E.I. Jury

Stability of multidimensional scalar and matrix polynomials

A comprehensive study of multidimensional stability and related problems of scalar and matrix polynomials is presented in this survey paper. In particular, applications of this study to stability of multidimensional recursive digital and continuous filters, to synthesis of network with commensurate and noncommensurate transmission lines, and to numerical stability of stiff differential equations are enumerated. A novel approach to the multidimensional stability study is the classification of various regions of analyticity. Various computational tests for checking these regions are presented. These include the classical ones based on inners and symmetric matrix approach, table form, local positivity, Lyapunov test, the impulse response tests, the cepstral method and the graphical ones based on Nyquist-like tests. A thorough discussion and comparison of the computational complexities which arise in the various tests are included. A critical view of the progress made during the last two decades on multidimensional stability is presented in the conclusions. The latter also includes some research topics for future investigations. An extensive list of references constitutes a major part of this survey.

Stabilization of certain two-dimensional recursive digital filters

A possible extension of a well-known stabilization technique for one-dimensional recursive digital filters to the two-dimensional case was proposed by Shanks via a conjecture, stating that the planar least squares inverse of a two-dimensional filter polynomial is minimum phase and hence stable. In the present work, the conjecture has been verified first for a class of polynomials which are linear in one variable and quadratic in the other and then extended to a class of polynomials of higher degrees in the same variables. Though the conjecture is known to be false, in general, some conditions under which the conjecture is valid are explored.

The theory and applications of the inners

The notion of the inners of a matrix is fully discussed. The inners applications to control theory, stability theory, communication theory, circuit theory, network theory, digital filters, bioengineering, sparse matrix theory, quantum physics, and some topics in mathematics are enumerated and analyzed. It is shown that the inners concept offers a theoretical as well as computational unification for these applications. In addition, the historical background and motivation is presented for the inners approach. The importance of the inners notion to education, computation, and research in system theory is surveyed and evaluated. Future research problems using this concept are enumerated. Finally, this survey is documented by many past and recent references.

A note on the analytical absolute stability test

A simplified version of the conditions for the analytical absolute stability test is introduced. The absolute stability test requires that the polynomial

A note on new inner-matrix for stability

In this note a new inner-matrix for stability is introduced. The application of such a matrix for open left half-plane and inside the unit circle roots of complex polynomials is shown. Furthermore, the Hurwitz minors for left half-plane stability of real polynomial are given in terms of inners determinants of a resultant matrix. This connection is obtained by symmetrizing the Schur-Complement of the resultant (innerwise) matrix.

The inner formulation for the total square integral (SUM)

In this letter the value of the complex integrals I n Δ 1/2πj ∫ -j∞ j∞G(s)G(-s)ds and I n =1/2πj ∮ unit circle F(z)F(z-1)z-1dz which arise in obtaining the total square integral for the continuous case and the total square sum for discrete case are obtained in a unified approach based in the Inners. In both cases it is shown that I n is obtained as the ratio of two Inner determinants. The denominator determinants are related to the stability conditions. The computational algorithm recently obtained for the Inners determinants can be readily used for the evaluation of I n .

Remark on an equivalent relation in least square approximation

In this note it is shown that the least square inverse of the 2-D polynomial B(z<inf>1</inf>, z<inf>2</inf>) = b<inf>00</inf>+ b<inf>mt</inf>z<inf>1</inf>^mz<inf>2</inf>^t+ b<inf>ns</inf>z<inf>1</inf>ⁿz<inf>2</inf>^s+ b<inf>m+n, s+t</inf>ċ z<inf>1</inf>^m+nz<inf>2</inf>^s+twith (nt - ms) ≠ 0 is stable. This extends the Shanks conjecture for a larger class of 2-D polynomials.

Alternate coefficients constraints for stability test

Necessary constraints useful in the stability test of linear discrete-time systems are introduced. These constraints, which are obtained from the coefficients of the characteristic polynomial, are also used to obtain information on the root distribution within the unit disk.