Zettler M. and Garlo J. (1998), \Robustness analysis of polynomials with polynomial parameter dependency using Bernstein expansion," IEEE Trans. Automat. Contr. Vol. 43, 425-431.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... given a family of characteristic functions p(s,q) associated with an uncertain system, and a set D in the complex plane, provide computationally tractable techniques for determining the D stability of p(s,q) i.e. for checking whether the zeros of the functions in p(s,q) remain with the set D [1 12,1517, 19 26]. For a more detailed description of research in this broad field, see the excellent books [1,6,8] and the references therein. An interesting research area within this framework is to find the parametric stability margin, measured in a suitably defined norm, since it gives the maximal ....

M. Zettler and J. Garloff, "Robustness analysis of polynomials with polynomial parameter dependency using bernstein expansion," IEEE Trans. on Automatic Control, Vol. 43, No. 3, pp. 425-431, 1998.

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Zettler M. and Garlo J. (1998), \Robustness analysis of polynomials with polynomial parameter dependency using Bernstein expansion," IEEE Trans. Automat. Contr. Vol. 43, 425-431.

....are ane lower bound functions. In [5] we concentrate on such bound functions for multivariate polynomials. These bound functions are constructed from the coecients of the expansion of the given polynomial into Bernstein polynomials. For properties of Bernstein polynomials the reader is referred to [2, 4, 10, 13]. In the univariate case the computational work for constructing such bound functions is negligible, but in the multivariate case a linear programming problem has to be solved. In the branch and bound framework it may happen that one has to solve subproblems on numerous subboxes of the starting ....

Zettler M. and Garlo J. (1998), \Robustness analysis of polynomials with polynomial parameter dependency using Bernstein expansion," IEEE Trans. Automat. Contr. Vol. 43, 425-431. 10

....satisfactory results for polynomial terms of higher degree. Moreover, several properties of these bound functions are discussed. For properties of Bernstein polynomials the reader is referred to Cargo and Shisha [5] Farin [7] Garlo [11] Garlo , Jansson and Smith [12] and Zettler and Garlo [30]. By using Bernstein coecients, bounds for the range of a multivariate polynomial over a box can be computed. It was shown by Stahl [28] that in the univariate case these bounds are often tighter than bounds which are obtained by applying interval computation techniques (cf. Neumaier [21] ....

M. Zettler and J. Garlo. Robustness analysis of polynomials with polynomial parameter dependency using Bernstein expansion. IEEE Trans. Automat. Contr., 43:425-431, 1998. 17

.... on [a; a ] respectively [a ; a] can be found on the main diagonal of the resulting triangular scheme, respectively in its last column, however, in reverse order, i.e. a;a ] i) a ;a] l Gammai) i = 0; l: 7) The generalisation to the multivariate case is given in [10]. 3 Convex concave extensions Convex concave extensions were introduced and investigated in [5] A convex concave extension of a function f : S R, S R , is a mapping [f ; f ] which provides for each nonempty box X S a convex function f : X R (lower bound function) and a ....

Zettler, M., Garloff, J.: Robustness analysis of polynomials with polynomial parameter dependency using Bernstein expansion. IEEE Trans. Automat. Contr. 43, 425--431 (1998). Jurgen Garloff University of Applied Sciences / FH Konstanz

....lower bound functions for polynomials can be computed by using Bernstein coefficients. Moreover, several properties of these bound functions are discussed. For properties of Bernstein polynomials the reader is referred to Cargo and Shisha [5] Farin [7] and Garloff [11] and Zettler and Garloff [29]. By using Bernstein coefficients, bounds for the range of a multivariate polynomial over a box can be computed. It was shown by Stahl [27] that in the univariate case these bounds are often tighter than bounds which are obtained by applying interval computation techniques (cf. Neumaier [20] ....

M. Zettler and J. Garloff. Robustness analysis of polynomials with polynomial parameter dependency using Bernstein expansion. IEEE Trans. Automat. Contr., 43:425--431, 1998.

....test from O(n ) to nearly O(n 2 ) where n is the dimension of the problem. Also, a closer look of the effects of preconditioning is presented. The organisation of this paper is as follows: In the next section we briefly recall the Bernstein expansion from Garloff [9] and Zettler and Garloff [10], wherein the relevant references can be found. The solution method is presented in Section 3. The reduction of the computational complexity as well as preconditioning are discussed in Section 4. In the process of writing this paper we were also investigating a new Bernstein form which is the ....

.... values in this computation, since for k = 0; n r the following relation holds (Garloff [9] b i 1 ; n r Gammak; i l (D 1 ) b (k) i 1 ; n r ; i l (D) A sweep needs O(n l 1 ) additions and multiplications, where n = maxfn i : i = 1; lg; cf. Zettler and Garloff [10]. Note that by the sweep procedure the explicit transformation of the subboxes generated by the sweeps back to U is avoided. Fig. 1 illustrates the sweeping process for l = 2 and r = 1. B(D) B(D 0 ) B(D 1 ) 6 1=2 0,1 0,0 x 2 x 1 1,0 1,1 0,1 0,0 1 2,0 1 2,1 1,1 1,0 Fig. 1. Two new ....

M. Zettler, J. Garloff, Robustness analysis of polynomials with polynomial parameter dependency using Bernstein expansion, IEEE Trans. Automat. Contr. 43 (1998) 425--431.

....the efficiency of both algorithms we present in Ch. 5 numerical results to real world problems like that of the Fiat Dedra engine studied in [6] Brief conclusions and directions for future research are given in Ch. 6. The results of this paper are presented in greater detail in the report [37] which is available upon request. We note that the approach the first algorithm is based on can be applied to other stability regions as well as to matrix stability using the determinantal criteria listed in [30] e.g. Ch. 17 in [6] however often at the expense of an increase of dimensionality. ....

.... The new algorithm explores the value set of the family of polynomials (1) P( Omega Gamma = fp(j ; q) 2 Omega q 2 Qg; Omega ae R: Having found a stable member of the family it suffices by the continuous dependency of the zeros of a polynomial from its coefficients to test 0 = 2 P(R) In [37] we discuss how one can find a tight compact interval Omega ae [0; 1) such that from 0 = 2 P( Omega Gamma we can conclude the robust stability of the family (1) We split the polynomial p(j ; q) into its even and odd parts p(j ; q) p e ( 2 ; q) j p o ( 2 ; q) 17) where p e ( 2 ....

M. Zettler and J. Garloff, "Robustness analysis of polynomials with polynomial parameter dependency using Bernstein expansion", Konstanz: Fachhochschule Konstanz, Fachbereich Informatik, Tech. Rep. No. 9601, 1996. 10

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Zettler, M. and Garloff, J.: 1997, `Robustness Analysis of Polynomials with Polynomial Parameter Dependency Using Bernstein Expansion', to appear in IEEE Trans. Automatic Control, Nov. 1997 veroeff.tex; ; ; p.13

.... was demonstrated in [1] The number of operations required by quantifier elimination methods is still doubly exponential in the number of variables, so that only relatively small problems can actually be solved, whereas Bernstein 1 expansion has been applied to larger robust stability problems [2, 3, 4]. However, it should be noted that in contrast to symbolic methods Bernstein expansion requires a priori bounds on the design parameter range. This is not a hard restriction since the designer often can estimate the interesting parameter range. We mention a third approach, the probabilistic ....

....I 2 S 0 : Formula (7) follows immediately from (6) Property (8) relies on two fundamental 6 properties of the Bernstein polynomials, viz. their nonnegativity on the unit box U and the fact that they form a partition of unity. 3. 2 Sweep Procedure In this subsection we follow the exposition in [4]. We define a sweep in rth direction (1 r l) similarly to de Casteljau s algorithm in Computer Aided Geometric Design, e.g. 28] as recursively applied linear interpolation. Let D be any subbox of U generated by sweep operations (at the beginning we have D = U , then subsequently D is obtained ....

ZETTLER, M., and GARLOFF, J.: 'Robustness analysis of polynomials with polynomial parameter dependency using Bernstein expansion', IEEE Trans. Automat. Contr., 1998, 43, (3), pp. 425-431 20

....i 1 ; n r Gammak; i l (D 1 ) b (k) i 1 ; n r ; i l (D) By analogy with Computer Aided Geometric Design, we call the arrays of Bernstein coefficients B(D 0 ) and B(D 1 ) patches. A sweep needs O(n l 1 ) additions and multiplications, where n = maxfn i : i = 1; lg; cf. [21]. Note that by the sweep procedure the explicit transformation of the subboxes generated by the sweeps back to U is avoided. Fig. 1 illustrates the sweeping process for l = 2 and r = 1. B(D) B(D 0 ) B(D 1 ) 6 1=2 0,1 0,0 x 2 x 1 1,0 1,1 0,1 0,0 1 2,0 1 2,1 1,1 1,0 Figure 1. Two new ....

....B(D) B(D 0 ) B(D 1 ) 6 1=2 0,1 0,0 x 2 x 1 1,0 1,1 0,1 0,0 1 2,0 1 2,1 1,1 1,0 Figure 1. Two new patches are obtained by a sweep in the first direction. 2.3. Selection Procedures The sweep direction remains to be chosen in the computation of the Bernstein coefficients. Our rule, cf. [21], is based on an upper bound associated with the first partial derivative of a polynomial in Bernstein form and takes advantage of the easy calculation of the partial derivatives of a polynomial in this form, see, e.g. 22] In an attempt to split the box perpendicular to the direction of maximum ....

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M. Zettler and J. Garloff, "Robustness analysis of polynomials with polynomial parameter dependency using Bernstein expansion," IEEE Trans. Automat. Contr., vol. 43, 1998, pp. 425-431.

....in degree it is assumed throughout this paper that a 0 (q) 6= 0 for all q 2 Q. Unfortunately, most of the methods known from literature can only treat problems with polynomial dependency with only a few parameters and or polynomials of lower degree. For references the reader is referred to (Zettler and Garloff, 1997); more recent references include (Malan et al. 1996) and (Vehi et al. 1996) Zettler and Garloff, 1997) presented a new algorithm, the socalled Convex Hull Bernstein Algorithm, which can be used to solve larger robust stability problems. This algorithm is based on the expansion of the even and ....

....of the methods known from literature can only treat problems with polynomial dependency with only a few parameters and or polynomials of lower degree. For references the reader is referred to (Zettler and Garloff, 1997) more recent references include (Malan et al. 1996) and (Vehi et al. 1996) (Zettler and Garloff, 1997) presented a new algorithm, the socalled Convex Hull Bernstein Algorithm, which can be used to solve larger robust stability problems. This algorithm is based on the expansion of the even and odd parts of the family (1) into Bernstein polynomials and on the Zero ExclusionTheorem, e.g. Ackermann, ....

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Zettler, M. and J. Garloff (1997). Robustness analysis of polynomials with polynomial parameter dependency using Bernstein expansion. Fachhochschule Konstanz, Fachbereich Informatik, Techn. Rep. No. 9601, to appear in: IEEE Trans. Automat. Contr., Nov. 97.

....Bernstein expansion has proved to be a well established and important tool for solving robust stability problems such as checking a polynomial with polynomial parameter dependency for stability. Stability regions covered include the open left half of the complex plane (Hurwitz stability) [1], 2] and sectors centered around the negative real axis with vertices at the origin (damping) 2] Also, exponentional parameter dependency can be treated [2] A comparison of Bernstein expansion with methods based on interval analysis can be found in [3] In this paper the stability region is ....

....called the Zero Exclusion Principle, cf. p. 114 in [5] For simplicity, we concentrate here on real coefficient functions a k (q) in (2) so that we can restrict the verification of (3) to 2 [0; schur2.tex; p.2 3 The organization of our paper is as follows: In Sect. 2 we recall from [1] the Bernstein transformation of a polynomial and explain the sweep procedure which is fundamental for our algorithm. In Sect. 3 we use the decomposition of a polynomial into its symmetric and antisymmetric parts to recast the test for common zeros of both polynomials on the upper half of the unit ....

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M. Zettler and J. Garloff, "Robustness analysis of polynomials with polynomial parameter dependency using Bernstein expansion," IEEE Trans. Automat. Contr., vol. 43, 1998, pp. 425-431.

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