CC BY
9
MSC 60G35, 93E03, 94A12
DOI: 10.14529 /mmp 180212
STOCHASTIC MODEL OF OPTIMAL DYNAMIC MEASUREMENTS
A.A. Zamyshlyaeva1, A.V. Keller1, M.B. Syropiatov1 1South Ural State University, Chelyabinsk, Russian Federation E-mail: zamyshliaevaaa@susu.ru, kellerav@susu.ru, syr94@mail.ru
Under consideration is the stochastic model of optimal dynamic measurements. To solve this problem, the theory of optimal dynamic measurements which has actively been developing for the deterministic problems is extended to the stochastic case. The main purpose of the model is to restore a dynamically distorted input signal from a given observation using methods of the theory of dynamic measurements and the optimal control theory for Leontief type systems. Based on the results obtained by the authors earlier it is shown that optimal dynamic measurement as a minimum point of the cost functional doesn't depend on stochastic interference such as resonances in chains and random interference at the output of measuring transducer.
Keywords: stochastic problem; optimal dynamic measurement; cost functional.
Introduction
The stochastic problem of dynamic measurements can be considered within the framework of the general control theory, based, on one hand, on the deterministic description in the class of differential equations of the system itself, on the other hand -on the stochastic description of the control and perturbing effects applied to it. One of the main characteristics of the control system is the dynamic accuracy of signal transmission or conversion, which is determined either by the difference or functional of the difference between the required and the actual values of the signal in time. There is a resonant problem, to get rid of noise in the observed signal, and to receive the measured signal in the required form. Since it is not possible to get rid of noise in full (in practice) even with the use of modern measuring instruments, filters and technical methods, there arises a question about mathematical solution of the problem, in particular with the use of methods of optimal control theory.
Mathematical model of measuring transducer (MT) is represented by the Leontief type system of equations (descriptor system [1])
J Lx = Ax + Bu, , ,
I y = Cx, ^
LA
is possible that det L = 0 [2], x(t) and x(t) are vector functions of the state of the MT and the velocity of the state change, respectively; y(t) is a vector-function of observations; C
observation; u(t) is a vector-function of measurements; B is the matrix characterizing
L
system (1) can be reduced to
J x = Mx + Fu, , .
I y = Cx, ^
where M = L-1A, F = L-1B.
This system appeared in the dynamic measurements theory [3] from the remote control theory, where (2) was obtained in the study of the transfer function of the MT. Finally, in accordance to new approaches in the field of measurements [4], it is possible to replace the terms "input signal" to "measurement" and "output signal" to "observation", respectively. We emphasize that all observations and measurements in systems (1) and (2) are simulated or "virtual".
The initial Showalter - Sidorov condition
[(aL - A)"1 L]P+1 (x(0) - xo) = 0 (3)
for some xo £ Rra, a £ pL(M), reflects initial state of the MT. The initial Showalter -Sidorov condition is equivalent to the initial Cauchy condition x (0) = x0 in case det L = 0.
The main goal of the theory of optimal dynamic measurements [5] is the restoration of a dynamically distorted input signal (measurement) u(t) according to a given observation y0(t). When using this approach the key concept is the optimal dynamic measurement v(t), which is constructed as a minimum of the functional
J(v) = min J(x(u),u)
on a set of admissible measurements Ud, where the pair (x(u),u) satisfies system (1), and Ud contains a priori information about measurements. The functional J(x(u),u) reflects the evaluation of the proximity of actual observation y0(t) and virtual observation y(t), obtained from (1). At present, within the framework of the theory of optimal dynamic measurements, the deterministic case is well studied [5, 6] algorithms for solving such problems are constructed [7]. However, the deterministic problem does not take into account the effects of random interferences, which are always present in real processes, so it was suggested to use the stochastic model of MT
{
LC = AC + B(u + p), (4)
n = CC + v,
[(aL - A)"1 LP+1 (C(0) - Co) = 0. (5)
Here matrices L, A, B, C have the same sense as in (1). Random processes p and v determine noises in the circuits and at the output of the the MT respectively.
1. Stochastic Leontief Type System in Spaces of "Noises"
Let Q = (Q, A, P) be a complete probability space, R be a set of real numbers, endowed with the Boreal a-algebra. The measurable mapping C ■ Q ^ R is called a random variable. The set of random variables with EC = 0 and finite variance forms a Hilbert space L2 with an inner product < C^C2 >= E(^2)- Let I C R be some interval. The mapping n ■ I x Q ^ R of the form n = n(t,u) is called a (one-dimensional) stochastic process, thus for every fixed t £ I the value of the mapping n = n(t, ') is a random variable, i.e. n = n(t, ■) £ L2 and for every fixed u £ Q the value of a stochastic process n = n( ■ ,u) is called the (sample) trajectory. The random process n is called continuous, if
CL2
random processes. Continuous random process, which independent random variables are Gaussian, is called Gaussian. By V(l) denote the l-th Nelson - Gliklikh derivative of of the stochastic process n [8]. The set of continuous stochastic processes having continuous Nelson - Gliklikh derivatives up to order k E N at each point of the set I forms a space, which is denoted by CkL2.
Consider the stochastic Leontief type system
o
LC = AC + B (u + p). (6)
Where u : I ^ Rn is a vector-function, p is a stochastic process. Let the matrix A be (L,p)-regular, p E {0} U N, and initial states of (6) are described by the Showalter -Sidorov condition
[(aL - A)-1L]p+1 (C(0) - Co) = 0, (7)
where C0 = Yin=0 C0,kek, C0,k are the pairwise independent Gaussian random variables and {ek^'l=1 is an orthonormal basis in Rn.
Theorem 1. For any vector-function u E Cp+1(I, initial values C0 and stochastic process p E Cp+1 L2(I, Rn) independent for every t E I, there exists a unique solution C of (6), (7) given by
C(t) = Cu(t)+ CM Cu E C 1(I, Rn), C^ E C 1L2(I, Rn), (8)
where
t p Cu(t)= f Ut-sL-1Qu(s)ds + Y, (M-1 (In - Q) L)q M-1 (Q - In) u(q)(t) (9) 0 q=0
is the deterministic part and
t p
Ut) = UtC0 + i Ut-sL-1Qp(s)ds + (M-1 (In - Q) L)q M-1 (Q - In) p(q)(t) (10)
0 q=0
is the stochastic part of the solution.
Here Ut = lim ((L - 1M)-1 l)\ Q = lim (rLL(M))p, LL(M) = L (L - -1 M)-1,
r^-tx \ y r ' J r^x v ' v r '
and In is an identity matrix of order n.
Proof. Problem (6), (7) can be split into a deterministic
Un = ^ + Bu, (11)
[(aL - A)-1L]p+1 (Ш) = 0 (12)
and stochastic problems
o
L^ = A^ + Bp, (13)
[(aL - A)-1L]p+1 (U0) - &) = 0. (14)
Due to (L, p)-regularity of matrix A there exists a unique solution [9] given by (9) of (11), (12). Using results on solvability of stochastic Sobolev type equations in spaces of "noises" [10-12] we can conclude existence of unique solution of stochastic Leontief type system (13), (14) as a particular case.
□
2. Stochastic Model of Optimal Dynamic Measurements
The key concept is the optimal dynamic measurement simulated by solving the optimal control problem for (4), (5). For finding it introduce the space of measurements U = {u E L2(I, Rra) : u(p+1) E L2((0,t), Rra)} and allocate in it a closed convex set of admissible measurements Ud C U which contains a priori information about measurements Analogously to the deterministic case when investigating the problem of restoration a dynamically distorted signal by random interference in the circuits and at the output of MT, we consider the control problem
J(v) = min J(u),
u£Ug
where the cost functional
k=0
J(u) = J(f(u)) = £ Ie f (k)(t) - v0k)(t)
dt
(15)
(16)
reflects the closeness of the real observation no (t) and the virtual observation n(t), obtained on the basis of mathematical model of the MT.
The minimum point v(t) of the functional (16) on the set Udl being a solution of optimal control problem (15) is called an optimal dynamic measurement. In practice, there is only indirect information about v(t).
Theorem 2. Optimal dynamic measurement doesn't depend on random initial condition, noises in the circuits and at the output of the MT.
Proof. Since the input signal is a subject to noise in circuits and and at the output of MT, the virtual observation n(t) is a stochastic process, the real observation n0(t) als° varies from experiment to experiment and it can also be considered as a stochastic process with a finite expectation n0(t) at each po int t E /.Denote by rq0(t) a stochastic process n0(t) — n0(t) with zero expectation. Transform the cost functional
k=0
J(u) = Y, I E V (k)(t) - V0k)(t)
dt =
dt
= £ ÎE d(k)(t) + 0(k) - (no(t)(k) + n0k)(t)) k=00
£ / E OCik)(t) + clf(t) + 0(k) - (nik)(t) + °fjW(t)) k=00
i }
= £/ ceuk)(t) - nik)(t)
dt
dt+
k=0
¡E{c^ik)(t) - fik)(t),cl lk)(t) + o(k) - f ik)(t)^j dt+
k = 0 n
T
2
T
2
T
2
1 т
+ £ E d$\t) +°(k) -п0k)(t) k=0 {
dt
L eg)® -4k\t)2dt+ £ E dp(t) +0(k) -п0k)(t)
k=0
k=0
dt.
Thus, noises and random initial condition do not affect the optimal dynamic measurement as a minimum point of cost functional. They affect only the value of the optimality criterion, namely, it increases.
□
Remark 1. The theorem allows us to assert the possibility of applying numerical algorithms developed for the deterministic case [13] to solve the problem of restoration of the measured signal distorted by stochastic interference. In the future, it is planned to modify the methods of finding the optimal dynamic measurement in the presence of noise.
2
T
2
Acknowledgements. The work was supported by Act 211 Government of the Russian Federation, contract No. 02.AOS.21.0011.
References
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2. Khudyakov Yu.V. On Mathematical Modeling of the Measurement Transducers. Journal of Computational and Engineering Mathematics, 2016, vol. 3, no. 3, pp. 68-73. DOI: 10.14529/jceml60308
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5. Shestakov A.L., Keller A.V., Sviridyuk G.A. Optimal Measurements. XXI IMEKO World Congress "Measurement in Research and Industry", 2015, pp. 2072-2076.
6. Shestakov A.L., Sagadeeva M.A., Sviridyuk G.A. Reconstruction of a Dynamically Distorted Signal with Respect to the Measuring Transducer Degradation. Applied Mathematical Sciences, 2014, vol. 8, no. 41-44, pp. 2125-2130. DOI: 10.12988/ams.2014.312718
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10. Zamyshlyaeva A.A., Sviridyuk G.A. The Linearized Benney - Luke Mathematical Model with Additive White Noise. Springer Proceedings in Mathematics and Statistics, 2015, vol. 113, pp. 327-337. DOI: 10.1007/978-3-319-12145-l_21
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12. Favini A., Sviridyuk G.A., Manakova N.A. Linear Sobolev Type Equations with Relatively p-Sectorial Operators in Space of "Noises". Abstract and Applied Analysis, 2015, vol. 2015, p. 697410. DOI: 10.1155/2015/697410
13. Khudyakov Yu.V. The Numerical Algorithm to Investigate Shestakov - Sviridyuk's Model of Measuring Device with Inertia and Resonances. Mathematical Notes of YSU, 2013, vol. 20, no. 2, pp. 211-221. (in Russian)
Received April 17, 2018
УДК 517.9 DOI: 10.14529/mmpl80212
СТОХАСТИЧЕСКАЯ МОДЕЛЬ ОПТИМАЛЬНЫХ ДИНАМИЧЕСКИХ ИЗМЕРЕНИЙ
А.А. Замышляева1, А.В. Келлер1, М.Б. Сыропятов1
1 Южно-Уральский государственный университет, г. Челябинск, Российская Федерация
Рассматривается стохастическая математическая модель оптимальных динамических измерений, которая позволяет осуществить восстановление динамически искаженного входного сигнала по заданному наблюдению с использованием методов теории динамических измерений и теории оптимального управления для систем леонтьевско-го типа. Для ее исследования теория оптимальных динамических измерений, которая активно развивалась для детерминированных задач, распространяется на стохастический случай. На основе результатов, полученных авторами ранее, показано, что оптимальное динамическое измерение как точка минимума функционала не зависит от стохастических помех, таких, как резонансы в цепях и случайные помехи на выходе измерительного устройства.
Ключевые слова: стохастическая модель; оптимальное динамическое измерение; функционал качества.
Статья выполнена при поддержке Правительства РФ (Постановление № 211 от 16.03.2013 г.), соглашение № 02.А03.21.0011.
Литература
1. Белов, А.А. Дескрипторные системы и задачи управления / А.А. Белов, А.П. Курдюков. - М.: ФИЗМАТЛИТ, 2015. - 300 с.
2. Khudyakov, Yu.V. On Mathematical Modeling of the Measurement Transducers / Yu.V. Khudyakov // Journal of Computational and Engineering Mathematics. - 2016. -V. 3, № 3. - R 68-73.
3. Грановский, В.А. Динамические измерения: теория и метрологическое обеспечение вчера и сегодня / В.А. Грановский // Датчики и системы. - 2016. - № 3. - С. 57-72.
4. Ruhm, К.Н. Dynamics and Stability - A Proposal for Related Terms in Metrology from a Mathematical Point of View / K.H. Ruhm // Measurement: Journal of the International Measurement Confederation. - 2016. - V. 79. - P. 276-284.
5. Shestakov, A.L. Optimal Measurements / A.L. Shestakov, A.V. Keller, G.A. Sviridyuk // XXI IMEKO World Congress «Measurement in Research and Industry:». - 2015. - P. 2072-2076.
6. Shestakov, A.L. Reconstruction of a Dynamically Distorted Signal with Respect to the Measuring Transducer Degradation / A.L. Shestakov, M.A. Sagadeeva, G.A. Sviridyuk // Applied Mathematical Sciences. - 2014. - V. 8, № 41-44. - P. 2125-2130.
7. Keller, A.V. The Numerical Algorithms for the Measurement of the Deterministic and Stochastic Signals / A.V. Keller, A.L. Shestakov, G.A. Sviridyuk, Yu.V. Khudyakov // Springer Proceedings in Mathematics and Statistics. - 2015. - V. 113. - P. 183-195.
8. Gliklikh, Yu.E. Stochastic Leontieff Type Equations and Mean Derivatives of Stochastic Processes / Yu.E. Gliklikh, E.Yu. Mashkov // Вестник ЮУрГУ. Серия: Математическое моделирование и программирование. - 2013. - Т. 6, № 2. - С. 25-39.
9. Shestakov, A.L. Dynamical Measurements in the View of the Group Operators Theory / A.L. Shestakov, G.A. Sviridyuk, Yu.V. Khudyakov // Springer Proceedings in Mathematics and Statistics. - 2015. - V. 113. - P. 273-286.
10. Zamyshlyaeva, A. A. The linearized Benney - Luke Mathematical Model with Additive WThite Noise / A.A. Zamyshlyaeva, G.A. Sviridyuk // Springer Proceedings in Mathematics and Statistics. - 2015. - V. 113. - P. 327-337.
11. Загребина, С.А. Линейные уравнения соболевского типа с относительно р-ограниченными операторами и аддитивным белым шумом / С.А. Загребина, Е.А. Сол-датова // Известия Иркутского государственного университета. Серия: Математика. -2013. - Т. 6, № 1. - С. 20-34.
12. Favini, A. Linear Sobolev Type Equations with Relatively p-Sectorial Operators in Space of "Noises" //A. Favini, G.A. Sviridyuk, N.A. Manakova // Abstract and Applied Analysis. -2015. - V. 2015. - P. 697410.
13. Худяков, Ю.В. Алгоритм численного исследования модели Шестакова - Свиридюка измерительного устройства с инерционностью и резонансами / Ю.В. Худяков // Математические заметки ЯГУ. - 2013. - Т. 20, № 2. - С. 211-221.
Алена Александровна Замышляева, доктор физико-математических наук, доцент, кафедра «Прикладная математика и программирование», ЮжноУральский государственный университет (г. Челябинск, Российская Федерация), zamyshliaevaaa@susu. га.
Алевтина Викторовна Келлер, доктор физико-математических наук, доцент, кафедра «Математическое и компьютерное моделирование», Южно-Уральский государственный университет (г. Челябинск, Российская Федерация), kellerav@susu.ru.
Михаил Б орисович Сыропятов, «
»
Российская Федерация), syr94@mail.ru.
Поступила в редакцию 17 апреля 2018 г.
