Schur-Cohn Matrix Research Articles

Modified Schur-Cohn Criterion for Stability of Delayed Systems

A modified Schur-Cohn criterion for time-delay linear time-invariant systems is derived. The classical Schur-Cohn criterion has two main drawbacks; namely, (i) the dimension of the Schur-Cohn matrix generates some round-off errors eventually resulting in a polynomial ofswith erroneous coefficients and (ii) imaginary roots are very hard to detect when numerical errors creep in. In contrast to the classical Schur-Cohn criterion an alternative approach is proposed in this paper which is based on the application of triangular matrices over a polynomial ring in a similar way as in the Jury test of stability for discrete systems. The advantages of the proposed approach are that it halves the dimension of the polynomial and it only requires seeking real roots, making this modified criterion comparable to the Rekasius substitution criterion.

Distributed stabilisation of spatially invariant systems: positive polynomial approach

The paper gives a computationally feasible characterisation of spatially distributed controllers stabilising a linear spatially invariant system, that is, a system described by linear partial differential equations with coefficients independent on time and location. With one spatial and one temporal variable such a system can be modelled by a 2-D transfer function. Stabilising distributed feedback controllers are then parametrised as a solution to the Diophantine equation ax + by = c for a given stable bi-variate polynomial c. The paper is built on the relationship between stability of a 2-D polynomial and positiveness of a related polynomial matrix on the unit circle. Such matrices are usually bilinear in the coefficients of the original polynomials. For low-order discrete-time systems it is shown that a linearising factorisation of the polynomial Schur-Cohn matrix exists. For higher order plants and/or controllers such factorisation is not possible as the solution set is non-convex and one has to resort to some relaxation. For continuous-time systems, an analogue factorisation of the polynomial Hermite-Fujiwara matrix is not known. However, for low-order systems and/or controller, positivity conditions on the closed-loop polynomial coefficients can be invoked. Then the computational framework of linear matrix inequalities can be used to describe the stability regions in the parameter space using a convex constraint.

Modified Sturm Sequence Property for Damped Systems

This technical note presents a method of checking the number of complex eigenvalues in some interested regions or the multiplicity of some complex eigenvalues for nonproportionally damped system. A Schur-Cohn matrix is constructed from the coeffi- cients of the characteristic polynomial for the damped system, and LDL T factorized using some standard numerical algorithms. By observing signs of the diagonal elements of the above diagonal matrix D, we can determine the number of complex eigenvalues in some interested regions or the multiplicity of some complex eigenvalues, which is very similar to the well-known Sturm sequence property for undamped systems. To verify the applicability of the proposed method, two numerical examples are considered.

Zero location of polynomials with respect to the unit-circle unhampered by nonessential singularities

A method to determine the distribution of the zeros of a polynomial with respect to the unit-circle, proposed by this author in the past, is revisited and refined. The revised procedure remains recursive and nonsingular for polynomials whose Schur-Cohn matrix is not singular. Other nonessential singularities that previously caused interruption of the recursion are assimilated into a more general regular form of the three-term recursion of symmetric polynomials that underlies the method. The new form of the procedure does not compromise the simplicity of the rules to extract the information on the distribution that are proved using a different and more direct proof, based on the evaluation of the Cauchy index along the unit-circle. The low count of operations of the original procedure (recognized as the least cost solution for the problem) is maintained and actually gets better by the elimination of nonessential singularities. The improved features make the revised procedure a better all-around unit-circle zero location method for any real or complex polynomial. Its wider range of regularity should also benefit a variety of related signal processing and algebraic problems including some that were already affected by the original formulation.

Zero Location of Polynomials with Respect to the Unit-Circle without Nonessential Singularities

The author's method to determine the distribution of the zeros of a (real or complex) polynomials with respect to the unit-circle is revisited and refined. The three-term recursion of symmetric polynomials that the method uses is generalized such that it remains regular for all polynomials whose Schur-Cohn matrix is not singular. The refined version eliminates nonessential singularities without compromising the low computational cost and the simplicity of the zero location rules of the original procedure.

Reflections on Schur-Cohn matrices and Jury-Marden tables and classification of related unit circle zero location criteria

We use the so-calledreflection coefficients (RCs) to examine, review, and classify the Schur-Cohn and Marden-Jury (SCMJ) class of tests for determining the zero location of a discrete-time system polynomial with respect to the unit circle. These parameters are taken as a platform to propose a partition of the SCMJ class into four useful types of schemes. The four types differ in the sequence of polynomials (the table) they associate with the tested polynomials by scaling factors: (A) a sequence of monic polynomials, (B) a sequence of least arithmetic operations, (C) a sequence that produces the principal minors of the Schur-Cohn matrix, and (D) a sequence that avoids division arithmetic. A direct derivation of a zero location rule in terms of the RCs is first provided and then used to track a proper zero location rule in terms of the leading coefficients of the polynomials of the B, C, and D scheme prototypes. We review many of the published stability tests in the SCMJ class and show that each can be sorted into one of these four types. This process is instrumental in extending some of the tests from stability conditions to zero location, from real to complex polynomial, in providing a proof of tests stated without a proof, or in correcting some inaccuracies. Another interesting outcome of the current approach is that a byproduct of developing a zero location rule for the Type C test is one more proof for the relation between the zero location of a polynomial and the inertia of its Schur-Cohn matrix.

The observability gramian of the observer form and the Schur-Cohn matrix

The observer form is a standard minimal realization of the transfer-function matrix G based on an irreducible left matrix-fraction description G = U -1V . It is shown that (in the case of stable G ) there is a simple closed-form expression for the observability gramian of this realization in term of the polynomial matrix U and its ‘reflection’: in fact, gramian is the inverse of the schur-cohn of U .

Factorization and the Schur-Cohn matrix of a matrix polynomial

Factorization and the Schur-Cohn matrix of a matrix polynomial

A further simplification to Jury's stability test

Consideration is given to R.H. Raible's table (1974), which is a simplified version of E.I. Jury's discrete stability test. It is explicitly shown that Raible's table is none other than the classical Schur algorithm performed on a certain scattering function. This directly leads to an interpretation of Raible's table as a continued fraction expansion. The connection between Raible's table and the fast triangular factorization of the Schur-Cohn matrix is given.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Modified stability table for 2-D digital filters

A modified stability table is obtained for checking the stability of 2-D digital filters. In this table, the appropriate entries of the first column are equivalent to the appropriate minors of the Hermitian Schur-Cohn matrix. Thus the positivity of the latter matrix requires the positivity checking of the (n-1) appropriate entries for one point, and the last entry, which is equivalent to the Schur-Cohn determinant, is to be checked for positivity for all mod z mod =1. This test is simpler than several others appearing in the literature.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Lyapunov, Bézout, and Hankel

Given a polynomial f of degree n , we denote by C its companion matrix, and by S the truncated shift operator of order n . We consider Lyapunov-type equations of the form X − SXC => W and X − CXS = W . We derive some properties of these equations which make it possible to characterize Bezoutian matrices as solutions of the first equation with suitable right-hand sides W (similarly for Hankel and the second equation) and to write down explicit expressions for these solutions. This yields explicit factorization formulae for polynomials in C , for the Schur-Cohn matrix, and for matrices satisfying certain intertwining relations, as well as for Bezoutian matrices.

The discrete Lyapunov equation in controllable canonical form

A new method of treating Lyapunov equations is proposed. If D and D^{-} are two self-adjoint one-dimensional matrices related in a certain way, then X-C^{\ast}XC=D if and only if X^{-1}-CX^{-1} C^{\ast}=D^{-} . As an application, a generalization of a recent result is given. If f is the vector f=(0,0...,0,1)^{T} , then the solution of X-CXC^{\ast} = ff^{\ast} is shown to be inverse of the Schur-Cohn matrix.

On the solution of the discrete-time Lyapunov matrix equation in controllable canonical form

The solution to the discrete-time Lyapunov matrix equation in controllable canonical form is shown to be the inverse of the Schur-Cohn matrix. A simple constructive procedure of Berkhout, based on the backwards Levinson algorithm is discussed and an application of the result in stochastic control is mentioned.

On the reduced Hermite and reduced Schur-Cohn matrix relationships†

In this paper, a transformation which connects the matrices of the Hermite and Sehur—Cohn criteria– for root distribution is obtained. Based on this transformation, the relationship between the reduced Hermite and reduced Schur–Cohn criteria is also obtained. Furthermore, the transformation which connects the Lienard—Chipart stability criterion and the simplified determinantal stability criterion for the unit circle is derived. Finally, the connection between the inner form of the Hermite and Schur7ndash;Cohn criteria is established. The importance of the various transformations lies in the fact that their existence implies that a proof of one left7ndash;half–plane stability criterion immediately yields a proof of a corresponding unit circle criterion, and conversely. The various matrix transformations are obtained from the matrix transformation linking vectors of the coefficients of two polynomials whose respective root distributions are studied and which are related by a bilinear transformation of the underlying variables. The symmetry and skew symmetry of the matrix transformation are utilized to obtain the various transformations derived in this paper

Stability test for two-dimensional recursive filters

For deciding the stability of a two-dimensional filter, it is of interest to determine whether or not a prescribed polynomial in the variables z 1 and z 2 is nonzero in the region |z_{1}| \leq 1 \cap |z_{2}| \leq 1 . A new procedure for testing for this property is given, which does not involve the use of bilinear tranformations. Key parts of the test involve the construction of a Schur-Cohn matrix and the checking for positivity on the unit circle of a set of self-inversive polynomials.