Specific optimal discrete filtering mechanizations

An optimization procedure is developed for obtaining the best discrete representation of a specified form for modeling a continuous, steady-state, Kalman filter. This technique is particularly useful for digital processors in real-time filtering applications. The generalized optimization format is developed, an optimal digital mechanization is obtained, and numerical comparisons are made with the Euler method and the second-order Runge-Kutta numerical integration formula. Significant improvements are shown for several fixed values of discretization intervals.     

Partially‐observable stochastic hybrid systems (poshss) state estimation and optimal control

This paper discusses the state estimation and optimal control problem of a class of partially‐observable stochastic hybrid systems (POSHS). The POSHS has interacting continuous and discrete dynamics with uncertainties. The continuous dynamics are given by a Markov‐jump linear system and the discrete dynamics are defined by a Markov chain whose transition probabilities are dependent on the continuous state via guard conditions. The only information available to the controller are noisy measurements of the continuous state. To solve the optimal control problem, a separable control scheme is applied: the controller estimates the continuous and discrete states of the POSHS using noisy measurements and computes the optimal control input from the state estimates. Since computing both optimal state estimates and optimal control inputs are intractable, this paper proposes computationally efficient algorithms to solve this problem numerically. The proposed hybrid estimation algorithm is able to handle state‐dependent Markov transitions and compute Gaussian‐ mixture distributions as the state estimates. With the computed state estimates, a reinforcement learning algorithm defined on a function space is proposed. This approach is based on Monte Carlo sampling and integration on a function space containing all the probability distributions of the hybrid state estimates. Finally, the proposed algorithm is tested via numerical simulations.     

Digital Redesign Of Continuous‐Time Suboptimal Tracker For Two‐Dimensional Systems

In this paper, the digital redesign of a continuous suboptimal tracker for the two‐dimensional (2‐D) systems is proposed. This paper presents a new optimal digital redesign technique of 2‐D systems for finding a dynamic digital control law from the given continuous‐time 2‐D systems by minimizing a quadratic cost function (performance index). We directly convert the original continuous‐time 2‐D quadratic cost function into the discretized form and solve the optimization problem in the discrete‐time domain. The developed optimal digital redesign control law enables the output of the digitally controlled closed‐loop systems to closely match the reference signal for 2‐D systems, and it can be easily implemented using microcomputers. An illustrative example is presented to demonstrate the effectiveness of the proposed procedure.     

Compensatability and optimal compensation of systems with white parameters in the delta domain

Using the delta operator, the strengthened discrete-time optimal projection equations for optimal reduced-order compensation of systems with white stochastic parameters are formulated in the delta domain. The delta domain unifies discrete time and continuous time. Moreover, when formulated in this domain, the efficiency and numerical conditioning of algorithms improves when the sampling rate is high. Exploiting the unification, important theoretical results, algorithms and compensatability tests concerning finite and infinite horizon optimal compensation of systems with white stochastic parameters are carried over from discrete time to continuous time. Among others, we consider the finite-horizon time-varying compensation problem for systems with white stochastic parameters and the property mean-square compensatability (ms-compensatability) that determines whether a system with white stochastic parameters can be stabilised by means of a compensator. In continuous time, both of these appear to be new. This also holds for the associated numerical algorithms and tests to verify ms-compensatability. They are illustrated with three numerical examples that reveal several interesting theoretical and numerical issues. A fourth example illustrates the improvement of both the efficiency and numerical conditioning of the algorithms. This is of vital practical importance for digital control system design when the sampling rate is high.     

Properties of the magnitude terms of orthogonal scaling functions

The spectrum of the convolution of two continuous functions can be determined as the continuous Fourier transform of the cross-correlation function. The same can be said about the spectrum of the convolution of two infinite discrete sequences, which can be determined as the discrete time Fourier transform of the cross-correlation function of the two sequences. In current digital signal processing, the spectrum of the continuous Fourier transform and the discrete time Fourier transform are approximately determined by numerical integration or by densely taking the discrete Fourier transform. It has been shown that all three transforms share many analogous properties. In this paper we will show another useful property of determining the spectrum terms of the convolution of two finite length sequences by determining the discrete Fourier transform of the modified cross-correlation function. In addition, two properties of the magnitude terms of orthogonal wavelet scaling functions are developed. These properties are used as constraints for an exhaustive search to determine a robust lower bound on conjoint localization of orthogonal scaling functions.     

A Comparative Approach for Time‐Delay Fractional Optimal Control Problems: Discrete Versus Continuous Chebyshev Polynomials

This paper aims to demonstrate the superiority of the discrete Chebyshev polynomials over the classical Chebyshev polynomials for solving time‐delay fractional optimal control problems (TDFOCPs). The discrete Chebyshev polynomials have been introduced and their properties are investigated thoroughly. Then, the fractional derivative of the state function in the dynamic constraint of TDFOCPs is approximated by these polynomials with unknown coefficients. The operational matrix of fractional integration together with the dynamical constraints is used to approximate the control function directly as a function of the state function. Finally, these approximations were put in the performance index and necessary conditions for optimality transform the under consideration TDFOCPs into an algabric system. A comparison has been made between the required CPU time and accuracy of the discrete and continuous Chebyshev polynomials methods. The obtained numerical results reveal that utilizing discrete Chebyshev polynomials is more efficient and less time‐consuming in comparison to the continuous Chebyshev polynomials.     

Neural networks for solving multicriteria solid transportation problem

In this paper, we present neural networks for solving multicriteria solid transportation problems. The original problem is transformed into an equivalent continuous problem from the continuous-time dynamic system and its optimal solution can be got. The procedure and efficiency of this approach are shown with numerical simulations.     

Optimal activation policies for continuous scanning observations in parameter estimation of distributed systems

The problem of determining an optimal measurement scheduling for identification of unknown parameters in distributed systems described by partial differential equations is discussed. The discrete-scanning observations are performed by an optimal selection of measurement data from spatially fixed sensors. In the adopted approach, the sensor scheduling problem is converted to a constrained optimal control problem. In this framework, the control value represents the selected sensor configuration. Thus the control variable is constrained to take values in a discrete set and switchings between sensors may occur in continuous time. By applying the control parameterization enhancing transform technique, a computational procedure for solving the optimal scanning measurement problem is obtained. The numerical scheme is then tested on a computer example regarding an advection-diffusion problem.     

连续时间 Hopfield网络模型数值实现分析

讨论使用Euler方法和梯形方法在数值求解连续时间的Hopfield网络模型时,离散时间步长的选择和迭代停止条件问题.利用凸函数的定义研究了能量函数下降的条件,根据凸函数的性质分析它的共轭函数减去二次函数之差仍为凸函数的条件.分析连续时间Hopfield网络模型的收敛性证明,提出了一个广义的连续时间Hopfield网络模型.对于常用的Euler方法和梯形方法数值求数值实现连续时间Hopfield网络,讨论了离散时间步长的选择.由于梯形方法为隐式方法,分析了它的迭代求算法的停止条件.根据连续时间Hopfield网络的特点,提出改进的迭代算法,并对其进行了分析.数值实验的结果表明,较大的离散时间步长不仅加速了数值实现,而且有利于提高优化性能.     

Linear quadratic networked control of uncertain polytopic systems

This paper investigates the problem of designing robust linear quadratic regulators for uncertain polytopic continuous‐time systems over networks subject to delays. The main contribution is to provide a procedure to determine a discrete‐time representation of the weighting matrices associated to the quadratic criterion and an accurate discretized model, in such a way that a robust state feedback gain computed in the discrete‐time domain assures a guaranteed quadratic cost to the closed‐loop continuous‐time system. The obtained discretized model has matrices with polynomial dependence on the uncertain parameters and an additive norm‐bounded term representing the approximation residual error. A strategy based on linear matrix inequality relaxations is proposed to synthesize, in the discrete‐time domain, a digital robust state feedback control law that stabilizes the original continuous‐time system assuring an upper bound to the quadratic cost of the closed‐loop system. The applicability of the proposed design method is illustrated through a numerical experiment. Copyright © 2015 John Wiley & Sons, Ltd.     

Hybrid state-space fuzzy model-based controller with dual-ratesampling for digital control of chaotic systems

We develop a hybrid state-space fuzzy model-based controller with dual-rate sampling for digital control of chaotic systems. A Takagi-Sugeno (TS) fuzzy model is used to model the chaotic dynamic system and the extended parallel-distributed compensation technique is proposed and formulated for designing the fuzzy model-based controller under stability conditions. The optimal regional-pole assignment technique is also adopted in the design of the local feedback controllers for the multiple TS linear state-space models. The proposed design procedure is as follows: an equivalent fast-rate discrete-time state-space model of the continuous-time system is first constructed by using fuzzy inference systems. To obtain the continuous-time optimal state-feedback gains, the constructed discrete-time fuzzy system is then converted into a continuous-time system. The developed optimal continuous-time control law is finally converted into an equivalent slow-rate digital control law using the proposed intelligent digital redesign method. The main contribution of the paper is the development of a systematic and effective framework for fuzzy model-based controller design with dual-rate sampling for digital control of complex such as chaotic systems. The effectiveness and the feasibility of the proposed controller design method is demonstrated through numerical simulations on the chaotic Chua circuit     

Dispersion-bounded numerical integration of the elastodynamic equations with cost-effective staggered schemes

Known procedures for designing numerical schemes for the integration of elastodynamic equations with explicit control over numerical dispersion are reviewed. In the literature, the analysis of such schemes has concentrated on the discrete space differentiators, and has neglected the role played by time discretization in the overall accuracy. In this paper we define a computational cost for a given dispersion error bound which fully includes the effect of temporal differencing. For some representative schemes based on leap-frog time marching, we provide an optimal operating point (time sampling rate and number of grid points per shortest wavelength) which minimizes the computational cost for a given dispersion error threshold. Based on this notion of cost, we introduce new optimal operators for staggered grids. Additionally, we introduce the notion of composite differentiators to design still more cost-effective schemes. The cost of the proposed schemes is shown to be less than that of known finite difference (FD) operators and compares favorably with pseudo-spectral (PS) algorithms. Numerical simulations are presented to illustrate the effectiveness of the new operators.     

Optimal design for 2D wave equations

In this paper We consider a problem of optimal design in 2D for the wave equation with Dirichlet boundary conditions. We introduce a finite element discrete version of this problem in which the domains under consideration are polygons defined on the numerical mesh. We prove that, as the mesh size tends to zero, any limit, in the sense of the complementary-Hausdorff convergence, of discrete optimal shapes is an optimal domain for the continuous optimal design problem. We work in the functional and geometric setting introduced by V. ?veràk in which the domains under consideration are assumed to have an a priori limited number of holes. We present in detail a numerical algorithm and show the efficiency of the method through various numerical experiments.     

The optimum shape of cooling towers

Numerous computer optimization techniques have been developed and applied primarily to the design of structures composed of discrete elements. Continuous surface structures have been optimized primarily by methods based upon the differential or integral calculus (e.g. the calculus of variations). However, the determination of the optimal shape of continuous surface structures can also be approached by algebraic methods more suitable for digital computation. If the coordinates of the middle surface of a shell are expressed by a finite polynomial series, an optimization problem in a finite set of discrete variables results. In the present work, this method is applied to a particular example of a shell of revolution: a natural draft cooling tower. A simple preliminary design model is formulated in order to evaluate the potential savings due to numerical optimization, and the resulting nonlinear programming problem is solved by iterated linear programming. The results indicate that the method is feasible and that significant savings might be attainable by computerized shape optimization.     

Precise Tracking for Continuous‐Time Non‐Minimum Phase Systems

Stable inversion based precise tracking for continuous‐time square or nonsquare non‐minimum phase systems is studied. However, high precision trajectory tracking of non‐minimum phase systems can be obtained by the stable inversion method but requiring large enough extended time interval. In order to solve this problem of large extended time restriction, a novel approach to precise trajectory tracking of non‐minimum phase systems is proposed, it is called the improved stable inversion (ISI) method, using an optimal integration of the pre‐actuation and the optimal state transition (OST) techniques. The ISI method can obtain precise trajectory tracking with a smaller extended time interval as compared to the stable inversion method. The proposed method achieves better validation through numerical simulations for the non‐minimum phase system.     

Weight and cost oriented multi-objective optimisation of impact damage resistant stiffened composite panels

In this paper, a multi-objective optimisation procedure, based on the adoption of genetic algorithms, is presented. The optimal configuration with minimum weight and minimum cost of a damage resistant stiffened, composite panel with buckling constraints has been determined. The numerical procedure is based on an in-house optimisation code used in conjunction with the ANSYS FEM code. The presence of both continuous and high sensitivity discrete design variables, suggested for GA the adoption of a special bit-masking data structure able to increase the overall computational efficiency. Optimal configurations of the stiffened panel, are finally analysed and discussed focusing on the influence of the damage resistance constraint on the overall costs.     

Solution quality of random search methods for discrete stochastic optimization

In this paper, we propose a framework for selecting a high quality global optimal solution for discrete stochastic optimization problems with a predetermined confidence level using general random search methods. This procedure is based on performing the random search algorithm several replications to get estimate of the error gap between the estimated optimal value and the actual optimal value. A confidence set that contains the optimal solution is then constructed and methods of the indifference zone approach are used to select the optimal solution with high probability. The proposed procedure is applied on a simulated annealing algorithm for solving a particular discrete stochastic optimization problem involving queuing models. The numerical results indicate that the proposed technique indeed locate a high quality optimal solution.     

Error Estimation of a Class of Stable Spectral Approximation to the Cahn-Hilliard Equation

In this work, the initial-boundary value problem of two-dimensional Cahn-Hilliard equation is considered. A class of fully discrete dissipative Fourier spectral schemes are proposed. The existence of the numerical solution is proved by a series of a priori estimations and the Brower fixed point theorem. The uniqueness of the numerical solution is discussed. The optimal converge rate is obtained by the energy method. The numerical simulations are performed to demonstrate the effectiveness of the proposed schemes.     

The continuous non-linear approximation of procedurally defined curves using integral B-splines

This paper outlines an algorithm for the continuous non-linear approximation of procedurally defined curves. Unlike conventional approximation methods using the discrete L_2 form metric with sampling points, this algorithm uses the continuous L_2 form metric based on minimizing the integral of the least square error metric between the original and approximate curves. Expressions for the optimality criteria are derived based on exact B-spline integration. Although numerical integration may be necessary for some complicated curves, the use of numerical integration is minimized by a priori explicit evaluations. Plane or space curves with high curvatures and/or discontinuities can also be handled by means of an adaptive knot placement strategy. It has been found that the proposed scheme is more efficient and accurate compared to currently existing interpolation and approximation methods.     

Structural synthesis of frame reinforced,submersible, circular,cylindrical hulls

This paper describes a mathematical programming procedure for the automated optimal structural synthesis of frame stiffened, cylindrical shells. For a specified set of design parameters such as external pressure, shell radius and length and material properties, the method generates those values of the design variables that produce a minimum weight design. The skin, frame web and frame flange thicknesses and the flange width are treated as continuous variables. Frame spacing is considered a discrete variable. Constraint equations control local and general shell and frame instability and yield. Limits may be placed on the variable values, and certain geometric or space constraints can be applied. The mixed (continuous and discrete nonlinear programming problem is solved by a combination of a discrete ‘Golden Search’ for the optimal number of frames and the ‘Direct Search Design Algorithm’ which provides the optimum values of the continuous variables.