The sensitivity functionals in the Bolts problem for multivariate dynamic systems described by integral equations with d.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 analysis and syntheses of control laws, identification, optimization, stability [1-25]. 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 equations with delays and in partial derivatives [2, 10, 11, 13, 17, 18, 20, 23-25]. Consider a vector output y(t) of dynamic object model under continuous time t e [t0,t1], implicitly depending on vectors parameters a(t),a and functional I constructed on y(t) under t e [t0, t1] . The first variation 5I of functional I and variations 5a(t) are connected with each other with the help of a single-line t1 functional - SF with respect to variable parameters ~(t): S~(t) I = j V (t )Sa(t )dt. SC with respect to constant t0 parameters a are called a gradient of I on a : (dI/da)T = VaI. SC are a coefficients of single-line relationship between the first variation of functional SI and the variations 5a of constant parameters a : m r)J 5-1 = (v5I)T sa = (dI / da)5a = sa . j=1 da 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 with _ t1 _ parameter variations 5y(t) = W (t)5a. For instance, for functional I = j f0( y(t ),a, t )dt we have following SC t0 t1 _ vector (row vector): dI / da = j"[(d f0/ dy) W (t) + д f0/ da]dt. For obtaining the matrix W(t) it is necessary t0 to decide a bulky system equations - sensitivity equations. The j -th column of matrix W (t) is made of the sensitivity functions dy(t) / da7- with respect to component a. of vector а . They satisfy a vector equation (if y is a vector) resulting from dynamic model (for y) by derivation on a parameter a,j. 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 SF and SC. On the basis of this method it is an extension of quality functional by means of inclusion into it 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 [21] for dynamic systems described by ordinary continuous Volterra's of the second-kind integral and integro-differential equations (the Lagrange problem) and in [22] for dynamic systems described by ordinary continuous general Volterra's of the second-kind integral equations (the Bolts problem). In this article the variational method of account of SC and of SF develops more general (on a comparison with papers [23-25]) continuous many-dimensional nonlinear dynamic systems circumscribed by the vectorial non-linear continuous more common Volterra's of the second-kind integral equations with delay time. The more common quality functional (the Bolts problem) is used also. 1. Problem statement We suppose that the dynamic object is described by system of non-linear continuous Volterra's of the second-kind integral equations (IE) with delay time т [17. P. 75] t y(t) = r(a(t),a, t0, t) + jK(t,y(s),y(s -т), a(s),a,s) ds, t0 < t < t1 , (1) t0 y(t) = y(a(t),a, t), t0 -т< t < t0, 0 +2/00+,t (t) M)+V ут (s) -KM A] л+ • -n(t) 5a 5a 5a J ^ + ф(0: Y 40 I 5ri(0 5a(0 5a(0 dK(s,t) 5a(0 0rT (s) (11) (12) + '0 0 rT (t) 5y( t) 5a ФС1)[«КО - K(t1, t ) + i(t; - t - x)(K(t1, t0 + x- 0) - K(t1, t0 + x + 0))] dt + + 5t t0 -x /0(0 + jyTCX-^-K(t,t0))dt + 1(t: -10-x) J yT(t)[K(t,t0 +x-0)-K(t,t0 +x + 0)]dt dt da - + dt1 r5r(t1) к t;5K(t;,s)Jn 5I1(t1) 5n(t1) 5I1(t1) _ . t. ФС X-tt2 + K(t1,t1) + 0 Ii ds] + -T(Tf+ + /0(t1) 5t * -t1 -D(t ) 5t 5t + + da [ф(-1)[1(-1 -10 - t)(K(t1, t0 + x - 0) - K(t1, t0 +x + 0)) - J^yCi-1)ds] + t0 -y(s -x) d (s -x) -t +1C1 - t0 -x) J yT (t)(K(t, t0 + x - 0) - K(t, t0 + x + 0))dt t0 +x + '' ' 5K(t, s) dy(s -x) JrT (t )J ~~ fSa. d a J (13) ds dt -y(s -x) d(s -x) For union of integrals with identical variations 8y we shift back interval of an integration on magnitude - in integral with 8y(t - x) (in this connection the argument in integrand thus will increase on -) and obtain a following result: 1 1л -K(t1, t) r T - K(s, t) j [ф(0 -jyT (-)- -ds]S y(t -x)dt = -y(t--) , - y(t--) Д. 5- (t1,t + -) J T 5K(s,t + -) jyT (-) ds]8 y(t)d t - -y(t) 1(ji---1 )[0(t1) - y(t) + J 1(f - - -1)[Ф(0+ J yT(-)^ds]Sy(t)dt. J-- dy(t) J- Sy(t) Here for compact writing the single function 1(z) (which equals to zero by negative value z) is introduced. In this connection such variants are taken into account when instant t1 -x is found inside and outside of interval of system operating period [t0, t1]. We substitute this formula in the first variation (10), join components with identical variations and obtain that -K (-, t) -y(t) 1 --(t1,t) | -/(') -n(t) | ( ) -y(t) -n(t) -y(t) j t1 JyT (-) 8 y(J)1 = J JyT (-) ds + i t1 + Kt1 - - - t^Otf) -K(f,J + f yT (s) -K(-, J + -) ds] - yT (t) 8 y(t)dt - -y(t) - y(t) 1(t' - - - t)[Ф(t1) (; + + J yT (s) -+ ds] - y T (t) -y(t) J- -y(') In a variation (14) we equate with zero factors before variations of phase coordinates 8y and discover: the conjugate equations for basic Lagrange's multipliers y(t) u -K(t\ t + x) '1 T - K (s, t + x) 8 y(t )dt. (14) + . -K(t1, t) , -/0(t) -n(t) t ■JyT(s) -K (s, t) -y(t) yT(t) = Ф(0 ds - -y(t) -n(t) -y(t) 1 ti ^ -K(t ,t + x) V T . . -K(s,t + x) ,, ^ ^ 1 +1(t - - - ')[Ф(':) ( ' ) + J yT (s) ( ' + ) ds], '0 < t < t1, c>y(t) '+- 5 y(t) (15) and equation of account of Lagrange's multipliers y(t) appropriate to initial function of integral equations with delay time (1) ' 1 - K (s, t + x) -ds], t0 - x< t < t0 . (16) yT(t)=K'1 -т-1W)-K+ J yT)dy(t) L д y(t) These equations are decided in the opposite direction of time (from t1). From the conjugate equations (15), (16) it is possible to remove single function and to add them a customary aspect. If t0 < t1 - x

Скачать электронную версию публикации