The notion of normal positive electrical circuits is introduced and some their specific properties are investigated. New state matrices of positive linear systems and electrical circuits are proposed and their properties are analyzed. It is shown that positive electrical circuits with diagonal state matrices are normal for all values of resistances, inductances and capacitances
Słowa kluczowe: normal, positive, linear, system, electrical circuit.
W artykule zaproponowano pojęcie dodatniego obwodu elektrycznego oraz przeanalizowano specjalne własności dodatnich układów i obwodów elektrycznych. Wykazano, że dodatnie obwody elektryczne z diagonalnymi macierzami stanu są zawsze normalne dla wszystkich wartości rezystancji, indukcyjności i pojemności.
Keywords: układ normalny, dodatni, liniowy, obwód elektryczny
A dynamical system is called positive if its trajectory starting from any nonnegative initial state remains forever in the positive orthant for all nonnegative inputs. An overview of state of the art in positive systems theory is given in the monographs [2, 15]. Variety of models having positive behavior can be found in engineering, economics, social sciences, biology and medicine, etc. The notions of controllability and observability have been introduced by Kalman in [28, 29] and they are the basic concepts of the modern control theory [1, 7, 8, 11, 12, 20, 27, 31]. The controllability, reachability and observability of linear systems and electrical circuits have been investigated in [9, 10, 16, 18, 19, 30]. The asymptotic stability of positive standard and fractional linear systems has been addressed in [6, 15, 26]. Cholewicki has been the pioneer in Poland of the application of the theory of matrices in the analysis and synthesis of electrical circuits [3, 4, 5]. The specific duality and stability of positive electrical circuits have been analyzed in  and positive systems and electrical circuits with inverse state matrices in . The stability of continuous-time and discrete-time linear systems with inverse state matrices has been investigated in . The reduction of linear electrical circuits with complex eigenvalues to linear electrical circuits with real eigenvalues has been considered in . Standard and positive electrical circuits with zero transfer matrices have been investigated in  and the normal positive electrical circuits have been introduced in . In this paper the normal positive linear systems and electrical circuits are investigated. The paper is organized as follows. In section 2 some preliminaries concerning positive linear continuous-time systems are recalled. Some properties of the transfer matrices of positive linear systems are presented in section 3. Normal positive linear syste [...]
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