Distributed Control of the Stochastic Burgers Equation with Random Input Data

We discuss a control problem involving a stochastic Burgers equation with a random diffusion coefficient. Numerical schemes are developed, involving the finite element method for the spatial discretisation and the sparse grid stochastic collocation method in the random parameter space. We also use these schemes to compute closed-loop suboptimal state feedback control. Several numerical experiments are performed, to demonstrate the efficiency and plausibility of our approximation methods for the stochastic Burgers equation and the related control problem.     

Fast numerical methods for robust optimal design

A fast numerical approach for robust design optimization is presented. The core of the method is based on the state-of-the-art fast numerical methods for stochastic computations with parametric uncertainty. These methods employ generalized polynomial chaos (gPC) as a high-order representation for random quantities and a stochastic Galerkin (SG) or stochastic collocation (SC) approach to transform the original stochastic governing equations to a set of deterministic equations. The gPC-based SG and SC algorithms are able to produce highly accurate stochastic solutions with (much) reduced computational cost. It is demonstrated that they can serve as efficient forward problem solvers in robust design problems. Possible alternative definitions for robustness are also discussed. Traditional robust optimization seeks to minimize the variance (or standard deviation) of the response function while optimizing its mean. It can be shown that although variance can be used as a measure of uncertainty, it is a weak measure and may not fully reflect the output variability. Subsequently a strong measure in terms of the sensitivity derivatives of the response function is proposed as an alternative robust optimization definition. Numerical examples are provided to demonstrate the efficiency of the gPC-based algorithms, in both the traditional weak measure and the newly proposed strong measure.     

A polynomial chaos‐based approach to risk‐averse piezoelectric control of random vibrations of beams

This paper proposes a risk‐averse formulation for the problem of piezoelectric control of random vibrations of elastic structures. The proposed formulation, inspired by the notion of risk aversion in economy, is applied to the piezoelectric control of a Bernoulli‐Euler beam subjected to uncertainties in its input data. To address the high computational burden associated to the presence of random fields in the model and the discontinuities involved in the cost functional and its gradient, a combination of a nonintrusive anisotropic polynomial chaos approach for uncertainty propagation with a Monte Carlo sampling method is proposed. In the first part, the well‐posedness of the control problem is established by proving the existence of optimal controls. In the second part, an adaptive gradient‐based method is proposed for the numerical resolution of the problem. Several experiments illustrate the performance of the proposed approach and the significant differences that may occur between the classical deterministic formulation of the problem and its stochastic risk‐averse counterpart.     

Supersensitivity due to uncertain boundary conditions

We study the viscous Burgers' equation subject to perturbations on the boundary conditions. Two kinds of perturbations are considered: deterministic and random. For deterministic perturbations, we show that small perturbations can result in O(1) changes in the location of the transition layer. For random perturbations, we solve the stochastic Burgers' equation using different approaches. First, we employ the Jacobi‐polynomial‐chaos, which is a subset of the generalized polynomial chaos for stochastic modeling. Converged numerical results are reported (up to seven significant digits), and we observe similar ‘stochastic supersensitivity’ for the mean location of the transition layer. Subsequently, we employ up to fourth‐order perturbation expansions. We show that even with small random inputs, the resolution of the perturbation method is relatively poor due to the larger stochastic responses in the output. Two types of distributions are considered: uniform distribution and a ‘truncated’ Gaussian distribution with no tails. Various solution statistics, including the spatial evolution of probability density function at steady state, are studied. Copyright © 2004 John Wiley & Sons, Ltd.     

Dynamic response and reliability analysis of structures with uncertain parameters

An original approach for dynamic response and reliability analysis of stochastic structures is proposed. The probability density evolution equation is established which implies that incremental rate of the probability density function is related to the structural response velocity. Therefore, the response analysis of stochastic structures becomes an initial‐value partial differential equation problem. For the dynamic reliability problem, the solution can be derived through solving the probability density evolution equation with an initial value condition and an absorbing boundary condition corresponding to specified failure criterion. The numerical algorithm for the proposed method is suggested by combining the precise time integration method and the finite difference method with TVD scheme. To verify and validate the proposed method, a SDOF system and an 8‐storey frame with random parameters are investigated in detail. In the SDOF system, the response obtained by the proposed method is compared with the counterparts by the exact solution. The responses and the reliabilities of a frame with random stiffness, subject to deterministic excitation or random excitation, are evaluated by the proposed method as well. The mean, the standard deviation and the reliabilities are compared, respectively, with the Monte Carlo simulation. The numerical examples verify that the proposed method is of high accuracy and efficiency. Moreover, it is found that the probability transition of structural responses is like water flowing in a river with many whirlpools, showing complexity of probability transition process of the stochastic dynamic responses. Copyright © 2004 John Wiley & Sons, Ltd.     

单轴机械臂系统的H∞控制及精细算法研究

针对机械臂系统在Hamilton体系下,给出一种有效的H∞控制策略和精细的数值计算方法。首先建立了系统的动力学方程,进而利用控制理论中针对扰动衰减问题的H∞控制策略对动力系统进行了控制研究。在数值模拟过程中,摒弃了传统的差分类计算方法,采用建立在线性系统的时程精细积分算法对状态方程及代数黎卡提方程进行了求解,最后通过数值例题说明了文中方法的有效性。     

A robust local polynomial collocation method

In this paper, a robust local polynomial collocation method is presented. Based on collocation, this method is rather simple and straightforward. The present method is developed in a way that the governing equation is satisfied on boundaries as well as boundary conditions. This requirement makes the present method more accurate and robust than conventional collocation methods, especially in estimating the partial derivatives of the solution near the boundary. Studies about the sensitivity of the shape parameter and the local supporting range in the moving least square approach and the convergence of the nodal resolution are carried out by using some benchmark problems. This method is further verified by applying it to a steady‐state convection–diffusion problem. Finally, the present method is applied to calculate the velocity fields of two potential flow problems. More accurate numerical results are obtained.Copyright © 2012 John Wiley & Sons, Ltd.     

ROBUSTNESS OPTIMIZATION FOR CONSTRAINED NONLINEAR PROGRAMMING PROBLEMS∗

In realistic situations engineering designs should take into consideration random aberrations from the stipulated design variables arising from manufacturing variability. Moreover, many environmental parameters are often stochastic in nature. Traditional nonlinear optimization attempts to find a deterministic optimum of a cost function and does not take into account the effect of these random variations on the objective. This paper attempts to devise a technique for finding optima of constrained nonlinear functions that are robust with respect to such variations. The expectation of the function over a domain of aberrations in the parameters is taken as a measure of ‘robustness’ of the function value at a point. It is pointed out that robustness optimization is ideally an attempt to trade off between ‘optimality’ and ‘robustness’. A newly-developed multi-criteria optimization technique known as Normal-Boundary Intersection is used to find evenly-spaced points on the Pareto curve for the ‘optimality’ and ‘robustness’ criteria. This Pareto curve enables the user to make the trade-off decision explicitly, free of arbitrary ‘weighting’ parameters.

This paper also formulates a derivative-based approximation for evaluating the expected value of the objective function on the nonlinear manifold defined by the state equations for the system. Existing procedures for evaluating the expectation usually involve numerical integration techniques requiring many solutions of the state equations for one evaluation of the expectation. The procedure presented here bypasses the need for multiple solutions of the state equations and hence provides a cheaper and more easily optimizable approximation to the expectation. Finally, this paper discusses how nonlinear inequality constraints should be treated in the presence of random parameters in the design. Computational results are presented for finding a robust optimum of a nonlinear structural optimization problem.     

Intelligent dynamic control of stochastic economic lot scheduling by agent-based reinforcement learning

This paper addresses a stochastic economic lot scheduling problem (SELSP) for a single machine make-to-stock production system in which the demands and the processing times for N types of products are random. The sequence-independent setup times and costs are explicitly considered and may have different values for various types of products. The SELSP is to decide when, what, and how much (the lot size) to produce so that the long-run average total cost, including setup, holding and backorder costs, is minimised. We develop a mathematical model and propose two reinforcement learning (RL) algorithms for real-time decision-making, in which a decision agent is assigned to the machine and improves the accuracy of its action-selection decisions via a ‘learning’ process. Specifically, one is a Q-learning algorithm for a semi-Markov decision process (QLS) and another is a Q-learning algorithm with a learning-improvement heuristic (QLIH) to further improve the performance of QLS. We compare the performance of QLS and QLIH with a benchmarking Brownian policy and the first-come-first-served policy. The numerical results show that QLIH outperforms QLS and both benchmarking policies.     

A Dendrite Net based decoupled framework for the reliability-based control co-design problem

To improve the reliability of the dynamic system including physical and control design, the reliability-based control co-design (RB-CCD) problem has been studied to account for the uncertainty stemming from the random physical design. However, when encountering RB-CCD in the sophisticated system in which the dynamic model simulation is time-consuming or the state equation is expressed implicitly, the available RB-CCD methods will consume significant computational effort to perform numerous system simulations for the reliability analysis and deterministic optimization. Therefore, this work proposes a Dendrite Net-based decoupled framework for RB-CCD to alleviate the computational burden. Specifically, the Dendrite (DD) model constructed by the suggested training scheme integrated with an adaptive sampling strategy is used to approximate the state equation in the dynamic system. After that, the sequential optimization and reliability assessment method decouples RB-CCD into the control co-design (CCD) problem and time-dependent reliability assessment problem, which are solved sequentially based on the cheap estimations of DD model, rather than the expensive simulations of the original system. Furthermore, two numerical examples and an engineering example of 3-DOF robot system are applied to demonstrate the feasibility and efficiency of the proposed framework.     

A periodic-review inventory model in a fluctuating environment

We study a single-item periodic-review inventory model in a fluctuating environment with a fixed lead time of λ periods. The state of the environment at the beginning of each period is described by a homogeneous Markov chain. Ordering, holding, penalty costs and the distributions of random variables representing the customer's demand and the supplier's capacity level are state environment dependent. By using dynamic programming it is proved in the finite-horizon case that the optimal policy is of a base stock type. Its parameters are monotonic in the number of the periods making up the horizon and also in stochastically ordered random variables representing different supplier capacities. Similar results are proved for the infinite-horizon problem.     

ROBUST TOLERANCE ALLOCATION USING STOCHASTIC PROGRAMMING

Abstract

Tolerance allocation in manufacturing is a prominent industrial task for enhancing productivity and reducing manufacturing costs. The classical tolerance allocation problem can be formulated as a stochastic program to determine the assignment of component tolerances such that the manufacturing cost is minimized. However, tolerance design is a prerequisite to the overall quality and cost of a product; robust tolerance design is particularly important and should be considered. In this paper, robustness is considered in formulating the tolerance allocation problem by minimizing the manufacturing cost's sensitivity. Moreover, from a practical perspective, the process capability index for each component and the upper bound of the manufacturing cost are also considered. To effectively and efficiently resolve the robust tolerance allocation problem, a sequential quadratic programming algorithm embedded with a Monte Carlo simulation is developed. To demonstrate this design method's robustness, two commonly used test problems are solved. The designs devised in this paper have lower manufacturing costs and smaller variations in manufacturing costs than those in previous studies, indicating that the proposed method is highly promising in the robust tolerance design.     

Robust optimization applied to uncertain production loading problems with import quota limits under the global supply chain management environment

Global supply chain management presents some special challenges and issues for manufacturing companies in planning production: these challenges are different from those discussed in domestic production plans. Globally loading production among different plants usually involves substantial uncertainty and great risk because of uncertain market demand, fluctuating quota costs incurred in the global manufacturing process, and shortening lead times. This study proposes a dual-response production loading strategy for two types of plants—company-owned and contracted—to hedge against the short lead time and uncertainty, and to be as responsive and flexible as possible to cope with the uncertainty and risk involved. Three types of robust optimization models are presented: the robust optimization model with solution robustness, the robust optimization model with model robustness, and the robust optimization model with the trade-off between solution robustness and model robustness. A series of experiments are designed to test the effectiveness of the proposed robust optimization models. Compared with the results of the two-stage stochastic recourse programming model, the robust optimization models provide a more responsive and flexible system with less risk, which is particularly important in the current context of global competitiveness.     

Adaptive reduced order modeling for nonlinear dynamical systems through a new a posteriori error estimator: Application to uncertainty quantification

Multiquery problems such as uncertainty quantification (UQ), optimization of a dynamical system require solving a differential equation at multiple parameter values. Therefore, for large systems, the computational cost becomes prohibitive. This issue can be addressed by using a cheaper reduced order model (ROM) instead. However, the ROM entails error in the solution due to approximation in a lower dimensional subspace. Moreover, the ROM lacks robustness over a wide range of parameter values. To address these issues, first, an upper bound on the norm of the state transition matrix is derived. This bound, along with the residual in the governing equation, are then used to develop an error estimator for general nonlinear dynamical systems. Furthermore, this error estimator is used in conjunction with the modified greedy search algorithm proposed by Hossain and Ghosh (Int J Numer Methods Eng, 2018;116(12-13): 741-758) to adaptively construct a robust proper orthogonal decomposition-based ROM. This adaptive ROM is subsequently deployed for UQ by invoking it in a statistical simulation. Two numerical studies: (i) viscous Burgers' equation and (ii) beam on nonlinear Winkler foundation, showed an improved accuracy of the error estimator compared to the current literature. A significant computational speed-up in UQ is achieved.     

Robust topology optimization of optical cloaks under uncertainties in wave number and angle of incident wave

This article presents a robust topology optimization method for optical cloaks under uncertainties in the wave number and angle in the incident wave. We first discuss the governing equation derived from Maxwell's equation, and extend it to the entire domain including the dielectric material and air, based on the level set-based topology optimization method. Next, a robust optimization problem is formulated as a minimization problem of the weighted sum of the scattered wave norm and its standard deviation with respect to the wave number and angle of the incident wave. The standard deviation is mathematically expressed by the Taylor series approximation and the use of the adjoint variable method. The design sensitivity of the objective functional is also derived by the adjoint variable method. An optimization algorithm is then constructed, based on the proposed formulation for robust designs of optical cloaks. Several numerical examples are finally provided to demonstrate the validity and utility of the proposed method.     

Passivity-based control of linear switched reluctance motors with robustness consideration

A robust passivity-based control (PBC) algorithm is proposed for the position tracking system of a linear switched reluctance motor (LSRM). By using the modelling analysis of the drive system, a full-order nonlinear controlled model is first developed. Then, on the basis of the state error equation, the proposed robust PBC algorithm is derived from the view points of energy dissipation, control stability and algorithm robustness. The resultant design provides a total structural solution for the control law with the winding excitation scheme integrated into the algorithm. The proposed algorithm guarantees global stability of the whole servo system. It can also overcome the inherent nonlinear characteristics of the system and make the whole system robust to uncertainties and bounded disturbances. Both simulations and experimental implementations are carried out on the proposed LSRM drive system to investigate the performance of the proposed algorithm. The results show that the experimental and the simulation outputs match very well. The proposed algorithm is effective for the high precision position tracking of the LSRM, with high robustness to the system uncertainties and bounded disturbances.     

An exact solution of the first-exit time problem for a class of structural systems

The mean time to escape from a region of desired operations is one the basic reliability measures in stochastic dynamics. In general, a precise solution of the first-exit time problem is unavailable. This paper demonstrates an exact solution of the mean exit time problem for a multidimensional non-dissipative Lagrangian system excited by additive Gaussian white noise. We identify the Fokker–Planck equation whose solution characterizes the mean time needed to reach a critical energy and explicitly construct the solution. For illustration, we apply the developed theory to engineering examples. We calculate the mean time of the standard operation for a flexural nanotube with likely noise-induced buckling and analyze the mean time of the stable functioning for a gyroscope subjected to random and dissipation torques. It is demonstrated that the solution of the first-exit time problem for a non-dissipative system gives a quite good approximation to a numerical solution of a similar problem for a system with small dissipation.     

迟滞的磁流变阻尼器的随机最优控制力

用Bouc-Wen迟滞模型描述磁流变阻尼器的动力学特性,分离阻尼器控制力的半主动部分和被动部分,被动部分结合到受控系统.先将该系统变换成等价的非迟滞的非线性随机控制系统,再运用随机平均法导出关于能量的It随机微分方程.根据随机动态规划原理,建立控制总能量的动态规划方程,并由此确定非clip的最优控制力.最后通过数值结果表明该控制力的有效性.     

Robust production control policy for a multiple machines and multiple product-types manufacturing system with inventory inaccuracy

Inventory inaccuracy often exists in manufacturing systems, which has great negative impact on the performance of production control, e.g. very high work-in-process holding cost or backlog penalty. To hedge against inventory inaccuracy, the robust production control problems will be investigated for a multiple machines and multiple product-types manufacturing system with uncertain production capacity. The objective of our problem is to minimise the average production cost. To solve this problem, a robust production control policy is developed, which is insensitive to the inventory record errors, and whose robustness is better than the traditional hedging point policy for optimal production control. Finally, numerical experiments are conducted to examine the performance of the proposed robust production control policy against inventory inaccuracy. Based on the experimental results, the conditions of applying the proposed policy are also obtained.     

Incorporation of a positivity constraint into a Kalman-filter-basedalgorithm for correction of spectrometric data

Improving the resolution of spectrometric analyses by numerical processing of spectrometric data subject to systematic errors of an instrumental type, as well as to random errors, is addressed. It is assumed that the model of the spectrometric data has the form of an integral, convolution-type equation of the first kind. The method for improving the resolution consists of numerically solving this equation on the basis of the acquired data. A new algorithm for dealing with this problem is proposed; it is based on the Kalman filter constrained in such a way that the negative values of the solution are suppressed. The efficiency of this constrained algorithm is demonstrated using both synthetic and real-world data