Measuring volterra kernels III: How to estimate the highest significant order

Efficient methods were previously given for measuring the frequency-domain transfer functions, namely Volterra kernels of weakly non-linear systems. the success of these methods depends crucially on the assumption that the highest significant order of the system under test is known. Practical algorithms for determining the highest significant order of non-linear systems are presented in this paper. Our algorithm can be used not only to validate the measurement results obtained by the methods presented earlier but is also capable of determining whether a Volterra series representation actually exists for the system under test.     

Blind identification of sparse Volterra systems

This paper is concerned with blind identification for single‐input single‐output Volterra systems with finite order and memory with the second‐order and the third‐order statistics. For the full‐sized Volterra system (i.e. all its kernels are nonzero) excited by unknown independently and identically distributed stationary random sequences, it is shown that blind identifiability does not hold in the second‐order moment (SOM) and the third‐order moment (TOM) domain. However, under some sufficient conditions, a class of truncated sparse Volterra systems, where some kernels are restricted to being zero, can be identified blindly and more Volterra parameters can be estimated in TOM than in SOM. Numerical examples illustrate the effectiveness of the proposed methods. Copyright © 2007 John Wiley & Sons, Ltd.     

A Rule-Based Approach for Second Order Model Formulation of Linear Time Invariant Multivariable Systems

A Rule-based approach for formulating a second order MIMO model for a given absolutely stable higher order MIMO system in transfer function matrix form is proposed. The Rule base has been built based on the performance characteristics of general second order systems. An initial second order approximant is constructed for each of the transfer functions in the matrix, based upon a set of guidelines that reflects the characteristics of the corresponding higher order systems. This is taken as input for the Rule based algorithm. Rules are iteratively applied so that a better second order approximant is evolved. Cumulative error index computed from the integral square error is used as the indicator for selecting the best second order model. An algebraic scheme has been proposed for commonizing the denominators of the individual second order models so that a second order MIMO model can be declared in the transfer function matrix form. The analysis is carried out in the s-domain. For discrete systems, a linear transformation of z=s+1 is used to obtain an equivalent continuous model in the s-domain. The proposed methodology is illustrated with two concrete examples taken from the literature.     

Nonlinear system modeling and identification using Volterra‐PARAFAC models

Discrete‐time Volterra models are widely used in various application areas. Their usefulness is mainly because of their ability to approximate to an arbitrary precision any fading memory nonlinear system and to their property of linearity with respect to parameters, the kernels coefficients. The main drawback of these models is their parametric complexity implying the need to estimate a huge number of parameters. Considering Volterra kernels of order higher than two as symmetric tensors, we use a parallel factor (PARAFAC) decomposition of the kernels to derive Volterra‐PARAFAC models that induce a substantial parametric complexity reduction. We show that these models are equivalent to a set of Wiener models in parallel. We also show that Volterra kernel expansions onto orthonormal basis functions (OBF) can be viewed as Tucker models that we shall call Volterra‐OBF‐Tucker models. Finally, we propose three adaptive algorithms for identifying Volterra‐PARAFAC models when input–output signals are complex‐valued: the extended complex Kalman filter, the complex least mean square (CLMS) algorithm and the normalized CLMS algorithm. Some simulation results illustrate the effectiveness of the proposed identification methods. Copyright © 2011 John Wiley & Sons, Ltd.     

A novel nonlinear macromodeling using time piecewise volterra series representation

Accurate and effective nonlinear system‐level macromodeling has played a very important role in solving electromagnetic compatibility problems in modern circuit and system design. The technique based on Volterra series expansions provides the possibility to approximate complex nonlinear circuit models. However, the traditional Volterra series macromodel has limited ability to illustrate strong nonlinearities, resulting from the inconsistency between the complex nature and the required computing cost. In this paper, a novel nonlinear macromodeling scheme using time piecewise Volterra series representations is proposed. In this method, the whole training time series is first divided into several piecewise time series according to the nonlinear characteristics and the model fidelity required. We then process each piecewise time series by the lower order Volterra series, and collect the piecewise Volterra kernels by the Volterra kernel set. Finally, we obtain the output result by combining the Volterra kernel set and the weight function. This approach, which combines the piecewise idea with Volterra series methods, strives to deliver a macromodel that can describe strong and weak nonlinearities simultaneously. Computational results and performance data are presented for the examples of a memristor and selected nonlinear circuits. These examples demonstrate that the proposed method is efficient and precise. © 2014 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.     

Measuring volterra kernels (II)

Accurate measurement of Volterra kernels is an essential step in the modelling of weakly non-linear physical systems via the block box approach, and in model validation. This paper is a sequel to an earlier paper,¹ where practical methods for measuring Volterra kernels were presented along with a quick method for measuring the second-order Volterra kernels. This paper extends that quick method for measuring higher-order frequency-domain Volterra kernels of weakly non-linear systems. We further developed the quick method to measure directly the points of the Volterra kernels which are located along one or more axes in the frequency domain, thus making our method complete and more accurate than that presented in Reference 1. We illustrate our method by actually measuring a weakly non-linear circuit whose Volterra kernels can be accurately calculated. the experimentally measured and the theoretically predicted results agree remarkably well.     

A novel empirical study of the two‐dimensional non‐linear drain‐current behaviour in sub‐micrometre MOS transistors

This paper presents a simple, quasi‐static, non‐linear (saturated mode) NMOS drain‐current model for Volterra‐series analysis. The model is based on a linear transconductance, a linear drain‐source conductance and a purely non‐linear drain‐source current generator. The drain‐current dependency on both drain‐source and gate‐source voltages is included. Model parameters are then extracted from direct numerical differentiation of DC I/V measurements performed on a 160 × 0.25 µm NMOS device. This paper presents the Volterra analysis of this model, including algebraic expressions for intercept points and output spectrum. The model has been verified by comparing measured two‐tone iIP₂ and iIP₃ with the corresponding model predictions over a wide range of bias points. The correspondence between the modelled and measured response is good. Copyright © 2005 John Wiley & Sons, Ltd.     

Z+ fading memory and extensions of input–output maps

An Erratum for this article has been published in the International Journal of Circuit Theory and Applications 30(4) 2002, 179. Much is known about time‐invariant non‐linear systems with inputs and outputs defined on Z₊ that possess approximately‐finite memory. For example, under mild additional conditions, they can be approximated arbitrarily well by the maps of certain interesting simple structures. An important fact that gives meaning to results concerning such systems is that the approximately‐finite‐memory condition is known to be often met. Here we consider the known proposition that if a causal time‐invariant discrete‐time input–output map H has fading memory on a set of bounded functions defined on all of the integers Z, then H can be approximated arbitrarily well by a finite Volterra series operator. We show that in a certain sense, involving the existence of extensions of system maps, this result too has wide applicability. Copyright © 2001 John Wiley & Sons, Ltd.     

Finite‐model adaptive control using an LS‐like algorithm

Adaptive control problem of a class of discrete‐time nonlinear uncertain systems, of which the internal uncertainty can be characterized by a finite set of functions, is formulated and studied by using an least squares (LS)‐like algorithm to design the feedback control law. For the finite‐model adaptive control problem, this algorithm is proposed as an extension of counterpart of traditional LS algorithm. Stability in sense of pth mean for the closed‐loop system is proved under a so‐called linear growth assumption, which is shown to be necessary in general by a counter‐example constructed in this paper. The main results have been also applied to parametric cases, which demonstrate how to bridge the non‐parametric case and parametric case. Copyright © 2006 John Wiley & Sons, Ltd.     

Structure generation and performance comparison of elliptic Gm‐C filters

Starting from a set of matrices describing a general G_m‐C filter topology, a procedure is developed for generating structures of lowpass filters. As the matrices and the filter topologies have a one‐to‐one correspondence, an algebraic method is used to identify filter topologies with desired properties, here, transfer functions with finite jω‐axis transmission zeros, specifically elliptic filters. Sensitivity expressions for these structures are derived and a performance comparison based on a set of chosen criteria is made. For a specified elliptic transfer function, filters with only grounded capacitors and those containing also floating capacitors emerge as alternative realizations, as are filters with a single input and those with distributed inputs. For third‐order functions, a detailed comparison is performed of leapfrog (LF) and inverse follow‐the‐leader‐feedback (IFLF), the most popular special cases, and of topologies that have also floating capacitors (LF_f, IFLF_f), as well as of a novel configuration that uses also distributed inputs (DI_f) and leads to a reduced element count. Design guidelines and restrictions are given, which follow from the derived results with focus on the circuits' sensitivity performance and other properties important for IC implementation. Copyright © 2004 John Wiley & Sons, Ltd.     

A closed‐form equation approach to multi‐port/multi‐zone behavioral modeling of GaN FET Amplifiers

In this paper, a systematic method for the simulation of weakly and mildly nonlinear GaN FET amplifiers is reported. The core of the proposal is a third‐order Volterra‐based behavioral model with multi‐spectral and multi‐node capabilities that is formally derived from a circuit‐level representation. Starting with the equivalent circuit of a typical FET device with thermal power feedback and fading memory, described in terms of its large‐signal functions, closed‐form expressions for the kernels at the gate, drain and thermal nodes are developed up to the third order. The use of these kernels allows the calculation of the responses in the dc, first‐, second‐ and third‐harmonic zones, which are shown to be dependent on the frequency response of the amplifier circuit terminating impedances and thermal filter. The simulation approach has been applied to calculate the nonlinear response of a typical power amplifier circuit, showing the ability of the proposed approach to provide an accurate prediction of multi‐spectral, multi‐node, multi‐bias characteristics, including AM/AM‐AM/PM conversion, spectral regrowth, intermodulation, and temperature rise, under diverse input signal waveforms and bandwidths. These results have been successfully compared with commercial CAD tools based on harmonic balance or envelope simulation. Copyright © 2014 John Wiley & Sons, Ltd.     

Poles–zeros analysis and broadband equivalent circuit for on‐chip spiral inductors

Poles and zeros of the transfer function determine the performance and the key features of the circuit network, such as phase, gain, and bandwidth. In this paper, the contribution of the poles and zeros to the transfer function has been shown. The factors η_p and η_z are proposed to account for the poles contribution to the peak and the zeros contribution to the valley of transfer function, respectively. A novel broadband equivalent circuit combining the physics‐based circuit model and behavioral macro‐model (black‐box) network is proposed for accurately characterizing on‐chip spiral inductors. The physics‐based elements are extracted using the linear dependence of a set of characteristic functions on variables (such as ω²) or other functions in a certain frequency range. The macro‐model network described by rational functions is determined using vector fitting approaches. The proposed modeling method is validated by the on‐chip spiral inductor fabricated with 0.13‐µm SiGe BiCMOS aluminum process. Excellent agreements are obtained between the measured data and calculation for the proposed model up to 40 GHz. Copyright © 2015 John Wiley & Sons, Ltd.     

An approximation for the input conductivity function of the non‐linear resistive grid

The 2‐term approximations for the input conductivity functions, i_in=f_in(v_in), of a grid of similar weakly non‐linear (parabolic) conductors, and the grid's symmetric cuts, measured between two close nodes, are derived, using a semi‐empirical method; the results of a relevant PSpice simulation are presented. The functions f_in(v_in) of the grid's symmetric cuts possess a common analytical feature. Simulation results show that the error in the calculation of the non‐linear terms in the input functions is less than 1 per cent. Copyright © 2001 John Wiley & Sons, Ltd.     

Fault detection filter design for discrete‐time switched linear systems with mode‐dependent average dwell‐time

This paper is concerned with the problem of the fault detection (FD) filter design for discrete‐time switched linear systems with mode‐dependent average dwell‐time. The switching law is mode‐dependent and each subsystem has its own average dwell‐time. The FD filters are designed such that the augmented switched systems are asymptotically stable, and the residual signal generated by the filters achieves a weighted l₂‐gain for some disturbances and guarantees an H _? performance for the fault. By the aid of multiple Lyapunov functions combined with projection lemma, sufficient conditions for the design of the FD filters are formulated by linear matrix inequalities, furthermore, the filters gains are characterized in terms of the solution of a convex optimization problem. Finally, an application to boost convertor is given to illustrate the effectiveness and the applicability of the proposed design method. Copyright © 2013 John Wiley & Sons, Ltd.     

Towards a performance theory of robust adaptive control

We consider standard robust adaptive control designs based on the dead‐zone and projection modifications, and compare their performance w.r.t. a worst case transient cost functional penalizing the ??^∞ norm of the output, control and control derivative. If a bound on the ??^∞ norm of the disturbance is known, it is shown that the dead‐zone controller outperforms the projection controller if the a priori information on the uncertainty level is sufficiently conservative. The second result shows that the projection controller is superior to the dead‐zone controller when the a priori information on the disturbance level is sufficiently conservative. For conceptual clarity the results are presented on a non‐linear scalar system with a single uncertain parameter and generalizations are briefly discussed. Copyright © 2004 John Wiley & Sons, Ltd.     

A DC stabilized log‐domain nth‐order multifunction filter based on the decomposition of nth‐order HP filter function to FLF topology

The design of high‐order log‐domain filters can be easily accomplished by transposing already known linear‐domain G_m‐C filter topologies to their counterparts in the log‐domain through the employment of a set of complementary operators. To achieve the G_m‐C filter topologies, the multiple feedback approach is widely used due to its accrued advantages. In this paper a synthesis approach for the development of an nth‐order multifunction log‐domain filter comprising lowpass (LP), highpass (HP) and bandpass (BP) filter functions is proposed. The approach is based on the decomposition of nth‐order HP filter function to follow‐the‐leader‐feedback (FLF) topology. The design is simple and simultaneously achieves nearly all of the chief advantages. The design offers superior performance factors vis‐à‐vis the ones recently reported. To verify the high‐order behavior of the topology, a 5th‐order multifunction filter was designed and the achieved simulated results verify the theory. Copyright © 2008 John Wiley & Sons, Ltd.     

Adaptive control design for uncertain polynomial nonlinear systems with parametric uncertainties

In this paper, we will develop an adaptive ??_∞ control approach for a class of polynomial nonlinear systems with parametric uncertainties. Motivated by the dissipation theory and the vector projection technique, we propose a nonlinear adaptive ??_∞ controller and its associated parameter adaptation law. The proposed adaptive control strategy is capable of identifying unknown parameter values quickly and minimizing the effect of estimation error. To further improve adaptive controlled performance, the Lyapunov function will be relaxed from quadratic to higher‐order forms and the controller gains are generalized from constant to parameter‐dependent. All of the synthesis conditions are formulated in the framework of polynomial/constant linear matrix inequalities and solvable using available software packages. Copyright © 2010 John Wiley & Sons, Ltd.     

Threshold voltage extraction techniques adaptable from sub‐micron CMOS to large‐area oxide TFT technologies

This paper proposed simple and accurate threshold voltage (V _TH ) extraction techniques, which can be directly adaptable to various semiconductor technologies ranging from deep sub‐micron complementary metal–oxide–semiconductor to large‐area thin‐film transistor devices. These techniques are developed using multiple circuits, namely, a dynamic source follower, an inverter with a diode‐connected load and a current mirror topology, which allow a direct determination of V _TH . As the proposed techniques are experimented with large‐area emerging technologies, which have a stable single type (n‐type) transistor, all the designs employed in this work are confined to only n‐type transistors for a fair comparison. The semiconductor technologies under consideration are standard complementary metal–oxide–semiconductor (65 and 130 nm) and oxide (indium–gallium–zinc–oxide and zinc–tin–oxide) thin‐film transistors. In order to validate the accuracy of the proposed techniques, extracted V _TH from these methods are compared against the value from linear transfer characteristics. The resulting relative error is within 5%, reinforcing proposed techniques suitability to different semiconductor technologies ranging from deep sub‐micron to large‐area transistors. Copyright © 2017 John Wiley & Sons, Ltd.     

Positive‐real property of passive fractional circuits in W‐domain

Fractional circuits have attracted extensive attention of scholars and researchers for their superior performance and potential applications. Fractional circuits constitute a new challenge for the analysis and synthesis methods of traditional circuits theory. Passivity is the fundamental property of traditional circuits (integer order electric circuits). As is known to all, passivity is equivalent to positive realness in traditional linear circuits. However, this equivalence is broken down by introducing fractional elements into electrical networks in s‐domain. To address this issue, on the basis of s‐W transformation, we study the passive criteria of fractional circuits with rational order elements in this paper. Definitions of positive‐real (matrix) function in W‐domain are given, and the equivalence conditions of positive realness are derived. In addition, a conclusion is proposed in which the immittance (matrix) function of passive fractional circuits with rational order elements is positive real in W‐domain. The applications of passive criteria in circuit synthesis are shown.