Recursive algorithm is proposed for identification single-input-single-output (SISO) fractional order Hammerstein systems with noise in output signal. The estimates are proved to be convergent to the true values with probability one. The results of a simulated example indicate that the proposed algorithm provides good estimates

Ivanov, D.V.

Katsyuba, O.A.

Samara University

RECURSIVE IDENTIFICATION FRACTIONAL ORDER HAMMERSTEIN SYSTEMS WITH NOISE IN OUTPUT SIGNAL С D.V. Ivanov, O.A. Katsyuba Samara State University of Transport, Samara, Russia Recursive algorithm is proposed for identification single-input-single-output (SISO) fractional order Hammer-stein systems with noise in output signal.  The estimates are proved to be convergent to the true values with probability one. The results of a simulated example indicate that the proposed algorithm provides good esti-mates/  Keywords: consistency, fractional order system, Hammerstein system, nonlinear least square, output-error, par-ametric identification, error in equation. Introduction Throughout the last few decades, the identification methods of the linear systems have been ex-plored intensively. But in the field of nonlinear systems, the modelling issue is more complicat-ed and harder.  For many systems, when there is a wide operating area rather than a unique operating point, a linear model cannot be used. In this case non-linear models such as Volterra series, neural net-works, Hammerstein and Wiener-type models can be used. Hammerstein models are composed of a memoryless static nonlinearity followed by a linear dynamical system. They have been used e.g., for modeling biological processes [1], [2], chemical processes [3], and in signal processing applications [4]. Hammerstein models have also been shown to be use-ful for control problems  [5]. As far as the applications of fractional calculus are concerned there is a large number of re-search on viscoelasticity/damping, [6, 7] and chaos/fractals [8], dielectric materials [9], electro-chemical processes and flexible robot [10], traffic in information networks [11]. Due to their long memory behavior, the identification of fractional order models is more diffi-cult compared with those of the integer order models.  In this paper, a Hammerstein model, with a static non-linear part connected to a fractional linear part, is considered. Non-recursive identification methods considered in [12,13,14]. The aim of this paper is to present new method recursive identification the output-error Hammerstein sys-tem with error in the equation.  The rest of the paper is organized as follows. Section 1 presents problem statement and intro-duces some notational conventions. Recursive algorithms are given and their strong consistency is proved in Section 2. To justify theoretical assertions some numerical simulation results are demonstrated in Section 3. Some short concluding remarks are finally given in Section 4. Информационные технологии и нанотехнологии-2016 642 Problem Statement  The structure of Hammerstein systems is shown in figure (1). It consists of a static non-linear part connected to a dynamic linear part:  Fig. 1. Hammerstein system consists of a memoryless static non-linearity followed by a linear dynamical system Let’s consider the fractional Hammerstein system described by the stochastic equations                                        (1) where    the Euler’s function  is defined as , Newton’s binomial is generalized to non-integer orders using the Euler’s function    nonlinear function. When  is unknown, it can be approximated with a polynomial expansion  This class of models includes the class of Hammerstein integer order models. The following assumptions are introduced: 1.  The dynamic system (1) is asymptotically stable. 2. Noises  and  are statistically independent sequences with   a.s., where  is the expectation operator.  3. The sequences  and  are mutually do not depend and do not depend with , , re-spectively.   ,11)(011)(0 iimrmmirmmi xfazbzmm ,iii zy ,)1(0ijjimji zjzm ,0 1 r     ,)1(0ijjimmjim xfjxfm ,011 r  01)( dtte t,)1()1()1(jjj mmm ,)1()1()1(jjj mmm mf mf  .)(Mkkimkim xcxf}{ i }{ i ,0}{ iE ,0}{  iE ,22  iE   22iE E}{ i }{ i iz ixИнформационные технологии и нанотехнологии-2016 643 4. Sequence are random signals with ,   and  a.s., matrix  is restricted positive defi-nite. 5. Ratio     is known apriori. It is required to recursive estimate unknown vector of coefficients  fractional order Ham-merstein system, described by the equation (1) in observable sequences , . Recursive identification algorithm Equation (1) is represented  in the vector form as a linear regression:                                    (2) where      From requirement 2 it follows that  generalized error  has zero mean. We obtain that  variance of generalized error equal to:   }{ ix   )(2 im xfE )( im xf    HNNiTixfTizixfizN 1)()()()()()(1lim,)1(,,)1(01011)(Tijjirjijjijiz zjzj       ,)1(,,)1(0011)()( 11Tijjirrjijjijixf xfjxfj   H22 ab, iy  ix,0 iTiiy     ,)( )()(TTixfTiyi  ,)1(,,)1(01011)(Tijjirjijjijiy yjyj       ,,...,,..., )(0)1(0)(0)1(0000 1TrrTTT aabbab ,)(0iTiii b  .)1(,,)1(01011)(Tijjirjijjijijj   i  ),(1 02002002222bbHbbHbTT  ,)()1()1()11(21)()( rrrrNiTiihhhhEHИнформационные технологии и нанотехнологии-2016 644  , . Let’s define the estimates  of the unknown true value from requirement of minimum consid-ered square variance  weighted [14]:  Theorem1. The estimates of unknown vector  can be obtained by means of the stochastically gradient algorithm of the functional minimization:        (3) where  sequence for which the requirements:  6.   и  at    7. ,  a.s. are met, then the estimates defined by algorithm (3) or  a.s., or  Proof. Function (2) can be represented as:                 , as follows from 1, 5 [15].                                                  In this case the asymptotic continuous deterministic model can be represented as [16]:   . Suppose Lyapunov function is equal to as    . To go from (3) to a continuous model to show that for ,  and  condition hold [16]:    1010)( )1()1( jiijkjjiijmjmkjjEh   NjNjjNNjkmN101limrm ,1 rk ,1̂2 ),( ab  .1min12~   NiTTiibNHby  ,)(ˆ1)(ˆ1)(ˆ)(ˆ)1(ˆ211ibiHibiyiiTTiiii,0ii1 ii0ili.1l1iii 1iii0)(ˆ  ii .)(ˆ  ii    bHbHJTT1002  J   , JV        2  JJVVTT}{ i }{ i }{ iИнформационные технологии и нанотехнологии-2016 645             (4) where for    euclidean norm. For equality (4) a priori requires the boundedness of , which is an implicit requirement for the growth rate of as , and in our case                                                                               From the boundedness sum provided in the 7 and the sequence implies that the sum of which implies the validity of (4). Of the theorems given in [16], it follows that at 1-7, the sequence  is bounded and tends to the points of the set  consists of stationary points of functional [17]. However, theorem 3.15 [17] implies that probable accumulation points of algorithm (3) are the points of set  Let’s show that , i.e. set  consists of a single point . For this purpose let’s take functional                        , where     It is obvious that           ,0,),(ˆ1suplimlim),(0iiiTnkniinTiT,0T ,:max),(knii TkTnk )(ˆ i J   .0lim J )(ˆ i  ,)(),(ˆ1iii ii )(ˆ i,i  )(ˆ i   0:1   VRB rr  J  0)(,0:21 JVRBrr  0:1   rrRB B 0uHuuHuuJTT )(,),...,(11 1Trruuu  ,limTiiiiiyyEH ,0000000111111111111 rrrrrrrrrrHHИнформационные технологии и нанотехнологии-2016 646                                       (4)   where  is a minimal eigenvalue of the regular pencil of matrices [18], i.e.  is the mini-mal root of equation  Assume that  and  are their eigenvectors. Then, , where  are stationary values of function , which can be obtained at , equal to  respectively. Hence, stationary values of func-tion ;  are obtained at points    It follows from (4) that   It remains to show that                                                       (5)                                                 only at one stationary point  The problem of determining minimum of function  is equivalent to the constrained extre-mum problem                                             (6) Problem (6) can be solved by means of Lagrange multiplier method. Then, the required condi-tions will be written as follows                                   (7) where  is Lagrange undetermined multiplier. Solution set of system (7), are  and their eigenvectors . Let’s test matrix  for its positive definiteness. From (4)  it follows that          where  and   are minimal eigenvalue of matrices  and  respectively.  Again, Sturm’s theorem [19] implies that  or                 (8)       ,minmin min0 JuJJuminmin.0)det(   HH max1)1(min1... rr11 1,...,rruuk 1,1 1  rrk  uJ u11 1,...,rruu J   0J  ,...,,..., )1(1)1(1)1(1)2(11 1Trruuuu  .,..., )1(1)1(1)1(1)2(11 111111Trrrrrrrrrrrruuuu.1     112 1,...,:,0  rrJ.01  J.1,minuHuuHuTT,1,0)(uHuuHHT  11 1,...,rr11 1,...,rruu  HH,)1()1(  HH  H)1( H)1(H H  HH)2()1(.)2()1(  HHИнформационные технологии и нанотехнологии-2016 647 From (8) it follows that matrix  is non-negatively defined only at  and (5) is performed , i.e. for all  matrix has negative eigenvalues, hence (3). The main feature of the algorithm which enables us to prove global convergence of the simple sto-chastically gradient algorithm is that function  is limited both from above and below. Among all stationary points of function  only point  is the minimum point, whereas all other points are saddle points and one point is a maximum point. Simulation results The offered algorithm has been realized in Matlab and compared to recursive least square (RLS) method.  The dynamic system is defined  by the equations     The noise-free input is defined as   where  - is white noise. Initial parameters are equal to 0. Fig. 2-4 gives plots of the root mean squared error (RMSE) which is defined by                      Conclusion In this paper, the recursive algorythm of consistent estimation of Hammerstein system has been studied. Simulation results indicate that the RLS method gives biased results. Hammerstein sys-tems have received much attention because of its important applications in signal processing, communications and control systems.  Fig. 2. RMSE of parameter estimates for , : 1-proposed algorithm; 2-RLS;   HH min1 min   HH J J 0    ,2.02.05.0 4.17.017.119.0iiiiii xfxfzzz    ,tanh)( iii xxxf  .iii zy ,75.02.05.0 4211   iiiiii xxi.100ˆ 002020  ii2.0 z 2.0 zИнформационные технологии и нанотехнологии-2016 648  Fig. 3. RMSE of parameter estimates for , : 1-proposed algorithm; 2-RLS;  Fig. 1. RMSE of parameter estimates for , : 1-proposed algorithm; 2-RLS. References  1. Korenberg, M.J. Recent advances in the identification of nonlinear systems: Minimum-variance approxi-mation by Hammerstein models. Proc. IEEE EMBS. – 1991. – Vol.13. – P. 2258-2259. 2. Westwick,  D., Kearney, R.,  Identification of a Hammerstein model of the stretch reflex EMG using sep-arable least squares. Proc. World Congr. Medical Physics and Biomedical Engineering, Chicago, IL. – 2000. – Vol. 3. – P. 1901-1904. 3. Eskinat, E. , Johnson,  S.H., Luyben, W.L. Use of Hammerstein models in identification of nonlinear sys-tems, AIChE Journal. – 1991. – Vol. 37(2). – P. 255–268. 4. Stapleton,  J.C., Bass, S.C. Adaptive noise cancellation for a class of nonlinear dynamic reference chan-nels. IEEE Transactions on Circuits and Systems. – 1985. – Vol.  32(2). – P. 143-150. 5. Lang, Z.H. Controller design oriented model identification method for Hammerstein systems. Automatica. – 1993. – Vol. 29. – P. 767-771. 6. Stiassnie, M. On the application of fractional calculus for the formulation of viscoelastic models.  Applied Mathematical Modelling. – 1979. – Vol. 3. – P. 300-302. 7. Bagley, R.L. Fractional calculus – a different approach to the analysis of viscoelastically damped struc-tures. AIAA Journal. – 1983. Vol. 21. – P. 741-748. 8. Le Mehaute,  A. Fractal Geometries, Theory and Applications, Penton Press, New York, 1991. 9. Reyes-Melo, M.E., Martinez-Vega, J.J., Guerrero-Salazar, C.A., Ortiz-Mendez, U. Application of frac-tional calculus to modelling of relaxation phenomena of organic dielectric materials. Proc Int Conf Solid Dielectrics. – 2004. – Vol. 2.  – P. 530-533. 10. Vinagre BM, FeliuV. Modeling and control of dynamic system using fractional calculus: Application to electrochemical processes and flexible structures. Proc 41-st IEEE Conf Decision Control, Las Vegas, NV; 2002: 214–239. 11. Zaborovsky V., Meylanov R. Informational network traffic model based on fractional calculus. Proc Int Conf Info-tech and Info-net, ICII 2001. –  2001. – Vol.  1. – P.  58–63. 12. Aoun, М., Malti, R., Cois, O., Oustaloup, A. System identification using fractional Hammerstein models. Proc. 15th IFAC World Congress, Barcelona, Spain. – 2002. P. 265-270. 5.0 z 2.0 z7.0 z 2.0 zИнформационные технологии и нанотехнологии-2016 649 13. Li Y, Zhai L, Chen Y, Ahn H. Fractional-order iterative learning control and identification for fractional-order Hammerstein system. Proc. of the 11th World Congresson Intelligent Control and Automation. – 2014. – P. 840–845. 14.  Ivanov DV. Identification discrete fractional order Hammerstein systems. Proc. International Siberian Conference on Control and Communications (SIBCON'2015). – 2015. – P. 1-4.  15. Wilks, S. Mathematical statistics. Princeton University Press, NJ, 1943. 16. Chen, H.F. Stochastic Approximation and Its Applications.  Kluwer, Dordrecht. 2005. 17. Derevitskiy,  D.P., Fradkov,  A.L.  Applied theory of discrete adaptive control systems.  Мoscow: Nauka, 1991 (in Russian). 18. Gantmacher, F.R. The theory of matrices. AMS Chelsea Publishing: Reprinted by American Mathemati-cal Society. 2000. 19. Bellman, R. Introduction to matrix analisys. McGraw-Hill, NJ, 1960.  

Recursive identification fractional order Hammerstein systems with noise in output signal

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Recursive identification fractional order Hammerstein systems with noise in output signal

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