Solution of optimal control problem of linear diffusion equations via Walsh functions

An attempt is made in this note to illustrate the use of Walsh functions in solving Riccati matrix equations arising in optimal control studies of linear diffusion equations with quadratic performance index.     

Analysis of piecewise constant delay systems via block-pulse functions

The solution of linear piecewise constant delay systems has been derived by using the block pulse series expansion method. A matrix called the delay operational matrix is introduced first to manipulate the time delay. Then the solution is obtained through using the matrix and the properties of block pulse functions.     

Anti-windup synthesis via sampled-data piecewise affine optimal control

Discrete-time receding horizon optimal control is employed in model-based anti-windup augmentation. The optimal control formulation enables designs that minimize the mismatch between the unconstrained closed-loop response with a given controller and the constrained closed-loop response with anti-windup augmentation. Recently developed techniques for off-line computation of the constrained linear regulator's solution, which is piecewise affine, facilitate implementation. The resulting sampled-data, anti-windup closed-loop system's properties are established and its performance is demonstrated on a simulation example.     

A computational method for simultaneous LQ optimal control designvia piecewise constant output feedback

The paper is concerned with simultaneous linear-quadratic (LQ) optimal control design for a set of LTI systems via piecewise constant output feedback. First, the discrete-time simultaneous LQ optimal control design problem is reduced to solving a set of coupled matrix inequalities and an iterative LMI algorithm is presented to compute the feedback gain. Then, simultaneous stabilization and simultaneous LQ optimal control design of a set of LTI continuous-time systems are considered via periodic piecewise constant feedback gain. It is shown that the design of a periodic piecewise constant feedback gain simultaneously minimizing a set of given continuous-time performance indexes can be reduced to that of a constant feedback gain minimizing a set of equivalent discrete-time performance indexes. Explicit formulas for computing the equivalent discrete-time systems and performance indexes are derived. Examples are used to demonstrate the effectiveness of the proposed method.     

Time-varying reactivity reconstruction via Walsh functions

The problem of computing and reconstructing the time-varying reactivity input of nuclear reactor systems by using neutron level data records is considered. The integral equation representation of the nuclear reactor is used, and an expansion of the reactivity in a Walsh function series is made. The method provides simple formulas for computing the coefficients of this series. By using these coefficients and the Walsh functions provided by existing generators the time-varying reactivity can be easily reconstructed in a step-like form. A simple example shows the effectiveness of the method.     

Solving integral equations via Walsh functions

A new method for solving integral equations based on Walsh functions is established. The solution of a convolution integral via the Walsh transform is studied first. The extension of the technique to the integral equation of the first kind is naturally obtained. In literature, the integral equation of the second kind is considered to be much more difficult to solve than the first. In this new approach, these two kinds are solved with nearly the same ease. Several numerical examples are included for illustration.     

Non-fragile control of fuzzy affine dynamic systems via piecewise Lyapunov functions

This paper addresses the robust H_∞ static output feedback (SOF) controller design problem for a class of uncertain fuzzy affine systems that are robust against both the plant parameter perturbations and controller gain variations. More specifically, the purpose is to synthesize a non-fragile piecewise affine SOF controller guaranteeing the stability of the resulting closed-loop fuzzy affine dynamic system with certainH_∞ performance index. Based on piecewise quadratic Lyapunov functions and applying some convexification procedures, two different approaches are proposed to solve the robust and non-fragile piecewise affine SOF controller synthesis problem. It is shown that the piecewise affine controller gains can be obtained by solving a set of linear matrix inequalities (LMIs). Finally, simulation examples are given to illustrate the effectiveness of the proposed methods.     

Spherical piecewise constant basis functions for all-frequency precomputed radiance transfer

This paper presents a novel basis function, called spherical piecewise constant basis function (SPCBF), for precomputed radiance transfer. SPCBFs have several desirable properties: rotatability, ability to represent all-frequency signals, and support for efficient multiple product. By smartly partitioning the illumination sphere into a set of subregions, and associating each subregion with an SPCBF valued 1 inside the region and 0 elsewhere, we precompute the light coefficients using the resulting SPCBFs. Efficient rotation of the light representation in SPCBFs is achieved by rotating the domain of SPCBFs. We run-time approximate the BRDF and visibility coefficients using the set of SPCBFs for light, possibly rotated, through fast lookup of summed-area-table (SAT) and visibility distance table (VDT), respectively. SPCBFs enable new effects such as object rotation in all-frequency rendering of dynamic scenes and on-the-fly BRDF editing under rotating environment lighting. With graphics hardware acceleration, our method achieves real-time frame rates.     

Note on the design of linear systems with piecewise constant feedback gains

The purpose of this correspondence is to report some relations that can be used to extend the scope of a computational algorithm for the suboptimal design of linear regulator systems. This algorithm was presented in [1], and can be used to determine suboptimal piecewise constant feedback gains to be used instead of the optimal continuously time-varying ones [3]. This correspondence deals with the extension to the case where, in addition to the piecewise constant gains, the switching times are also optimized.     

Identification of time-lag systems via Walsh functions

This note successfully combines the two techniques of Prasada Rao and Sivakumar, namely the iterative shift algorithm for identifying time-lag systems [5] and the basic technique of system identification via Walsh functions [1] and illustrates how Walsh functions may be employed to identify systems with unknown time lags.     

Minimum energy control of time delay systems via piecewise linear polynomial functions

The piecewise linear polynomial function approach to the minimum energy control of linear systems with time delay, is presented in this paper. The concepts of a delay shift matrix and an operational matrix for integration are employed in solving the related state and costate equations containing terms with advanced and delayed arguments. An attractive feature of the present method is its ultimate simplicity and convenience. The differential equations with delay and advance terms are converted into a set of linear algebraic equations using a recurrence algorithm. An example demonstrates the accuracy of the method.     

Driving a linear constant system by a piecewise constant control

Recent studies have shown that complete controllability by means of all admissible controls is equivalent to complete controllability by means of piecewise constant controls with n preassigned switching times. This paper addresses the problem of determining the switching times and provides a simple algorithm for the derivation of a piecewise constant control function which accomplishes the transfer of the system from its current position to a desired reachable target. A discussion on the applications of the results obtained in this study to design problems, especially to computer-controlled systems, is given.     

Fitting optimal piecewise linear functions using genetic algorithms

Constructing a model for data in R² is a common problem in many scientific fields, including pattern recognition, computer vision, and applied mathematics. Often little is known about the process which generated the data or its statistical properties. For example, in fitting a piecewise linear model, the number of pieces, as well as the knot locations, may be unknown. Hence, the method used to build the statistical model should have few assumptions, yet, still provide a model that is optimal in some sense. Such methods can be designed through the use of genetic algorithms. We examine the use of genetic algorithms to fit piecewise linear functions to data in R². The number of pieces, the location of the knots, and the underlying distribution of the data are assumed to be unknown. We discuss existing methods which attempt to solve this problem and introduce a new method which employs genetic algorithms to optimize the number and location of the pieces. Experimental results are presented which demonstrate the performance of our method and compare it to the performance of several existing methods, We conclude that our method represents a valuable tool for fitting both robust and nonrobust piecewise linear functions     

A concise derivation of optimal constant limited state feedback gains

A concise derivation of the necessary conditions that must be satisfied by optimal constant limited state feedback gains is given. The approach taken to obtain these conditions is applicable to other limited state feedback controller design problems as well.     

A simple derivation of the gradient conditions for optimal constant output feedback gains

A simple and direct derivation of necessary conditions for optimality of output feedback gains for a linear time-invariant system is presented. The dependence of the cost on the initial statex(0)is eliminated by the well-known technique of averaging the initial state over the surface of then-dimensional unit sphere.     

分段线性系统最优控制设计的一种混合算法

将分段线性系统的最优控制设计问题转化成以反馈增益为寻优参数，以最优控制性能上界为目标的一组双线性矩阵不等式(BMI)问题．将遗传算法与内点法相结合设计出一种混合算法，对BMI问题进行求解．算例仿真表明该算法是简便而有效的．     

Functional data clustering via piecewise constant nonparametric density estimation

In this paper, we present a novel way of analyzing and summarizing a collection of curves, based on piecewise constant density estimation. The curves are partitioned into clusters, and the dimensions of the curves points are discretized into intervals. The cross-product of these univariate partitions forms a data grid of cells, which represents a nonparametric estimator of the joint density of the curves and point dimensions. The best model is selected using a Bayesian model selection approach and retrieved using combinatorial optimization algorithms. The proposed method requires no parameter setting and makes no assumption regarding the curves; beyond functional data, it can be applied to distributional data. The practical interest of the approach for functional data and distributional data exploratory analysis is presented on two real world datasets.     

Model predictive control of continuous-time nonlinear systems with piecewise constant control

A new model predictive control (MPC) algorithm for nonlinear systems is presented. The plant under control, the state and control constraints, and the performance index to be minimized are described in continuous time, while the manipulated variables are allowed to change at fixed and uniformly distributed sampling times. In so doing, the optimization is performed with respect to sequences, as in discrete-time nonlinear MPC, but the continuous-time evolution of the system is considered as in continuous-time nonlinear MPC.