A parameterization of reduced stable models and controllers

This note addresses the problem of instability in both model and controller reduction methods based on similarity transformations. For a given model (controller) a general framework is proposed that parameterizes a large set of reduced models (controllers) that preserve the stability of the original model (closed loop system). In addition, a sufficient condition for the existence of such a framework is derived. As an application of the main results, it is shown how different reduction methods can be modified, if they fail to maintain stability.     

A new parameterization of digital straight lines

A 1:1 correspondence is established between digital straight lines which start at a fixed point and a simple set of quadruples of integer parameters. Such a representation by parameters is useful for enumeration, First, the authors show a 1:1 correspondence between point pairs in a planar set of points and the linear dichotomies of this set. Then, from the equivalence between digital lines and linear dichotomies of points on the digitization grid, they prove a 1:1 and 'onto' correspondence between digital straight lines starting at a fixed point and a well-defined set of pairs of grid points. It follows that four parameters uniquely represent any given digital line with a fixed starting point. An O(N) algorithm is given for determining the parameters from the digital line, as well as O(log N) algorithms for transforming between these parameters and the parameters suggested by L. Dorst and A.W.M. Smeulders (1984)     

Decentralized stable factors and a parameterization ofdecentralized controllers

Decentralized stable factors (DSF) are used to extend the stable factorization approach to the design of decentralized controllers. The DSF allow a parameterization of all the stabilizing decentralized controllers in the form of a Youla parameterization. A decentralized controller serves as the central controller, and the Youla parameter has a particular structure and must satisfy an interaction constraint. This parameterization encompasses several other parameterization results on decentralized control systems     

Constructing convex directions for stable polynomials

New constructions of convex directions for Hurwitz stable polynomials are obtained. The technique is based on interpretations of the phase-derivative conditions in terms of the sensitivity of the root-locus associated with the even and odd parts of a polynomial     

A hierarchy of LMI inner approximations of the set of stable polynomials

Exploiting spectral properties of symmetric banded Toeplitz matrices, we describe simple sufficient conditions for the positivity of a trigonometric polynomial formulated as linear matrix inequalities (LMIs) in the coefficients. As an application of these results, we derive a hierarchy of convex LMI inner approximations (affine sections of the cone of positive definite matrices of size m) of the nonconvex set of Schur stable polynomials of given degree n<m. It is shown that when m tends to infinity the hierarchy converges to a lifted LMI approximation (projection of an LMI set defined in a lifted space of dimension quadratic in n) already studied in the technical literature. An application to robust controller design is described.     

Local convex directions for Hurwitz stable polynomials

A condition for a polynomial p(s) to be a local convex direction for a Hurwitz stable polynomial q(s) is derived. The condition is in terms of polynomials associated with the even and odd parts of p(s) and q ( s), and constitutes a generalization of Rantzer's phase-growth condition for global convex directions. It is used to determine convex directions for certain subsets of Hurwitz stable polynomials     

A new thickness parameterization for Discrete Material and Thickness Optimization

In this work, a new thickness parameterization which allows for internal ply-drops without intermediate voids is introduced in the Discrete Material and Thickness Optimization (DMTO) method. With the original DMTO formulation, material had to be removed from the top in order to prevent non-physical intermediate voids in the structure. The new thickness formulation relies on a relation between density variables and ply-thicknesses rather than constitutive properties. This new formulation allows internal ply-drops which is essential for composite structures as it is common practice to cover dropped plies as to avoid delaminations. Furthermore, it is demonstrated how the new thickness formulation in some cases improves the convergence characteristics. Finally, it is also shown how solid-shell elements can be utilized within the DMTO method for structural optimization of tapered laminated composite structures.     

A new method for T-spline parameterization of complex 2D geometries

We present a new strategy, based on the idea of the meccano method and a novel T-mesh optimization procedure, to construct a T-spline parameterization of 2D geometries for the application of isogeometric analysis. The proposed method only demands a boundary representation of the geometry as input data. The algorithm obtains, as a result, high quality parametric transformation between 2D objects and the parametric domain, the unit square. First, we define a parametric mapping between the input boundary of the object and the boundary of the parametric domain. Then, we build a T-mesh adapted to the geometric singularities of the domain to preserve the features of the object boundary with a desired tolerance. The key of the method lies in defining an isomorphic transformation between the parametric and physical T-mesh finding the optimal position of the interior nodes by applying a new T-mesh untangling and smoothing procedure. Bivariate T-spline representation is calculated by imposing the interpolation conditions on points sited both in the interior and on the boundary of the geometry. The efficacy of the proposed technique is shown in several examples. Also we present some results of the application of isogeometric analysis in a geometry parameterized with this technique.     

A new algorithm for sparse interpolation of multivariate polynomials

To reconstruct a black box multivariate sparse polynomial from its floating point evaluations, the existing algorithms need to know upper bounds for both the number of terms in the polynomial and the partial degree in each of the variables. Here we present a new technique, based on Rutishauser’s qd$qd$-algorithm, in which we overcome both drawbacks.     

A stable algorithm for Hankel transforms using hybrid of Block-pulse and Legendre polynomials

A new numerical method, based on hybrid of Block-pulse and Legendre polynomials for numerical evaluation of Hankel transform is proposed in this paper. Hybrid of Block-pulse and Legendre polynomials are used as a basis to expand a part of the integrand, rf(r), appearing in the Hankel transform integral. Thus transforming the integral into a Fourier-Bessel series. Truncating the series, an efficient algorithm is obtained for the numerical evaluations of the Hankel transforms of order ν>−1. The method is quite accurate and stable, as illustrated by given numerical examples with varying degree of random noise terms εθi added to the data function f(r), where θi is a uniform random variable with values in [−1,1]. Finally, an application of the proposed method is given in solving the heat equation in an infinite cylinder with a radiation condition.     

A new algorithm for stable assignments

A new algorithm for solving stable assignment problems is presented. By using a weight table, the overall situation of a problem is considered. The algorithm can deal with not only the one-to-one type assignments but also multi-function assignments for both equal and unequal sets. Some examples are given to illustrate the algorithm.     

A new form of stable adaptive observer

A new form of adaptive observer which is guaranteed to be globally stable via hyperstability theories is presented. Its main advantage over existing designs is that asymptotic unbiased estimates are obtained in the presence of output noise. Experimental results are given to demonstrate the performance of this observer.     

A new approach to the symbolic factorization of multivariate polynomials

A heuristic factorization scheme that uses learning and other heuristic programming techniques to improve the efficiency of determining the symbolic factorization of multivariate polynomials with integer coefficients and an arbitrary number of variables and terms is described. The learning program, POLYFACT, in which the factorization scheme is implemented is also described. POLYFACT uses learning through the dynamic construction and manipulation of first-order predicate calculus heuristics to reduce the amount of searching for the irreducible factors of a polynomial.Tables containing the results of factoring randomly generated multivariate polynomials are presented: (1) to demonstrate that learning does improve considerably the efficiency of factoring polynomials, and (2) to show that POLYFACT does learn from previous experience.The factorization times of polynomials factored by both the scheme implemented in POLYFACT and Wang's implementation of Berlekamp's algorithm are given. The two algorithms are compared, and two situations where POLYFACT'S algorithm can be used to improve the efficiency of Wang's algorithm are discussed.     

A parameterization of minimal plants

We present the input-normal Riccati parameterization, a generalization of Kabamba's input-normal form, which allows the continuous parameterization of the set of minimal linear systems of a given order that have distinct singular values; the input-normal Riccati form has no requirement that the underlying system be stable. We also present formulas for the use of the input-normal Riccati parameterization for the synthesis of systems via gradient methods     

A new result on the p-irreducibility of binding polynomials

Binding polynomials and p-irreducibility are important concepts in biochemistry. In this paper, we study the p-irreducibility of binding polynomials with degree four and present, for the first time, an explicit criterion for p-irreducibility, which is expressed by polynomials in the coefficients of a given binding polynomial.     

A parameterization of minimal stochastic realizations

Given a rational spectral density and a minimal spectral factor, a parameterization of all minimal spectral factors is derived. A minimal realization of each minimal spectral factor is also provided. This parameterization is shown to hold under the condition of nonintersection between the spectra of two matrices. This condition, first introduced in previous work by Ferrante et al. (1993), is proved here to be also necessary. Finally these results are applied to the stochastic realization problem     

A new stable tracking control scheme for robotic manipulators

The paper considers tracking control of robots in joint space. A new control algorithm is proposed based on the well known computed torque method and a compensating controller. The compensating controller is realized by using a switch type structure and an RBF neural network. It is shown that stability of the closed loop system and better tracking performance can be established based on Lyapunov theory. Simulation results are also provided to support our analysis.     

The root-locus method of synthesis of stable polynomials by adjustment of all coefficients

The problem of synthesis of an asymptotically stable polynomial on the basis of the initial unstable polynomial is solved. For the purpose of its solution, the notion of the extended (complete) root locus of a polynomial is introduced, which enables one to observe the dynamics of all its coefficients simultaneously, to isolate the root-locus trajectories, along which values of each coefficient change, to establish their interrelation, which provides a way of using these trajectories as “conductors” for the movement of roots in the desired domains. Values of the coefficients that ensure the stability of a polynomial are chosen from the stability intervals found on the stated trajectories as the nearest values to the values of appropriate coefficients of the unstable polynomial or by any other criterion, for example, the criterion of provision of the required stability reserve. The sphere of application of the root locus, which is conventionally used for synthesis of characteristic polynomials through the variation of only one parameter (coefficient) of the polynomial, is extended for the synthesis of polynomials by way of changing all coefficients and with many changing coefficients. Examples of application of the developed algorithm are considered for the synthesis of stable polynomials with constant and interval coefficients.     

A convex parameterization of robustly stabilizing controllers

This paper treats synthesis of robust controllers for linear time-invariant systems. Uncertain real parameters are assumed to appear linearly in the closed loop characteristic polynomial. The main contribution is to give a convex parameterization of all controllers that simultaneously stabilize the system for all possible parameter combinations. With the new parameterization, certain robust performance problems can be stated in terms of quasi-convex optimization