Chzhan, E.

Karavanov, A.

Kukartsev, V.

Lonina Nonparametric Dual Control Algorithm For Discrete Linear Dynamic Systems 3, A.

Raskina, A.

English

Siberian Federal University Digital Repository

Applied Methods of Statistical AnalysisNonparametric dual control algorithm for discretelinear dynamic systemsA.V. Raskina1, E. A. Chzhan1, V.V. Kukartsev1,2, A.V. Karavanov1,A.V. Lonina11 Siberian Federal University, Krasnoyarsk, Russia2 Siberian State University of Science and Technology, Krasnoyarsk, Russiae-mail: raskina.1012@gmail.comAbstractThe article devotes to the problem of controlling discrete linear dynamicsystems under non-parametric uncertainty. Control action are calculated inwhich the diﬀerence equation degree of a dynamic process model is reﬁned basedon the rule of selection of signiﬁcant variables. The computational experimentsconﬁrmed the eﬃciency of using non-parametric algorithms to control dynamicsystems in comparison with the PID algorithm and the quasi-optimal controlsystem.Keywords: non-parametric algorithm, discrete dynamic system, priory in-formation.IntroductionDesigning intelligent systems for controlling dynamic objects is one of the importanttask of system analysis. Previously, algorithms of control dynamic objects were de-veloped, in particular, the most widely used standard control algorithms. In somecases, their use is not eﬀective enough. Into contemporary scientiﬁc approaches, op-timal control algorithms are used. However, for their application, as a rule, a prioriknowledge of the structure and parameters of the controlled object is necessary.In the conditions when there is no prior information, the development of newcontrol algorithms is a signiﬁcant scientiﬁc problem. One of the ways to solve thistask to use non-parametric methods [1]. For the application of non-parametricmethods, it is necessary to know only about the quality characteristics of the objectunder study.1 Dual controlDual control was suggested by A.A. Feldbaum [2] and developed on the basis of thetheory of statistical solutions. The theory of dual control was further developed inthe studies of various authors [3], in particular B. Wittenmark [4]. It should benoted that a system in which dual control algorithms are used is an adaptive system,because as current information is received from an object, the quality of functioningincreases.1Novosibirsk, 18-20 September, 20192 Problem set-up and algorithmThe paper considers classes of control objects that can be described by linear diﬀer-ence equations of the form (1).xt = F (xt−1, ..., xt−k, ut, ξt). (1)where F is an unknown linear functional, k is the degree of a diﬀerence equation,which is limited k ≤ kmax. The input ut and output xt of a dynamic object arerepresented by measurements that form a sample of the form xi, ui, i = ¯1, s, where sis the sample size, ui, xi are the measurements of the input and output of the objectat a time instant ti.For a dynamic object that can be described by diﬀerence equation (1), the controlproblem is to ﬁnd the control functions ut. The control function translates the outputof an object xt to a speciﬁed value x∗t in some ﬁnite time tp. In this case, the functionalF is assumed to be unknown from a priori information, but there is a sample ofobservations xi, ui, i = ¯1, s. Non-parametric dual control algorithm has the followingform (2) [5]:us+1 = u∗s + ∆us+1, (2)where u∗s is the component accumulating information about the object under study,∆us+1 = ε(x∗s+1−xs) is the studying search step. The dual control scheme is shownin Figure 1.Figure 1: Dual control scheme of a dynamic objectIn this case, we use the following estimation to get the value u∗s from equation(3):2Applied Methods of Statistical Analysisu∗s =∑si=1 ui · Φ(x∗s+1−xics)·∏kj=1 (xs−j−xi−jcs )∑si=1 Φ(x∗s+1−xics)·∏kj=1 (xs−j−xi−jcs ) , (3)where u∗s is a kernel function, cus , cx[j]s are bandwidths. The optimal bandwidths arefound by minimizing a quadratic error function by using the sliding exam method.3 Essential variablesThe control algorithm for dynamic systems is constructed as follows. The diﬀerentialequation degree of the dynamic process model k is determined on the bases of therule of selection of essential variables. The vale k is further used in the calculation ofcontrol actions in (3), where only selected variables are present.Formulation of the rule: in formula (3), each variable xs−1, ..., xs−k is assigned itsown bandwidths cx[1]s , ..., cx[j]s , the greater bandwidths, the less inﬂuence this variablehas on the output of the object.The algorithm for calculating signiﬁcant variables xi−j is based on the followingscheme. First, the initial value of k is given. The model is constructed by equation.xts =∑si=1 xi · Φ(us−uicus)·∏kj=1 (xs−j−xi−jcs )∑si=1 Φ(us−uics)·∏kj=1 (xs−j−xi−jcs ) , (4)and the relative error W0 is calculated:W =√√√√1ss∑i=1(xi − xsi )2 /s∑i=11s− 1 (mx − xi)2 (5)where mx is an expected value.For each i - th iteration, the following set of actions is performed:1. For each coeﬃcient cx1s , ..., cxks the optimal value is found: cx1s = c∗x1s , cx2s =c∗x2s ,...,cxks = c∗xks ;2. The maximum of all the values obtained is found: cxjmaxs ;3. The model is constructed by the equation (4). The multiplier Φ(us−uicus)isexcluded, taking into account that j is a number for cxjmaxs .4. A relative error Wi is calculated.These actions will be repeated until Wi ≥ Wi−1.We choose a non-gradient multidimensional optimization the Nelder-Mead methodas an optimization algorithm, since this method is eﬀective at a low speed of cal-culation of the minimized function. To select the initial vertices of a deformable3Novosibirsk, 18-20 September, 2019polyhedron, a region of possible values of bandwidths of the kernel functions was set,from which n+ k + 1 points were chosen arbitrarily, where n is the number of inputvariables, k is the degree of the diﬀerence equation, which form the simplex n+ k.In the case of the relation of an object to a class of linear, the algorithm allowsone to determine the structure of the model with an accuracy of parameters.4 Computational experimentTo illustrate the performance of the proposed algorithm, an example will be consid-ered.The results of controlling a dynamic object using a non-parametric dual controlalgorithm (3) were compared with the control results of a typical control algorithm(PID), and with the results of using an quasi-optimal control system (criterion iscontrol time).The control quality was estimated by the control time (tp) the time from thebeginning of the control to the moment when the output quantity diﬀers from of nomore than some given value α. (α = 5%). As an example, we give the work of threecontrol algorithms. The control object is a series connection of three aperiodic links.Detailed control results are shown in table.Table 1: Comparison results of the non-parametric dual control algorithm (NDCA)with the PID controller and the quasi-optimal control system (QOCS)Type of controlsystemstp1 PID 7.92 NDCA 8.1 (at the stage of information accumulation)1.2 (after passing the stage of accumulation of information)3 QOCS 3.5The dualism of the algorithm (3) is as follows. At the ﬁrst control cycles, the mainrole in the formation of control actions is played by the term ∆us+1 from formula(3). But with the accumulation of information about the object, the role of the termu∗s increases. Thus, the use of the non-parametric algorithm after passing throughthe stage of accumulation of information alows it possible to reduce the control timeunder equal conditions of noise and sample size compared to a typical PID controllerand a quasi-optimal control system.ConclusionsNonparametric dual control algorithms for dynamic objects have been developed. Adistinctive feature of the algorithms is the use of information about the order of adiﬀerence equation of a dynamic object when calculating control actions. The taskof controlling dynamic objects is most eﬀectively solved by the proposed algorithm,4Applied Methods of Statistical Analysisas compared with typical control algorithms, in particular, the PID algorithm andthe quasi-optimal control system.References[1] Medvedev A.V. (2015). Fundamentals of the theory of adaptive systems. Pub.SibSAU, Krasnoyarsk.[2] Feldbaum A.A. (1963). Fundamentals of the theory of optimal automaticsystems. Pub. Fizmatgiz, Moskow.[3] Lia Duan, Qianb Fucai, Fuc Peilin. (2008). Optimal nominal dual control fordiscrete-time linear-quadratic Gaussian problems with unknown parameters. Au-tomatica Vol. 44, Issue 1, pp. 119127.[4] Wittenmark B. (1995). Adaptive dual control methods: An overview. In5th IFAC Symposium on Adaptive Systems in Control and Signal Processingpp. 6772.[5] Medvedev A.V. (2013). Theory of non-parametric systems. Control - I Vestnikof the Siberian State University named after Academician M.F. Reshetnev Vol.2 (48), pp. 57-63.5

Nonparametric dual control algorithm for discrete linear dynamic systems

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Nonparametric dual control algorithm for discrete linear dynamic systems

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