Combined active learning Kriging with optimal saturation nonlinear vibration control for uncertain systems with both aleatory and epistemic uncertainties
TL;DR: In this article , the optimal saturation nonlinear control (OSNC) and active learning Kriging (ALK) method were combined to solve the vibration control problems of uncertain systems with both random and multidimensional parallelepiped (MP) convex variables.
Abstract: This paper addresses the vibration control problems of uncertain systems with both random and multidimensional parallelepiped (MP) convex variables by uniting the optimal saturation nonlinear control (OSNC) and an active learning Kriging (ALK) method. This method can be named ALK-MP-OSNC. The dynamic equations of the controlled systems can be written in ODE forms, and the functions containing saturation nonlinearities on the right side of each of ODE equation can be approximately replaced via using the Kriging model. The efficiency of the Kriging model can be improved through combining the differential evolution (DE) global optimal algorithm with the distance constraint condition. A three-pendulum system, a satellite motion and a moving-mass beam system are employed to demonstrate the performance of the improved ALK-MP-OSNC. Results indicates that the proposed method can efficiently drive the uncertain pendulum system to a chaotic behavior and the other two uncertain systems to a periodic motion. The efficiency and the accuracy of the proposed method can be researched through comparing with the original ALK and the Monte Carlo simulation. In conclusion, the proposed method can be applied to complex engineering fields such as aerospace engineering, civil engineering, ocean engineering, and space deployable engineering.
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TL;DR: In this article , the uncertain sandwich impulsive control system with impulsive time windows is presented, in which the linear entry matrix of the system is indeterminate and linear matrix inequalities (LMIs) and inequalities techniques are used to investigate the exponential stability of the considered system.
Abstract: In this paper, we formulate a new system, named the uncertain sandwich impulsive control system with impulsive time windows. The presented system shows that the linear entry matrix of the system is indeterminate. We first investigate the exponential stability of the considered system by linear matrix inequalities (LMIs) and inequalities techniques, then extend the considered system to a more general one and further study the stability of the general system. Finally, numerical simulations are delivered to demonstrate the effectiveness of the theoretical results.
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TL;DR: This paper introduces the reader to a response surface methodology that is especially good at modeling the nonlinear, multimodal functions that often occur in engineering and shows how these approximating functions can be used to construct an efficient global optimization algorithm with a credible stopping rule.
Abstract: In many engineering optimization problems, the number of function evaluations is severely limited by time or cost. These problems pose a special challenge to the field of global optimization, since existing methods often require more function evaluations than can be comfortably afforded. One way to address this challenge is to fit response surfaces to data collected by evaluating the objective and constraint functions at a few points. These surfaces can then be used for visualization, tradeoff analysis, and optimization. In this paper, we introduce the reader to a response surface methodology that is especially good at modeling the nonlinear, multimodal functions that often occur in engineering. We then show how these approximating functions can be used to construct an efficient global optimization algorithm with a credible stopping rule. The key to using response surfaces for global optimization lies in balancing the need to exploit the approximating surface (by sampling where it is minimized) with the need to improve the approximation (by sampling where prediction error may be high). Striking this balance requires solving certain auxiliary problems which have previously been considered intractable, but we show how these computational obstacles can be overcome.
6,914 citations
TL;DR: An iterative approach based on Monte Carlo Simulation and Kriging metamodel to assess the reliability of structures in a more efficient way and is shown to be very efficient as the probability of failure obtained with AK-MCS is very accurate and this, for only a small number of calls to the performance function.
Abstract: An important challenge in structural reliability is to keep to a minimum the number of calls to the numerical models. Engineering problems involve more and more complex computer codes and the evaluation of the probability of failure may require very time-consuming computations. Metamodels are used to reduce these computation times. To assess reliability, the most popular approach remains the numerous variants of response surfaces. Polynomial Chaos [1] and Support Vector Machine [2] are also possibilities and have gained considerations among researchers in the last decades. However, recently, Kriging, originated from geostatistics, have emerged in reliability analysis. Widespread in optimisation, Kriging has just started to appear in uncertainty propagation [3] and reliability [4] , [5] studies. It presents interesting characteristics such as exact interpolation and a local index of uncertainty on the prediction which can be used in active learning methods. The aim of this paper is to propose an iterative approach based on Monte Carlo Simulation and Kriging metamodel to assess the reliability of structures in a more efficient way. The method is called AK-MCS for Active learning reliability method combining Kriging and Monte Carlo Simulation. It is shown to be very efficient as the probability of failure obtained with AK-MCS is very accurate and this, for only a small number of calls to the performance function. Several examples from literature are performed to illustrate the methodology and to prove its efficiency particularly for problems dealing with high non-linearity, non-differentiability, non-convex and non-connex domains of failure and high dimensionality.
1,234 citations
TL;DR: This paper develops an efficient reliability analysis method that accurately characterizes the limit state throughout the random variable space and is both accurate for any arbitrarily shaped limit state and computationally efficient even for expensive response functions.
Abstract: Many engineering applications are characterized by implicit response functions that are expensive to evaluate and sometimes nonlinear in their behavior, making reliability analysis difficult. This paper develops an efficient reliability analysis method that accurately characterizes the limit state throughout the random variable space. The method begins with a Gaussian process model built from a very small number of samples, and then adaptively chooses where to generate subsequent samples to ensure that the model is accurate in the vicinity of the limit state. The resulting Gaussian process model is then sampled using multimodal adaptive importance sampling to calculate the probability of exceeding (or failing to exceed) the response level of interest. By locating multiple points on or near the limit state, more complex and nonlinear limit states can be modeled, leading to more accurate probability integration. By concentrating the samples in the area where accuracy is important (i.e., in the vicinity of the limit state), only a small number of true function evaluations are required to build a quality surrogate model. The resulting method is both accurate for any arbitrarily shaped limit state and computationally efficient even for expensive response functions. This new method is applied to a collection of example problems including one that analyzes the reliability of a microelectromechanical system device that current available methods have difficulty solving either accurately or efficiently.
804 citations
TL;DR: It can be deduced from the results given in this paper that using the most common approach in the literature to find the kriging parameters does not guarantee a good result for the structural reliability problems.
Abstract: Approximation methods are widely used to alleviate the computational burden of engineering analyses For structural reliability analyses, the common approach is to use the response surface method (RSM) based on the least square regression However, another approximation method based on kriging has gained popularity especially in the field of deterministic optimization However, the application of the kriging method to structural reliability problems has not been realized until recently Therefore, this paper investigates the use of the kriging method for structural reliability problems by comparing it with the most common RSM The effects of the kriging parameters are also examined on the basis of the β computation and fitting behavior It can be deduced from the results given in this paper that using the most common approach in the literature to find the kriging parameters does not guarantee a good result for the structural reliability problems As a result, some advantages as well as disadvantages of the kriging method are reported, based on the results obtained from the application of the kriging method to the examples from the literature Finally, this paper suggests the points for which the kriging model could be improved to get better results for structural reliability problems
542 citations
TL;DR: In this article, a linear feedback control for nonlinear systems has been formulated under an optimal control theory viewpoint, where the stability of the closed-loop nonlinear system is guaranteed by means of a Lyapunov function which can clearly be seen as the solution of the Hamilton-Jacobi-Bellman equation.
Abstract: This paper presents the control and synchronization of chaos by designing linear feedback controllers. The linear feedback control problem for nonlinear systems has been formulated under optimal control theory viewpoint. Asymptotic stability of the closed-loop nonlinear system is guaranteed by means of a Lyapunov function which can clearly be seen to be the solution of the Hamilton–Jacobi–Bellman equation thus guaranteeing both stability and optimality. The formulated theorem expresses explicitly the form of minimized functional and gives the sufficient conditions that allow using the linear feedback control for nonlinear system. The numerical simulations were provided in order to show the effectiveness of this method for the control of the chaotic Rossler system and synchronization of the hyperchaotic Rossler system.
253 citations