LQG/LTR loop shaping design with an application to position control

Abstract

In contrast to the conventional LQG method using calculus of variations, this paper presents an alternative approach based on operator theory and spectrum factorization. This approach, which works toward transfer function domain, has merits in its simple and straightforward development. The novelty of this method is not the results achieved, but rather in their derivation. A loop shaping design procedure which incorporates the classical loop shaping into LQG/LTR synthesis is developed such that the robustness properties of the optimal full state feedback are guaranteed at both the plant input and the plant output. The design procedure is applied to a position controller design in a servo system, and the performance is evaluated by a digital signal processor, the TMS320C30. Experimental results have shown effectiveness of this design method.     

频域加权的LQG振动主动控制实验研究

针对实际工程领域中存在的伴有大量宽频干扰/噪声的复杂振动环境,直接影响了常规振动主动控制(AVC)的性能甚至稳定性。本文提出了频域加权的线性二次高斯型(LQG)控制方法,实现对特定振动主频重点衰减的同时,保证宽频域的干扰/噪声控制作用。最后利用压电陶瓷贴片的悬臂梁,振动主动控制试验验证了该方法的有效性。     

一种确定车辆悬架LQG控制加权系数的方法

针对车辆悬架LQG控制,提供一种确定与车辆平顺性各评价指标相关加权系数的方法.根据特定工况下对车辆平顺性各评价指标的具体要求,利用层次分析法,确定它们的主观加权比例系数;根据特定工况下原型悬架的性能统计数据,确定各评价指标的同尺度量化比例系数;结合各评价指标的主观加权比例系数与同尺度量化比例系数,确定LQG控制中与平顺性各评价指标相关的最终加权系数.运用实例表明:使用该方法确定加权系数的LQG控制效果明显,且适用性强.     

Development of a simple and efficient method for robust optimization

Robust optimization problems are newly formulated and an efficient computational scheme is proposed. Both design variables and design parameters are considered as random variables about their nominal values. To ensure the robustness of objective performance, we introduce a new performance index bounding the performance together with a constraint limiting the performance variation. The constraint variations are regulated by considering the probability of feasibility. Each probability constraint is transformed into a sub‐optimization problem by the advanced first‐order second moment (AFOSM) method for computational efficiency. The proposed robust optimization method has the advantages that the mean value and the variation of the performance function are controlled simultaneously and rationally and the second‐order sensitivity information is not required even in case of gradient‐based optimization process. The suggested method is examined by solving three examples and the results are compared with those for the deterministic case and those available in the literature. Copyright © 2002 John Wiley & Sons, Ltd.     

Gearbox design for uncertain load requirements using active robust optimization

Design and optimization of gear transmissions have been intensively studied, but surprisingly the robustness of the resulting optimal design to uncertain loads has never been considered. Active Robust (AR) optimization is a methodology to design products that attain robustness to uncertain or changing environmental conditions through adaptation. In this study the AR methodology is utilized to optimize the number of transmissions, as well as their gearing ratios, for an uncertain load demand. The problem is formulated as a bi-objective optimization problem where the objectives are to satisfy the load demand in the most energy efficient manner and to minimize production cost. The results show that this approach can find a set of robust designs, revealing a trade-off between energy efficiency and production cost. This can serve as a useful decision-making tool for the gearbox design process, as well as for other applications.     

SMA迟滞系统的PID控制及参数优化研究

形状记忆合金(shape memory alloy,SMA)是一类新型功能材料,作为电热驱动器,广泛应用于工程领域。但由于材料内在的非线性性质和迟滞效应使其在控制系统的设计和热力学建模上较为困难,不利于简单控制系统和实时控制的实现。基于Preisach理论给出SMA驱动器系统的温度-位移迟滞模型,分析系统的热传导过程,将传统的比例-积分-微分PID(proportional-integral-differential,PID)策略应用于控制器的设计,并引入蚁群和粒子群两类智能算法优化控制器参数,讨论优化过程和控制效果;与传统PID参数整定方法相比,系统的阶跃响应和正弦信号跟踪性能取得比较理想的效果,且智能算法调节控制器参数的实时性更强,克服非线性和迟滞效应的影响。     

动力总成悬置系统频率和解耦率的稳健优化方法

由于测量误差、安装误差及老化等原因,动力总成悬置的刚度存在一定程度的误差或波动,从而悬置系统的频率和解耦率必然有一定程度的不确定性。考虑到通常容易得到悬置刚度的变化范围,在不需了解其统计特性的情况下,采用区间数描述悬置刚度、悬置系统的频率及解耦率的不确定性。给出了计算悬置系统频率和解耦率变化范围的改进区间截断方法,并验证了其计算精度。为提高悬置系统频率和解耦率的稳健性,提出一种区间型稳健优化方法（简称区间优化）对悬置刚度进行稳健设计。对某轿车悬置系统的频率和解耦率进行了稳健优化,结果表明,对于该悬置系统,稳健优化方法可以较大幅度地提高悬置系统侧倾和俯仰方向频率的稳健性,避免了悬置系统与其它零部件产生共振。与确定性优化相比,悬置系统在垂直方向和绕发动机曲轴扭转方向解耦率稍有降低,但能够满足悬置系统解耦布置的要求     

Simulated variance optimization for robust design

A method to aid robust design in the presence of design parameter uncertainty is described. For a given relationship between a performance measure (or output parameter) and the uncertain design parameters a probabilistic simulation is used to obtain the variance of the performance measure as a function of the nominal design parameter values. The optimum values of the latter are then obtained as those corresponding to a minimum of the computed variance, determined by means of a particular non-linear optimization algorithm in the presence of constraints. The latter are in the form of limits on the nominal values of the design parameters and a specified value for the performance measure at the nominal design point, i.e. the deterministic design target. Some problems inherent in this type of procedure are discussed and methods of solution are described. A specific example is studied and the results from the present method are compared with those previously obtained by use of another procedure. © 1998 John Wiley & Sons, Ltd.     

A new algorithm for the robust optimization of rotor-bearing systems

This article presents a new algorithm for the robust optimization of rotor-bearing systems. The goal of the optimization problem is to find the values of a set of parameters for which the natural frequencies of the system are as far away as possible from the rotational speeds of the machine. To accomplish this, the penalization proposed by Ritto, Lopez, Sampaio, and Souza de Cursi in 2011 is employed. Since the rotor-bearing system is subject to uncertainties, such a penalization is modelled as a random variable. The robust optimization is performed by minimizing the expected value and variance of the penalization, resulting in a multi-objective optimization problem (MOP). The objective function of this MOP is known to be non-convex and it is shown that its resulting Pareto front (PF) is also non-convex. Thus, a new algorithm is proposed for solving the MOP: the normal boundary intersection (NBI) is employed to discretize the PF handling its non-convexity, and a global optimization algorithm based on a restart procedure and local searches are employed to minimize the NBI subproblems tackling the non-convexity of the objective function. A numerical analysis section shows the advantage of using the proposed algorithm over the weighted-sum (WS) and NSGA-II approaches. In comparison with the WS, the proposed approach obtains a much more even and useful set of Pareto points. Compared with the NSGA-II approach, the proposed algorithm provides a better approximation of the PF requiring much lower computational cost.     

Experiments with hybrid Bernstein global optimization algorithm for the OPF problem in power systems

This article presents an algorithm based on the Bernstein form of polynomials for solving the optimal power flow (OPF) problem in electrical power networks. The proposed algorithm combines local and global optimization methods and is therefore referred to as a ‘hybrid’ Bernstein algorithm in the context of this work. The proposed algorithm is a branch-and-bound procedure wherein a local search method is used to obtain a good upper bound on the global minimum at each branching node. Subsequently, the Bernstein form of polynomials is used to obtain a lower bound on the global minimum. The performance of the proposed algorithm is compared with the previously reported Bernstein algorithm to demonstrate its efficacy in terms of the chosen performance metrics. Furthermore, the proposed algorithm is tested on the OPF problem for several benchmark IEEE power system examples and its performance is compared with generic global optimization solvers such as BARON and COUENNE. The test results demonstrate that the hybrid Bernstein global optimization algorithm delivers satisfactory performance in terms of solution optimality.     

Second‐order optimal control algorithm for complex systems

The solution to large‐scale optimal control problems, characterized by complex dynamics and extended time periods, is often computationally demanding. We present a solution algorithm with favourable local convergence properties as a way to reduce simulation times. This method is based on using a trapezoidal direct collocation to convert the differential equations into algebraic constraints. The resulting constrained minimization problem is then solved with an augmented Lagrangian formulation to accommodate both equality and inequality constraints. In contrast to the prevalent optimal control software implementations, we calculate analytical first and second derivatives. We then apply a generalized Newton method to the augmented Lagrangian formulation, solving for all unknowns simultaneously. The computational costs of the Hessian formation and matrix solution remain manageable as the system size increases due to the sparsity of all tensor quantities. Likewise, the total iterations for convergence scale well due to the local quadratic convergence of the generalized Newton method. We demonstrate the method with an inverted pendulum problem and a neuromuscular control problem with complex dynamics and 18 forcing functions. The optimal control solutions are successfully found. In both examples, we obtain quadratic convergence rates in the neighbourhood of the solution. Copyright © 2002 John Wiley & Sons, Ltd.     

Parametric optimal control of uncertain systems under an optimistic value criterion

It is well known that the optimal control of a linear quadratic model is characterized by the solution of a Riccati differential equation. In many cases, the corresponding Riccati differential equation cannot be solved exactly such that the optimal feedback control may be a complex time-oriented function. In this article, a parametric optimal control problem of an uncertain linear quadratic model under an optimistic value criterion is considered for simplifying the expression of optimal control. Based on the equation of optimality for the uncertain optimal control problem, an approximation method is presented to solve it. As an application, a two-spool turbofan engine optimal control problem is given to show the utility of the proposed model and the efficiency of the presented approximation method.     

约束阻尼结构的改进准则法拓扑减振动力学优化

旨在为减振设计提供理论基础,研究约束阻尼结构拓扑动力学优化。以阻尼材料用量、振动特征方程、模态频率为约束,以多模态损耗因子倒数的加权和最小为目标,建立了约束阻尼结构拓扑优化模型,引入MAC因子控制结构的振型跃阶。在引入质量阵惩罚因子基础上推导出优化目标灵敏度。考虑到优化目标函数的非凸性,采用常规准则法(OC)寻优可能会使拓扑变量出现负值或陷入局部优化,故引入数学规划移动渐近技术对OC法进行改进,从而将全体拓扑变量纳入改进算法的优化迭代全过程。编程实现了约束阻尼板改进OC法拓扑动力学优化并对改进法性能进行了仿真。结果显示,改进算法可得到更合理的约束阻尼层构形,可使结构取得更佳减振效果。研究表明,改进算法迭代稳定性更好、寻优效率更高、更具全域最优性。     

Efficient generation of pareto‐optimal topologies for compliance optimization

In multi‐objective optimization, a design is defined to beit pareto‐optimal if no other design exists that is better with respect to one objective, and as good with respect to other objectives. In this paper, we first show that if a topology is pareto‐optimal, then it must satisfy certain properties associated with the topological sensitivity field, i.e. no further comparison is necessary. This, in turn, leads to a deterministic, i.e. non‐stochastic, method for efficiently generating pareto‐optimal topologies using the classic fixed‐point iteration scheme. The proposed method is illustrated, and compared against SIMP‐based methods, through numerical examples. In this paper, the proposed method of generating pareto‐optimal topologies is limited to bi‐objective optimization, namely compliance–volume and compliance–compliance. The future work will focus on extending the method to non‐compliance and higher dimensional pareto optimization. Copyright © 2011 John Wiley & Sons, Ltd.     

最优鲁棒极不灵敏时滞滤波器消除小阻尼柔性系统残留振动

对于频率大范围变化的小阻尼柔性系统，以零振动时滞滤波器的幅频特性为目标函数，基于优化理论和矢量图法，设计最优鲁棒极不灵敏时滞滤波器。该滤波器的特点是考虑了系统参数的变化范围，系统残留振动的最大幅值在参数变化区间内相等，从而提高了时滞滤波器的鲁棒性，特别是当系统参数大范围变化时，时滞滤波器有更强的鲁棒性，能有效消除柔性系统的残留振动。将这一控制技术用于桥式起重机实验系统中，可以有效抑制载荷的残留摆动。     

Fixed-point algorithms for optimal ascent trajectories of launch vehicles

A fixed-point formulation and the associated algorithms for rapid generation of optimal ascent trajectories of launch vehicles through the atmosphere are presented. The complete ascent trajectory is tacitly divided into an endo-atmospheric and an exo-atmospheric portion. Since the exo-atmospheric trajectory can be obtained reliably and rapidly, a special form of two-point boundary value problem (TPBVP) is formulated in the endo-atmospheric portion. This special TPBVP leads to a sequence of nested fixed-point problems. Several algorithmic options to solve these problems using modified fixed-point iterations are discussed. The main advantages of the fixed-point formulation and algorithm are software simplicity and more robust convergence. Numerical results, both in open-loop trajectory generation and closed-loop simulations, are presented to demonstrate the discussions. The algorithm is validated and verified by comparing the results with the results obtained from existing method.     

Adaptive optimal tuning of a general class of stable LTI systems with restricted inputs

The problem addressed is one of model reference adaptive control (MRAC) of asymptotically stable plants of unknown order with zeros located anywhere in thes-plane except at the origin. The reference model is also asymptotically stable and lacking zero(s) ats = 0. The control law is to be specified only in terms of the inputs to and outputs of the plant and the reference model. For inputs from a class of functions that approach a non-zero constant, the problem is formulated in an optimal control framework. By successive refinements of the sub-optimal laws proposed here, two schemes are finally designed. These schemes are characterized by boundedness, convergence and optimality. Simplicity and total time-domain implementation are the additional striking features. Simulations to demonstrate the efficacy of the control schemes are presented. This paper is dedicated to Dr R M Umesh.     

Practical considerations in the optimization of flow production systems

In this paper we consider the problems faced by an industrial planner who is responsible for the design of real-life asynchronous production lines under stochastic conditions that may be due to breakdowns or random processing times. Based on real-life system data, it is shown that a number of available algorithms for the performance evaluation of a given system configuration as well as an algorithm for determining the optimum buffer configuration can be successfully applied in industrial practice.     

Multiobjective optimization by genetic algorithms: application to safety systems

When attempting to optimize the design of engineered systems, the analyst is frequently faced with the demand of achieving several targets (e.g. low costs, high revenues, high reliability, low accident risks), some of which may very well be in conflict. At the same time, several requirements (e.g. maximum allowable weight, volume etc.) should also be satisfied. This kind of problem is usually tackled by focusing the optimization on a single objective which may be a weighed combination of some of the targets of the design problem and imposing some constraints to satisfy the other targets and requirements. This approach, however, introduces a strong arbitrariness in the definition of the weights and constraints levels and a criticizable homogenization of physically different targets, usually all translated in monetary terms.The purpose of this paper is to present an approach to optimization in which every target is considered as a separate objective to be optimized. For an efficient search through the solution space we use a multiobjective genetic algorithm which allows us to identify a set of Pareto optimal solutions providing the decision maker with the complete spectrum of optimal solutions with respect to the various targets. Based on this information, the decision maker can select the best compromise among these objectives, without a priori introducing arbitrary weights.