An algorithm for modeling the sinusoidal input/steady-state response behavior of nonlinear systems over a set of frequencies and amplitudes

An algorithm for constructing a black box model of the sinusoidal input/steady-state response behavior of nonlinear time-invariant systems over a set of frequencies and amplitudes is presented. It is assumed that the steady-state response is periodic of the same fundamental frequency as the excitation, and that the Fourier coefficients are continuous functions of amplitude and square-integrable functions of frequency. The algorithm converges, in a mean-square sense, to an exact representation of the first N harmonics of the steady-state response minus its d.c. component. The model constructed by the algorithm admits a relatively simple physical realization characterized by 2NM+1 linear dynamic elements, and N(2M+1)+1 nonlinear static elements. The underlying mathematical structure of the model is an orthogonal series expansion relative to time whose coefficients are themselves truncated orthogonal expansions relative to frequency. Here M, the number of harmonics used for frequency interpolation, is determined by the algorithm. Of the N(2M+1)+1 memoryless nonlinearities which characterize the model, N of these are specified ahead of time (Tchebysheff polynomials), and 2NM+1 are parameters which mold the representation to the specific system being modeled. Each of these functions of a single variable can be obtained in a pointwise manner directly from steady-state measurements. The algorithm was implemented on a digital computer, and forced versions of the classic equations of van der Pol and Duffing were run as examples. An additional analytic example of a frequency multiplier of prescribed bandwidth was also presented.     

Powers of SPR functions and preservation properties

Basic properties of a new class of strictly positive real (SPR) functions are stated. Four problems are studied. The first deals with SPR preservation of transfer functions, obtained under the composition of polynomials with SPR0 functions. The second deals with Hurwitz stability preservation of the numerator of transfer functions, obtained under the composition of polynomials with SPR0 functions. The third deals with making a Hurwitz closed-loop plant, an SPR0 function by substituting s by SPR0 functions. The four deals with the synthesis of simultaneous SPR feedback plants. For the new class of SPR0 functions, a characterization is presented. For the first and second problems, sufficient conditions are presented using the new class of SPR0 functions. For the third and four problems, two examples are presented, the first being for simultaneous SPR closed-loop systems via constant controllers. The second is for simultaneous stabilization via universal feedback adaptive control.     

Polynomial Root Clustering

A sequence of tests on derived polynomials to be strictly Hurwitz polynomials is shown to be equivalent to a given (typically real) polynomial having all its zeros in an open sector, symmetric with respect to the real axis, in the left half-plane. The number of tests needed is at most 1 + ?(ln k)/(ln 3)?, where k is the integer associated with the central angle π/k of the sector. An extension of this result on the sector as a region of root clustering is given which shows that only a limited number of tests are needed to verify that the roots are clustered in a region composed as the intersection of a set of primative (sector-like) regions. The results reported evolve from application of a collection of mappings on the complex plane defined by a particular collection of Schwarz-Christoffel transformations.     

Event-triggered non-fragile H∞ fault detection for discrete time-delayed nonlinear systems with channel fadings

In this paper, the event-triggered non-fragile H_∞ fault detection filter is designed for a class of discrete-time nonlinear systems subject to time-varying delays and channel fadings. The Lth Rice fading model is utilized to reflect the actual received measurement signals, and its channel coefficients own arbitrary probability density functions on interval [0,1]. The event-based filter is constructed to reduce unnecessary data transmissions in the communication channel, which only updates the measurement signal to the filter when the prespecified “event” is triggered. Multiplicative gain variations are utilized to describe the phenomenon of parameter variations in actual implementation of the filter. Based on Lyapunov stability theory, stochastic analysis technology along with linear matrix inequalities (LMIs) skills, sufficient conditions for the existence of the non-fragile fault detection filter are obtained which make the filtering error system stochastically stable and satisfy the H_∞ constraint. The gains of the filter can be calculated out by solving the feasible solution to a certain LMI. A simulation example is given to show the effectiveness of the proposed method.     

On the construction of transitional filter nomographs

The construction of nomographs for transitional classical filters is described. Gain functions of classical filters are related to filter requirements resulting in a formulation for the general gain nomograph. The transitional filters that are products of approximating polynomials are incorporated into the general gain nomograph resulting in transitional filter nomographs that are sums of the individual nomographs. Nomographs for transitional filters using alternative forms where poles are interpolated are also considered. The resulting nomographs allow for quick optimization of transitional filter frequency response in many cases. Design examples are submitted and discussed. The proposed transitional filter nomographs provide the engineer with increased insight into the selection of classical transitional filters with optimum frequency response.     

Solution of delay fractional optimal control problems using a hybrid of block-pulse functions and orthonormal Taylor polynomials

This paper introduces an efficient direct approach for solving delay fractional optimal control problems. The concepts of the fractional integral and the fractional derivative are considered in the Riemann–Liouville sense and the Caputo sense, respectively. The suggested framework is based on a hybrid of block-pulse functions and orthonormal Taylor polynomials. The convergence of the proposed hybrid functions with respect to the L²-norm is demonstrated. The operational matrix of fractional integration associated with the hybrid functions is constructed by using the Laplace transform method. The problem under consideration is transformed into a mathematical programming one. The method of Lagrange multipliers is then implemented for solving the resulting optimization problem. The performance and computational efficiency of the developed numerical scheme are assessed through various types of delay fractional optimal control problems. Our numerical findings are compared with either exact solutions or the existing results in the literature.     

Networked H∞ filtering for Takagi–Sugeno fuzzy systems under multi-output multi-rate sampling

This paper studies networked H_∞ filtering for Takagi–Sugeno fuzzy systems with multi-output multi-sensor asynchronous sampling. Different output variables in a dynamic system are sampled by multiple sensors with different sampling rates. To estimate the signals of such a system, a continuous multi-rate sampled-data fusion method is proposed to design a novel networked filter. By considering a class of decentralized event-triggered transmission schemes, multi-channel network-induced delays, and the updating modes of the MOMR sampled-data, a networked jumping fuzzy filter is proposed to estimate system signals based on the transmitted multi-rate sampled-data of fuzzy system and the multi-rate sampled states of filter, and the jumping among filter modes is governed by a Markov process which depends on the arrival times of sampled output sub-vectors. To deal with asynchronous membership functions, the networked fuzzy filtering system is modeled as an uncertain fuzzy stochastic system with membership function deviation bounds. Based on stability and H_∞ performance analysis, several membership-function-dependent conditions are presented to co-design the event-triggered transmission schemes and the fuzzy filter such that the filtering error system is robustly mean-square exponentially stable with a prescribed H_∞ attenuation level. Finally, the improvement in estimation performance and comparison with the existing filtering methods are discussed through simulation examples.     

Design of 2-D stable analog and recursive digital filters using properties of the derivative of even or odd Parts of Hurwitz polynomials

In this paper, a method for the design of 2-D analog and recursive digital filters is presented. Starting from a structure in the analog domain, suitable even or odd parts of two-variable Hurwitz polynomials are generated. This enables 2-variable very strictly Hurwitz polynomials (VSHP) to be obtained,² thus avoiding non-essential singularities of the second kind. Thus it will ensure a stable 2-D recursive digital filter obtained by the use of bilinear transformations. Examples are given to illustrate the method.     

Finite-time asynchronous filtering for switched linear systems with an event-triggered mechanism

This paper investigates the event-triggered finite-time H_∞ filtering for a class of continuous-time switched linear systems. Considering that the system may switch within an inter-event interval, the asynchronous problem is taken into account for the system and filter modes. By adopting the average dwell time (ADT) technique and multiple Lyapunov functions, new conditions are obtained to guarantee that the filtering error system is finite-time bounded with a prescribed disturbance attenuation performance. Further, the finite-time H_∞ filter together with event-triggered mechanism is co-designed for the switched linear systems. Finally, a numerical example is provided to demonstrate the effectiveness of the method proposed in this paper.     

Parameter identification of a class of time-varying systems via orthogonal shifted legendre polynomials

A method of using orthogonal shifted Legendre polynomials for identifying the parameters of a process whose behaviour can be modelled by a linear differential equation with time-varying coefficients in the form of finite-order polynomials is presented. It is based on the repeated integration of the differential equation and the representations of $\text{∫}\text{0}\text{t}$s(τ) dτ = Ps(t) and ts(t) = Rs(t), where P and R are constant matrices and s(t) is a shifted Legendre vector whose elements are shifted Legendre polynomials. The differential input-output equation is converted into a set of overdetermined linear algebraic equations for a least squares solution. The results of simulation studies are included to illustrate the applicability of the method.     

Approximation based on orthogonal and almost orthogonal functions

In this paper, we define a class of almost orthogonal rational functions of Legendre type in a new manner. Relations of these functions with classical exponentional functions orthogonal over interval (0, ∞), as well as classical polynomials orthogonal over (0, 1) are explained. Defining relations of these functions can be used for designing almost orthogonal filters. These filters are generators of orthogonal signals and can be successfully applied in finding the best signal approximation in the sense of the mean square error. The filters orthogonal property enables building of physical (in this case electrical) models of dynamical systems (the sources of signals to be approximated) either with less components for the same model accuracy or higher accuracy for the same number of components than the other known models. New filters represent further improvement of previously designed filters, by the same authors, in the sense of simplicity, higher accuracy, lesser approximation time and even a possibility to approximate signals generated by systems with built-in imperfections. Series of experiments were performed to analyze the dependence of approximation accuracy and the number of filters sections.     

Adaptive event-triggered H∞ fuzzy filtering for interval type-2 T–S fuzzy-model-based networked control systems with asynchronously and imperfectly matched membership functions

This paper is concerned with the network-based H_∞ fuzzy filtering for non-linear systems with parameter uncertainties under a novel adaptive discrete event-triggered communication scheme (DETCS). Based on interval type-2 (IT2) Takagi–Sugeno (T–S) fuzzy model, the non-linear systems with parameter uncertainties are represented as a class of IT2 T–S fuzzy systems. In the design process, a novel adaptive DETCS is proposed to reduce the usage of system resources and adapt the variation of plant output, and a novel networked IT2 T–S fuzzy filter is applied to improve the flexibility of filter design. By employing the time-delay systems modeling method, the filtering-error-system is modeled as a class of interval time-varying delayed IT2 T–S fuzzy systems with asynchronously and imperfectly matched membership functions, and further conditionally expressed as a favorable form. Then, some relaxed stability criteria are established to determine that this class of delayed IT2 T–S fuzzy systems is asymptotically stable with a prescribed H_∞ disturbance attenuation performance. Also, the co-design of parameter matrices of adaptive DETCS and filter is implemented. Finally, two numerical examples are provided to demonstrate the effectiveness of the proposed method.     

(1+λu)-Constacyclic codes over Fp[u]/〈u〉

Motivated by the work in [1] of Abualrub and Siap (2009), we investigate (1+λu)-constacyclic codes over F_p[u]/〈u^m〉 of an arbitrary length, where λ is a nonzero element of F_p. We find the generator polynomials of (1+λu)-constacyclic codes over F_p[u]/〈u^m〉, and determine the number of (1+λu)-constacyclic codes over F_p[u]/〈u^m〉 for a given length, as well as the number of codewords in each such code. Some optimal linear codes over F₃ and F₅ are constructed from (1+λu)-constacyclic codes over F_p+uF_p under a Gray map.     

A test for robust Hurwitz stability of convex combinations of complex polynomials

In this paper we present a method for testing the Hurwitz property of a segment of polynomials (1−λ)p₀(s)+λp₁(s), where λ∈[0,1] and p₀(s) and p₁(s) are nth-degree polynomials with complex coefficients. The method consists in constructing a parametric Routh-like array with polynomial entries and generating Sturm sequences for checking the absence of zeros of two real λ-polynomials of degrees 2 and 2n in the interval (0,1). The presented method is easy to implement. Moreover, it accomplishes the test in a finite number of arithmetic operations because it does not invoke any numerical root-finding procedure.     

Eigenvalue localization for the tikhonova model of crystal growth

Matrix A with characteristic polynomial Q(z) is defined positive or negative Hurwitz according to whether Q(z) or Q(-z) is a Hurwitz polynomial. Leading principle sections of the Tikhonova growth matrix have associated characteristic polynomials P_n(-z) which satisfy the recursion

$\text{P}{}_{\mathrm{n+1}}\text{(z)=zP}{}_{\mathrm{n}}\text{(z)+}\text{1}\text{n(n+1)}\text{P}{}_{\mathrm{n-1}}\text{(z),P}{}_0\text{(z)=1,P}{}_1\text{(z)=1+z}$

That the Tikhonova growth matrix is negative Hurwitz is established through applying the Wall-Stieltjes theory of continued fraction expansions to show the P_n(-z) are Hurwitz polynomials. The Kayeya-Enestrom theorem and a procedure for refinement of the Gerschgorin estimate are used to obtain analytical bounds on spectral radii for the Tikhonova model, which provides estimates of maximal growth rates. The theory allows generalization to more complicated growth models.     

A simple design method for the cosine-modulated filter banks using weighted constrained least square technique

This paper presents a simple and efficient design method for cosine-modulated filter banks with prescribed stopband attenuation, passband ripple, and channel overlap. The method casts the design problem as a linear minimization of filter coefficients such that their value at ω=π/2M is 0.707, which results in a simpler, more direct design procedure. The weighted constrained least squares technique is exploited for designing the prototype filter for cosine modulation (CM) filter banks. Several design examples are included to show the increased efficiency and flexibility of the proposed method over the exiting methods. An application of the proposed method is considered in the area of sub-band coding of the ECG and speech signals.     

Some properties of a class of biased regression estimators

For the linear statistical model y = Xb + e, X of full column rank estimates of b of the form (C + X′X)⁺X′y are studied, where C commutes with X′X and Q⁺ is the Moore-Penrose inverse of Q. Such estimators may have smaller mean square error, component by component than does the least squares estimator. It is shown that this class of estimators is equivalent to two apparently different classes considered by other authors. It is also shown that there is no C such that (C + X′X)⁺X′Y = My, in which My has the smallest mean square error, component by component. Two criteria, other than tmse, are suggested for selecting C. Each leads to an estimator independent of the unknown b and σ². Subsequently, comparisons are made between estimators in which the C matrices are functions of a parameter k. Finally, it is shown for the no intercept model that standardizing, using a biased estimate for the transformed parameter vector, and retransforming to the original units yields an estimator with larger tmse than the least squares estimator.     

Central suboptimal H∞ filtering for nonlinear polynomial systems with multiplicative noise

This paper presents the central finite-dimensional H_∞ filter for nonlinear polynomial systems with multiplicative noise, that is suboptimal for a given threshold γ with respect to a modified Bolza-Meyer quadratic criterion including the attenuation control term with the opposite sign. In contrast to the previously obtained results, the paper reduces the original H_∞ filtering problem to the corresponding optimal H₂ filtering problem, using the technique proposed in [1]. The paper presents the central suboptimal H_∞ filter for the general case of nonlinear polynomial systems with multiplicative noise, based on the optimal H₂ filter given in [31]. The central suboptimal H_∞ filter is also derived in a closed finite-dimensional form for third (and less) degree polynomial system states. Numerical simulations are conducted to verify performance of the designed central suboptimal filter for nonlinear polynomial systems against the central suboptimal H_∞ filters available for polynomial systems with state-independent noise and the corresponding linearized system.     

H2/H∞ filtering for networked systems with data transmission time-varying delay,data packet dropout and sequence disorder

This paper investigates an H₂/H_∞ filter designing for networked systems perturbed by multiple noises. The measurement transmission from the sensor to the remote filter is completed via a communication network in simultaneously presenting of data transmission time-varying delays, data packet dropout and data sequence disorder. Since the filter will receive delayed and disordered information, a zero-order-hold (ZOH) or a logical-ZOH (LZOH) is firstly employed for resorting the chaos data sequence. Afterwards, a hybrid H₂/H_∞ filtering scheme is designed for accurately estimating the target output. By Itô formula and a novel free-weight method, the almost surely mean square exponentially stable (ASMSES) condition of the error system is conveniently obtained and the corresponding filter design method is finally presented. By the proposed method, not only the ASMSES with a pre-scheduled H₂/H_∞ performance can be achieved, but also the convergence rate of overall system is pre-regulable. In addition, it has been point out the dynamic filtering performance of LZOH scheme should be better than ZOH ones due to less time-varying delays are introduced and more latest measurement information are employed. Numerical examples are provided to demonstrate the effectiveness of the proposed methods.