Irrational Transfer Functions Research Articles

A novel approach to stability analysis of a wide class of irrational linear systems

This paper presents a general methodology for finding stability equivalence regions, of a wide class of linear time-invariant systems with irrational transfer functions, inside a parametric space. The proposed methodology can be applied to distributed-parameter, time-delay and fractional systems. Unlike rational transfer functions which have only a finite number of poles, irrational transfer functions may generally possess an infinite number of poles, branch points and even essential singularities. Due to this, stability of such systems is more difficult to analyze. Two variants of the new methodology are presented. The first one analyzes stability equivalence along a curve in the parametric space, starting from a given parametric point. The second one finds the maximal stability equivalence region in the parametric space around a given parametric point. Both methodologies are based on iterative application of Rouché’s theorem. They are illustrated on several examples, including heat diffusion equation and generalized time-fractional telegrapher’s equation, which exhibit special functions such as \(\sinh \) and \(\cosh \) of \(\sqrt{s}\), the Laplace variable of order 0.5.

Physics-based irrational transfer function of IPMC sensor output voltage⋆

Ionic Polymer-Metal Composite (IPMC) sensors are a kind of soft and wet polymeric sensors and are expected to be used in soft robotics. Zhu's model is one of the most promising physical models for IPMC sensors and is represented by a set of partial differential equations for cation concentration, water concentration, and potential. This paper first derives the exact solution of Zhu's sensor voltage model using the Laplace transform. Specifically, this paper derives irrational (infinite-dimensional) transfer functions from the deformation curvature to the outputs such as the cation concentration, the water concentration, the electric potential, and the sensor output voltage. The obtained frequency response of the sensor voltage is confirmed to be consistent with that of the finite element method in a previous study. Furthermore, this paper derives the DC gain and the peak gain of the transfer function of the open-circuit sensor voltage.

Frequency response-based decoupling tuning for feedforward compensation ADRC of distributed parameter systems

A practical tuning method is proposed via frequency response or even cut-off frequency for active disturbance rejection control (ADRC) of distributed parameter systems (DPSs), whose models are hard to get but frequency characteristics are easy to measure. Moreover, a novel control structure, feedforward compensation ADRC (FC-ADRC), is presented to decouple the tuning of tracking and disturbance rejection (DR). The frequency response-based tuning method initializes the parameters and then iterates them to the optimal solution by following a simple rule. The basic idea of FC-ADRC is skillful. The closed-loop system is tuned to the desired dynamic via ADRC. Such a control system is a platform for a compensation block to be combined in cascade. ADRC only rejects disturbance while the feedforward compensation purely tracks. Simulation of six accepted irrational transfer functions (ITFs) shows the frequency response-based tuning for FC-ADRC performs better than compared control methods in robustness and performance of tracking and DR. The stability is proved based on the Nyquist criterion for DPSs. Two experiments verify the effectiveness of decoupling tuning and frequency response-based tuning. Therefore, frequency response-based tuning for FC-ADRC is not only a skillful and effective control structure but also a practical and compact method.

An Approximate Transfer Function Model for a Double-Pipe Counter-Flow Heat Exchanger

The transfer functions G(s) for different types of heat exchangers obtained from their partial differential equations usually contain some irrational components which reflect quite well their spatio-temporal dynamic properties. However, such a relatively complex mathematical representation is often not suitable for various practical applications, and some kind of approximation of the original model would be more preferable. In this paper we discuss approximate rational transfer functions G^(s) for a typical thick-walled double-pipe heat exchanger operating in the counter-flow mode. Using the semi-analytical method of lines, we transform the original partial differential equations into a set of ordinary differential equations representing N spatial sections of the exchanger, where each nth section can be described by a simple rational transfer function matrix Gn(s), n=1,2,…,N. Their proper interconnection results in the overall approximation model expressed by a rational transfer function matrix G^(s) of high order. As compared to the previously analyzed approximation model for the double-pipe parallel-flow heat exchanger which took the form of a simple, cascade interconnection of the sections, here we obtain a different connection structure which requires the use of the so-called linear fractional transformation with the Redheffer star product. Based on the resulting rational transfer function matrix G^(s), the frequency and the steady-state responses of the approximate model are compared here with those obtained from the original irrational transfer function model G(s). The presented results show: (a) the advantage of the counter-flow regime over the parallel-flow one; (b) better approximation quality for the transfer function channels with dominating heat conduction effects, as compared to the channels characterized by the transport delay associated with the heat convection.

Fractional‐order controllers for irrational systems

In this contribution, fractional-order controllers of the type PDμ and PIλ are applied to a class of irrational transfer function models that appear in large-scale systems, such as networks of mechanical/electrical elements and distributed parameter systems. More precisely, by considering the fractional-order controller k p + k η s α in the Laplace domain with − 1 ≤ α ≤ 1 , a stability analysis in the parameter-space ( k p , k η , α ) is presented. Furthermore, as a way to measure the controller robustness, the controller's fragility analysis using the parameter-space ( k p , k η , α ) is derived. Finally, several applications that demonstrate the utility of our results are included.

Rational Transfer Function Model for a Double-Pipe Parallel-Flow Heat Exchanger

Transfer functions of typical heat exchangers, resulting from their partial differential equations, usually contain irrational functions which quite accurately describe the spatio-temporal nature of the processes occurring therein. However, such an accurate but complex mathematical representation is often not convenient from the practical point of view, and some approximation of the original model would be more useful. This paper discusses approximate rational transfer functions for a typical thick-walled double-pipe heat exchanger working in the parallel-flow configuration. Using the method of lines with the backward difference scheme, the original symmetric hyperbolic partial differential equations describing the heat transfer phenomena are transformed into a set of ordinary differential equations and expressed in the form of N subsystems representing spatial sections of the exchanger. Each section is described by a rational transfer function matrix and their cascade interconnection results in the overall approximation model expressed by a matrix of rational transfer functions of high order. Based on the rational transfer function representation, the frequency and steady-state responses of the approximate model are evaluated and compared with those resulting from its original irrational transfer function model. The presented results show better approximation quality for the “crossover” input–output channels where the in-domain heat conduction effects prevail as compared to the “straightforward” channels, where the transport delay associated with the heat convection dominates.

Computational Algorithms for Reducing Rational Transfer Functions’ Order

A problem of reducing a linear time-invariant dynamic system is considered as a problem of approximating its initial rational transfer function with a similar function of a lower order. The initial transfer function is also assumed to be rational. The approximation error is defined as the standard integral deviation of the transient characteristics of the initial and reduced transfer function in the time domain. The formulations of two main types of approximation problems are considered: a) the traditional problem of minimizing the approximation error at a given order of the reduced model; b) the proposed problem of minimizing the order of the model at a given tolerance on the approximation error.
 Algorithms for solving approximation problems based on the Gauss-Newton iterative process are developed. At the iteration step, the current deviation of the transient characteristics is linearized with respect to the coefficients of the denominator of the reduced transfer function. Linearized deviations are used to obtain new values of the transfer function coefficients using the least-squares method in a functional space based on Gram-Schmidt orthogonalization. The general form of expressions representing linearized deviations of transient characteristics is obtained.
 To solve the problem of minimizing the order of the transfer function in the framework of the least squares algorithm, the Gram-Schmidt process is also used. The completion criterion of the process is to achieve a given error tolerance. It is shown that the sequence of process steps corresponding to the alternation of coefficients of polynomials of the numerator and denominator of the transfer function provides the minimum order of transfer function.
 The paper presents an extension of the developed algorithms to the case of a vector transfer function with a common denominator. An algorithm is presented with the approximation error defined in the form of a geometric sum of scalar errors. The use of the minimax form for error estimation and the possibility of extending the proposed approach to the problem of reducing the irrational initial transfer function are discussed.
 Experimental code implementing the proposed algorithms is developed, and the results of numerical evaluations of test examples of various types are obtained.

New Design Method for Fractional Order Proportional Integral (FO-PI) Controller of 3x3 Multivariable Systems

This paper presents a new design method of Fractional Order Proportional Integral Controller (FO-PI) for 3x3 multivariable system (three-input-three-output). The Optimal parameters of the FO-PI controllers are tuning by minimizing performance index criterion as objective function. The irrational transfer function of the fractional operator is performed by means of diffusive representation and allows to formulate the optimization problem as a function of fractional order. The simulation results show that the performance of the response obtained by diffusive approach -based FO-PI are better than whose obtained by the classical controllers.

Adaptive unit-vector law with time-varying gain for finite-time parameter estimation in LTI systems

A continuation of previous authors' work on adaptive parameter estimation for linear dynamical systems having irrational transfer function is presented in this work. An original modification of the gradient algorithm, inspired by the variable structure control techniques and additionally featuring a time-varying adaptation gain, is presented and analyzed using Lyapunov techniques. The exposition is illustrated by several numerical examples which illustrate the effectiveness of the proposed algorithm.

Closed-loop time response analysis of irrational fractional-order systems with numerical Laplace transform technique

Irrational transfer function has been widely used in modelling and identification. But time response analysis of systems with irrational transfer functions is hard to be achieved in comparison with the rational ones. One of the main reasons is that irrational transfer function generally has infinite poles or zero. In this paper, the closed-loop time response of fractional-system with irrational transfer function is analyzed based on numerical inverse Laplace transform. The numerical solutions and stability evaluation of irrational fractional-order systems are presented. Several examples of fractional-order systems with irrational transfer functions are shown to verify the effectiveness of the proposed algorithm in both time and frequency domain analysis. The MATLAB codes developed to solve the fractional differential equations using numerical Laplace transform are also provided. The results of this paper can be used on analysis and design of control system described by irrational fractional-order or integer-order transfer function.

Інтегральний метод розв’язування диференціальних рівнянь при моделюванні об’єктів із розподіленими параметрами

The article deals with the method of obtaining one-dimensional integral dynamic models of systems with distributed parameters in the integral form on the basis of applying the differential equations with fractional derivatives obtained by transformations of irrational transfer functions. Such transfer functions occur in the description of the problems of heat conductivity, diffusion, oscillatory processes and other problems, which are described by differential equations with partial derivatives of a parabolic and hyperbolic type. Transfer functions that describe the semi-integral or semi-inertial links in which the Laplace variable is under the root may be the typical examples. The received Cauchy problem for an ordinary differential equation with fractional derivatives is given in the form of the Volterra integral equation of the second kind of convolution type. The applying of this approach is considered in solving various differential equations: the ordinary differential equation of the order 0 < a <1, the ordinary differential equation of the fractional order a > 1, the differential equation of the n-th order with fractional derivatives. Solving of the last equation is based on the compilation of the characteristic equation, which leads to solving of ordinary differential equations of order a. An important example is the researched approach to solving a system of differential equations with fractional derivatives. Due to the transition to equivalent integral equations, the researched problem can be solved by various approximate methods, which are based on quadrature methods with applying the approximation models of polynomial form, in particular, Chebyshev polynomials. The suggested approach makes it possible to construct one-dimensional integral dynamic models of systems of interconnected objects with distributed parameters that can provide high accuracy and stability of solving.

Robust stability conditions for feedback interconnections of distributed-parameter negative imaginary systems

Sufficient and necessary conditions for the stability of positive feedback interconnections of negative imaginary systems are derived via an integral quadratic constraint (IQC) approach. The IQC framework accommodates distributed-parameter systems with irrational transfer function representations, while generalising existing results in the literature and allowing exploitation of flexibility at zero and infinite frequencies to reduce conservatism in the analysis. The main results manifest the important property that the negative imaginariness of systems gives rise to a certain form of IQCs on positive frequencies that are bounded away from zero and infinity. Two additional sets of IQCs on the DC and instantaneous gains of the systems are shown to be sufficient and necessary for closed-loop stability along a homotopy of systems.

Boundary control of a Rijke Tube using irrational transfer functions with experimental validation

This paper is concerned with boundary stabilization of thermoacoustic oscillations in the Rijke tube. This system consists of a vertical tube open in both ends and a heater placed in the lower half of the tube. A speaker placed under the tube is used as actuator while a microphone placed near the top of the tube provides the pressure measurement. To study this problem we consider that the mathematical model takes the form of two interconnected compartments: one for the cold zone and other for the hot zone. The control input is applied on the left boundary condition of the cold zone. From this model we derive an irrational transfer function to design a stabilizing boundary control in the frequency domain. In particular, we derive necessary and sufficient conditions for the input-output stability through the use of Nyquist-type test. Experimental results show the effectiveness and real-life applicability of the method.

Stability analysis of a single-link flexible arm driven by a motor of speed reference type

The stability analyses for motion and vibration control of a single-link flexible arm having a tip payload are investigated using a linear distributed parameter model. The controller design in this paper includes the joint angle proportional–integral–derivative (PID), the bending strain (S) and bending strain integration (SI) feedbacks, and their combinations. The paper presents a new method applying the properties of an irrational passive transfer function and root locus to study the effect of various feedbacks on the infinitely many open loop poles, necessary and sufficient conditions in terms of the dimensionless positive gain constants are reported such that the closed-loop is asymptotically stable. The exact solutions of the infinite-dimensional closed-loop system are obtained using the infinite product formulation. Since the controller design is based on the infinite-dimensional system and is independent of system parameters, the proposed system promises stability robustness to parameter uncertainties and is also free of spillover problems. The validity of the controllers is assessed and compared with the experimental results in the existing literature.

Two-stage adaptive estimation of irrational linear systems

In the present paper, an adaptive parameter estimation algorithm applicable to linear systems with transfer functions of arbitrary structure is proposed. The approach can be applied to a wide class of linear processes, including non-linearly parameterized ones. The proposed method is applicable to fractional-order systems, distributed-parameter and delayed systems, and other classes of systems described by irrational transfer functions. In the first stage of the proposed procedure, values of the transfer function at specific frequencies are pinpointed by means of the Recursive Least Square algorithm with forgetting factor. In the second stage, the unknown parameters are found by numerically inverting complex non-linear relations linking them to the quantities estimated in the first stage. The inversion is performed by means of an iterative, gradient-based scheme. The method is illustrated by several detailedly explained numerical examples.

Identification of models of long lines on the basis of adaptive filters with fractional-order differences

In many applications, objects are distributed and described by equations of mathematical physics. Such objects are often described by models with lumped parameters. Therefore, improving the accuracy of the description of a distributed object on the basis of a model with lumped parameters is an urgent problem. The aim of this paper is to develop a method for adaptive identification of long-line models on the basis of equations with fractional-order differences in the presence of noise. In many applications, the spatial coordinate can be considered constant and measurements can be performed at the beginning and at the end of the line. For such cases, the object transfer function is an irrational function of variable p. It is shown that an object with an irrational transfer function can be approximated by equations with fractional-order differences. A mathematical model of a long line in the form of a finite impulse-response filter with fractional-order differences is proposed. Such a filter combines the advantages of the known infinite impulse-response and finite impulse-response filters: it has a comparatively small number of coefficients and remains stable at any limited values of the coefficients. On the basis of the recurrent least-squares method, an algorithm for adaptive identification of long-line models with fractional-order differences was developed. A computer experiment showed a high accuracy of the proposed model as compared with the known infinite impulse-response and finite impulse-response filters. In addition, the identification algorithm on the basis of the proposed model showed the higher noise immunity. The obtained results can be used in the development of adaptive-filtering algorithms for long lines (communication channels, track circuits, power lines).

A Innovation Identification Approach of Control System by Irrational (Fractional) Transfer

<p>In this paper, an innovative algorithm of identification of control system, described by irrational transfer function with distributed parameter characteristics - with irrational components, is proposed. Algorithm is based on real interpolation method (RIM). Parameters of irrational transfer function can be identified by its experimental transient responses. Each of them can be represented by an analytic expression, table or graph. The proposed method is computationally efficient, simple and practical, as is illustrated by numerical examples. In the furure, the method can be used for tuning the controller and for direct application construction of adaptive controllers, working on the identification principle.</p>

Output Feedback Direct Adaptive Fuzzy Controller Based on Frequency-Domain Methods

This paper describes a design method that guarantees the stability of an adaptive fuzzy control system. The system consists of an unstable plant with an irrational transfer function, a Takagi–Sugeno fuzzy adaptive controller, a nonlinear reference model, and an adaptation mechanism. The gradient-based adaptation mechanism changes the consequents of the controller rules in such a way that the closed-loop system behaves like the reference model. The proposed design procedure utilizes frequency-domain methods in the forms of the Nyquist and circle criteria. It is assumed that the function of the fuzzy controller is a nonlinearity described by a sector condition. This condition means that the controller function lies between two lines passing through the origin. In this method, the function of the fuzzy controller is verified during online adaptation; therefore, it stays in the allowed sector. This method is illustrated by an example of a control system containing an unstable plant with a transport delay. The main contribution of this paper is to use frequency methods combined with adaptive fuzzy control. These methods have previously been used for nonadaptive fuzzy control, but in this paper, they are used for the first time for adaptive fuzzy control.

電磁トランスデューサのインピーダンスのモデル化に関する一考察

Electromagnetic transducers have the characteristics of actuators, sensors, or both. By using these characteristics, there is the vibration control technique that called electromagnetic shunt damping. This technique is passive damping, and therefore, modeling accuracy of electromagnetic transducer is very important, in especially high frequency. However, we observed that conventional model of electromagnetic transducer does not match the experimental results of frequency responses in high frequency. This paper proposes a novel model which has higher accuracy compared with conventional model in high frequency. To improve the modeling accuracy in high frequency, this paper derives the novel model of electromagnetic transducer considering eddy current generated in electromagnetic transducer. The derived model is represented as an irrational transfer function. The validity of the proposed model is demonstrated by performing experiments. The experimental result shows that the accuracy of the proposed model is higher than that of the conventional model.

Travelling waves in a multi-agent system with general graph topology

The paper presents a novel approach for the analysis of a multi-agent system with arbitrary interaction topology and identical agents. The approach is based on an irrational transfer function, the Wave transfer function, that decomposes the interactions between the agents and identifies the travelling waves in the multi-agent system. The approach based on the travelling waves describes the behaviour of the system from the local perspective, which is complementary to the traditional overall approach. The proposed approach allows us, for example, to describe the local effect of the agents that have more than two neighbours.