Stability of stochastic self-adjusting automatic control systems with after effect. I. mean square asymptotic stability
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STABILITY OF STOCHASTIC SELF-ADJUSTING AUTOMATIC CONTROL SYSTEMS WITH AFTEREFFECT. I. MEAN SQUARE ASYMPTOTIC STABILITY OF SYSTEMS OF LINEAR STOCHASTIC DIFFERENTIAL-DIFFERENCE EQUATIONS A. V. Nikitin,a I. V. Yurchenko,a and V. K. Yasinskiya
UDC 519.217:519.718:519.837
The stability of stochastic self-adjusting automatic control systems with aftereffect is investigated. Theorems on the mean square stability of stochastic differential difference equations are proved, which allow the stability analysis of stochastic self-adjusting systems. Keywords: stochastic self-adjusting systems, automatic control, aftereffect, stability.
A great amount of studies in technical cybernetics is related to designing automatic control systems (ACS) for objects with delay. One-loop and two-loop ACSs for objects with delay are distinguished. The dynamics of a closed-loop ACS for objects with delay can be described by equations of both delay and neutral types. Given a transfer function and using the inverse Laplace transform, it is possible to derive exact analytic expressions for processes in closed ACSs. A quite extensive literature (see [1] and the bibliography therein, 603 references) deals with the stability of ACSs of objects with delay. Studying the domains of stability of ACSs allows choosing the type of controller based on the requirements to the closed-loop ACS. Two-loop systems produce better results in designing ACSs [2, 3]. Researchers face widely scattering parameters of the object of constructing satisfactory ACSs. It appears that when parameters of the object are subject to significant variation, “good” ACS can be designed if self-adjusting systems (SASs) are used [2, 4, 5] (their block diagram is presented in Fig. 1). As is seen from the diagram, SASs for objects with delay are nonlinear and nonstationary. That is why it is impossible to analyze it well within the framework of the theory of linear stationary systems. In [1, Sec. 6.3], an SAS is analyzed based on the second Lyapunov method. The development of the theory of stochastic differential, differential-difference, and functional-differential equations [6–14] allows considering stochastic SASs (SSASs), which are more accurate mathematical models of well-known deterministic SASs. MEAN SQUARE STABILITY OF LINEAR STOCHASTIC DIFFERENTIAL-DIFFERENCE EQUATIONS IN THE SCALAR CASE Let there be given a probability space (W , F , Å, Ï ), Å = {Ft Ì F , t ³ 0}; Ñ1 be a real Euclidean space [6, 15]. The Case of Several Delays. A random process {x( t ) º x( t , w ), t ³ 0} Ì Ñ1 is defined as a strong solution of the stochastic differential-difference equation [6, 16] dx( t ) +
N
N
k=0
k=0
å a k x( t - D k )dt = å bk x( t - D k )dwk ( t )
a
(1)
Yu. Fedkovych National University, Chernivtsi, Ukraine, †[email protected]; ‡[email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 90–104, January–February 2010. Original article submitted April 30, 2009. 80
1060-0396/10/4601-0080
©
2010 Springer Science+Business Media, Inc.
y( t ) ACS MODEL
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