M. Mansour

Stability and the matrix Lyapunov equation for discrete 2-dimensional systems

The stability of two-dimensional, linear, discrete systems is examined using the 2-D matrix Lyapunov equation. While the existence of a positive definite solution pair to the 2-D Lyapunov equation is sufficient for stability, the paper proves that such existence is not necessary for stability, disproving a long-standing conjecture.

On robust Hurwitz polynomials

In this note, Kharitonovs theorem on robust Hurwitz polynomials is simplified for low-order polynomials. Specifically, for n = 3, 4 , and 5, the number of polynomials required to check robust stability is one, two, and three, respectively, instead of four. Furthermore, it is shown that for n \geq 6 , the number of polynomials for robust stability checking is necessarily four, thus further simplification is not possible. The same simplifications arise in robust Schur polynomials by using the bilinear transformation. Applications of these simplifications to two-dimensional polynomials as well as to robustness for single parameters are indicated.

Robust stability of interval matrices

Sufficient conditions for the Hurwitz and Schur stability of interval matrices are reviewed with some simplifications and some new results. The necessary and sufficient conditions for the Hurwitz and Schur stability of 2*2 matrices are determined. For general m*n interval matrices the pseudodivision method is applied for (2n-4)-dimensional faces for continuous systems and for (2n-2)-dimensional faces for discrete systems as well as for the corresponding lower dimensional faces. The method is applicable to low-dimensional matrices, e.g., n=3 or n=4. For higher dimensions numerical and computational problems arise.<<ETX>>

Robust Schur polynomial stability and Kharitonov's theorem

The paper considers robust stability properties for Schur polynomials of the form f(z) = ¿i=0 nan-izi By plotting coefficient variations in planes defined by variable pairs ai, an-i for each i and requiring in each such plane the region of obtained coefficients to be bounded by lines of slope 45°, 90° and 135°, we show that stability for all polynomials defined by comer points is necessary and sufficient for stability of all polynomials defined by any points in the region. Using this idea, one can construct several necessity and differing sufficiency conditions for the stability of polynomials where each ai can vary independently in an interval [ai, a- i]. As the sufficiency conditions become closer to necessity conditions the number of distinct polynomials for which stability has to be tested increases.

Frequency domain conditions for the robust stability of linear and nonlinear dynamical systems

The authors establish a generalized frequency-domain criterion for checking families of polynomials for root confinement in open subsets of the complex plane. The authors show how this criterion reduces to checking certain curves in the complex plane for zero confinement. Moreover, in some special cases, it further reduces to some complex functions with pointwise phase differences that are always less than pi in magnitude. Most of the currently available results on the robust stability of linear systems with parametric uncertainties can be viewed within the unifying frequency-domain framework presented. The framework encapsulates not just finite-dimensional systems, but any linear-time-invariant (LTI) system that can be characterized by transfer functions of a single variable. It also covers robust stability of LTI systems under passive feedback. >

Stability conditions for a class of delay differential systems

In this paper necessary and sufficient conditions of asymptotic stability independent of delay of a certain class of both the retarded and neutral types of delay differential systems are obtained. The stability conditions reduce to positivity of a quartic equation. Conditions for the latter in explicit form are given by the authors in an earlier publication. In addition, sufficient conditions for stability for a general order delay differential system independent of delay are also obtained. Such conditions are ascertained from the fact that a sufficient condition for any algebraic equation to have no positive real roots, is that all its coefficients be positive.

Strong Kharitonov theorem for discrete systems

Necessary and sufficient conditions for the stability of discrete systems with parameters in a certain domain of the parameter space are derived. The result is the analog of Kharitonovs strong theorem. Two methods are used to arrive at this result, one by projecting the roots of the symmetric and the asymmetric part of the polynomial f(z) on the (-1, +1) line. The resulting Chebyshev and Jacobi polynomials give certain intervals on the (-1, +1) line. In each interval it is necessary to check the four corner polynomials corresponding to Kharitonovs strong theorem for continuous systems. The number of intervals increases with the degree of the polynomial. The other method is the frequency-domain method where the intervals are easily obtained through the roots of trigonometric functions. A recursion formula is derived and the number of intervals is shown to be a sum of Euler functions.<<ETX>>

On the robustness of low-order Schur polynomials

Robust Schur stability conditions are obtained for polynomials of orders two to five. For n=2 and 3, the conditions obtained are related to stability of the corner points, while for n=4 and 5, the conditions are related to stability of corner and possible supplementary points. The number of points increases substantially as the polynomial order increases. The obtained results are of importance in the robust design of control systems. The difficulty in extending the approach to higher-order polynomials is discussed. Special cases for such extensions are mentioned. Further applications of the results obtained may be of use in the stability study of two-dimensional systems. Some examples for the application of the stability conditions are given and, in particular, the two counterexamples presented in the literature are discussed. >

The margin of stability of 2-D linear discrete systems

In this paper the margin of stability for a 2-D discrete system is considered. The definition of the margin of stability for the 1-D case is extended to the 2-D case in terms of analytic regions of rational functions in two variables. The relationship between the newly defined stability margin and the impulse response is established. A method to compute the stability margin is presented and illustrated with some examples.

Simplified sufficient conditions for the asymptotic stability of interval matrices

It will be shown that the sufficient conditions for the asymptotic stability of interval matrices proved in Mansour (1988) can be further simplified by considering only a part of the extreme matrices.