M. Sebek, H. Kwakernaak, D. Henrion and S. Pejchov'a "New Algorithms for Polynomial Matrices used in the Polynomial Toolbox 2.0 for Matlab", Proceedings of the IEEE Conference on Decision and Control, pp. 3661--3668, Tampa, Florida, 1998.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the finite structure of a polynomial matrix. The algorithm is designed while keeping with our main impetus, which is the development of reliable numerical methods for dealing with polynomial matrices [8, 9] and their implementation in a user friendly Matlab package called the Polynomial Toolbox [20, 18]. In this regard, all the routines used in the algorithm are numerically stable. Moreover, the routines take advantage of the special structure of the matrices so as to reduced the overall computational cost. The outline of the paper is as follows. In Section 2 we recall the notions of zeros and ....

....R(s) 1 2s s 2 3s 2s 2 2s 3 Gammas Gamma s 2 1 Gamma 2s Gamma s 2 Gamma 2s 3 # : One can check that R(s) is unimodular, as expected. Left factor L(s) such that A(s) L(s)R(s) is readily retrieved with the numerically stable macro xab of the Polynomial Toolbox, see [20]. We obtain L(s) 1 Gamma 2s s 2 0 1 Gamma 2s s 2 1 Gamma 2s s 2 # which appears to be a triangular form of A(s) see [9] All the infinite zeros of A(s) are captured by R(s) All the finite zeros of A(s) are captured by L(s) Now if we apply the QZ decomposition to the ....

M. Sebek, H. Kwakernaak, D. Henrion and S. Pejchov'a "New Algorithms for Polynomial Matrices used in the Polynomial Toolbox 2.0 for Matlab", Proceedings of the IEEE Conference on Decision and Control, pp. 3661--3668, Tampa, Florida, 1998.

....trying to develop a series of numerical algorithms to deal directly with polynomial matrices and solve various control problems via an algebraic, or polynomial approach [14] The algorithms will be part of the new release 3. 0 of a comprehensive package called the Polynomial Toolbox for Matlab 1 [20, 17]. Prior to describing the new algorithms, we shortly review available state space methods for H1 norm computation. Before 1989, a popular way to compute the H1 norm was to search for the maximum singular value over frequency either graphically (using the so called singular value plot, or Bode ....

.... transform theory and the evaluation of a polynomial at a point lying on the complex unit circle can be employed [2, 13] As a result the efficient fast Fourier transform algorithm [2] can be utilized for computing the values of the involved transfer function over selected complex frequencies [20, 17] which makes the whole procedure much more effective. As pointed out in [3] obvious shortcomings are (a) determining the range and spacing of the frequencies to be checked, and (b) the large number of computations involved. Motivated by these issues, Boyd et al. 3] and Robel [19] independently ....

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M. Sebek, H. Kwakernaak, D. Henrion and S. Pejchov'a "New Algorithms for Polynomial Matrices used in the Polynomial Toolbox 2.0 for Matlab", Proceedings of the IEEE Conference on Decision and Control, pp. 3661--3668, Tampa, Florida, 1998.

....ScalSyl and MatSyl can be considered as numerically reliable alternatives to Algorithms ScalRed and MatRed, respectively. Matlab implementations of algorithms ScalRed, MatRed, ScalSyl and MatSyl are currently under development and will be included to the next version of the Polynomial Toolbox [20]. 7 Concluding Remarks After a generalization of classical polynomial matrix notions to the complex case, we proposed conditions of existence of a unique solution to scalar and matrix symmetric polynomial equations. Our proofs were constructive and resulted in a first family of resolution ....

....numerical unstability, we also developed alternative Sylvester matrix algorithms only relying upon well known and reliable tools from numerical linear algebra. It must be underlined that all these algorithms will soon become available in the next version of the Polynomial Toolbox for Matlab, see [20]. Possible directions for further research are now mentioned. The extension to the complex case must be generalized to other types of algorithms on polynomial matrices. In particular, interpolation techniques as presented in [6, 7] would undoubtedly benefit from such an extension. Indeed, it is ....

M. Sebek, D. Henrion and H. Kwakernaak, New Algorithms for Polynomial Matrices used in the Polynomial Toolbox for Matlab, to be presented at the IEEE Conference on Decision and Control, Tampa, Florida, 1998.

....for polynomial matrix factor extraction. Illustrating numerical examples are collected in Section 5 and we conclude the paper with some important remarks. The numerical routines described in this paper are implemented in the new release 2. 0 of the Polynomial Toolbox for Matlab 1 presented in [10]. 2 Preliminaries We describe two standard techniques of numerical algebra that are the backbone of the routines developed in this paper: reduction of a constant matrix into Column Echelon Form (CEF) by stable Householder transformations and extraction of a basis in Row Echelon Form (REF) for the ....

.... 5 Illustration Consider the polynomial matrix A(s) 2 6 4 1 s 0 2 s Gamma s 2 Gamma1 s Gamma 3s 2 s 3 Gamma2s 3s 2 Gamma s 3 Gamma2 Gamma s s 2 Gamma s 3 2s Gamma s 2 Gamma2s s 2 s 2 3 7 5 : With the help of the Polynomial Toolbox for Matlab [10], we found that its zeros are located at 0, Gamma1 and 2. 5.1 Algorithm GCV First, we illustrate the results of Section 3 and Algorithm GCV. We compute the GCVs of A(s) at s = 0. Using Algorithm CEF, Toeplitz matrix T 0 is transformed into CEF as follows T 0 = 2 6 4 1 Gamma1 0 0 0 0 2 Gamma2 ....

M. Sebek, D. Henrion and H. Kwakernaak "New Algorithms for Polynomial Matrices used in the Polynomial Toolbox 2.0 for Matlab", to be presented at the IEEE Conference on Decision and Control, Tampa, Florida, December 1998.

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