H. Kwakernaak, M. Sebek "Polynomial J-Spectral Factorization", IEEE Transactions on Automatic Control, Vol. 39, No. 2, pp. 315--328, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....matrices that are sign definite on the imaginary axis play an important role in systems and control theory. They can represent spectral density function of stochastic processes and show up in spectral factorization, a fundamental ingredient in the polynomial approach to H 2 and H1 optimization [36, 17]. In the same vein, it is shown in [16] that the set of continuous time (resp. discrete time) two sided polynomial matrices non negative on the real axis (resp. on the unit circle) is described by a convex LMI. Two sided polynomial matrices arise when studying linearization schemes for solving ....

H. Kwakernaak and M. Sebek "Polynomial J-Spectral Factorization", IEEE Transactions on Automatic Control, Vol. 39, No. 2, pp. 315--328, 1994.

.... explored in [1, 3] The links established between J spectral factors and their realizations in terms of solutions of Riccati equations, are all based on the canonical factorization theorem developed in 1979 [4] Recently some papers appeared devoted solely to the J spectral factorization problem [10, 11]. In [10] conjugation methods are explored avoiding the use of the difficult canonical factorization theorem. In [11] a review on polynomial methods for J spectral factorization is given. It is the aim of this paper to show that the canonical factorization theorem has an easy proof. The proof ....

....equations, are all based on the canonical factorization theorem developed in 1979 [4] Recently some papers appeared devoted solely to the J spectral factorization problem [10, 11] In [10] conjugation methods are explored avoiding the use of the difficult canonical factorization theorem. In [11] a review on polynomial methods for J spectral factorization is given. It is the aim of this paper to show that the canonical factorization theorem has an easy proof. The proof hinges on the so called equalizing vectors. Equalizing vectors turn out to be effective in determining the existence or ....

Kwakernaak H. and M. Sebek, "Polynomial J-spectral factorization," accepted for publication in IEEE Trans. Aut. Control .

....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 the spectral factorization problem. Spectral factorization of polynomial matrices is a fundamental ingredient in the polynomial approach to LQ, H 2 and H1 control [11, 4]. In [17, Prop. 5.1] it is shown that the whole set of spectral factors of a continuous time polynomial matrix positive definite on the real axis can be parametrized through an LMI in some symmetric matrix P . Once the LMI is solved for some P , the spectral factor is recovered by extracting the ....

H. Kwakernaak, M. Sebek "Polynomial J-Spectral Factorization", IEEE Transactions on Automatic Control, Vol. 39, No. 2, pp. 315--328, 1994.

.... and zeros computation are ffl interpolation of the determinant via FFT (see [10] for a detailed treatment) and computation of the roots of the determinant via Schur decomposition of a companion matrix [5]# ffl direct computation of the zeros via the QZ decomposition of a pencil companion matrix [12,5]. One of the main stumbling blocks when computing the determinant and the zeros is the presence of undesirable infinite zeros, namely ffl when interpolating the determinant of a polynomial matrix, infinite zeros will introduce artificial non zero leading coefficients in the determinant. An ....

....are chosen appropriately. 5.2 Extraction of Infinite Zeros Let A(s) 1 ; s 2 ; 2s 3 2s 4 3s ; 4s 2 s 3 ; 2s 4 2s 5 1 ; s ; s 2 s 3 1 ; s ; s 3 s 4 be a non singular polynomial matrix. Wewould liketo compute the zeros of A(s) Following the approachproposed in [12], we build the singular matrix pencil P (s) whose generalized eigenvalues are zeros of polynomial matrix A(s) These eigenvalues are obtained as ratios of diagonal elements of the matrices obtained via the QZ decomposition [5] of the above pencil. With the Matlab built in macro qz,we obtained the ....

H. Kwakernaak and M. Sebek, "Polynomial J-Spectral Factorization", IEEE Transactions on Automatic Control,Vol. 39, No. 2, pp. 315--328, 1994.

.... zeros computation are ffl interpolation of the determinant via FFT (see [12] for a detailed treatment) and computation of the roots of the determinant via Schur decomposition of a companion matrix [7] ffl direct computation of the zeros via the QZ decomposition of a pencil companion matrix [15, 7]. One of the main stumbling blocks when computing the zeros is the presence of indesirable infinite zeros, namely ffl when interpolating the determinant of a polynomial matrix, infinite zeros will introduce artificial non zero leading coefficients in the determinant. An ad hoc technique called ....

....Let A(s) 1 Gamma s 2 Gamma 2s 3 2s 4 3s Gamma 4s 2 s 3 Gamma 2s 4 2s 5 1 Gamma s Gamma s 2 s 3 1 Gamma s Gamma s 3 s 4 # 10 be a non singular polynomial matrix. We would like to compute the finite zeros of A(s) Following the approach proposed in [15], we build the singular matrix pencil P (s) 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 Gamma1 0 0 Gamma3 1 4 2 Gamma1 Gamma2 2 ....

H. Kwakernaak and M. Sebek, "Polynomial J-Spectral Factorization", IEEE Transactions on Automatic Control, Vol. 39, No. 2, pp. 315--328, 1994.

....interpretations of well known linear system theory results. Applications include the resolution of matrix polynomial equations like for instance the symmetric matrix polynomial equation [7] or the Diophantine equation, pole assignment by state or output feedback, J spectral factorization [13], or also H1 formulation of the optimal control problem. The above are just a few of the many examples of the strong presence of interpolation results in the system and control literature. It should thus seem surprising that, up to the authors knowledge, no results are available so far concerning ....

H. Kwakernaak and M. Sebek "Polynomial J-Spectral Factorization", IEEE Transactions on Automatic Control, Vol. 39, No. 2, pp. 315--328, 1994.

.... has been reported recently [20, 17] Second, use one of the two methods depicted above on the resulting scalar polynomial; ffl Compute the zeros of the polynomial matrix [30] Here also, a standard technique is to compute the eigenvalues of the pencil matrix associated to the polynomial matrix [25], using for example the numerically stable QZ decomposition [14] As pointed out in [8] some numerical difficulties may be expected with high multiplicity zeros. Considering rectangular polynomial matrices, the only method we are aware of was reported in [24] and makes indirect use of state space ....

H. Kwakernaak and M. Sebek "Polynomial J-Spectral Factorization", IEEE Transactions on Automatic Control, Vol. 39, No. 2, pp. 315--328, 1994.

....interpretations of well known linear system theory results. Applications include the resolution of matrix polynomial equations like for instance the symmetric matrix polynomial equation [6] or the Diophantine equation, pole assignment by state or output feedback, J spectral factorization [10], or also H1 formulation of the optimal control problem. The above are just a few of the many examples of the strong presence of interpolation results in the system and control literature. It should thus seem surprising that, up to the authors knowledge, no results are available so far concerning ....

H. Kwakernaak and M. Sebek "Polynomial J-Spectral Factorization", IEEE Transactions on Automatic Control, Vol. 39, No. 2, pp. 315--328, 1994.

....and the Smith form of a polynomial matrix. Some important remarks are in order. If the main assumption of this paper is relaxed, i.e. if the zeros to be extracted have not been computed at a previous stage, then an alternative approach to factorization can still be pursued. It is described in [9] in the special case of symmetric factor extraction for spectral factorization. Based on the Schur or QZ decomposition of a matrix pencil, this method computes the zeros of the polynomial matrix to be factorized together with invariant subspaces spanned by its characteristic vectors. By suitable ....

....polynomial matrix factor. However, when pursuing this later approach, the column degrees of the extracted factor must be known in advance. For instance, when used for polynomial J spectral factorization, this necessarily involves diagonal reduction of the input matrix as a preprocessing step [9]. This is in sharp contrast with the method described in this paper, where column reducedness of the input matrix is not required. Indeed, our factorization algorithm is designed to overcome the problem of degree predictability by simultaneously computing column degrees and factor coefficients. As ....

H. Kwakernaak and M. Sebek "Polynomial J-Spectral Factorization", IEEE Transactions on Automatic Control, Vol. 39, No. 2, pp. 315--328, 1994.

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