ALGEBRAIC CONDITION FOR DECOMPOSITION OF LARGE–SCALE LINEAR DYNAMIC SYSTEMS

Abstract

evelopment. In its contemporary formulation, the theory grows directly from advances in understanding complex and nonlinear systems in physics and mathematics, but it also follows a long and rich tradition of systems thinking in biology and psychology. The term dynamic system, in its most generic form, means systems of elements that change over time. The more technical use, dynamical systems, refers to a class of mathematical equations that describe time-based systems with particular properties. These branches of mathematics include control theory, homotopy theory, and matroid theory. As a complement to these varied perspectives, the approach of this paper is the study of algebraic Conditions for decomposition of large–scale linear dynamic systems.There are many articles concerning decompositions of polynomials (Alagar and Thanh, 1985; Bartoni and Zippel, 1985; Borodin et al., 1985; Coulter et al., 2001; 1998; Gathen, 1990; Giesbrecht and May, 2005; Kozenet al., [1, 2, 3, 4 & 5]1989; Kozen and Landau, 1989; Watt, 2008).

These methods deal with polynomials in numerical forms. The main idea is to find a decomposition f(x) = g[h(x)]. For example, let f(x) =  +6 + +9 + 3x− 5. It is possible to decompose this polynomial, where y = h(x) = + 3x and g(x) = + y − 5. For this type of decomposition, if possible, various algorithms are presented. In this article, a general method for decomposition of polynomials in the symbolic form f(x) = , ai∈is presented. Using computer algebra systems, it is possible to decompose polynomials of very high degrees.

Decomposition of large-scale linear systems into smaller dynamic systems is always possible if the eigenvalues of the state matrices which characterize the systems are known (Górecki and Popek, 1987). On the other hand, it is well known that, in general, algebraic equations of the degree n ≥ 5 cannot be solved in radicals when their coefficients belong to the rational field; see the theorem of Ruffini and Abel.

evelopment. In its contemporary formulation, the theory grows directly from advances in understanding complex and nonlinear systems in physics and mathematics, but it also follows a long and rich tradition of systems thinking in biology and psychology. The term dynamic system, in its most generic form, means systems of elements that change over time. The more technical use, dynamical systems, refers to a class of mathematical equations that describe time-based systems with particular properties. These branches of mathematics include control theory, homotopy theory, and matroid theory. As a complement to these varied perspectives, the approach of this paper is the study of algebraic Conditions for decomposition of large–scale linear dynamic systems.There are many articles concerning decompositions of polynomials (Alagar and Thanh, 1985; Bartoni and Zippel, 1985; Borodin et al., 1985; Coulter et al., 2001; 1998; Gathen, 1990; Giesbrecht and May, 2005; Kozenet al., [1, 2, 3, 4 & 5]1989; Kozen and Landau, 1989; Watt, 2008).

These methods deal with polynomials in numerical forms. The main idea is to find a decomposition f(x) = g[h(x)]. For example, let f(x) =  +6 + +9 + 3x− 5. It is possible to decompose this polynomial, where y = h(x) = + 3x and g(x) = + y − 5. For this type of decomposition, if possible, various algorithms are presented. In this article, a general method for decomposition of polynomials in the symbolic form f(x) = , ai∈is presented. Using computer algebra systems, it is possible to decompose polynomials of very high degrees.

Decomposition of large-scale linear systems into smaller dynamic systems is always possible if the eigenvalues of the state matrices which characterize the systems are known (Górecki and Popek, 1987). On the other hand, it is well known that, in general, algebraic equations of the degree n ≥ 5 cannot be solved in radicals when their coefficients belong to the rational field; see the theorem of Ruffini and Abel.