CC BY
6
MSC 93E12, 68W25, 65G40
DOI: 10.14529/ mmp 180114
APPROXIMATION OF THE SOLUTION SET FOR A SYSTEM OF NONLINEAR INEQUALITIES FOR MODELLING A ONE-DIMENSIONAL CHAOTIC PROCESS
A.S. Sheludko, South Ural State University, Chelyabinsk, Russian Federation, sheludkoas@susu.ru
The paper is focused on the modelling of a one-dimensional chaotic process which dynamics is described by a one-parameter nonlinear map. The problem is to estimate the initial condition and model parameter from measurements corrupted by additive errors. The considered guaranteed (set-membership) approach assumes that the prior information about the unknown variables (initial condition, model parameter and measurement errors) is presented as interval estimates. In this context, the estimation problem can be stated as a problem of solving a system of nonlinear inequalities. Due to the nonlinearity, it is not possible to obtain an exact characterization of the solution set. The developed algorithm computes an outer approximation as a union of non-overlapping boxes.
Keywords: chaotic process; nonlinear modelling; guaranteed approach; interval estimate; outer approximation.
Introduction. Consider the model of a one-dimensional chaotic process
Xk = f (xk-i,X), (1)
where f is a chaotic map [1]. The problem is to estimate the unknown initial condition xo and model parameter A from measurements
Vk = Xk + vk,k =1, 2,..., N, (2)
where vk are measurement errors. The optimization approach is based on the minimization of a cost function that measures the similarity between the data obtained from the model equation (1) and measurements (2) [2]. The most common technique is the least squares method and its modifications (see review in [3]). The main difficulty of the optimization approach is that the cost function becomes extremely complex and has a large number of local minima [4]. Thus it is necessary to use global optimization algorithms, e.g., particle swarm optimization and differential evolution [5]. One of the promising approaches is preprocessing of measurements to specify the set of possible values of the unknown variables (the search set). It decreases the number of local minima of the cost function.
Guaranteed estimation [6-10] assumes that the prior information about the initial condition xo, parameter A and measurement errors vk is presented as interval estimates:
xo e Xo = [ Xo , Xo ], A e Ao = [ Ao , Ao ] , vk e Vk = [ Vk , Vk ]. (3)
The equations (1), (2) and restrictions (3) lead to the following system of nonlinear inequalities:
' Vi - vi < f (xo,A) < Vi - vi,
V2 - v2 < f2(xo,A) < V2 - v2, < ... (4)
,VN - vn < fN(xo, A) < VN - vn,
where fk is the k-fold composition of the map f with itself:
fk (xo, A) = Kff_f(xo,A)...)).
k
Let P* C X0 x A0 be the set of all solutions (x0,A) of the system (4). Due to the
nonlinearity, it is not possible to obtain an exact characterization of the solution set P*
P* Xk Ak
xk A
Xk e Xk = [ Xk , Xk ], A e Ak = [Ak , Ak] .
f
considered as an interval function. In this section, the following notations will be used:
f (X, A) = {u | u = f (x, A),x e X, A e A} , f (x, A) = {u I u = f (x, A), A e A} , f (X, A) = {u I u = f (x, A),x e X} .
The GA is a recurrent procedure that can be used in the forward and backward time directions (denoted by the superscripts H^d " —", respectively).
Forward GA recursions. Suppose that X+, A+ are the prior interval estimates for k = 0. The following steps represent the computation of the interval estimates X+, A+ for k = 1, 2,..., N.
Step 1. The predicted state set Xk/k-i is defined by the interval estimates X+_v A+1 found at the previous time step:
Xk/k-i = f (X+-i, A+-i)-
Step 2. The consistent state set Yk is defined by the observation yk and the interval estimate Vk of the measurement error vk:
Yk = {x I x = yk — v,v e Vk} = [ yk — Vk , yk — Vk ] •
Step 3. The interval estimate X+ of the state xk is the intersection of the predicted state set Xk/k-1 and the consistent st ate set Yk\
X+ = Xk/k-i n Yk.
Step 4. The interval estimate A+ C A+_1 of the parameter A is given by
A+ = {A e A+-i | f (A,X+-i) n X+ = 0} .
Backward. GA recursions. For the last time step, let X— = X+, A~N = AN■ For k = N — 1,N — 2,..., 0 the interval estimates X—, A- are defined by the following steps. Step 1. The interval estimate X— C X+ of the state xk is given by
X— = {x e x+ I f (x, A-+i) n X—+i = 0}.
Step 2. The interval estimate Afc C Afc+1 of the parameter A is given by
A- = {A e A-+1 | f (A,X-) n X-+1 = 0} .
If the prior information (3) is correct, the result of the forward and backward computations is guaranteed: at every time step k found interval estimates always contain the true values of the unknown variables. In this way. the GA can be used to specify the prior interval estimates X+, A+ for the initial condition x0 and parameter A: X- C X+, A- C A+. In the case of incorrect prior information (x0 e X+ or A e A+), the result is the empty sets: X- = 0, A- = 0. Thus, the GA is a procedure to verify if the box P + = X+ x A+ contains any solutions of the system (4) and to compute a more accurate outer approximation P- = X- x A- such that P* C P- C P+.
2. Numerical Example. Consider the chaotic process xk given by the logistic map
Xk
Axk-i(l - Xk-i)
with the initial condition x0 = 0, 3 ^^d parame ter A = 3, 7. The measurement errors vk are pseudo-random numbers with normal distribution, zero mean and standard deviation a = 0,1 (the values are generated by the function randn of Matlab). The available N = 30 measurements yk are shown in Fig. 1. The prior interval estimates are taken as follows:
X0 = [0, 0, 5] , A0 = [3, 4] , Vk = [-3a, 3a].
xk yk
P*
P = X0 x A0
P
P
|Jpp(i) = Xf> x a,
.(0
iei
As mentioned on the previous section, the GA allows to verify if the box
P(i)
contains any
solutions of the system (4) and to compute the box P(i = xq x A 0i) such th at p(i) C p(i),
(i)
0C
(i) 0
A0i) C A0J. As a result, the current approximation can be specified. Xew outer
(i)
approximation is the union
PP
PP
iei
(i),
3,6
3,4
3,2
И II
ОД 0,2 0,3 0,4
Fig. 2. Evolution of the outer approximation
where I C I. Then the same procedure can be recursively applied to the boxes P(l\ i e I. In the considered example, each box is divided into four equal ones. The evolution of the outer approximation is shown in Fig. 2.
Conclusion. Guaranteed approach for modelling a one-dimensional chaotic process leads to solving a system of nonlinear inequalities. The developed algorithm can be applied to construct an outer approximation of the solution set. If the parameter estimation problem is solved by the least squares method, obtained approximation can be used as a set of possible values of the unknown variables. It decreases the number of local minima of the cost function.
Acknowledgements. The work was supported by Act 211 Government of the Russian Federation, contract no. 02.AOS.21.0011.
References
1. Devaney R.L. An Introduction to Chaotic Dynamical Systems. Addison-Wesley, 1989.
2. Bezruchko B.P., Smirnov D.A. Extracting Knowledge from Time Series. An Introduction to Nonlinear Empirical Modelling. Springer, 2010. DOI: 10.1007/978-3-642-12601-7
3. Aguirre L.A., Letellier C. Modeling Nonlinear Dynamics and Chaos: A Review. Mathematical Problems in Engineering, 2009. DOI: 10.1155/2009/238960
4. Jafari S., Sprott J.C., Pham V.-T. et al. A New Cost Function for Parameter Estimation of Chaotic Systems Using Return Maps as Fingerprints. International Journal of Bifurcation and Chaos, 2014, vol. 24, no. 10, 18 p. DOI: 10.1142/S021812741450134X
5. Gotmare A., Bhattacharjee S.S., Patidar R., George N.V. Swarm and Evolutionary Computing Algorithms for System Identification and Filter Design: A Comprehensive Review. Swarm and Evolutionary Computation, 2017, vol. 32, pp. 68-84. DOI: 10.1016/j.swevo.2016.06.007
6. Kurzhanski A.B. Identification - A Theory of Guaranteed Estimates. From Data to Model, 1989, pp. 135-214. DOI: 10.1007/978-3-642-75007-6_4
7. Jaulin L., Kieffer M., Didrit O., Walter E. Applied Interval Analysis. Springer, 2001. DOI: 10.1007/978-1-4471-0249-6
8. Shary S.P. A New Technique in Systems Analysis under Interval Uncertainty and Ambiguity. Reliable Computing, 2002, vol. 8, no. 5, pp. 321-418. DOI: 10.1023/A:1020505620702
9. Raissi T., Ramdani N., Candau Y. Set Membership State and Parameter Estimation for Systems Described by Nonlinear Differential Equations. Automatica, 2004, vol. 40, no. 10, pp. 1771-1777. DOI: 10.1016/j.automatica.2004.05.006
10. Paulen R., Villanueva M., Fikar M., Chachuat B. Guaranteed Parameter Estimation in Nonlinear Dynamic Systems Using Improved Bounding Techniques. European Control Conference, 2013, pp. 4514-4519.
11. Sheludko A.S., Shiryaev V.I. Guaranteed State and Parameter Estimation for One-Dimensional Chaotic System. 2nd International Conference on Industrial Engineering, Applications and Manufacturing, 2016, 4 p. DOI: 10.1109/ICIEAM.2016.7911580
Received December 12, 2017
УДК 519.7 DOI: 10.14529/mmpl80114
АППРОКСИМАЦИЯ МНОЖЕСТВА РЕШЕНИЙ СИСТЕМЫ НЕЛИНЕЙНЫХ НЕРАВЕНСТВ ПРИ МОДЕЛИРОВАНИИ ОДНОМЕРНОГО ХАОТИЧЕСКОГО ПРОЦЕССА
A.C. Шелудько, Южно-Уральский государственный университет, г. Челябинск, Российская Федерация
В работе рассматривается класс моделей одномерных хаотических процессов, заданных в виде однопараметрических нелинейных отображений. Решается задача оценивания начального условия и параметра модели по зашумленным измерениям.
Применение гарантированного (теоретико-множественного) подхода предполагает, что априорная информация о неизвестных переменных (начальном условии, параметре модели и эддитивных ошибках измерений) представлена в виде интервальных оценок. При данных предположениях задачу оценивания можно интерпретировать как задачу поиска решений системы нелинейных неравенств. При этом вследствие нелинейности точное описание множества решений системы невозможно. Результатом разрабатываемого алгоритма является внешняя аппроксимация множества решений в виде объединения непересекающихся подмножеств.
Ключевые слова: хаотический процесс; нелинейная модель; гарантированный подход; интервальная оценка; внешняя аппроксимация.
Литература
1. Devaney, R.L. An Introduction to Chaotic Dynamical Systems / R.L. Devaney. - Addison-Wesley, 1989.
2. Bezruchko, B.P. Extracting Knowledge from Time Series. An Introduction to Nonlinear Empirical Modeling / B.P. Bezruchko, D.A. Smirnov. - Springer, 2010.
3. Aguirre, L.A. Modeling Nonlinear Dynamics and Chaos: a Review / L.A. Aguirre, C. Letellier // Mathematical Problems in Engineering. - 2009. - Article ID 238960.
4. Jafari, S. A New Cost Function for Parameter Estimation of Chaotic Systems Using Return Maps as Fingerprints / S. Jafari, J.C. Sprott, V.-T. Pham et al. // International Journal of Bifurcation and Chaos. - 2014. - V. 24, № 10. - 18 p.
5. Gotmare, A. Swarm and Evolutionary Computing Algorithms for System Identification and Filter Design: a Comprehensive Review / A. Gotmare, S.S. Bhattacharjee, R. Patidar, N.V. George // Swarm and Evolutionary Computation. - 2017. - V. 32. - P. 68-84.
6. Kurzhanski, A.B. Identification - a Theory of Guaranteed Estimates / A.B. Kurzhanski // From Data to Model. - Springer, 1989. - P. 135-214.
7. Jaulin, L. Applied Interval Analysis / L. Jaulin, M. Kieffer, O. Didrit, E. Walter. - Springer, 2001.
8. Shary, S.P. A New Technique in Systems Analysis under Interval Uncertainty and Ambiguity / S.P. Shary // Reliable Computing. - 2002. - V. 8, № 5. - P. 321-418.
9. Raissi, T. Set Membership State and Parameter Estimation for Systems Described by Nonlinear Differential Equations / T. Raissi, N. Ramdani, Y. Candau // Automatica. -2004. - V. 40, № 10. - P. 1771-1777.
10. Paulen, R. Guaranteed Parameter Estimation in Nonlinear Dynamic Systems Using Improved Bounding Techniques / R. Paulen, M. Villanueva, M. Fikar, B. Chachuat // European Control Conference. - 2013. - P. 4514-4519.
11. Sheludko, A.S. Guaranteed State and Parameter Estimation for One-Dimensional Chaotic System / A.S. Sheludko, V.I. Shiryaev // 2nd International Conference on Industrial Engineering, Applications and Manufacturing. - 2016. - 4 p.
Антон Сергеевич Шелудько, ассистент, кафедра «Прикладная математика и программирование:», Южно-Уральский государственный университет (г. Челябинск, Российская Федерация), sheludkoas@susu.ru.
Поступила в редакцию 12 декабря 2017 г.
