TO NONPARAMETRIC IDENTIFICATION OF DYNAMIC SYSTEMS UNDER NORMAL OPERATION

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Introduction. The main objective of identification theory is the model construction based on input and out- put process variables’ observations while the data about the object is incomplete [1-3]. The article is devoted to dynamic objects identification under nonparametric uncertainty [4; 5], when the dynamical model cannot be identified up to parameters vector due to the lack of priori data. In this case receiving of transient response and following estimation of an object weight function are reasonable. The basis of this paper is Duhamel integral use, due to the principle of superposition [6; 7]. Identification algo- rithms of the object in normal operation conditions are described. Three methods of obtaining weight function estimation using Heaviside function [8; 9], delta-shaped input and arbitrary input are analyzed. Problem formulation. Suppose that object is a dy- namic system and described by the equation [1] tion of unknown order. In this case, for zero initial condi- tions, x(t) is found as t x(t) = ò h(t - t)u(t)d t , (1) 0 where h(t - t) - weight function, that is derivative of transition function h(t) = k¢(t) . xt = f (xt -1, xt -2 ,ut ) , where f (×) - is unknown function; ut - control input variable; xt - output variable. In fig. 1, a block diagram of the dynamic process is illustrated [2], with following notations: xˆt - output of model; ut - control variable; (t) - continuous time; t - Fig. 1. Identification scheme discrete time; xt , ht - random noise acting on the object and output variable measuring channel, with zero mathe- matical expectation and limited dispersion. Variables control is carried out through time inter- val Dt . Thus, it is possible to obtain initial input - output variables sample{xi , ui ,i = 1, s}, where s - sample size. Non-parametric identification algorithm when standard signals can be inputted to the object. Suppose that the object is described by a linear differential equa- Рис. 1. Блок-схема системы идентификации This problem reduces to the weight function estima- tion, so, firstly, it is needed to obtain the transition func- tion. As it was mentioned, weight function can be obtained by various means. First case. Suppose that the object is described by lin- ear differential equation of unknown order. In zero initial conditions, x(t) is found as (1). Transition function is an object reaction on input impact, namely as Heaviside function u(t) = 1(t) . í 1(t) = ì0, u(t) < 0, î1, u(t) ³ 0. (2) After obtaining transition function, it is needed to find its nonparametric estimation [10; 11]: T s æ t - ti ö k (t) = sc å ki H ç c ÷ , (3) s i=0 è s ø where ki - transition function estimate; ki - transition function; ti - discrete time of measurements; s - sample size; cs - kernel smoothing; H - kernel function; T - time observation period [2]. We note that kernel function and kernel smoothing satisfy the following terms [10; 11]: Fig. 2. Delta-shaped function example Рис. 2. Пример дельтообразного входного воздействия ò i 1 ¥ æ t - t ö H ç ÷ dt = 1, lim 1 ¥ æ t - t ö ò i j(t)H ç ÷dt = Third case. If control action and object output are known, weight function may be described by (1). cs -¥ è cs ø s®0 cs -¥ è cs ø In a discrete form: = j(t ), H æ t - ti ö ³ 0, (4) æ s s ö i ç ÷ è cs ø hi = xt - ç åui Dt + å h0 ÷, i = 1, s , (8) è i=1 i=1 ø cs > 0, lim scs ® ¥, lim cs ® 0, where s - sample size; Dt - variables control time intercs ®¥ s®¥ where φ(t ) - an arbitrary function. val; ui - control variable; xt - object output; h0 - value i In particular, kernel function would be considered as Sobolev function (5): of the weight function on previous iteration steps. Thus, nonparametric process model is following: T t s æ t - t ö ì0, t - t > c x (t) = å k H ' ç i ÷ u(t)d t i s s sc ò i c í H = ï æ -(t -ti )2 ö , t - t £ c . (5) s 0 i=1 è s ø è ø ç 2 2 ÷ i s ï 0.827 eç (t -ti ) -cs ÷ or ï cs T t s Since weight function h(t) is derivative of transition xs (t) = sc òå hiu(t)d t, (9) function k(t), then s 0 i=1 where ki - transition function; hi - weight function; sc h(t) = T s k H ' æ t - ti ö c - kernel smoothing; s - sample size; T - observation c å i s i=0 ç ÷ . (6) è s ø s period. Second case. The weight function could be obtained when a delta-shaped function is inputted. It has a step function type (7), Dt - discretization interval (fig. 2): íDt dD (t) = ì 1 , t Î Dt , (7) î where Dt , for example, is an equation Dt = t¢ - 0 , or Dt = t¢ - t ¢ . Identification algorithm under normal object op- eration. Constructing an adaptive object model often re- quires identification of measuring channels under normal object operation [2; 12]. This means that inputted impacts must be small enough so that the effect on production would be minimal. This is necessary for keeping the proc- ess in acceptable limits [8]. Thus, the third case has the priority in solving the problem of nonparametric identification [4; 6]. The fol- lowing algorithm when input impact has sinusoidal type function (as an example) is analyzed below. Computer experiment. Suppose that dynamical ob- ject is described by third-order differential equation. It can be represented as: xt = 0.5xt -3 - xt -2 + xt -1 - 0.5ut . (10) Let us note that the equation (10) is used for obtaining sampling points. Nonparametric algorithm does not as- sume the known form of the differential equation, only information on the linearity of an object is known, in con- trast with [13; 14]. The first method of obtaining weight function is to take the derivative of transition function (fig. 3), if Heaviside function is submitted to the object, then object output is a transitional feature: x(t) = k (t) , further it is necessary to find the value of transition function and weight function according to formulas (3) and (6): In fig. 3: k (t) - transition function, h(t) - weight function. Put known values of transition and weight functions into Duhamel integral (1) and get an object model, fig. 4. Fig. 3. Weight and transition response when u(t) = 1(t) Рис. 3. Весовая и переходная характеристика процесса при u(t) = 1(t) Fig. 4. Weight response when input is a delta-shaped function Рис. 4. Весовая характеристика процесса при подаче на вход объекта дельтообразного входного воздействия Let us change the order of differential equation Consider the case when delta-shaped function integral that describes the object and conduct computer experi- ments. Suppose that the object is described by differential equation of the second order represented as follows: dD (t) differs from 1. As a result, delta-shaped function becomes “pseudo-delta-shaped”, in particular integral of delta function does not equal 1 (fig. 6). Fig. 5 illustrates discretization interval Dt = 0.1, intext = 0.25xt -1 - 0.33xt -2 + 0.33ut . (11) Suppose the integral of delta-shaped function differs gral of delta-shaped function dD (t) equals 1, recovery error w = 4.2 %. from 1. In fig. 6 discretization interval Dt = 0.1, integral of Fig. 5 illustrates the experiment when the integral of delta-shaped function dD (t) > 1, recovery error w = 40 %. delta-shaped dD (t) equals 1, u(t) = 1 △t delta-shaped in- Hence, in order to construct the appropriate model, the following term should be kept - integral of delta-shaped put, x(t) - object output, Dt - discretization interval, function must be equal 1. xˆ(t) - output object model. Note that when Dt Î[0.1; 1] Dt Î[0.1; 1] , input u(t) In conditions of normal object operation as an arbi- trary input signal we take the following function: takes values from 1 to 10, it can conform to the techno- logical requirements. ut = t - t / 2 - A*sin (0.5t) , (12) where A - oscillation amplitude. Fig. 5. Algorithm work with delta-shaped input Рис. 5. Результат работы алгоритма при дельтообразном входном воздействии Fig. 6. Algorithm work with “pseudo-delta-shaped” input Рис. 6. Результат работы алгоритма при «псевдодельтообразном» входном воздействии Let us add a random noise that arising in the channel of output signal measurement x(t) Noise level = 5 %, recovery error w - 0.067, according to the chart and recovery error, this model could be considht = lxt xt , where xt Î[-1; 1] , noise level l = 5 %, 10 % . (13) ered as satisfactory. Thus, table illustrates that lowering oscillation ampli- tude leads to model accuracy decreasing. Calculate the recovery error - w according to the for- Dependence between recovery error mula (14), where 1 s = s å x xt i=1 - arithmetical mean, xˆ(t) - and oscillation amplitude Подпись: s A W 10.5 0.5 % 3.5 1.4 % 2.5 2 % 1.5 3.3 % 1 4.9 % 0.5 9.8 % 0.1 53.4 % object model output: å| xt - xˆt | s w = i=1 , å| xt - x | (14) i=1 Fig. 7 appeals to the following definitions: u(t) - input impact, x(t) - object output, xˆ(t) - model output. Fig. 7. Object output when input is an arbitrary signal Рис. 7. Результаты выхода объекта при произвольном входном воздействии Let us change the input signal and answer the question of how the quality of constructed model depends on the oscillation amplitude: ut = A*sin(0.1t) , (14) where A - oscillation amplitude. We conduct computer experiments, in table following descriptions are analyzed A - oscillation amplitude, w - recovery error. Conclusion. The problem of nonparametric identifica- tion of linear dynamical objects in conditions of incom- plete data is analyzed. The main result of this paper is resolving of identification problem in an object’s normal operation conditions. The nonparametric linear dynamical system models that based on Duhamel integral estimation by means of Nadaraya-Watson statistics are submitted. The main conclusions that could be made on the basis of extensive numerical research of nonparametric models are as follows: although in practice delta function cannot be submitted to the object input, sometimes it is possible to submit delta-shaped input signal and then construct a satisfactory model. Certainly, noise increase in input- output variables measurement and increase in discreteness of input-output variables control, in natural way, worsen accuracy of nonparametric models [15-17]. In addition, it is important to note that the algorithm does not require particular object equation and known differential equation order, all equations that have been described are analyzed as the examples. Thus, algorithm is not dependent on the type of input impact, the main condition is observance of the superposition principle.

Авторлар туралы

M. Kornet

Siberian Federal University Space and Information Technology Institute

26b, Kirensky Str., Krasnoyarsk, 660074, Russian Federation

A. Shishkina

Siberian Federal University Space and Information Technology Institute

Email: nastya.shishkina9666@mail.ru
26b, Kirensky Str., Krasnoyarsk, 660074, Russian Federation

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