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This article is focused on the design of an output tracking control scheme for a class of continuous-time periodic piecewise time-varying systems (PPTVSs) with actuator saturation and nonlinear perturbations. The energy-to-peak tracking performance is studied based on an equivalent condition on the definiteness property of matrix polynomials. Considering the actuator saturation and nonlinear perturbation, matrix polynomial-based sufficient conditions are derived through the Lyapunov method using periodic matrix functions. From a perspective of subinterval segmentation aimed at PPTVSs, the proposed conditions can achieve less conservatism for tracking the output of a periodic time-varying reference system, while the controller gains can be computed using convex optimization. Moreover, a heuristic algorithm is constructed to simultaneously guarantee the closed-loop state convergence and the output tracking performance. The reduction in conservatism and the effectiveness of algorithm are demonstrated by illustrative case studies.

In this article, we generalize previously reported results for linear, time-invariant, stabilizable multivariable systems described by a strictly proper transfer function matrix $P(s)$ with number of outputs greater than or equal to the number of inputs. By making use of a special kind of a left generalized inverse $P(s)_{\alpha }^{\oplus }$ of $P(s)$ , we define and examine the equivalent relation $\mathcal {R}$ relating $P(s)$ with the members of the equivalence class $[P(s)]_{R}$ of the closed loop-transfer function matrices $P_{C}(s)$ obtainable from $P(s)$ by the use of a proper compensator $C(s)$ in the feedback path. For $\mathcal {R}$ , we establish a set of complete invariants and a canonical form. These results give rise to a simple algorithmic procedure for the computation of proper internally stabilizing and denominator assigning compensators $C(s)$ for the class of plants with $p=m$ and having no zeros in the closed right half complex plane: $\mathbb {C}^{+}$ and in the case when $p>m$ plants characterized by right polynomial matrix fraction descriptions with a polynomial matrix numerator having at least one subset of $m$ rows that give rise to a nonsingular polynomial matrix with no zeros in $\mathbb {C}^{+}$ .

This paper focuses on stability analysis and stabilization of a switched system under average dwell time criteria in the continuous-time domain. The matrix polynomial approach is applied to the switched system to construct a continuous Lyapunov function and design a more effective controller, which can not only improve the system performance but also lessen the system conservativeness. The stability problem is studied with square matricial representation for the first time. The new method is more applicable than previous work under average dwell time switching with the time-varying controller gains. In addition, new sufficient conditions of stability and stabilization are derived to guarantee the global uniform exponential stability of the switched system. A numerical example is provided to show the advantages of the new method.

In this article, a novel nonfragile controller design scheme is developed for a class of periodic piecewise systems with linear time-varying subsystems. Two types of norm-bounded controller perturbations, including additive and multiplicative ones, are considered and partially characterized by periodic piecewise time-varying parameters. Using a new matrix polynomial lemma, the problems of nonfragile controller synthesis for periodic piecewise time-varying systems (PPTVSs) are made amenable to convex optimization based on the favorable property of a class of matrix polynomials. Depending on selectable divisions of subintervals, sufficient conditions of the stability and nonfragile controller design are proposed for PPTVSs. Case studies based on a multi-input multi-output PPTVS and a mass-spring-damper system show that the proposed control schemes can effectively guarantee the close-loop stability and accelerate the convergence under controller perturbations, with more flexible periodic time-varying controller gains than those obtained by the existing methods.

This paper investigates the secure synchronization control problem for a class of cyber-physical systems &#x0028 CPSs &#x0029 with unknown system matrices and intermittent denial-of-service &#x0028 DoS &#x0029 attacks. For the attack free case, an optimal control law consisting of a feedback control and a compensated feedforward control is proposed to achieve the synchronization, and the feedback control gain matrix is learned by iteratively solving an algebraic Riccati equation &#x0028 ARE &#x0029 . For considering the attack cases, it is difficult to perform the stability analysis of the synchronization errors by using the existing Lyapunov function method due to the presence of unknown system matrices. In order to overcome this difficulty, a matrix polynomial replacement method is given and it is shown that, the proposed optimal control law can still guarantee the asymptotical convergence of synchronization errors if two inequality conditions related with the DoS attacks hold. Finally, two examples are given to illustrate the effectiveness of the proposed approaches.

In this paper, the stability, stabilization, and $L_{2}$ -gain problems are investigated for periodic piecewise systems with time-varying subsystems. Continuous Lyapunov function with time-varying Lyapunov matrix is adopted. A condition guaranteeing the negative definiteness of a matrix polynomial, deriving from the Lyapunov derivative, is first obtained. Based on such a condition, an exponential stability condition is provided. Moreover, a state-feedback controller with time-varying gain is developed to stabilize the unstable periodic piecewise time-varying system. The $L_{2}$ -gain criterion for periodic piecewise time-varying system is also studied. Numerical examples are given to show the validity of the proposed techniques.

Based on unique decoding of the polynomial residue code with non-pairwise coprime moduli, a polynomial with degree less than that of the least common multiple (lcm) of all the moduli can be accurately reconstructed when the number of residue errors is less than half the minimum distance of the code. However, once the number of residue errors is beyond half the minimum distance of the code, the unique decoding may fail and lead to a large reconstruction error. In this paper, assuming that all the residues are allowed to have errors with small degrees, we consider how to reconstruct the polynomial as accurately as possible in the sense that a reconstructed polynomial is obtained with only the last $\tau$ number of coefficients being possibly erroneous, when the residues are affected by errors with degrees upper bounded by $\tau$. In this regard, we first propose a multi-level robust Chinese remainder theorem (CRT) for polynomials, namely, a trade-off between the dynamic range of the degree of the polynomial to be reconstructed and the residue error bound $\tau$ is formulated. Furthermore, a simple closed-form reconstruction algorithm is also proposed., Comment: 5 pages

In this paper, we propose a new framework for uplink–downlink duality in multi-cell multi-user multiple-input multiple-output systems suffering from residual hardware impairments (HWIs) at the base stations and the user terminals. We apply the proposed uplink–downlink duality framework to derive a multi-cell interference and HWI aware minimum mean square error (MCHA-MMSE) precoder which also takes channel state information estimation errors into account. We use results from random matrix theory to derive an analytical expression for the downlink power allocation for the proposed MCHA-MMSE precoder in the large system limit, which only depends on the channel’s second order statistics. In contrast to the conventional MMSE precoder, the proposed MCHA-MMSE precoder takes inter-cell interference, pilot contamination, and residual HWIs into account and therefore achieves substantially higher sum rates. The proposed MCHA-MMSE precoder exploits statistical channel knowledge, but does not require data exchange between base stations via backhaul links. In order to reduce the computational complexity, the matrix inversion required for the computation of the MCHA-MMSE precoder is approximated by a matrix polynomial leading to a new polynomial-expansion MCHA-MMSE precoder. Using results from the random matrix theory, we derive closed-form expressions for the asymptotically optimal coefficients of the matrix polynomial, which only depend on the channel’s second order statistics.

We study the optimal design of graph filters (GFs) to implement arbitrary linear transformations between graph signals. GFs can be represented by matrix polynomials of the graph-shift operator (GSO). Since this operator captures the local structure of the graph, GFs naturally give rise to distributed linear network operators. In most setups, the GSO is given so that GF design consists fundamentally in choosing the (filter) coefficients of the matrix polynomial to resemble desired linear transformations. We determine spectral conditions under which a specific linear transformation can be implemented perfectly using GFs. For the cases where perfect implementation is infeasible, we address the optimization of the filter coefficients to approximate the desired transformation. Additionally, for settings where the GSO itself can be modified, we study its optimal design as well. After this, we introduce the notion of a node-variant GF, which allows the simultaneous implementation of multiple (regular) GFs in different nodes of the graph. This additional flexibility enables the design of more general operators without undermining the locality in implementation. Perfect and approximate designs are also studied for this new type of GFs. To showcase the relevance of the results in the context of distributed linear network operators, this paper closes with the application of our framework to two particular distributed problems: finite-time consensus and analog network coding.

Binary sequences with large linear complexity have been found many applications in communication systems. We determine the linear complexity of a family of p2-periodic binary sequences derived from polynomial quotients modulo an odd prime p. Results show that these sequences have high linear complexity, which means they can resist the linear attack method.

We consider a low complexity polynomial expansion (PE) detector in a massive multiple-input multiple-output (MIMO) uplink channel. 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. Moreover, the channel between a base station (BS) and a UE is a jointly correlated Rician fading channel. The PE detector reduces the computational complexity of the minimum mean square error (MMSE) detector by replacing the matrix inversion with an approximate matrix polynomial. The coefficients of the approximate matrix polynomial are computed from the deterministic equivalents of the moments of the channel Gram matrix. We use operator-valued free probability, which is a more general version of free probability, to derive the deterministic equivalents. In particular, we use the operator-valued moment-cumulant formula. The proposed low complexity PE detector is easy to compute. Simulation results show that the proposed detector can achieve performance close to the MMSE detector.

For parahermitian polynomial matrices, which can be used, for example, to characterise space-time covariance in broadband array processing, the conventional eigenvalue\ud decomposition (EVD) can be generalised to a polynomial matrix EVD (PEVD). In this paper, a new iterative PEVD algorithm based on sequential matrix diagonalisation (SMD) is introduced. At every step the SMD algorithm shifts the dominant column or row of the polynomial matrix to the zero lag position and eliminates the resulting instantaneous correlation. A proof of convergence is provided, and it is demonstrated that SMD establishes diagonalisation faster and with lower order operations than existing PEVD algorithms.

In this paper, we show that the conditional expectation operators corresponding to a family of source-channel models, defined by natural exponential families with quadratic variance functions and their conjugate priors, have orthonormal polynomials as singular vectors. These models include the Gaussian channel with Gaussian source, the Poisson channel with gamma source, and the binomial channel with beta source. To derive the singular vectors of these models, we prove and employ the equivalent condition that their conditional moments are strictly degree preserving polynomials.

Robust synchronization problem is a key issue in chaotic circuits and nonlinear systems. This paper is concerned with robust synchronization problem of polynomial nonlinear system affected by time-varying uncertainties on topology, i.e., structured uncertain parameters constrained in a bounded-rate polytope. Via partial contraction analysis, novel conditions, both for robust exponential synchronization and for robust asymptotical synchronization, are proposed by using parameter-dependent contraction matrices. In addition, for polynomial nonlinear system, this paper introduces a new class of contraction matrix, i.e., homogeneous parameter-dependent polynomial contraction matrix (HPD-PCM), by which tractable conditions of linear matrix inequalities (LMIs) are provided via affine space parametrizations. Furthermore, the variant rate margin for robust asymptotical synchronization is, for the first time, proposed and investigated via handling generalized eigenvalue problems (GEVPs). A set of representative examples demonstrate the effectiveness of proposed method.

We consider reachability of discrete-time polynomial systems. The reachability of such systems can be interpreted as the surjectivity of the polynomial mappings defined by the systems. For a polynomial mapping between suitable spaces of the same dimensions, it has been shown that the injectivity of the mapping also implies the surjectivity. By utilizing this fact, we derive sufficient conditions for the reachability of the polynomial systems, which essentially check the injectivity of the polynomial mappings.

We present a method for solving systems of polynomial equations appearing in computer vision. This method is based on polynomial eigenvalue solvers and is more straightforward and easier to implement than the state-of-the-art Gröbner basis method since eigenvalue problems are well studied, easy to understand, and efficient and robust algorithms for solving these problems are available. We provide a characterization of problems that can be efficiently solved as polynomial eigenvalue problems (PEPs) and present a resultant-based method for transforming a system of polynomial equations to a polynomial eigenvalue problem. We propose techniques that can be used to reduce the size of the computed polynomial eigenvalue problems. To show the applicability of the proposed polynomial eigenvalue method, we present the polynomial eigenvalue solutions to several important minimal relative pose problems.

Following a polynomial approach, many robust fixed-order controller design problems can be formulated as optimization problems whose set of feasible solutions is modeled by parametrized polynomial matrix inequalities (PMIs). These feasibility sets are typically nonconvex. Given a parametrized PMI set, we provide a hierarchy of linear matrix inequality (LMI) problems whose optimal solutions generate inner approximations modeled by a single polynomial superlevel set. Those inner approximations converge in a well-defined analytic sense to the nonconvex original feasible set, with asymptotically vanishing conservatism. One may also impose the hierarchy of inner approximations to be nested or convex. In the latter case, they do not converge any more to the feasible set, but they can be used in a convex optimization framework at the price of some conservatism. Finally, we show that the specific geometry of nonconvex polynomial stability regions can be exploited to improve convergence of the hierarchy of inner approximations.

In the present paper, we propose a novel method for estimating one-dimensional damped and undamped harmonics. Our method utilizes the multiple shift-invariance property comprised in the signal model. We develop a new rank-reduction estimator which is formed as the weighted sum of the individual matrix polynomials obtained from individual shift-invariance equations. The uniqueness conditions for the proposed rank-reduction criteria are derived under the assumption that all samples are available. Moreover, a novel technique for the incomplete data case, where some samples are missing, is presented. In this case, the rank-reduction estimator may suffer from ambiguities. To overcome this problem, we propose an extension of the rank-reduction estimator that is based on polynomial intersection and the properties of the Sylvester matrix. The latter algorithm yields unique estimates of the damped harmonics. The proposed high-resolution techniques are search-free and therefore, they enjoy moderate computational complexity.

A cubic polynomial inter/extrapolation method is investigated to improve the inter/extrapolation accuracy of the matrix over a frequency band in the method of moments (MoM). In the method, the error of the MoM matrix in the Frobenius norm can be expressed as a product of the error coefficient and the polynomial component. The error coefficient is insensitive to the positions of the frequency samples and the operating frequency, and hence it is practical to minimize the amplitude of the polynomial component rather than the error of matrix by optimizing the frequency samples. Actually, the amplitude of the polynomial component attains the minimum when the frequency samples are analytically expressed in terms of the roots of the Chebyshev polynomial of degree 4. Numerical examples are presented to validate the proposed method.

This brief studies prime factorization problems of a multivariate polynomial matrix. We mainly investigate the Lin-Bose problem proposed by Lin and Bose in the multidimensional system theory context. A simple proof to the Lin-Bose problem is presented in a much easier way to understand. As a by-product, we obtain some properties on modules generated by all the rows of a matrix.

This paper presents stability analysis of polynomial fuzzy-model-based (FMB) control systems using the sum-of-squares (SOS) approach. Recently, stability analysis of the polynomial fuzzy-control systems, which is a generalized form of the well-known Takagi-Sugeno (T-S) FMB control systems, has been reported in the form of SOS-based stability conditions. Lack of information on the relations between membership functions and premise variables, in the existing stability analysis approaches, causes conservatism of their results. In this paper, to derive relaxed stability conditions for polynomial FMB control systems, membership functions which are approximated with polynomials and carrying relations between membership functions and premise variables are brought into the stability analysis. Considering a polynomial FMB control system and based on the Lyapunov stability theory, stability conditions in the form of fuzzy summations are derived, where each term contains product of membership functions of the polynomial fuzzy model and polynomial fuzzy controller. Each product term is approximated by a polynomial. In order to obtain better approximation, the operating domain of membership functions is partitioned to subregions. Then, SOS-based stability conditions for all subregions are derived. Unlike some published stability-analysis approaches, the proposed one can be employed for stability analysis of polynomial fuzzy-control systems under imperfect premise matching of which the fuzzy model and fuzzy controller do not share the same membership functions. The solution of the SOS-based stability conditions can be found numerically using SOSTOOLS, which is a free third-party MATLAB Toolbox. Numerical examples are given to illustrate the effectiveness of the proposed stability conditions.

In this paper, a new algorithm for calculating the QR decomposition (QRD) of a polynomial matrix is introduced. This algorithm amounts to transforming a polynomial matrix to upper triangular form by application of a series of paraunitary matrices such as elementary delay and rotation matrices. It is shown that this algorithm can also be used to formulate the singular value decomposition (SVD) of a polynomial matrix, which essentially amounts to diagonalizing a polynomial matrix again by application of a series of paraunitary matrices. Example matrices are used to demonstrate both types of decomposition. Mathematical proofs of convergence of both decompositions are also outlined. Finally, a possible application of such decompositions in multichannel signal processing is discussed.

We deal with the problem of choosing a piecewise polynomial estimator of a regression function s mapping [0,1] p into R. In a first part of this paper, we consider some collection of piecewise polynomial models. Each model is defined by a partition M of [0,1] p and a series of degrees d = (d J)J?M ? NM. We propose a penalized least squares criterion which selects a model whose associated piecewise polynomial estimator performs approximately as well as the best one, in the sense that its quadratic risk is close to the infimum of the risks. The risk bound we provide is nonasymptotic. In a second part, we apply this result to tree-structured collections of partitions, which look like the one constructed in the first step of the CART algorithm. And we propose an extension of the CART algorithm to build a piecewise polynomial estimator of a regression function.

A given polynomial is transformed herein into a rational function. All the roots and multiplicities of the polynomial are then easily obtained from the poles and residues of this rational function, instead of solving for them directly using the original, high-degree multiple-root polynomial. The derived program, using only basic MATLAB built-in routines and existing double-precision arithmetic, amazingly gives the expected results for test polynomials of very high degree and multiplicity, even as high as p(x) = (x + 98.765)1234.

Classical Takagi-Sugeno (T-S) fuzzy models are formed by convex combinations of linear consequent local models. Such fuzzy models can be obtained from nonlinear first-principle equations by the well-known sector-nonlinearity modeling technique. This paper extends the sector-nonlinearity approach to the polynomial case. This way, generalized polynomial fuzzy models are obtained. The new class of models is polynomial, both in the membership functions and in the consequent models. Importantly, T-S models become a particular case of the proposed technique. Recent possibilities for stability analysis and controller synthesis are also discussed. A set of examples shows that polynomial modeling is able to reduce conservativeness with respect to standard T-S approaches as the degrees of the involved polynomials increase.

There is a fair amount of research about privacy, but few empirical studies about its cost have been conducted. In the area of secure multiparty computation, the scalar product has long been reckoned as one of the most promising alternatives to classic logic gates. The reason for this is that the scalar product is not only complete, which is as good as logic gates, but also much more efficient than logic gates. As a result, we set out to study the computation and communication resources needed for some of the most well-known and frequently referenced secure scalar product protocols, including the composite residuosity, the invertible matrix, the polynomial sharing, and the commodity-based approaches. In addition to the implementation details of these approaches, we analyze and compare their execution time, computation time, and memory and random number consumption. Moreover, Fairplay, the benchmark approach that implements Yao's circuit evaluation protocol, is also included in our experiments in order to demonstrate the potential for the scalar products to replace logic gates.

In this paper, we present a novel approach to robust semidefinite programs, whose coefficient matrices depend polynomially on uncertain parameters. The procedure to construct an approximate problem for a given robust semidefinite program is based on the sum-of-squares representation of a positive semidefinite polynomial matrix. In contrast to the conventional sum-of-squares approach, quality of the approximation is improved by dividing the parameter region into several subregions. The optimal value of the approximate problem converges to that of the original problem as the resolution of the division becomes finer. An advantage of this approach is that an upper bound on the approximation error can be explicitly obtained in terms of the resolution of the division. Numerical examples on polynomial optimizations are presented to show usefulness of the present approach. We also consider exploitation of sparsity of the given problem with respect to the uncertain parameters, in order to construct a reduced-size approximate problem.

This note deals with a computational approach to an optimal control problem for input-affine polynomial systems based on a state-dependent linear matrix inequality (SDLMI) from the Hamilton-Jacobi inequality. The design follows a two-step procedure to obtain an upper bound on the optimal value and a state feedback law. In the first step, a direct usage of the matrix sum of squares relaxations and semidefinite programming gives a feasible solution to the SDLMI. In the second step, two kinds of polynomial annihilators decrease the conservativeness of the first design. The note also deals with a control-oriented structural reduction method to reduce the computational effort. Numerical examples illustrate the resulting design method.

In this note, we study the controllability and observability problem for a class of controlled switching impulsive systems. By proposing several formula of variation of parameters for this type of time-varying systems and employing the characteristic polynomial theory of matrix, we establish necessary and sufficient conditions for controllability and controlled observability with respect to a given switching time sequence. Specializing the obtained results to the case of time-invariant linear switching impulsive systems, we derive some simple algebraic criteria, which include the results reported in the literature for time-invariant linear switching systems, linear impulsive systems and classical linear systems. One example is worked out for illustration.

In this note, a sufficient condition is derived for the stability of a spatially invariant distributed dynamical system, based on the geometrical structure of the null space of a matrix polynomial. This condition is less conservative than the available computationally feasible criteria. Moreover, using the idea of parameter dependent linear matrix inequalities, a necessary and sufficient condition is obtained. Both of these two conditions are expressed by LMIs. While the necessity of the latter is lost if the degree of the related matrix polynomials is small, its conservativeness can be sequentially reduced.

In this paper, three small classes of finite fields GF(2m) are found for which low complexity bit-parallel multipliers are proposed. The proposed multipliers have lower complexities compared to those based on the irreducible pentanomials. It is also shown that there does not always exist an irreducible all-one polynomial, equally-spaced polynomial, or trinomial for the new classes of fields.

We propose using (bilinear) sum-of-squares programming for obtaining inner bounds of RoAs for dynamical systems with polynomial vector fields. We search for polynomial as well as composite Lyapunov functions comprised of pointwise maximums of polynomial functions. Results for several examples from the literature are presented using the proposed methods and the PENBMI solver.

Multidimensional (n-D) polynomial matrix factorizations are intimately linked to many problems of multidimensional systems and signal processing. This paper gives a new result for a n-D polynomial matrix to have an minor prime factorization using methods from computer algebra. This result may be regarded as a generalization of a previous criterion under a special restriction [IEEE Trans. Circuits Syst. II: Exp Briefs, vol. 52, no. 9 (2005)]. Examples are given to illustrate results using computer algebra software system Singular.

This paper presents a general class of fast algorithms for computing the polynomial time-frequency transform (PTFT) of length-apb, where a, b, and p are positive integers. The process of derivation shows some interesting properties that are effectively used for minimization of the computational complexity. By assigning values of a, b, and p, various algorithms, for example, radix-o and split-radix-2/(2a), can be easily obtained to provide the flexibility supporting polynomial time-frequency transforms of various sequence lengths. The detailed analysis on the computational complexities needed by these algorithms is also presented in terms of the numbers of additions and multiplications. It is shown that the proposed algorithms significantly reduce the computational complexity for applications that deal with polynomial phase signals.

This paper investigates the problem of factor prime factorizations for n-D polynomial matrices and presents a criterion for the existence of factor prime factorizations for an important class of n-D polynomial matrices. As a by-product, we also obtain an algebraic algorithm to check n-D factor primeness in some important cases which partially solves the long-standing open problem of recognizing n-D factor prime matrices. Some problems related to the factorization methods are also studied. Several examples are given to illustrate the results. The results presented in this paper are true over any coefficient field.

The concepts of Kronecker maps and Sylvester-polynomial matrix equations are proposed. Some properties of the Kronecker maps are deduced. It is shown that many kinds of matrix equations can be formulated into the so-called Sylvester-polynomial matrix equations. With the help of the Kronecker maps, general explicit solutions of the Sylvester-polynomial matrix equations are established when the polynomial matrices possess F-primeness. For the nonsquare case, the proposed solutions can offer all the degrees of freedom represented by a free parameter matrix

An algorithm for computing the eigenvalue decomposition of a para-Hermitian polynomial matrix is described. This amounts to diagonalizing the polynomial matrix by means of a paraunitary "similarity" transformation. The algorithm makes use of "elementary paraunitary transformations" and constitutes a generalization of the classical Jacobi algorithm for conventional Hermitian matrix diagonalization. A proof of convergence is presented. The application to signal processing is highlighted in terms of strong decorrelation and multichannel data compaction. Some simulated results are presented to demonstrate the capability of the algorithm

We prove that equivalence of deterministic top-down tree-to-string transducers is decidable, thus solving a long-standing open problem in formal language theory. We also present efficient algorithms for subclasses: for linear transducers or total transducers with unary output alphabet (over a given top-down regular domain language), as well as for transducers with the single-use restriction. These results are obtained using techniques from multi-linear algebra. For our main result, we introduce polynomial transducers and prove that for these, validity of a polynomial invariant can be certified by means of an inductive invariant of polynomial ideals. This allows us to construct two semi-algorithms, one searching for a certificate of the invariant and one searching for a witness of its violation. Via a translation into polynomial transducers, we thus obtain that equivalence of general y dt transducers is decidable. In fact, our translation also shows that equivalence is decidable when the output is not in a free monoid but in a free group.

This paper proposes a novel algorithm for robust partial eigenvalue assignment (RPEVA) problem for a cubic matrix pencil arising from modeling of vibrating systems with aerodynamic effects. The RPEVA problem for a cubic pencil is the one of choosing suitable feedback matrices to reassign a few (say k < 3n) unwanted eigenvalues while leaving the remaining large number (3n - k) of them unchanged, in such a way that the the eigenvalues of the closed-loop matrix are as insensitive as possible to small perturbation of the data. The latter amounts to minimizing the condition number of the closed-loop eigenvector matrix. The problem is solved directly in the cubic matrix polynomial setting without making any transformation to a standard first-order state-space system. This allows us to take advantage of the exploitable structures such as the sparsity, definiteness, bandness, etc., very often offered by large practical problems. The major computational requirements are: (i)solutions of a small Sylvester equation, (ii) QR factorizations, and (iii) solutions or some standard least squares problems. The least-squares problems result from matrix rank-one and rank-two update techniques used in the algorithm for reassigning, respectively, simple and complex eigenvalues. The practical effectiveness of the method is demonstrated by implementational results on simulated data provided by the Boeing company.

Using a moment interpretation of recent results on sum-of-squares decompositions of nonnegative polynomial matrices, we propose a hierarchy of convex linear matrix inequality (LMI) relaxations to solve nonconvex polynomial matrix inequality (PMI) optimization problems, including bilinear matrix inequality (BMI) problems. This hierarchy of LMI relaxations generates a monotone sequence of lower bounds that converges to the global optimum. Results from the theory of moments are used to detect whether the global optimum is reached at a given LMI relaxation, and if so, to extract global minimizers that satisfy the PMI. The approach is successfully applied to PMIs arising from static output feedback design problems.

A tractable criterion is presented for the existence of minor prime factorizations for a class of multidimensional (n-D) (n>2) polynomial matrices whose reduced minors and greatest common divisors have some common zeros. We also present a constructive method for carrying out the minor prime factorizations when they exist. The proposed method is further extended to a larger class of n-D polynomial matrices by an invertible variable transformation. Three illustrative examples are given to show the effectiveness of the proposed method.

Traditionally, when approaching controller design with the Youla-Kuc/spl caron/era parametrization of all stabilizing controllers, the denominator of the rational parameter is fixed to a given stable polynomial, and optimization is carried out over the numerator polynomial. In this note, we revisit this design technique, allowing to optimize simultaneously over the numerator and denominator polynomials. Stability of the denominator polynomial, as well as fixed-order controller design with H/sub /spl infin// performance are ensured via the notion of a central polynomial and linear matrix inequality (LMI) conditions for polynomial positivity.

The polynomial expansion (PE) multiuser detector can iteratively approximate the linear decorrelating and MMSE multiuser detectors. This is a very promising approach since the complexity of the PE detector is considerably less than that of the decorrelating and MMSE detectors. The concept of the PE multiuser detector is a weighted matrix polynomial for which optimal weighting has been suggested in the literature. Unfortunately, the optimal weights apply only to a specific correlation matrix. As soon as the correlation between the users changes (and hence also the correlation matrix), a new set of optimal weights has to be calculated. The calculation of these weights is computationally very intense. In this paper, an approach is presented where the weights are predefined and apply for all matrices. Furthermore, a normalization factor is needed to ensure convergence. It will be shown how this factor has to be chosen to obtain the optimal convergence speed. For the optimal normalization factor the minimum and maximum eigenvalues are needed. A low complexity and accurate method to estimate these eigenvalues is derived which can be applied to all correlation matrices and therefore to any code division multiple access (CDMA) scenario. Hence, the matrix-dependent optimal normalization factor can easily be calculated and ensures a good bit error rate (BER) performance, even if the correlation between the users changes quickly (e.g., in time-variant channels). Furthermore, it will be shown how to enhance the PE detector such that it has a high near-far resistance. Additionally, for the first time it can be proven that the approximation error of the proposed detector diminishes exponentially with the number of iterations. Finally, simulations verify the fast convergence of the proposed PE detector and its flexible usage in a variety of scenarios.

In this paper, we analyze a linear multiuser receiver for code-division multiple-access systems that is based on a matrix polynomial expansion. We focus on the receiver where the polynomial coefficients are chosen to minimize the mean squared error at the output and observe that the resultant coefficients are also signal-to-interference ratio maximizing. We present a simple derivation for the (known) large system coefficients and signal-to-interference ratio of this optimal multistage receiver and make a significant step toward a direct derivation of Honig and Xiao's recursive expression for this large system signal-to-interference ratio. Finally, we extend these results to take into account arbitrary power distributions.

The Cayley-Hamilton theorem is extended for real polynomial matrices in n variables. The known extensions of the classical Cayley-Hamilton theorem are particular cases of the proposed extension.

On the basis of the relationship of the mth power of a polynomial and its modular form (polynomial whose coefficients are the moduli of the coefficients of that polynomial), we derive a necessary and sufficient condition for the modulus of the mth power of a polynomial for contacting its modular form on the boundary of a disc. Combined with the result about distribution of zeros of analytic function, some new sufficient conditions are derived which give bounds of the absolute values of the roots of a quasi-critical polynomial. These results extend certain earlier similar tests for linear discrete-time systems. Finally, four examples are given to demonstrate the results, Example 2.1 gives a state feedback application, Examples 2.2 and 2.4 deal with r-stability, and Example 2.3 display that our theorems give better results when m increases but at the cost of increasing complexity.

This paper undertakes the study of multidimensional finite impulse response (FIR) filterbanks. One way to design a filterbank is to factorize its polyphase matrices in terms of elementary building blocks that are fully parameterized. Factorization of one-dimensional (1-D) paraunitary (PU) filterbanks has been successfully accomplished, but its generalization to the multidimensional case has been an open problem. In this paper, a complete factorization for multichannel, two-dimensional (2-D), FIR PU filterbanks is presented. This factorization is based on considering a two-variable FIR PU matrix as a polynomial in one variable whose coefficients are matrices with entries from the ring of polynomials in the other variable. This representation allows the polyphase matrix to be treated as a one-variable matrix polynomial. To perform the factorization, the definition of paraunitariness is generalized to the ring of polynomials. In addition, a new degree-one building block in the ring setting is defined. This results in a building block that generates all two-variable FIR PU matrices. A similar approach is taken for PU matrices with higher dimensions. However, only a first-level factorization is always possible in such cases. Further factorization depends on the structure of the factors obtained in the first level.

In code-division multiple-access (CDMA) transmissions, computing the multiuser minimum mean-squared error (MMSE) detector coefficients requires the inversion of the covariance matrix associated to the received vector signal, an operation usually difficult to implement when the spreading factor and the number of users are large. It is therefore interesting to approximate the inverse by a matrix polynomial. In this correspondence, means for computing the polynomial coefficients are proposed in the context of CDMA downlink transmissions on frequency-selective channels, the users having possibly different powers. Derivations are made in the asymptotic regime where the spreading factor and the number of users grow toward infinity at the same rate. Results pertaining to the mathematics of large random matrices, and in particular to free probability theory, are used. Spreading matrices are modeled as isometric random matrices (spreading vectors orthonormality is a natural assumption in downlink) and also as random matrices with independent and identically distributed (i.i.d.) elements.

Multivariate polynomials are often used to model nonlinear behavior, e.g., in parallel Wiener models. These multivariate polynomials are mostly hard to interpret due to the presence of cross terms. These polynomials also have a high amount of coefficients, and the calculation of an inverse of a multivariate polynomial with cross terms is cumbersome. This paper proposes a method to eliminate the cross terms of a multivariate polynomial using a linear input transformation. It is shown how every homogeneous polynomial described using tensors can be transformed to a canonical form using multilinear algebraic decomposition methods. Such tensor decomposition methods have already been used in nonlinear system modeling to reduce the complexity of Volterra models. Since every polynomial can be written as a sum of homogeneous polynomials, this method results in a decoupled description of any multivariate polynomial, allowing a model description that is easier to interpret, easier to use in a design, and easier to invert. This paper first describes a method to represent and decouple multivariate polynomials using tensor representation and tensor decomposition techniques. This method is applied to a parallel Wiener model structure, where a multiple-input-single-output polynomial is used to describe the static nonlinearity of the system. A numerical example shows the performance of the proposed method.

A sufficient condition for checking the robust stability of a polytope of polynomial matrices is proposed in this note. A simple feasibility test performed in a convex set of linear-matrix inequalities defined at the vertices of the polytope yields sufficient conditions for the robust stability of the entire domain. Both continuous-time (left half-plane) and discrete-time stability (unit disk) are investigated. Numerical comparisons with quadratic stability results and with another method recently appeared in the literature show that the conditions proposed provide, in general, less conservative results.