On nonparametric control of a dynamic system Текст научной статьи по специальности «Физика»

UDC 52-601

Siberian Journal of Science and Technology. 2017, Vol. 18, No. 4, P. 711-718 ON NONPARAMETRIC CONTROL OF A DYNAMIC SYSTEM E. D. Agafonov1, A. V. Shishkina2*

1Reshetnev Siberian State University of Science and Technology 31, Krasnoyarsky Rabochy Av., Krasnoyarsk, 660037, Russian Federation

2Siberian Federal University 26b, Academica Kirenskogo Str., Krasnoyarsk, 660074, Russian Federation E-mail: nastya.shishkina9666@mail.ru

The paper considers the problem of dual control of an inertia-free object whose input is affected by a control variable and an observable but uncontrollable variable. The idea of dual control belongs to A. Feldbaum and was developed on the basis of Bayesian approach. In this case, the probability densities of interference, as well as input output variables are known. In particular, the case of Gaussian probability densities was investigated. As a result, algorithms of dual control with the simplest objects of the inertia-free class were obtained. In the case of combined control systems, these studies were carried out by Feldbaum's followers.

Further development of dual control theory was described by Y. Tsypkin. The probability density of the interference was unknown, but the stage of selecting the structure of the control device and the equation of the describing object were necessary. In Tsypkin's works, the corresponding parametric algorithms for dual control are given. In this case, a simultaneous estimation of the coefficient of the model and the regulator was made on the basis of the method of stochastic approximations.

Later on nonparametric dual control algorithms were proposed. It is this way of control design that is discussed in the current paper. The parametric model of the object and the parametric structure of the controller were unknown in the problem statement. Nevertheless, it was known from a priori information that the characteristic of the object was one-to-one in control. Below we consider nonparametric control algorithms that combine the processes of simultaneous control and the study of the object with an accuracy of the structure unknown to the parameters, but a priori with a given number of delayed elements of the output variable of the object. In other words, the depth of the memory of the managed object was determined. In this case, a nonparametric algorithm for dual control can function in conditions of passive and active accumulation of information. The technique for representing a one-dimensional inertial-free dynamic system to a multidimensional static one is presented in the paper. Some results of numerical investigation of nonparametric algorithms of dual control are also presented.

Keywords: object with memory, dual control, combined system, nonparametric algorithms, bandwidth parameter, parameter setting.

Сибирский журнал науки и технологий. 2017. Т. 18, № 4. С. 711-718 О НЕПАРАМЕТРИЧЕСКОМ УПРАВЛЕНИИ ДИНАМИЧЕСКОЙ СИСТЕМОЙ

Е. Д. Агафонов1, А. В. Шишкина2*

Сибирский государственный университет науки и технологий имени академика М. Ф. Решетнева Российская Федерация, 660037, г. Красноярск, просп. им. газ. «Красноярский рабочий», 31

2Сибирский федеральный университет Российская Федерация, 660074, г. Красноярск, ул. Академика Киренского, 26б E-mail: nastya.shishkina9666@mail.ru

Рассматривается задача дуального управления безынерционным объектом, на вход которого действует управляющая переменная и неуправляемая, но контролируемая переменная. Идея дуального управления принадлежит А. А. Фельдбауму и была развита на основе байесовского подхода, в этом случае плотности вероятности помех, а также входных-выходных переменных известны, в частности, исследовался случай гауссовых плотностей вероятностей. В результате были получены алгоритмы дуального управления с самыми простыми объектами класса безынерционных. На случай комбинированных систем управления эти исследования проводились его последователями.

Дальнейшее развитие теории дуального управления описывается в книге Я. З. Цыпкина, где плотность вероятности помех уже неизвестна, но также необходим был этап выбора структуры управляющего устройства и уравнения описывающего объекта. В работах Цыпкина приведены соответствующие параметрические алгоритмы дуального управления. В этом случае осуществлялась одновременная оценка коэффициента модели и регулятора на основе метода стохастических аппроксимаций.

Несколько позже были предложены непараметрические алгоритмы дуального управления. Именно этот путь и рассматривается в настоящей статье. Здесь уже не известна ни параметрическая модель объекта, ни параметрическая структура регулятора. Тем не менее, из априорных сведений известно, что характеристика объекта является взаимно однозначной по управлению. Рассматриваются непараметрические алгоритмы управления, совмещающие процессы одновременного управления объектом и его изучения с точностью неизвестной до параметров структуры, но априори с заданным числом запаздывающих элементов выходной переменной объекта. Иными словами, глубина памяти управляемого объекта определена. В этом случае непараметрический алгоритм дуального управления может функционировать в условиях пассивного и активного накопления информации. Приведена методика представления одномерной безынерционной динамической системы к многомерной статической. Представлены также некоторые результаты численного исследования рассматриваемых непараметрических алгоритмов дуального управления.

Ключевые слова: объект с памятью, дуальное управление, комбинированная система, непараметрические алгоритмы, параметр размытости, настройка параметров.

Introduction. The problem to be solved in the paper is the design and implementation of nonparametric control algorithms. An object under control is assumed to be inertia-free and it is described by an equation with an unknown structure with respect to its parameters. Together with the control effect the object is influenced by uncontrollable but observable input. The nonparametric dual control algorithms under consideration were investigated for various tactics of determining the bandwidth parameters at each clock cycle. A concept of dual control was created by A. Feldbaum [1] and developed by Y. Tsypkin [2]. It was originally intended for Bayes control problem statement when the object under control was inertia-free. The main idea of the concept was simultaneous control and learning of the object. That involves procedures of parametric identification [3] together with implementation of control methods in the parametric formulation [4-7]. It should be noted that a dual control system is an example of a control device with memory.

Problem statement. The following notations are introduced: let x = (x1 ... xn) e Rn be output of the object, u = (u1 ... uk) e Rk be controlled input effect, 1 = (| ... |m) e Rm be uncontrollable but observable input, and x* = (x* ... x*) e Rn be reference of the object under control (fig. 1) [8].

In fig. 1 the control device is denoted by 'yy', the object under control is denoted by 'О', and random stationary noise effects influencing both the object itself and the object measurement channels are denoted by "t,t, ht. We assume these noise effects to be unbiased and to have limited variance. The unknown model curve of the object x = f (u, |) is assumed to be one-to-one with respect to

control effect u eQ(u)eRk for the fixed vector

|efi(|) e Rm in the feasible domain of u e Q(u).

Fig. 1. The scheme of a nonparametric control system, where (t) denotes continuous time, and the subscript t indicates discreet time moments of measurements

Рис. 1. Схема непараметрической системы управления: t - непрерывное время; t в качестве индекса - дискретное время контроля измерения

Nonparametric combined control algorithms.

A nonparametric control algorithm is based upon Nadaraya and Watson [9] nonparametric estimates of regression. A nonparametric dual control algorithm in the form [10] for a single-input case is represented by the expression:

ut+i =

t S u Ф i=1 z * xt+1 xi л Ф J z л ц,+1 _ц,

V cx cц V ct J

t SФ i=1 V * л xt+1 xi z Ф V ц,+1 _ц, л

ctx J ¿ц ct J

+ Aut+l, i = 1, t.

(1)

For the multiple-input object a nonparametric dual control algorithm can be expressed by

S

i=1 j=1

( * j j A x J _ xi

V "t j

ПФ

j=1

( j j Л

ц _nu

V ct J

t n Л.*j л m Л. j ..1 л +Aut+1,

SПФ

i=1 j=1

V "t J

ПФ

j=1

ц j

V ut J

(2)

•n = 1, k,

where ui are learning sample elements, ®() is a kernel function with the following features:

0 <Ф(-) < œ ; — [фГ^-ct V c J

dx = 1 ; lim — Ф

tc.

z x л

= 8(x );

cx = l x* _ x0

= /2 ||+1 -|p|, (5)

where /2 is a coefficient to be found experimentally: l0 is found from the optimization process

l2 >1.

1^0 = min Ijj-jj |, i = 1, p . Finally we get a new reduced

data set [xf, ui, jj}, i = 1, s, whose size 5 satisfies the inequality s < p .

Judging from our experience, the algorithm (1) is in-

and

bandwidth

sensitive to the sequence of parameters fitting.

Let us pay attention to learning process of the nonparametric dual control algorithm. It was discussed in [10], and is represented by the iterative scheme:

ut+i = u* + Aut+1 . (6) Information about the object under control is contained in u*, and learning capabilities are fulfilled by the search additive Auf_

t+1

Aut+1 = a ( x*+l _ xt+1 ;

(7)

v is an arbitrary argument; cxt, cJ are bandwidth parameters that satisfy the following convergence conditions [8; 10]:

ct > 0 ; lim tct ^ to ; lim ct ^ 0.

t^w t

Estimate (1) is control algorithm for a combined system described by the following equation:

xt = f (ut, J, ), (3)

where the shape of the function f (•) is unknown, but we

know that f (•) is a continuous one-to-one stationary

function up to its arguments [11].

A single-dimension nonparametric dual algorithm functioning can be explained as follows. A learning process begins with the first ternary of observed variables:

x , Uj, Jtj.

The initial stage of the control process is mostly concentrated on active accumulation of information in order to bring the object to the target state.

Here two key problems should be solved: how to fit bandwidth parameters cxt and cJ , and which of them should be found first.

The bandwidth parameter ctx is found by means of the

weighted residual calculation [12-14]:

' ' (4)

where a is a coefficient defining search amplitude that should be fitted. We require that Aut+1 ^ 0 with the growth of t.

It is a well-known fact that the output of a discrete dynamic object can be represented as follows [3]:

xt = f ( xt_l

vt _k

■ut).

(S)

In this case xt_

x_ ... xt_k can be interpreted as supplementary incontrollable inputs in terms of previously introduced static object modeling routine. Fig. 2 describes the approach.

In fig. 2 the following notations are given: x* is a reference output variable of the object; t with round brackets is a continuous time variable; subscript t denotes discrete time indices; h, h¡x are random noise in measurement channels corresponding to the variables of the object; £ (t) is an unobservable random effect.

Thus, for the uncontrollable but observable variable xt_, i = 1, k as the input effect of the object the algorithm (1) can be rewritten in the following form [15]:

ut

t+1

t S u Ф i=1 z * xt+1 xi к ПФ j=1 xt_ j xi_j

V cx V cx

tz S Ф i=1 V * л xt+1 xi к ПФ j=1 V л xt_j xi_j

x ct J x ct J

л

(9)

where x° = min| x* - xi |, i = 1, t, and coefficient /1 > 1. After repeating the control process p times one receives the data set {xi, ui, |}, i = 1, p .

Further we reduce the set {x;-, ui, |i}, i = 1, p taking

into consideration only measurements that satisfy the following condition:

+ Aut+1, i = 1, t.

A general control theory of similar objects control is explicated in [2]. Implementation of the corresponding dual control algorithms can be found in [16; 17]. Below we will focus on numerical experiments with a nonpara-metric dual control algorithm (9). During the experiments the object under control will be substituted by either an inertia-free (memory-free) operator or a dynamic operator.

И, , =

It should be noted that the control algorithm does not possess information on the equation (operator) of the object under control excepting the type of the operator. The use of dual control algorithms is presented in [16; 17].

Numerical experiments. To perform the first batch of numerical experiments the object was substituted by the expression

(10)

where x(t) is an output variable; u(t) is a controllable input; ^(t) is observable but uncontrollable effect, taken as the process

^ = 0.5 + 0.3sin(0.2t).

(11)

The control procedure starts with the first point (x, u1, |j). Further data accumulation results in active learning of the control algorithm. As a consequence the object can be more effectively driven to a reference state or follow a reference trajectory.

Functioning of a nonparametric control algorithm is illustrated by fig. 3 and 4. A particular experiment was conducted to demonstrate the ability to follow the stepwise reference trajectory х*(/).

Fig. 4, a shows an enlarged scale of reference and control x(t) processes depicted in fig. 4, b. A nonparametric dual control algorithm can effectively solve the control problem even for a noticeable level of noise. The case when random noise with amplitude up to 3 % of the output value is presented in fig 5.

Let the reference to the control process be given by the expression x* = 2 + sin(0.1/). For the case the corresponding control process is depicted in fig. 6.

Except stepwise functions and continuous reference functions one can construct other references using even random functions. To illustrate capabilities of the control algorithm (1) to follow random reference the following

experiment was carried out (fig. 7). x* is defined here as a sequence of sine function and purely random effect evenly distributed in the interval [0.5; 2.5].

Fig. 2. A control scheme for an object with memory (dynamic object) Рис. 2. Управление объектом с памятью

Fig. 3. Uncontrollable effect ¡¡(t) Рис. 3. Неуправляемое входное воздействие

b

Fig. 4. Control process for a stepwise reference value Рис. 4. Управление при задающем воздействии в виде ступенчатой функции

Fig. 5. Control process in case of 3 % random noise applied to the object output Рис. 5. Управление при задающем воздействии с помехой

Fig. 6. Control process for continuous reference Рис. 6. Управление при задающем воздействии в виде траектории

Experiments with the control algorithm operation make it evident that it is able to deal even with random references. On the contrary, standard P, PI, PID controllers cannot reach the level of control quality, because they are not based on data accumulation and analysis. Moreover, settling time is expected to be much worse for the controllers.

The control process in fig. 7 demonstrates satisfactory quality. That is an exceptional functionality of the control algorithm (1) can be noticed. It should be noted that none

of already existing controllers can reach the same level of control accuracy and velocity.

Let us take into consideration another case when the object is represented by the dynamic operator (8). The equation is accepted in the form of the first-order discrete operator:

xt = f (xt-J, ut). (12)

Particularly, the linear first-order object is described by

xt =P1ut +P2 xt-\, (13)

where p1 and p2 are finite constants.

a

For the case we describe peculiarities of the control procedure. The learning process begins with a pair of measurements, namely (x0, u0) and (x1, u1). The initial phase of control is devoted to data accumulation needed to bring the object to the target state. Further, time to reach the target diminishes to a great extent.

Let us demonstrate functioning of the algorithm (9). Let control reference be a stepwise function. Control process for the reference is depicted in fig. 8.

Fig. 9 demonstrates functioning of the algorithm when reference is a combination of a sine function and a random function. Again, the control process can be qualified as highly effective.

Fig. 7. Control procedure for combined reference containing random noise Рис. 7. Управление при задающем воздействии в виде траектории и случайного задания

Fig. 8. Control process in case of stepwise reference Рис. 8. Управление при задающем воздействии в виде ступенчатой функции

Fig. 9. A random reference test for dynamic system dual control Рис. 9. Результаты управления при случайном задании

Thus, the algorithm (9) is able to control a dynamic object with memory, providing good quality due to data accumulation and proper model-based control synthesis.

Conclusion. The problem of dynamic system control in case of nonparametric uncertainty conditions is discussed in the paper. After re-designation of object variables this problem can be reformulated in terms of multidimensional inertia-free object control. Bandwidth determination techniques for both controllable and uncontrollable input effects are proposed. Two variants of nonparametric control algorithm learning are discussed. Illustrations of some numerical experiments with the algorithm prove that it can be used in various computer-added systems of adaptive control. The key point of the algorithm is the capability to control continuous production processes with discrete-time measurement equipment.

References

1. Feldbaum A. A. Osnovy teorii optimal 'nykh avtomaticheskikh system [Fundamentals of the Theory of Optimal Automatic Systems]. Moscow, Fizmatgiz Publ., 1963, 552 p.

2. Tsypkin Ya. Z. Adaptatsiya i obuchenie v avtomaticheskikh sistemakh [Adaptation and Training in Automatic Systems]. Moscow, Nauka Publ., 1968, 400 p.

3. Eykhoff P. System Identification: Parameter and State Estimation. Chester, England: Wiley, 1974, 555 p.

4. Voronov A. A. Osnovy teorii avtomaticheskogo upravleniya. Chast' 1: Lineynye sistemy regulirovaniya odnoy velichiny [Fundamentals of Automatic Control Theory. Part 1: Linear Control Systems of Single Variable]. Moscow-Leningrad, Energy Publ., 1965, 396 p.

5. Voronov, A. A. Osnovy teorii avtomaticheskogo upravleniya. Chast' 2: Spetsial'nye lineynye i nelineynye sistemy avtomaticheskogo regulirovaniya odnoy velichiny [Fundamentals of Automatic Control Theory. Part 2: Special Linear and Nonlinear Automatic Control Systems of Single Variable]. Moscow-Leningrad, Energy Publ., 1966, 364 p.

6. Metody klassicheskoy i sovremennoy teorii avtomaticheskogo upravleniya. T. 3: Sintez regulyatorov sistem avtomaticheskogo upravleniya [Methods of Classical and Modern Automation Control Theory. Vol. 3: Synthesis of Controllers in Automation Control Systems]. Moscow, MSTU Publ., 2004, 616 p.

7. Metody klassicheskoy i sovremennoy teorii avtomaticheskogo upravleniya. T. 4: Teoriya optimizatsii sistem avtomaticheskogo upravleniya [Methods of Classical and Modern Automation Control Theory. Methods of the classical and modern theory of automatic control. Vol. 4: The Optimization Theory of Automation Control Systems]. Moscow, MSTU Publ., 2004, 742 p.

8. Medvedev A. V. Adaptatsiya v usloviyakh nepara-metricheskoy neopredelennosti Adaptivnye sistemy i ikh prilozheniya [Adaptation in Conditions of Nonparametric Uncertainty. Adaptive Systems and Their Applications]. Novosibirsk, Nauka Publ., 1978, P. 4-34.

9. Nadaraya E. A. Neparametricheskoe otsenivanie plotnosti veroyatnostey i krivoy regressii [Nonparametric Estimation of the Probability Density and the Regression Function]. Tbilisi, Tbil. Univ. Publ., 1983, 194 p.

10. Medvedev A. V. Osnovy teorii adaptivnykh sistem [Fundamentals of Adaptive Systems Theory]. Krasnoyarsk, SibSAU Publ., 2015, 526 p.

11. Vasil'ev V. A., Dobrovidov A. V., Koshkin G. M. Neparametricheskoe otsenivanie funktsionalov ot raspre-deleniy statsionarnykh posledovatel'nostey [Nonparametric Estimation of Functionals from Distributions of Stationary Sequences]. Moscow, Nauka Publ., 2004, 512 p.

12. Bannikova A. V., Medvedev A. V. [On Nonparametric Control Algorithms for a Dynamic System]. Prob-lemy upravleniya i modelirovaniya v slozhnykh sistemakh [Control Problems and Modeling in Complex Systems]. Samara, 2014, P. 15-21 (In Russ.).

13. Medvedev A. V. [The Theory of Nonparametric Systems. Control - I]. Vestnik SibGAU. 2013, No. 2 (48), P. 57-63 (In Russ.).

14. Medvedev A. V. [The Theory of Nonparametric Systems. Control - II]. Vestnik SibGAU. 2013, No. 3 (49), P. 85-90 (In Russ.).

15. Medvedev A. V. [Elements of the theory of Nonparametric Control Systems]. Aktual'nye problemy infor-matiki, prikladnoy matematiki i mekhaniki. V 3 ch. Ch. III Informatika. Novosibirsk-Krasnoyarsk: Sib. Brunch of Ros. Acad. Sci., 1996, P. 87-112.

16. Wenk C. J., Bar-Shalom Y. A. Multiple Model Adaptive Dual Control Algorithm for Stochastic Systems With Unknown Parameters. Automatic Control, IEEE Transactions. 2003, Vol. 25, Iss. 4, P. 703-710.

17. Tse E. Bar-Shalom, Y. An Actively Adaptive Control for Linear Systems With Random Parameters via the Dual Control Approach. Automatic Control, IEEE Transactions. 2003, Vol. 18, Iss. 2, P. 109-117.

Библиографические ссылки

1. Фельдбаум А. А. Основы теории оптимальных автоматических систем. М. : Физматгиз, 1963. 552 с.

2. Цыпкин Я. З. Адаптация и обучение в автоматических системах. М. : Наука, 1968. 400 с.

3. Эйкхофф П. Основы идентификации систем управления. М. : Мир, 1975. 683 с.

4. Воронов А. А. Основы теории автоматического управления. Ч. 1. Линейные системы регулирования одной величины. М. ; Л. : Энергия, 1965. 396 с.

5. Воронов А. А. Основы теории автоматического управления. Ч. 2. Специальные линейные и нелинейные системы автоматического регулирования одной величины. М. ; Л. : Энергия, 1966. 364 с.

6. Методы классической и современной теории автоматического управления : учебник. В 5 т. Т. 3. Синтез регуляторов систем автоматического управления. 2-е изд., перераб. и доп. М. : МГТУ им. Н. Э. Баумана, 2004. 616 с.

7. Методы классической и современной теории автоматического управления. В 5 т. Т. 4. Теория оптимизации систем автоматического управления. 2-е изд., перераб. и доп. М. : МГТУ им. Н. Э. Баумана, 2004. 742 с.

8. Медведев А. В. Адаптация в условиях непараметрической неопределенности. Адаптивные системы и их приложения. Новосибирск : Наука. СО АНССР, 1978. С. 4-34.

9. Надарая Э. А. Непараметрическое оценивание плотности вероятностей и кривой регрессии. Тбилиси : Изд-во Тбил. ун-та, 1983.

10. Медведев А. В. Основы теории адаптивных систем. Красноярск, 2015. 526 с.

11. Васильев В. А., Добровидов А. В., Кошкин Г. М. Непараметрическое оценивание функционалов от распределений стационарных последовательностей. М. : Наука, 2004.

12. Банникова А. В., Медведев А. В. О непараметрических алгоритмах управления динамической системой // Проблемы управления и моделирования в сложных системах : Тр. XVI Междунар. конф. Самара, 2014. С. 15-21.

13. Медведев А. В. Теория непараметрических систем. Управление -I// Вестник СибГАУ. 2013. Вып. 2 (48). С. 57-63.

14. Медведев А. В. Теория непараметрических систем. Управление - II // Вестник СибГАУ. 2013. Вып. 3 (49). С. 85-90.

15. Медведев А. В. Элементы теории непараметрических систем управления // Актуальные проблемы информатики, прикладной математики и механики. В 3 ч. Ч. 3. Информатика. Новосибирск ; Красноярск : Изд-во СО РАН, 1996. С. 87-112.

16. Wenk C. J., Bar-Shalom Y. A multiple model adaptive dual control algorithm for stochastic systems with unknown parameters Automatic Control // IEEE Transactions. 2003. Vol. 25, iss. 4. P. 703-710.

17. Tse E. Bar-Shalom. Y. An actively adaptive control for linear systems with random parameters via the dual control approach Automatic Control // IEEE Transactions. 2003. Vol. 18, iss. 2. P. 109-117.

© Agafonov E. D., Shishkina A. V., 2017