Dynamical modelling of linear discrete-continuous systems

Dynamic modelling of a non-conservative discrete-continuous system

“…Hence, relations (3.4) and (3.3) are satisfied in the above-mentioned cases and, according to the wellknown definition [4,5], the quasi-rational fractions Wj(~,) are physically possible. Moreover, the functions D(k), Qj(~,) are analytic on the imaginary axis and in the right half of the complex plane (k).…”

“…It follows from the proof of the above-mentioned theorem presented in [4] that, in the case of an unstable quasi-polynomial D(k) when N of its roots are located in the right half-plane of (~,), the vector…”

“…Consequently, according to the theorems in [4,5] on the stability of quasi-rational fractions, the dynamic model ( if, as to increases monotonically from 0 to o% the vector D(ito) turns in the (u, ia~)-plane from the positive real semi-axis in a positive direction through an angle of (n + X)g/2, that is, an increment of argument ¢ = A argD(ito) = (n +X)g/2 (3.7)…”

“…If just one of the roots of the quasi-polynomial D(~,) lies to the right of the imaginary axis of the complex plane (~,), then the dynamic model (2.6) is unstable. Since the function D(~,) is analytic on the imaginary axis and in the right half of the complex plane ~, = ~ + ito and, according to relations (3.O~(-o*,**): D(i(o) = u(oo)+ix)(oo)*O, u(-oo) = u(~), 1)(~) =-o(o))are satisfied, then, according to the theorem on the stability of a quasi-polynomial[4], all of the roots of the quasi-polynomial D(k) will be located to the left of the imaginary axis of the complex plane (~)…”