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This paper studies the problem of finite-time non-fragile extended dissipative control for the periodic piecewise time-varying systems. First, based on the time-varying Lyapunov matrix and the matrix polynomial condition, a sufficient condition of finite-time boundedness for periodic piecewise time-varying subsystem is derived, and the finite-time extended dissipative performance is then analyzed. Furthermore, considering two types of time-varying controller perturbations, the non-fragile controllers are developed, which can guarantee the finite-time boundedness of periodic piecewise time-varying systems and satisfy the extended dissipative performance at the same time. Finally, numerical examples demonstrate the effectiveness of the proposed methods.

In this paper, new conditions of the stability, stabilization and L2-gain performance of periodic piecewise systems are proposed. Both the continuous and discontinuous Lyapunov functions with dwell-time related time-varying Lyapunov matrix polynomial are adopted, and methods guaranteeing the positive and negative definiteness of a matrix polynomial are introduced. Exponential stability conditions are derived based on continuous and discontinuous Lyapunov functions, respectively. A stabilizing controller with time-varying controller gain is designed with continuous Lyapunov function and the weighted L2-gain performance based on the discontinuous Lyapunov matrix polynomial is studied as well. Numerical examples are used to verify the effectiveness of the proposed methods.

Alternating least squares (ALS) and its variations are the most commonly used algorithms for the PARAFAC decomposition of a tensor. However, it is still troubled for one how to accelerate the ALS algorithm with the reduced computational complexity. In this paper, a new acceleration method for the ALS with a matrix polynomial predictive model (MPPM) is proposed. In the MPPM, a matrix-valued function is first approximated by a matrix polynomial. It is shown that the future value of the function can be predicted by an FIR filter with the coefficients determined offline. By viewing each factor matrix of a tensor as a matrix-valued function, a new ALS algorithm, the ALS-MPPM algorithm, is then given. Analyses show that our ALS-MPPM algorithm is of low computational complexity and a close relation with the existing ALS algorithms. Moreover, to further accelerate the convergence of the proposed algorithm, a new technique called the multi-model (MM) prediction is also introduced. While the analytical results are verified by the numerical simulations, it is also shown that our ALS-MPPM outperforms the existing ALS-based algorithms in terms of the rate of convergence.

When applying polynomial methods for the synthesis of multichannel controllers, there is a need for polynomial matrix calculus. However, when using this method, plants with the number of output channels equal to the number of input channels are mainly considered. This is necessary for the convenience of solving a system of linear algebraic equations in a matrix polynomial calculation. A fairly large number of real technical systems have an unequal number of input and output channels. At the same time, the issue of the synthesis of controllers by the polynomial method for multi-channel plants with an unequal number of input and output effects is not worked out in depth enough. In this paper, we consider an example of a linear model of an unstable plant consisting of three standard links with three channels for the input action and two channels for the output action. It is necessary to achieve certain quality indicators of the output vector value, while the control is carried out in the feedback of the system and is summed up with the input action. The plant feature is to limit the task to the second output, since it is essentially a derivative of the first output. The plant was represented as a left - hand polynomial matrix fractional description, and the controller was represented as a right-hand one. For the formation of the characteristic matrix of a closed system, with this variant of the plant and controller decomposition, some structural system transformations are demonstrated. The plant simplicity under consideration is related to the convenience of demonstrating a polynomial synthesis method for such a plants class.

In downlink massive multiple input multiple output (MIMO) systems, matrix polynomial expansion-based (MPEB) precoder suffers either slow convergence performance or complicated computation of polynomial coefficients. To address this issue, in this paper, we proposed a novel Metropolis-Hasting (MH) Sampling-based precoder which can improve the convergence significantly with simple calculation process. The matrix polynomial coefficients of the proposed precoding are designed to boost the precision of matrix inversion approximation. The optimal polynomial coefficients can be derived by an eigenvalues estimation algorithm based on MH Sampling approach. Simulation results exhibit that compared with the benchmark approximate precodings, the proposed MH precoding is able to achieve a significant enhancement performance with lower complexity. Meanwhile, the MH precoding shows low cost and simplicity in implementation.

In this paper, low-complexity channel estimators are proposed for massive multiple-input multiple-output generalized frequency division multiplexing (MIMO-GFDM) systems. In order to combat the effect of non-orthogonality in GFDM, interference-free pilots are used in frequency-domain minimum mean square error (MMSE) channel estimation. Polynomial expansion is used to approximately compute the matrix inverse in conventional MMSE estimation, consequently reducing the cubic computational complexity to square order. The degree of the matrix polynomial can be properly selected to get a required trade-off between complexity and estimation performance. Different weights can be assigned to the terms in the polynomial expansion and be optimized to achieve a minimal mean square error (MSE). Performance limits on the MSE of the proposed estimators are derived. The computational complexity of the proposed MMSE estimators is analyzed and the impacts of the polynomial degree and the pilot subcarrier spacing are also investigated. Numerical results show the accuracy of the proposed channel estimators.

An optimal multivariable state-feedback controller (OSFC) design is presented based on a set of diagonally form block-roots which are the solvent of matrix polynomial, using the new Harris Hawks Optimizer (HHO), right block Vandermonde matrix and block transformations, respectively. The best static matrix gains $K$ of the OSFC is selected based on solving a nonlinear convex optimization problem using HHO. A Bidirectional Inductive Power Transfer system (IPT), is introduced as a good case of study due to its weak-controllability problem. A comparative investigation is performed, with two well famous controllers: optimal LQR and Pole Placement controller, for proving the reliableness of the proposed algorithm.

We consider the problem of computing (approximate) greatest common divisors for matrix polynomials and we present some related facts and applications in system and control theory. The main application is to compute the distance of a controllable multi-input multi-output system to the set of uncontrollable ones; then we describe some related results. Index Terms— Matrix polynomial common factor - Distance to uncontrollability - Behavioral approach

The method of synthesizing linear multichannel regulators for multichannel objects using matrix polynomial expansion, along with classical synthesis methods, such as synthesis in the state space, using full and reduced order observers, and others, is finding increasingly widespread. In the polynomial method of synthesizing multichannel regulators, as a rule, a transition from polynomial representations to matrix numeric ones is used, which in some cases leads to equations with a degenerate Sylvester matrix. In this paper, based on the results obtained by Chen, Kailath and other authors, a detailed formalized synthesis algorithm is presented based on the formalization of algorithms for the synthesis of multichannel systems, analysis of calculations of numerous examples of multichannel regulators. The proposed algorithm allows us to find the general solution of a system of linear equations. As a result, we obtained polynomial regulator matrices containing free parameters that can be set arbitrarily. In other words, when setting free parameters of the controller, it becomes possible to obtain a variety of polynomial matrices of the “numerator” and “denominator” of the regulator, which makes it possible to provide additional properties of the automatic control system.The proposed algorithm synthesizes a two-channel regulator that stabilizes the three-mass system and provides exactly the desired characteristic polynomial matrix of the system.

Is considered bringing of the matrix polynomial decomposition of the transfer function of the object to a coprime form. Violation requirements coprime decomposition leads to the appearance in the same roots “numerator” and “denominator” as in single-channel systems and in multi-channel systems. To check on the relatively coprime compiled of the system of linear homogeneous equations: in other words, a matrix of Sylvester and the desired matrix vector of the “elements” coprime decomposition.Then, using the $[q,r]=qr(R)$ command of the Matlab package, where $R$ is the Sylvester matrix, the linearly dependent columns are checked in the direction from left to right. If the diagonal elements of $r$ are non-zero, then the original polynomial matrices are coprime, and if there are zeros on the diagonal, we look for the right coprime polynomial matrices. Is proposed an algorithm for calculating a coprimedecom-position is presented and one example illustrating algorithm.

Closed form expressions for solutions to MIMO systems are typically derived by means of Laplace transforms. Central to these expressions is the matrix exponential $e^{At}$ and they usually include the eigenvalues and the coefficients of the characteristic equation. In this paper, we derive such expressions by focusing, instead, on the fundamental solution of the underlying differential equation. The advantage of this is threefold. First, it simplifies the derivation of such expressions. Second, it provides a useful insight into the nature of these expressions. Third, since we present an effective procedure for the evaluation of the fundamental solution and its derivatives, it can be used as a basis for procedures to evaluate these expressions both numerically and symbolically. We first consider expressions for $e^{At}$ , when $A$ is in the controller form, where the coefficients of the characteristic equation appear explicitly. We then make use of these to derive general $\mathcal{O}(n^{4})$ matrix polynomial expressions for $e^{At}$ that leads to a recursive $\mathcal{O}(n^{3})$ expression for $e^{At}b$ . Finally, we use these to make expressions for solutions to state equations for the SISO as well as the MIMO cases.

In this paper we have described a new design algorithm for the whole set of latent-structure assignment via the approaches of matrix polynomial placement with output noise rejection for a Quadruple tank process, of course the mathematical dynamic model of the process is obtained by MIMO (Simplified Refined Instrumental Variable) SRIV and/or MIMO (Linear Multi Stage Auto Regressive Moving Average with eXogenous input) LMS-ARMAX identification algorithms and then is handled and used in the control procedure which will provide us instead of placing only a set of desired eigenvalues we are able to assign both latent-vectors and the corresponding latent-values or more generally it is more efficient to assign the latent structure via the approach of Block pole placement.

The modified algorithm for calculation of the six matrix polynomial of the linear models of bipolar and field-effect transistors in the electrical and wave parameter systems is developed and realized on the base of the analytical two-port theory. This is enough for analytical description in these systems of all six matrix forms for any (from three) circuits of transistor connection. Two methods are suggested to reduce the matrix order (in operator and time domains) depending on the frequency application area for the transistor, which simplifies designing of radio electronic and information-communication systems.

In this paper aggregation functions whose expressions are given by polynomial functions are investigated. A detailed study focused on binary polynomial aggregation functions of degree one and two is given not only in general, but also requiring some additional properties like idempotency, commutativity, associativity, one-side neutral (or absorbing) element and so on, leading to some families of binary polynomial aggregation functions. The results concerning polynomials of degree one are generalized to n-ary polynomial aggregation functions whereas, with respect to those of degree two, the commutative case is also generalized obtaining the characterization of all commutative n-ary polynomial aggregation functions of degree two in general and jointly with the idempotent property.

The root locus method is proposed in the paper for calculating intervals of uncertainty for coefficients of the given (initial) polynomial with constant coefficients under perturbations, ensuring its robust stability regardless of whether the given polynomial is hurwitz or not. The method is based on introduction and application of the “extended root locus” notion and can be used for both, synthesis of interval robustly stable polynomials by setting up (adjusting) the unstable ones and analysis of the polynomial behavior under coefficient perturbations. Polynomial adjustment is performed by setting up each one of its coefficients separately and sequentially and determining values of coefficient variation intervals (intervals of uncertainty). The influence of each coefficient variation upon the polynomial roots dynamics (behavior) is considered and analyzed separately, and this influence could be observed in the root locus portraits. Root locus method is thus generalized to the cases when the number of polynomial variable coefficients is arbitrary.

State-of-the-art narrowband noise cancellation techniques utilise the generalised eigenvalue decomposition (GEVD) for multi-channel Wiener filtering, which can be applied to independent frequency bins in order to achieve broadband processing. Here we investigate the extension of the GEVD to broadband, polynomial matrices, akin to strategies that have already been developed by McWhirter et. al on the polynomial matrix eigenvalue decomposition (PEVD). In our approach we extend the Cholesky method for calculating the scalar GEVD to polynomial matrices. In this paper we outline our Cholesky-like approach, which utilises recently developed techniques for polynomial matrix spectral factorisation and polynomial matrix inversion.

Support Vector Machine (SVM) as one of the most popular machine learning methods is playing a significant role in statistical learning theory. Smooth Support Vector Machine (SSVM) is one of new formulation to improve the SVM. In SSVM, smoothing method is used to optimize the unconstrained model. Smoothing function can be used to replace plus function in SVM. In this paper we evaluate eight smoothing functions including quadratic polynomial, fourth order polynomial, piecewise polynomial, spline function, sixth order polynomial, advanced fourth order polynomial function, quadratic Bezier function, third order Bezier function, and fourth order Bezier function. Some of those functions have been studied previously in order to find better accuracy in SVM. In this research, we evaluate and analyze the performance of all those eight smoothing functions based on their infinity-norm values. We compare the error approximation between smoothing function and plus function (as the standard kernel function in SVM) where the best smoothing function has the minimum error of its infinity-norm. Based on theoretical analysis and numerical approximation, our results show that piecewise polynomial function is better than quadratic polynomial function, fourth polynomial function, and third order spline function. While the advanced fourth order polynomial function and sixth order polynomial function have error approximation value almost the same as that of plus function. However, since the piecewise polynomial function has less control parameters than those of the advanced fourth order polynomial function and the sixth order polynomial function, we could not conclude which one is the best. Furthermore, based on values of infinity-norm of all the smoothing functions, we found that the quadratic Bezier and quadratic polynomial show the same error values. Meanwhile the forth order Bezier function shows the smallest error approximation value among the other functions which have been tested. In conclusion, based on our results in this study we found that the forth order Bezier function is the best choice among the eight smoothing function for SSVM.

We study the problem of computing the largest root of a real rooted polynomial p(x) to within error 'z' given only black box access to it, i.e., for any x, the algorithm can query an oracle for the value of p(x), but the algorithm is not allowed access to the coefficients of p(x). A folklore result for this problem is that the largest root of a polynomial can be computed in O(n log (1/z)) polynomial queries using the Newton iteration. We give a simple algorithm that queries the oracle at only O(log n log(1/z)) points, where n is the degree of the polynomial. Our algorithm is based on a novel approach for accelerating the Newton method by using higher derivatives.

In the downlink of Massive Multiple-Input-Multiple-Output (MIMO) systems, the high computational cost of precoding is a major challenge for real-time data transmission. In this paper, we propose a simplified matrix polynomial-aided block diagonalization (SMP-BD) precoding scheme to simplify the conventional block diagonalization (BD) type precoding schemes for users with multiple antennas. By replacing the singular value decomposition (SVD) operation with a matrix inversion, which is then approximated by matrix polynomials with optimized coefficients, SMP-BD is shown to be hardware-efficient, reduce the transmission overhead and obtain high performance. The optimized coefficients can be calculated offline and the convergence of the matrix polynomials is guaranteed. Simulations show that SMP-BD with polynomial order L = 3 performs close to previously reported algorithms based on matrix inversions, while being much simpler.

This paper reviews and provides perspectives on the design of nonlinear model predictive control systems for polynomial systems. General nonlinear systems can often be rewritten exactly as polynomial systems or approximated as polynomial systems using Taylor series. This paper discusses the application of model predictive control (MPC) to these types of systems. After MPC problem for discrete-time polynomial systems is formulated as a polynomial program, moment-based and dual-based sum-of-squares (SOS) algorithms and their relationship are described as two promising methods for solving the polynomial programs to global optimality. Finally, future directions for research are proposed, including real-time, output-feedback, and robust/stochastic polynomial MPC.

General partial differential equations, which can describe any complex functions, may be solved by an adapted method of the similarity analysis that models polynomial data relations of discrete observations. The proposed new differential polynomial networks define and substitute for a selective form of the general partial differential equation using fraction derivative units to model an unknown system or pattern. Convergent series of relative derivative substitution terms, produced in all network layers, describe the partial derivative changes of some combinations of input variables to generalize elementary polynomial data relations. The general differential equation is decomposed into polynomial network backward structure, which defines simple and composite sum derivative terms in respect of previous layers variables. The proposed method enables to form more complex and varied derivative selective series models than standard soft-computing techniques. The sigmoidal function, commonly employed as an activation function in artificial neurons, may improve the abilities of the polynomial networks and substituting derivative terms to approximate complicated periodic multi-variable or time-series functions in a system model.

A method is provided for constructing upper triangular square matrices over the univariate polynomial ring over a finite field, under certain constraints on the eigenvalues of the matrices. In some cases of interest, the degree of the determinant of such matrices is shown to be the smallest possible. The method is then applied to construct generator polynomial matrices of quasi-cyclic codes for correcting phased burst errors. Finally, an interpolation-based list decoding algorithm is presented for these codes, which, for a wide range of code parameters, is shown to outperform existing list decoding schemes.

In this paper, we present a new multichannel spectral factorization algorithm which can be utilized to calculate the approximate spectral factor of any para-Hermitian polynomial matrix. The proposed algorithm is based on an iterative method for polynomial matrix eigenvalue decomposition (PEVD). By using the PEVD algorithm, the multichannel spectral factorization problem is simply broken down to a set of single channel problems which can be solved by means of existing one-dimensional spectral factorization algorithms. In effect, it transforms the multichannel spectral factorization problem into one which is much easier to solve.

This paper deals with control of decomposable systems using polynomial theory and linear matrix inequalities (LMIs). The synthesized controller has fixed structure and is independent from the order of plant and the weighing functions. The design procedure is based on positive polynomial matrices and considers optimization. The controller is computed by formulating and solving the resultant synthesis LMIs. To demonstrate the efficacy of approach, the technique is illustrated with simulation example of multi agent system (MAS), in which number of agents can be chosen arbitrarily.

In this paper, a low complexity polynomial expansion (PE) detector for massive multi-input multi-output (MIMO) uplink transmissions is proposed. In contrast to most massive MIMO systems in the literature, where single-antenna user equipments (UEs) are assumed, multiple-antenna UEs are employed in this paper. Furthermore, each link between a user and the base station forms a jointly correlated Rician fading channel. The PE detector reduces the complexity of the minimum mean square error (MMSE) detector by replacing the matrix inversion with an approximate matrix polynomial. In the design of the low complexity PE detector, the approximations of the moments of the channel Gram matrix are needed. We use operator- valued free probability to derive these approximations. The low complexity PE detector is easy to compute. Simulation results show that it can achieve performance close to the MMSE detector.

This paper is concerned with obtaining the inverse of polynomial functions using semidefinite programming (SDP). Given a polynomial function and a nominal point at which the Jacobian of the function is invertible, the inverse function theorem states that the inverse of the polynomial function exists at a neighborhood of the nominal point. In this work, we show that this inverse function can be found locally using convex optimization. More precisely, we propose infinitely many SDPs, each of which finds the inverse function at a neighborhood of the nominal point. We also design a convex optimization to check the existence of an SDP problem that finds the inverse of the polynomial function at multiple nominal points and a neighborhood around each point. This makes it possible to identify an SDP problem (if any) that finds the inverse function over a large region. As an application, any system of polynomial equations can be solved by means of the proposed SDP problem whenever an approximate solution is available. The method developed in this work is numerically compared with Newton's method and the nuclear-norm technique.

This paper proposes a technique for the design of full and reduced-order linear filters for continuous-time polynomial systems satisfying a prescribed induced ℒ2-gain. The filter variables appear affinely in a polynomial inequality and are computed by solving a semidefinite program (SDP). Bounds on the filter performance, established in terms of the induced ℒ2 norms between the input noise and the filtering error are then assessed via polynomial storage functions. Results from numerical experiments are presented.

This paper studies a control synthesis problem to enlarge the domain of attraction (DA) for non-polynomial systems by using polynomial Lyapunov functions. The basic idea is to formulate an uncertain polynomial system with parameter ranges obtained form the truncated Taylor expansion and the parameterizable remainder of the non-polynomial system. A strategy for searching a polynomial output feedback controller and estimating the lower bound of the largest DA is proposed via an optimization of linear matrix inequalities (LMIs). Furthermore, in order to check the tightness of the lower bound of the largest estimated DA, a necessary and sufficient condition is given for the proposed controller. Lastly, several methods are provided to show how the proposed strategy can be extended to the case of variable Lyapunov functions. The effectiveness of this approach is demonstrated by numerical examples.

We present a novel method to compute all stationary points of optimization problems, of which the objective function and equality constraints are expressed as multivariate polynomials, in the linear algebra setting. It is shown how Stetter-Moller matrix methods can be obtained through a parameterization of the objective function, subsequently manipulated using Macaulay matrices. An algorithm is provided to extend this framework to circumvent the necessity of a Grobner basis. The generalized eigenvalue problem is obtained through a sequence of unitary transformations and rank tests operating directly on the polynomial coefficients (data-driven). The proposed method is illustrated by means of a structured total least squares (STLS) example.

Algorithms of finding polynomials roots have been proposed and they are continuously suggesting in different works [5] which are devoted for solving as theoretical so application problems because of theirs complexity and sufficient perfection (accuracy) in series of certain situations. A method of finding polynomials roots which is reduced to obtaining analytical representation of polynomial roots, and for this purpose the methods of eigenvectors of associated to this polynomial a companion (Frobenius or similar to it) matrix is proposed.

Cloud radio access networks (C-RANs) have attracted considerable attention because of the capability of meeting the exponential increasing traffic demand in the future communication systems. In this paper, we consider the segment training based channel estimation in C-RANs. As the classical minimum mean-square-error estimator has cubic complexity in the dimension of the covariance matrices, due to the inversion operation, we propose a low-complexity channel estimator by means of the \emph{L}-degree matrix polynomial expansion, which can significantly reduce the computational complexity without degrading much performance. The numerical results are presented to verify the proposed channel estimators, and the simulation results show there are significant performance gains from our proposal.

This paper proposes a new efficient multi-antenna expansion method for Vandermonde-subspace frequency division multiplexing (VFDM). VFDM is a technique for interference cancellation in overlay networks that allows a secondary network to operate simultaneously with a primary network on the same frequency band; however, conventional VFDM is not applicable to multi-antenna systems in general, such as the multiple-input multiple-output system. Our proposed method expands on conventional VFDM without losing its advantage of computational efficiency by adopting an expanded Vandermonde matrix for a multi-antenna system. In the new method, the null-space precoder can be obtained directly from the solution matrices of the “matrix coefficient matrix polynomial” equation, and the solution matrices are obtained by Newton's method for a nonlinear system of equations. Moreover, we also show the particular characteristics of solutions of the matrix polynomial, which are important for the efficient implementation of the multi-antenna VFDM.

© 2015 EUCA.This paper provides a numerically tractable approach for long-time average cost control of nonlinear dynamical systems with polynomials of system state on the right-hand side. First, a recently-proposed method of obtaining rigorous bounds of long-time average cost is outlined for the uncontrolled system, where the polynomial constraints are strengthened to be sum-of-squares and formulated as semi-definite programs. As such, it allows to use any general (polynomial) functions to optimize the bound. Then, a polynomial type state feedback controller design scheme is presented to further suppress the long-time average cost. The derivation of state feedback controller is given in terms of the solvability conditions of state-dependent bilinear matrix inequalities. Finally, the mitigation of oscillatory vortex shedding behind a cylinder is addressed to illustrate the validity of the proposed approach.

We formulate a method for designing dynamic average consensus estimators with optimal worst-case asymptotic convergence rate over a large set of undirected graphs. The estimators achieve average consensus for constant inputs and are robust to both initialization errors and changes in network topology. The structure of a general class of polynomial linear protocols is characterized and used to find global optimal parameters using polynomial matrix inequalities (PMIs). For the case of the PI estimator, these conditions are converted into convex linear matrix inequalities (LMIs) and solved efficiently.

The analysis of a homogeneous system of algebraic-differential equations is related to the algebraic structure of the polynomial matrix that describes the system. A decomposable pair containing this algebraic structure exists for the regular case. Extending the results to the nonregular case, we construct a decomposable pair containing information from the right minimal indices of the polynomial matrix, in addition to the information coming from its finite and infinite elementary divisors. Once again a 'decomposable linearization' is introduced.

In this paper, impulsive mode problem of polynomial matrix systems (PMSs) is investigated by analyzing the infinite zero structure of polynomial matrix. Some useful definitions and lemmas associated with linearization are introduced first, then a sufficient condition for PMSs having no impulsive modes is derived by using strong linearization approach. Based on dual polynomial matrix, the structure of infinite zeros for matrix pencil is analyzed and extended to general polynomial matrices. An algorithm is presented to compute the infinite zeros structure for arbitrary polynomial matrices. Finally, the numerical examples are given to illustrate the main results.

This paper investigates the filtering problem of polynomial fuzzy-model-based (PFMB) nonlinear systems. A novel polynomial fuzzy filter is designed to guarantee that the filter error system is asymptotically stable and satisfies a desired performance. Polynomial approximated membership functions obtained by Taylor series are employed for filtering analysis. Furthermore, sufficient conditions represented as sum of squares (SOS), which can be solved by SOSTOOLS, are obtained based on a polynomial Lyapunov function. A numerical example is provided to demonstrate the effectiveness of the proposed method.

It is difficult for least square method (LS) to deal with the ill-conditioned matrix of nonlinear polynomial model. In the case of the higher order of system, the matrix inversion is very complicated. A new approach based on LS is present which is combined with mirror-injection algorithm in order to obtain polynomial parameters identification of nonlinear system model. The columns of coefficient matrix of the inconsistent equations of nonlinear polynomial model are orthogonalized. The novel method avoids the high-order matrix inversion and ill-conditioned matrix problem. The precision and velocity of identification are improved, while the computation load is low simultaneously. Performance analysis is carried out using MATLAB simulation. The results prove the effectiveness of the proposed approach.

The solution of polynomial equation systems is a problem frequently encountered by researchers in solving equations in specific derivatives, algebraic geometry and in optimization tasks. There exist various realizations of the Grobner basis building method, but their serious disadvantage is the high complexity of calculations. Therefore, the algorithms currently employed for symbol-aided solutions are effective only for lower order polynomial equations systems. The article offers a method based on tables individually corresponding to a polynomial which makes it possible to forgo the solution of the problem of dividing the matrix into parts in distributing the calculations on the systems enabling the parallel execution of the program. The tables corresponding to individual polynomials of the initial system or the basis can be distributed among the processors without decomposition.

Methods of modern algebra have appeared to be extremely useful in system theory and digital signal processing. New classes of linear filters and fast linear convolution algorithms have been developed based on the algebraic approach. In this paper, we apply this approach to the description of a class of polynomial (Volterra) filters of signals and fields defined over finite groups. Since the polynomial filters can be considered as a direct extension of linear filters, it is reasonable to apply the algebraic methods to a nonlinear case too. After a brief introduction to the abstract signal theory, a matrix representation of generalized polynomial filter is presented. Finally, we discuss the construction of fast nonlinear convolutions algorithms on the basis of their linear counterparts.

The attractive practical stability of the linear continuous control systems with time delay was investigated in this paper. The Lyapunov's direct method was used to obtain the time-dependent conditions for the system stability. For this purpose, two matrix equations were presented: the matrix polynomial equation and the continuous algebraic Lyapunov matrix equation. The solution of the time delay differential equation was combined with the well-known Lyapunov approach to obtain the conditions for the attractive practical stability. Using this approach, we derived a sufficient novel delay-dependent condition for the practical stability of these control systems. As a result, the fulfillment of the stability conditions guarantees the attractivity of the system.

For a locally stable polynomial dynamical system its region of attraction can be estimated by a polynomial Lyapunov function level set. We formulate this problem in terms of global minimization of a polynomial function over single polynomial constraint. We describe a simple Matlab tool, which employs numerical algebraic geometry methods for computing all local solutions of the optimization problem and therefore its global solution.

In this paper, we propose a multi-cell aware (MCA) detector for uplink multi-cell massive multiple-input multiple-output (MIMO) systems. The proposed detector exploits knowledge of the channel statistics but data exchange between different base stations over backhaul links is not required. A correlated channel model is considered and the adopted channel state information (CSI) acquisition model includes the effects of estimation errors and pilot contamination. In contrast to the conventional minimum mean square error (MMSE) detector, which mitigates only the multiple-access interference (MAI) in the target cell, the proposed detector takes the interference from neighboring cells and pilot contamination into account and therefore achieves substantially higher sum rates. Moreover, in order to reduce the computational complexity, the matrix inversion required for the MCA detector is approximated by a matrix polynomial leading to a new polynomial-expansion MCA (PEMCA) detector. Using results from random matrix theory, we derive closed-form expressions for the optimal coefficients of the matrix polynomial, which only depend on the channel statistics but not on the channel realizations. Our simulation results show that the PEMCA detector with only a few terms in the matrix polynomial achieves a considerably higher sum rate than the conventional MMSE detector while having a lower computational complexity.