Optimization Of Nonlinear Systems Research Articles

A framework of adaptive fuzzy control and optimization for nonlinear systems with output constraints

This paper presents a framework for adaptive fuzzy control and optimization of nonlinear systems subject to uncertainties and disturbances. The barrier Lyapunov function (BLF) technique was adopted to determine output constraints. To enhance the tracking performance of the system, fuzzy logic systems (FLSs) were introduced to approximate nonlinear terms. Subsequently, a combination of Bayesian optimization and particle swarm optimization (BO-PSO) was employed for gains optimization to further improve the control performance. Furthermore, the multilayer neural networks (MNNs) were applied as surrogate models of nonlinear systems with interval parameters to improve the computational efficiency of the optimization process. Finally, two simulations were conducted to demonstrate the effectiveness of the proposed framework.

Distributed optimization for uncertain nonlinear interconnected multi-agent systems

Distributed optimization for uncertain nonlinear interconnected multi-agent systems is considered in this paper. The objective is to design distributed algorithms by using local information such that all the agents’ outputs converge to the global optimal output. The problem is challenging due to unknown modeling uncertainties and nonlinear interconnections. First, a distributed algorithm composed of a distributed state observer and a distributed optimal output observer is presented for heterogeneous outputs. Next, it is shown that the objective can be achieved by employing solely a distributed optimal output observer if the agents’ outputs are homogeneous. Finally, the algorithms’ effectiveness is validated by an electrical power system composed of four-machine subsystems.

Impact of Optimally Growing Initial Errors on the Mesoscale Predictability of Heavy Precipitation Events along the Mei-Yu Front in China

Abstract Based on the conditional nonlinear optimal perturbation (CNOP) approach, the impact of the optimally growing initial errors on the mesoscale predictability of typical mei-yu front heavy precipitation events over eastern China was explored. First, based on a nonlinear optimization system built using the high-resolution Weather Research and Forecasting Model and particle swarm optimization algorithm, the CNOPs for three heavy precipitation cases were obtained. The CNOPs as the optimally growing initial errors caused the largest forecast errors and made the 24-h accumulated precipitation stronger than any other kind of initial errors. Moreover, the CNOPs showed faster growth and saturation than the random errors in space, highlighting the importance of the initial error with specific spatial structure in the meso- and convective-scale processes. Despite different CNOPs having particular spatial structures, the large amplitudes of the CNOPs at lower levels were mainly located in the rainband along the mei-yu front. Although the spectral energies of the CNOPs increased with increasing scales, the forecast error growth for the CNOPs generally followed the well-known three-stage conceptual model. Moreover, the large-scale and large-amplitude initial errors in the CNOPs were the most influential in terms of the forecast quality. This suggests that reducing large-scale initial errors can potentially improve the forecast accuracy. However, the mesoscale predictability of the mei-yu front heavy precipitation events is inherently limited, for which the moist convection was found to be the main reason.

Decomposition-based multiobjective optimization for nonlinear equation systems with many and infinitely many roots

Although the development of Pareto-dominance-based multiobjective optimization algorithms has enabled the solution of nonlinear equation systems, few studies have been conducted on the use of multiobjective optimization techniques for the solution of nonlinear equation systems. Accordingly, in this study, we use decomposition-based multiobjective optimization to solve nonlinear equation system, an attempt is made at deploying the decomposition-based multiobjective optimization to solve nonlinear equation systems, including those with infinite roots. In our novel approach, a given system is transformed into a bi-objective optimization problem using reference points. An improved decomposition-based multiobjective optimization is then applied to solve a transformed bi-objective optimization problem. To ensure this optimization suits the characteristics of the problem, we develop an adaptive multiobjective differential evolution and local search approach. The roots of the original nonlinear equation system can then be identified, together with the Pareto-optimal solutions of the transformed problem. We conducted extensive experiments to compare the performance of our novel approach with that of 16 state-of-the-art algorithms in solving 30 nonlinear equation systems derived from industrial applications. The results clearly show that our novel approach achieves better performance with a shorter execution time than that of the selected single-objective-based evolutionary approaches or Pareto-dominance-based multiobjective optimization algorithms.

Loss landscapes and optimization in over-parameterized non-linear systems and neural networks

The success of deep learning is due, to a large extent, to the remarkable effectiveness of gradient-based optimization methods applied to large neural networks. The purpose of this work is to propose a modern view and a general mathematical framework for loss landscapes and efficient optimization in over-parameterized machine learning models and systems of non-linear equations, a setting that includes over-parameterized deep neural networks. Our starting observation is that optimization landscapes corresponding to such systems are generally not convex, even locally around a global minimum, a condition we call essential non-convexity . We argue that instead they satisfy PL ⁎ , a variant of the Polyak-Łojasiewicz condition [32] , [25] on most (but not all) of the parameter space, which guarantees both the existence of solutions and efficient optimization by (stochastic) gradient descent (SGD/GD). The PL ⁎ condition of these systems is closely related to the condition number of the tangent kernel associated to a non-linear system showing how a PL ⁎ -based non-linear theory parallels classical analyses of over-parameterized linear equations. We show that wide neural networks satisfy the PL ⁎ condition, which explains the (S)GD convergence to a global minimum. Finally we propose a relaxation of the PL ⁎ condition applicable to “almost” over-parameterized systems.

Sampled-data extremum-seeking framework for constrained optimization of nonlinear dynamical systems

Most extremum-seeking control (ESC) approaches focus solely on the problem of finding the extremum of some unknown, steady-state input–output map, providing parameter settings that lead to optimal steady-state system performance. However, many industrial applications also have to deal with constraints on operating conditions due to, e.g., actuator limitations, limitations on tunable system parameters, or constraints on measurable variables. In particular, constraints on measurable variables are typically unknown in terms of their relationship with the tunable system parameters. In addition, the constraints on system inputs as a result of the constraints on measurable variables may conflict with the otherwise optimal operational condition, and hence should be taken into account in the data-based optimization approach. In this work, we propose a sampled-data extremum-seeking framework for the constrained optimization of a class of nonlinear dynamical systems with measurable constrained variables. In this framework, barrier function methods are employed, exploiting both the objective function and constraint functions which are available through output measurement only. We show, under the assumption that the parametric initialization yield operating conditions that do not violate the constraints, that (1) the resulting closed-loop dynamics is stable, (2) constraint satisfaction of the inputs is guaranteed for all iterations of the optimization process, and (3) constrained optimization is achieved. We illustrate the working principle of the proposed framework by means of an industrial case study of the constrained optimization of extreme ultraviolet light generation in a laser-produced plasma source within a state-of-the-art lithography system.

Comparison of meta-heuristic approaches for the optimization of non-linear base-isolation systems considering the influence of superstructure flexibility

The main aim of this study is to investigate the meta-heuristic approaches called Harmony Search (HS), Grey Wolf Optimizer (GWO), Teaching-Learning Based Optimization (TLBO), and Jaya Algorithm (JA) in the optimization process of non-linear base isolation systems under near-fault earthquakes considering the effect of the superstructure flexibility. The optimization processes were performed by achieving the objective function set as minimizing the peak top floor acceleration to peak ground acceleration ratio with and without base displacement limits. The context of the optimization process, the analytical model of the superstructure, and the number of isolators are considered as the design constants. The mechanical parameters of the non-linear isolation systems, such as the isolation system period, the total characteristic strength ratio, and the yield displacement, are determined as the independent design variables. According to the results obtained in this study, GWO, JA, and TLBO algorithms can be more suitable for solving such design problems among the considered algorithms. Although the objective function values are amplified with the increase in the superstructure flexibility, these values remain around 1.0, even for the most stringent base displacement limit. It can be expressed as a successful seismic performance, especially considering such strong near-fault ground motions, which are the most challenging types for seismically isolated buildings.

Dynamic real-time optimization for nonlinear systems with Lyapunov stabilizing MPC

The present work addresses the problem of stabilizing dynamic real-time optimization and control of nonlinear systems. Dynamic real-time optimization (DRTO) is a framework for computing an optimal operating trajectory for a plant and providing corresponding set-points for an underlying control algorithm to track. This technique can be enhanced by directly modeling the underlying control algorithm (such as an MPC) in a closed-loop DRTO (CL-DRTO). This allows the CL-DRTO to predict both the controller and plant responses to set-point changes, improving performance of the overall system. Certain systems are additionally complicated by the existence of unstable steady states, which make control of the system substantially more difficult. A number of stabilizing MPC techniques have been developed in order to handle this, and a closed-loop DRTO formulation was recently proposed to handle linear systems. This work seeks to bring the improved performance of a CL-DRTO to stabilizing control systems by formulating a CL-DRTO which utilizes and explicitly models an underlying Lyapunov stabilizing MPC to achieve stabilization for nonlinear systems. The proposed formulation is compared to the previously developed formulation to demonstrate the improved closed-loop control and performance.

Analytical optimization of wideband nonlinear optical fiber communication systems.

In the design of fiber links for both continental and transoceanic optical communication systems, the optimization of span length is of high importance from both performance and cost perspectives. In this work, the maximization of signal-to-noise ratio (SNR) is investigated by optimizing the span length in wideband (up to 4.5-THz) Nyquist-spaced optical fiber communication systems. A simple and accurate closed-form expression of the optimal span length is provided, and a quick estimation of SNR is also described for practically feasible and cost-effective span length values.

Distributed Optimization of High-Order Nonlinear Systems: Saving Computation and Communication via Prefiltering

This brief investigates a distributed optimization problem for high-order strict-feedback nonlinear multiagent systems. A virtual system is built for each agent using a distributed proportional-integral (PI) optimization algorithm to estimate the global optimal solution online. The estimate is then input into a prefilter to generate an alternative estimate and its high-order derivatives. Based on these signals, a backstepping controller is used to make the actual system track the alternative estimate asymptotically, which thereby realizes exact global consensus optimization. Compared to the existing result, the proposed algorithm is fully distributed under undirected topologies with much weaker assumptions on local and global objective functions, lower computation and communication cost. Exact global optimal consensus can also be reached in the cases with directed topologies and intermittent communications. The effectiveness of the method is verified by simulations.

Distributed convex optimization for nonlinear multi-agent systems disturbed by a second-order stationary process over a digraph

Distributed convex optimization for nonlinear multi-agent systems disturbed by a second-order stationary process over a digraph

A Model-Free Approach for Online Optimization of Nonlinear Systems

This paper proposes a strategy to search optimal control parameters of a complex nonlinear system using a metaheuristic optimization algorithm in a computationally efficient manner. The proposed algorithm, called BAS-swarm (Beetle Antennae Search-swarm), is a gradient-free optimizer based on the BAS algorithm, inspired by mimicking the food foraging behavior of beetles. BAS-swarm takes advantage of the fact that the antennae of insects are not single sensory organs. The antennae contain an array of tiny fiber. Antennae fiber enables the insects to have a microscopic insight into the environment when moving toward the source of food smell. BAS-swarm uses this insight to improve the performance of BAS by approximating the gradient direction at each iteration with the help of a swarm of antenna fiber. Since the proposed algorithm approximates gradient by mimicking the behavior of beetle antenna fiber located at random locations, it does not require the numerical computation of the actual gradient, making it very efficient for optimization of nonlinear non-convex systems. We verified the accuracy and efficiency of the proposed algorithm by training single-layer neural networks with nonlinear activation function and compared its performance with Particle Swarm Optimizer (PSO), a well-studied extremum seeking algorithm, and the original BAS algorithm. The experiment shows that the proposed algorithm provides several-fold improvement and faster convergence as compared to other metaheuristic algorithms.

Teaching-Learning Based Optimization of Nonlinear Isolation Systems under Far Fault Earthquakes

Seismic isolation systems exposed to far-fault earthquakes can reduce floor accelerations and story drift ratios to acceptable levels. However, they exhibit different structural performances in each earthquake due to different excitation frequency contents. By optimizing the isolation system parameters, their performance may be maintained at the best level under different far-fault earthquakes. In this study, the optimization of the parameters of the nonlinear isolation system of a 5-story benchmark building is performed by Teaching-Learning Based Optimization (TLBO) algorithm to minimize peak floor accelerations under historical far-fault earthquakes with and without exceeding a specified base displacement limit. According to the results of the analyses, it can be said that TLBO algorithm is a robust algorithm with low standard deviations for determining optimum nonlinear isolation system parameters.

Optimizing TEG Dehydration Process under Metamodel Uncertainty

Natural gas processing requires the removal of acidic gases and dehydration using absorption, mainly conducted in tri-ethylene glycol (TEG). The dehydration process is accompanied by the emission of volatile organic compounds, including BTEX. In our previous work, multi-objective optimization was undertaken to determine the optimal operating conditions in terms of the process parameters that can mitigate BTEX emission using data-driven metamodeling and metaheuristic optimization. Data obtained from a process simulation conducted using the ProMax® process simulator were used to develop a metamodel with machine learning techniques to reduce the computational time of the iterations in a robust process simulation. The metamodels were created using limited samples and some underlying phenomena must therefore be excluded. This introduces the so-called metamodeling uncertainty. Thus, the performance of the resulting optimized process variables may be compromised by the lack of adequately accounting for the uncertainty introduced by the metamodel. In the present work, the bias of the metamodel uncertainty was addressed for parameter optimization. An algorithmic framework was developed for parameter optimization, given these uncertainties. In this framework, metamodel uncertainties are quantified using real model data to generate distribution functions. We then use the novel Better Optimization of Nonlinear Uncertain Systems (BONUS) algorithm to solve the problem. BTEX mitigation is used as the objective of the optimization. Our algorithm allows the determination of the optimal process condition for BTEX emission mitigation from the TEG dehydration process under metamodel uncertainty. The BONUS algorithm determines optimal process conditions compared to those from the metaheuristic method, resulting in BTEX emission mitigation up to 405.25 ton/yr.

Open-Loop Dynamic Optimization for Nonlinear Multi-Input Systems. Application to Recombinant Protein Production

This paper proposes a novel strategy for dynamic open-loop optimization of multivariable nonlinear systems. The methodology is based on the Fourier series and orthonormal polynomials for the control vector parameterization in a sequential direct solution approach. The advantages of this technique are that a few number of parameters is required for optimization and a smooth control profile is obtained. The proposed strategy is evaluated in the case study of recombinant protein production, that is a nonlinear system with two control actions, the substrate and inhibitor feed flow rate. The algorithms are tested through simulations and the results are compared with those published in the bibliography.

Static gain estimation for nonlinear dynamic systems from steady-state values hidden in historical data

Static gains are often required for control, diagnosis and optimization of nonlinear dynamic systems. This paper proposes a new approach to estimate static gains for nonlinear dynamic systems from steady-state values hidden in historical data. First, steady-state values of system inputs and outputs are extracted by automatically finding data segments in steady-state conditions. Second, static gains of nonlinear dynamic systems in different operating conditions are estimated via linear regression from these steady-state values. The proposed approach has two practical features: (i) estimated static gains can be verified in a convincing way, because the validness of extracted steady-state values is confirmed by visualizing data segments in steady-state conditions, and the accuracy of estimated static gains is verified by comparing the extracted steady-state values and their estimates; (ii) the proposed approach is simple to understand and implement in practice, since it only involves a linear equation between steady-state values and static gains, as well as a basic technique of linear regression. Numerical simulation and industrial application demonstrate the effectiveness of the proposed approach.

Mathematical modeling to standardize times in assembly processes: Application to four case studies

Purpose: This paper proposes model-based standard times estimates, using multiple linear regression, nonlinear optimization, and fuzzy systems in four real cases assembly lines. The work includes a description of the models and a comparison of their performance with values obtained using the conventional chronometer method. These models allow estimating standard times without reconducting field studies.Design/methodology/approach: For the development of the time study, the methodology applied by the International Labour Organization (ILO) was used as a baseline. This methodology is structured in three phases: selection of the case study, registration of the process by direct observation, and calculation/estimation of the standard time. The selected case studies belong to real assembly lines of motorcycles, television sets, printed circuit boards, and bicycles.Findings: In the motorcycle’s assembly case, the study allowed constructing seven linear regression models to estimate standard times for assembling the front parts, and seven linear regression models to predict standard times for the rear parts of the different motorcycle types. Compared to the classical chronometer method, the results obtained never exceeded 10%. Regarding the case studies of assembling TV sets and PCBs, the study considered the construction of nonlinear optimization models that allow making appropriate predictions of the standard times in their assembly lines. Finally, for the bicycle assembly line, a fuzzy logic model to represent the standard time was constructed and validated.Research limitations/implications: For reasons of confidentiality of information, this work omitted the names of companies, services, and models of manufactured products.Originality/value: The literature consulted does not refer to the representation of standard time on assembly lines using mathematical models. The construction of these models with empirical data from actual assembly lines was a valuable aid to the companies involved in supporting activity planning.

Dynamic optimization of nonlinear systems with guaranteed feasibility of inequality-path-constraints

An algorithm is proposed for locating an optimal solution satisfying the KKT conditions to a specified tolerance to inequality-path-constrained dynamic optimization (PCDO) problem within finite iterations. This algorithm solves the optimization problem by iteratively approximating the original optimization problem through adaptive convexification of its lower level programs. In the process of convexification of the lower level programs, αBB technique is used adaptively to construct convex relaxations of a path constraint in each time subinterval. Compared to the result in (Fu, Faust, Chachuat, & Mitsos, 2015), the distinguishing feature is that the proposed algorithm avoids numerically solving the non-convex lower level program to global optimality at each iteration. Two numerical examples are shown to demonstrate the performance of the algorithm.

Parametric Optimization of Nonlinear Systems Represented by Models Using the Extended Linearization Method

We state an optimal control problem for a class of dynamical systems whose nonlinear objects can be represented as objects with linear structure and state-dependent parameters. The linearity of the structure of the transformed nonlinear system and the quadratic performance functional allow one to move from the need to search for solutions of the Hamilton–Jacobi equation to an equation of the Riccati type with state-dependent parameters when synthesizing the optimal control, i.e., the controller parameters. The main problem of implementing the optimal control is related to the problem of being capable of finding solutions of such an equation online, at the object operation rate. An algorithmic method for the parametric optimization of the controller is proposed. The method is based on using the necessary optimality conditions for the control system in question. Our algorithms can be used both to optimize the time-varying objects themselves given an appropriate choice of the parameters for this purpose and to optimize the entire control system using an appropriate parametric adjustment of the controllers. The efficiency of the algorithms is demonstrated by the example of drug treatment of patients with HIV.

The Performance Analysis of a New Modification of Conjugate Gradient Parameter for Unconstrained Optimization Models

Conjugate Gradient (CG) method is the most prominent iterative mathematical technique that can be useful for the optimization of both linear and non-linear systems due to its simplicity, low memory requirement, computational cost, and global convergence properties. However, some of the classical CG methods have some drawbacks which include weak global convergence, poor numerical performance both in terms of number of iterations and the CPU time. To overcome these drawbacks, researchers proposed new variants of the CG parameters with efficient numerical results and nice convergence properties. Some of the variants of the CG method include the scale CG method, hybrid CG method, spectral CG method, three-term CG method, and many more. The hybrid conjugate gradient (CG) algorithm is among the efficient variant in the class of the conjugate gradient methods mentioned above. Some interesting features of the hybrid modifications include inherenting the nice convergence properties and efficient numerical performance of the existing CG methods. In this paper, we proposed a new hybrid CG algorithm that inherits the features of the Rivaie et al. (RMIL*) and Dai (RMIL+) conjugate gradient methods. The proposed algorithm generates a descent direction under the strong Wolfe line search conditions. Preliminary results on some benchmark problems show that the proposed method efficient and promising.