decomposition of generalized frequency response functions for nonlinear systems using symbolic computation a symbolic manipulation procedure which both computes and automatically decomposes the generalized frequency response functions of nonlinear rational polynomial and integrodifferential equation models into a set of closed form nth order transfer functions is presented the symbolic representation exposes the explicit relationship between the model parameters and the nonlinear transfer functions in the frequency domain and leads to important insights into the characterization of nonlinear systems examples for each model type are included to demonstrate the symbolic transfer function approach

block triangular decoupling for linear systems over principal ideal domains this paper studies in the framework of the so called geometric approach the block triangular decoupling problem with state feedback for linear systems defined over a principal ideal domain with identity first various properties of feedback reachability submodules are discussed and then under certain assumptions necessary and sufficient conditions for its solvability are obtained further the pole assignability for decoupled systems is investigated finally a simple example is given to illustrate the results

gain bounds of higher order nonlinear transfer functions bounds on the gain for high order nonlinear transfer functions of a wide class of nonlinear systems are derived it is shown that the gain bounds can be expressed explicitly in terms of the coefficients of discrete time nonlinear models the bounds can be used to check the convergence of the transfer function series and hence to determine the truncation order in practical applications

brief paper some new results for system decoupling and pole assignment problems in a related article we derived a canonical decomposition of the right invertible system c a b and applied this canonical decomposition to study the smith form of the matrix pencil p s a sibc0 and findout the finite zeros and infinite zeros of p s the range of the ranks of p s for email protected c and the controllability of the right invertible system in this paper we will apply this canonical decomposition of the right invertible system c a b to deduce the triangular decouple upon to row permutation provide some new results of the row by row decoupling and associated pole assignment problems

efficient computation of higher order frequency response functions for nonlinear systems with and without a constant term the behaviour of a nonlinear system can be profoundly affected by the presence of a constant or dc term in the system governing equation these changes are reflected in the nonlinear frequency response characteristics of the system which provide a powerful insight into the system s dynamics in this article a new and efficient algorithm is presented for computing the higher order volterra frequency response functions from nonlinear time domain models that may contain a constant term a comparison with previous methods is included to demonstrate the significant gains in computational efficiency that are achieved using the new method the algorithm is applicable to systems modelled by nonlinear differential or difference equations and is easily automated several examples are used to illustrate the method and to highlight the importance of dc terms in nonlinear system analysis

the design of energy transfer filters for energy focus filtering this paper is concerned with extending the energy transfer filters etfs designs to a more general case where the design requirement is to focus signal energy from different input frequency bands into a desired output frequency range etfs are a class of non linear filters recently proposed by the authors to employ non linear effects to transfer signal energy from one frequency band to a different frequency location the new etf designs considered in the present study involve the development of an algorithm for determining output frequencies of non linear systems subject to an input the frequency components of which are located in two separate frequency intervals and a new procedure for the implementation of the filter design the results achieved provide an effective approach to the solution to a more general etf design problem

discrete time subharmonic modelling and analysis traditionally the volterra time and frequency domain analysis tools cannot be applied to severely non linear systems in this paper a new method of building a time domain narx miso model for a class of severely siso non linear systems that exhibit subharmonics is introduced and it is shown how this allows the volterra time and frequency domain analysis to be extended to this class of non linear systems the new approach is based on decomposing the original single input based on a fourier analysis to provide a set of modified input signals which have the same period as the output signal a miso narx model can then be constructed from the decomposed multiple inputs and the single output signal the resulting miso model is shown to meet the basic requirement for the existence of a volterra series representation from which important frequency domain properties can be derived explained and discussed this is done by first introducing the derivation of generalized frequency response functions gfrfs from tim

reconstruction of linear and non linear continuous time system models from input output data using the kernel invariance algorithm abstract a new kernel invariance algorithm kia is introduced to determine both the significant model terms and estimate the unknown parameters in non linear continuous time differential equation models of unknown systems

frequency response functions for nonlinear rational models a recursive algorithm which maps a general class of nonlinear rational model defined as the ratio of two polynomial functions into the frequency domain is derived using the harmonic expansion method the new algorithm provides for the first time a direct analytic map from the time domain rational model parameters to the higher order frequency response functions complex nonlinear time domain behaviours can be analysed and interpreted in the frequency domain and simulated examples are included to illustrate the concepts involved

non linear output frequency response functions for multi input non linear volterra systems the concept of non linear output frequency response functions nofrfs is extended to the non linear systems that can be described by a multi input volterra series model a new algorithm is also developed to determine the output frequency range of non linear systems from the frequency range of the inputs these results allow the concept of nofrfs to be applied to a wide range of engineering systems the phenomenon of the energy transfer in a two degree of freedom non linear system is studied using the new concepts to demonstrate the significance of the new results

solution to morgan s problem a necessary and sufficient condition is presented for the solution of the row by row decoupling problem known as morgan s problem in the general case that is without any restrictive assumption added to the system to the feedback law u fx gy g may be noninvertible this is a structural condition in terms of invariant lists of integers which are easily computable from a given state realization of the system these integers are the infinite zero orders morse s list i sub 4 and the essential orders of the system which only depend on the input output behavior and morse s list i sub 2 of the system which depends on the choice of a particular state realization

analysis of output frequencies of nonlinear systems in this paper an algorithm is derived for the determination of the output frequency ranges of nonlinear systems which extends previous results on the output frequencies of nonlinear systems to a more general situation the new results are significant for the analysis of the output frequency response of a wide class of nonlinear systems

solution of the static state feedback decoupling problem for linear systems with two outputs we give for linear systems with two outputs a necessary and sufficient condition for the solvability of the restricted decoupling problem rdp in terms of some structural invariants a geometric characterization is also given which provides the least dynamical extension needed when rdp is not solvable

improved structure selection for nonlinear models based on term clustering in this paper the concepts of term clusters and cluster coefficients are defined and used in the context of system identification it is argued that if a certain type of term in a nonlinear model is spurious the respective cluster coefficient is small compared with the coefficients of the other clusters represented in the model once the spurious clusters have been detected the corresponding terms can be deleted from the set of candidate terms the consequences of doing this are i a drastic reduction in the size of the set of candidate terms and consequently a substantial gain in computation time is achieved ii the final estimated model is more likely to reproduce the dynamics of the original system and iii the final model is more robust to overparametrization numerical examples are included to illustrate the new procedure

generalized frequency response functions and output response synthesis for mimo non linear systems the generation of generalized frequency response functions for multi imput multi output non linear system models is investigated with particular emphasis on the cross kernel transforms a unified framework for the output response of non linear systems in the frequency domain is also introduced

new bound characteristics of narx model in the frequency domain new results about the bound characteristics of both the generalized frequency response functions gfrfs and the output frequency response for the narx non linear autoregressive model with exogenous input model are established it is shown that the magnitudes of the gfrfs and the system output spectrum can all be bounded by a polynomial function of the magnitude bound of the first order gfrf and the coefficients of the polynomial are functions of the narx model parameters these new bound characteristics of the narx model provide an important insight into the relationship between the model parameters and the magnitudes of the system frequency response functions reveal the effect of the model parameters on the stability of the narx model to a certain extent and provide a useful technique for the magnitude based analysis of nonlinear systems in the frequency domain for example evaluation of the truncation error in a volterra series expression of non linear systems and the highest order needed in the

on the convergence of the volterra series representation of the duffing s oscillators subjected to harmonic excitations abstract this paper is concerned with the investigation of the convergence issue of the volterra series representations of the duffing s oscillator subjected to harmonic inputs a simple criterion is proposed to determine the upper limit of the magnitude of the harmonic inputs a comparison between the new proposed criterion and a criterion suggested by tomlinson et al g r tomlinson g manson g m lee a simple criterion for establishing an upper limit to the harmonic excitation level of the duffing oscillator using the volterra series journal of sound and vibration 190 1996 751 762 was carried out the results show that the new criterion can provide a more accurate prediction about the convergence of the volterra series representation of the duffing s oscillator

mapping non linear integro differential equations into the frequency domain a recursive algorithm is derived to compute the generalized frequency response functions for a large class of non linear integro differential equations applications to duffing s equation and a modified van der pol model are discussed

the parametric characteristic of frequency response functions for nonlinear systems the characteristic of the frequency response functions of nonlinear systems can be revealed and analyzed by analyzing of the parametric characteristics of these functions to achieve these objectives a new operator is defined and several fundamental and important results about the parametric characteristics of the frequency response functions of nonlinear systems are developed these theoretical results provide a significant and novel insight into the frequency domain characteristics of nonlinear systems and circumvent a large amount of complicated integral and symbolic calculations which have previously been required to perform nonlinear system frequency domain analysis several new results for the analysis and synthesis of nonlinear systems are also developed examples are included to illustrate potential applications of the new results

analysis of nonlinear oscillators using volterra series in the frequency domain the volterra series representation is a direct generalisation of the linear convolution integral and has been widely applied in the analysis and design of nonlinear systems both in the time and the frequency domain the volterra series is associated with the so called weakly nonlinear systems but even within the framework of weak nonlinearity there is a convergence limit for the existence of a valid volterra series representation for a given nonlinear differential equation barrett 1965 1 proposed a time domain criterion to prove that the volterra series converges within a given region for a class of nonlinear systems with cubic stiffness nonlinearity in this paper this time domain criterion is extended to the frequency domain to accommodate the analysis of nonlinear oscillators subject to harmonic excitation a common and severe nonlinear phenomenon called jump a behavior associated with the duffing oscillator and the multi valued properties of the response solution is investigated using the new frequency domain criterion of establishing the upper limits of the nonlinear oscillators to predict the onset point of the jump and the volterra time and frequency domain analysis of this phenomenon are carried out based on graphical and numerical techniques

recursive algorithm for computing the frequency response of a class of non linear difference equation models abstract a recursive algorithm is developed which is both computationally compact and enables the nth order frequency response functions of a large class of non linear difference equation models to be found without restriction on the order n

mapping nonlinear integro differential equations to a generalized describing function form an algebraic relationship is derived which enables the generalized amplitude dependent describing function to be expressed in terms of the parameters of a general nonlinear integro differential equation the algorithm is not restricted to a specific input but treats the whole class of harmonically related phase independent sinusoidal inputs and is illustrated by a number of examples

analysis of the effects of time delay in nonlinear systems using generalised frequency response functions abstract a recursive algorithm for computing the generalised frequency response functions gfrfs of nonlinear time delay systems described by nonlinear differential difference equation models is derived using the operator e n and used to analyse the effects of time delay in nonlinear systems the algorithm shows an explicit relationship between the model parameters and the gfrfs such a relationship provides important insight into the behaviour of nonlinear systems the effect of delay on several properties of nonlinear systems such as harmonic generation gain compression expansion and desensitisation is studied by considering the example of a duffing oscillator with retarded damping results of the studies convincingly demonstrate that the system delay has a significant effect on several important properties of nonlinear systems

frequency domain analysis for non linear volterra systems with a general non linear output function for non linear volterra systems which include a non linear state equation and a general non linear output function the system frequency response functions and some related frequency response characteristics are developed and discussed in this study these new results establish the frequency response functions for this general form of non linear systems by extending some existing theory and provide an analytical insight into the relationship between model parameters and frequency response functions and the relationship between model parameters and the magnitude bound of frequency response functions several examples are given to illustrate the new results

mapping from parametric characteristics to generalized frequency response functions of non linear systems based on the parametric characteristic of the nth order generalized frequency response function gfrf for non linear systems described by a non linear differential equation nde model a mapping function from the parametric characteristics to the gfrfs is established by which the nth order gfrf can be directly written into a more straightforward and meaningful form in terms of the first order gfrf i e an n degree polynomial function of the first order gfrf the new expression has no recursive relationship between different order gfrfs and demonstrates some new properties of the gfrfs which can explicitly unveil the linear and non linear factors included in the gfrfs and reveal clearly the relationship between the nth order gfrf and its parametric characteristic as well as the relationship between the nth order gfrf and the first order gfrf the new results provide a novel and useful insight into the frequency domain analysis and design of non linear systems based on the gfrfs several examples are given to illustrate the theoretical results

output frequency response function of nonlinear volterra systems an expression for the output frequency response function ofrf which defines the explicit analytical relationship between the output spectrum and the system parameters is derived for nonlinear systems which can be described by a polynomial form differential equation model an effective algorithm is developed to determine the ofrf directly from system simulation or experimental data simulation studies demonstrate the significance of the ofrf concept and verify the effectiveness of the algorithm which evaluates the ofrf numerically these new results provide an important basis for the analytical study and design of a wide class of nonlinear systems in the frequency domain

estimation of generalized frequency response functions for quadratically and cubically nonlinear systems a new method of estimating the volterra kernels in the frequency domain is introduced based on a non parametric algorithm unlike the traditional non parametric methods using the frequency domain formulations based on discrete fourier transform dft data this new approach uses the time domain measurements directly to estimate the frequency domain response functions

a novel characteristic parameter approach for analysis and design of linear components in nonlinear systems nonlinearity always brings difficulties to the analysis and design of practical systems this study shows that for a wide class of nonlinear systems described by nonlinear differential equations the nonlinear output spectrum can be expressed as a polynomial function with respect to linear model parameters based on the recently developed concepts of parametric characteristic analysis and nonlinear characteristic output spectrum ncos function the coefficients of the new ncos function are independent of the linear parameters of interest and can be directly determined with proposed procedures this reveals an explicit and analytical relationship between the output spectrum of a nonlinear system and its linear components and therefore provides a convenient tool for the analysis and design of nonlinear systems using only linear components or linear feedback control while simultaneously fully considering inherent nonlinear influence this result is a useful complement to the ncos function based analysis and design recently developed in the literature the effectiveness and usefulness of the results are demonstrated through an application to the analysis of an amplifier with a multi tone input for suppressing intermodulation distortion it is shown that the new results reveal a straightforward insight into the parametric optimization of the amplifier for suppression of intermodulation distortion

a volterra series representation for a class of nonlinear infinite dimensional systems with periodic boundary conditions abstract this paper proves the existence of a volterra series representation for the mild solutions of a class of nonlinear infinite dimensional systems more specifically given the evolutionary system operator u t s 0 s t associated with a semilinear evolution equation u t 2 u x 2 f u u 0 u 0 x with periodic boundary conditions it is proved that under suitable conditions the unique mild solution u t u t 0 u 0 t 0 can be expanded by a volterra series a recursive algorithm is given to construct the volterra kernels series terms and a nonlinear heat equation is discussed to illustrate the proposed method

input output parametric models for non linear systems part ii stochastic non linear systems recursive input output models for non linear multivariate discrete time systems are derived and sufficient conditions for their existence are defined the paper is divided into two parts the first part introduces and defines concepts such as nerode realization multistructural forms and results from differential geometry which are then used to derive a recursive input output model for multivariable deterministic non linear systems the second part introduces several examples compares the derived model with other representations and extends the results to create prediction error or innovation input output models for non linear stochastic systems these latter models are the generalization of the multivariable arm ax models for linear systems and are referred to as narmax or non linear autoregressive moving average models with exogenous inputs

nonlinear influence in the frequency domain alternating series the nonlinear influence on system output spectrum is studied for a class of nonlinear systems which have volterra series expansion it is shown that under certain conditions the system output spectrum can be expressed in an alternating series with respect to some model parameters which define system nonlinearities the magnitude of the system output spectrum can therefore be suppressed by exploiting the properties of alternating series sufficient and necessary conditions in which the output spectrum can be cast into an alternating series are studied these results reveal a novel frequency domain insight into the nonlinear influence on a system and provide a new method for the analysis and design of nonlinear systems in the frequency domain examples are given to illustrate the results

input output triangular decoupling and data sensitivity the problem of input output triangular decoupling and data sensitivity is studied for the case of linear time invariant systems using a static feedback law the necessary and sufficient conditions for the problem to have a solution are established the conditions are derived in terms of the variations of the system matrices the general solution of the controller matrices is also derived the necessary and sufficient conditions for triangular decoupling and data sensitivity with simultaneous asymptotic stability are established the structural property of the perturbation disturbance equivalence is also established

comparisons between harmonic balance and nonlinear output frequency response function in nonlinear system analysis by using the duffing oscillator as a case study this paper shows that the harmonic components in the nonlinear system response to a sinusoidal input calculated using the nonlinear output frequency response functions nofrfs are one of the solutions obtained using the harmonic balance method hbm a comparison of the performances of the two methods shows that the hbm can capture the well known jump phenomenon but is restricted by computational limits for some strongly nonlinear systems and can fail to provide accurate predictions for some harmonic components although the nofrfs cannot capture the jump phenomenon the method has few computational restrictions for the nonlinear damping systems the nofrfs can give better predictions for all the harmonic components in the system response than the hbm even when the damping system is strongly nonlinear

a bound for the magnitude characteristics of nonlinear output frequency response functions part 1 analysis and computation a bound for the magnitude frequency domain characteristics associated with the outputs of a wide class of nonlinear systems is derived as a relatively simple function of the generalized frequency response functions and properties of the system inputs it is shown how the practical computation of the new bound can be easily performed for nonlinear systems with finite but arbitrary order nonlinearities and worked examples are included the paper is divided into two parts in part 1 an expression for the output magnitude bound is derived properties of the result are discussed and general procedures for the practical computation of the bound are developed in part 2 the practical computations associated with applying the bound to the polynomial nonlinear autoregressive model with exogenous input are discussed

a numerical solution for the simultaneous disturbance decoupling and row by row decoupling problem the numerical computation of the simultaneous disturbance decoupling and row by row decoupling problem is considered based on a condensed form for the system matrices the solvability of the simultaneous disturbance decoupling and row by row decoupling problem is reduced to the verification of an integer equality and the nonsingularity of a lower dimensional constant matrix the condensed form we use is computed using only orthogonal transformations which can be implemented in a numerically stable way hence it leads directly to an effective numerical method for solving the disturbance decoupling and row by row decoupling problem using existing tools such as lapack and matlab our result complements existing geometric and structural approaches which are of a theoretical nature and are not suitable for numerical computation

automatic computation of polyharmonic balance equations for non linear differential systems an automated algorithm is presented which enables the harmonic balance equations for any polynomial type pure or cross product non linear differential system to be written down directly in terms of the coefficients of the governing equation and the complex amplitudes of a general harmonic waveform the system frequency response in the form of a multi input amplitude dependent describing function is therefore readily computed the method is illustrated by means of an example and the results validated against detailed numeric simulation

the row by row decoupling via state feedback a polynomial approach necessary and sufficient conditions for the row by row and integrator decoupling of a linear system x ax bu y cx via nonregular static state feedback are established a procedure is outlined to calculate one such gain

new results about morgan s problem some aspects of the nonsquare morgan s problem are considered in particular it is shown that the condition given by j descusse et al 1986 to solve this problem is not sufficient it is also shown that the essential orders are not necessarily preserved in the static decoupling problem a necessary and sufficient condition for static decoupling of a system without modifying the essential orders is given for the case k p 1 where p is the number of outputs and k is the rank at infinity of the proper part of the interactor

determination of the analytical parametric relationship for output spectrum of volterra systems based on its parametric characteristics the output frequency response function ofrf of volterra systems can be described as a polynomial function of model parameters however the analytical determination of the ofrf is very computationally intensive especially for higher order ofrf to circumvent this problem a numerical method can be adopted provided that a series of simulation or experimental data for this polynomial function are given in this study it is theoretically shown that the analytical parametric relationship of ofrf up to any order can be determined accurately by using a simple least square method and every specific component of the output spectrum can also be determined explicitly based on the ofrf s parametric characteristics practical techniques to obtain a unique and accurate solution for the least square method are discussed this study provides a fundamental result for the determination of the analytical parametric relationship for this kind of system polynomial functions by using numerical methods

output frequency characteristics of nonlinear systems some interesting properties of output frequencies of volterra type nonlinear systems are particularly investigated these results provide a very novel and useful insight into the super harmonic and inter modulation phenomena in output frequency response of nonlinear systems with consideration of the effects incurred by different nonlinear components in the system the new properties theoretically demonstrate several fundamental output frequency characteristics and unveil clearly the mechanism of the interaction or coupling effects between different harmonic behaviors in system output frequency response incurred by different nonlinear components these results have significance in the analysis and design of nonlinear systems and nonlinear filters in order to achieve a specific output spectrum in a desired frequency band by taking advantage of nonlinearities they can provide an important guidance to modeling identification control and signal processing by using the volterra series theory in practice

frequency domain analysis and identification of block oriented nonlinear systems abstract block oriented nonlinear models including wiener models hammerstein models and wiener hammerstein models etc have been extensively applied in practice for system identification signal processing and control in this study analytical frequency response functions including generalized frequency response functions gfrfs and nonlinear output spectrum of block oriented nonlinear systems are developed which can demonstrate clearly the relationship between frequency response functions and model parameters and also the dependence of frequency response functions on the linear part of the model the nonlinear part of these models can be a more general multivariate polynomial function these fundamental results provide a significant insight into the analysis and design of block oriented nonlinear systems effective algorithms are therefore proposed for the estimation of nonlinear output spectrum and for parametric or nonparametric identification of nonlinear systems compared with some existing frequency domain identification methods the new estimation algorithms do not necessarily require model structure information not need the invertibility of the nonlinearity and not restrict to harmonic inputs simulation examples are given to illustrate these new results

describing functions volterra series and the analysis of non linear systems in the frequency domain a relationship between the higher order frequency response functions of the volterra series model and the well known describing function representation for non linear systems is derived it is shown that a large class of generalized describing functions can be derived from the volterra model and that this approach removes the restriction to specific inputs the concept of a worst case transfer function is introduced and examples are included to illustrate the results

approximate observability of infinite dimensional bilinear systems using a volterra series expansion in this paper the approximate observability of a class of infinite dimensional bilinear systems is investigated the observability conditions are discussed based on a volterra series representation of this class of systems a testable observability criterion is derived for the case where the infinitesimal generator is self adjoint or riesz spectral operator the theoretical results are illustrated by examples

orthogonal parameter estimation algorithm for non linear stochastic systems an orthogonal parameter estimation algorithm is derived which allows each parameter in a linear in the parameters non linear difference equation model to be estimated sequentially and quite independently of the other parameters in the model the algorithm can be applied for any persistently exciting input and provides both unbiased estimates in the presence of correlated noise and an indication of which terms to include in the model several simulated examples are included to demonstrate the effectiveness of the algorithm

frequency domain truncation of volterra series for weakly non linear systems subject to harmonic excitation new results for model order reduction for weakly non linear systems in the frequency domain are derived based on a parametric modelling approach

robust feedback synthesis for nonlinear integro differential equation models using generalized describing functions we propose a technique for synthesizing robust feedback systems for plants described by nonlinear integro differential equations nide with considerable coefficient uncertainties the technique is based on the concept of generalized frequency response functions gfrf and exploits recent results that enable ready derivation of these from the nide description of the plant a powerful synthesis procedure that combines the gfrf and quantitative feedback theory qft technique is given the proposed procedure has several advantages over the existing nonlinear qft methods a nonlinear chemical reactor example is solved using the procedure and is found to yield satisfactory results

interpretation of non linear frequency response functions analytical and graphical methods of interpreting generalised frequency response functions for non linear systems are derived it is shown that non linear phenomena can be classified into intra kernel and inter kernel interference and that worst case responses can be computed the results are illustrated using several discrete and continuous time non linear systems