Modified Realization Technique for Digital Filters Derived from Analog Counterparts

A simple realization scheme for one-dimensional and two-dimensional recursive digital filters derived from analog reference transfer functions is presented. The method is based on proper predistortion of the analog transfer function to obtain a new Hurwitz polynomial. Analog-to-digital transformations, such as the bilinear transformation, are then applied to the resulting predistorted reference transfer function to obtain the discrete version of the system. It is shown that a proper choice of the predistorting function will yield digital realizations which are free of the delay-free loops and in most cases are near minimal. To illustrate the simplicity and efficiency of the technique, examples of 1-D and 2-D cases are worked out. The proposed scheme can readily be extended to include the multi-dimensional case.     

On the Algebra of Multiple Bilinear Transformations

In this paper a closed-form relationship between the original coefficients of a 2-D analog transfer function and the coefficients of the transformed function after application of the double bilinear transformation in an alternative manner is given. The extension to the n-dimensional case, as well as the 1-dimensional case are presented. Examples are given to illustrate the usefulness of the derived relationship.     

Transfer function realization of a class of doubly-terminated two-variable lossless networks and their application in linear-phase 2-dimensional digital filter design

A method is presented for the design of a class of two-dimensional (2-D) stable digital filters satisfying prescribed magnitude and constant group delay specifications. The design method generates a 2-D digital transfer function which is a product of two transfer functions H₁(z₁,z₂) and H₂(z₁,z₂), corresponding to a recursive filter and a nonrecursive filter, respectively, Component H₁(z₁,z₂) ensures a wave-digital realization, that is, the design method guarantees the generation of a corresponding analog function H_1A(s₁,s₂) which is realizable as the transfer function of a doubly-terminated two-variable lossless network. Thus the design technique ensures that not only is a given frequency response achieved, but also the generated transfer function is realizable as a cascade of a wave-digital filter and a nonrecursive digital filter. The class of filters considered here is one in which the doubly-terminated analog network used to realize the wave digital filter is a cascade of s₁- and s₂-variable lossess two-ports with all their transmission zeros at infinity.     

Stable simplification of z-transfer functions by squared magnitude continued-fraction expansion

A full computer-oriented procedure is presented for simplifying the rational z-transfer function of a stable and minimum-phase discrete-time system. The simplification is based on truncating the u-domain (where u = z+z^-1) squared magnitude continued-fraction expansion and using the factorization technique to obtain simplified models. An example is given to illustrate the procedure.     

Approximation based on orthogonal and almost orthogonal functions

In this paper, we define a class of almost orthogonal rational functions of Legendre type in a new manner. Relations of these functions with classical exponentional functions orthogonal over interval (0, ∞), as well as classical polynomials orthogonal over (0, 1) are explained. Defining relations of these functions can be used for designing almost orthogonal filters. These filters are generators of orthogonal signals and can be successfully applied in finding the best signal approximation in the sense of the mean square error. The filters orthogonal property enables building of physical (in this case electrical) models of dynamical systems (the sources of signals to be approximated) either with less components for the same model accuracy or higher accuracy for the same number of components than the other known models. New filters represent further improvement of previously designed filters, by the same authors, in the sense of simplicity, higher accuracy, lesser approximation time and even a possibility to approximate signals generated by systems with built-in imperfections. Series of experiments were performed to analyze the dependence of approximation accuracy and the number of filters sections.     

A method for the integer-order approximation of fractional-order systems

A procedure for approximating fractional-order systems by means of integer-order state-space models is presented. It is based on the rational approximation of fractional-order operators suggested by Oustaloup. First, a matrix differential equation is obtained from the original fractional-order representation. Then, this equation is realized in a state-space form that has a sparse block-companion structure. The dimension of the resulting integer-order model can be reduced using an efficient algorithm for rational L₂ approximation. Two numerical examples are worked out to show the performance of the suggested technique.     

Finite-time dynamic coverage for mobile sensor networks in unknown environments using neural networks

This paper addresses the finite-time dynamic coverage problem for mobile sensor networks in unknown environments. By introducing a condition where dynamic coverage of all points within the sensing range of each sensor exceeds the desired coverage level by a positive constant, a switching control strategy is developed to guarantee the achievement of desired coverage of the whole mission domain in finite time. The environment is modeled by a density function and neural networks are introduced to learn the function. Due to the approximation capability of neural networks, the proposed control scheme can learn the environment without a priori knowledge on the structure of the density function.     

Non-weighted H∞ performance for 2-D FMLSS switched system with maximum and minimum dwell time

This paper investigates the H_∞ performance of two-dimensional (2-D) switched system represented by Fornasini–Marchesini local state-space (FMLSS) model with maximum and minimum dwell time approach. By using the multiple Lyapunov function approach, and designing a set of switching signals subject to maximum and minimum dwell time characteristic, respectively, for all stable subsystems or both stable and unstable subsystems exist, we give the sufficient condition on exponential stability of the given switched system, and propose the sufficient condition which can guarantee that the given switched system is exponentially stable and has a specified H_∞ disturbance attenuation level γ. All the results obtained are on normal noise attenuation index of strictly non-weighted form, which are better than the existing results on weak one of weighted form from the physical point of view. Finally, numerical examples are presented to display the effectiveness of the proposed results.     

Design of Stable 2-D Recursive Filters by Generation of VSHP Using Terminated n-Port Gyrator Networks

An n-port gyrator, terminated by s₁-type capacitors in m₁-ports, by s₂-type capacitors in the next m₂-ports and by resistors in the remaining (n-m₁-m₂)-ports is considered. The determinant of the admittance matrix of the network can be made to yield VSHP (a two-variable Hurwitz polynomial without non-essential singularities of the second kind) under certain conditions involving the sub-determinants of the gyrator matrix. With the gyrator constants as variables and the above conditions as constraints, some 2-D stable low-pass filters have been designed using a suitable optimization procedure. The method is illustrated by examples.     

A new set of orthogonal functions and its application to the analysis of dynamic systems

The present work proposes a complementary pair of orthogonal triangular function (TF) sets derived from the well-known block pulse function (BPF) set. The operational matrices for integration in TF domain have been computed and their relation with the BPF domain integral operational matrix is shown. It has been established with illustration that the TF domain technique is more accurate than the BPF domain technique as far as integration is concerned, and it provides with a piecewise linear solution.As a further study, the newly proposed sets have been applied to the analysis of dynamic systems to prove the fact that it introduces less mean integral squared error (MISE) than the staircase solution obtained from BPF domain analysis, without any extra computational burden. Finally, a detailed study of the representational error has been made to estimate the upper bound of the MISE for the TF approximation of a function f(t) of Lebesgue measure.     

Stable denominators for the simplification of z-Transfer Functions

A well-known discrete stability test is used to derive from the denominator D(z) of a given stable high-order transfer function G(z), the denominator of a low-order approximant of G(z). The proposed method, based on the truncation and inversion of a continued fraction formed with the coefficients of D(z), yields a reduced denominator d(z) of degree, say m, which is always stable. Furthermore, depending on the neglected parts of the continued fraction, d(z) approximates m₁ and m₂ = m−m₁ zeros of D(z), located very near the points z=1 and z=-1, respectively. In the special case m₁=m, d(z) is identical to the polynomial obtained by applying to D(z) the indirect technique, which combines the bilinear transformation with the Routh or the Schwarz approximation method.     

Delay-dependent guaranteed cost control for uncertain 2-D discrete systems with state delay in the FM second model

This paper is concerned with the problem of delay-dependent guaranteed cost control for uncertain two-dimensional (2-D) state delay systems described by the Fornasini and Marchesini (FM) second state-space model. Given a scalar α∈(0,1), a sufficient condition for the existence of delay-dependent guaranteed cost controllers is given in terms of a linear matrix inequality (LMI) based on a summation inequality for 2-D discrete systems. A convex optimization problem is proposed to design a state feedback controller stabilizing the 2-D state delay system as well as achieving the least guaranteed cost for the resulting closed-loop system. Finally, the simulation example of thermal processes is given to illustrate the effectiveness of the proposed result.     

A simplified stability test for 1-D discrete systems

This paper shows that the stability tests for 1-D discrete systems using the transformation p=(z+z⁻¹) and properties of Chebyshev polynomials developed previously can be directly obtained from the z-domain continued fraction expansion based on the functions (z+1) and (z⁻¹+1) on an alternate basis. Furthermore, it is shown that the root distribution of a polynomial with real coefficient can be determined by the same algorithm.     

Replacing points by compacta in neural network approximation

It is shown that cartesian product and pointwise-sum with a fixed compact set preserve various approximation-theoretic properties. Results for pointwise-sum are proved for F-spaces and so hold for any normed linear space, while the other results hold in general metric spaces. Applications are given to approximation of L_p-functions on the d-dimensional cube, 1?p<∞, by linear combinations of half-space characteristic functions; i.e., by Heaviside perceptron networks.     

基于函数值的二元混合插值格式

运用插值与逼近方法解决曲线,曲面造型问题是计算机辅助几何设计最基础的课题。3／1型有理插值具有单调性、连续性、收敛性及保凸性的性质,但它的导数参数一般是未知的。利用3／1型有理插值函数与标准的三次Hermite插值进行类似于张量积的处理,并用插值节点处差商代替参数导数,构造了二元混合有理差值格式,并通过数据实例说明它在计算机辅助设计中的灵活性、有效性。     

Application of the G/G-expansion method for nonlinear diffusion equations with nonlinear source

The generalized diffusion equations, with nonlinear source terms which encompasses the Fisher, Newell-Whitehead, FitzHugh-Nagumo and Allen-Cahn equations as particular forms are solved by the G^′/G-expansion method. The exact solutions are in terms of hyperbolic, trigonometric and rational functions with external parameters. This paper concludes with the stationary topological soliton solution of the FitzHugh-Nagumo equation that is obtained by the Ansatz method.     

Backstepping-based decentralized adaptive neural H∞ tracking control for a class of large-scale nonlinear interconnected systems

The decentralized tracking control methods for large-scale nonlinear systems are investigated in this paper. A backstepping-based robust decentralized adaptive neural H_∞ tracking control method is addressed for a class of large-scale strict feedback nonlinear systems with uncertain disturbances. Under the condition that the nonlinear interconnection functions in subsystems are unknown and mismatched, the decentralized adaptive neural network H_∞ tracking controllers are designed based on backstepping technology. Neural networks are used to approximate the packaged multinomial including the unknown interconnections and nonlinear functions in the subsystems as well as the derivatives of the virtual controls. The effect of external disturbances and approximation errors is attenuated by H_∞ tracking performance. Whether the external disturbances occur or not, the output tracking errors of the close-loop system are guaranteed to be bounded. A practical example is provided to show the effectiveness of the proposed control approach.     

Sampling method for linear system reduction

A new method for obtaining reduced order models for single-input-single-output, continuous-time systems is presented. The proposed algorithm matches the transfer functions of the original and the reduced system at 2M points where M is the order of the reduced model. The location of these points depends on a parameter which can be selected to control the accuracy of the approximation and stability. Numerical examples and comparisons with other methods of model reduction are given.     

Feedback characteristic polynomial controller design of 3-D systems in state-space

For a general state-space model of three-dimensional (3-D) systems the characteristic polynomial (eigenvalue) control problem via state and output feedback is considered. A frequency domain approach is employed which in the scalar input case leads to a set of necessary and sufficient conditions. The multi-input problem is treated by assuming that the state or output feedback gain matrix is expressed as the dyadic product ⊙F = ⊙ ⊙f^T of a column vector ⊙β and a row vector ⊙f^T. This assumption leads to an equivalent scalar input problem β which is directly solved by using the scalar input results. Concerning the dynamic feedback compensator design problem, the important particular case of proportional plus integral plus derivative (PID) control is considered and treated by essentially the same algorithm, which leads to a linear algebraic system in the unknown parameters, along with some constraint equations upon the closed-loop characteristic polynomial sought.