value is acceptable within a given user-specified range. Previous approaches for general nonlinear robust optimization problems under interval uncertainty involve nested optimization and are not computationally tractable. The overall objective in this paper is to develop an efficient robust optimization method that is scalable and does not contain nested optimization. The proposed method is applied to a variety of numerical and engineering examples to test its applicability. Current results show that the approach is able to numerically obtain a locally optimal robust solution to problems with quasi-convex constraints (≤ type) and an approximate locally optimal robust solution to general nonlinear optimization problems.     

Optimal interval controller design for uncertain systems

The interval controller design is a hot issue for uncertain systems, whereas how to design an optimal interval controller under the premise of ensuring system stability is a difficult problem that needs further study. This paper mainly aims at the single input single output uncertain system to propose an optimal interval controller based on the Kharitonov theorem and an interval optimization algorithm, which can guarantee the stability and optimization of a closed-loop interval system. According to the Kharitonov theorem, the optimal interval controller design can be transformed into an optimal controller synthesis issue of multiple vertex objects. An interval particle swarm optimization (IPSO) algorithm is then used to optimize the quadratic performance index with interval variables for each vertex object to obtain the solution domains of the controller parameters, and the vertex method is utilized to prevent interval width expansion or divergence in the iteration. Finally, the intersections of the solution domains for all vertex objects are obtained as the optimal interval solution of interval controller parameters. In addition, the stability verification approach of the closed-loop system and the empirical rule to select the interval particle width are given. Simulation results for typical examples demonstrate that the designed interval controller not only performs optimally but also can robustly stabilize the interval system.     

Pitch Angle Control of Unmanned Air Vehicle with Uncertain System Parameters

In this paper a new algorithm of trajectory tracking based on closest radius solution of the interval equations system is proposed. The design procedure is given and applied to the pitch angle control of unmanned testing rocket with uncertain parameters. The proposed algorithm gives a framework to design a control for a wide range of different linear time-invariant processes with uncertain parameters and can be implemented also in the case of non-convex problems. The algorithm gives the analytical way of finding the nearly optimal solution of model reference trajectory tracking in the case of general time-invariant systems with uncertain parameters and can be used when optimization method fails due to the complexity of the problem.     

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。     

Hydroelastic optimization of a keel fin of a sailing boat: a multidisciplinary robust formulation for ship design

The paper presents a formulation for multidisciplinary design optimization of vessels, subject to uncertain operating conditions. The formulation couples the multidisciplinary design analysis with the Bayesian approach to decision problems affected by uncertainty. In the present context, the design specifications are no longer given in terms of a single operating design point, but in terms of probability density function of the operating scenario. The optimal configuration is that which maximizes the performance expectation over the uncertain parameters variation. In this sense, the optimal solution is “robust” within the stochastic scenario assumed. Theoretical and numerical issues are addressed and numerical results in the hydroelastic optimization of a keel fin of a sailing yacht are presented.     

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．     

Uncertain Method for Optimal Control Problems With Uncertainties Using Chebyshev Inclusion Functions

In this paper, a new uncertain analysis method is developed for optimal control problems, including interval variables (uncertainties) based on truncated Chebyshev polynomials. The interval arithmetic in this research is employed for analyzing the uncertainties in optimal control problems comprising uncertain‐but‐bounded parameters with only lower and upper bounds of uncertain parameters. In this research, the Chebyshev method is utilized because it generates sharper bounds for meaningful solutions of interval functions, rather than the Taylor inclusion function, which is efficient in handling the overestimation derived from the wrapping effect due to interval computations. For utilizing the proposed interval method on the optimal control problems with uncertainties, the Lagrange multiplier method is first applied to achieve the necessary conditions and then, by using some algebraic manipulations, they are converted into the ordinary differential equation. Afterwards, the Chebyshev inclusion method is employed to achieve the solution of the system. The final results of the Chebyshev inclusion method are compared with the interval Taylor method. The results show that the proposed Chebyshev inclusion function based method better handle the wrapping effect than the interval Taylor method.     

服务率不确定的单站点传送带给料加工站系统鲁棒优化控制

研究了服务率不确定情况下的单站点传送带给料加工站(CSPS)系统的鲁棒优化控制问题。在仅知服务率区间的条件下,以CSPS系统的前视距离作为控制变量,将鲁棒优化控制问题建模成不确定参数的半马尔可夫决策过程(SMDP)的极大极小优化问题,在状态相关的情况下,给出全局优化算法进行鲁棒控制策略求解。首先,运用遗传算法求解固定策略下的最差性能值;其次,根据求解得到的最差性能值,运用模拟退火算法求解最优鲁棒控制策略。仿真结果表明,服务率不确定的CSPS系统的最优鲁棒性能代价与服务率固定为区间中值系统的最优性能代价相差不大,并且随着不确定区间的缩小,两者的差值越小,说明了全局优化算法的有效性。     

Digital redesign of uncertain interval systems based on extremal gain/phase margins via a hybrid particle swarm optimizer

In this paper, a hybrid optimizer incorporating particle swarm optimization (PSO) and an enhanced NM simplex search method is proposed to derive an optimal digital controller for uncertain interval systems based on resemblance of extremal gain/phase margins (GM/PM). By combining the uncertain plant and controller, extremal GM/PM of the redesigned digital system and its continuous counterpart can be obtained as the basis for comparison. The design problem is then formulated as an optimization problem of an aggregated error function in terms of deviation on extremal GM/PM between the redesigned digital system having an interval plant and its continuous counterpart, and subsequently optimized by the proposed optimizer to obtain an optimal set of parameters for the digital controller. Thanks to the performance of the proposed hybrid optimizer, frequency-response performances of the redesigned digital system using the digital controller evolutionarily derived by the proposed approach bare a far better resemblance to its continuous-time counter part in comparison to those obtained using existing open-loop discretization methods.     

Topology optimization of structure for dynamic properties considering hybrid uncertain parameters

In the design and manufacturing of mechanical components, the dynamic properties of continuum structure are one of the most significant performances. At the same time, the uncertainty is widespread in these dynamic problems. This paper presents a robust topology optimization methodology of structure for dynamic properties with consideration of hybrid uncertain parameters. The imprecise probability uncertainties including materials, geometry and boundary condition are treated as an interval random model, in which the probability distribution parameters of random variables are modeled as the interval variables instead of given precise values. Two dynamic properties, including dynamic-compliance and eigenvalue, are chosen as the objective function. In addition, different excitation frequency or eigenvalue is discussed. In this work, the bi-directional evolutionary structural optimization (BESO) method is adopted to find the optimal robust layout of the structure. A series of numerical examples is presented to illustrate the optimization procedure, and the effectiveness of the proposed method is demonstrated clearly.     

Interval analysis based robust truss optimization with continuous and discrete variables using mix-coded genetic algorithm

The problem of optimizing truss structures in the presence of uncertain parameters considering both continuous and discrete design variables is studied. An interval analysis based robust optimization method combined with the improved genetic algorithm is proposed for solving the problem. Uncertain parameters are assumed to be bounded in specified intervals. The natural interval extensions are employed to obtain explicitly a conservative approximation of the upper and lower bounds of the structural response, and hereby the bounds of the objective function and the constraint function. This way the uncertainty design may be performed in a very efficient manner in comparison with the probabilistic analysis based method. A mix-coded genetic algorithm (GA), where the discrete variables are coded with binary numbers while the continuous variables are coded with real numbers, is developed to deal with simultaneously the continuous and discrete design variables of the optimization model. An improved differences control strategy is proposed to avoid the GA getting stuck in local optima. Several numerical examples concerning the optimization of plane and space truss structures with continuous, discrete or mixed design variables are presented to validate the method developed in the present paper. Monte Carlo simulation shows that the interval analysis based optimization method gives much more robust designs in comparison with the deterministic optimization method.     

Reliability-based design optimization for problems with interval distribution parameters

Traditional reliability-based design optimization (RBDO) generally describes uncertain variables using random distributions, while some crucial distribution parameters in practical engineering problems can only be given intervals rather than precise values due to the limited information. Then, an important probability-interval hybrid reliability problem emerged. For uncertain problems in which interval variables are included in probability distribution functions of the random parameters, this paper establishes a hybrid reliability optimization design model and the corresponding efficient decoupling algorithm, which aims to provide an effective computational tool for reliability design of many complex structures. The reliability of an inner constraint is an interval since the interval distribution parameters are involved; this paper thus establishes the probability constraint using the lower bound of the reliability degree which ensures a safety design of the structure. An approximate reliability analysis method is given to avoid the time-consuming multivariable optimization of the inner hybrid reliability analysis. By using an incremental shifting vector (ISV) technique, the nested optimization problem involved in RBDO is converted into an efficient sequential iterative process of the deterministic design optimization and the hybrid reliability analysis. Three numerical examples are presented to verify the proposed method, which include one simple problem with explicit expression and two complex practical applications.     

Maximum variation analysis based analytical target cascading for multidisciplinary robust design optimization under interval uncertainty

Analytical target cascading (ATC) is a generally used hierarchical method for deterministic multidisciplinary design optimization (MDO). However, uncertainty is almost inevitable in the lifecycle of a complex system. In engineering practical design, the interval information of uncertainty can be more easily obtained compared to probability information. In this paper, a maximum variation analysis based ATC (MVA-ATC) approach is developed. In this approach, all subsystems are autonomously optimized under the interval uncertainty. MVA is used to establish an outer-inner framework which is employed to find the optimal scheme of system and subsystems. All subsystems are coordinated at the system level to search the system robust optimal solution. The accuracy and validation of the presented approach are tested using a classical mathematical example, a heart dipole optimization problem, and a battery thermal management system (BTMS) design problem.     

Quadratic optimization of motion coordination and control

Algorithms for continuous-time quadratic optimization of motion control are presented. Explicit solutions to the Hamilton-Jacobi equation for optimal control of rigid-body motion are found by solving an algebraic matrix equation. The system stability is investigated according to Lyapunov function theory and it is shown that global asymptotic stability holds. How optimal control and adaptive control may act in concert in the case of unknown or uncertain system parameters is shown. The solution results in natural design parameters in the form of square weighting matrices, as known from linear quadratic optimal control. The proposed optimal control is useful both for motion control, trajectory planning, and motion analysis     

Interval optimization of rotor-bearing systems with dynamic behavior constraints using an interval genetic algorithm

A new interval optimization algorithm is presented in this paper. In engineering, most optimization algorithms focus on exact parameters and optimum objectives. However, exact parameters are not easy to be manufactured to because of manufacturing errors and expensive manufacturing cost. To account for these problems, it is necessary to estimate interval design parameters and allowable objective error. This is the first paper to propose a new interval optimization algorithm within the context of Genetic Algorithms. This new algorithm, the Interval Genetic Algorithm (IGA), can neglect interval analysis and determines the optimum interval parameters. Furthermore, it can also effectively maximize the design scope. The optimizing ability of the IGA is tested through the interval optimization of a two-dimensional function. Then the IGA is applied to rotor-bearing systems. The results show that the IGA is effective in deriving optimal interval design parameters within the allowable error when minimizing shaft weight and/or transmitted force of rotor-bearing systems.     

Interval-model-based global optimization framework for robust stability and performance of PID controllers

PID controller structure is regarded as a standard in the control-engineering community and is supported by a vast range of automation hardware. Therefore, PID controllers are widely used in industrial practice. However, the problem of tuning the controller parameters has to be tackled by the control engineer and this is often not dealt with in an optimal way, resulting in poor control performance and even compromised safety. The paper proposes a framework, which involves using an interval model for describing the uncertain or variable dynamics of the process. The framework employs a particle swarm optimization algorithm for obtaining the best performing PID controller with regard to several possible criteria, but at the same time taking into account the complementary sensitivity function constraints, which ensure robustness within the bounds of the uncertain parameters’ intervals. Hence, the presented approach enables a simple, computationally tractable and efficient constrained optimization solution for tuning the parameters of the controller, while considering the eventual gain, pole, zero and time-delay uncertainties defined using an interval model of the controlled process. The results provide good control performance while assuring stability within the prescribed uncertainty constraints. Furthermore, the controller performance is adequate only if the relative system perturbations are considered, as proposed in the paper. The proposed approach has been tested on various examples. The results suggest that it is a useful framework for obtaining adequate controller parameters, which ensure robust stability and favorable control performance of the closed-loop, even when considerable process uncertainties are expected.     

考虑区间约束的物流网络双层规划模型及算法

考虑物流网络需求的不确定性,利用区间参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双层规划模型,设计了一种含区间参数与变量的递阶优化遗传算法,通过定义问题求解的风险系数与最大决策偏差,给出适合物流网络结构的区间运算准则,实现模型的确定性转化。以区间松弛变量与0-1决策变量定义初始种群,通过两阶遗传操作运算,求解不同情景下双层规划目标的区间最优解与节点决策方案。算例测试表明算法求解的可操作性更强,求解结果具有区间最优解与情景决策的优越性。     

基于相似度的多粒度不确定语言群决策方法

针对具有多粒度不确定语言评价信息的多属性群决策问题,提出了一种基于区间二元语义信息处理和矢量相似度的群决策方法,弥补了基于距离测度的决策方法易造成信息混淆的不足。该方法首先使用二元语义转换函数对多粒度区间语言评价信息进行一致化处理;然后通过建立使备选方案对正理想解相似度最大、负理想解相似度最小的最优化模型来获得相应的属性权重;最后利用区间二元语义的集结算子对评价信息进行加权集成,并通过优序数排序法实现对各方案的排序。实例分析说明了该方法的可行性和有效性。     

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．     

Structural optimal design of systems with imprecise properties: a possibilistic approach

The optimization problem of structural systems with imprecise properties on the basis of a possibilistic approach is considered. System imprecisions are defined by fuzzy numbers and characterized by membership functions. A methodology for the efficient solution of the optimization process is presented. A two-step method is used to include the imprecision within the optimization, where high quality approximations are used for the evaluation of structural responses. The approximations are constructed using concepts of intermediate response quantities and intermediate variables. The approach is basically an algebraic process which can be implemented very efficiently for the optimal design of general structural systems with imprecise parameters. The method provides more information to the designer than is available using conventional design tools. The effectiveness of the methodology and the interpretation of the results are illustrated by the solution of two example problems.