Fine Structure of Pareto Front of Multi-Objective Optimal Feedback Control Design

2013 ◽  
Author(s):  
Yousef Naranjani ◽  
Yousef Sardahi ◽  
Jian-Qiao Sun ◽  
Carlos Hernandez ◽  
Oliver Schuetze
Keyword(s):  
Fine Structure ◽  
Pareto Front ◽  
Optimization Problems ◽  
Tracking Error ◽  
Control Design ◽  
Pareto Set ◽  
Optimal Feedback Control ◽  
Step Response ◽  
Peak Time ◽  
Multi Objective

Recently, we have proposed the simple cell mapping method (SCM) for global solutionsof multi-objective optimization problems (MOPs). We have applied the SCM method to the multi-objective optimal time domain design of PID control gains for linear systems to simultaneously minimize the overshoot, peak time and integrated absolute tracking error of the closed-loop step response. The SCM method can efficiently obtain the Pareto set and Pareto front globally, which represent the optimal control gains and performance measures, respectively. The Pareto set and Pareto front contain a complete set of control designs with various compromises in the tracking performance, and give the system designer a much wider range of choices and flexibility. Furthermore, we have discovered a fine structure of the Pareto front of the MOP solution, which was notseen before in the literature. In this paper, we further examine the implication of the fine structure with regard to the vibration control design and expected performance of the controller, and compare our findings with the dominant method in MOP studies, i.e. the genetic algorithm.

scholarly journals Numerical Computation of Lightly Multi-Objective Robust Optimal Solutions by Means of Generalized Cell Mapping

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1959
Author(s):  
Carlos Ignacio Hernández Castellanos ◽  
Oliver Schütze ◽  
Jian-Qiao Sun ◽  
Guillermo Morales-Luna ◽  
Sina Ober-Blöbaum
Keyword(s):  
Optimization Problems ◽  
Global Analysis ◽  
Tracking Error ◽  
Benchmark Problems ◽  
Peak Time ◽  
Optimal Solutions ◽  
Cell Mapping ◽  
Generalized Cell Mapping ◽  
Multi Objective ◽  
Low Dimensional

In this paper, we present a novel algorithm for the computation of lightly robust optimal solutions for multi-objective optimization problems. To this end, we adapt the generalized cell mapping, originally designed for the global analysis of dynamical systems, to the current context. This is the first time that a set-based method is developed for such kinds of problems. We demonstrate the strength of the novel algorithms on several benchmark problems as well as on one feed-back control design problem where the objectives are given by the peak time, the overshoot, and the absolute tracking error for the linear control system, which has a control time delay. The numerical results indicate that the new algorithm is well-suited for the reliable treatment of low dimensional problems.

scholarly journals The dilemma between eliminating dominance-resistant solutions and preserving boundary solutions of extremely convex Pareto fronts

2021 ◽  
Author(s):  
Zhenkun Wang ◽  
Qingyan Li ◽  
Qite Yang ◽  
Hisao Ishibuchi
Keyword(s):  
Evolutionary Algorithms ◽  
Coping Strategies ◽  
Test Problem ◽  
Pareto Front ◽  
Optimization Problems ◽  
Feasible Region ◽  
Multi Objective Optimization ◽  
Multi Objective ◽  
Pareto Fronts ◽  
Boundary Solutions

AbstractIt has been acknowledged that dominance-resistant solutions (DRSs) extensively exist in the feasible region of multi-objective optimization problems. Recent studies show that DRSs can cause serious performance degradation of many multi-objective evolutionary algorithms (MOEAs). Thereafter, various strategies (e.g., the $$\epsilon $$ ϵ -dominance and the modified objective calculation) to eliminate DRSs have been proposed. However, these strategies may in turn cause algorithm inefficiency in other aspects. We argue that these coping strategies prevent the algorithm from obtaining some boundary solutions of an extremely convex Pareto front (ECPF). That is, there is a dilemma between eliminating DRSs and preserving boundary solutions of the ECPF. To illustrate such a dilemma, we propose a new multi-objective optimization test problem with the ECPF as well as DRSs. Using this test problem, we investigate the performance of six representative MOEAs in terms of boundary solutions preservation and DRS elimination. The results reveal that it is quite challenging to distinguish between DRSs and boundary solutions of the ECPF.

Toward the Use of Pareto Performance Solutions and Pareto Robustness Solutions for Multi-Objective Robust Optimization Problems

2012 ◽  
Author(s):  
Weijun Wang ◽  
Stéphane Caro ◽  
Fouad Bennis ◽  
Oscar Brito Augusto
Keyword(s):  
Robust Optimization ◽  
Pareto Front ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Multi Objective Optimization ◽  
Robust Solutions ◽  
Multi Objective ◽  
Robust Optimization Problem ◽  
Design Solutions ◽  
Further Development

For Multi-Objective Robust Optimization Problem (MOROP), it is important to obtain design solutions that are both optimal and robust. To find these solutions, usually, the designer need to set a threshold of the variation of Performance Functions (PFs) before optimization, or add the effects of uncertainties on the original PFs to generate a new Pareto robust front. In this paper, we divide a MOROP into two Multi-Objective Optimization Problems (MOOPs). One is the original MOOP, another one is that we take the Robustness Functions (RFs), robust counterparts of the original PFs, as optimization objectives. After solving these two MOOPs separately, two sets of solutions come out, namely the Pareto Performance Solutions (PP) and the Pareto Robustness Solutions (PR). Make a further development on these two sets, we can get two types of solutions, namely the Pareto Robustness Solutions among the Pareto Performance Solutions (PR(PP)), and the Pareto Performance Solutions among the Pareto Robustness Solutions (PP(PR)). Further more, the intersection of PR(PP) and PP(PR) can represent the intersection of PR and PP well. Then the designer can choose good solutions by comparing the results of PR(PP) and PP(PR). Thanks to this method, we can find out the optimal and robust solutions without setting the threshold of the variation of PFs nor losing the initial Pareto front. Finally, an illustrative example highlights the contributions of the paper.

A New Deterministic Approach Using Sensitivity Region Measures for Multi-Objective Robust and Feasibility Robust Design Optimization

2005 ◽  
Vol 128 (4) ◽  
pp. 874-883 ◽  
Author(s):  
Mian Li ◽  
Shapour Azarm ◽  
Art Boyars
Keyword(s):  
Pareto Front ◽  
Optimization Problems ◽  
Continuous Functions ◽  
Robust Design Optimization ◽  
Worst Case ◽  
New Approach ◽  
Multi Objective ◽  
Tolerance Region ◽  
Parameter Variations ◽  
Gradient Based

We present a deterministic non-gradient based approach that uses robustness measures in multi-objective optimization problems where uncontrollable parameter variations cause variation in the objective and constraint values. The approach is applicable for cases that have discontinuous objective and constraint functions with respect to uncontrollable parameters, and can be used for objective or feasibility robust optimization, or both together. In our approach, the known parameter tolerance region maps into sensitivity regions in the objective and constraint spaces. The robustness measures are indices calculated, using an optimizer, from the sizes of the acceptable objective and constraint variation regions and from worst-case estimates of the sensitivity regions’ sizes, resulting in an outer-inner structure. Two examples provide comparisons of the new approach with a similar published approach that is applicable only with continuous functions. Both approaches work well with continuous functions. For discontinuous functions the new approach gives solutions near the nominal Pareto front; the earlier approach does not.

Many-Objective Optimal Design of Sliding Mode Controls

2016 ◽  
Vol 139 (1) ◽  
Author(s):  
Yousef Sardahi ◽  
Jian-Qiao Sun
Keyword(s):  
Optimal Design ◽  
Pareto Front ◽  
Sliding Mode ◽  
Design Method ◽  
Tracking Error ◽  
Control Design ◽  
Processing Algorithm ◽  
Order System ◽  
Post Processing ◽  
Control Effort

This paper presents a many-objective optimal (MOO) control design of an adaptive and robust sliding mode control (SMC). A second-order system is used as an example to demonstrate the design method. The robustness of the closed-loop system in terms of stability and disturbance rejection are explicitly considered in the optimal design, in addition to the typical time-domain performance specifications such as the rise time, tracking error, and control effort. The genetic algorithm is used to solve for the many-objective optimization problem (MOOP). The optimal solutions known as the Pareto set and the corresponding objective functions known as the Pareto front are presented. To assist the decision-maker to choose from the solution set, we present a post-processing algorithm that operates on the Pareto front. Numerical simulations show that the proposed many-objective optimal control design and the post-processing algorithm are promising.

scholarly journals A New Method for Dynamic Multi-Objective Optimization Based on Segment and Cloud Prediction

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 465 ◽  
Author(s):  
Peng Ni ◽  
Jiale Gao ◽  
Yafei Song ◽  
Wen Quan ◽  
Qinghua Xing
Keyword(s):  
Optimization Problems ◽  
Pareto Set ◽  
Benchmark Problems ◽  
Small Range ◽  
Optimal Solutions ◽  
Multi Objective Optimization ◽  
Good Convergence ◽  
Multi Objective ◽  
The Law ◽  
Cloud Theory

In the real world, multi-objective optimization problems always change over time in most projects. Once the environment changes, the distribution of the optimal solutions would also be changed in decision space. Sometimes, such change may obey the law of symmetry, i.e., the minimum of the objective function in such environment is its maximum in another environment. In such cases, the optimal solutions keep unchanged or vibrate in a small range. However, in most cases, they do not obey the law of symmetry. In order to continue the search that maintains previous search advantages in the changed environment, some prediction strategy would be used to predict the operation position of the Pareto set. Because of this, the segment and multi-directional prediction is proposed in this paper, which consists of three mechanisms. First, by segmenting the optimal solutions set, the prediction about the changes in the distribution of the Pareto front can be ensured. Second, by introducing the cloud theory, the distance error of direction prediction can be offset effectively. Third, by using extra angle search, the angle error of prediction caused by the Pareto set nonlinear variation can also be offset effectively. Finally, eight benchmark problems were used to verify the performance of the proposed algorithm and compared algorithms. The results indicate that the algorithm proposed in this paper has good convergence and distribution, as well as a quick response ability to the changed environment.

Optimization of Vehicle-Borne Radar Antenna Pedestal Based on Modified NSGA-II

2014 ◽  
Vol 945-949 ◽  
pp. 2241-2247
Author(s):  
De Gao Zhao ◽  
Qiang Li
Keyword(s):  
Pareto Front ◽  
Optimization Problems ◽  
Population Diversity ◽  
Multi Objective Optimization ◽  
Nsga Ii ◽  
Multi Objective ◽  
Sorting Process ◽  
Definition Of ◽  
Radar Antenna ◽  
Modified Algorithm

This paper deals with application of Non-dominated Sorting Genetic Algorithm with elitism (NSGA-II) to solve multi-objective optimization problems of designing a vehicle-borne radar antenna pedestal. Five technical improvements are proposed due to the disadvantages of NSGA-II. They are as follow: (1) presenting a new method to calculate the fitness of individuals in population; (2) renewing the definition of crowding distance; (3) introducing a threshold for choosing elitist; (4) reducing some redundant sorting process; (5) developing a self-adaptive arithmetic cross and mutation probability. The modified algorithm can lead to better population diversity than the original NSGA-II. Simulation results prove rationality and validity of the modified NSGA-II. A uniformly distributed Pareto front can be obtained by using the modified NSGA-II. Finally, a multi-objective problem of designing a vehicle-borne radar antenna pedestal is settled with the modified algorithm.

A Two-Phase Complete Algorithm for Multi-Objective Distributed Constraint Optimization

2014 ◽  
Vol 18 (4) ◽  
pp. 573-580
Author(s):  
Alexandre Medi ◽  
◽  
Tenda Okimoto ◽  
Katsumi Inoue ◽  
◽  
...  
Keyword(s):  
Branch And Bound ◽  
Pareto Front ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Branch And Bound Algorithm ◽  
Constraint Optimization ◽  
Distributed Constraint Optimization ◽  
Multi Objective ◽  
Multi Agent ◽  
Distributed Constraint Optimization Problem

A Distributed Constraint Optimization Problem (DCOP) is a fundamental problem that can formalize various applications related to multi-agent cooperation. Many application problems in multi-agent systems can be formalized as DCOPs. However, many real world optimization problems involve multiple criteria that should be considered separately and optimized simultaneously. A Multi-Objective Distributed Constraint Optimization Problem (MO-DCOP) is an extension of a mono-objective DCOP. Compared to DCOPs, there exists few works on MO-DCOPs. In this paper, we develop a novel complete algorithm for solving an MO-DCOP. This algorithm utilizes a widely used method called Pareto Local Search (PLS) to generate an approximation of the Pareto front. Then, the obtained information is used to guide the search thresholds in a Branch and Bound algorithm. In the evaluations, we evaluate the runtime of our algorithm and show empirically that using a Pareto front approximation obtained by a PLS algorithm allows to significantly speed-up the search in a Branch and Bound algorithm.

Expected-Improvement-Based Methods for Adaptive Sampling in Multi-Objective Optimization Problems

2016 ◽  
Author(s):  
Jesper Kristensen ◽  
You Ling ◽  
Isaac Asher ◽  
Liping Wang
Keyword(s):  
Convex Hull ◽  
Design Optimization ◽  
Adaptive Sampling ◽  
Pareto Front ◽  
Performance Metrics ◽  
Sampling Methods ◽  
Optimization Problems ◽  
Expected Improvement ◽  
Multi Objective Optimization ◽  
Multi Objective

Adaptive sampling methods have been used to build accurate meta-models across large design spaces from which engineers can explore data trends, investigate optimal designs, study the sensitivity of objectives on the modeling design features, etc. For global design optimization applications, adaptive sampling methods need to be extended to sample more efficiently near the optimal domains of the design space (i.e., the Pareto front/frontier in multi-objective optimization). Expected Improvement (EI) methods have been shown to be efficient to solve design optimization problems using meta-models by incorporating prediction uncertainty. In this paper, a set of state-of-the-art methods (hypervolume EI method and centroid EI method) are presented and implemented for selecting sampling points for multi-objective optimizations. The classical hypervolume EI method uses hyperrectangles to represent the Pareto front, which shows undesirable behavior at the tails of the Pareto front. This issue is addressed utilizing the concepts from physical programming to shape the Pareto front. The modified hypervolume EI method can be extended to increase local Pareto front accuracy in any area identified by an engineer, and this method can be applied to Pareto frontiers of any shape. A novel hypervolume EI method is also developed that does not rely on the assumption of hyperrectangles, but instead assumes the Pareto frontier can be represented by a convex hull. The method exploits fast methods for convex hull construction and numerical integration, and results in a Pareto front shape that is desired in many practical applications. Various performance metrics are defined in order to quantitatively compare and discuss all methods applied to a particular 2D optimization problem from the literature. The modified hypervolume EI methods lead to dramatic resource savings while improving the predictive capabilities near the optimal objective values.

Multi-objective particle swarm optimization using Pareto-based set and aggregation approach

2017 ◽  
Vol 31 (19-21) ◽  
pp. 1740073 ◽  
Author(s):  
Song Huang ◽  
Yan Wang ◽  
Zhicheng Ji
Keyword(s):  
Particle Swarm Optimization ◽  
Local Search ◽  
Real World ◽  
Optimization Problems ◽  
Particle Swarm ◽  
Pareto Set ◽  
Multi Objective Optimization ◽  
Particle Swarm Optimizer ◽  
Swarm Optimization ◽  
Multi Objective

Multi-objective optimization problems (MOPs) need to be solved in real world recently. In this paper, a multi-objective particle swarm optimization based on Pareto set and aggregation approach was proposed to deal with MOPs. Firstly, velocities and positions were updated similar to PSO. Then, global-best set was defined in particle swarm optimizer to preserve Pareto-based set obtained by the population. Specifically, a hybrid updating strategy based on Pareto set and aggregation approach was introduced to update the global-best set and local search was carried on global-best set. Thirdly, personal-best positions were updated in decomposition way, and global-best position was selected from global-best set. Finally, ZDT instances and DTLZ instances were selected to evaluate the performance of MULPSO and the results show validity of the proposed algorithm for MOPs.

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