A convenient solver for solving optimal control problems

Abstract

This paper focuses on the development of a solver for solving optimal control problems. A developed numerical optimal control module integrated with the Sequential Quadratic Programming method is introduced. An optimal control problem solver based on the proposed method is implemented to solve optimal control problems efficiently in engineering applications. In addition, a systematic procedure for solving optimal control problems by using the optimal control problem solver is also proposed. A time‐optimal benchmark problem presented in the literature is used to illustrate for the capability and facility of solving optimal control problems. The numerical results demonstrate the proposed method and the procedure suggested in this paper are helpful to engineers in solving optimal control problems in a systematic and efficient manner.     

An Enhanced Transcribing Scheme for the Numerical Solution of a Class of Optimal Control Problems

An enhanced scheme of transcribing the system dynamics for the numerical solution of optimal control problems is proposed. This new scheme is based on the standard method of direct collocation that converts an optimal control problem into a nonlinear programming problem via simultaneous state and control discretization. When compared with the standard method, the enhanced scheme has the advantage of higher solution accuracy with minimal additional computational effort. It is particularly suited for systems with states that are related to each other in a special form. For such systems, the ensuing nonlinear programming problem has the same number of constraints as those using the standard method. Numerical results on several optimal control problems using the enhanced scheme are presented, together with comparisons with the results obtained from the standard scheme.     

Combined interior‐point method and semismooth Newton method for frictionless contact problems

In the present paper, a solution scheme is proposed for frictionless contact problems of linear elastic bodies, which are discretized using the finite element method with lower order elements. An approach combining the interior‐point method and the semismooth Newton method is proposed. In this method, an initial active set for the semismooth Newton method is obtained from the approximate optimal solution by the interior‐point method. The simplest node‐to‐node contact model is considered in the present paper, that is, pairs of matching nodes exist on the contact surfaces. However, the discussions can be easily extended to a node‐to‐segment or segment‐to‐segment contact model. In order to evaluate the proposed method, a number of illustrative examples of the frictionless contact problem are shown. The proposed combined method is compared with the interior‐point method and the semismooth Newton method. Two numerical examples that are difficult to solve using the semismooth Newton method are solved effectively using the proposed combined method. It is shown that the proposed method converges within far fewer iterations than the semismooth Newton methods or the interior‐point method. Copyright © 2009 John Wiley & Sons, Ltd.     

Optimal tracking control of flow velocity in a one-dimensional magnetohydrodynamic flow

This article considers a dynamic optimization problem arising in a one-dimensional magnetohydrodynamic flow, which can be modelled by coupled partial differential equations, where the external control input (external induction of magnetic field) takes a multiplicative effect on the system states (momentum and magnetic components). The aim is to drive the flow velocity to within close proximity of a desired target of flow velocity at the pre-indicated terminal time. First, the Galerkin method is utilized to reduce the original dynamic optimization problem to a lower finite-dimensional dynamic optimization problem governed by a set of ordinary differential equations. Then the control parameterization method is employed to parameterize the finite-dimensional optimization problem and thus obtain an approximate optimal parameter selection problem, which can be solved using gradient-based optimization techniques, such as sequential quadratic programming. The exact gradients of the cost functional with respect to the decision parameters, which are the key advantages of this approach, are computed using the analytical equations. Finally, numerical examples are illustrated to verify the effectiveness of the proposed method. The novel scheme of this article is to develop an effective computational optimal control method for realizing the optimal tracking control of flow velocity in a one-dimensional magnetohydrodynamic flow.     

Feedback minimization of the first-passage failure of a hysteretic system under random excitations

The stochastic optimal bounded control of a hysteretic system for minimizing its first-passage failure is presented. The hysteretic system subjected to random excitation is firstly replaced by an equivalent nonlinear non-hysteretic system. The controlled non-hysteretic system is reduced to a one-dimensional controlled diffusion process by using the stochastic averaging of the energy envelope method. The dynamical programming equations and their associated boundary and final-time conditions for the problems of maximization of reliability and mean first-passage time are formulated. The optimal control law is derived from the dynamical programming equations and the control constraints. The dynamical programming equations for the maximum reliability problem and the mean first-passage time problem are finalized and solved numerically. Finally, numerical results are worked out to illustrate the application and effectiveness of the proposed method.     

异步切换下切换时变时滞系统的保成本控制

切换系统是一种重要的混杂系统,由若干子系统及决定子系统之间切换的切换信号组成。在工程应用中,控制器切换与子系统的切换会存在时延,即异步切换。研究了异步切换下的时变时滞系统的保成本控制问题,利用分段李雅普诺夫函数法和平均驻留时间法,得到保成本控制器存在的充分条件,并从线性矩阵不等式的角度,设计了一个异步切换保成本控制器,使得系统具有鲁棒性能。最后,给出了一个数值例子验证提出的方法的有效性。     

冷冻外科中一维相变问题的数值解

用变时间步长法求解了冷冻外科手术时模拟生物组织中的一维相变问题。计算方法经与文献比较,简便可行。运用本文的计算方法计算并讨论了不同边界条件下,相变区的温度场分布和无量纲相界面降温速率随Ｓｔｅ数的变化,计算分析表明,相变区的温度基本呈线性分布,无量纲相界面降温速率随Ｓｔｅ数增大而降低,所得结论可对冷刀设计和冷冻外科手术提供帮助。     

异步切换下切换系统的动态输出反馈保成本控制

作为一类重要的混杂系统,切换系统由若干个子系统以及一个协调各子系统之间切换的切换信号构成,在自然科学、工程控制和社会系统等方面有着广泛的应用。在对切换系统的控制问题进行研究时,一般假设子系统和控制器同步运行。然而,在实际工程控制中,控制器的切换相对于子系统的切换存在切换时延,从而产生异步切换。因此,对异步切换下的切换系统研究是十分必要的。针对一类异步切换下切换系统的动态输出反馈保成本控制问题进行了研究。报告了切换系统的研究现状以及异步切换下切换系统的最新研究成果。利用分段李雅普诺夫函数法和平均驻留时间法,得到了使得异步切换下的闭环切换系统稳定的动态输出反馈控制器存在的充分条件。从线性矩阵不等式的角度,提出了异步切换动态输出反馈保成本控制器设计方案,并给出成本上界的优化方法。最后,通过一个数值算例说明了提出的方法的有效性。     

Computing Switching Surfaces in Optimal Control Based on Triangular Decomposition

Various algorithms for optimal control require the explicit determination of switching surfaces. However, switching strategies may be very complicated, such that the computation of switching surfaces is quite challenging. General methods are proposed here to compute switching surfaces systematically, based on algebraic computational tools such as triangular decomposition. Our methods are highly complex compared to some widely-used numerical options, but they can be made feasible for real-time applications by moving the computational burden off-line. The tutorial-style presentation is intended to introduce potentially powerful symbolic computation methods to system scientists in particular, and an illustrative example of time-optimal control is given to show the effectiveness and generality of our approach.     

Solution of Inverse Boundary Optimization Problem by Trefftz Method and Exponentially Convergent Scalar Homotopy Algorithm

The inverse boundary optimization problem, governed by the Helmholtz equation, is analyzed by the Trefftz method (TM) and the exponentially convergent scalar homotopy algorithm (ECSHA). In the inverse boundary optimization problem, the position for part of boundary with given boundary condition is unknown, and the position for the rest of boundary with additionally specified boundary conditions is given. Therefore, it is very difficult to handle the boundary optimization problem by any numerical scheme. In order to stably solve the boundary optimization problem, the TM, one kind of boundary-type meshless methods, is adopted in this study, since it can avoid the generation of mesh grid and numerical integration. In the boundary optimization problem governed by the Helmholtz equation, the numerical solution of TM is expressed as linear combination of the T-complete functions. When this problem is considered by TM, a system of nonlinear algebraic equations will be formed and solved by ECSHA which will converge exponentially. The evolutionary process of ECSHA can acquire the unknown coefficients in TM and the spatial position of the unknown boundary simultaneously. Some numerical examples will be provided to demonstrate the ability and accuracy of the proposed scheme. Besides, the stability of the proposed meshless method will be validated by adding some noise into the boundary conditions.     

H(curl)空间中椭圆最优控制问题的自适应有限元算法

针对H(curl)空间椭圆型最优控制问题提出了一种自适应有限元方法。首先将H(curl)空间Maxwell方程的最优控制模型转化为偏微分方程组,给出了偏微分方程组的解得正则性。其次利用自适应有限元方法求解此偏微分方程组,同时讨论了方法的后验误差及收敛性。最后通过数值算例给出了该方法的数值结果,验证了有限元方法的有效性和可靠性。这一方法可以应用于更复杂的最优控制问题的求解。     

A direct boundary element approach for unsteady non-linear heat transfer problems

Unsteady non-linear problems are usually solved based on the time marching scheme together with iterative methods. In this paper, we employed an example to illustrate a new direct boundary element technique, which can provide accurate solutions of unsteady non-linear problems without the need for iterations. The principle and validity of the technique is demonstrated by considering an unsteady non-linear hyperbolic heat transfer problem, together with a fully implicit difference scheme in the time domain. It illustrates that the proposed non-iterative boundary element approach is numerically rather efficient for unsteady non-linear problems. Thus, it provides an alternative to traditional iterative methods for unsteady non-linear problems.     

脉冲风洞测力系统建模与载荷辨识方法研究

将载荷辨识技术应用于脉冲燃烧风洞模型测力。用子结构综合法建立了测力试验系统的动力学模型,在时域内将动力学方程进行离散,建立起天平测量信号与模型气动载荷历程之间的线性关系,作为载荷辨识的模型。采用Tikhonov正则化和子空间投影法相结合的混合正则化方法,将高维的、不适定的载荷辨识问题转化为低维的适定问题,以利于快速求解。提出了一种新方法来确定合适的投影子空间维数,然后应用L曲线准则来寻找低维正则化问题的最优正则化参数。最后通过算例验证了系统建模方法的精度和载荷辨识算法的有效性与稳定性。     

An exact method for designing Shewhart and S2 control charts to guarantee in-control performance

The in-control performance of Shewhart and S² control charts with estimated in-control parameters has been evaluated by a number of authors. Results indicate that an unrealistically large amount of Phase I data is needed to have the desired in-control average run length (ARL) value in Phase II. To overcome this problem, it has been recommended that the control limits be adjusted based on a bootstrap method to guarantee that the in-control ARL is at least a specified value with a certain specified probability. In this article we present simple formulas using the assumption of normality to compute the control limits and therefore, users do not have to use the bootstrap method. The advantage of our proposed method is in its simplicity for users; additionally, the control chart constants do not depend on the Phase I sample data.     

The method of fundamental solutions for inverse source problems associated with the steady‐state heat conduction

This paper presents the use of the method of fundamental solutions (MFS) for recovering the heat source in steady‐state heat conduction problems from boundary temperature and heat flux measurements. It is well known that boundary data alone do not determine uniquely a general heat source and hence some a priori knowledge is assumed in order to guarantee the uniqueness of the solution. In the present study, the heat source is assumed to satisfy a second‐order partial differential equation on a physical basis, thereby transforming the problem into a fourth‐order partial differential equation, which can be conveniently solved using the MFS. Since the matrix arising from the MFS discretization is severely ill‐conditioned, a regularized solution is obtained by employing the truncated singular value decomposition, whilst the optimal regularization parameter is determined by the L‐curve criterion. Numerical results are presented for several two‐dimensional problems with both exact and noisy data. The sensitivity analysis with respect to two solution parameters, i.e. the number of source points and the distance between the fictitious and physical boundaries, and one problem parameter, i.e. the measure of the accessible part of the boundary, is also performed. The stability of the scheme with respect to the amount of noise added into the data is analysed. The numerical results obtained show that the proposed numerical algorithm is accurate, convergent, stable and computationally efficient for solving inverse source problems in steady‐state heat conduction. Copyright © 2006 John Wiley & Sons, Ltd.     

A new numerical method for the boundary optimal control problems of the heat conduction equation

A new numerical method is developed for the boundary optimal control problems of the heat conduction equation in the present paper. When the boundary optimal control problem is solved by minimizing the objective function employing a conjugate‐gradient method, the most crucial step is the determination of the gradient of objective function usually employing either the direct differentiation method or the adjoint variable method. The direct differentiation method is simple to implement and always yields accurate results, but consumes a large amount of computational time. Although the adjoint variable method is computationally very efficient, the adjoint variable does not have sufficient regularity at the boundary for the boundary optimal control problems. As a result, a large numerical error is incurred in the evaluation of the gradient function, resulting in premature termination of the conjugate gradient iteration. In the present investigation, a new method is developed that circumvents this difficulty with the adjoint variable method by introducing a partial differential equation that describes the temporal and spatial dynamics of the control variable at the boundary. The present method is applied to the Neumann and Dirichlet boundary optimal control problems, respectively, and is found to solve the problems efficiently with sufficient accuracy. Copyright © 2001 John Wiley & Sons, Ltd.     

Job ordering and management of wearing tools

This article considers a scheduling problem arising in flexible manufacturing systems. It is assumed that a computer numerical control machine processes a set of jobs with a set of wearing tools. The tool magazine of the machine has a given capacity and each job requires some subset of tools. The goal is to minimize the average completion time of the jobs by choosing their processing order and the tool management decisions intelligently. Previous studies concerning this problem have either omitted the tool wearing or assumed only one tool type. This study gives a mathematical formulation for the problem when the tool lifetimes are deterministic. It is shown that problems of a practical size cannot be solved to optimality within a reasonable time. Therefore genetic algorithms and local search methods are considered to resolve the problem. When the solutions of these new algorithms are compared against the optimal solutions and lower bounds, they are nearly optimal.     

A Robust Phase I Exponentially Weighted Moving Average Chart for Dispersion

A Phase I estimator of the dispersion should be efficient under in‐control data and robust against contaminations. Most estimation methods proposed in the literature are either efficient or robust against either sustained shifts or scattered disturbances. In this article, we propose a new estimation method of the dispersion parameter, based on exponentially weighted moving average charting, which is efficient and robust to both types of unacceptable observations in Phase I. We compare the method with various existing estimation methods and show that the proposed method has the best overall performance if it is unknown what type of contaminations are present in Phase I. We also study the effect of the robust estimator from Phase I on the Phase II exponentially weighted moving average control chart performance. Copyright © 2014 John Wiley & Sons, Ltd.     

Fourth-Order Compact Difference Scheme for the Backward Heat Conduction Problem

Abstract

In this paper, the backward heat conduction problem is solved numerically using the fourth-order compact difference scheme. For regularization of this ill-posed problem, the quasi-reversibility technique is used. Compact finite difference approximation of the second derivative has been compared with the fourth-order standard finite difference approximation via Fourier analysis. The comparison shows that the resolution ability of the compact difference approximation is better than the standard finite difference approximation. The discrete dispersion relation for the compact difference scheme and finite difference scheme is obtained. Two test problems are considered for the numerical experiments. The CPU time taken by a fourth-order compact difference scheme is obtained and compared with the fourth-order standard finite difference scheme which shows that the fourth-order compact difference scheme takes lesser CPU time.     

A parallel bi-level multidisciplinary design optimization architecture with convergence proof for general problem

Quite a number of distributed Multidisciplinary Design Optimization (MDO) architectures have been proposed for the optimal design of large-scale multidisciplinary systems. However, just a few of them have available numerical convergence proof. In this article, a parallel bi-level MDO architecture is presented to solve the general MDO problem with shared constraints and a shared objective. The presented architecture decomposes the original MDO problem into one implicit nonlinear equation and multiple concurrent sub-optimization problems, then solves them through a bi-level process. In particular, this architecture allows each sub-optimization problem to be solved in parallel and its solution is proven to converge to the Karush–Kuhn–Tucker (KKT) point of the original MDO problem. Finally, two MDO problems are introduced to perform a comprehensive evaluation and verification of the presented architecture and the results demonstrate that it has a good performance both in convergence and efficiency.