Jürgen Garloff

Robustness analysis of polynomials with polynomial parameter dependency using Bernstein expansion

This paper considers the robust stability verification of polynomials with coefficients depending polynomially on parameters varying in given intervals. Two algorithms are presented, both rely on the expansion of a multivariate polynomial into Bernstein polynomials. The first one is an improvement of the so-called Bernstein algorithm and checks the Hurwitz determinant for positivity over the parameter set. The second one is based on the analysis of the value set of the family of polynomials and profits from the convex hull property of the Bernstein polynomials. Numerical results to real-world control problems are presented showing the efficiency of both algorithms.

Stability of polynomials under coefficient perturbation

Let the real polynomial a_{0}x^{n} + a_{1}x^{n-1} + ... + a_{n-1}x + a_{n} be stable and let the real numbers b_{k}, c_{k} \geq 0, 0 \leq k \leq n , be given. We present a simple determinant criterion for finding the largest t_{0} \geq 0 such that the polynomial \alpha_{0}x^{n} + \alpha_{1}x^{n-1}+ ... +\alpha_{n-1}x + \alpha_{n} is stable for all \alpha_{k} \in (a_{k} - b_{k}t_{0}, a_{k} + C_{k}t_{0}) \cup {a_{k}}, 0 \leq k \leq n . Several further observations allow us to reduce the computational cost considerably.

Convex combinations of stable polynomials

Abstract Sufficient conditions are given under which convex combinations of stable (complex and real) polynomials are stable.

Bounds for the eigenvalues of the solution of the discrete Riccati and Lyapunov equations and the continuous Lyapunov equation

We present some bounds for the eigenvalues and certain sums and products of the eigenvalues of the solution of the discrete Riccati and Lyapunov matrix equations and the continuous Lyapunov matrix equation. Nearly all of our bounds for the discrete Riccati equation are new. The bounds for the discrete and continuous Lyapunov equations give a completion of some known bounds for the extremal eigenvalues and the determinant and the trace of the solution of the respective equation.

Investigation of a subdivision based algorithm for solving systems of polynomial equations

A method for enclosing all solutions of a system of polynomial equations inside a given box is investigated. This method relies on the expansion of a multivariate polynomial into Bernstein polynomials and constitutes a domain-splitting approach. After a pruning step, a collection of subboxes remain which undergo an existence test provided by Mirandas theorem. In this paper, the complexity of this test is reduced from O(n!) to nearly O(n 2 ). Also, some observations on the e ects of preconditioning of the system and results on the application of Bernstein expansion to the mean value form are presented.

Lower bound functions for polynomials

Relaxation techniques for solving nonlinear systems and global optimisation problems require bounding from below the nonconvexities that occur in the constraints or in the objective function by affine or convex functions. In this paper we consider such lower bound functions in the case of problems involving multivariate polynomials. They are constructed by using Bernstein expansion. An error bound exhibiting quadratic convergence in the univariate case and some numerical examples are given.

Application of Bernstein Expansion to the Solution of Control Problems

We survey some recent applications of Bernstein expansion to robust stability, viz. checking robust Hurwitz and Schur stability of polynomials with polynomial parameter dependency by testing determinantal criteria and by inspection of the value set. Then we show how Bernstein expansion can be used to solve systems of strict polynomial inequalities.

Criteria for sign regularity of sets of matrices

Abstract We consider matrix intervals with respect to a certain “checkerboard” partial ordering. We show that every real matrix contained in a matrix interval is sign-regular if two special matrices taken from that matrix interval are sign-regular.

Boundary implications for stability properties: present status

Publisher Summary This chapter discusses the current status of boundary implications for stability properties and presents certain problems in the robust and adaptive control theory. In feedback control problems, it is sometimes necessary to design a robust compensator that stabilizes a family of plants. Using Kharitonovs result for real interval polynomials, it has been possible to obtain sufficiency conditions on the problems of simultaneous stabilization and simultaneous pole assignment on a family of single-input single-output plants by a single non-switching compensator. Each of these sufficiency conditions is expressed in terms of the Hurwitz stability of four polynomials. In the control of flexible large space structures such as solar power satellites and orbiting telescopes, the objective is to achieve robust control of a plant of high order with a controller of low order. The robust control is necessary because of elastic mode truncation and the presence of parameter uncertainty.

Interval Gaussian Elimination with Pivot Tightening

We present a method by which the breakdown of the interval Gaussian elimination caused by division of an interval containing zero can be avoided for some classes of matrices. These include the inverse nonnegative matrices, the totally nonnegative matrices, and the inverse