A method of time‐domain approximation using the transfer function whose attenuation poles are restricted on the imaginary axis

We consider a system described by the one-dimensional linear wave equation in a bounded domain with appropriate boundary conditions. To stabilize the system, we propose a dynamic boundary controller applied at the free end of the system. The transfer function of the proposed controller is a proper rational function of the complex variable and may contain a single pole at the origin and a pair of complex conjugate poles on the imaginary axis, provided that the residues corresponding to these poles are nonnegative; the rest of the transfer function is required to be a strictly positive real function. We then show that depending on the location of the pole on the imaginary axis, the closed-loop system is asymptotically stable. We also consider the case where the output of the controller is corrupted by a disturbance and show that it may be possible to attenuate the effect of the disturbance at the output if we choose the controller transfer function appropriately. We also present some numerical simulation results which support this argument.

Two Pade methods are discussed for constructing low-degree Hurwitz polynomials from a given high-degree Hurwitz polynomial to approximate its argument. Using the Hurwitz polynomial approximants as characteristic polynomials, the numerator dynamics of reduced-order (matrix) transfer-function models are then easily determined by partial Pade approximation of a given large-order model. Stability of such reduced models is always assured. By suitable linear fractional transformations the methods are made applicable to discrete-time systems. The methods are compared in simulation examples for both continuous and discrete-time systems.

A new tuning method for Fractional Order Controllers (FOC) based on Bode’s ideal transfer function as a reference model is proposed in this paper. The proposed fractional order controller parameters are obtained by a simple tuning technique in two steps. In the first step, the open loop reference fractional integrator is designed according to the desired performances, and then approximated by a rational transfer function using Charef’s or Oustaloup’s approximation method. In the second step, a standard pole placement technique is used to align each pole of the plant transfer function with the nearest one of the Rational Function Approximation (RFA). The FOC transfer function is then obtained by gathering the remained poles and zeros of the RFA. The most innovative character of the proposed method is its simplicity and its remarkable performances in terms of robustness towards the variation of the static gain. Simulation of some illustrative examples confirms and validates the proposed method.

Transfer functions are input–output relationships in the Laplace domain. They are multiplicative operators. Multiplying a transfer function by the Laplace transform of an input variable yields the corresponding output variable. A block diagram represents input–output relationships graphically. The operators are contained within rectangular “blocks.” The input and output variables, or “signals,” are the lines which connect blocks. Block diagrams can combine transfer functions representing mathematical operations, such as differentiation, with transfer functions created from system equations, to predict the response of systems created by interconnection. Feedback loops allow calculation of the difference between the commanded value and the response of the system, which is termed the error signal, and is the basis of feedback control. The s-plane is a complex plane, in which the real and imaginary axes are the components of the eigenvalues of a system. The association between regions of the s-plane with the homogeneous response of a system allows the s-plane to be used as a graphical design tool.

Bivariate analysis is essential in several applied fields, such as medicine, engineering, biology, and econometrics. Many parametric and non-parametric procedures can be used for bivariate analysis. Non-parametric procedures require fewer conditions than their parametric counterparts. Therefore, this article proposes an accurate approximation of the p-values and the probability mass functions of some non-parametric bivariate tests, including bivariate sign tests and medians. The proposed approximation, saddlepoint approximation, is compared to the traditional approximation method, asymptotic normal approximation method, and Edgeworth expansion through three real data examples and a simulation study. The numerical results show that the proposed approximation method is more accurate than the asymptotic method and Edgeworth expansion. Furthermore, it is computationally less demanding than the simulation method, which is permutation-based and time-consuming.

System identification is an important methodology used in control theory and constitutes the first step of control design. It is known that many real systems can be better characterized by fractional-order models. However, it is often quite complex and difficult to apply classical control theory methods analytically for fractional-order models. For this reason, integer-order models are generally considered in classical control theory. In this study, an alternative approximation method is proposed for fractional-order models. The proposed method converts a fractional-order transfer function directly into an integer-order transfer function. The proposed method is based on curve fitting that uses a quadratic system model and Equilibrium Optimizer (EO) algorithm. The curve fitting is implemented based on the unit step response signal. The EO algorithm aims to determine the optimal coefficients of integer-order transfer functions by minimizing the error between general parametric quadratic model and objective data. The objective data are unit step response of fractional-order transfer functions and obtained by using the Grünwald-Letnikov (GL) method in the Fractional-Order Modeling and Control (FOMCON) toolbox. Thus, the coefficients of an integer-order transfer function most properly can be determined. Some examples are provided based on different fractional-order transfer functions to evaluate the performance of the proposed method. The proposed method is compared with studies from the literature in terms of time and frequency responses. It is seen that the proposed method exhibits better model approximation performance and provides a lower order model.

This paper introduces building blocks for modular design of elliptic, pseudoelliptic, and self-equalized filters. The first building block is of second order and generates two transmission zeros (TZs), which are either on the real or imaginary axis. Moving the zeros from the real axis (linear phase response) to the imaginary axis (attenuation poles) requires changing the sign of one coupling coefficient. The second building block is a structure of order three, called extended doublet, which allows the generation of two TZs practically anywhere in the complex plane. An important property of this block is its ability to move the two TZs from the real axis to the imaginary axis of the complex s-plane without changing the signs of its coupling coefficients. The third building block is of third order and generates two TZs, which can be moved from the real to the imaginary axis by changing the sign of one coupling coefficient. Simple waveguide structures to implement these blocks are introduced for validation, although this general approach is feasible for all resonator filter types. Higher order filters are designed modularly by cascading an arbitrary number of these building blocks. A novel concept, which allows the independent control of each pair of TZs in higher order filters, is then introduced. It is shown that the new concept allows a filter of order N to generate N TZs without directly coupling the source to the load even when the coupling coefficients are all assumed frequency independent. The same approach can be used to design and reduce the sensitivity of higher order elliptic and pseudoelliptic filters using other building blocks such as doublets or a mixture of building blocks of different orders and properties. Measured results and extensive computer simulation are presented to demonstrate the validity of the concept and the performance of the designed filters.

The conversion of the concept and the teaching of approximation function for analog low pass filter to computer aided numerical class project as an active learning strategy is presented. Governing equations for five approximation functions are presented and discussed. It is recommended that students are tasked to develop computer programs in any convenient computer programming language to make the pole-zero plot, obtain the transfer function, and plot the frequency response of low pass filters. Steps for such programs are suggested. Typical programs were developed and tested. The pole-zero plots and frequency response graphs generated are good illustrative and innovative teaching aids. Linear-phase response over the passband is a characteristic of Bessel approximation function but suffers less amplitude discrimination in the stopband. Chebyshev and Inverse Chebyshev approximation functions have the same order requirement which is usually greater than that of Elliptic approximation function but less than that of Butterworth approximation function. Elliptic approximation function is the best choice although it is more complex than the others. Through this computer aided numerical class project, the authors themselves gained more insight in the subject matter; certainly, students will learn actively.

A new approximation method is presented for directly minimizing a composite nonsmooth function that is locally Lipschitzian. This method approximates only the generalized gradient vector, enabling us to use directly well-developed smooth optimization algorithms for solving composite nonsmooth optimization problems. This generalized gradient vector is approximated on each design variable coordinate by using only the active components of the subgradient vectors; then, its usability is validated numerically by the Pareto optimum concept. In order to show the performance of the proposed method, we solve four academic composite nonsmooth optimization problems and two dynamic response optimization problems with multicriteria. Specifically, the optimization results of the two dynamic response optimization problems are compared with those obtained by three typical multicriteria optimization strategies such as the weighting method, distance method, and min-max method, which introduces an artificial design variable in order to replace the max-value cost function with additional inequality constraints. The comparisons show that the proposed approximation method gives more accurate and efficient results than the other methods.

Another approach for generating reduced order discrete time system using composite method is being proposed. The proposed method is validated by considering a Linear Time Invariant Discrete System (LTIDS) having single input single output(SISO) of higher order being represented in frequency domain. Eigen Spectrum Analysis and Least Squares moment matching methods form the proposed composite method to obtain the reduced numerator and denominator polynomials. The suitability of the proposed method is judged by comparing the step, impulse responses of the original and reduced systems. The performances of the systems are measured in terms of Steady State Error (SSE). It is seen from the numerical examples that the results are quite competitive as compared to other methods thus justifying the applicability of the proposed method.

<p style='text-indent:20px;'>In this paper, an inequality for a transfer function is obtained assuming that its residues at the poles located on the imaginary axis in the right half plane. In addition, the extremal function of the proposed inequality is obtained by performing sharpness analysis. To interpret the results of analyses in terms of control theory, root-locus curves are plotted. According to the results, marginally and asymptotically stable transfer functions can be determined using the obtained extremal function in the proposed theorem.</p>

Digital filters are now key components in many modern digital systems. This paper proposed a novel evolutionary method to design structure-diversified digital filters. On the investigation of the existing evolution-based digital filter design methods, most state-of-the-art works concentrate on the evolution of appropriate transfer functions for digital filters. However, a transfer function is not equivalent to a practical digital filter that has proper structure and can be implemented by a hardware directly. Some researchers proposed a synthesis method to generate the structure of digital filter from a specified transfer function. However, this method needs an existing transfer function as the prior knowledge. Compared with existing works on the evolution of digital filters, the proposed method is novel at the following aspect: the proposed method can directly evolve the structure and parameters of digital filter without the pre-definition of the transfer function. The only prior knowledge we need is the specification of the design target, such as the frequency range of the passband and stop-band. In the experimental study, a significant characteristic is revealed that the proposed method is able to evolve structure-diversified filters with approximate frequency response. The proposed method is a prototype, and it is demonstrated to be a promising way of digital filter design.

Multichannel arma spectral estimation by the modified Yule-Walker method

It is shown that magnitude hyperspherically symmetric transfer functions of multidimensional (MD) causal recursive digital filters must have numerator and denominator polynomials that are separately magnitude hyperspherically symmetric. Further, the exact reference-domain magnitude hyperspherically symmetric denominator polynomial is of infinite order, possessing only one free parameter, and the magnitude hyperspherically symmetric numerator polynomial itself has to be a radial even function. The corresponding MD design problem is shown to be essentially a one-dimensional design problem. Filter transfer functions having good symmetry and moderate degree can be designed by using the presented procedure. >

Strong Stabilization of the Acrobot at the Down–Up Equilibrium Point⋆