Tuning of PID controller based on a multiobjective genetic algorithm applied to a robotic manipulator

Most controllers optimization and design problems are multiobjective in nature, since they normally have several (possibly conflicting) objectives that must be satisfied at the same time. Instead of aiming at finding a single solution, the multiobjective optimization methods try to produce a set of good trade-off solutions from which the decision maker may select one. Several methods have been devised for solving multiobjective optimization problems in control systems field. Traditionally, classical optimization algorithms based on nonlinear programming or optimal control theories are applied to obtain the solution of such problems. The presence of multiple objectives in a problem usually gives rise to a set of optimal solutions, largely known as Pareto-optimal solutions. Recently, Multiobjective Evolutionary Algorithms (MOEAs) have been applied to control systems problems. Compared with mathematical programming, MOEAs are very suitable to solve multiobjective optimization problems, because they deal simultaneously with a set of solutions and find a number of Pareto optimal solutions in a single run of algorithm. Starting from a set of initial solutions, MOEAs use iteratively improving optimization techniques to find the optimal solutions. In every iterative progress, MOEAs favor population-based Pareto dominance as a measure of fitness. In the MOEAs context, the Non-dominated Sorting Genetic Algorithm (NSGA-II) has been successfully applied to solving many multiobjective problems. This paper presents the design and the tuning of two PID (Proportional–Integral–Derivative) controllers through the NSGA-II approach. Simulation numerical results of multivariable PID control and convergence of the NSGA-II is presented and discussed with application in a robotic manipulator of two-degree-of-freedom. The proposed optimization method based on NSGA-II offers an effective way to implement simple but robust solutions providing a good reference tracking performance in closed loop.     

AN ANALYSIS OF MULTIOBJECTIVE EVOLUTIONARY ALGORITHMS FOR OPTIMIZATION PROBLEMS WITH TIME CONSTRAINTS

Many multiobjective optimization problems in the engineering field are required to be solved within more or less severe time restrictions. Because the optimization criteria, the parameters, and/or constraints might change with time, the optimization solutions must be recalculated when a change takes place. The time required by the optimization procedure to arrive at the new solutions should be bounded accordingly with the rate of change observed in these dynamic problems. This way, the faster the optimization algorithm is to obtain solutions, the wider is the set of dynamic problems to which that algorithm can be applied. Here, we analyze the performance of the nondominated sorting algorithm (NSGA-II), strength Pareto evolutionary algorithm (SPEA2), and single front genetic algorithms (SFGA, and SFGA2) on two different multiobjective optimization problems, with two and three objectives, respectively. For these two studied problems, the single front genetic algorithms have obtained adequate quality in the solutions in very little time. Moreover, for the second and more complex problem approached, SFGA2 and NSGA-II obtain the best hypervolume in the found set of nondominated solutions, but SFGA2 employs much less time than NSGA-II. These results may suggest that single front genetic algorithms, especially SFGA2, could be appropiate to deal with optimization problems with high rates of change, and thus stronger time constraints.     

MOEA/D: A Multiobjective Evolutionary Algorithm Based on Decomposition

Decomposition is a basic strategy in traditional multiobjective optimization. However, it has not yet been widely used in multiobjective evolutionary optimization. This paper proposes a multiobjective evolutionary algorithm based on decomposition (MOEA/D). It decomposes a multiobjective optimization problem into a number of scalar optimization subproblems and optimizes them simultaneously. Each subproblem is optimized by only using information from its several neighboring subproblems, which makes MOEA/D have lower computational complexity at each generation than MOGLS and nondominated sorting genetic algorithm II (NSGA-II). Experimental results have demonstrated that MOEA/D with simple decomposition methods outperforms or performs similarly to MOGLS and NSGA-II on multiobjective 0-1 knapsack problems and continuous multiobjective optimization problems. It has been shown that MOEA/D using objective normalization can deal with disparately-scaled objectives, and MOEA/D with an advanced decomposition method can generate a set of very evenly distributed solutions for 3-objective test instances. The ability of MOEA/D with small population, the scalability and sensitivity of MOEA/D have also been experimentally investigated in this paper.     

An elitist non-dominated sorting genetic algorithm enhanced with a neural network applied to the multi-objective optimization of a polysiloxane synthesis process

This paper presents an original software implementation of the elitist non-dominated sorting genetic algorithm (NSGA-II) applied and adapted to the multi-objective optimization of a polysiloxane synthesis process. An optimized feed-forward neural network, modeling the variation in time of the main parameters of the process, was used to calculate the vectorial objective function of NSGA-II, as an enhancement to the multi-objective optimization procedure. An original technique was utilized in order to find the most appropriate parameters for maximizing the performance of NSGA-II. The algorithm provided the optimum reaction conditions (reaction temperature, reaction time, amount of catalyst, and amount of co-catalyst), which maximize the reaction conversion and minimize the difference between the obtained viscometric molecular weight and the desired molecular weight. The algorithm has proven to be able to find the entire non-dominated Pareto front and to quickly evolve optimal solutions as an acceptable compromise between objectives competing with each other. The use of the neural network makes it also suitable to the multi-objective optimization of processes for which the amount of knowledge is limited.     

不确定环境下的危险品运输车辆路径优化

针对非满载危险品运输车辆路径优化问题,通过模糊变量刻画运输过程中的人口密度、行驶速度与运输时间以及客户需求量等方面的不确定因素,考虑载货量变化对风险评估的影响,建立基于动态载货量的风险评估模型,以运输总风险、车辆总行程、车辆使用数最小为优化目标,同时兼顾时间窗、事故概率、载货量等约束构建了不确定环境下的危险品运输车辆路径多目标优化模型。将NSGA-Ⅱ算法与LNS算法相结合,设计混合NSGA-Ⅱ算法求解模型。结果表明,混合NSGA-Ⅱ算法可以获得空间分布均匀且收敛性较好的Pareto解集,不同运输参与者可根据自身偏好在解集中选择相应的配送方案;该算法得到的最优总风险、总行程及车辆使用数目分别比NSGA-Ⅱ算法优化了11.5%、1.0%和14.3%,算法搜索性能和求解精度明显提高。     

An Investigation on Preference Order Ranking Scheme for Multiobjective Evolutionary Optimization

It may be generalized that all Evolutionary Algorithms (EA) draw their strength from two sources: exploration and exploitation. Surprisingly, within the context of multiobjective (MO) optimization, the impact of fitness assignment on the exploration-exploitation balance has drawn little attention. The vast majority of multiobjective evolutionary algorithms (MOEAs) presented to date resort to Pareto dominance classification as a fitness assignment methodology. However, the proportion of Pareto optimal elements in a set P grows with the dimensionality of P. Therefore, when the number of objectives of a multiobjective problem (MOP) is large, Pareto dominance-based ranking procedures become ineffective in sorting out the quality of solutions. This paper investigates the potential of using preference order-based approach as an optimality criterion in the ranking stage of MOEAs. A ranking procedure that exploits the definition of preference ordering (PO) is proposed, along with two strategies that make different use of the conditions of efficiency provided, and it is compared with a more traditional Pareto dominance-based ranking scheme within the framework of NSGA-II. A series of extensive experiments is performed on seven widely applied test functions, namely, DTLZ1, DTLZ2, DTLZ3, DTLZ4, DTLZ5, DTLZ6, and DTLZ7, for up to eight objectives. The results are analyzed through a suite of five performance metrics and indicate that the ranking procedure based on PO enables NSGA-II to achieve better scalability properties compared with the standard ranking scheme and suggest that the proposed methodology could be successfully extended to other MOEAs     

MOEA/D + uniform design: A new version of MOEA/D for optimization problems with many objectives

To extend multiobjective evolutionary algorithm based on decomposition (MOEA/D) in higher dimensional objective spaces, this paper proposes a new version of MOEA/D with uniform design, named the uniform design multiobjective evolutionary algorithm based on decomposition (UMOEA/D), and compares the proposed algorithm with MOEA/D and NSGA-II on some scalable test problems with three to five objectives. UMOEA/D adopts the uniform design method to set the aggregation coefficient vectors of the subproblems. Compared with MOEA/D, distribution of the coefficient vectors is more uniform over the design space, and the population size neither increases nonlinearly with the number of objectives nor considers a formulaic setting. The experimental results indicate that UMOEA/D outperforms MOEA/D and NSGA-II on almost all these many-objective test instances, especially on problems with higher dimensional objectives and complicated Pareto set shapes. Experimental results also show that UMOEA/D runs faster than NSGA-II for the problems used in this paper. In additional, the results obtained are very competitive when comparing UMOEA/D with some other algorithm on the multiobjective knapsack problems.     

A portfolio optimization model with three objectives and discrete variables

We formulate the portfolio selection as a tri-objective optimization problem so as to find tradeoffs between risk, return and the number of securities in the portfolio. Furthermore, quantity and class constraints are introduced into the model in order to limit the proportion of the portfolio invested in assets with common characteristics and to avoid very small holdings. Since the proposed portfolio selection model involves mixed integer decision variables and multiple objectives finding the exact efficient frontier may be very hard. Nevertheless, finding a good approximation of the efficient surface which provides the investor with a diverse set of portfolios capturing all possible tradeoffs between the objectives within limited computational time is usually acceptable. We experiment with the current state of the art evolutionary multiobjective optimization techniques, namely the Non-dominated Sorting Genetic Algorithm II (NSGA-II), Pareto Envelope-based Selection Algorithm (PESA) and Strength Pareto Evolutionary Algorithm 2 (SPEA2), for solving the mixed-integer multiobjective optimization problem and provide a performance comparison among them using metrics proposed by the community.     

Pareto frontier for simultaneous structural and manufacturing optimization of a composite part

Optimized design of composite structures requires simultaneous optimization of structural performance and manufacturing process. Such a challenge calls for a multi-objective optimization. Here, a generating multi-objective optimization method called normalized normal constraint method, which attains a set of optimal solutions and allows the designer to explore design alternatives before making the final decision, is coupled with a local-global search called constrained globalized bounded Nelder–Mead method. The proposed approach is applied to the design of a Z-shaped composite bracket for optimization of structural and manufacturing objectives. Comparison of the results with non-dominated sorting genetic algorithm (NSGA-II) shows that when only a small number of function evaluations are possible and a few Pareto optima are desired, the proposed method outperforms NSGA-II in terms of convergence to the true Pareto frontier. The results are validated by an enumeration search and by exploring the neighbourhood of the final solutions.     

Multi-objective optimization of shield construction parameters based on random forests and NSGA-II

Metro shield construction will inevitably cause changes in the stress and strain state of the surrounding soil, resulting in stratum deformation and surface settlement (SS), which will seriously endanger the safety of nearby buildings, roads and underground pipe networks. Therefore, in the design and construction stage, optimizing the shield construction parameters (SCP) is the key to reducing the SS rate and increasing the safe driving speed (DS). However, optimization of existing SCP are challenged by the need to construct a unified multiobjective model for optimization that are efficient, convenient, and widely applicable. This paper innovatively proposes a hybrid intelligence framework that combines random forest (RF) and non-dominant classification genetic algorithm II (NSGA-II), which overcomes the shortcomings of time-consuming and high cost for the establishment and verification of traditional prediction models. First, RF is used to rank the importance of 10 influencing factors, and the nonlinear mapping relationship between the main SCP and the two objectives is constructed as the fitness function of the NSGA-II algorithm. Second, a multiobjective optimization framework for RF-NSGA-II is established, based on which the optimal Pareto front is calculated, and reasonable optimized control ranges for the SCP are obtained. Finally, a case study in the Wuhan Rail Transit Line 6 project is examined. The results show that the SS is reduced by 12.5% and the DS is increased by 2.5% with the proposed framework. Meanwhile, the prediction results are compared with the back-propagation neural network (BPNN), support vector machine (SVM), and gradient boosting decision tree (GBDT). The findings indicate that the RF-NSGA-II framework can not only meet the requirements of SS and DS calculation, but also used as a support tool for real-time optimization and control of SCP.     

基于分解和多策略变异的多目标差分进化算法*

差分进化是一种有效的优化技术,已成功用于多目标优化问题。但也存在Pareto最优集合的收敛慢和多样性差等问题。针对上述不足,本文提出了一种基于分解和多策略变异的多目标差分进化算法(MODE/DMSM)。该算法利用基于分解的方法将多目标优化问题分解为多个单目标优化问题;通过高效的非支配排序方法选择具有良好收敛性和多样性的解来指导差分进化过程;采用了多策略变异方法来平衡进化过程中收敛性和多样性。在ZDT和DTLZ的10个测试函数上的仿真结果表明,本文算法在Parato最优集合的收敛性和多样性优于其他六种代表性多目标优化算法。     

Adaptive genetic algorithms applied to dynamic multiobjective problems

This paper describes an adaptive genetic algorithm (AGA) with dynamic fitness function for multiobjective problems (MOPs) in a dynamic environment. In order to see performance of the algorithm, AGA was applied to two kinds of MOPs. Firstly, the algorithm was used to find an optimal force allocation for a combat simulation. The paper discusses four objectives that need to be optimized and presents a fuzzy inference system that forms an aggregation of the four objectives. A second fuzzy inference system is used to control the crossover and mutation rates based on statistics of the aggregate fitness. In addition to dynamic force allocation optimization problem, a simple example of a dynamic multiobjective optimization problem taken from Farina et al. [M. Farina, K. Deb, P. Amato, Dynamic multiobjective optimization problems: test cases, approximations, and applications, IEEE Trans. Evol. Comput. 8 (5) (2004) 425–442] is presented and solved with the proposed algorithm. The results obtained here indicate that performance of the fuzzy-augmented GA is better than a standard GA method in terms of improvement of convergence to solutions of dynamic MOPs.     

WBMOAIS: A novel artificial immune system for multiobjective optimization

This study presents a novel weight-based multiobjective artificial immune system (WBMOAIS) based on opt-aiNET, the artificial immune system algorithm for multi-modal optimization. The proposed algorithm follows the elementary structure of opt-aiNET, but has the following distinct characteristics: (1) a randomly weighted sum of multiple objectives is used as a fitness function. The fitness assignment has a much lower computational complexity than that based on Pareto ranking, (2) the individuals of the population are chosen from the memory, which is a set of elite solutions, and a local search procedure is utilized to facilitate the exploitation of the search space, and (3) in addition to the clonal suppression algorithm similar to that used in opt-aiNET, a new truncation algorithm with similar individuals (TASI) is presented in order to eliminate similar individuals in memory and obtain a well-distributed spread of non-dominated solutions. The proposed algorithm, WBMOAIS, is compared with the vector immune algorithm (VIS) and the elitist non-dominated sorting genetic system (NSGA-II) that are representative of the state-of-the-art in multiobjective optimization metaheuristics. Simulation results on seven standard problems (ZDT6, SCH2, DEB, KUR, POL, FON, and VNT) show WBMOAIS outperforms VIS and NSGA-II and can become a valid alternative to standard algorithms for solving multiobjective optimization problems.     

基于alpha支配的高维目标进化算法研究

基于Pareto支配的多目标进化算法能够很好地处理2～3维的多目标优化问题。但在处理高维多目标问题时,随着目标维数的增大,支配受阻解的数量急剧增加,导致现有的多目标算法存在选择压力不够、优化效果较差的问题。通过引入α支配提供严格的Pareto分层,在同层中挑选相对稀疏的解作为候选解,同时详细分析不同α对算法性能的影响,提出一种新的基于α偏序和拥塞距离抽样的高维目标进化算法。将该算法在DTLZ上进行性能测试,并采用世代距离(GD)、空间评价(SP)、超体积(HV)等多个指标评估算法的性能。实验结果表明,引入α支配能去除绝大部分支配受阻解(DRSs),提高算法的收敛性。与快速非支配排序算法(NSGA-II)、基于分解的多目标进化算法(MOEA/D)、基于距离更新的分解多目标进化算法(MOEA/D-DU)相比,该算法的整体解集的质量 有明显提高。     

Multiobjective immune algorithm with nondominated neighbor-based selection

Abstract Nondominated Neighbor Immune Algorithm (NNIA) is proposed for multiobjective optimization by using a novel nondominated neighbor-based selection technique, an immune inspired operator, two heuristic search operators, and elitism. The unique selection technique of NNIA only selects minority isolated nondominated individuals in the population. The selected individuals are then cloned proportionally to their crowding-distance values before heuristic search. By using the nondominated neighbor-based selection and proportional cloning, NNIA pays more attention to the less-crowded regions of the current trade-off front. We compare NNIA with NSGA-II, SPEA2, PESA-II, and MISA in solving five DTLZ problems, five ZDT problems, and three low-dimensional problems. The statistical analysis based on three performance metrics including the coverage of two sets, the convergence metric, and the spacing, show that the unique selection method is effective, and NNIA is an effective algorithm for solving multiobjective optimization problems. The empirical study on NNIA's scalability with respect to the number of objectives shows that the new algorithm scales well along the number of objectives.     

Dominance-Based Multiobjective Simulated Annealing

Simulated annealing is a provably convergent optimizer for single-objective problems. Previously proposed multiobjective extensions have mostly taken the form of a single-objective simulated annealer optimizing a composite function of the objectives. We propose a multiobjective simulated annealer utilizing the relative dominance of a solution as the system energy for optimization, eliminating problems associated with composite objective functions. We also propose a method for choosing perturbation scalings promoting search both towards and across the Pareto front. We illustrate the simulated annealer's performance on a suite of standard test problems and provide comparisons with another multiobjective simulated annealer and the NSGA-II genetic algorithm. The new simulated annealer is shown to promote rapid convergence to the true Pareto front with a good coverage of solutions across it comparing favorably with the other algorithms. An application of the simulated annealer to an industrial problem, the optimization of a code-division-multiple access (CDMA) mobile telecommunications network's air interface, is presented and the simulated annealer is shown to generate nondominated solutions with an even and dense coverage that outperforms single objective genetic algorithm optimizers.     

Multiobjective Optimization Problems With Complicated Pareto Sets, MOEA/D and NSGA-II

Partly due to lack of test problems, the impact of the Pareto set (PS) shapes on the performance of evolutionary algorithms has not yet attracted much attention. This paper introduces a general class of continuous multiobjective optimization test instances with arbitrary prescribed PS shapes, which could be used for studying the ability of multiobjective evolutionary algorithms for dealing with complicated PS shapes. It also proposes a new version of MOEA/D based on differential evolution (DE), i.e., MOEA/D-DE, and compares the proposed algorithm with NSGA-II with the same reproduction operators on the test instances introduced in this paper. The experimental results indicate that MOEA/D could significantly outperform NSGA-II on these test instances. It suggests that decomposition based multiobjective evolutionary algorithms are very promising in dealing with complicated PS shapes.     

多目标遗传算法求解认知无线电性能优化问题

认知无线电的性能优化是一个动态多目标优化问题。现有的Bio-CR模型基于遗传算法优化认知无线电的性能,它使用线性加权方法将此多目标优化问题简化为了一个单目标优化问题。针对Bio-CR很难确定每个适应度函数的权值和容易漏掉一些最优解的问题,提出了基于多目标遗传算法的认知无线电性能优化算法CREA。CREA能够根据信道条件和用户服务需求的变化动态地调整传输参数以优化性能,不仅克服了Bio-CR的两个缺点,而且通过保存计算结果进一步减少了遗传算法的运行次数。CREA首先根据信道条件的变化动态确定一组适应度函数,然后运行多目标遗传算法获得一个Pareto-optimal set,最后根据用户服务需求从中选出一个最满意解,并通知认知无线电更新自己的传输参数。Matlab仿真实验证明了CREA的正确性和有效性。     

Multiobjective firefly algorithm for continuous optimization

Design problems in industrial engineering often involve a large number of design variables with multiple objectives, under complex nonlinear constraints. The algorithms for multiobjective problems can be significantly different from the methods for single objective optimization. To find the Pareto front and non-dominated set for a nonlinear multiobjective optimization problem may require significant computing effort, even for seemingly simple problems. Metaheuristic algorithms start to show their advantages in dealing with multiobjective optimization. In this paper, we extend the recently developed firefly algorithm to solve multiobjective optimization problems. We validate the proposed approach using a selected subset of test functions and then apply it to solve design optimization benchmarks. We will discuss our results and provide topics for further research.     

Pareto set analysis: local measures of objective coupling in multiobjective design optimization

Multiobjective optimization focuses on the explicit trade-offs between competing criteria. A particular case is the study of combined optimal design and optimal control, or co-design, of smart artifacts where the artifact design and controller design objectives compete. In the system-level co-design problem, the objective is often the weighted sum of these two objectives. A frequently referenced practice is to solve co-design problems in a sequential manner: design first, control next. The success of this approach depends on the form of coupling between the two subproblems. In this paper, the coupling vector derived for a system problem with unidirectional coupling is shown to be related to the alignment of competing objectives, as measured by the polar cone of objective gradients, in the bi-objective programming formulation. Further, it is shown that a measure describing the case where a range of objective weighting values for the system objective result in identical design solutions can be normalized when the system problem is considered as a bi-objective one. Changes to the mathematical structure and input parameter values of a bi-objective programming problem can lead to changes in the shape of the attainable set and its Pareto boundary. We illustrate the link between the coupling and alignment measures and the outcomes of the Pareto set. Systematically studying changes to coupling and alignment measures due to changes to the multiobjective formulation can yield deeper insights into the system-level design problem. Two examples illustrate these results.