Proper Rational Functions Research Articles

Rational Functions as Sums of Reciprocals of Polynomials

An algorithm is given to express a proper rational function (i.e., the degree of the denominator exceeds that of the numerator) over any field as an alternating sum of reciprocals of polynomials of increasing degree. This is analogous to existing algorithms for representing proper rational numbers.

Derivative Interpolating Subspace Frameworks for Nonlinear Eigenvalue Problems

We first consider the problem of approximating a few eigenvalues of a proper rational matrix-valued function closest to a prescribed target. It is assumed that the proper rational matrix-valued function is expressed in the transfer function form $H(s) = C (sI - A)^{-1} B$, where the middle factor is large, whereas the number of rows of $C$ and the number of columns of $B$ are equal and small. We propose a subspace framework that performs two-sided projections on the state-space representation of $H(\cdot)$, commonly employed in model reduction and giving rise to a reduced transfer function. At every iteration, the projection subspaces are expanded to attain Hermite interpolation conditions at the eigenvalues of the reduced transfer function closest to the target, which in turn leads to a new reduced transfer function. We prove in theory that, when a sequence of eigenvalues of the reduced transfer functions converges to an eigenvalue of the full problem, it converges at least at a quadratic rate. In the second part, we extend the proposed framework to locate the eigenvalues of a general square large-scale nonlinear meromorphic matrix-valued function $T(\cdot)$, where we exploit a representation ${\mathcal R}(s) = C(s) A(s)^{-1} B(s) - D(s)$ defined in terms of the block components of $T(\cdot)$. The numerical experiments illustrate that the proposed framework is reliable in locating a few eigenvalues closest to the target point, and that, with respect to runtime, it is competitive to established methods for nonlinear eigenvalue problems.

New Method of Solving the Proper Rational Function and Application to Electrical Control System

New Method of Solving the Proper Rational Function and Application to Electrical Control System

Auotomodeling rational mnemofunctions and their link to an analytical representation of distributions

Mnemofunctions of the form f(x/ε), where f is the proper rational function without singularities on the real line, are considered in this article. Such mnemofunctions are called automodeling rational mnemofunctions. They possess the following fine properties: asymptotic expansions in the space of distributions can be written in explicit form and the asymptotic expansion of the product of such mnemofunctions is uniquely determined by the expansions of multiplicands.Partial fraction decomposition of automodeling rational mnemofunctions generates the so-called sloped analytical representation of a distribution, i.e. the representation of a distribution by a jump of the boundary values of the functions analytical in upper and lower half-planes. Sloped analytical representation is similar to the classical Cauchy analytical representation, but its structure is more complicated. The multiplication rule of such representations is described in this article.

Rational mnemofunctions on R

The subspace of rational distributions was considered it this paper. Distribution is called rational if it has analytical representation f = (f+, f-) where functions f+ and f- are proper rational functions. The embedding of the rational distributions subspace into the rational mnemofunctions algebra on was built by the mean of mapping Ra(f)=fε(x)=f+(x+iε)-f-(x-iε). A complete description of this algebra was given. Its generators were singled out; the multiplication rule of distributions in this algebra was formulated explicitly. Known cases when product of distributions is a distribution were analyzed by the terms of rational mnemofunctions theory. The conditions under which the product of arbitrary rational distributions is associated with a distribution were formulated.

Stability and Stabilization With Additive Freedom for Delayed Takagi–Sugeno Fuzzy Systems by Intermediary-Polynomial-Based Functions

This paper is devoted to the stability and stabilization for Takagi-Sugeno fuzzy systems with time-varying delays. First, an improved matrix inequality is presented to bound both strictly and nonstrictly proper rational functions, which is more general than the existing versions of reciprocally convex lemmas. Second, by suitable operations on parameter-dependent polynomial multiplied by state rate, a couple of novel intermediary-polynomial-based functions (IPFs) are developed in delay-product types. Benefitting from slack matrices of IPFs, a certain degree of flexibility is furnished. More importantly than that, by feat of adjustment of the variable parameter, the resulting conditions will be further endowed with additive freedom, which relaxes the feasible space in a distinctive manner. Third, by utilizing IPFs along with triple integrals, the stability criteria and the controller design approach are derived by some advanced integral inequalities. Resorting to elaborate construction of IPFs, the strengths of bounding techniques are sufficiently exploited, and the information on delay derivative is adequately reflected. Consequently, more desirable performances are achieved, while without excessive computational complexity. Finally, the effectiveness of the proposed methods is verified by numerical examples.

H-Adic Polynomials and Partial Fraction Decomposition of Proper Rational Functions over R or C

The partial fraction decomposition technique is very useful in many areas including mathematics and engineering. In this paper we present a new and simple method on the partial fraction decomposition of proper rational functions which have completely factored denominators over R or C. The method is based on a recursive computation of the h-adic polynomial in commutative algebra which is a generalization of the Taylor polynomial. Since its computation requires only simple algebraic operations, it does not require a computer algebra system to be programmed.

Nonlinear Sequence Transformation-Based Continuous-Time Wavelet Filter Approximation

Time-domain synthesis is used to design continuous-time filters when their time-domain response for a given excitation is known. A case in point is the design of a continuous-time wavelet filter which requires a chosen wavelet as the filter’s impulse response. A key step in the design of a continuous-time wavelet filter is the approximation of its transfer function to a proper rational function. This approximation problem is addressed using methods that can be broadly categorized as closed-form methods and numerical optimization methods. While optimization methods are very effective, closed-form solutions are attractive because they are easy to design. In this work, we propose variants of nonlinear sequence transformation (a closed-form method) that obtain proper rational approximations of wavelet functions which are stable and also perform better in terms of mean square error when compared to those obtained by other closed-form methods. It is shown that the model-order reduction of the proposed variants leads to either similar or better performance compared to \(L_2\) optimization method. These variants are also shown to act as good starting points for optimization methods that use local search routines.

Improved stability and stabilization design for networked control systems using new quadruple-integral functionals

This paper investigates stability analysis and stabilization for networked control systems. By a refined delay decomposition approach, slightly different Lyapunov–Krasovskii functionals (LKFs) with quadruple-integral terms and augmented vectors containing triple-integral forms of state are constructed. New integral inequalities are proposed to estimate the cross terms from derivatives of the LKFs, which can be proved to offer tighter bounds than what the Jensen one produces theoretically. Moreover, the non-strictly proper rational functions in deriving process are fully handled via reciprocally convex approach. A state feedback controller design approach is also developed. Numerical examples and applications to practical power and oscillator systems demonstrate the superiority of the proposed criteria in conservatism reduction compared to some existing ones.

Skew‐selfadjoint Dirac systems with rational rectangular Weyl functions: explicit solutions of direct and inverse problems and integrable wave equations

In this paper we study direct and inverse problems for discrete and continuous skew‐selfadjoint Dirac systems with rectangular (possibly non‐square) pseudo‐exponential potentials. For such a system the Weyl function is a strictly proper rational matrix function and any strictly proper rational matrix function appears in this way. In fact, extending earlier results, given a strictly proper rational matrix function we present an explicit procedure to recover the corresponding potential using techniques from mathematical system and control theory. We also introduce and study a nonlinear generalized discrete Heisenberg magnet model, extending earlier results for the isotropic case. A large part of the paper is devoted to the related discrete systems of which the pseudo‐exponential potential depends on an additional continuous time parameter. Our technique allows us to obtain explicit solutions for the generalized discrete Heisenberg magnet model and evolution of the Weyl functions.

A frequency-independent and parallel algorithm for computing the zeros of strictly proper rational transfer functions

We develop an algorithm for computation of the zeros of a strictly proper rational transfer function in partial fraction form, by transforming the problem of finding the roots of the determinant of a frequency-dependent matrix into one of finding the eigenvalues of a companion matrix comprised of the determinants of a binomial-based set of frequency-independent matrices. The proposed algorithm offers a fundamentally new approach that avoids solving severely ill-conditioned system of linear equations, where condition numbers increase rapidly with frequency. The developed algorithm is straightforward, and enables parallel computation of the characteristic polynomial coefficients an′ that comprise the companion matrix to the characteristic polynomial ∑nansn of the frequency-dependent matrix. Additionally, the algorithm allows for relatively inexpensive computation of asymptotically accurate approximants of the transfer function, such that an′ need be computed only for selected powers of s=jω, where the number of required determinant operations are shown to be relatively small. Additionally, limitations of the developed algorithm are highlighted, where the computational cost is shown to be on the order O(2Np) determinant operations on matrices of dimensions (Np+1)×(Np+1), and Np is the number of poles. Illustrative numerical examples are selected and discussed to provide further insight about pros/cons of the proposed method, and to identify potential areas for further research.

Finite and infinite structures of rational matrices: a local approach

The structure of a rational matrix is given by its Smith-McMillan invariants. Some properties of the Smith-McMillan invariants of rational matrices with elements in different principal ideal domains are presented: In the ring of polynomials in one indeterminate (global structure), in the local ring at an irreducible polynomial (local structure), and in the ring of proper rational functions (infinite structure). Furthermore, the change of the finite (global and local) and infinite structures is studied when performing a Mobius transformation on a rational matrix. The results are applied to define an equivalence relation in the set of polynomial matrices, with no restriction on size, for which a complete system of invariants are the finite and infinite elementary divisors.

Rheological properties of a straw layer

The experiment was performed on a layer of straw whose thickness was equivalent to the load of 4 kg m−2. Every straw sample was subjected to 15 strain cycles. A preliminary analysis of the results revealed that compression and decompression curves resemble the proper rational function, where selected parameters of that function were determined by the direction of strain and the number of strain cycles. Straw was described with the use of a rheological model where it was divided into layers subjected to elastic strain and internal friction proportional to the rate of displacement and deformation. Unknown model parameters were estimated based on an analysis of the equation describing the results of the experiment and the results produced by the model simulating the rheological properties of straw. The elastic modulus was directly proportional to load, and its value decreased in successive strain cycles. Internal friction was determined based on the loss of mechanical energy. In fluid dynamics, internal friction is associated with viscosity, whereas in fibrous materials such as straw, mechanical energy is lost mainly due to sliding friction. The losses were minimized in successive strain cycles.

Improved criteria on robust stability and [formula omitted] performance for linear systems with interval time-varying delays via new triple integral functionals

This paper analyzes delay-dependent robust stability and H∞ performance of linear systems with an interval time-varying delay, based on a new Lyapunov–Krasovskii functional containing new triple integral terms. The time derivative of the Lyapunov–Krasovskii functional produces not only the strictly proper rational functions but also the non-strictly proper rational functions of the time-varying delays with first-order denominators. The combinations of the rational functions are directly handled via the Jensen inequality lemma and the lower bound lemma for reciprocal convexity, whereas such combinations were approximated in the literature. The proposed criteria become less conservative with the significantly smaller number of decision variables than the existing criteria, which will be demonstrated by some numerical examples.

Solving Burgersʼ equation using optimal rational approximations

We solve viscous Burgersʼ equation using a fast and accurate algorithm—referred to here as the reduction algorithm—for computing near optimal rational approximations.Given a proper rational function with n poles, the reduction algorithm computes (for a desired L∞-approximation error) a rational approximation of the same form, but with a (near) optimally small number m≪n of poles. Although it is well known that (nonlinear) optimal rational approximations are much more efficient than linear representations of functions via a fixed basis (e.g. wavelets), their use in numerical computations has been limited by a lack of efficient, robust, and accurate algorithms. The reduction algorithm presented here computes reliably (near) optimal rational approximations with high accuracy (e.g., ≈10−14) and a complexity that is essentially linear in the number n of original poles. A key tool is a recently developed algorithm for computing small con-eigenvalues of Cauchy matrices with high relative accuracy, an impossible task for standard algorithms without extended precision.Using the reduction algorithm, we develop a numerical calculus for rational representations of functions. Indeed, while operations such as multiplication and convolution increase the number of poles in the representation, we use the reduction algorithm to maintain an optimally small number of poles.To demonstrate the efficiency, robustness, and accuracy of our approach, we solve Burgersʼ equation with small viscosity ν. It is well known that its solutions exhibit moving transition regions of width O(ν), so that this equation provides a stringent test for adaptive PDE solvers. We show that optimal rational approximations capture the solutions with high accuracy using a small number of poles. In particular, we solve the equation with local accuracy ϵ=10−9 for viscosity as small as ν=10−5.

Semiseparable Integral Operators and Explicit Solution of an Inverse Problem for a Skew-Self-Adjoint Dirac-Type System

Inverse problem to recover the skew-self-adjoint Dirac-type system from the generalized Weyl matrix function is treated in the paper. Sufficient conditions under which the unique solution of the inverse problem exists, are formulated in terms of the Weyl function and a procedure to solve the inverse problem is given. The case of the generalized Weyl functions of the form $\phi(\lambda)\exp\{-2i\lambda D\}$, where $\phi$ is a strictly proper rational matrix function and $D=D^* \geq 0$ is a diagonal matrix, is treated in greater detail. Explicit formulas for the inversion of the corresponding semiseparable integral operators and recovery of the Dirac-type system are obtained for this case.

An improved Heaviside approach to partial fraction expansion and its applications†

In this note, we present an improved Heaviside approach to compute the partial fraction expansions of proper rational functions. This method uses synthetic divisions to determine the unknown partial fraction coefficients successively, without the need to use differentiation or to solve a system of linear equations. Examples of its applications in indefinite integration, inverse Laplace transforms and linear ordinary differential equations are included. †This work is an extended version of my paper presented at the International Conference of Applied and Engineering Mathematics (ICAEM) held at London, UK, on 2–4 July 2008.

Sum-of-Squares Decomposition via Generalized KYP Lemma

The Kalman-Yakubovich-Popov (KYP) lemma establishes the equivalence between a frequency domain inequality (FDI) of a proper rational function and a linear matrix inequality (LMI). A recent result generalized the KYP lemma to characterize an FDI of a possibly nonproper rational function on a portion of a curve on the complex plane. This note examines implications of the generalized KYP result to sum-of-squares (SOS) decompositions of matrix-valued nonnegative polynomials of a single complex variable on a curve in the complex plane. Our result generalizes and unifies some existing SOS results, and also establishes equivalences among FDI, LMI, and SOS.

Linear systems with locally integrable trajectories

The locally integrable function space L 1 loc is made a module over the ring of proper rational functions. Using this module structure, the notion of relative dimension is defined naturally for every linear subspace in ( L 1 loc ) q . It is shown that the property of having finite relative dimension together with the property of having sufficiently many smooth trajectories and the evident property of differentiation-invariance characterize the weak solutions sets of linear constant coefficient differential systems among all linear subspaces.

An algebraic approach to control design for systems with periodically time-varying parameters

This paper deals with the application of simple PI and PID algorithms to control of systems with periodically time-varying parameters. The proposed algebraic synthesis method is based on general solutions of Diophantine equations in the ring of proper and stable rational functions and subsequently on selection of appropriate controller from the set defined by Youla-Kucera parameterisation. An advantage of the technique is that the controllers can be tuned by a single positive scalar parameter. The capabilities of obtained control laws are demonstrated on three simulation examples.