Event-triggered control for sampled-data set stabilization of switched delayed boolean control networks

This paper considers the event-triggered control design for the uniform sampled-data set stabilization of switched delayed Boolean control networks (SDBCNs). First, using the algebraic state space representation method, SDBCNs are converted into the equivalent algebraic form. Second, using the algebraic form, the uniform sampled-data reachable sets are constructed, based on which, a necessary and sufficient condition is obtained for the uniform sampled-data set stabilization of SDBCNs. Finally, the event-triggered mechanism is presented, and a sufficient condition is proposed to design the time-variant state feedback event-triggered controller for the uniform sampled-data set stabilization of SDBCNs.     

PIλDμ controller design for integer and fractional plants using piecewise orthogonal functions

This paper deals with the design of fractional PID controller for integer and fractional plants. A new analytic method is proposed, the developments are based on the expansion of the control loop signals as well as a chosen reference model input and output over a piecewise orthogonal functions, namely, Block pulse, Walsh and Haar wavelets. The generalized operational matrices of differentiation related to these bases which are fitting the Riemann–Liouville definition accurately are used to replace the fractional differential calculus by an algebraic one easier to solve. Thereafter, the controller tuning is elaborated simply with a matrix manipulation manner. At first, a least square is drawn to find only the controller gains, then a nonlinear function defined as a matrix norm is minimized to optimize the whole parameters. A variety of examples covering both integer and fractional systems and reference models are presented to show the validity of the technique.     

On the properties of a new tensor product of matrices

Previously, the author introduced a new tensor product of matrices according to which the matrix of the discrete Walsh-Paley transform can be represented as a power of the second-order discrete Walsh transform matrix H with respect to this product. This power is an analogue of the representation of the Sylvester-Hadamard matrix in the form of a Kronecker power of H. The properties of the new tensor product of matrices are examined and compared with those of the Kronecker product. An algebraic structure with the matrix H used as a generator element and with these two tensor products of matrices is constructed and analyzed. It is shown that the new tensor product operation proposed can be treated as a convenient mathematical language for describing the foundations of discrete Fourier analysis.     

Robust digital controllers for uncertain chaotic systems: A digital redesign approach

In this paper, a new and systematic method for designing robust digital controllers for uncertain nonlinear systems with structured uncertainties is presented. In the proposed method, a controller is designed in terms of the optimal linear model representation of the nominal system around each operating point of the trajectory, while the uncertainties are decomposed such that the uncertain nonlinear system can be rewritten as a set of local linear models with disturbed inputs. Applying conventional robust control techniques, continuous-time robust controllers are first designed to eliminate the effects of the uncertainties on the underlying system. Then, a robust digital controller is obtained as the result of a digital redesign of the designed continuous-time robust controller using the state-matching technique. The effectiveness of the proposed controller design method is illustrated through some numerical examples on complex nonlinear systems––chaotic systems.     

On the arithmetic Walsh coefficients of Boolean functions

We generalize to the arithmetic Walsh transform (AWT) some results which were previously known for the Walsh–Hadamard transform of Boolean functions. We first generalize the classical Poisson summation formula to the AWT. We then define a generalized notion of resilience with respect to an arbitrary statistical measure of Boolean functions. We apply the Poisson summation formula to obtain a condition equivalent to resilience for one such statistical measure. Last, we show that the AWT of a large class of Boolean functions can be expressed in terms of the AWT of a Boolean function of algebraic degree at most three in a larger number of variables.     

Walsh函数的一个迭代方程体系

Shanks曾用迭代方程产生离散佩利编号Walsh函数。作者在[4]中,给出了产生离散沃尔什编号Wa1sh函数的迭代方程. 本文在上述基础上,提出了一个产生离散哈德玛编号Walsh函数的迭代方程,推出了离散哈德玛编号Walsh函数的表示式及其变换(FWHT)的快速计算公式. 上述三个极为类似的迭代方程,已构成了离散Walsh函数的迭代方程体系.连续的Walsh函数,也能用迭代方程这种形式来描述.     

A novel algorithm enumerating bent functions

Based on the relationship between the Walsh spectra of a Boolean function at partial points and the Walsh spectra of its subfunctions, and on the binary Möbius transform, a novel algorithm is developed, which can theoretically construct all bent functions. Practically we enumerate all bent functions in 6 variables. With the restriction on the algebraic normal form, the algorithm is also efficient in more variables case. For example, enumeration of all homogeneous bent functions of degree 3 in 8 variables can be done in one minute with a P4 1.7 GHz computer; the nonexistence of homogeneous bent functions in 10 variables of degree 4 is computationally proved.     

A new time-discretization for delay multiple-input nonlinear systems using the Taylor method and first order hold

A new discretization method is proposed for multi-input-driven nonlinear continuous systems with time-delays, based on a combination of the Taylor series expansion and the first-order hold (FOH) assumption. The mathematical structure of the new discretization scheme is explored. On the basis of this structure, the sampled-data representation of the time-delayed multi-input nonlinear system is derived. First the new approach is applied to nonlinear systems with two inputs, and then the delayed multi-input general equation is derived. The resulting time discretization method provides a finite-dimensional representation for multi-input nonlinear systems with time-delays, thereby enabling the application of existing controller design techniques to such systems. The performance of the proposed method is evaluated using a nonlinear system with time-delays (maneuvering an automobile). Various sampling rates, time-delay values and control inputs are considered to evaluate the proposed method. The results demonstrate that the proposed discretization scheme can meet the system requirements even when using a large sampling period with precision limitations. The discretization results of the FOH method are also compared with those of the zero order hold (ZOH) method. The precision of the FOH method in the discretization procedure combined with the Taylor series expansion is much higher than that of the ZOH method except in the case of constant inputs.     

Distribution semigroups and control systems

We introduce the new concept of a distributional control system. This class of systems is the natural generalization of distribution semigroups to input/state/output systems. We showthat, under the Laplace transform, this new class of systems is equivalent to the class of distributional resolvent linear systems which we introduced in an earlier article. There we showed that this latter class of systems is the correct abstract setting in which to study many non-well-posed control systems such as the heat equation with Dirichlet control and Neumann observation. In this article we further show that any holomorphic function defined and polynomially bounded on some right half-plane can be realized as the transfer function of some exponentially bounded distributional resolvent linear system.     

Design PD controller for master–slave synchronization of chaotic Lur’e systems with sector and slope restricted nonlinearities

In this paper, design PD controller for master–slave synchronization of chaotic Lur’e systems with sector and slope restricted nonlinearities is presented. A new synchronization criterion is proposed based on Lyapunov functions with quadratic form of states and nonlinear functions of the systems. Sector and slope bounds are employed to the Lyanunov–Krasovskii functional through convex representation of the nonlinearities so that less conservative stability conditions are obtained. The criteria is given in terms of linear matrix inequalities (LMIs) by using Finsler’s lemma. A numerical example is provided to illustrate the effectiveness of the method.     

An algebraic algorithm for parameter identification in a class of systems described by linear partial differential equations

The identification of parameters in systems described by one-dimensional linear partial differential equations with constant coefficients is addressed. The algebraic approach used for identification has previously been applied to different infinite dimensional systems. Here, an algebraic algorithm is presented generalizing the underlying ideas from these examples to the class of systems mentioned before. It derives simple polynomial equations relating the concentrated measurements and the unknown parameters by using the Laplace transform and applying methods of commutative and differential algebra such as the Ritt algorithm. In the end, the identification of parameters requires only the calculation of convolution products of measurement signals. A vibrating string serves to illustrate the theory. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Doubly coprime representation of linear systems and its application to simultaneous stabilization

This paper develops a new representation theory of multivariabletime-invariant linear systems based on the well-known coprimefactorizations of their transfer matrices. This representationfully uses the algebraic structure of coprime factorizations,and hence has a number of advantages for describing variousproperties of linear systems in simpler and/or more compactforms than the usual transfer matrix representation. In theframework of this new representation theory, various stabilityand stabilizability properties of linear systems are characterizedand finally a simultaneous stabilization problem for a givenset of linear systems is examined.     

On the functional properties of the confluent hypergeometric zeta-function

A zeta-function associated with Kummer’s confluent hypergeometric function is introduced as a classical Dirichlet series. An integral representation, a transformation formula, and relation formulas between contiguous functions and one generalization of Ramanujan’s formula are given. The inverse Laplace transform of confluent hypergeometric functions is essentially used to derive the integral representation.     

Acceleration of Convergence of Static and Dynamic Iterations

A new technique for acceleration of convergence of static and dynamic iterations for systems of linear equations and systems of linear differential equations is proposed. This technique is based on splitting the matrix of the system in such a way that the resulting iteration matrix has a minimal spectral radius for linear systems and a minimal spectral radius for some value of a parameter in Laplace transform domain for linear differential systems.This revised version was published online in October 2005 with corrections to the Cover Date.     

A controller design for a special class of nonlinear systems

In this paper, we propose a new controller design approach for a special class of nonlinear systems. The controller design is simple and systematic and is based on the construction of a similarity transformation which is used in finding canonical forms for linear controllable systems. In addition, the design does not require any coordinate transformation and is directly applied on the original structure of the system. The performance of the proposed controller is shown via a simulation example dealing with a typical synchronous generator     

Application of high order numerical quadratures to numerical inversion of the Laplace transform

In this paper, new algorithms are proposed for Fredholm integral equations of the first kind corresponding to the inverse Laplace transform. We apply high order numerical quadratures to the truncated integral equation and apply regularization to the discretized linear systems. The resulted regularized least square problems are then solved by the reduced QR factorization method. Several examples taken from the literature are tested. Numerical results show that the approximate inverse Laplace transform obtained by our approach can be very accurate.     

An approximation method using approximate approximations

The aim of this article is to extend the method of approximate approximations to boundary value problems. This method was introduced by V. Maz'ya in 1991 and has been used until now for the approximation of smooth functions defined on the whole space and for the approximation of volume potentials. In the present article we develop an approximation procedure for the solution of the interior Dirichlet problem for the Laplace equation in two dimensions using approximate approximations. The procedure is based on potential theoretical considerations in connection with a boundary integral equations method and consists of three approximation steps as follows. In the first step, the unknown source density in the potential representation of the solution is replaced by approximate approximations. In the second, the decay behavior of the generating functions is used to gain a suitable approximation for the potential kernel, and in the third, Nyström's method leads to a linear algebraic system for the approximate source density. For every step a convergence analysis is established and corresponding error estimates are given.     

A coupled method of Laplace transform and Legendre wavelets for nonlinear Klein–Gordon equations

Klein–Gordon equation models many phenomena in both physics and applied mathematics. In this paper, a coupled method of Laplace transform and Legendre wavelets, named (LLWM), is presented for the approximate solutions of nonlinear Klein–Gordon equations. By employing Laplace operator and Legendre wavelets operational matrices, the Klein–Gordon equation is converted into an algebraic system. Hence, the unknown Legendre wavelets coefficients are calculated in the form of series whose components are computed by applying a recursive relation. Block pulse functions are used to calculate the Legendre wavelets coefficient vectors of nonlinear terms. The convergence analysis of the LLWM is discussed. The results show that LLWM is very effective and easy to implement. Copyright © 2013 John Wiley & Sons, Ltd.     

Controller design for stabilizing bilinear systems with state-based switching

Some nonlinear systems can be approximated by switching bilinear systems. In this paper, we proposed a method to design state-based stabilizing controller for switching bilinear systems. Based on the similarity between switching bilinear systems and switching linear systems, corresponding switching linear systems are obtained for switching bilinear systems by applying state-based feedback control laws. Instead, we consider asymptotically stabilizing the corresponding switching linear system through solving a number of relaxed LMI conditions. Stabilizing controllers for switching bilinear systems can be derived based on the results of the corresponding switching linear systems. The stability of the controller is proved step by step through the decreasing of the multiple Lyapunov functions along the state trajectory. The effectiveness of the method is demonstrated by both a theoretical example and an example of urban traffic network with traffic signals.     

Numerical solutions of distributed order fractional differential equations in the time domain using the Müntz–Legendre wavelets approach

In this paper, a numerical method is presented to obtain and analyze the behavior of numerical solutions of distributed order fractional differential equations of the general form in the time domain with the Caputo fractional derivative. The suggested method is based on the Müntz–Legendre wavelet approximation. We derive a new operational vector for the Riemann–Liouville fractional integral of the Müntz–Legendre wavelets by using the Laplace transform method. Applying this operational vector and collocation method in our approach, the problem can be reduced to a system of linear and nonlinear algebraic equations. The arising system can be solved by the Newton method. Discussion on the error bound and convergence analysis for the proposed method is presented. Finally, seven test problems are considered to compare our results with other well‐known methods used for solving these problems. The results in the tabulated tables highlighted that the proposed method is an efficient mathematical tool for analyzing distributed order fractional differential equations of the general form.