Orthogonal Function Approximation Research Articles

Approximation of optimal control surfaces for 2 × 2 skew-symmetric evolutionary game dynamics

In this paper we study the problem of approximating the general solution to an optimal control problem whose dynamics arise from a 2 × 2 skew-symmetric evolutionary game with arbitrary initial condition. Our approach uses a Fourier approximation method and generalizes prior work in the use of orthogonal function approximation for optimal control. At the same time we cast the fitting problem in the context of a non-standard feedforward neural network and derive the back-propagation operator in this context. An example of the efficacy of this approach is provided and generalizations are discussed.

Design of sliding mode adaptive DC-DC converter based on function approximation

The output power and voltage of solar photovoltaic power generation are easily affected by climate and load changes, so it is necessary to use DC-DC converter at its output to provide stable output voltage. DC-DC converter is a time-varying nonlinear system. When the output load of the converter varies widely, the output waveform distortion and dynamic response speed of the DC-DC converter will slow down. The essence of sliding mode control is a kind of switch control method, and the DC-DC converter needs the switch of switch tube to realize voltage regulation. In this paper, a sliding mode adaptive control strategy based on orthogonal polynomial function approximation is proposed, and Laguerre polynomial is selected as the basis function of DC-DC transformation system. The output voltage error and its first derivative of the system are taken as the input of the network, and the output of the network is taken as the approximation term of the system uncertainty, and the network weight is updated according to the adaptive law. Through simulation and comparison, the adaptive controller designed in this paper is superior to the ordinary sliding mode controller in reducing chattering, adaptive load change and reducing the influence of system interference.

Adaptivno upravljanje unazad bazirano na tehnici aproksimacije funkcije kod elektromehaničkog sistema sa nepoznatim ulaznim koeficijentom

This paper is focused on orthogonal function approximation technique FAT-based adaptive backstepping control of a geared DC motor coupled with a rotational mechanical component. It is assumed that all parameters of the actuator are unknown including the torque-current constant (i.e., unknown input coefficient) and hence a control system with three motor control modes is proposed: 1) motor torque control mode, 2) motor current control mode, and 3) motor voltage control mode. The proposed control algorithm is a powerful tool to control a dynamic system with an unknown input coefficient. Each uncertain parameter/term is represented by a linear combination of weighting and orthogonal basis function vectors. Chebyshev polynomial is used as a strong approximator for estimation of uncertainty. The designed control law includes three terms: a feedforward term, a feedback term and a robust term for compensation of modeling error. Lyapunov stability is used to prove the validity of the proposed controller and to derive the update laws for the weighting vectors of orthogonal Chebyshev approximators. A case study of a geared DC motor in connection with a rotating output load is simulated to prove the effectiveness of the proposed controller structure.

Adaptive Trajectory Tracking Control of Dynamics of Nonholonomic Mobile Robot based on Orthogonal Function Approximation Technique

Adaptive Trajectory Tracking Control of Dynamics of Nonholonomic Mobile Robot based on Orthogonal Function Approximation Technique

Modified Chebyshev-Picard Iteration Methods for Solution of Initial Value Problems

Modified Chebyshev-Picard iteration methods are presented for solving boundary value problems. Chebyshev polynomials are used to approximate the state trajectory in Picard iterations, while the boundary conditions are maintained by constraining the coefficients of the Chebyshev polynomials. Using Picard iteration and Clenshaw-Curtis quadrature, the presented methods iteratively refine an orthogonal function approximation of the entire state trajectory, in contrast to step-wise, forward integration approaches, which render the methods well-suited for parallel computation because computation of force functions along each path iteration can be rigorously distributed over many parallel cores with negligible cross communication needed. The presented methods solve optimal control problems through Pontryagin’s principle without requiring shooting methods or gradient information. The methods are demonstrated to be computationally efficient and strikingly accurate when compared with Battin’s method for a classical Lambert’s problem and with a Chebyshev pseudospectral method for an optimal trajectory design problem. The reported simulation results obtained on a serial machine suggest a strong basis for optimism of using the presented methods for solving more challenging boundary value problems, especially when highly parallel architectures are fully exploited.

Sensor Location Optimization for Distributed Parameter Systems Based on Wavelet Transforms

The sensor location optimization is the special problem of distributed parameter systems(DPS) for parameter identification. It has a great influence on the accuracy of parameter identification and it is important to optimal control of DPS. Therefore, it has a great significance to research the strategy of sensor optimal measurement locations for DPS. In this paper, based on the orthogonal function approximation method, Haar wavelet transforms method is applied to research the problem of sensor measurement locations of DPS. The method based on Fisher information matrix derived from the differential sensitivity equations is proposed to determine the sensor optimal measurement locations. The identification examples prove that the proposed method based on wavelet transforms is feasible and effective to locate the sensor in order to obtain the optimal measurement values. The algorithm makes the calculation simple, easy and accurate.

Numerical solution of mixed problem of parabolic equation with an integral conditions by using finite difference and orthogonal function approximation

In this work the combined finite difference and orthogonal function approximation methods have been proposed for the numerical solution of the one-dimensional parabolic equation with two integrals conditions. The time variable is approximated using a finite difference scheme. But the orthogonal function approximation is employed for discretizing the space variable by using orthogonal cosine function.

Modified Chebyshev-Picard Iteration Methods for Orbit Propagation

Modified Chebyshev-Picard Iteration methods are presented for solving high precision, long-term orbit propagation problems. Fusing Chebyshev polynomials with the classical Picard iteration method, the proposed methods iteratively refine an orthogonal function approximation of the entire state trajectory, in contrast to traditional, step-wise, forward integration methods. Numerical results demonstrate that for orbit propagation problems, the presented methods are comparable to or superior to a state-of-the-art 12th order Runge-Kutta-Nystrom method in a serial processor as measured by both precision and efficiency. We have found revolutionary long solution arcs with more than eleven digit path approximations over one to three lower-case Earth orbit periods, multiple solution arcs can be patched continuously together to achieve very long-term propagation, leading to more than ten digit accuracy with built-in precise interpolation. Of revolutionary practical promise to much more efficiently solving high precision, long-term orbital trajectory propagation problems is the observation that the presented methods are well suited to massive parallelization because computation of force functions along each path iteration can be rigorously distributed over many parallel cores with negligible cross communication needed.

PRACTICAL ASPECTS IN MOVING LOAD IDENTIFICATION

Several methods have been developed in recent years to identify moving loads on top of a continuous beam using measured vibration responses. The methods can identify the forces with some accuracy, but they have not been tested under field measurement conditions with a bridge–vehicle system. This paper discusses the weaknesses and merits of two methods when applied to a single-span bridge deck. The influence, on the moving load identification, of practical aspects such as measurement noise, sampling frequency, a small number of measured response modes, a small number of measuring points, road surface roughness and non-uniform velocity or braking of vehicle is studied in simulations and experiment. Results show that finite element approach with orthogonal function approximation of the responses give more accurate results, in general, than the exact solution approach for all the studies presented in this paper. The road surface roughness and a large variation in the speed are identified as the two main obstacles leading to erroneous results.

Using Taguchi's Method and Orthogonal Function Approximation to Design Optimal Manipulated Trajectory in Batch Processes

A novel experimental scheme is proposed to integrate orthogonal function approximation with Taguchi's method (orthogonal array) for designing the optimal manipulated trajectory of a batch process. The orthogonal function approximation finds a set of orthonormal functions as the basis to represent the batch trajectory. The optimal trajectory can be obtained if the location of the coefficients of the orthonormal functions is properly adjusted in the function space. The Taguchi approach is used to design and analyze the effect of each coefficient on reaching the optimal objective (quality) function. Because the coefficients are implicitly related to the objective function, they simply vary over two levels in a systematic way, and they would be moved into the optimal design condition. A search procedure for the optimal design coefficients is also proposed. As opposed to model-based design, the proposed method utilizes the simplicity of Taguchi methods to determine the potentially available knowledge of dynamic batch processes. For its potential applications, this combinational method is demonstrated through two simulation case studies.

Design of the pH profile for asymmetric bioreduction of ethyl 4-chloro-3-oxobutyrate on the basis of a data-driven method.

The goal of this paper was to design the optimal time-varying operating pH profile in the asymmetric reduction of ethyl 4-chloro-3-oxobutyrate by baker's yeast. Ethyl (S)-4-chloro-3-hydroxybutyrate was produced to reach two important quality indices: reaction yield and product optical purity. The method integrated an orthogonal function approximation and an orthogonal array. The technique used a set of orthonormal functions as the basis for representing the possible profile. The optimal profile could be obtained if the orthogonal coefficients were properly adjusted. The orthogonal array was used to design and analyze the effect of each orthogonal coefficient in order to reach the optimal objective (quality) function. The performance based on the proposed strategy was significantly improved by over 10% compared with the traditional fixed pH or uncontrolled pH values during the reaction. The proposed method can be applied to the required dynamic profile in the bioreactor process to effectively improve the product quality, given good design directions and the advantage of the traditional statistical approach.

On-line Piecewise Monitoring for Batch Processes

Abstract In this work, a technique of multivariable statistical process control for on-line piecewise monitoring of the progress of the batch process is developed. The proposed method is the combination of the orthogonal function approximation and principal component analysis (PCA). This method has been developed in our previous research (Chen and Liu, 1999). In this paper, the method is further exended into on-line monitoring of a batch process with fixed duration. First, the proper number of orthogonal function coefficients based on the approximation theory should be computed in order to capture the information of the whole batch profiles. Based on the process characters, several monitoring points are set in one batch. The information collected from the beginning up to each monitoring point is modeled by the orthogonal function approximation and PCA to extract the correlated feature. Accordingly, the control charts (Q and T) are constructed. Like the philosophy of statistical process control in the traditional MPCA, this method leads to simple monitoring charts, easy tracking of the progress in each batch run and monitoring the occurrence of observable upsets. Comparing with some of the traditional on-line MPCA algorithms, the proposed method not only does correct analysis but also saves a lot of computational load. The effectiveness of the proposed method is demonstrated through a simulation batch reactor to show potential applications.

An Improved Two-Dimensional Micromechanical Theory for Brittle Solids with Randomly Located Interacting Microcracks

This paper presents applications of accurate orthogonal function approximation methods to the two-dimensional problems of brittle solids containing many randomly located (and possibly randomly oriented) and strongly interacting microcracks within the framework of micromechanics and ensemble-volume average approach. The randomly located and oriented two-crack interaction problems are solved by using the highly accurate Legendre and Tchebycheff orthogonal polynomials to any desired order. The complex stress potential method is subsequently employed to micromechanically derive microcrack opening displacements under complicated loadings due to microcrack interaction effects-including concentrated loadings, arbitrary loadings and polynomial loadings. Improved local ensemble-averaged and overall effective elastic compliances of brittle solids due to microcracks and their interactions are systematically constructed by using the pairwise microcrack interaction mechanism and the ensemble-volume average approach. A number of interesting analytical-numerical examples containing different random microcrack configurations are also presented to illustrate the capabilities of the proposed framework.