PolyX, Ltd. The Polynomial Toolbox for Matlab. Version 2.5. Prague, Czech Republic, 2001.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....system design. 1 Introduction Polynomial methods o#er an alternative to state space methods when designing computeraided control system design algorithms. Amongst other software products, the Fortran SLICOT library [N98] incorporates mostly state space methods, whereas the Polynomial Toolbox [P98] can be viewed as an alternative, or complementary product including various routines to deal with algebraic entities such as polynomials and polynomial matrices, based on the theory pioneered in [K79] Keeping with our endeavour of designing e#cient and reliable numerical routines for polynomial ....

....rotation, and process again column j 1 . 7 The above algorithm has been implemented in a documented, stand alone Matlab m file available at http: www.laas.fr #henrion software hqr This function will be included to the next release 3. 0 of the commercial Polynomial Toolbox for Matlab, see [P98]. 5 Examples 5.1 No solution to problem HQR With this basic example we want to show that matrices A and J may exist for which assumption (7) is not satisfied. Just take . It holds rank A = 1 yet rank A JA = 0. As a result, problem HQR cannot be solved. 5.2 Cancelling all remaining ....

PolyX, Ltd. The Polynomial Toolbox for Matlab. Prague, Czech Republic. Version 2.5 released in 2000. See www.polyx.cz

....on system signals are then translated onto the internal state. Finding the appropriate open loop control stategy then amounts to solving a polynomial interpolation problem with additional bound constraints. Using results on positive polynomials recently exposed in [N00] based on conic duality) [P00] (based on sum of squares decompositions and algebraic geometry results around Hilbert s 17th problem) and [L01] based on the theory of moments) it turns out that this constrained polynomial interpolation problem can be easily formulated as a convex optimization problem over linear matrix ....

....only if there exist positive semidefinite G i x = H(Q i , R i ) Fx = f. 9) 5 Numerical example The numerical example was treated with the help of Matlab 6.5 running under SunOS release 5.8 on a SunBlade 100 workstation. Operations on polynomials were performed with the Polynomial Toolbox 2. 5 [P00]. The LMI problems were solved with SeDuMi 1.05 [S99] with default tuning parameters. We consider the motorized base stage high precision positioning system described in [LN03] System polynomial matrices in description (1) are given by: A l (s) 25s 0 0 450s 1.1875 s ....

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PolyX, Ltd. The Polynomial Toolbox for Matlab. Prague, Czech Republic. Version 2.5 released in 2000. See www.polyx.cz

.... of a polytope of polynomial matrices (Section 3) and simultaneous stabilization of a family of linear systems (Section 4) The algorithms described in this paper will soon be added to the already existing and quite extensive set of numerical routines available in the Polynomial Toolbox for Matlab [15, 16, 14]. 2 Spectral Factorization Let B be a para Hermitian polynomial matrix positive definite on the stability boundary. Then there exists a polynomial matrix A such that A is stable and A A = B, where the star denotes the adjoint. Given B, the problem of finding spectral factor A is referred to as ....

....and LMIs on three key control problems: spectral factorization in both continuous time and discrete time, robust stability analysis and robust design via simultaneous stabilization. The algorithms 9 described in this paper will be part of the next version of the Polynomial Toolbox for Matlab [14]. ....

Polyx, Inc. "Polynomial Toolbox for Matlab", Version 2.0, 1999. See the home page www.polyx.com or www.polyx.cz.

....examples found in the literature, and a basic Matlab 1 SeDuMi [Sturm, 1999] implementation of the robust SPR design algorithm is given in the Appendix. A more sophisticated version of the design algorithm will be implemented into the next release 3. 0 of the Polynomial Toolbox for Matlab [Polyx Ltd. 2001]. 1 Matlab is a trademark of The MathWorks, Inc. 2 2 Main Result We go straight to the point and state our main result which provides a FIR solution of given degree to the robust SPR design problem. Theorem 1 Suppose we are given a stable polytope of discrete time polynomials whose monic ....

Polyx, Ltd. "The Polynomial Toolbox for Matlab", Version 2.5 released in 2001. Version 3.0 to be released soon. See www.polyx.cz.

....have shown that the problem of simultaneously stabilizing a set of scalar plants by a static output feedback can be solved very easily with standard tools of numerical linear algebra. The algorithm described in the paper will be implemented in the next release of the Polynomial Toolbox for Matlab [Polyx Ltd. 2001]. Due to the bilinearity of the Hermite matrix in the design parameters, there is unfortunately no direct extension of these results to dynamic output feedback controllers design. The lack of a polynomial matrix version of the Hermite criterion also prevents us from generalizing our results to ....

Polyx, Ltd. "The Polynomial Toolbox for Matlab", Version 2.5 released in 2001. Version 3.0 to be released soon. See www.polyx.cz.

....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 [6,7]and their implementation in a user friendly Matlab package called the Polynomial Toolbox [15]. In this regard, all the routines used in the algorithm are numerically stable in the sense that they are based on backward stable (orthogonal) transformations. Moreover, the routines takeadvantage of the special structure of the matrices so as to reduce the overall computational cost. In this ....

....as a polynomial approach alternative to the state space approach algorithms proposed in [18]and recently implemented in the Fortran library Slicot, see [21] The numerical routines described in this paper are implemented in the new release 3. 0 of the Polynomial Toolbox for Matlab [15]. 2 Infinite Zeros and Infinite Structure In this section, we recall standard facts on the structure of polynomial matrices at infinity. Consider the n Theta n non singular polynomial matrix A(s) A 0 sA 1 : s d;1 A d;1 s d A d and define as in [20, x4.2.1] the dual polynomial ....

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Polyx, Ltd. "The Polynomial Toolbox for Matlab", 2000. See the web page www.polyx.com or www.polyx.cz.

....criterion. This point is emphasized in the next section. 6 Numerical Experiments 6.1 Parameters We wrote a macro for solving the static output feedback problem with the algorithm described in section 5. The macro will be part of the next release of the Polynomial Toolbox for Matlab, see [Polyx, 2000]. The LMI problems are solved with the LMI Toolbox for Matlab [The Mathworks, 1998] There are several input parameters to the macro: ffl the relative accuracy ffl used by the LMI solver ffl the maximum allowed number of iterations k max of the algorithm: when k k max then we consider that the ....

Polyx, Inc., "Polynomial Toolbox for Matlab", see the web page www.polyx.cz, 2000.

....and polynomial approaches are known to be closely related. A Matlab 1 package has recently been developed to solve problems in control but also signal processing with the polynomial approach. As a part of the MathWorks partner program, the package is commercialized by the small Czech company PolyX [PolyX, 2000]. Macros in the toolbox systematically rely on well spread numerical procedures, the main emphasis being numerical reliability and efficiency. The objective of this paper is to give a brief overview of some algorithms used in the Polynomial Toolbox for Matlab. We first describe in section 2 how ....

PolyX, Inc., "Polynomial Toolbox for Matlab", see the web page www.polyx.cz, 2000.

....the way for the development of a whole bunch of new numerical tools for dealing with polynomial matrices and solving standard control problems. These new routines are to be inserted to the already existing and quite extensive set of numerical routines available in the Polynomial Toolbox for Matlab [41, 42, 40]. The outline of this survey paper is as follows. In Section 2 we first mention standard and easy control problems that can be approached using polynomial methods and convex LMI optimization. In Section 3, we enumerate a series of inherently difficult control problems that can be dealt with ....

Polyx, Inc. "Polynomial Toolbox for Matlab", Version 2.0, 1999. See the home page www.polyx.com or www.polyx.cz.

....Let ff = 0:625; fi = 0:8333333. We illustrate the polynomial method and the state space method to design the controller transfer function that minimizes the H 2 norm of the transfer function between v and z. The corresponding Matlab scripts based on macros of the Polynomial Toolbox 2. 0 see (Polyx, Ltd. 1998) are given below. Polynomial solution (Algorithm 3) Step 1 (A = AL = AR, B = BL = BR) BLs,AL] ss2lmf(F,eye(4) H2) BL = BLs G2; NL = BLs G1 AL J21; BRs,AR] ss2rmf(F,G2,eye(4) BR = H2 BRs; NR = H1 BRs J12 AR; Step 2 First spectral ....

Polyx, Ltd. (1998). The Polynomial Toolbox for Matlab.

....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 ....

....of the subspace of impulsive solutions of a linear differential equation (Section 5.1) and computation of finite zeros of a polynomial matrix (Sections 5.2 and 5.3) The numerical routines described in this paper are implemented in the new release 3. 0 of the Polynomial Toolbox for Matlab [18]. 2 Infinite Zeros and Infinite Structure In this section, we recall standard facts on the structure of polynomial matrices at infinity. Consider the n Theta n non singular polynomial matrix A(s) A 0 sA 1 : s d Gamma1 A d Gamma1 s d A d and define as in [21, x4.2.1] the dual ....

Polyx, Ltd. "The Polynomial Toolbox for Matlab", 2000. See the web page at www.polyx.com or www.polyx.cz.

....an example of such a procedure) ffl Numerical aspects of solving the LMI conditions must also be carefully checked. Intensive numerical experiments are currently performed and will result in several new macros to be implemented in the next release 3. 0 of the Polynomial Toolbox for Matlab, see [23]. ffl Another interesting extension could eventually be to consider robust synthesis problems and Diophantine equations over polynomial matrices [20] Acknowledgments Most of this work was done when the first and third authors were visiting the Department of Aerospace Engineering at the ....

Polyx, Inc., "Polynomial Toolbox for Matlab", Release 2.0.0, 1999. See the web page www.polyx.cz.

....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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Polyx, Ltd. "The Polynomial Toolbox for Matlab", 2000. See the web page at www.polyx.cz.

....of the Edge Theorem [2, 3] We randomly generated 100 polytopes of polynomials for different values of degree d and number of vertices N . Each vertex of each polytope was chosen as a random stable monic continuous time polynomial generated with macro prand of the Polynomial Toolbox for Matlab [14]. Of course, stability of each vertex does not imply robust stability of the whole polytope. In Table 1 we reported the number of polytopes that were found robustly stable ffl with quadratic stability condition (5) columns labeled QS) ffl with our new stability condition (1) columns labeled ....

Polyx, Inc., "Polynomial Toolbox for Matlab", Release 2.0.0, 1999. See the web page www.polyx.cz.

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PolyX, Ltd. The Polynomial Toolbox for Matlab. Version 2.5. Prague, Czech Republic, 2001.

No context found.

PolyX, Ltd. The Polynomial Toolbox for Matlab. Prague, Czech Republic. Version 2.5 released in 2001. See www.polyx.cz

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Polyx, Ltd. The Polynomial Toolbox for Matlab. Version 2.5 released in 2001. Version 3.0 to be released soon. See www.polyx.cz.

No context found.

Polyx, Ltd. The Polynomial Toolbox for Matlab. Version 2.5 released in 2000. See www.polyx.cz

No context found.

PolyX, Ltd. The Polynomial Toolbox for Matlab. Prague, Czech Republic. Version 2.5 released in 2000. See www.polyx.cz

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