Optimization Of Nonlinear Systems Research Articles

Controller Parameter Optimization for Nonlinear Systems Using Enhanced Bacteria Foraging Algorithm

An enhanced bacteria foraging optimization (EBFO) algorithm-based Proportional + integral + derivative (PID) controller tuning is proposed for a class of nonlinear process models. The EBFO algorithm is a modified form of standard BFO algorithm. A multiobjective performance index is considered to guide the EBFO algorithm for discovering the best possible value of controller parameters. The efficiency of the proposed scheme has been validated through a comparative study with classical BFO, adaptive BFO, PSO, and GA based controller tuning methods proposed in the literature. The proposed algorithm is tested in real time on a nonlinear spherical tank system. The real-time results show that, EBFO tuned PID controller gives a smooth response for setpoint tracking performance.

Fuzzy Model Reference Adaptive Controller Design for Multivariable Nonlinear System

This paper describes the fuzzy model reference adaptive controller design for nonlinear system based on TSK model. The stability of closed-loop system can be assured by Lyapunov function and the adaptive law is developed. The parameters of the controller can be regulated by the adaptive rules and the control scheme is applied on the automatic gauge control system and spherical motor system as two examples. The analysis of dynamic performance for ordinary PID controller and fuzzy adaptive controller is performed in detail with simulation software. Simulation results verify the systems with strong adaptive ability and can adapt to the changes and disturbances of the controlled object. This study provides a theoretical guide for the configuration design, optimization and control research of multivariable nonlinear systems.

An efficient stochastic optimization framework for studying the impact of seasonal variation on the minimum water consumption of pulverized coal (PC) power plants

Coal-fired power plants use water in large amounts second only to agricultural irrigation. Water restrictions become more influential when water-expensive carbon capture technologies are added to the process. Therefore, national efforts to study the reduction of water withdrawal and consumption in existing and future plants have been intensified. Water consumption in these plants is strongly associated with the dry bulb temperature and humidity of air. There is significant variability in these parameters. In this work, we characterized the uncertainty and variability in these parameters in terms of probability distributions. These distributions change with seasons. We obtained optimal operating conditions for each season in the face of uncertainties using water minimization as the objective. It has been found that the water consumption is reduced from 3.2 to 15.4% depending on the season. In order to solve this large scale real world optimization problem, we had to resort to an efficient stochastic nonlinear programming (SNLP) algorithm called Better Optimization of Nonlinear Uncertain Systems (BONUS). It has been found that BONUS reduces the computationally intensity of this problem by 98.7%, making it possible to obtain optimal operating conditions in reasonable computational time.

A quasi-optimal sliding mode control scheme based on control Lyapunov function

In this paper, the sliding mode control (SMC) method is integrated with a nonlinear suboptimal control method based on control Lyapunov function (CLF). According to the system nominal part, a CLF is first constructed in general to facilitate the nonlinear optimal system design and Sontag's formula is used in particular to generate a suboptimal controller. To take system uncertainties into account, the SMC mechanism is designed based on the CLF. By integration, the suboptimal control and SMC are made to function in a complementary manner. When the system state is far away from the equilibrium and the system nominal part is predominant, the nonlinear optimal control part will govern the system response as well as drive the system state approach the equilibrium in an optimal fashion. On the contrary, when approaching the equilibrium such that system perturbations become the main factor, the SMC will take over the control task to warrant the desired robustness property and achieve precise control.

Second-Order Switching Time Optimization for Nonlinear Time-Varying Dynamic Systems

This technical note gives a method for calculating the first and second derivatives of a cost function with respect to switching times for systems with piecewise second-differentiable dynamics. Differential equations governing the linear and bilinear operators required for calculating the derivatives are presented. Example optimizations of linear and nonlinear systems are presented as evidence for the value of second-order optimization methods. One example converges in 33 iterations using second-order methods whereas the first-order algorithm requires over 30 000 iterations.

Minimization of Water Consumption under Uncertainty for a Pulverized Coal Power Plant

Coal-fired power plants are large water consumers. Water consumption in thermoelectric generation is strongly associated with evaporation losses and makeup streams on cooling and contaminant removal systems. Thus, minimization of water consumption requires optimal operating conditions and parameters, while fulfilling the environmental constraints. Several uncertainties affect the operation of the plants, and this work studies those associated with weather. Air conditions (temperature and humidity) were included as uncertain factors for pulverized coal (PC) power plants. Optimization under uncertainty for these large-scale complex processes with black-box models cannot be solved with conventional stochastic programming algorithms because of the large computational expense. Employment of the novel better optimization of nonlinear uncertain systems (BONUS) algorithm, dramatically decreased the computational requirements of the stochastic optimization. Operating conditions including reactor temperatures and pressures; reactant ratios and conditions; and steam flow rates and conditions were calculated to obtain the minimum water consumption under the above-mentioned uncertainties. Reductions of up to 6.3% in water consumption were obtained for the fall season when process variables were set to optimal values. Additionally, the proposed methodology allowed the analysis of other performance parameters like gas emissions and cycle efficiency which were also improved.

A Krylov enhanced proper orthogonal decomposition method for efficient nonlinear model reduction

A model order reduction method is proposed for approximating nonlinear partial differential equations (PDEs). The method attempts to combine the desirable attributes of Krylov reduction and proper orthogonal decomposition (POD) and is entitled Krylov enhanced POD (KPOD). The method approximates nonlinear input/output (I/O) behavior using a sequence of state dependent Krylov subspaces which are obtained via simulation of a nonlinear finite element discretized system. POD is then implemented to extract, from the sequence of subspaces, a projection basis that describes the state dependent variance of the Krylov subspaces. Due to Krylov's moment matching property, the variance describes how the I/O of a sequence of Krylov reduced models would vary as the original model's state varies. Galerkin projection is performed using the extracted basis to yield a low dimensional I/O approximation of the original nonlinear model. Reduced order models of an electro-thermal switch generated using the conventional POD and KPOD are presented and used to compare the I/O approximation capabilities of each method. Our results indicate that KPOD reduced models are capable of accurately modeling I/O using a sequence of Krylov subspaces generated from a single nonlinear discretized model simulation while POD can require multiple nonlinear discretized model simulations to generate a reduced model with similar I/O accuracy. The proposed method can be a potential asset for practical applications in the area of I/O behavior prediction and design optimization of nonlinear systems.

Influence of Number of Gene Loci on Prediction of Animal Phenotype Value Using Back-Propagation Artificial Neural Network

Vast amount of bioinformation immerged in the past, HapMap Project had genotyped more than 3.1 million Single Nucleotide Polymorphisms (SNPs) information by 2007, a prediction equation based on SNPs was derived to calculate genomic breeding values. However, the simple mathematical function could not reflect the complex relation between genome and phenotypes. Unlike the methods of regression, artificial neural networks could perform well for optimization in complex non-linear systems; artificial neural networks have not been used to calculate genomic breeding values. In this paper, back-propagation neural network is used to simulate and predict the genomic breeding values or polygenic genotype value, and the different numbers of gene loci and hidden neurons were used to discuss the influence of the learning rate on estimating the polygenic genotype value. The result showed normalization was very important for training prediction model. After phenotype value normalized, optimum neural network for estimating the animal phenotype could be established without considering the gene number, but the optimum neural network should be selected from amount of neuronal networks with different hidden neuron number. No matter what the gene number is, as well as the number of hidden neurons is right, BP networks could be used to predict the animal phenotypes.

A Mumford-Shah level-set approach for the inversion and segmentation of SPECT/CT data

This paper presents a level-set based approach for the simultaneous reconstruction and segmentation of the activity as well as the density distribution from tomography data gathered by an integrated SPECT/CT scanner. Activity and density distributions are modeled as piecewise constant functions. The segmenting contours and the corresponding function values of both the activity and the density distribution are found as minimizers of a Mumford-Shah like functional over the set of admissible contours and -- for fixed contours -- over the spaces of piecewise constant density and activity distributions which may be discontinuous across their corresponding contours. For the latter step a Newton method is used to solve the nonlinear optimality system. Shape sensitivity calculus is used to find a descent direction for the cost functional with respect to the geometrical variables which leads to an update formula for the contours in the level-set framework. A heuristic approach for the insertion of new components for the activity as well as the density function is used. The method is tested for synthetic data with different noise levels.

Improving the shuffled complex evolution scheme for optimization of complex nonlinear hydrological systems: Application to the calibration of the Sacramento soil‐moisture accounting model

An innovative algorithm, shuffled complexes with principal components analysis (SP‐UCI), is developed to overcome a critical deficiency of the shuffled complex evolution scheme: population degeneration. Population degeneration means that, during the evolutionary search process, the population of search particles may degenerate into a subspace of the full parameter space, thereby missing the capacity of fully exploring the parameter space. Being confined in a subspace may even lead the particle population to converge to nonstationary points, which is a fatal malfunction. To overcome this problem, SP‐UCI employs the principal components analysis to detect the occurrence of population degeneration and remedy the adverse effects. The ensemble of calibrations of the Sacramento soil moisture accounting model with the SP‐UCI method over the Leaf River basin, Mississippi, retrieves the optimal parameter values with the lowest recorded root‐mean‐squared error of simulated daily runoff against the observation. Moreover, the result also provides consistent (narrow ranges) model parameter distribution, which results in a better understanding of the model's behavior, given the watershed's hydrologic features.

Receding Horizon Cost Optimization and Control for Nonlinear Plants

Abstract Control performance and cost optimization can be conflicting goals in the management of industrial processes. Even when optimal or optimization-based control synthesis tools are applied, the economic cost associated with plant operation is often only optimized according to static criteria that pick, among all feasible equilibria, those associated to minimal cost. This note collects and illustrates some recent advances in Receding Horizon optimization of nonlinear systems that allow to simultaneously optimize transient and steady-state economic performance.

Simultaneous identification of electric permittivity and magnetic permeability

In this paper we investigate the simultaneous reconstruction of the electric permittivity and magnetic permeability. The two physical parameters are allowed to be highly discontinuous in the concerned physical domain. The ill-posed inverse problem is formulated into an output least-squares nonlinear minimization with BV-regularization. The regularizing effect and mathematical properties of the regularized system are justified and analysed. A fully discrete Nedelec's edge element method is applied to approximate the regularized nonlinear optimization system, and its convergence is demonstrated.

Reliability-based synthesis of non-linear stochastic dynamical systems: a global approximation approach

The paper presents an efficient methodology to carry out reliability-based structural optimisation of non-linear systems under stochastic loading. The optimisation problem is formulated as the minimisation of an objective function subject to multiple reliability constraints. The reliability constraints are given in terms of first excursion probabilities with large state-space dimensions. A sequential approximate optimisation scheme based on global approximations of the reliability constraints is implemented in the proposed formulation. The approximations of the first excursion probabilities in terms of the design variables are constructed by combining a global mixed linearisation approach with a reliability sensitivity analysis. The sensitivity of the first excursion probabilities are estimated by considering some statistics of an augmented reliability problem. The number of reliability estimations required during the optimal design process is generally very small. Two numerical examples are presented to illustrate the effectiveness of the method.

Constructive methods of control optimization in nonlinear systems

The paper investigates methods of optimal program and point-to-point control in nonlinear differential systems. The proposed methods are based on the idea of discretizing the continuous-time problem. In case of program control, the method of projected Lagrangian is used, which involves solution of an auxiliary problem with linearized constraints by the reduced gradient method; in case of point-to-point control, Bellman's optimality principle is employed for a "grid" over an approximating solvability tube of the system for a specified goal set with approximation of the point-to-point control by families of controlling function or parameters. An example is given, where the optimal point-to-point control is calculated for a model of maneuvering aircraft.

Tradeoff analysis of non-linear dynamical systems under stochastic excitation

An efficient methodology to carry out multi-objective optimization of non-linear structural systems under stochastic excitation is presented. Specifically, an efficient determination of particular Pareto or non-inferior solutions is implemented. Pareto solutions are obtained by compromise programming which is based on the minimization of the distance between the point that contains the individual optima of each of the objective functions and the Pareto set. The response of the structural system is characterized in terms of the first two statistical moments of the response process, i.e. the mean and variance. An efficient sensitivity analysis of non-inferior solutions with respect to the design variables becomes possible with the proposed formulation. Such information is used for decision making and tradeoff analysis. The compromise programming problem is solved by an efficient procedure that combines a local statistical linearization approach, modal analysis, global approximation concepts, and a sequential optimization scheme. Numerical results show that the total number of stochastic analyses required during the multi-objective optimization process is in general very small. Hence, different compromise solutions including the design that best represents the outcome that the designer considers potentially satisfactory are obtained in an efficient manner. In this way, the analyst can conduct a decision-making analysis through an efficient interactive procedure.

Dynamic Optimization of Nonlinear Processes with an Enhanced Scatter Search Method

An enhanced scatter search method for the global dynamic optimization of nonlinear processes using the control vector parametrization (CVP) approach is presented. Sharing some features of the scatter search metaheuristic, this new method presents a simpler but more effective design which helps to overcome typical difficulties of nonlinear dynamic systems optimization such as noise, flat areas, nonsmoothness, and/or discontinuities. This new algorithm provides a good balance between robustness and efficiency in the global phase, and couples a local search procedure to accelerate the convergence to optimal solutions. Its application to four multimodal dynamic optimization problems, compared with other state-of-the-art global optimization algorithms, including an advanced scatter search design, proves its efficiency and robustness, showing a very good scalability.

On the linear convergence of Newton–Krylov methods

Solution of large-scale nonlinear optimization problems and systems of nonlinear equations continues to be a difficult problem. One of the main classes of algorithms for solving these problems is the class of Newton–Krylov methods. In this article the number of iterations of a Krylov subspace method needed to guarantee at least a linear convergence of the respective Newton–Krylov method is studied. The problem is especially complicated if the matrix of the linear system to be solved is nonsymmetric. In the article, explicit estimates for the number of iterations for the Krylov subspace methods are obtained.

Numerical estimation of the Robin coefficient in a stationary diffusion equation

A finite-element method is proposed for the nonlinear inverse problem of estimating the Robin coefficient in a stationary diffusion equation from boundary measurements of the solution and the heat flux. The inverse problem is formulated as an output least squares optimization problem with an appropriate regularization, then the finite-element method is employed to discretize the nonlinear optimization system. Mathematical properties of both the continuous and the discrete optimization problems are investigated. The conjugate gradient method is employed to solve the optimization problem, and an efficient preconditioner via the Sobolev inner product is also suggested. Numerical results for several two-dimensional problems are presented to illustrate the efficiency of the proposed algorithm.

Comparative Study Between Levenberg Marquardt And Genetic Algorithm For Parameter Optimization Of An Electrical System

Parameter optimization is an important problem in control field of power system. In this paper, a comparative study between Levenberg Marquardt (LM) and genetic algorithm (GA) for parameter optimization of nonlinear systems is presented. Some basic steps and technologies used for implementing these approaches, as well as its advantages and disadvantages are discussed. The main goal is to show the merits of genetic algorithm optimization and to determine its suitability in the area of numerical field analysis. Since the DC series motor is widely used in several industrial sectors, the algorithm developed is potentially useful in order to implement a robust closed-loop control. Accordingly, the application of these two approaches to the estimation of the field winding and the armature resistance and inductance armature of DC motor show that the genetic algorithm have a better dynamic performance then the traditional one (LM).

L-Shaped BONUS Algorithm with Application to Water Pollutant Trading

Applications of optimization in the field of process systems engineering often involve dealing with nonlinearity and uncertainty, necessitating the solution of stochastic nonlinear programming (SNLP) problems. However, the existing algorithms to solve such problems suffer from various limitations. The L-shaped BONUS algorithm, which is an integration of the BONUS (Better Optimization of Nonlinear Uncertain Systems) algorithm and the sampling-based L-shaped method, has been recently proposed to overcome some of these problems. It is shown to have desirable computational properties. This work further investigates the properties of the algorithm by applying it to the environmental problem of pollutant (nutrient) trading in the Christina River Basin. With environmental concerns being heightened, pollution abatement-related decisions are important for industry, making efficient solution techniques invaluable. The results confirm the computational efficiency of the L-shaped BONUS algorithm. Simultaneously, interesting aspects of the environmental trading problem are also explored, pointing toward the scope of implementing stochastic programming techniques for better decision making in the field of industrial ecology.