The sensitivity coefficients for dynamic systems described by interconnected difference ordinary equations and equations.pdf The sensitivity coefficients (SC) are components of vector gradient from quality functional according to parameters. The problem of calculation of SC for dynamic systems is principal in the analysis and syntheses of control laws, identification, optimization, stability [1-16]. The first-order sensitivity characteristics are mostly used. Later on we shall examine only SC of the first-order. Consider two vectorial outputs x(t) and y(t) of dynamic object model: interconnected ordinary difference equations and difference equations with distributed memory on phase coordinates under discrete time t e[0,1,..., N +1] implicitly depending on vector a parameters and I(a) is functional constructed on x(t), y(t), a under t e[0,1,..., N +1]: N +1 I(a) = £ fo (x(t), y(t), a, t). t=o SC with respect to constant a parameters are called a gradient from I(a) on a vector: (dl(a)/da)T , VaI(a). SC are a coefficients of single-line relationship between the first variation 5aI(a) of functional I(a) and the variations 5a of constant a parameters: 5.1 ( a) = dfal 5a, £ *№ 5a j=1 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: Wf (t) = dx(t)/da, Wf(t) = dy(t)/da. For instance, for functional I(a) we have following SC dI( a) da N +1 £ t=o dfo( x(t),y(t), a, t) dx(t) wax (t)+ dfo( x(t), y(t), a, t) dy(t) way (t))+ dfo( x(t), y(t), a, t) da For variable parameters such method essentially becomes complicated and practically is not applicable. At a choice of good initial approach of parameters at identification of objects and also at consecutive calculation of control actions on object often apply a gradient algorithm. It appears that for calculation of components of a gradient from an optimized functional to required variables and constant parameters, it is convenient to apply the conjugate equations (in relation to the dynamic equations of object). Variational method [6] makes possible to simplify the process of determination of conjugate equations and formulas of account of SC. On the basis of this method it is an extension of quality functional by means of inclusion into it dynamic equations of object 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 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 sensitivity functional on concerning parameters. In difference from other papers devoted to calculation of SC in given paper the generalized difference models are used: interconnected ordinary difference equations and difference equations with distributed memory on phase coordinates and variable parameters. Besides variables and constant parameters enter into the right parts of difference equations of dynamic object, in an indicator of quality of system work and initial values of phase coordinates depend on constant parameters. At the right part of the equations of object model there are also phase coordinates and variable parameters during the previous moments of time. Such discrete equations are similar numerical decisions of integro-differential Volterra’s equations. It is proved that both methods to calculation of SC (with use of Lagrange's functions or with use of sensitivity functions) yield the same result, but the first method it is essential more simple in the computing relation. 67 Обработка информации / Data processing 1. Problem statement We suppose that the dynamic system is described by non-linear interconnected difference equations x(t +1) = f (x(t), y (t), a(t), a, t), t = 0, 1, 2, ..., N, x(0) = x0(a) . (1) t y(t +1) = XK(t,x(s),y(s),a(s),a,s), t = 0,1,2, ..., N, y(0) = Уо(а). s=0 Here: x,y are a vector-columns of phase coordinates; a(t), a are a vector-columns of interesting variable and constant parameters; f (),K(•) are known continuously differentiated limited vector-functions. The quality of functioning of system it is characterised of functional N I (a, a) = ^f, (x(t), y(t), a(t), a, t) + f (x(N +1), y( N +1), a( N +1), a, N + 1), (2) t=0 depending on a(t) and a . The conditions for function f0() are the same as for f (•), K(•). With use of a functional (2) the optimization problem (in the theory of optimal control) are named as the Bolts’s problem. With the purpose of simplification of appropriate deductions with preservation of a generality in all transformations (1), (2) there are two vectors of parameters a(t), a . If in the equations (1), (2) parameters are different then it is possible formally to unit them in two vectors a(t), a , to use obtained outcomes and then to make appropriate simplifications, taking into account a structure of a vectors a(t), a . It is shown also that the variation method allows to receive SC in relation to variable and constant parameters: N+1 81

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