ON NONPARAMETRIC CONTROL OF A DYNAMIC SYSTEM

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Introduction. The problem to be solved in the paper Problem statement. The following notations are inis the design and implementation of nonparametric con- trol algorithms. An object under control is assumed to be inertia-free and it is described by an equation with an untroduced: let u = (u1 ... uk x = ( x1 ... xn ) Î Rn be output of the object, ) Î Rk be controlled input effect, known structure with respect to its parameters. Together m = (m1 ... mm ) Î Rm be uncontrollable but observable with the control effect the object is influenced by unconinput, and x* = (x* ... x*) Î Rn be reference of the object trollable but observable input. The nonparametric dual 1 n control algorithms under consideration were investigated for various tactics of determining the bandwidth parame- ters 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 exam- ple of a control device with memory. under control (fig. 1) [8]. In fig. 1 the control device is denoted by ‘yy’, the ob- ject under control is denoted by ‘О’, and random station- ary noise effects influencing both the object itself and the object measurement channels are denoted by xt , ht . We assume these noise effects to be unbiased and to have limited variance. The unknown model curve of the object x = f (u, m) is assumed to be one-to-one with respect to control effect u ÎW(u ) Î Rk for the fixed vector mÎW(m) Î Rm in the feasible domain of u ÎW (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 cm = l m - m0 , (5) t 2 t +1 r 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: where l2 l2 > 1. is a coefficient to be found experimentally: r 0 is found from the optimization process ut +1 = m m0 = min m - m , i = 1, r . Finally we get a new reduced t æ * ö æ ö r i i åu F çç xt +1 - xi ÷÷ F çç mt +1 - mi ÷÷ data set {x , u , m }, i = 1, s, whose size s satisfies the inei cx cm (1) i i i = i=1 è t ø è t ø + Du , i = 1, t. quality s < r . t æ x* - x ö æ m - m ö t +1 åF ç t +1 i ÷F ç t +1 i ÷ Judging from our experience, the algorithm (1) is inc c c c ç i=1 è x ÷ ç m ÷ t t t ø è t ø sensitive to the sequence of x and m bandwidth For the multiple-input object a nonparametric dual control algorithm can be expressed by u = h t +1 parameters fitting. Let us pay attention to learning process of the nonpara- metric dual control algorithm. It was discussed in [10], and t n æ x* j - x j ö m æ m j - m j ö is represented by the iterative scheme: i x Õ m åuhÕF çç i ÷÷ F çç u ÷÷ t = i=1 j =1 è ct ø j =1 è ct ø + Du , (2) ut +1 = u* + Dut +1 . (6) Information about the object under control is cont n æ x* j - x j ö m æ m j - m j ö t +1 * åÕF ç i ÷ÕF ç u ÷ tained in ut , and learning capabilities are fulfilled by the c c ç x i=1 j =1 ÷ ç m ÷ j =1 Du : è t ø è t ø search additive t +1 where ui h= 1, k, t +1 are learning sample elements, F(×) is a kernel Dut +1 = a (x* - xt +1 ) , (7) function with the following features: where a is a coefficient defining search amplitude that 0 £ F(×) < ¥ ; 1 òF æ x ö = 1 ; lim 1 F æ x ö = d( x) ; should be fitted. We require that growth of t. Dut +1 ® 0 with the ç ÷dx ct è ct ø t ®¥ ct è ct ø It is a well-known fact that the output of a discrete ç ÷ v is an arbitrary argument; cx, cm are bandwidth paramedynamic object can be represented as follows [3]: t t x = f (x ... x , u ). (8) ters that satisfy the following convergence conditions [8; 10]: t t -1 t -k t c > 0 ; lim tc ®¥ ; lim c ® 0 . In this case xt -1 ... xt -k can be interpreted as supplet t®¥ t t®¥ t mentary incontrollable inputs in terms of previously Estimate (1) is control algorithm for a combined sys- tem described by the following equation: introduced static object modeling routine. Fig. 2 describes the approach. In fig. 2 the following notations are given: x* is a refxt = f (ut ,mt , xt ), (3) t where the shape of the function f (×) is unknown, but we know that f (×) is a continuous one-to-one stationary erence output variable of the object; t with round brackets is a continuous time variable; subscript t denotes discrete t h t time indices; hu , x are random noise in measurement 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: channels corresponding to the variables of the object; x(t ) is an unobservable random effect. Thus, for the uncontrollable but observable variable x1 , u1 , m1 . The initial stage of the control process is xt -i , i = 1, k as the input effect of the object the algomostly concentrated on active accumulation of informa- tion in order to bring the object to the target state. Here two key problems should be solved: how to fit rithm (1) can be rewritten in the following form [15]: ut +1 = bandwidth parameters cx and cm , and which of them t æ x* - x ö k æ xt - j - xi- j ö t t åuiF ç t +1 i ÷ÕF ç ÷ c should be found first. i=1 ç cx ÷ ç x ÷ j =1 = è t ø è t ø + (9) The bandwidth parameter cx is found by means of the t æ x* - x ö k æ x - x ö t åF ç t +1 i ÷ÕF ç t - j i- j ÷ c c weighted residual calculation [12-14]: ç x i =1 ÷ ç x ÷ j =1 cx = l x* - x0 , (4) è t ø è t ø t 1 t t + Dut +1, i = 1, t. where x0 = min x* - xi , i = 1, t, and coefficient l1 > 1 . A general control theory of similar objects control is t i After repeating the control process r times one receives the data set {xi , ui , mi}, i = 1, r . Further we reduce the set {xi , ui , mi}, i = 1, r taking into consideration only measurements that satisfy the fol- lowing condition: 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. 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 xt +1 = ut +1 + mt +1 , (10) where x(t) is an output variable; u(t) is a controllable input; μ(t) is observable but uncontrollable effect, taken as the process 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 step- wise reference trajectory х*(t). Fig. 4, a shows an enlarged scale of reference х*(t) and control x(t) processes depicted in fig. 4, b. A non- parametric 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. t Let the reference to the control process be given by the expression x* = 2 + sin(0.1t) . For the case the corremt = 0.5 + 0.3sin(0.2t) . (11) The control procedure starts with the first point (x1, u1, m1) . 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. sponding 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 x t experiment was carried out (fig. 7). * 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 μ(t) Рис. 3. Неуправляемое входное воздействие a 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 control- lers cannot reach the level of control quality, because they are not based on data accumulation and analysis. More- over, 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 -1, ut ) . (12) Particularly, the linear first-order object is described by xt = b1ut + b2 xt -1 , (13) where b and b are finite constants. 1 2 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 ran- dom 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 multidimen- sional inertia-free object control. Bandwidth determina- tion 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.

About the authors

E. D. Agafonov

E. D. Agafonov1, A. V. ShishkinaReshetnev Siberian State University of Science and Technology

31, Krasnoyarsky Rabochy Av., Krasnoyarsk, 660037, Russian Federation

A. V. Shishkina

Siberian Federal University

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

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