Pade Approximants Research Articles

Novel IMC filter design-based PID controller design for systems with one right half plane pole and dead-time

In this article, design of PID controller using a modified internal model control (IMC) filter for right half plane (RHP) pole process with dead time is proposed. To possess H2 optimal behaviour, the derived IMC controller minimises the integral square error (ISE) for step input disturbances by defining the Blaschke product of unstable poles of the specific input and the model. Then it is converted into a single feedback loop controller as either PID or PID with first order filter on the basis of proposed underdamped IMC filter to improve the integral action and thereby providing fast response which is not feasible with critically damped filter. Maclaurin series approximation is used to design PID controller and Pade's approximation is used to design PID with first order lead lag filter. Various first order plus dead time (FOPDT) examples are taken and simulation is executed on diverse unstable processes and compared with some of the developed methods available in the literature. The two proposed controllers provide improved responses with respect to both nominal and perturbed conditions. The robustness studies have also been carried out for uncertainties in the plant dynamics and it is apparent that the proposed tuning method is robust.

Novel IMC filter design-based PID controller design for systems with one right half plane pole and dead-time

In this article, design of PID controller using a modified internal model control (IMC) filter for right half plane (RHP) pole process with dead time is proposed. To possess H2 optimal behaviour, the derived IMC controller minimises the integral square error (ISE) for step input disturbances by defining the Blaschke product of unstable poles of the specific input and the model. Then it is converted into a single feedback loop controller as either PID or PID with first order filter on the basis of proposed underdamped IMC filter to improve the integral action and thereby providing fast response which is not feasible with critically damped filter. Maclaurin series approximation is used to design PID controller and Pade's approximation is used to design PID with first order lead lag filter. Various first order plus dead time (FOPDT) examples are taken and simulation is executed on diverse unstable processes and compared with some of the developed methods available in the literature. The two proposed controllers provide improved responses with respect to both nominal and perturbed conditions. The robustness studies have also been carried out for uncertainties in the plant dynamics and it is apparent that the proposed tuning method is robust.

Numerical Simulation of Modified Kortweg-de Vries Equation by Linearized Implicit Schemes

In this paper, we are going to derive four numerical methods for solving the Modified Kortweg-de Vries (MKdV) equation using fourth Pade approximation for space direction and Crank Nicolson in the time direction. Two nonlinear schemes and two linearized schemes are presented. All resulting schemes will be analyzed for accuracy and stability. The exact solution and the conserved quantities are used to highlight the efficiency and the robustness of the proposed schemes. Interaction of two and three solitons will be also conducted. The numerical results show that the interaction behavior is elastic and the conserved quantities are conserved exactly, and this is a good indication of the reliability of the schemes which we derived. A comparison with some existing is presented as well.

Двовимірні узагальнені моментні зображення та апроксиманти типу Паде функції двох змінних

Generalized instantaneous image were introduced by V.K. Dzyaduk [1] in 1981 and proved to be a convenient tool for constructing and studying the Padé approximants and their generalizations (see [2]). The method of generalized instantaneous images proposed by Dzyadyk made it possible to construct and study rational Padé approximants and their generalizations for many classes of special functions from a single position. As an example, the Padé approximants is constructed for a class of basic hypergeometric series, which includes a q-analogue of the exponential function. In this paper the construction of the Pade approximants for the function of two variables is investigated. A two-dimensional functional sequence is constructed, which has a generalized instantaneous image, and rational approximants are determined, which will be generalizations of one-dimensional Padé approximants. The function of the two variables is entirely related to the basic hypergeometric series.

Adaptive Feedforward Compensator Based on Approximated Causal Transfer Function for CACC with Communication Delay

Abstract In this paper, we propose an adaptive feedforward compensator using parameter estimation to compensate for the communication time delay in cooperative adaptive cruise control (CACC). When CACC uses a feedforward controller to improve tracking performance and satisfy the string stability, it should take into account the communication delay between vehicles. Pade approximation of the time delay can be used for the design of feedforward compensator, but there is a limitation since the approximated system becomes the non-minimum system. To cope with this inherent non-causality problem, we propose an approximated causal transfer function for the feedforward compensator. Then, we apply extended Kalman filter as one of parameter estimation methods with a nonlinear model from the state augmented by a parameter of the approximated causal transfer function. Numerical simulation results show that the proposed system not only mitigates spacing error in time-varying communication delay but also satisfies string stability in a platoon.

Quartic Padé approximation to the exponential function and a class of local analytical difference schemes

This paper investigates the quartic non-diagonal algebraic Hermite-Pade approximation to the exponential function. Explicit formulas and differential equations are obtained for the polynomial coefficients. An exact asymptotic expression is obtained for the error function. As an application, a class of local analytical difference schemes based on quartic Pade approximation for diffusion-convection equation with constant coefficients are proposed. A numerical example is provided to demonstrate the effectiveness of the theoretical results.

On subdiagonal rational Padé approximations and the Brenner-Thomée approximation theorem for operator semigroups

The computational powers of Mathematica are used to prove polynomial identities that are essential to obtain growth estimates for subdiagonal rational Pade approximations of the exponential function and to obtain new estimates of the constants of the Brenner-Thomee Approximation Theorem of Semigroup Theory.

Numerical Solution of Kortweg-de Vries Equation

In this paper, we are going to derive numerical methods for solving the KdV equation using Pade approximation for space direction, trapezoidal and implicit mid-point rule in the time direction. The schemes will be analyzed for accuracy and stability. The exact solution and the conserved quantities will be used to display the efficiency and the robustness of the proposed schemes. Interaction of two and three solitons will be conducted. The numerical results showed, interaction behavior is elastic and the conserved quantities are conserved which is a good indication of the reliability of the schemes under consideration.

Natural resonance frequency identification for sea waves and EEG epileptic patients using Pade approximation and neural network

Pade method is used in the spectral domain estimation to classify both the remote sensing and the biomedical signals. The first example presented from remote sensing is the sea wave classification while the second example depicted from biomedical engineering field is the Epilepsy seizure type classification. Feature extractions of both the Global Navigation Satellite Systems (GNSS) signal and the epilepsy seizure from a human Electroencephalograph (EEG) signal are based on the poles location of the signal.

A Nonlinear $q$-Axis Inductance Modeling of a 12-Slot 10-Pole IPM Using Approximate Analytical Methods

This article introduces a novel nonlinear model for the $q$ -axis inductance of a Fractional Slot Concentrated Winding Interior Permanent Magnet (FSCW-IPM) machine in order to decrease the time needed to obtain a solution. The complicated geometry and nonlinear magnetic behavior of the core make analytical modeling difficult. So numerical methods are usually used to model machines in which magnetic saturation plays an important role. The winding function of a 12-slot 10-pole FSCW-IPM machine is analyzed and a magnetic equivalent circuit is obtained using the Lumped Parameter Method (LPM) according to the MMF distribution analysis. A mathematical function which represents the magnetic behavior of the core material is appropritely selected to reduce the equations, which show the relationship between the flux and MMF of every lumped parameter into the one equivalent equation. After that, perturbation theory and Pade approximants are used to solve the nonlinear equivalent equation. Finally, a new nonlinear analytical model and solution method is introduced for 12-slots 10-pole FSCW-IPM and the simulation time is decreased to under 0.5 s while it takes about 10 mins with FEA because the proposed method does not need any iteration. The proposed method is compared with FEA and experimental results and the comparison shows the proposed method has sufficient accuracy.

Asymptotic of Cauchy Biorthogonal Polynomials

We consider sequences of biorthogonal polynomials with respect to a Cauchy-type convolution kernel and give the weak and ratio asymptotic of the corresponding sequences of biorthogonal polynomials. The construction is intimately related with a mixed type Hermite–Pade approximation problem whose asymptotic properties is also revealed.

On Differentiation with Respect to Parameter of a Hypergeometric Function of a Special Type

We propose an effective construction of Pade approximation for the hypergeometric function of special type and its derivative with respect to parameter and this parameter is included both in numerator and denominator of the general term of the corresponding series. The aforementioned construction has been realized by means of a modification of the method that was applied earlier in simpler situations. Thus obtained approximation is made use of afterwards to establish the lower estimates of the numerical linear forms in the values of the functions under consideration. One could obtain similar estimates by the known in the theory of transcendental numbers Siegel’s method, however the estimates obtained by this method would be of less exactness.

Rational Approximation of the Analytical Model of Nonlaminated Cylindrical Magnetic Actuators for Flux Estimation and Control

Eddy currents can significantly hamper the electromagnetic dynamics, especially those of nonlaminated actuators, and have thus been of great interest in actuator control for many years. Although there are high-fidelity models describing phenomena inside the solid core, such as the magnetic skin effect, these models have never been applied directly within the control circuit. The main reason for this is the lack of an appropriate approximation suitable for time-discrete digital implementation on a microcontroller. This article aims to provide a complete review about possible approaches to obtain a rational approximation of the frequency-dependent effective reluctance and inductance of an electromagnetic actuator. The rational approximation is then used to design a common infinite impulse response filter (IIR-filter). Using the example of a flux-controlled magnetic thrust bearing, we conclude with two novel approximations: a mathematically complex Pade approximation of a sum of continued fractions and a simple Matsuda approximation of an implicit equivalent fractional-order system. On the other hand, the former is the most accurate, and the latter allows wider approximation bandwidths. Depending on possible sampling and switching frequencies as well as the bandwidth of the actuator and computing capacity of the control unit, the most appropriate approximation for almost any use case can be selected.

On the solution of time-fractional coupled system of partial differential equations

In this research, a newly proposed Laplace Pade reduced differential transform method (LPRDTM) has been applied for solving a coupled system of partial differential equations with time-fractional derivatives. Here the fractional derivatives are defined in the Caputo sense. This is a hybrid technique which is the coupling of laplace transform, Pade approximant, and the reduced differential transform method. Two examples are solved using the present method, and the obtained results are compared with the analytical solutions in terms of the tables, which recommend that LPRDTM provides exact solutions with a few iterations. The main benefit of applying this method is that it does not require any assumption, perturbation, and discretization for solving the governing fractional PDEs. Also, the computation time of this method is less compared to other techniques.

The asymptotic approximations to linear weakly singular Volterra integral equations via Laplace transform

In this paper, the asymptotic expansions for the solution about zero and infinity are formulated via Laplace transform for linear Volterra integral equation with weakly singular convolution kernel. The expansions about zero and infinity, as well as their Pade approximations, are used to approximate the solution when the argument is small and large, respectively, and the Pade approximations are more accurate. The methods are also valid to solve some other Volterra type integral equations including linear Volterra integro-differential equations, fractional integro-differential equations, and system of singular Volterra integral equations of the second kind with convolution kernels. Some examples confirm the correctness of the methods and the effectiveness of the asymptotic expansions. They show that numerical methods are only necessary in a small interval in practical computation when uniform high precision evaluations are needed for solving these kinds of Volterra integral equations.

Development of analytical solution for a generalized Ambartsumian equation

Based on the conformable derivative, a generalized model of the Ambartsumian equation is analyzed in this paper. The solution is expressed as a power series of arbitrary powers. In addition, the convergence of the obtained series solution is theoretically proven. Furthermore, it is shown that the current series reduces to the corresponding one in the literature as the conformable derivative tends to one. It is also revealed that the obtained results are of acceptable accuracy. It is found that the residuals tend to zero in specific sub-domains. The diagonal Pade approximants are implemented to extend the domain of converge to include the whole domain.

Multivariate homogeneous two-point Padé approximants and continued fractions

The multivariate homogeneous two-point Pade approximants have been defined and studied recently. In the current work, we consider higher-order approximants and derive error formulas of these approximants using orthogonality conditions. Diverse three-term recurrence relations satisfied by the monic orthogonal polynomials are presented. Various continued fractions provided by these relations and the quotient-difference algorithm applied to a power series (positive or negative exponents) are described in terms of their relationships with the multivariate homogeneous two-point Pade table. Numerical examples are furnished to illustrate our results.

Müntz–Legendre neural network construction for solving delay optimal control problems of fractional order with equality and inequality constraints

In this paper, an artificial intelligence approach using neural networks is described to solve a class of delay optimal control problems of fractional order with equality and inequality constraints. In the proposed method, a functional link neural network based on the Muntz–Legendre polynomial is developed. The problem is first transformed into an equivalent problem with a fractional dynamical system without delay, using a Pade approximation. According to the Pontryagin’s minimum principle for optimal control problems of fractional order and by constructing an error function, the authors then define an unconstrained minimization problem. The authors use trial solutions for the states, Lagrange multipliers and control functions where these trial solutions are constructed by a single-layer Muntz–Legendre neural network model. The authors then exploit an unconstrained optimization scheme for adjusting the network parameters (weights and bias) and to minimize the computed error function. Some numerical examples are given to illustrate the effectiveness of the proposed method.

Designing of internal model control proportional, integral, and derivative controller with second‐order filtering for lag‐ and delay‐dominating processes based on suitable dead time approximation

AbstractIn practice, majority of the industrial chemical processes contain considerable dead time and significant process lag. Depending on the value of dead time to time constant ratio, industrial processes can be classified as lag dominating and delay dominating in nature. Here, an attempt has been made to find out an improved dead time estimation technique from the available methodologies like Taylor series and Pade's approximation relations so that a more accurate process model can be obtained and, consequently, an enhanced model‐based internal model control based proportional, integral, and derivative (IMC‐PID) controller may be designed. In addition, to ensure an improved closed loop performance, a second‐order filter is incorporated in IMC‐PID structure where damping term of the filter is calculated from dead time to time constant ratio of the concerned process. Efficacy of the proposed scheme is verified through simulation study on first‐order plus dead time, second‐order plus dead time, higher order, and nonminimum phase processes in comparison with recently reported model‐based PID settings. In addition to simulation study, a real‐time experimental verification is also made on a level control loop under transient and steady state operating phases.

The research of the influence of digital exchange in the stabilization system of the unmanned aerial vehicle on the dynamic characteristics of the steering actuator

When designing a stabilization system for highly maneuverable unmanned aerial vehicles (UAVs), one of the relevant tasks is to impose requirements on the dynamic characteristics and control methods of the steering actuators, which will ensure the required stability margins of the stabilization system as part of the UAV control system. Currently, there is an increasing preference for microcontroller method of electric actuator control and digital exchange between the control system and the steering actuators. One of the reasons for the reduction of stability margins of the stabilization system is the delay introduced by the digital exchange between the elements of the stabilization system. In the process of solving the problem of transition to digital exchange between the elements of the stabilization system, a research was conducted of the influence of amplitude and phase distortions arising in the path "data transmission interface steering actuator" on the dynamic characteristics of the steering actuator. As an actuator of the stabilization system, the real electric drive used on highly maneuverable UAVs is considered. For this drive, extremely stringent requirements for bandwidth and phase delays are introduced, which complicates the problem of ensuring the stability of the stabilization system, taking into account the delays in the digital exchange. As a result of the research, a frequency model has been proposed that allows to estimate the minimum possible exchange rate in the path "data interface steering actuator", taking into account ensuring the required dynamic characteristics of the actuator. In the proposed model, the data transfer interface is represented as a zero-order hold, the transfer function of which is replaced by Pade approximations of the second order. In the course of the research, a comparison was made of the results obtained on the proposed model with the results of experiments on a real electric actuator and its complete nonlinear time model. The main advantage of the proposed frequency model is the simplicity of obtaining the transfer function of the path "data interface steering actuator". This allows at the initial stage of the research to quickly and accurately determine the minimum possible rate of exchange, which will ensure the fulfillment of the requirements imposed on the drive dynamics.