Realization Of Transfer Function Research Articles

State Observers/Kalman Filters are Generally Unsuitable for Feedback Control While a Far More Rational and Superior Solution is Available

-An observer that estimates system state vector x(t) is a state observer. Although a state observer can generate state feedback control signal Kx(t) where the constant gain K can bese parately designed before its realizing observer, the actual observer feedback system cannot have its loop transfer function equal K(sI – A)-1B, the loop transfer function of the direct state feedback control, for a great majority of plant systems. Because loop transfer function determines the sensitivity function and robust properties of the corresponding feedback system, this implies that the robust properties of the Kx(t)-control are failed to be realized by state observers for a great majority of plant systems.Because robustness against system model uncertainty and terminal disturbance is foremost of feedback control even above performance, state observers are unsuitable for feedback control. Although this problem has initiated a vibrant robust control research in the past 40+ years, the result of that research has been unsatisfactory if parameter K is designed separately and prior to observer design. As a result, control theory remains essentially stagnant and a large amount of works still applied Kalman filters and state observers to feedback control applications. Fortunately, this vital problem has found a fundamentally novel yet decisively satisfactory solution, that is to design parameter K based on the key observer parameters! These new observers are not restricted to be state observers, and can have freely designed and reduced observer order for the first time, and thus can guarantee the full realization of loop transfer function and robust properties of their corresponding Kx(t)-control. Such an observer exits for a great majority of plant systems!In addition, this new design principle is very simple to be learned, and adjusted very easily. Thus, a design methodology that can achieve high performance and robustness for general L-T-I systems is finally developed

On a general structure for adaptation/learning algorithms. — Stability and performance issues

The paper introduces a general structure for parameter adaptation/learning algorithms (PALA). This structure is characterized by the presence of an embedded ARMA (poles-zeros) filter in the PALA. The key question is how to select the coefficients of this filter in order, on the one hand, to guarantee the stability of the parameter estimator for any (positive) value of the adaptation gain/learning rate and for any initial conditions and on the other hand to accelerate the adaptation transient. In order to achieve this, it is shown that on one hand the embedded ARMA filter should be characterized by a positive real transfer function and on the other hand the filter acting on the correcting term (the dynamic adaptation gain) should be characterized by a strictly positive real transfer function. Specific conditions for the design of a second order ARMA embedded filter (ARIMA2 algorithm) are provided.It is shown in the paper that many parameter adaptation/learning algorithms (PALA) used in adaptive control, system identification and neural networks (Nesterov, Conjugate gradients, Momentum back propagation, Averaged gradient, Integral+proportional+derivative, …) are particular cases of the PALA structure introduced in this paper and specific conditions for the stable operation of these algorithms are given.Performance of the ARIMA2 algorithm as well as of the other algorithms reviewed in the paper will be comparatively evaluated by simulations and experimental results obtained on an active noise control system.

Hyperstability of Linear Feed-Forward Time-Invariant Systems Subject to Internal and External Point Delays and Impulsive Nonlinear Time-Varying Feedback Controls

This paper investigates the asymptotic hyperstability of a single-input–single-output closed-loop system whose controlled plant is time-invariant and possesses a strongly strictly positive real transfer function that is subject to internal and external point delays. There are, in general, two controls involved, namely, the internal one that stabilizes the system with linear state feedback independent of the delay sizes and the external one that belongs to an hyperstable class and satisfies a Popov’s-type time integral inequality. Such a class of hyperstable controllers under consideration combines, in general, a regular impulse-free part with an impulsive part.

Current-Mode First-Order Versatile Filter Using Translinear Current Conveyors with Controlled Current Gain

This paper offers a new current-mode first-order versatile filter employing two translinear current conveyors with controlled current gain and one grounded capacitor. The proposed filter offers the following features: realization of first-order transfer functions of low-pass, high-pass, and all-pass current responses from single topology, availability of non-inverting and inverting transfer functions for all current responses, electronic control of current gain for all current responses, no requirement of component-matching conditions for realizing all current responses, low-input impedance and high-output impedance which are required for current-mode circuits, and electronic control of the pole frequency for all current responses. The proposed first-order versatile filter is used to realize a quadrature sinusoidal oscillator to confirm the advantage of the new topology. To confirm the functionality and workability of new circuits, the proposed circuit and its application are simulated by the SPICE program using transistor model process parameters NR100N (NPN) and PR100N (PNP) of bipolar arrays ALA400-CBIC-R from AT&T.

A low‐dimensional realization algorithm for periodic input/output behavioral systems

The state‐space realization of linear systems is of utmost importance in linear systems theory. After the realization problem for the time‐invariant case was solved, particular attention was paid to the case of linear periodic systems. Recently, such systems have regained importance, for instance, in the context of coding theory where periodic convolutional encoders play an important role. The majority of the contributions within this area only concern the realization of transfer functions (or impulse responses), thus excluding the case of input/output linear systems without coprime representations. By the end of the eighties of the last century, Jan C. Willems suggested an approach (nowadays known as the behavioral approach) that considers a wider class of systems and allows to overcome this drawback. According to this approach, the central object in a system is its behavior which consists of all the signals that satisfy the system laws (also called system trajectories). Consequently, the behavior of a system with an input/output representation that is not coprime contains more trajectories than the set of input/output signals defined by the system transfer function. Our work takes this fact into account. In previous research conducted by the authors, some results on the realization of periodic behavioral systems were obtained, which, despite their importance, are conservative with respect to the dimension of the obtained realizations. In this paper, we revisit the behavioral periodic realization problem and give further insight into this question. This allows to derive new results and set up an algorithm to compute low‐dimensional state‐space realizations.

Designing Optimal Linear Detectors: A Bottom-Up Approach

This paper develops a systematic approach to realizing linear detectors with an optimized sensitivity, allowing for the detection of extremely weak signals. First, general constraints are derived on a specific class of input-output transfer functions of a linear detector. Then a physical realization of transfer functions in that class is found using the quantum network synthesis technique, which allows for the inference of the physical setup directly from the input-output transfer function. By exploring a minimal realization which has the minimum number of internal modes, it is shown that the optimal such detectors are internal squeezing schemes. Then, investigating nonminimal realizations, which is motivated by parity-time symmetric systems, a quantum nondemolition measurement is systematically recovered.

Comments on “New Results on the Characterization of Strictly Positive Real Matrix Transfer Functions”

In this note, we address an inconsistency in our recent article (Hakimi-Moghaddam and Ferrante, 2021). In particular, we provide an example showing that Lemma III.1 of that article is not correct. We show how the statement of this lemma can be modified in order to resolve this inconsistency. We also describe how this modification impact the other results of the article. Notwithstanding the fact that Lemma III.1 plays a central role in the proof of the main result of (Hakimi-Moghaddam and Ferrante, 2021), we show that such a result is indeed correct as it may be derived by using the modified (and correct) version of the lemma. An illustrative example is provided to support the results.

Linear time-continuous state-space realization of flame transfer functions by means of a propagation equation.

Low-order network models, commonly used to assess the thermo-acoustic stability of combustors, can be cast in a linear, time-continuous state-space representation. A standard linear eigenvalue problem for the system modes results, which can be solved in a robust and efficient manner. To represent the linear dynamics of any time-invariant flame in the state-space framework, this study presents an approximation of the distributed time-delayed flame response to acoustic velocity perturbations based on a spatially discretized propagation equation (PE). We derive the rational flame transfer function of a first-order-upwind-PE state-space model and discuss its relation to the Tustin approximation of flame transfer functions. For an exemplary discrete finite impulse response of a flame, a third-order-upwind-PE state-space model is shown to match the discrete flame frequency response with comparable accuracy as a rational approximation found by non-linear optimization. The numerical dissipation introduced by discretization of the PE ensures zero gain above the Nyquist frequency of the underlying discrete flame impulse response. Finally, we apply the PE state-space flame model to a generic Rijke tube and show that the predicted thermo-acoustic modes agree well with results obtained from a classical non-linearly optimized rational approximation of the frequency response function of the flame.

Calculation of Band-Pass Filters with Fixed Frequencies of Infinite Attenuation

Introduction. When calculating band-pass filters (BPF) with infinite attenuation frequencies using the frequency conversion method, the parameters of the prototype – an inverse or quasi-elliptic low-pass filter (LPF) – are recalculated into BPF parameters according to conventional formulas. Using the selected low-pass filter cutoff frequency and the Q-factor of the band-pass filter, one can select at their discretion only one infinite attenuation frequency (attenuation pole). In order to suppress a pair of concrete frequencies in the attenuation band, the synthesis of BPF should initially fix the frequencies of maximum attenuation and the central frequency of the filter. The reverse transition toward the frequency response parameters of a low-frequency prototype is carried out using frequency conversion formulas.Aim. To develop of a method for calculating band-pass filters with fixed attenuation poles.Materials and methods. Odd-order filters with an additional capacitor in the transverse branch of the П-link and an inductance in the longitudinal branch of the Т-link were used as low-frequency prototypes of the BPF with attenuation poles. Approximation of the frequency response of a low-frequency prototype (inverse or quasi-elliptical LPF) was performed by methods based on solving systems of nonlinear equations.Results. A realizable transfer function (TF) of an n-th order LPF with attenuation poles was written as the ratio of the product of binomials and a polynomial of power n with real coefficients. Systems of equations were derived to determine amplitude-frequency response coefficients with a given frequency of maximum attenuation interference for both types of filters. Analytical expressions for the TF of the low frequency prototypes of 3th and 5th orders were recorded through the capacitances of the BPF circuits tuned to the central and suppressed frequencies, thus allowing the desired capacitances to be directly calculated. The BPF inductances were determined by formulas expressing the dependences of the BPF central frequency on the circuits parameters, taking into account the relationships given in the article. An example of calculating a 10th order quasi-elliptic BPF was provided.Conclusion. The proposed method can be used to determine the BPF parameters directly, without an intermediate calculation and subsequent transformation of the LPF prototype parameters. The given analytical expressions for the frequency response of the П- and Т-shaped BPFs of the 6th and 10th orders make it possible to verify the performed calculations and to correct the frequency response using inductances, when replacing the calculated capacitance values with their standard values.

Asymptotic Hyperstability and Input–Output Energy Positivity of a Single-Input Single-Output System Which Incorporates a Memoryless Non-Linear Device in the Feed-Forward Loop

This paper visualizes the role of hyperstable controllers in the closed-loop asymptotic stability of a single-input single-output system subject to any nonlinear and eventually time-varying controller within the hyperstable class. The feed-forward controlled loop (or controlled plant) contains a strongly strictly positive real transfer function in parallel with a non-linear and memory-free device. The properties of positivity and boundedness of the input–output energy are examined based on the “ad hoc” use of the Rayleigh energy theorem on the truncated relevant signals for finite time intervals. The cases of minimal and non-minimal state-space realizations of the linear part are characterized from a global asymptotic stability (asymptotic hyperstability) point of view. Some related extended results are obtained for the case when the linear part is both positive real and externally positive and for the case of incorporation of other linear components which are stable but not necessarily positive real.

Passivity-Based Output Feedback Control for Systems With Nonlinear Outputs Via High-Gain Observers

Abstract In this paper, a high-gain observer with nonlinear output is designed. The scaled estimation error system is constructed with a passive function based on the nonlinear output function and a strictly positive real transfer function of boundary-layer system. The ultimate boundedness and exponential stability of the estimation error for the global and regional two cases are demonstrated, as long as high-gain observer's decay rate is fast enough. The extended results under passivity of the output function and strictly positive realness of the transfer function are presented. The performance recovery property of the output feedback using high-gain observer with nonlinear output is validated. Some examples are applied in the simulation to illustrate the proposed results in this paper.

On Discrete-Time Output Negative Imaginary Systems

This letter introduces the notion of linear Discrete-time Output Negative Imaginary (D-ONI) systems. The D-ONI class is defined in the z-domain and it includes the systems having poles on the unit circle. The proposed definition involves a real parameter δ ≥ 0, which indicates the strictness properties. δ > 0 specifies the strict subset, Discrete-time Output Strictly Negative Imaginary (D-OSNI), within the stable D-ONI class. Interestingly, the new D-ONI class captures the existing D-NI systems while restricted to discrete-time LTI systems having a real, rational and proper transfer function. However, the D-OSNI systems are not identical to the existing strictly D-NI (D-SNI) subset. Instead, these two subsets intersect each other. An LMI-based state-space characterisation is derived to check the strict/non-strict D-ONI properties of a given system relying on the value of δ. The paper also establishes the connections between the discrete-time Passive and discrete-time NI systems. Finally, a closed-loop stability result is proposed for a positive feedback interconnection of two D-ONI systems without poles at z = -1 and z = +1.

FPAA Implementations of Fractional-Order Chaotic Systems

In this paper, fractional-order chaotic systems in an analog-based platform are realized using field programmable analog arrays (FPAA) hardware. With the help of this work, we aim to increase the complexity of chaotic systems. Approximated transfer functions in frequency domain are obtained by analyzing different values of fractional-order integrator with the Charef approximation method. In this study, fractional-order numerical calculation of Rssler and Sprott type-H chaotic systems is carried out. MATLAB Simulink model for chaotic systems that satisfy the conditions of chaos in the boundaries of fractional order value is schematically presented. Moreover, CAM designs and analysis that facilitate the realization of fractional-order transfer functions in FPAA platforms are introduced. The analog-based FPAA experimental and numerical outcomes for fractional order chaotic systems are demonstrated. The comparison of the results obtained in the numerical analysis and simulation study with the experimental results is given. This study confirms that the unpredictability of the chaos carrier signals realized by digital-based can be increased with analog-based FPAA hardware and fractional-order structures so as to provide safer transfer of information signals.

Revisiting Minimal Realizations [Lecture Notes

Although realizations of transfer functions are a standard topic in textbooks about linear systems theory, this article focuses on several points that are not widely treated. First, it is shown that, for a given dynamics matrix A, the smallest number of sensors and actuators that can be used to form a minimal realization is determined by properties of the Jordan form of A. This leads to the notions of minimally sensed and minimally actuated systems. Next, the rank of a transfer function is related to properties of the Jordan form of A and the rank of the Rosenbrock system matrix.

Transfer function analysis and implementation of active disturbance rejection control

To support the adoption of active disturbance rejection control (ADRC) in industrial practice, this article aims at improving both understanding and implementation of ADRC using traditional means, in particular via transfer functions and a frequency-domain view. Firstly, to enable an immediate comparability with existing classical control solutions, a realizable transfer function implementation of continous-time linear ADRC is introduced. Secondly, a frequency-domain analysis of ADRC components, performance, parameter sensitivity, and tuning method is performed. Finally, an exact implementation of discrete-time ADRC using transfer functions is introduced for the first time, with special emphasis on practical aspects such as computational efficiency, low parameter footprint, and windup protection.

Kummert's approach to realization on the bidisk

We give a simplified exposition of Kummert's approach to proving that every matrix-valued rational inner function in two variables has a minimal unitary transfer function realization. A slight modification of the approach extends to rational functions which are isometric on the two-torus and we use this to give a largely elementary new proof of the existence of Agler decompositions for every matrix-valued Schur function in two variables. We use a recent result of Dritschel to prove two variable matrix-valued rational Schur functions always have finite-dimensional contractive transfer function realizations. Finally, we prove that two variable matrix-valued polynomial inner functions have transfer function realizations built out of special nilpotent linear combinations.

Attasi nD Systems and Polynomial System Solving

Firstly it will be shown that, using the concept of monomial orderings, the classical 1D theory for linear systems generalizes in a very natural way to (autonomous) Attasi nD-systems, giving the Attasi-Hankel matrix, the Attasi transfer function and Attasi state-space realizations. Secondly we will explain how one can associate an Attasi system to any set of polynomial equations having a finite number of solutions and how Attasi realization theory can be used to (1) describe a commutative matrix solution to the set of polynomial equations (this generalizes the Cayley-Hamilton theorem) and (2) find the (scalar) solutions to the set of polynomial equations by computing the joint eigenvalues and eigenvectors of the commuting matrices. Some remarks will be made about how the (discrete time) Attasi system equations can be solved recursively and how that can be used in large-scale eigensolvers. Using the associated Attasi system to a set of polynomial equations also helps to understand various different approaches to polynomial equation solving and multivariate polynomial and rational function minimization.

Function Space Optimization: A Symbolic Regression Method for Estimating Parameter Transfer Functions for Hydrological Models.

Estimating parameters for distributed hydrological models is a challenging and long studied task. Parameter transfer functions, which define model parameters as functions of geophysical properties of a catchment, might improve the calibration procedure, increase process realism, and can enable prediction in ungauged areas. We present the function space optimization (FSO), a symbolic regression method for estimating parameter transfer functions for distributed hydrological models. FSO is based on the idea of transferring the search for mathematical expressions into a continuous vector space that can be used for optimization. This is accomplished by using a text generating neural network with a variational autoencoder architecture that can learn to compress the information of mathematical functions. To evaluate the performance of FSO, we conducted a case study using a parsimonious hydrological model and synthetic discharge data. The case study consisted of two FSO applications: single‐criteria FSO, where only discharge was used for optimization, and multicriteria FSO, where additional spatiotemporal observations of model states were used for transfer function estimation. The results show that FSO is able to estimate transfer functions correctly or approximate them sufficiently. We observed a reduced fit of the parameter density functions resulting from the inferred transfer functions for less sensitive model parameters. For those it was sufficient to estimate functions resulting in parameter distributions with approximately the same mean parameter values as the real transfer functions. The results of the multicriteria FSO showed that using multiple spatiotemporal observations for optimization increased the quality of estimation considerably.

New Results on the Characterization of Strictly Positive Real Matrix Transfer Functions

In this article, we analyze the high-frequency condition that discriminates strictly positive real systems from weakly strictly positive real ones. Different versions of this condition have been proposed in the literature. We consider the general case of multi-input–multi-output (MIMO) systems and derive new equivalent conditions. A graphic interpretation is also presented. In comparison to the existing literature, the conditions derived here are easier to check, especially for large dimension systems and high-order systems. Several illustrative examples are provided to support the results.

Transfer functions between teleseismic data components

SUMMARYDifferent data components of teleseismic waveforms are related by transfer functions that depend only on receiver-side structure. This is the common basis of a number of teleseismic techniques, including receiver functions and shear wave splitting analysis. Common trace misfits used in these analysis techniques are shown to be equivalent to band-limited comparisons of real and synthetic transfer functions. The data deconvolution used in receiver function analysis leads to reduced structural resolution compared to direct trace-based misfits such as cross-convolution, with direct transfer function modelling of a data trace having the particular advantage of a physically meaningful misfit. Having established that the intertrace transfer function contains all available structural information, the sensitivity of transfer functions to structure is examined for a series of teleseismic scenarios. Transfer functions for the teleseismic P coda show a strong sensitivity to shallow low-velocity structures such as sedimentary basins; the Sp precursors used in S receiver functions are less affected. Examination of transfer functions for shear wave splitting shows that response complexities occur at frequencies too high to be observable in teleseismic studies, and that the dominant control on the response is the splitting intensity.