The sensitivity coefficients for dynamic systems described by nonlinear difference interconnected ordinary equations and generalized equations with the distributed Текст научной статьи по специальности «Математика»

ВЕСТНИК ТОМСКОГО ГОСУДАРСТВЕННОГО УНИВЕРСИТЕТА

2023 Управление, вычислительная техника и информатика № 63

Tomsk StateUniversity Journalof Control and Computer Science

Original article

doi: 10.17223/19988605/63/10

The sensitivity coefficients for dynamic systems described by nonlinear difference interconnected ordinary equations and generalized equations with the distributed

memory

Anatoly I. Rouban

Siberian Federal University, Krasnoyarsk, Russian Federation, [email protected]

Abstract. The variation method of calculation of sensitivity coefficients connecting first variation of quality functional with variations of variable and constant parameters for multivariate non-linear dynamic systems described by nonlinear interconnected ordinary difference equations and generalized difference equations with the distributed memory on phase coordinates and variable parameters is developed. The nonlinear quality functional has also a generalized form. Sensitivity coefficients are components of variation of generalized sensitivity functional and they are before variations of variable and constant parameters. The base of sensitivity coefficients calculation are the decision of object equations in the forward direction of discrete time and corresponding difference conjugate equations for Lagrange's multipliers in the opposite direction of discrete time.

Keywords: variational method; sensitivity coefficient; difference equation; conjugate equation; Lagrange's multiplier.

For citation: Rouban, A.I. (2023) The sensitivity coefficients for dynamic systems described by nonlinear difference interconnected ordinary equations and generalized equations with the distributed memory. Vestnik Tomskogo gosu-darstvennogo universiteta. Upravlenie, vychislitelnaja tehnika i informatika - Tomsk State University Journal of Control and Computer Science. 63. pp. 84-91. doi: 10.17223/19988605/63/10

Научная статья УДК 62-50

doi: 10.17223/19988605/63/10

Коэффициенты чувствительности для динамических систем, описываемых нелинейными разностными взаимосвязанными обыкновенными уравнениями и обобщенными уравнениями с распределенной памятью

Анатолий Иванович Рубан

Сибирский Федеральный университет, Красноярск, Россия, [email protected]

Аннотация. Вариационный метод применен для расчета коэффициентов чувствительности, которые связывают первую вариацию функционала качества работы системы с вариациями переменных и постоянных параметров, для многомерных динамических систем, описываемых разностными нелинейными взаимосвязанными обыкновенными уравнениями и обобщенными уравнениями с распределенной памятью по фазовым координатам и переменным параметрам. Показатели качества работы систем также являются обобщенными нелинейными функционалами.

Ключевые слова: вариационный метод; коэффициент чувствительности; разностное уравнение; распределенная память; функционал качества работы системы; сопряженное уравнение; множитель Лагранжа.

Для цитирования: Рубан А.И. Коэффициенты чувствительности для динамических систем, описываемых нелинейными разностными взаимосвязанными обыкновенными уравнениями и обобщенными уравнениями с распределенной памятью // Вестник Томского государственного университета. Управление, вычислительная техника и информатика. 2023. № 63. С. 84-91. doi: 10.17223/19988605/63/10

© A.I. Rouban, 2023

Introduction

For dynamic systems the problem of calculation of sensitivity functions and of sensitivity coefficients (SC) are central at the analysis and syntheses of control laws, optimization, identification, stability [1-11]. The first-order sensitivity characteristics mostly are used. Later on we shall examine only SC of the firstorder.

The sensitivity functional connects the first variation of quality functional with variations of variable and constant parameters. The SC are components of vector gradient from quality functional according to parameters.

Consider a vector output z(t) of dynamic object model under discrete time t e[0,1,...,N +1] implicitly depending on vector constant a parameters and functional I(a) constructed on a basis of z(t) under t e[0,1,...,N +1] and on a basis of a parameters:

I(a) = fo(z(N +1),z(N),...,z(1),z(0),a)- / (•,a) .

SC with respect to constant a parameters are called a gradient from I(a) on a: (dl (a)/ da)T =Va I (a). SC are a coefficients of single-line relationship between the first variation of functional 5a I (a) and the variations da of constant parameters:

Sa I(a) = (VaI(a))Tda = dI(a) da^ Y ^^ da, .

da Y 5a 1

Ul V_A. •_| \J V_A. j

The known method of SC calculation inevitably requires a solution of cumbersome sensitivity equations to sensitivity functions W(t). For instance, for functional I(a) we have following row vector for SC:

dI(a) _ y 5/0 (•, + 5/00, a)

da dz(t) 5a

W (t) is the matrix of single-line relationship of the first variation of dynamic model output with parameter variations: Sz(t) = W(t)da . For obtaining the matrix W(t) it is necessary to decide bulky equation systems -sensitivity equations. The j-th column of matrix W(t) is made of the sensitivity functions 5z(t)/da - with respect to a j component of a vector. They satisfy a vector equation (if z is a vector) resulting from dynamic model (for z) by derivation on a parameter a ..

For variable parameters such method in practice is not applied because of the complexity.

Variational method [4], 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 according to concerning parameters.

In difference from other papers devoted to calculation of SC in given paper the generalized nonlinear difference models and the generalized nonlinear purposeful functional are used. Also variables and constant parameters enter into the right parts of difference equations of dynamic object, in an indicator of quality of system work, in the measuring device model and initial values of phase coordinates depend on constant parameters. At the right part of the nonlinear equations of object model there are also phase coordinates and variable parameters during the previous moments of time.

It is proved that both methods to calculation of SC (either 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.

1. Problem definition

We suppose that the dynamic object is described by system of non-linear difference interconnected ordinary equations and generalized equations with the distributed memory on phase coordinates and variable parameters:

x(t + Y) = fx(x(t),y(t),a(t),a,t), t = 0,1, 2, ..., N, x(0) = x0(a), (1)

y(t +1) = fy(x (t),... ,x (l),x (0); j (0, ... ,y (1), J (0);a (0,... ,á (1),5 (0);a, t), t = 0,1,..., N,

y(0) = yo(a) .

Here: a(t), a are a vector-columns of variable and constant parameters; x,y are a vector-columns of phase coordinates; fx(•), f (•),x0(a)y0(a) are known continuously differentiated limited vector-functions.

The quality of functioning of system it is characterised of generalized functional

I (a, a) = fo( x( N +1),..., x(1), x(0); y(N +1),... , y(1), y(0); a(N +1),... , 5(1), a(0);a) (2)

depending on a(t) and a parameters. The conditions for function f0 (•) are the same as for fx (•), f (•).

Functional (2) are used for solution of a the optimization problems.

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 .

Is shown also that the variation method without basic modifications allows to receive SC in relation to variable and constant parameters:

N+1

SI (а, а) = ^

ôI (а,а) 8S(t )+ô^ 5а.

t=0

ôx(t )

Va(t )I (a, а) =

ôI (а, а) ôxj (t )

ôI (а, а)

ô« m (t)

ôa

t = 0,1, 2,

(3)

, N, N +1,

Va I (а, а) =

ôI (а, а) ô^

ôI (а, а) ô^

By obtaining of results it is used the obvious designations:

fx(t) - fx(xit),y(t),a(t),a,t), t = 0,1,2,..., N, (4)

fy(t) - fy(x(t),...,x(1),x(0);y(t),...,y(1),y(0);a(t), ...,5(1),5(0);a, t), t = 0,1, ..., N,

fo(") - f0(x(N +1),... ,x(1),x(0);y(N +1),... ,y(1),y(0);5(N +1),... ,5(1),5(0);a). The indexe t in functions (4) also reflects not only obvious dependence on step number, but also that the kind of functions from a step to a step can change.

Let's receive the conjugate equations for calculation of Lagrange's multipliers and on the basis of them formulas for SC calculation.

2. Conjugate equations and sensitivity coefficients

Complement a quality functional (2) by restrictions-equalities (1) by means of Lagrange's multipliers Xx (t), X (t), t = 0,1, 2,..., N +1 (column vectors) and get the extended functional

I = I (а, а) + ^Àrx (t +1)[- x(t +1) + fx (t )]+ ÀTx (0)[-x(0) + Xo (а)]

t=0

N

+ ^ ÀTy (t +1)[- y (t +1) + fy ( t)]+ ÀTy (0)[- y (0) + У0 (а)] =

t=0

T

T

2 У

N

= /0 (•) - AT (N +1)x(N +1) - Ay (N +1)y (N +1) +

N

+X[(t)x(t) - AT(t)y(t) + AT(t + 1)/x(t) + AT(t + 1)/y(t)] + AT(0)x0(a) + AT(0)y0(a)) .

t=0

Functional (5) complies with I (a, a) when the dynamic object moves on a trajectory corresponding to the equations (1).

We calculate the first variation of extended functional, caused by a variation of phase coordinates and also by a variation of variables and constant parameters:

N+1 fîr N+1 fîj N+1 fîr fîr

SI = SXI + S,I + SsI + SaI - ^--dx(t) + X—dKO + X^da(t) +—d

y t=0 fîx(t) f=0 fîy(t) 7=0 fîa(t) fîa

Here:

S, I =

S yI =

fî/o(0 fîx(N +1)

fî/QQ

fîy( N +1)

-ATX ( N +1)

dx( N +1) + X

t=o

fî/o(-) ^fî/x (t )

fîx(t )

-A; (t ) + ATx (t +1).

+

N t

X X Ary (t +1)

t=Q î=o

fî/y (t)

fîx(î)

dX(î).

-Arv ( N +1)

N

dy ( N +1) + X

fî/°°-A3: (t )+Ax (t+1)

fîy(t )

fîx(t )

fîfx (t)

fîy(t )

dx(t ) +

(6)

(7)

dy (t) +

N t

+

fî/y (t)

We consider equality [11]:

N t

Nt

t=0 x=0

assumes the following form:

N t

XXATy (t +1) ff^rdy (î).

^^ fîy(î)

Nî=N

XX At, î)= XX a(Î, t ).

t=0 î=Q Q t

N 1 fî/ (t) <î) and XX A3 (t +1) fî^

fîx(î) ) XX y ( ) fîy(î)

(8)

/ (t) " ' / (t) For summands YYAt(t +1)—y—dx(s) and Y >A(t +1)——-dy(s) in formulas (7) the equality (8)

fî/ (t) t=N î=N r fî/y (î)

XXATy (t +1) dx(î) = X XATy (î +1) /-)dx(t ),

t=0 î=o

N t

fîx(î)

fî/y (t )

o t

t=Nî=N

fîx(t)

fî/y (î)

XXATy (t+(î)=XXAT (î+1)^>(t).

t=0 7=0 y fîy(î) 0 t y fîy(t)

Variations SxI, S I in (7) becomes now:

S. I =

fî/o(-) fîX(N +1)

-Arx ( N +1)

t=N

+ X

0

t = N

+ X

0

dx(N +1) +

/0-ç (t ) + AX (t +1) /t) + X AT, (î +1) ^

cX(t) cX(t) X cX(t)

(9)

dx(t) ;

S / =

—fî/0(^--A3 ( N +1)

fîy( N +1) '

dy( N +1) +

-A, (t ) + ATx (t +1)

fîy(t) y

/t) + X AT (î +1) /î)

fîy(t) X ,V ' fîy(t)

dy(t )

The factors standing in the formula (9) before variations of phase coordinates look like:

fîI = -AT(N +1) + fî/o°

fîI

fîx( N +1)

fîx(N +1)' fîy(N +1)

= -A3, ( N +1) +

fî/o(-)

fîy( N +1)'

(10)

t=0

dI Т T df (t) s=N T dfy (s) QL (•) = -XT (t) + XTx (t +1)dfx(t) + I XTv (s ++ df0(). dx(t) x x dx(t) t v dx(t) dx(t) '

dI - = -ÀTv (t) + XTx (t + 1) ^ +

*=N „ dfv (s) dfn (•)

(s +1)4^ + f(-). t = N.N -1.

.1.0.

dy(t) ^ ' dy(t) ^ yv ' 5y(t) cy(t)'

From equality to zero of these factors we receive the equations for Lagrange's multipliers:

Ъ ( N +1) =

df0(- )

dx( N +1)

XTV ( N +1) =

XT (t )=xx (t+1)

dfx (t)

dx(t)

+

III xy (s+1)

dfo(-)

dv( N +1).

dfy(s) , dfo(-) dx(t) dx(t):

XT (t ) =XTx (t + 1)

dfx (t)

s=N

+

(s +1) t = N. N -1.

.1.0.

dy(t) f dy(t) dy(t)

These equations are decided in the opposite direction changes of an independent integer variable t. Let's calculate now the sensitivity coefficients.

In the equation (6) SC concerning variables and constant parameters look like:

dI _ foO

da(N +1) da(N +1) ' dI dfo(-) dfx(t) . ^„x, . (s)

+ XTx (t +1)

da(t) da(t) — f + I X'x(t +1):

da da „

da(t )

- +

IT (s +1)

. t = N. N -1. ... .1.0.

ndfx(t) , ^dfy(t)

da

■ + XTy (t +1)-

da

da(t )

+ XT (0) ад. (0) dyo(a)

da

da

This result is more common in relation to appropriate results of monograph [4] and paper [11]. At reception SC (12) it was used extended functional (7) and equality (8). We prove equivalence of sensitivity coefficients for initial (2) and extended (5) functionals. We take extended functional, presented in an initial part of the formula (5):

I = I (a. a) + IXT (t +1)[- x(t +1) + fx (t )] + XTx (0)[-x(0) + x0(a)]

t=0

+1 Xy (t +1)[- y(t +1) + fy ( t )]+ Xy (0)[- y(0) + V0(a)].

+

(11)

(12)

Before Ax (•), Ay (•) in brackets there are the dynamic equations of the dynamic system which have

been written down in the form of the equations of equality type. Hence, values of functions in brackets are always equal to zero.

Let's calculate from both parts of the previous equation derivatives in the beginning on a vector of constant a parameters:

"Л "Л / ' x v '

da da t=0

-Wx (t +1)+wx (t)+wy (t) ' dfx (t)

N

+Ixy(t+1)

t=0

-wy (t +1) + I

dfy ( t )

dx(s)

dx(t) wax (s) -

dfy ( t )

dy(s)

dy(t) way (s)

da

-Km-w: (0)+-

dfy ( t )

da

xy (0)

-way (0)-

d a

dy 0(a) d a

Here wax (t ) =

dx(t ) = dy(t )

da ' a da

there are the sensitivity functions.

Before XTx (t +1), A^ (t +1) now there are sensitivity equations for a matrixes of sensitivity functions.

These equations are written down as in the form of restriction of equality type. Values of functions in brackets also are always equal to zero.

t=0

Hence, SC rather both for initial functional and for its extended variant have identical values. For reception of the sensitivity equations it is necessary in equations (1) to impose a condition of differentiability for /x (t) and /, ( t ) on phase coordinates and on considered parameters. On a parameters

should be differentiated initial functions x0 (a) and y0 (a).

We can receive the same result for SC on relation to variable parameters. The sensitivity equations for each fixed value of argument of variable parameters a(j), j = 0,1,...,N +1 have more complex form. They demand special consideration. Important that such sensitivity equations are objectively exist.

At additional use of model of the measuring device it is necessary to make changes to problem statement:

/o (•) - /o (n( N+1),..., n(1), n(o); a( N+1),..., a(1), a(o); a),

n(t) - n(x(t),y(t), a(t), a, t), t = 0,1, 2, ..., N +1. In the received results it is necessary to execute small replacements:

^ to replace on ^ Ml ; Ml to replace on ^ Ml ;

fîx(t) fî-n(t) fîx(t) fîy(t) fîn(t) fîy(t)

/•> to replace on + M> ; M> to replace on /•> + X^.

fîa(t) fîa(t) fîn(t) fîa(t) fîa fîa ^fîn(t) fîa

3. Examples

Indicator of a quality functional of system has the same appearance (2).

Example 1. We consider that in dynamic system there is a memory on phase coordinates, but there is no extra memory on variable parameters, i.e.:

/y(t) - /y(x(t),...,x(1),x(0);y(t),...,y(1),y(0);a(t);a, t), t = 0,1, ... , N.

Structures of the conjugate equations (11) and SC (12) for constant parameters remain, but structure SC (12) for variable parameters becomes simpler:

JL =/i + AT (t +1)/W + A (t +1) M>, t = N,N -1, ... ,1,0.

fîa(t) fîa(t) x fîa(t) y fîa(t)

Example 2. In mathematical model of dynamic system there is no additional memory on phase coordinates x, but there is a memory on phase coordinates y and on variable parameters. In this case

/,(t) = /,(x(t);y(t),...,y(1),y(0);a(t),...,a(1),a(0);a, t), t = 0,1,..., N. The structure of the conjugate equations (11) becomes simpler:

3 (N +1) = , AT (N +1) = Al (t) = Al (t +1) VM + Al (t +1) ^ +/(l'

fîx(N +1) y c,( N +1) fîx(t) y cx(t) cx(t)

Al (t ) = Al (t +1) ^ + X Al (î +1) ^^ + ^^ t = N, N -1, ... ,1,0. y x ( ) fîy(t) Xy ( ) fîy(t) fîy(t), , , "

The SC structures (12) for variables and constant parameters remain.

Example 3. In dynamic system there is no additional memory on phase coordinates y . Then

/y(t) - /y(x(t),...,x(1),x(0);y(t);a(t),...,a(1),a(0);a, t), t = 0,1,..., N.

A quality functional of system has the same appearance (2).

In this variant the conjugate equations become more simple

Ax(N +1) = /-, Ay(N +1) = 3W , AT(t) = Alx(t +1)+ YA(s +1)^ + /i.

cx(N +1) y 5y(N +1) x x cx(t) Y Cx(t) cx(t)'

ATV (t) = AT (t +1) + AT (t +1) + 5/a(l, t = N, N -1, ... ,1,0. y x ( ) 5y(t) y ( ) cy(t) Cy(t), , , ,,

The SC to variable and constant parameters remain former.

Example 4. In dynamic system there is no additional memory on phase coordinates x, y . Then

/,(t) = /,(x(t);y(t);a(t),...,5(1),a(0);a, t), t = 0,1, ..., N, In this variant the conjugate equations become more simpler

+ t = N, dy(t) dy(t )

The SC to variable and constant parameters keep the kind.

Conclusion

Variational method can be used for calculation of SC for multivariate dynamic systems described by difference nonlinear interconnected ordinary equations and generalized equations with the distributed memory on phase coordinates and on variable parameters. Variables and constant parameters are present at object model, at model of the measuring device and at generalized quality functional for system.

In a basis of calculation of SC the decision of the difference equations of object model in a forward direction of time and the decision of obtained difference equations for Lagrange's multipliers in the opposite direction of time lies.

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.

Results of present paper are applicable at design of high-precision systems and devices.

This paper continues research in [4, 11].

It is possible generalization of the received results on the dynamic systems described by the difference equations with late argument.

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References

Information about the author:

Rouban Anatoly I. (Doktor of Technical Sciences, Professor of Computer Science Department of Institute of Space and Information Technologies, Siberian Federal University, Krasnoyarsk, Russian Federation). E-mail: [email protected]

The author declares no conflicts of interests.

Информация об авторе:

Рубан Анатолий Иванович - профессор, доктор технических наук, профессор кафедры информатики Института космических и информационных технологий Сибирского Федерального университета (Красноярск, Россия). E-mail: [email protected]

Автор заявляет об отсутствии конфликта интересов.

Received 27.08.2022; accepted for publication 09.06.2023 Поступила в редакцию 27.08.2022; принята к публикации 09.06.2023