序列线性规划移动限搜索策略及结合输入整形的PD控制研究
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摘要
随着科技不断发展,结构优化设计已经成为现代固体力学的一个重要分支,在航空航天、机械制造、汽车等众多工程领域中有着广泛应用。
工程实际中结构优化问题通常为大型、高阶、非线性隐式问题,其分析过程复杂且需花费大量时间,能否有效的解决此类问题除了需要建立合理的优化问题数学模型之外,关键在于选择合理、有效的优化算法。以序列线性规划方法为代表的序列近似规划方法由于具有坚实的理论基础和普适性,目前已被广泛应用于工程结构优化领域及通用结构优化软件中。序列线性规划方法通过泰勒展开式将非线性规划问题转化为一系列的线性规划问题,从寻优的角度出发近似求解此线性规划问题,即可获得原问题的近似最优解。然而,应用传统序列线性规划法求解结构优化问题时效率较低,并受限于问题规模。为有效解决大型工程结构优化问题,研究适用序列线性规划并具备良好收敛性、稳定性、高效性的算法类,具有重要的理论意义和实际应用价值。
在实际工程问题中含有刚性模态的弹性机构在机械结构中占有很大的比重,快速加、减速运动时将会引起载荷摆动,造成载荷难以快速就位。特别是飞行器、大型起重机械、弹性机器人等具有挠性机械结构的系统,在受到阶跃输入信号作用的情况下,将导致刚性体运动与弹性体运动相互耦合,这是因为弹性体在发生刚性体运动时将产生弹性形变,同时产生残留振动,从而使整个系统难以精确定位。为达到抑制弹性机构残留振动的目的,通常采用的输入整形是合理地、计划地利用时滞加以改善弹性机构的动力学特性。针对含有刚性模态的弹性机构,本文提出了基于PD结合输入整形控制策略,通过对PD控制器参数与系统振动频率和阻尼比的关系进行分析,研究了输入整形参数对系统控制力的作用规律以及降低系统能耗的机理。在反馈增益相同的情况下,PD结合输入整形控制策略与PD反馈控制策略相比,能够更好的使系统产生较小的控制力,有效地降低了系统的能耗,大大提高了系统的响应速度。
本文主要研究内容:
1.概述了序列线性规划方法的基本理论、优化流程。随后,通过分析序列线性规划方法的特点,扬长避短,为解决算法收敛慢、计算量大等问题,研究在可行性空间中约束系列线性搜索的移动限概念。根据移动限控制变量的变化范围,设计提高算法的收敛性,并通过数值算例对此进行验证,为下一章如何精确调整移动限提供了依据。
2.提出一种基于线性误差和信赖域思想的改进序列线性规划算法(D-SLP)来解决移动限对线性近似精度的影响。在迭代过程中,算法基于线性近似误差概念评估近似模型与真实模型之间的差异,随着线性搜索,值函数近似程度不断提高,使得优化空间内设计变量的移动限范围不断扩大,有效地提高了计算精度;同时,根据信赖域的原理构造自适应调整步长的方案,用以判别移动限是否被接受,该方法大大加快了算法的收敛速度,节省了计算成本。此算法原理简单直观,具有较强的收敛性和鲁棒性。
3.设计了基于信赖域思想的内点序列线性规划算法(IP-TR-SLP)。目前内点法和单纯形法是有效求解线性规划问题的两种主要途径,其中单纯形法沿着约束边界移动寻找最优解,而内点法则沿着中心路径寻优。当优化问题约束条件较多、维数较高时,内点法求解效率和收敛性要明显优于单纯形法,更适于解决工程实际问题。本章围绕如何进一步提高内点法的计算效率这一目标,应用补偿间隙能够反映控制变量对不等式约束是否满足这一判定条件,将补偿间隙作为罚函数引入到目标函数当中;同时,通过引入信赖域思想强制限定每次迭代时新的迭代点与当前迭代点之间的距离,保证整个优化过程中近似模型的可靠。虽然每次迭代时计算量并未减少,但是基于信赖域控制的改进内点法有效缩短了整个优化过程的计算时间,且不需要给定初始点的可行性,从而拓宽了算法在大型工程实际优化问题中的应用范围。
4.采用本文提出的D-SLP算法和IP-TR-SLP算法,对经典桁架结构算例进行位移、应力以及频率约束下的尺寸优化,以验证算法的精度及效率。通过对10杆、25杆、72杆桁架结构算例的计算,证明了本文提出的两种改进算法求解结构优化问题是可行的,且计算精度优于已有文献提出的算法的计算结果;200杆桁架结构算例验证了本文提出算法对求解大型结构优化问题时具有高效性。数值优化结果表明,在10杆和25杆小型桁架结构算例中,基于内点法寻优的IP-TR-SLP算法的效率略低于基于单纯形法寻优的D-SLP算法;而在求解200杆的大型桁架结构时,IP-TR-SLP算法的运算效率明显优于D-SLP算法。这是由于基于内点法的IP-TR-SLP算法虽然在每次迭代过程中耗费大量计算时间,但算法结合信赖域思想自适应地调整步长,从而使得整体计算效率大大提高,这种情况在大型结构优化时尤为突出。最后,将本文提出的改进算法应用于结构拓扑优化问题,并对二维平面结构进行有效的拓扑优化设计,拓展了序列线性规划算法的求解领域。
5.在研究含有刚性模态弹性机构的动力学特性的基础上,将时滞环节引入到控制系统当中,提出PD结合输入整形控制策略。并且在阶跃输入信号作用下研究PD结合输入整形控制系统的振动特性,分析了输入整形对系统控制力和能耗的作用规律。在反馈增益相同的情况下,PD结合输入整形控制策略与PD反馈控制相比,使系统控制力降低56%,能耗降低47%,有效的提高了系统的响应速度。
最后对全文研究内容进行总结,并为今后的研究提出展望。
As along the development of technology, structural optimization has been an important branch of solid mechanics nowadays which is widely applied in many engineering fields, such as aerospace, mechanical manufacturing, vehicle and so on.
Generally, there are always large-scale, high-order and nonlinear problems in the optimization actual structures which cost lots of time to analyze. Whether we can solve these problems except depending on constructing the reasonable mathematic model, choosing an available optimization method is also a key point. As a kind of most effective and popular sequential approximate programming, sequential linear programming (SLP) has been widely used in universal optimization software and structural optimization field. The main idea of SLP is transforming the nonlinear problems into linear ones by Taylor expansion. By means of solving the linear problems parelly, one can finally obtain the optimum solution of the original problem. However, traditional SLP method is restricted by the scales of problem so that it converges slowly. For solving the large-scale structural optimization problems, it is necessary to research an improved SLP method in order to enhance the convergence, stability and validity of original SLP. The research will be very meaningful and valuable in the engineering application.
In practical engineering problems, the elastic mechanism with rigid modes accounts for a large proportion in mechanical structure. The movement of acceleration and deceleration will cause the load swing, resulting that the load is difficult to replace. Particularly in the system of flexible mechanical structure such as aircraft, large scale hoisting machinery and flexible robots, when the step input signal role under control, the system are coupling between rigid body motion and elastic motion, which is due to the elastic stiffness in the event of body movement transforming to elastic deformation and producing residual vibration, resulting that the entire system is difficult to pinpoint. Commonly, we use input-shaping to suppress residual vibration of elastic bodies on purpose, which is reasonable and well-planned to make use of the time-delay dynamics of elastic bodies. The dynamical behavior of PD combined with input-shaping control system, subject to step inputs, is investigated. The actuator efforts and energy usage of PD combined with input-shaping control system are described in general formula. Given the condition of the same feedback as PD, the PD combined with input shaping control minimizes actuator effort, saves energy, and speeds system response.
The main work of this paper is presented followed:
1. First of all the paper reviews the basic theory and the optimization process of SLP. Then it studies the moving limit concept of constraint in feasibility space to improve the convergence of SLP through sequence linear programming method. According to the size of move limit, it is designed to control the convergence of SLP. Numerical examples are verified the affective of move limit which is the foundation of the theory in next chapter.
2. In order to overcome the approximate accuracy of SLP, we improves SLP algorithm affected by move limit, based on linear error and proposed trust region. At the iteration, the error between approximate model and true model is evaluated by approximately linear concept. Along with the linearly searching, the approximate accuracy is enhanced greatly, resulting in the size of move limit in the design space broadened and the accuracy improved naturally. Additionally, step size is adaptively adjusted, which can be used to choose whether accept the move limit, therefore the convergence process is speeded up. This algorithm is simple with strong convergence and robust.
3. In this paper, it is proposed an improved SLP method (IP-TR-SLP), which solves the linear optimization problems by interior-point method basing on the concept of trust region. Nowadays, interior-point method and simplex method are the two main ways to solve the linear optimization problems. Simple method moves alone the boundary to search the optimum solution, while the interior-point method searches along the central path. When the problems are large-scale, including many constraint conditions, the efficiency and convergence of interior-point method will be better than the simple method. Thereby the interior-point method is fitting for solving practical optimization problems. This paper focus on how to improve the efficiency of interior-point method, using complementary interval as a penalty function into the objective function. Meanwhile, one can restrict the distance between the new iterative point and current iterative point based on the trust region in each iteration to ensure the reliability of approximate models during the iteration. Although the computational cost in each iteration does not decrease, (IP-TR-SLP) method shortens the whole computational time during optimization and there is no necessary constraint the initial feasible point that (IP-TR-SLP) method will applied expanded to solve the practical optimization problems, especially in large scale cases.
4. The proposed D-SLP and IP-TR-SLP algorithms are introduced to verify the size optimization problem of classical truss structure with displacement, stress, and frequency constraints. Through the calculation of 10-bar, 25-bar and 72-bar, it is showed that these two improved algorithms are feasible and the accuracy of the numerical examples is not lower than that of those existed references. The efficiency of these two methods are tested by 200-bar examples. The data optimization of results shows that, in the calculation of 10-bar and 25-bar, the efficiency of IP-TR-SLP is lower than that of D-SLP, however, for large-scale structure, IP-TR-SLP is superior to D-SLP. Although IP-TR-SLP algorithm spent much time at every iterative step, it do speed up the whole optimization process due to the adaptive adjustment of step size based on trust region, apparently in large-scale structure optimization. As a result, the solution field of SLP is broadened by these two algorithms applying in topology optimization.
5. Based on researching the dynamical performances of the flexible mechanisms with a rigid mode, it is introduced time-delay in a control system and presented PD combined with input-shaping control for the mechanical structures. The dynamical behavior of PD combined with input-shaping control system, subject to step inputs, is investigated. The actuator efforts and energy usage of PD combined with input-shaping control system are described in general formula. Given the condition of the same feedback gains as the PD feedback control, the PD combined with input shaping control minimizes actuator effort by 56%, saves energy by 47%, and speeds up the system-response.
At last, the paper summaries all the study and prospects for the whole thesis.
工程实际中结构优化问题通常为大型、高阶、非线性隐式问题,其分析过程复杂且需花费大量时间,能否有效的解决此类问题除了需要建立合理的优化问题数学模型之外,关键在于选择合理、有效的优化算法。以序列线性规划方法为代表的序列近似规划方法由于具有坚实的理论基础和普适性,目前已被广泛应用于工程结构优化领域及通用结构优化软件中。序列线性规划方法通过泰勒展开式将非线性规划问题转化为一系列的线性规划问题,从寻优的角度出发近似求解此线性规划问题,即可获得原问题的近似最优解。然而,应用传统序列线性规划法求解结构优化问题时效率较低,并受限于问题规模。为有效解决大型工程结构优化问题,研究适用序列线性规划并具备良好收敛性、稳定性、高效性的算法类,具有重要的理论意义和实际应用价值。
在实际工程问题中含有刚性模态的弹性机构在机械结构中占有很大的比重,快速加、减速运动时将会引起载荷摆动,造成载荷难以快速就位。特别是飞行器、大型起重机械、弹性机器人等具有挠性机械结构的系统,在受到阶跃输入信号作用的情况下,将导致刚性体运动与弹性体运动相互耦合,这是因为弹性体在发生刚性体运动时将产生弹性形变,同时产生残留振动,从而使整个系统难以精确定位。为达到抑制弹性机构残留振动的目的,通常采用的输入整形是合理地、计划地利用时滞加以改善弹性机构的动力学特性。针对含有刚性模态的弹性机构,本文提出了基于PD结合输入整形控制策略,通过对PD控制器参数与系统振动频率和阻尼比的关系进行分析,研究了输入整形参数对系统控制力的作用规律以及降低系统能耗的机理。在反馈增益相同的情况下,PD结合输入整形控制策略与PD反馈控制策略相比,能够更好的使系统产生较小的控制力,有效地降低了系统的能耗,大大提高了系统的响应速度。
本文主要研究内容:
1.概述了序列线性规划方法的基本理论、优化流程。随后,通过分析序列线性规划方法的特点,扬长避短,为解决算法收敛慢、计算量大等问题,研究在可行性空间中约束系列线性搜索的移动限概念。根据移动限控制变量的变化范围,设计提高算法的收敛性,并通过数值算例对此进行验证,为下一章如何精确调整移动限提供了依据。
2.提出一种基于线性误差和信赖域思想的改进序列线性规划算法(D-SLP)来解决移动限对线性近似精度的影响。在迭代过程中,算法基于线性近似误差概念评估近似模型与真实模型之间的差异,随着线性搜索,值函数近似程度不断提高,使得优化空间内设计变量的移动限范围不断扩大,有效地提高了计算精度;同时,根据信赖域的原理构造自适应调整步长的方案,用以判别移动限是否被接受,该方法大大加快了算法的收敛速度,节省了计算成本。此算法原理简单直观,具有较强的收敛性和鲁棒性。
3.设计了基于信赖域思想的内点序列线性规划算法(IP-TR-SLP)。目前内点法和单纯形法是有效求解线性规划问题的两种主要途径,其中单纯形法沿着约束边界移动寻找最优解,而内点法则沿着中心路径寻优。当优化问题约束条件较多、维数较高时,内点法求解效率和收敛性要明显优于单纯形法,更适于解决工程实际问题。本章围绕如何进一步提高内点法的计算效率这一目标,应用补偿间隙能够反映控制变量对不等式约束是否满足这一判定条件,将补偿间隙作为罚函数引入到目标函数当中;同时,通过引入信赖域思想强制限定每次迭代时新的迭代点与当前迭代点之间的距离,保证整个优化过程中近似模型的可靠。虽然每次迭代时计算量并未减少,但是基于信赖域控制的改进内点法有效缩短了整个优化过程的计算时间,且不需要给定初始点的可行性,从而拓宽了算法在大型工程实际优化问题中的应用范围。
4.采用本文提出的D-SLP算法和IP-TR-SLP算法,对经典桁架结构算例进行位移、应力以及频率约束下的尺寸优化,以验证算法的精度及效率。通过对10杆、25杆、72杆桁架结构算例的计算,证明了本文提出的两种改进算法求解结构优化问题是可行的,且计算精度优于已有文献提出的算法的计算结果;200杆桁架结构算例验证了本文提出算法对求解大型结构优化问题时具有高效性。数值优化结果表明,在10杆和25杆小型桁架结构算例中,基于内点法寻优的IP-TR-SLP算法的效率略低于基于单纯形法寻优的D-SLP算法;而在求解200杆的大型桁架结构时,IP-TR-SLP算法的运算效率明显优于D-SLP算法。这是由于基于内点法的IP-TR-SLP算法虽然在每次迭代过程中耗费大量计算时间,但算法结合信赖域思想自适应地调整步长,从而使得整体计算效率大大提高,这种情况在大型结构优化时尤为突出。最后,将本文提出的改进算法应用于结构拓扑优化问题,并对二维平面结构进行有效的拓扑优化设计,拓展了序列线性规划算法的求解领域。
5.在研究含有刚性模态弹性机构的动力学特性的基础上,将时滞环节引入到控制系统当中,提出PD结合输入整形控制策略。并且在阶跃输入信号作用下研究PD结合输入整形控制系统的振动特性,分析了输入整形对系统控制力和能耗的作用规律。在反馈增益相同的情况下,PD结合输入整形控制策略与PD反馈控制相比,使系统控制力降低56%,能耗降低47%,有效的提高了系统的响应速度。
最后对全文研究内容进行总结,并为今后的研究提出展望。
As along the development of technology, structural optimization has been an important branch of solid mechanics nowadays which is widely applied in many engineering fields, such as aerospace, mechanical manufacturing, vehicle and so on.
Generally, there are always large-scale, high-order and nonlinear problems in the optimization actual structures which cost lots of time to analyze. Whether we can solve these problems except depending on constructing the reasonable mathematic model, choosing an available optimization method is also a key point. As a kind of most effective and popular sequential approximate programming, sequential linear programming (SLP) has been widely used in universal optimization software and structural optimization field. The main idea of SLP is transforming the nonlinear problems into linear ones by Taylor expansion. By means of solving the linear problems parelly, one can finally obtain the optimum solution of the original problem. However, traditional SLP method is restricted by the scales of problem so that it converges slowly. For solving the large-scale structural optimization problems, it is necessary to research an improved SLP method in order to enhance the convergence, stability and validity of original SLP. The research will be very meaningful and valuable in the engineering application.
In practical engineering problems, the elastic mechanism with rigid modes accounts for a large proportion in mechanical structure. The movement of acceleration and deceleration will cause the load swing, resulting that the load is difficult to replace. Particularly in the system of flexible mechanical structure such as aircraft, large scale hoisting machinery and flexible robots, when the step input signal role under control, the system are coupling between rigid body motion and elastic motion, which is due to the elastic stiffness in the event of body movement transforming to elastic deformation and producing residual vibration, resulting that the entire system is difficult to pinpoint. Commonly, we use input-shaping to suppress residual vibration of elastic bodies on purpose, which is reasonable and well-planned to make use of the time-delay dynamics of elastic bodies. The dynamical behavior of PD combined with input-shaping control system, subject to step inputs, is investigated. The actuator efforts and energy usage of PD combined with input-shaping control system are described in general formula. Given the condition of the same feedback as PD, the PD combined with input shaping control minimizes actuator effort, saves energy, and speeds system response.
The main work of this paper is presented followed:
1. First of all the paper reviews the basic theory and the optimization process of SLP. Then it studies the moving limit concept of constraint in feasibility space to improve the convergence of SLP through sequence linear programming method. According to the size of move limit, it is designed to control the convergence of SLP. Numerical examples are verified the affective of move limit which is the foundation of the theory in next chapter.
2. In order to overcome the approximate accuracy of SLP, we improves SLP algorithm affected by move limit, based on linear error and proposed trust region. At the iteration, the error between approximate model and true model is evaluated by approximately linear concept. Along with the linearly searching, the approximate accuracy is enhanced greatly, resulting in the size of move limit in the design space broadened and the accuracy improved naturally. Additionally, step size is adaptively adjusted, which can be used to choose whether accept the move limit, therefore the convergence process is speeded up. This algorithm is simple with strong convergence and robust.
3. In this paper, it is proposed an improved SLP method (IP-TR-SLP), which solves the linear optimization problems by interior-point method basing on the concept of trust region. Nowadays, interior-point method and simplex method are the two main ways to solve the linear optimization problems. Simple method moves alone the boundary to search the optimum solution, while the interior-point method searches along the central path. When the problems are large-scale, including many constraint conditions, the efficiency and convergence of interior-point method will be better than the simple method. Thereby the interior-point method is fitting for solving practical optimization problems. This paper focus on how to improve the efficiency of interior-point method, using complementary interval as a penalty function into the objective function. Meanwhile, one can restrict the distance between the new iterative point and current iterative point based on the trust region in each iteration to ensure the reliability of approximate models during the iteration. Although the computational cost in each iteration does not decrease, (IP-TR-SLP) method shortens the whole computational time during optimization and there is no necessary constraint the initial feasible point that (IP-TR-SLP) method will applied expanded to solve the practical optimization problems, especially in large scale cases.
4. The proposed D-SLP and IP-TR-SLP algorithms are introduced to verify the size optimization problem of classical truss structure with displacement, stress, and frequency constraints. Through the calculation of 10-bar, 25-bar and 72-bar, it is showed that these two improved algorithms are feasible and the accuracy of the numerical examples is not lower than that of those existed references. The efficiency of these two methods are tested by 200-bar examples. The data optimization of results shows that, in the calculation of 10-bar and 25-bar, the efficiency of IP-TR-SLP is lower than that of D-SLP, however, for large-scale structure, IP-TR-SLP is superior to D-SLP. Although IP-TR-SLP algorithm spent much time at every iterative step, it do speed up the whole optimization process due to the adaptive adjustment of step size based on trust region, apparently in large-scale structure optimization. As a result, the solution field of SLP is broadened by these two algorithms applying in topology optimization.
5. Based on researching the dynamical performances of the flexible mechanisms with a rigid mode, it is introduced time-delay in a control system and presented PD combined with input-shaping control for the mechanical structures. The dynamical behavior of PD combined with input-shaping control system, subject to step inputs, is investigated. The actuator efforts and energy usage of PD combined with input-shaping control system are described in general formula. Given the condition of the same feedback gains as the PD feedback control, the PD combined with input shaping control minimizes actuator effort by 56%, saves energy by 47%, and speeds up the system-response.
At last, the paper summaries all the study and prospects for the whole thesis.
引文
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