Функционалы чувствительности в задаче Больца для многомерных динамических систем, описываемых интегро-дифференциальными уравнениями с запаздывающим аргументом | Вестн. Том. гос. ун-та. Управление, вычислительная техника и информатика. 2019. № 48. DOI: 10.17223/19988605/48/4

Функционалы чувствительности в задаче Больца для многомерных динамических систем, описываемых интегро-дифференциальными уравнениями с запаздывающим аргументом

Вариационный метод применен для расчета функционалов чувствительности, которые связывают первую вариацию функционалов качества работы систем (функционалов Больца) с вариациями переменных и постоянных параметров, для многомерных нелинейных динамических систем, описываемых обобщенными интегро-дифференциальными уравнениями Вольтерра второго рода с запаздывающим аргументом.

The sensitivity functionals in the Bolts's problem for multivariate dynamic systems described by integro-differenti.pdf The sensitivity functional (SF) connect the first variation of quality functional with variations of variable and constant parameters and the sensitivity coefficients (SC) are components of vector gradient from quality functional according to constant parameters. Sensitivity coefficients are components of SF. The problem of calculation of SF and SC of dynamic systems is principal in the syntheses and analysis of control laws, of identification and optimization algorisms, in the stability criterions [1-27]. The first-order sensitivity characteristics are mostly used. Later on we shall examine only SC and SF of the first-order. The most difficult are the distributed objects which are described by the dynamic (differential, integral, integrodifferential, difference) equations with delays and by the dynamic equations in partial derivatives [2, 10, 11, 13, 17, 18, 20-23, 27]. Consider a vector output x(t) of dynamic object model under continuous time t ∈ [tO , t1], implicitly depending on parameters vectors a(t), a and functional I constructed on x(t) under t ∈ [tO, t^] . The first variation δI of functional I and variations δa(t) are connected with each other with the help of a single-t1 line functional - SF with respect to variable parameters a~(t) : δa~(t) I = ∫V (t)δa~(t)dt . SC with respect to con-tO stant parameters a are called a gradient of I on a : (dI/ da)τ I. SC are a coefficients of single-line relationship between the first variation of functional δI and the variations δa of constant parameters a : δ-1 = (^-I)t δa = (d^I / da)δa ≡ δa j . J=1 ^a j The direct method of SC calculation (by means of the differentiation of quality functional with respect to constant parameters) inevitably requires a solution of cumbersome sensitivity equations to sensitivity functions W (t) . W (t) is the matrix of single-line relationship of the first variation of dynamic model output _ t1 _ with parameter variations: δx(t) = W(t)δa. For instance, for functional I = ∫ f^(x(t),a,t^dt we have following tO 31 A.I. Rouban t1 _ SC vector (row vector): dI/ dα = ∫[(5f / 5x) (t) + 5f / 5α]dt. For obtaining the matrix W(t) it is neces-t0 sary to decide a bulky system equations - sensitivity equations. The j-th column of matrix W (t) is made of the sensitivity functions dx(t~) / dα,j with respect to component аj of vector α . They satisfy a vector equation (if y is a vector) resulting from dynamic model (for x) by derivation on a parameter α . To variable parameters such a method is inapplicable because the sensitivity functions exist with respect to constant parameters. For relatively simply classes of dynamic systems it is shown that in the SC calculation it is possible to get rid of deciding the bulky sensitivity equations due to the passage of deciding the conjugate equations -conjugate with respect to dynamic equations of object. Method of receipt of conjugate equations (it was offered in 1962) is cumbersome, because it is based on the analysis of sensitivity equations, and it does not get its developments. Variational method [7], ascending to Lagrange's, Hamilton's, Euler's memoirs, makes possible to simplify the process of determination of conjugate equations and formulas of account of SC and SF. On the basis of this method it is an extension of quality functional by means of inclusion into it an object dynamic equations by means of Lagrange's multipliers and obtaining the first variation of extended functional on phase coordinates of object and on interesting parameters. Dynamic equations for Lagrange's multipliers are obtained due to set equal to a zero (in the first variation of extended functional) the functions before the first variations of phase coordinates. Given simplification first variation of extended functional brings at presence in the right part only parameter variations, i.e. it is got the SF. If all parameters are constant that the parameters variations are carried out from corresponding integrals and at the final result in obtained functional variation the coefficients before parameters variations are the required SC. Given method was used in [2123, 25-27] for calculation of sensitivity coefficients and sensitivity functionals in the Bolts's problem for multivariate dynamic systems described by the differential, integral, integro-differential ordinary equations and equations with delay time under various initial conditions. In [21] the sensitivity coefficients for many-dimensional dynamic systems described by the continuous and discontinuous differential equations with delay time are calculated. In [22. 23] for dynamic systems described by ordinary continuous Volterra's of the second-kind integral equations [22] with delay time and integro-differential equations with delay time [23] the SC are received. Calculation possibility also SF for variable parameters is noted. In [25] the SF and SC for multivariate dynamic systems described by generalized ordinary integral equations are calculated. In [26] the same problem for multivariate dynamic systems described by generalized ordinary integrodifferential equations is solved. In [27] the SF and SC for multivariate dynamic systems described by generalized integral equations with delay time are calculated. In this paper the variational method of account of SF is developed to more general continuous manydimensional non-linear dynamic systems circumscribed by the vectorial non-linear continuous Volterra's integro-differential equations of the second genus with delay time, with variable α(t) and constant α parameters and with reviewing of generalised quality functional (the Bolts problem) and registration of dependencies: 1) disturbing actions of a object model from initial instant; 2) of initial t0 and final t1 instants and of dead time from constant parameters α. 1. Problem statement We suppose that the dynamic object is described by system of non-linear continuous Volterra's of the second-kind integro-differential equations (I-DE) with delay time τ 32 The sens∕t∕v∕ty funct∕onals ∕n the Bolts's problem for mult∕var∕ate dynam∕c systems x(t) = f(x(t),x(t-τ),y(t),y(t-τ),a(t),a,t), t0 , γ y y) , y) -y(t) y ( ’ gy(t) -η(t) s-^,(,) J 'y -,.-(t) + 1(t1 - τ -1 )∣λ t (t + τ)-f0t+τl + φy(t1)-K(,1•t + у) + J γT(,)-K-^s ], 0 ≤, ≤,1; YT {t) = 1(t' - τ -1)[λτ,(t + τ) -f(t±2) + Φy (,1) ^k(^,t + τ) + ∫ yt (s) ^K*s,t + τ) -s ], t0 - τ ≤ t ≤ t0, ∂x(t) y ∂x(t) 2τ y -,(t) γτy (t) = 1(t1 -τ-t)[λτχ (t + τ) + Φy (t1) -k (/ ,, +τ) + ∫ γτy (s) -k ∖ ÷τ) -s ], t^-τ≤ t ≤ t^ . ^y(t) y ∂y(t) y ∂y(t) ,+τ y 38 The sensitivity functionals in the Bolts's problem for multivariate dynamic systems 1 dI1(t1) dη(t1) 1 dI1(t1) dη(t1) Here Φx(t') , Φy(t‘) ≡ 1'' / / x dη(t1) dx(t1) y dη(t1) dy(t1) SF are calculated under the formula: t1 δI = δ~(t)i + δi + δai; δa(t)i = ∫ '^^dfo(t) dη(tk df(tk + Юl dη(t) da(t) ∂a(t') x da(t) +γTy (t)dr^ +Φy(t') ,1+∫γy(s)dK(s,i ds δa(t)dt + f [^γT (t) y da(t) y da(t) J y da(t) J J x dI1(t ‘) dη(t 4 (15) tO d^xd:! + γ T (tk ]δa(tk dt; t dα(f) d>att) δ 11 = [dIi^ + Φ y (t') -d~^k]δa(t'); a(t1k dη(t1) da(t1) y da(t') 1 (dI1(,1k - i + В,1(,1у +λ, (^^^k dxo^ +1∫(^,kdf(tk j, + dη(t^k da da da t∫ da dK (t', s) da t∫ da + '∫ [γ T (tkd⅞x^tk+γ T (tkd¾z(tk da da t0 - τ δa I = I 1 Γ dr(t1) + Φ y (t ^) '' ’ ds + + ∫ [ 11 dfO^i 8η(Zk + dfOGtk + γy (tk dr(tk + ∫γy (sk y da ∖ y dn(tk 3a aa ∂K (s, t) J., J ---- ds ] dt + da λ - f (t0 ) ]I] + J fdxO( a, to) к dtO +1(t - t0 - τ)λx (t0 + τ)( f (t0 + τ - 0) - f (t0 + τ + 0)) + + Φy (t *k^dr(L2 - K(t^, IoI + 1(t1 - tO-τk(K(t^, ,o +τ-O) - K(t1,1o +τ + O))]y d t0 0 0 0 0 ] dt + + λτx (t O) 11 - fO(1o) + ∫γ'τy (t)(^rrt'^ - K(t,tO^')dt + +1(11 - tO - τk ∫γ'Ty(t)[K(t, tO + τ - Ok - K(t, tO + τ + O)]dt tO y tO tO +τ dt -+ da Γ 1 1 1 dr(t1) 1 1 t dK (t1, s) dI1 ( t1 ) dη(t1) dI1 ( t1 ) + Φx(t‘)f(t‘) + Φy(t1)^√ + K(t‘,t‘k + o---ds] + + 1ξ ’ y^L dt' ’ ,∫ dt^ dη(t’k dt' dt^ , K dt^ + fO(t ) + da ⅛J , , ,y t!, , f,, , , . IV df(tk dx(da≡ d^da . da Conclusion In this paper the variational method of calculation SF and SC for the multivariate nonlinear dynamic systems described by general continuous vectorial Volterra's integro-differential equations of the second genus with dead time is developed. The variational method is based on invariant expansion of initial functional for system due to inclusion in it of the dynamic equations of object model and of measuring device model with the help of Lagrange's 39 A.I. Rouban multipliers and on computation of the first variation expanded functional on phase coordinates of model and in required parameters. The equating with a zero of the functions facing to variations of phase coordinates gives the dynamic equations for Lagrange's multipliers. The simplified first variation represents required sensitivity functional. Novelty and generality of results consists in a generality of dynamic object model, of the measuring device model and of quality functional. In models both variables and constant parameters are present. In a basis of calculation of sensitivity indexes the decision of the integro-differential equations of object model in a forward direction of time and obtained integro-differential equations for Lagrange's multipliers in the opposite direction of time lays. Received in paper SF are more general in comparison with known in the scientific literature. Results are applicable at the decision of problems of identification, adaptive optimal control and optimization of dynamic systems, i.e. they allow to create precision systems and devices. This paper continues research in [17, 21-23. 25-27∣. Integro-differential models structurally include separately differential and integrated models, and also 4 kinds of more simple integro-differential models which differ character of interaction of phase coordinates of integrated and differential parts. Examples of reception of these results will be presented in special paper. Variational method of calculation of SC and SF allows to do a generalization on more complex dynamic system classes, described by: differential, integral and integro-differential equations with additional some dead times and different classes of discontinuous dynamic equations.

Ключевые слова

variational method, sensitivity functional, sensitivity coefficient, integro-differential equation, conjugate equation, delay time, вариационный метод, функционал чувствительности, интегро-дифференциальное уравнение с запаздывающим аргументом, функционал качества работы системы, задача Больца, сопряженное уравнение

Авторы

ФИООрганизацияДополнительноE-mail
Рубан Анатолий ИвановичСибирский Федеральный университетпрофессор, доктор технических наук, профессор кафедры информатики Института космических и информационных технологийai-rouban@mail.ru
Всего: 1

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 Функционалы чувствительности в задаче Больца для многомерных динамических систем, описываемых интегро-дифференциальными уравнениями с запаздывающим аргументом | Вестн. Том. гос. ун-та. Управление, вычислительная техника и информатика. 2019. № 48. DOI: 10.17223/19988605/48/4

Функционалы чувствительности в задаче Больца для многомерных динамических систем, описываемых интегро-дифференциальными уравнениями с запаздывающим аргументом | Вестн. Том. гос. ун-та. Управление, вычислительная техника и информатика. 2019. № 48. DOI: 10.17223/19988605/48/4