E. Jury

Output feedback stabilization and related problems-solution via decision methods

Given an unstable finite-dimensional linear system, the output feedback problem is, first, to decide whether it is possible by memoryless linear feedback of the output to stabilize the system, and, second, to determine a stabilizing feedback law if such exists. This paper shows how this and a number of other linear system theory problems can be simply reformulated so as to allow application of known algorithms for solution of the existence question, with the construction problem being solved by some extension of these known algorithms. The first part of the output feedback problem is solvable with a finite number of rational operations, and the second with a finite number of polynomial factorizations. Other areas of application of the algorithm are described: Stability and positivity tests, low-order observer and controller design, and problems related to output feedback. Alternative computational procedures more or less divorced from the known algorithms are also proposed.

On the absolute stability of nonlinear sample-data systems

A general form of area inequality is used to develop a frequency domain criterion for absolute stability of single-nonlinear sampled-data (SNSD) systems. By reductions of the general form to specific area inequalities, it is suggested that all known and some new frequency domain criteria applicable to different subclasses of SNSD systems may be derived as special cases. One of the latter involving q is investigated to determine its significance and general applicability to SNSD systems. An example of a SNSD system with dead zone is used to illustrate some of the results presented.

A stability analysis of two-dimensional nonlinear digital state-space filters

Two different methods for a stability analysis of 2-D nonlinear discrete state-space systems under zero input conditions are provided. The first method reduces the task of testing a 2-D discrete nonlinear or shift-varying system to a single 2-D linear stability test of a system matrix with nonnegative system matrix entries. The second method is based on a number of norm tests for products of extreme matrices, and can be considered the 2-D counterpart of the method for 1-D systems described by K.T. Erickson and A.N. Michel (1985). Both of the introduced methods are based on the sector description of the nonlinearity and can be used to analyze digital filter stability under finite-word-length effects. >

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.

Stability analysis of multidimensional (m-D) direct realization digital filters under the influence of nonlinearities

A stability test of m-D direct realization digital filters with nonlinearities such as quantization, overflow, saturation, etc., is presented. The method presented reduces a given nonlinear m-D system to a nonlinear 1-D system such that the stability of the latter is sufficient for the stability of the former. To check stability of the nonlinear 1-D system, some new results on 1-D recursive systems with positive coefficients are applied. The case of multiple nonlinearities in multidimensions is also addressed. Examples are given illustrating the application of the method. >

Positivity and nonnegativity conditions of a quartic equation and related problems

In this paper explicit conditions for positivity (no real roots), nonnegativity on positive real axis (no positive real roots with odd multiplicity), and stability aperiodicity (all roots are real, and, negative and simple) of a quartic (or biquadratic) equation are given. The derived conditions from the known solution of the quartic equation are not only complete, but simpler than those derived from Sturm, extended Hurwitz, inners, and Hankel methods. Because of Abels Theorem (no explicit solution in terms of the roots of an equation higher than quartic exists), similar simplification for higher degree polynomial equations may not be possible. Furthermore, explicit conditions for positivity and nonnegativity of equations of higher degree than four are extremely difficult to obtain and may not be possible. The results of the paper will hopefully shed some light on a century old problem and thus enhance the engineering application of the derived condition to higher order systems.

Schwarz matrix properties for continuous and discrete time systems

New properties of the Schwarz matrix associated with a prescribed polynomial are derived. A matrix analogous to the Schuarz matrix is described which is useful in studying the root distribution of a prescribed polynomial relative to the unit circle.

Some new results on discrete system stability

New properties of the denominator polynomial of a stable discrete system transfer function are introduced along with two forms of stability test for such systems.

Estimation of the margin of stability for linear continuous and discrete systems

In this paper estimates of the abscissa of stability for stable linear continuous systems are determined based on the value of the integral of the time-weighted quadratic performance index. Similar results are obtained for linear discrete systems using the quadratic sum and the time-weighted quadratic sum. It is shown how the use of the Schwarz matrix may simplify the results.

On the theory of discrete systems

The authors in this paper have joined their efforts in writing a survey paper on the theory of discrete systems. In this combined effort, several important topics are covered: description of pulsed systems, analysis of pulsed systems, synthesis of pulsed systems, analysis of non-linear pulsed systems, synthesis of optimal non-linear systems, random sampling, adaptive pulsed systems, digital filtering, sequential machines, and biocontrol systems. For each of the mentioned topics, sections giving a brief evaluation of the work done are presented, combined with references to research performed both in the East and the West. Because of the limited space available, the authors have omitted many other references which are pertinent to this survey. However, the references which are included are familiar to the authors and can be used for further study and research. The aim of this paper is to present to control scientists the wide spectrum of the research done in this important area within the last two decades. It is hoped that the objectives of the authors will be fulfilled.