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Fractional Hammerstein system identification based on two decomposition principles
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IFAC PapersOnLine 52-13 (2019) 206–210
Fractional Fractional Hammerstein Hammerstein system system Fractional Hammerstein system identification based on two decomposition Fractional Hammerstein system identification based on two decomposition identification based on two decomposition principles identification based on two decomposition principles principles principles ∗∗ Karima Hammar ∗∗ Tounsia Djamah ∗∗ Maamar Bettayeb ∗∗
Karima Hammar ∗∗ Tounsia Djamah ∗∗ Maamar Bettayeb ∗∗ Karima ∗∗ Karima Hammar Hammar ∗ Tounsia Tounsia Djamah Djamah ∗ Maamar Maamar Bettayeb Bettayeb ∗∗ Karima Hammar ∗ Tounsia Djamah ∗ Maamar Bettayeb ∗∗ Karima Hammar Tounsia Djamah Maamar Bettayeb ∗ ∗ University M. Mammeri of Tizi Ouzou, U.M.M.T.O. (L2CSP), ∗ ∗ University M. Mammeri of Tizi Ouzou, U.M.M.T.O. (L2CSP), ∗ University M. Mammeri U.M.M.T.O. Algeria (e-mail: [email protected]). M. [email protected], Mammeri of of Tizi Tizi Ouzou, Ouzou, U.M.M.T.O. (L2CSP), (L2CSP), ∗ University Algeria (e-mail: [email protected], [email protected]). ∗∗ University M. Mammeri of Tizi Ouzou, U.M.M.T.O. (L2CSP), ∗∗ Algeria (e-mail: [email protected], [email protected]). of Sharjah, Dept of Electr. & Comp. Engineering, Algeria (e-mail: [email protected], [email protected]). ∗∗ University of Sharjah, of & ∗∗ Algeria (e-mail: [email protected]). ∗∗ University University [email protected], Sharjah, Dept Dept of Electr. Electr. & Comp. Comp. Engineering, Engineering, U.A.E(e-mail: [email protected]) University of Sharjah, Dept of Electr. & Comp. Engineering, ∗∗ U.A.E(e-mail: [email protected]) UniversityU.A.E(e-mail: of Sharjah, Dept of Electr. & Comp. Engineering, U.A.E(e-mail: [email protected]) [email protected]) U.A.E(e-mail: [email protected]) Abstract: This This paper paper deals deals with with fractional fractional non-linear non-linear system system identification, identification, where where the the HamHamAbstract: Abstract: This paper paper deals with with fractional fractional non-linear system identification, identification, where the the HamAbstract: This deals non-linear system where Hammerstein Controlled Auto-Regression (HCAR) model is considered. The identification process is merstein Controlled Auto-Regression (HCAR) model is considered. The identification process is Abstract: This paper deals with fractional non-linear system identification, where the Hammerstein Controlled Auto-Regression (HCAR) model is considered. The identification process is merstein Controlled Auto-Regression (HCAR) model is considered. The identification process is derived for the regression form of the HCAR system based on the Over-parametrization principle derived for the regression form of the HCAR system based on the Over-parametrization principle merstein Controlled Auto-Regression (HCAR) model is considered. The identification process is derived for the regression form of the HCAR system based on the Over-parametrization principle derived for the regression form of the HCAR system based on the Over-parametrization principle and the Key-term separation principle. Levenberg-Marquardt algorithm combined with each of and the Key-term separation principle. Levenberg-Marquardt algorithm combined with each of derived for the regression formto ofidentify the HCAR system based on the Over-parametrization and the Key-term separation principle. Levenberg-Marquardt algorithm combined withprinciple eachtest of these principles is developed the fractional HCAR system. Various simulations and the Key-term separation principle. Levenberg-Marquardt algorithm combined with each of thesethe principles is separation developed to to identifyLevenberg-Marquardt the fractional fractional HCAR HCAR algorithm system. Various Various simulations test and Key-term principle. combined withIFAC eachtest of these principles is developed identify the system. simulations test these principles is developed to identify the fractional HCAR system. Various simulations the efficiency of the optimization method based on these principles. Copyright © 2019 the efficiency of the optimization method based on these principles. Copyright © 2019 IFAC these principles is developed to identify the fractional HCAR system. Various simulations test the efficiency of the optimization method based on these principles. Copyright © 2019 IFAC the efficiency of the optimization method based on these principles. Copyright © 2019 IFAC © 2019, IFAC (International Federation of Automatic by Elsevier Ltd. All the efficiency of the optimization method based Control) on theseHosting principles. Copyright © rights 2019 reserved. IFAC 1. INTRODUCTION INTRODUCTION for both both principles. principles. 1. for 1. INTRODUCTION INTRODUCTION for both principles. Theboth paper is organized organized as as follows: follows: Section Section 22 provides provides some some 1. for principles. The paper is 1. INTRODUCTION for both principles. The paper is organized as follows: Section 22 provides some theoretical concepts on fractional calculus. In Section 3, The paper is organized as follows: Section provides some The class of nonlinear systems can be conveniently detheoretical concepts onasfractional fractional calculus. In Section Section 3, The class of nonlinear systems can be conveniently deThe paper isconcepts organized follows: Section 2along provides some theoretical on calculus. In 3, The class of nonlinear systems can be conveniently dethe fractional HCAR system is presented with the theoretical concepts on fractional calculus. In Section 3, The class of nonlinear systems can be conveniently described using block-oriented models such as Hammerstein the fractional fractional HCARon system is presented presented along with the the scribed using block-oriented models such as Hammerstein concepts fractional calculus.along In Section 3, The class of nonlinear systems can(Ding be conveniently de- theoretical the HCAR system is along with scribed using block-oriented models such as problem definition, and the identification identification principles are the fractional HCAR system is presented with the scribed using block-oriented models such aset Hammerstein (Wang (2016)), Wiener models etHammerstein al. (2016)), (2016)), problem definition, and the principles are (Wang (2016)), Wiener models (Ding al. the fractional HCAR system is presented along with the scribed using block-oriented models such as Hammerstein problem definition, and the identification principles are (Wang (2016)), WienerHammerstein-Wiener models (Ding (Ding et et al. al.(Wang (2016)), describeddefinition, in Section Section and 4. In Inthe Section 5, the the proposed proposed method identification principles are and their their(2016)), combinations et problem (Wang Wiener models (2016)), described in 4. Section 5, method and combinations Hammerstein-Wiener (Wang et problem definition, and the identification principles are (Wang WienerHammerstein-Wiener models(Giordano (Ding etetal. (2016)), described in Section 4. In Section 5, the proposed method and their(2016)), combinations Hammerstein-Wiener (Wang et is developed. Section 6 compares the method effciency for described in Section 4. In Section 5, the proposed method and their combinations (Wang et al. (2008)), Wiener-Hammerstein al. (2015)), is developed. developed. Section4.66 In compares the method effciency for al. (2008)), Wiener-Hammerstein (Giordano et al. (2015)), described in Section Section 5, the proposed method and their combinations Hammerstein-Wiener (Wang et is Section compares the method effciency for al. (2008)), Wiener-Hammerstein (Giordano et al. (2015)), both principles, and finally, some conclusions and perspecis developed. Section 6 compares the method effciency for al. (2008)), Wiener-Hammerstein (Giordano et al. (2015)), etc... They separate the linear part and the nonlinear one both principles, and finally, finally, some conclusions conclusions and perspecetc... They separate the linear part and the nonlinear one developed. Section 6 compares the method and effciency for al. Wiener-Hammerstein (Giordano et al. (2015)), both principles, and some and perspecetc... They and one tives are drawn. drawn. both principles, and finally, some conclusions perspecinto(2008)), different blocks.the etc... They separate separate the linear linear part part and the the nonlinear nonlinear one is tives are into different blocks. both principles, and finally, some conclusions and perspecetc... They separate the linear part and the nonlinear one tives are are drawn. drawn. into different blocks. On the otherblocks. hand, most most real real systems exhibiting exhibiting long long tives into different 2. PRELIMINARIES PRELIMINARIES On other hand, into different blocks. 2. On the the other hand, and mostinfinite real systems systems exhibiting long tives are drawn. 2. On the other hand, most real systems exhibiting long memory transients dimensional structure PRELIMINARIES memory transients and infinite dimensional structure 2. PRELIMINARIES On the other hand, mostfractional real systems exhibiting long memory transients and infinite dimensional structure memory transients and infinite dimensional structure are well described using models. For the last 2. have PRELIMINARIES are well described using fractional models. For the last Different definitions definitions been proposed proposed in in the the literature literature memory transients and infinite dimensional structure are well described using fractional models. For the last Different have been decades, great effort has been devoted for the fractional are well described using fractional models. For the last Different definitions have been proposed in the literature decades, great effort has been devoted for the fractional for the differintegral operator of fractional (non-integer) Different definitions have been proposed in the literature are well described using fractional models. For the last decades, greatidentification effort has has been been devoted for the the Cui fractional for the the differintegral differintegral operator of fractional (non-integer) linear system system (Dai et al. al. (2016); (2016); et al. al. for decades, great effort devoted for fractional Different definitions have been of proposed in the literature operator fractional (non-integer) linear (Dai et et order. The Grünwald-Letnikov (GL) difference defined in for theThe differintegral operator of fractional (non-integer) decades, greatidentification effort has been devoted for the Cui fractional linear system identification (Dai et al. (2016); Cui et al. order. Grünwald-Letnikov (GL) difference defined in linear system identification (Dai et al. (2016); Cui et al. (2017); Djamah et al. (2008, 2013)) for the differintegral operator of fractional (non-integer) order. The Grünwald-Letnikov (GL) difference defined in (2017); Djamah et al. (2008, 2013)) Miller et al. (1993); Dzieliński (2008); Kilbas et al. (2006) is order. The Grünwald-Letnikov (GL) difference defined in linear system identification (Dai et al. (2016); Cui et al. (2017); Djamah et al. (2008, 2013)) Miller et et al. Grünwald-Letnikov (1993); Dzieliński Dzieliński (2008); (2008); Kilbas et al. al.defined (2006) in is (2017); Djamah et al.which (2008,consists 2013)) of Hammerstein model aa static nonlinear order. The (GL) difference Miller al. (1993); Kilbas et (2006) is Hammerstein model which consists of static nonlinear mostly used in the discrete case, it is expressed as follows: Miller et al. (1993); Dzieliński (2008); Kilbas et al. (2006) is (2017); Djamah et al.which (2008, 2013)) Hammerstein model consists of static mostly used in the discrete discrete case, it is Kilbas expressed as follows: follows: part followed followed by a linear linear block has been been popular in many many mostly Hammerstein model which consists of a a popular static nonlinear nonlinear Miller al. (1993); Dzieliński (2008); et al. is part by block has in in case, it as mostlyetused used in the the discrete case, it is is expressed expressed as(2006) follows: Hammerstein model which consists of a popular static nonlinear part followedincluding by aa linear linear block has been popular in many applications control, signal processing and compart followed by a block has been in many mostly used in the discrete case, it is expressed as follows: applications including control, signal processing and compart followed by a linear block hasal.been popular in many ( k applications including control, signal processing and comk ) (α) applications including control, signal processing and communication (Vörös (2010); Cao et (2014)). The identifik 1 ∑ munication (Vörös (2010); Cao et al. (2014)). The identifiα k (−1)jjj ( )x((k − j)h), (α) α x(kh) = 1 ∑ applications including control, signal processing and comk munication (Vörös (2010); Cao et al. (2014)). The identifi∑ (1) ∆ α munication (Vörös (2010); Cao et al. (2014)). The identifi∑ α 1 cation of Hammerstein system has focused in major studies α − j)h), (1) x(kh) = (−1) ∆ ( α 1 α α k cation of Hammerstein system has focused in major studies jj j )x((k α munication (Vörös (2010); Cao et al. (2014)). The identifi∑ x((k − j)h), (1) x(kh) = =h (−1) ∆α cation of Hammerstein system has focused in(2015); major studies j h x((k − j)h), (1) x(kh) (−1) ∆ α 1 j=0 on the classical integer case as Chen et al. Liu et cation of Hammerstein system has focused in major studies α j=0 j=0 α j j α h on the classical integer case as Chen et al. (2015); Liu et (1) = hα j=0 (−1) j x((k − j)h), cation Hammerstein system has focused major studies j=0 on (2010); the of classical integer case as asand Chen etfew al.inwork (2015); Liu et et where ∆∆is x(kh) on the classical integer case Chen et al. (2015); Liu the discrete fractional-order difference operaal. Ding et al. (2013), only considers j h j=0 al. (2010); Ding et al. (2013), and only few work considers where ∆ is the discrete fractional-order difference operaon the classical integer case as Chen et al. (2015); Liu et ∗+ al. (2010); Ding et al. (2013), and only few work considers ∗+ where ∆ Ris is∗+ the discrete fractional-order difference period, operaal. (2010); Ding et al. (2013), and only few work considers where the fractional nonlinear case. ∆ the discrete fractional-order difference operator, α ∈ the fractional order, h the sampling the fractional nonlinear case. tor, α ∈ the fractional order, h the sampling period, ∗+ al. (2010); Ding et al. (2013), and only an few iterative work considers where ∆R isnumber the discrete fractional-order difference opera∗+ the fractional nonlinear case. tor, α ∈ R the fractional order, h the sampling period, the fractional nonlinear case. and k the of samples for which the approximation Recently, in Rahmani et al. (2017), linear tor, α ∈ R the fractional order, h the sampling period, ∗+ Recently, in Rahmani et al. (2017), iterative linear and k the number of samples for which the approximation the fractional nonlinearwith case. tor, α ∈ R the fractional order, h the sampling period, Recently, in algorithm Rahmani et al.a Lyapunov (2017), an anmethod iterative linear and k the number of samples for which the approximation optimization has been Recently, in Rahmani et al. (2017), an iterative linear of the derivative Throughout this paper, we k the numberis ofcalculated. samples for which the approximation optimization algorithm with Lyapunov method haslinear been and of derivative Throughout this paper, Recently, in algorithm Rahmani et al. (2017), anmethod iterative and k the numberis ofcalculated. samples for which the approximation optimization algorithm with aa Lyapunov Lyapunov method has been of the the derivative is calculated. Throughout this paper, we we optimization with a has been assume, without loss of generality, that h = 1. proposed for the fractional Hammerstein system identifiof the derivative is calculated. Throughout this paper, we proposed for the fractional Hammerstein system identifiassume, without loss of generality, that h = 1. optimization algorithm with a iterative Lyapunovalgorithm method has been of the derivative is calculated. Throughout paper, we proposed for the fractional Hammerstein system identifiassume, without loss of generality, that h h from =this 1. the proposed for the fractional Hammerstein system identifiThe binomial term in (1) can be obtained followcation; in Ivanov (2015) the is used, assume, without loss of generality, that = 1. cation; in Ivanov (2015) the iterative algorithm is used, The binomial term in (1) can be obtained from the followproposed for the fractional Hammerstein system identifiassume, without loss of generality, that h = 1. cation; in Ivanov (2015) the iterative algorithm is used, The binomial term in (1) can be obtained from the followcation; inthe Ivanov (2015) theis iterative algorithm is Lia used, however, fractional order not estimated and in et Therelation: binomial term in (1) can be obtained from the following however, fractional order is not estimated and Lia et relation: cation; inthe Ivanov (2015) the is used, The binomial term in (1) can be obtained from the followhowever, the fractional order isiterative noton estimated and in in Lia et ing ing al. (2012) (2012) the identification based thealgorithm subspace method however, the fractional order is not estimated and in Lia et ing relation: relation: al. the identification based on the subspace method however, the orderbased is noton estimated andalgorithm in Lia et ing relation: al. (2012) (2012) thefractional identification based on the subspace subspace method al. the identification the method was developed. The particle swarm optimization ) ( was developed. The particle swarm optimization algorithm 11 f or jj = 0 al. (2012) theinidentification based on the subspace method ) ( was developed. The particle swarm optimization algorithm was developed. The particle swarm optimization algorithm is also used Hammar et al. (2015). α ) ( (α) = 1 − j + 1) fff or or jj = =0 0 α(α − 1)...(α is also used in Hammar et al. (2015). 1 or = 0 (2) was developed. The particle swarm optimization algorithm is also used in Hammar et al. (2015). α α(α − 1)...(α − j + 1) (2) (α is alsoobjective used in Hammar et al. (2015). The of this paper is to contribute to the fffor 1 − or jjj > = 0. 0 jj ) = α(α − − 1)...(α 1)...(α − j + 1) The objective of this paper is to contribute to the = (2) or > 0. α(α j + 1) = (2) is also used in Hammar et al. (2015). α j! The objectiveofof ofthethis this paper is isHammerstein to contribute contribute to the the ff or jj > 0. identification fractional Controlled The objective paper to to jj = j! or > 0. α(α − 1)...(α − j + 1) (2) identification of the fractional Hammerstein Controlled j! The objective of this paper is to contribute to the j! f or j > 0.gives an identification of of the the fractional Hammerstein Controlled j The fractional difference operator of Equation (1) identification fractional Hammerstein Controlled Auto-Regression (HCAR) model. It is performed based The fractional difference operator of Equation (1) gives an j! Auto-Regression (HCAR) model. It is performed based identification of the fractional Hammerstein Controlled Auto-Regression (HCAR) model. It is based The fractional difference operator of (1) gives overall characterization this function at each k, Auto-Regression (HCAR) model. Itprinciples: is performed performed based The fractional differenceof operator of Equation Equation (1)instant gives an an on the regression form using two the Overoverall characterization of this function at each instant k, on the regression form using two principles: the OverAuto-Regression (HCAR) model. Itprinciples: is performed based The fractional difference operator offunction. Equation (1)instant gives an on the regression form using two the Overoverall characterization of this function at each k, parametrization principle and the Key-term separation by considering all past values of this For practical on the regression form using two principles: the Overoverall characterization of this function at each instant k, by considering all past values of this function. For practical parametrization principle and the Key-term separation on the regression form using two principles: the Overoverall characterization of this function at each instant k, parametrization principle and the the Key-termalgorithm separation by considering considering allnumber past values values of this this function. function. Foraccount practical principle. The robust Levenberg Marquardt is realizations, the of samples taken into is parametrization principle and Key-term separation by all past of For practical realizations, the number of samples taken into account is principle. The robust Levenberg Marquardt algorithm is parametrization principle and the Key-term separation by considering all past values of this function. For practical principle. The The robust Levenberg Levenberg Marquardt algorithm of is realizations, realizations, the number number of samples samples takenmemory into account account is limited to a number L, called the system length. principle. robust Marquardt algorithm is the of taken into is developed for identifiying the HCAR model parameters limited to a number L, called the system memory length. developed for identifiying the HCAR model parameters of principle. The robust Levenberg Marquardt algorithm is realizations, the Equation number of(1) samples taken into account is developed for identifiying the HCAR model parameters of limited to aa number L, called the system memory length. In what follows, is used for the simulation of developed for identifiying the HCAR model parameters of limited to number L, called the system memory length. the linear part, the nonlinear part and its fractional order. In what follows, Equation (1) is used for the simulation of the linear part, the nonlinear part and its fractional order. developed for identifiying the HCAR model parameters of limited to a number L, called the system memory length. the linear part, the nonlinear part and its fractional order. In what follows, Equation (1) is used for the simulation of Numerical simulations investigate the method efficiency fractional order systems. the linear part, the nonlinear part and fractional order. In what follows, Equation (1) is used for the simulation of Numerical simulations investigate theits method efficiency order the linear part, the nonlinear part and itsmethod fractional order. fractional In what follows, Equation (1) is used for the simulation of Numerical simulations investigate the method efficiency fractional order systems. systems. Numerical simulations investigate the efficiency fractional order systems. Numerical simulations investigate the method efficiency fractional order systems. 2405-8963 © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
Copyright © 2019 IFAC 211 Copyright 2019 IFAC 211 Control. Peer review© responsibility of International Federation of Automatic Copyright © under 2019 IFAC IFAC 211 Copyright © 2019 211 10.1016/j.ifacol.2019.11.106 Copyright © 2019 IFAC 211
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Karima Hammar et al. / IFAC PapersOnLine 52-13 (2019) 206–210
3. PROBLEM DESCRIPTION A fractional Hammerstein CAR (HCAR) is represented in Fig.1. It consists of a static non-linear block followed by a fractional linear dynamical block and it is described by the following equation: A (z) y (k) = B (z) u ˜ (k) + n (k) (3) where y(k) is the measured output, u ˜(k) is the output of the non-linear block assumed to be a non-linear function of a known basis of the system input u(k), and n(k) is a disturbance noise, with: f (u(k)) = [ f1 (u(k)), f2 (u(k)), ..., fnp (u(k)) ] ∈ Rnp u ˜(k) = f (u(k)) = p1 f1 (u(k)) + p2 f2 (u(k)) + · · · + pnp fnp (u(k)) =
np ∑
pj fj (u(k))
j=1
(4)
The linear block transfer function of the CAR model is described by the fractional commensurate order polynomials A(z) and B(z) of the shift operator z −1 as follows: A (z) = 1 + a1 z −α + a2 z −2α + · · · + ana z −na α B (z) = b1 z −α + b2 z −2α + · · · + bnb z −nb α
n(k)
u(k)
f (.)
u˜(k)
(5)
1 A(z)
B(z) A(z)
+
y(k)
form. The objective of this work is to develop an identification method to estimate the parameter vectors a, b, p, and the fractional order α. 4. IDENTIFICATION PRINCIPLES Different identification models can be derived for fractional HCAR system output of equation (6) based on regression models Wang et al. (2009); Vörös (2003). Two identification principles have been reported for the identification of classical nonlinear systems; the first one is the Over-parameterization principle and the second one is the key term separation principle. In this study, these last are investigated for the fractional case. For the Over-parametrization identification principle Wang et al. (2009), the system output in (6) can be written under the form: (7) y(k) = ξ(α, k)T θ˜1 + n(k) Where the parameter vector θ˜1 is defined as follows: θ˜1 = [ a b p2 b · · · pnp b ]T ∈ Rnθ1 , nθ1 = na + nb np (8) and the information vector ξ(α, k) can be written as: ξ(α, k) = [φ(α, k)T ∆α f (u(k − 1)) · · · ∆α f (u(k − nb ))]T (9) where φ(α, k) = [−∆α y(k − 1) − ∆α y(k − 2) · · · − ∆α y(k − na )]T In this case, the model parameter vector θ˜1 to be estimated contains the cross-product of the coefficients of the nonlinear block and the linear block; this results in a large dimension of the unknown parameter vector (nθ1 = na + nb np ) and a redundancy of the estimation of the parameters . In order to solve this problem, the Key-term separation principle is adopted Vörös (2003); Wang et al. (2013). This last allows the separation of the linear part and nonlinear part and at the same time, avoids the occurence of the cross-products coefficients. Let us choose ∆α f1 (u(k − i)) as the key term and let us write the Equation (6) as a new regression model as follows:
Fig. 1. Fractional Hammerstein system
y(k) = φks (α, k)T θ˜2 + n(k) φks (α, k) = [φ0 (α, k), ψ(α, b, k)]T
Let us define the parameter vectors a, b and p of the HCAR system with: b = [ b1 , b2 , · · · , bnb ], a = [ a1 , a2 , · · · , ana ], p = [ p1 , p2 , · · · , pnp ].
In order to get unique parameter estimates, we assume that p1 = 1. The substitution of the polynomials A(z) and B(z) in equation (3) leads to a recursive equation as follows: nb na ∑ ∑ y(k) = − ai ∆α y(k − i) + bi ∆α f (u((k − i)) + n(k) i=1
= −
na ∑ i=1
+ n(k)
i=1
α
ai ∆ y(k − i) +
np ∑ j=1
pj
nb ∑ i=1
bi ∆α fj (u((k − i))
(6) The overall output of the HCAR system contains the parameters of the linear and the nonlinear part and their cross-products. It can be expressed under a regression 212
207
−∆α y(k − 1) −∆α y(k − 2) .. .
(10) (11)
α −∆ y(k − na ) φ0 (α, k) = α ∈ Rna +nb ∆ f1 (u(k − 1)) ∆α f (u(k − 2)) 1 .. . ∆α f1 (u(k − nb ))
(12)
bj ∆α f2 (u(k − j)) α bj ∆ f3 (u(k − j)) j = 1, · · · , nb (13) ψj (α, b, k) = .. . α bj ∆ fnp (u(k − j))
θ˜2 = [a
b
p2 · · · pnp ]T ∈ Rnθ˜2 , nθ˜2 = na + nb + np − 1 (14)
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Karima Hammar et al. / IFAC PapersOnLine 52-13 (2019) 206–210
The difficulty here is that the information vector φks (α, k) in (10) contains unknown variables, in occurence the coefficients bj (j = 1, 2, · · · , nb ) and the fractional order α. This difficulty is circumvented by replacing the coefficients bj and α at iteration l, by their respective estimates bˆj and α ˆ obtained at iteraion (l − 1). These two principles are combined with Levenberg Marquardt algorithm in the following section and their performances are compared.
Levenberg-Marquardt (LM) is a nonlinear optimization method, based on the computation of the Gradient and the Hessian (Marquardt (1963); Djamah et al. (2013)); the optimal value of θ is obtained by minimizing the quadratic criterion:
J(θ) =
ε(k)
(15)
2
k=1
where ε(k) = y(k) − yˆ(k) is the output prediction error and yˆ(k) is the corresponding output estimate. The LM update rule is given as follows: θ
(l+1)
=θ
(l)
−
{[
′′
l
J (θ ) + λI
]−1
′
l
J (θ )
}
(16)
with J , J are respectively the gradient and the Hessian, λ is a monitoring parameter. Let us combine this algorithm (LM) with the described principles previously. For the Over parametrization principle, θ = θ1 = [ θ˜1 , α ], and the quadratic criterion J1 is computed as follows: ′
′′
J1 =
N ∑
k=1
ˆ [ξ(α, k)T θ˜1 − y(k)]2
k=1
The LM algorithm is applied to equations (16)-(19) by replacing θ˜1 by θ˜2 and using the key term principle equations (10)-(14). 6. SIMULATION EXAMPLE
5. LEVENBERG MARQUARDT ALGORITHM
K ∑
For the key-term separation principle, the optimization is performed by minimizing the quadratic criterion function J2 : N ∑ ˆ J2 = [φTks (α, k)θ˜2 − y(k)]2 .
(17)
′ ˜ α) with respect to The gradient J1 of the criterion J1 (θ, ˜ θ1 and α is given by the following equations: ) (N ∑ ′ J1 θ˜1 = −2ξ(α, k)T y(k) − ξ(k, α)T θ˜1 ( N k=1 ) ∑ ′ J1 α = −2σyk /α y(k) − ξ(α, k)T θ˜1 k=1 T ˜ σy /α = ∂(ξ(α, k) θ1 ) is the output sensitivity function k ∂α (18) ′′ We compute the Hessian matrix J1 of the cost function with respect to θ˜1 and α as follows: ′′ k)T ξ(α, k) J1 θ˜1 = 2ξ(α, (N ) ∑ (19) ′′ T σyk /α σyk /α J1 α = 2
k=1
For the fractional order α, its sensitivity is calculated numerically using: ∂y = δα σyk /α (20) y(k, α + δα) − y(k, α) ≈ δα ∂α
213
Consider the following fractional HCAR system: y(k) = 0.1∆0.3 y(k − 1) + 0.3∆0.3 y(k − 2) + 0.2∆0.3 f (u((k − 1)) − 0.5∆0.3 f (u((k − 2)) (21) + n(k) The nonlinear block is described by the following polynomial f (u(k)) = u(k) + 0.5u2 (k) + 0.25u3 (k) The input u(k) is a persistent excitation sequence of zero mean and unit variance, and the disturbance n(k) is a white noise sequence of zero mean. The simulations have been carried out for noise-free data, then for data corrupted with noise for different signal-tonoise ratios SN R = 26 dB and SN R = 50 dB. Applying LM algorithm based on the described principles, the system identification results for the noise free case and for SN R = 50 dB and SN R = 26 dB are reported respectively in Table 1, Table 2 and in Table 3. They show that the method based on both principles gives satisfactory results, however the key-term principle is more accurate when the noise level is increased. In order to compare the convergence of the method for each principle, the normalized estimation error δ with δ = ∥θˆ − θ∥/∥θ∥ is computed and its evolution versus the number of iterations is depicted in Fig. 2 for the noise free case, in Fig. 3 for SN R = 50 dB and in Fig. 4 for SN R = 26 dB. From these results we can draw the conclusion that the developed method gives consistent parameters estimates for both principles. L-M algorithm based on the Key-term separation principle estimates the non-linear parameters more accurately than the over-parametrization principle and converge more rapidly, moreover it give satisfactory results even in presence of a high level of noise. The performed simulations illustrate the satisfactory performance of the LM algorithm and the obtained results, show that it the 7. CONCLUSION This paper presents the identification problem of a fractional HCAR system modeled under a regression form. Two principles combined with LM algorithm are investigated. The model based on the over-parameterization principle includes the product terms of the parameters of the static non-linear block and the dynamic linear block which induce redundancy in the estimation process. To
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Table 1. Simulation results for the noise free case key − term 0.100 0.300 0.200 −0.500 0.500 0.250 0.300 1.2e − 31
a1 a2 b1 b2 p2 p3 α J
over − parameterization 0.100 0.300 0.200 −0.500 0.500 0.250 0.300 5.8e − 30
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Table 3. Simulation results for SN R = 26 dB key − term 0.116 0.3126 0.2052 −0.5315 0.4636 0.2213 0.2999 0.4540
a1 a2 b1 b2 p2 p3 α J
exact values 0.100 0.300 0.200 −0.500 0.500 0.250 0.300 −
over − parameterization 0.132 0.415 0.266 −0.613 0.491 0.236 0.198 0.498
exact values 0.100 0.300 0.200 −0.500 0.500 0.250 0.300 −
2 1.6
1.2
Over−parameterization principle δ
Key−term principle
1
δ
Key−term principle
1.5
1.4
Over−parameterization principle
1
0.8
0.5
0.6 0.4
0 0
0.2 0 0
20
40
60 Iteration
80
100
Fig. 2. Evolution of the error versus number of iterations for the noise free case Table 2. Simulation results for SN R = 50 dB key − term 0.101 0.300 0.200 −0.501 0.499 0.249 0.300 3.8e − 4
a1 a2 b1 b2 p2 p3 α J
over − parameterization 0.100 0.4157 0.202 −0.499 0.499 0.248 0.299 6.70e − 4
exact values 0.100 0.300 0.200 −0.500 0.500 0.250 0.300 −
Key−term principle
δ
Over−parameterization principle
0.5 0 0
20
40
60
80 100 Iteration
120
140
160
100
150 Iteration
200
250
300
Fig. 4. Evolution of the error versus number of iterations for SN R = 26dB enhance the efficiency of this algorithm when the level of noise is increased. REFERENCES
1.5 1
50
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Fig. 3. Evolution of the error versus number of iterations for SN R = 50dB overcome this drawback, the key-term principle can be used. This last allows the separation of the linear part and nonlinear part and at the same time, avoids the occurence of the cross-products coefficients. This study confirms the fact that LM algorithm can approximate any non linear system with a good accuracy for both principles and the key-term separation principle 214
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Fractional Fractional Hammerstein Hammerstein system system Fractional Hammerstein system identification based on two decomposition Fractional Hammerstein system identification based on two decomposition identification based on two decomposition principles identification based on two decomposition principles principles principles ∗∗ Karima Hammar ∗∗ Tounsia Djamah ∗∗ Maamar Bettayeb ∗∗
Karima Hammar ∗∗ Tounsia Djamah ∗∗ Maamar Bettayeb ∗∗ Karima ∗∗ Karima Hammar Hammar ∗ Tounsia Tounsia Djamah Djamah ∗ Maamar Maamar Bettayeb Bettayeb ∗∗ Karima Hammar ∗ Tounsia Djamah ∗ Maamar Bettayeb ∗∗ Karima Hammar Tounsia Djamah Maamar Bettayeb ∗ ∗ University M. Mammeri of Tizi Ouzou, U.M.M.T.O. (L2CSP), ∗ ∗ University M. Mammeri of Tizi Ouzou, U.M.M.T.O. (L2CSP), ∗ University M. Mammeri U.M.M.T.O. Algeria (e-mail: [email protected]). M. [email protected], Mammeri of of Tizi Tizi Ouzou, Ouzou, U.M.M.T.O. (L2CSP), (L2CSP), ∗ University Algeria (e-mail: [email protected], [email protected]). ∗∗ University M. Mammeri of Tizi Ouzou, U.M.M.T.O. (L2CSP), ∗∗ Algeria (e-mail: [email protected], [email protected]). of Sharjah, Dept of Electr. & Comp. Engineering, Algeria (e-mail: [email protected], [email protected]). ∗∗ University of Sharjah, of & ∗∗ Algeria (e-mail: [email protected]). ∗∗ University University [email protected], Sharjah, Dept Dept of Electr. Electr. & Comp. Comp. Engineering, Engineering, U.A.E(e-mail: [email protected]) University of Sharjah, Dept of Electr. & Comp. Engineering, ∗∗ U.A.E(e-mail: [email protected]) UniversityU.A.E(e-mail: of Sharjah, Dept of Electr. & Comp. Engineering, U.A.E(e-mail: [email protected]) [email protected]) U.A.E(e-mail: [email protected]) Abstract: This This paper paper deals deals with with fractional fractional non-linear non-linear system system identification, identification, where where the the HamHamAbstract: Abstract: This paper paper deals with with fractional fractional non-linear system identification, identification, where the the HamAbstract: This deals non-linear system where Hammerstein Controlled Auto-Regression (HCAR) model is considered. The identification process is merstein Controlled Auto-Regression (HCAR) model is considered. The identification process is Abstract: This paper deals with fractional non-linear system identification, where the Hammerstein Controlled Auto-Regression (HCAR) model is considered. The identification process is merstein Controlled Auto-Regression (HCAR) model is considered. The identification process is derived for the regression form of the HCAR system based on the Over-parametrization principle derived for the regression form of the HCAR system based on the Over-parametrization principle merstein Controlled Auto-Regression (HCAR) model is considered. The identification process is derived for the regression form of the HCAR system based on the Over-parametrization principle derived for the regression form of the HCAR system based on the Over-parametrization principle and the Key-term separation principle. Levenberg-Marquardt algorithm combined with each of and the Key-term separation principle. Levenberg-Marquardt algorithm combined with each of derived for the regression formto ofidentify the HCAR system based on the Over-parametrization and the Key-term separation principle. Levenberg-Marquardt algorithm combined withprinciple eachtest of these principles is developed the fractional HCAR system. Various simulations and the Key-term separation principle. Levenberg-Marquardt algorithm combined with each of thesethe principles is separation developed to to identifyLevenberg-Marquardt the fractional fractional HCAR HCAR algorithm system. Various Various simulations test and Key-term principle. combined withIFAC eachtest of these principles is developed identify the system. simulations test these principles is developed to identify the fractional HCAR system. Various simulations the efficiency of the optimization method based on these principles. Copyright © 2019 the efficiency of the optimization method based on these principles. Copyright © 2019 IFAC these principles is developed to identify the fractional HCAR system. Various simulations test the efficiency of the optimization method based on these principles. Copyright © 2019 IFAC the efficiency of the optimization method based on these principles. Copyright © 2019 IFAC © 2019, IFAC (International Federation of Automatic by Elsevier Ltd. All the efficiency of the optimization method based Control) on theseHosting principles. Copyright © rights 2019 reserved. IFAC 1. INTRODUCTION INTRODUCTION for both both principles. principles. 1. for 1. INTRODUCTION INTRODUCTION for both principles. Theboth paper is organized organized as as follows: follows: Section Section 22 provides provides some some 1. for principles. The paper is 1. INTRODUCTION for both principles. The paper is organized as follows: Section 22 provides some theoretical concepts on fractional calculus. In Section 3, The paper is organized as follows: Section provides some The class of nonlinear systems can be conveniently detheoretical concepts onasfractional fractional calculus. In Section Section 3, The class of nonlinear systems can be conveniently deThe paper isconcepts organized follows: Section 2along provides some theoretical on calculus. In 3, The class of nonlinear systems can be conveniently dethe fractional HCAR system is presented with the theoretical concepts on fractional calculus. In Section 3, The class of nonlinear systems can be conveniently described using block-oriented models such as Hammerstein the fractional fractional HCARon system is presented presented along with the the scribed using block-oriented models such as Hammerstein concepts fractional calculus.along In Section 3, The class of nonlinear systems can(Ding be conveniently de- theoretical the HCAR system is along with scribed using block-oriented models such as problem definition, and the identification identification principles are the fractional HCAR system is presented with the scribed using block-oriented models such aset Hammerstein (Wang (2016)), Wiener models etHammerstein al. (2016)), (2016)), problem definition, and the principles are (Wang (2016)), Wiener models (Ding al. the fractional HCAR system is presented along with the scribed using block-oriented models such as Hammerstein problem definition, and the identification principles are (Wang (2016)), WienerHammerstein-Wiener models (Ding (Ding et et al. al.(Wang (2016)), describeddefinition, in Section Section and 4. In Inthe Section 5, the the proposed proposed method identification principles are and their their(2016)), combinations et problem (Wang Wiener models (2016)), described in 4. Section 5, method and combinations Hammerstein-Wiener (Wang et problem definition, and the identification principles are (Wang WienerHammerstein-Wiener models(Giordano (Ding etetal. (2016)), described in Section 4. In Section 5, the proposed method and their(2016)), combinations Hammerstein-Wiener (Wang et is developed. Section 6 compares the method effciency for described in Section 4. In Section 5, the proposed method and their combinations (Wang et al. (2008)), Wiener-Hammerstein al. (2015)), is developed. developed. Section4.66 In compares the method effciency for al. (2008)), Wiener-Hammerstein (Giordano et al. (2015)), described in Section Section 5, the proposed method and their combinations Hammerstein-Wiener (Wang et is Section compares the method effciency for al. (2008)), Wiener-Hammerstein (Giordano et al. (2015)), both principles, and finally, some conclusions and perspecis developed. Section 6 compares the method effciency for al. (2008)), Wiener-Hammerstein (Giordano et al. (2015)), etc... They separate the linear part and the nonlinear one both principles, and finally, finally, some conclusions conclusions and perspecetc... They separate the linear part and the nonlinear one developed. Section 6 compares the method and effciency for al. Wiener-Hammerstein (Giordano et al. (2015)), both principles, and some and perspecetc... They and one tives are drawn. drawn. both principles, and finally, some conclusions perspecinto(2008)), different blocks.the etc... They separate separate the linear linear part part and the the nonlinear nonlinear one is tives are into different blocks. both principles, and finally, some conclusions and perspecetc... They separate the linear part and the nonlinear one tives are are drawn. drawn. into different blocks. On the otherblocks. hand, most most real real systems exhibiting exhibiting long long tives into different 2. PRELIMINARIES PRELIMINARIES On other hand, into different blocks. 2. On the the other hand, and mostinfinite real systems systems exhibiting long tives are drawn. 2. On the other hand, most real systems exhibiting long memory transients dimensional structure PRELIMINARIES memory transients and infinite dimensional structure 2. PRELIMINARIES On the other hand, mostfractional real systems exhibiting long memory transients and infinite dimensional structure memory transients and infinite dimensional structure are well described using models. For the last 2. have PRELIMINARIES are well described using fractional models. For the last Different definitions definitions been proposed proposed in in the the literature literature memory transients and infinite dimensional structure are well described using fractional models. For the last Different have been decades, great effort has been devoted for the fractional are well described using fractional models. For the last Different definitions have been proposed in the literature decades, great effort has been devoted for the fractional for the differintegral operator of fractional (non-integer) Different definitions have been proposed in the literature are well described using fractional models. For the last decades, greatidentification effort has has been been devoted for the the Cui fractional for the the differintegral differintegral operator of fractional (non-integer) linear system system (Dai et al. al. (2016); (2016); et al. al. for decades, great effort devoted for fractional Different definitions have been of proposed in the literature operator fractional (non-integer) linear (Dai et et order. The Grünwald-Letnikov (GL) difference defined in for theThe differintegral operator of fractional (non-integer) decades, greatidentification effort has been devoted for the Cui fractional linear system identification (Dai et al. (2016); Cui et al. order. Grünwald-Letnikov (GL) difference defined in linear system identification (Dai et al. (2016); Cui et al. (2017); Djamah et al. (2008, 2013)) for the differintegral operator of fractional (non-integer) order. The Grünwald-Letnikov (GL) difference defined in (2017); Djamah et al. (2008, 2013)) Miller et al. (1993); Dzieliński (2008); Kilbas et al. (2006) is order. The Grünwald-Letnikov (GL) difference defined in linear system identification (Dai et al. (2016); Cui et al. (2017); Djamah et al. (2008, 2013)) Miller et et al. Grünwald-Letnikov (1993); Dzieliński Dzieliński (2008); (2008); Kilbas et al. al.defined (2006) in is (2017); Djamah et al.which (2008,consists 2013)) of Hammerstein model aa static nonlinear order. The (GL) difference Miller al. (1993); Kilbas et (2006) is Hammerstein model which consists of static nonlinear mostly used in the discrete case, it is expressed as follows: Miller et al. (1993); Dzieliński (2008); Kilbas et al. (2006) is (2017); Djamah et al.which (2008, 2013)) Hammerstein model consists of static mostly used in the discrete discrete case, it is Kilbas expressed as follows: follows: part followed followed by a linear linear block has been been popular in many many mostly Hammerstein model which consists of a a popular static nonlinear nonlinear Miller al. (1993); Dzieliński (2008); et al. is part by block has in in case, it as mostlyetused used in the the discrete case, it is is expressed expressed as(2006) follows: Hammerstein model which consists of a popular static nonlinear part followedincluding by aa linear linear block has been popular in many applications control, signal processing and compart followed by a block has been in many mostly used in the discrete case, it is expressed as follows: applications including control, signal processing and compart followed by a linear block hasal.been popular in many ( k applications including control, signal processing and comk ) (α) applications including control, signal processing and communication (Vörös (2010); Cao et (2014)). The identifik 1 ∑ munication (Vörös (2010); Cao et al. (2014)). The identifiα k (−1)jjj ( )x((k − j)h), (α) α x(kh) = 1 ∑ applications including control, signal processing and comk munication (Vörös (2010); Cao et al. (2014)). The identifi∑ (1) ∆ α munication (Vörös (2010); Cao et al. (2014)). The identifi∑ α 1 cation of Hammerstein system has focused in major studies α − j)h), (1) x(kh) = (−1) ∆ ( α 1 α α k cation of Hammerstein system has focused in major studies jj j )x((k α munication (Vörös (2010); Cao et al. (2014)). The identifi∑ x((k − j)h), (1) x(kh) = =h (−1) ∆α cation of Hammerstein system has focused in(2015); major studies j h x((k − j)h), (1) x(kh) (−1) ∆ α 1 j=0 on the classical integer case as Chen et al. Liu et cation of Hammerstein system has focused in major studies α j=0 j=0 α j j α h on the classical integer case as Chen et al. (2015); Liu et (1) = hα j=0 (−1) j x((k − j)h), cation Hammerstein system has focused major studies j=0 on (2010); the of classical integer case as asand Chen etfew al.inwork (2015); Liu et et where ∆∆is x(kh) on the classical integer case Chen et al. (2015); Liu the discrete fractional-order difference operaal. Ding et al. (2013), only considers j h j=0 al. (2010); Ding et al. (2013), and only few work considers where ∆ is the discrete fractional-order difference operaon the classical integer case as Chen et al. (2015); Liu et ∗+ al. (2010); Ding et al. (2013), and only few work considers ∗+ where ∆ Ris is∗+ the discrete fractional-order difference period, operaal. (2010); Ding et al. (2013), and only few work considers where the fractional nonlinear case. ∆ the discrete fractional-order difference operator, α ∈ the fractional order, h the sampling the fractional nonlinear case. tor, α ∈ the fractional order, h the sampling period, ∗+ al. (2010); Ding et al. (2013), and only an few iterative work considers where ∆R isnumber the discrete fractional-order difference opera∗+ the fractional nonlinear case. tor, α ∈ R the fractional order, h the sampling period, the fractional nonlinear case. and k the of samples for which the approximation Recently, in Rahmani et al. (2017), linear tor, α ∈ R the fractional order, h the sampling period, ∗+ Recently, in Rahmani et al. (2017), iterative linear and k the number of samples for which the approximation the fractional nonlinearwith case. tor, α ∈ R the fractional order, h the sampling period, Recently, in algorithm Rahmani et al.a Lyapunov (2017), an anmethod iterative linear and k the number of samples for which the approximation optimization has been Recently, in Rahmani et al. (2017), an iterative linear of the derivative Throughout this paper, we k the numberis ofcalculated. samples for which the approximation optimization algorithm with Lyapunov method haslinear been and of derivative Throughout this paper, Recently, in algorithm Rahmani et al. (2017), anmethod iterative and k the numberis ofcalculated. samples for which the approximation optimization algorithm with aa Lyapunov Lyapunov method has been of the the derivative is calculated. Throughout this paper, we we optimization with a has been assume, without loss of generality, that h = 1. proposed for the fractional Hammerstein system identifiof the derivative is calculated. Throughout this paper, we proposed for the fractional Hammerstein system identifiassume, without loss of generality, that h = 1. optimization algorithm with a iterative Lyapunovalgorithm method has been of the derivative is calculated. Throughout paper, we proposed for the fractional Hammerstein system identifiassume, without loss of generality, that h h from =this 1. the proposed for the fractional Hammerstein system identifiThe binomial term in (1) can be obtained followcation; in Ivanov (2015) the is used, assume, without loss of generality, that = 1. cation; in Ivanov (2015) the iterative algorithm is used, The binomial term in (1) can be obtained from the followproposed for the fractional Hammerstein system identifiassume, without loss of generality, that h = 1. cation; in Ivanov (2015) the iterative algorithm is used, The binomial term in (1) can be obtained from the followcation; inthe Ivanov (2015) theis iterative algorithm is Lia used, however, fractional order not estimated and in et Therelation: binomial term in (1) can be obtained from the following however, fractional order is not estimated and Lia et relation: cation; inthe Ivanov (2015) the is used, The binomial term in (1) can be obtained from the followhowever, the fractional order isiterative noton estimated and in in Lia et ing ing al. (2012) (2012) the identification based thealgorithm subspace method however, the fractional order is not estimated and in Lia et ing relation: relation: al. the identification based on the subspace method however, the orderbased is noton estimated andalgorithm in Lia et ing relation: al. (2012) (2012) thefractional identification based on the subspace subspace method al. the identification the method was developed. The particle swarm optimization ) ( was developed. The particle swarm optimization algorithm 11 f or jj = 0 al. (2012) theinidentification based on the subspace method ) ( was developed. The particle swarm optimization algorithm was developed. The particle swarm optimization algorithm is also used Hammar et al. (2015). α ) ( (α) = 1 − j + 1) fff or or jj = =0 0 α(α − 1)...(α is also used in Hammar et al. (2015). 1 or = 0 (2) was developed. The particle swarm optimization algorithm is also used in Hammar et al. (2015). α α(α − 1)...(α − j + 1) (2) (α is alsoobjective used in Hammar et al. (2015). The of this paper is to contribute to the fffor 1 − or jjj > = 0. 0 jj ) = α(α − − 1)...(α 1)...(α − j + 1) The objective of this paper is to contribute to the = (2) or > 0. α(α j + 1) = (2) is also used in Hammar et al. (2015). α j! The objectiveofof ofthethis this paper is isHammerstein to contribute contribute to the the ff or jj > 0. identification fractional Controlled The objective paper to to jj = j! or > 0. α(α − 1)...(α − j + 1) (2) identification of the fractional Hammerstein Controlled j! The objective of this paper is to contribute to the j! f or j > 0.gives an identification of of the the fractional Hammerstein Controlled j The fractional difference operator of Equation (1) identification fractional Hammerstein Controlled Auto-Regression (HCAR) model. It is performed based The fractional difference operator of Equation (1) gives an j! Auto-Regression (HCAR) model. It is performed based identification of the fractional Hammerstein Controlled Auto-Regression (HCAR) model. It is based The fractional difference operator of (1) gives overall characterization this function at each k, Auto-Regression (HCAR) model. Itprinciples: is performed performed based The fractional differenceof operator of Equation Equation (1)instant gives an an on the regression form using two the Overoverall characterization of this function at each instant k, on the regression form using two principles: the OverAuto-Regression (HCAR) model. Itprinciples: is performed based The fractional difference operator offunction. Equation (1)instant gives an on the regression form using two the Overoverall characterization of this function at each k, parametrization principle and the Key-term separation by considering all past values of this For practical on the regression form using two principles: the Overoverall characterization of this function at each instant k, by considering all past values of this function. For practical parametrization principle and the Key-term separation on the regression form using two principles: the Overoverall characterization of this function at each instant k, parametrization principle and the the Key-termalgorithm separation by considering considering allnumber past values values of this this function. function. Foraccount practical principle. The robust Levenberg Marquardt is realizations, the of samples taken into is parametrization principle and Key-term separation by all past of For practical realizations, the number of samples taken into account is principle. The robust Levenberg Marquardt algorithm is parametrization principle and the Key-term separation by considering all past values of this function. For practical principle. The The robust Levenberg Levenberg Marquardt algorithm of is realizations, realizations, the number number of samples samples takenmemory into account account is limited to a number L, called the system length. principle. robust Marquardt algorithm is the of taken into is developed for identifiying the HCAR model parameters limited to a number L, called the system memory length. developed for identifiying the HCAR model parameters of principle. The robust Levenberg Marquardt algorithm is realizations, the Equation number of(1) samples taken into account is developed for identifiying the HCAR model parameters of limited to aa number L, called the system memory length. In what follows, is used for the simulation of developed for identifiying the HCAR model parameters of limited to number L, called the system memory length. the linear part, the nonlinear part and its fractional order. In what follows, Equation (1) is used for the simulation of the linear part, the nonlinear part and its fractional order. developed for identifiying the HCAR model parameters of limited to a number L, called the system memory length. the linear part, the nonlinear part and its fractional order. In what follows, Equation (1) is used for the simulation of Numerical simulations investigate the method efficiency fractional order systems. the linear part, the nonlinear part and fractional order. In what follows, Equation (1) is used for the simulation of Numerical simulations investigate theits method efficiency order the linear part, the nonlinear part and itsmethod fractional order. fractional In what follows, Equation (1) is used for the simulation of Numerical simulations investigate the method efficiency fractional order systems. systems. Numerical simulations investigate the efficiency fractional order systems. Numerical simulations investigate the method efficiency fractional order systems. 2405-8963 © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
Copyright © 2019 IFAC 211 Copyright 2019 IFAC 211 Control. Peer review© responsibility of International Federation of Automatic Copyright © under 2019 IFAC IFAC 211 Copyright © 2019 211 10.1016/j.ifacol.2019.11.106 Copyright © 2019 IFAC 211
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3. PROBLEM DESCRIPTION A fractional Hammerstein CAR (HCAR) is represented in Fig.1. It consists of a static non-linear block followed by a fractional linear dynamical block and it is described by the following equation: A (z) y (k) = B (z) u ˜ (k) + n (k) (3) where y(k) is the measured output, u ˜(k) is the output of the non-linear block assumed to be a non-linear function of a known basis of the system input u(k), and n(k) is a disturbance noise, with: f (u(k)) = [ f1 (u(k)), f2 (u(k)), ..., fnp (u(k)) ] ∈ Rnp u ˜(k) = f (u(k)) = p1 f1 (u(k)) + p2 f2 (u(k)) + · · · + pnp fnp (u(k)) =
np ∑
pj fj (u(k))
j=1
(4)
The linear block transfer function of the CAR model is described by the fractional commensurate order polynomials A(z) and B(z) of the shift operator z −1 as follows: A (z) = 1 + a1 z −α + a2 z −2α + · · · + ana z −na α B (z) = b1 z −α + b2 z −2α + · · · + bnb z −nb α
n(k)
u(k)
f (.)
u˜(k)
(5)
1 A(z)
B(z) A(z)
+
y(k)
form. The objective of this work is to develop an identification method to estimate the parameter vectors a, b, p, and the fractional order α. 4. IDENTIFICATION PRINCIPLES Different identification models can be derived for fractional HCAR system output of equation (6) based on regression models Wang et al. (2009); Vörös (2003). Two identification principles have been reported for the identification of classical nonlinear systems; the first one is the Over-parameterization principle and the second one is the key term separation principle. In this study, these last are investigated for the fractional case. For the Over-parametrization identification principle Wang et al. (2009), the system output in (6) can be written under the form: (7) y(k) = ξ(α, k)T θ˜1 + n(k) Where the parameter vector θ˜1 is defined as follows: θ˜1 = [ a b p2 b · · · pnp b ]T ∈ Rnθ1 , nθ1 = na + nb np (8) and the information vector ξ(α, k) can be written as: ξ(α, k) = [φ(α, k)T ∆α f (u(k − 1)) · · · ∆α f (u(k − nb ))]T (9) where φ(α, k) = [−∆α y(k − 1) − ∆α y(k − 2) · · · − ∆α y(k − na )]T In this case, the model parameter vector θ˜1 to be estimated contains the cross-product of the coefficients of the nonlinear block and the linear block; this results in a large dimension of the unknown parameter vector (nθ1 = na + nb np ) and a redundancy of the estimation of the parameters . In order to solve this problem, the Key-term separation principle is adopted Vörös (2003); Wang et al. (2013). This last allows the separation of the linear part and nonlinear part and at the same time, avoids the occurence of the cross-products coefficients. Let us choose ∆α f1 (u(k − i)) as the key term and let us write the Equation (6) as a new regression model as follows:
Fig. 1. Fractional Hammerstein system
y(k) = φks (α, k)T θ˜2 + n(k) φks (α, k) = [φ0 (α, k), ψ(α, b, k)]T
Let us define the parameter vectors a, b and p of the HCAR system with: b = [ b1 , b2 , · · · , bnb ], a = [ a1 , a2 , · · · , ana ], p = [ p1 , p2 , · · · , pnp ].
In order to get unique parameter estimates, we assume that p1 = 1. The substitution of the polynomials A(z) and B(z) in equation (3) leads to a recursive equation as follows: nb na ∑ ∑ y(k) = − ai ∆α y(k − i) + bi ∆α f (u((k − i)) + n(k) i=1
= −
na ∑ i=1
+ n(k)
i=1
α
ai ∆ y(k − i) +
np ∑ j=1
pj
nb ∑ i=1
bi ∆α fj (u((k − i))
(6) The overall output of the HCAR system contains the parameters of the linear and the nonlinear part and their cross-products. It can be expressed under a regression 212
207
−∆α y(k − 1) −∆α y(k − 2) .. .
(10) (11)
α −∆ y(k − na ) φ0 (α, k) = α ∈ Rna +nb ∆ f1 (u(k − 1)) ∆α f (u(k − 2)) 1 .. . ∆α f1 (u(k − nb ))
(12)
bj ∆α f2 (u(k − j)) α bj ∆ f3 (u(k − j)) j = 1, · · · , nb (13) ψj (α, b, k) = .. . α bj ∆ fnp (u(k − j))
θ˜2 = [a
b
p2 · · · pnp ]T ∈ Rnθ˜2 , nθ˜2 = na + nb + np − 1 (14)
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The difficulty here is that the information vector φks (α, k) in (10) contains unknown variables, in occurence the coefficients bj (j = 1, 2, · · · , nb ) and the fractional order α. This difficulty is circumvented by replacing the coefficients bj and α at iteration l, by their respective estimates bˆj and α ˆ obtained at iteraion (l − 1). These two principles are combined with Levenberg Marquardt algorithm in the following section and their performances are compared.
Levenberg-Marquardt (LM) is a nonlinear optimization method, based on the computation of the Gradient and the Hessian (Marquardt (1963); Djamah et al. (2013)); the optimal value of θ is obtained by minimizing the quadratic criterion:
J(θ) =
ε(k)
(15)
2
k=1
where ε(k) = y(k) − yˆ(k) is the output prediction error and yˆ(k) is the corresponding output estimate. The LM update rule is given as follows: θ
(l+1)
=θ
(l)
−
{[
′′
l
J (θ ) + λI
]−1
′
l
J (θ )
}
(16)
with J , J are respectively the gradient and the Hessian, λ is a monitoring parameter. Let us combine this algorithm (LM) with the described principles previously. For the Over parametrization principle, θ = θ1 = [ θ˜1 , α ], and the quadratic criterion J1 is computed as follows: ′
′′
J1 =
N ∑
k=1
ˆ [ξ(α, k)T θ˜1 − y(k)]2
k=1
The LM algorithm is applied to equations (16)-(19) by replacing θ˜1 by θ˜2 and using the key term principle equations (10)-(14). 6. SIMULATION EXAMPLE
5. LEVENBERG MARQUARDT ALGORITHM
K ∑
For the key-term separation principle, the optimization is performed by minimizing the quadratic criterion function J2 : N ∑ ˆ J2 = [φTks (α, k)θ˜2 − y(k)]2 .
(17)
′ ˜ α) with respect to The gradient J1 of the criterion J1 (θ, ˜ θ1 and α is given by the following equations: ) (N ∑ ′ J1 θ˜1 = −2ξ(α, k)T y(k) − ξ(k, α)T θ˜1 ( N k=1 ) ∑ ′ J1 α = −2σyk /α y(k) − ξ(α, k)T θ˜1 k=1 T ˜ σy /α = ∂(ξ(α, k) θ1 ) is the output sensitivity function k ∂α (18) ′′ We compute the Hessian matrix J1 of the cost function with respect to θ˜1 and α as follows: ′′ k)T ξ(α, k) J1 θ˜1 = 2ξ(α, (N ) ∑ (19) ′′ T σyk /α σyk /α J1 α = 2
k=1
For the fractional order α, its sensitivity is calculated numerically using: ∂y = δα σyk /α (20) y(k, α + δα) − y(k, α) ≈ δα ∂α
213
Consider the following fractional HCAR system: y(k) = 0.1∆0.3 y(k − 1) + 0.3∆0.3 y(k − 2) + 0.2∆0.3 f (u((k − 1)) − 0.5∆0.3 f (u((k − 2)) (21) + n(k) The nonlinear block is described by the following polynomial f (u(k)) = u(k) + 0.5u2 (k) + 0.25u3 (k) The input u(k) is a persistent excitation sequence of zero mean and unit variance, and the disturbance n(k) is a white noise sequence of zero mean. The simulations have been carried out for noise-free data, then for data corrupted with noise for different signal-tonoise ratios SN R = 26 dB and SN R = 50 dB. Applying LM algorithm based on the described principles, the system identification results for the noise free case and for SN R = 50 dB and SN R = 26 dB are reported respectively in Table 1, Table 2 and in Table 3. They show that the method based on both principles gives satisfactory results, however the key-term principle is more accurate when the noise level is increased. In order to compare the convergence of the method for each principle, the normalized estimation error δ with δ = ∥θˆ − θ∥/∥θ∥ is computed and its evolution versus the number of iterations is depicted in Fig. 2 for the noise free case, in Fig. 3 for SN R = 50 dB and in Fig. 4 for SN R = 26 dB. From these results we can draw the conclusion that the developed method gives consistent parameters estimates for both principles. L-M algorithm based on the Key-term separation principle estimates the non-linear parameters more accurately than the over-parametrization principle and converge more rapidly, moreover it give satisfactory results even in presence of a high level of noise. The performed simulations illustrate the satisfactory performance of the LM algorithm and the obtained results, show that it the 7. CONCLUSION This paper presents the identification problem of a fractional HCAR system modeled under a regression form. Two principles combined with LM algorithm are investigated. The model based on the over-parameterization principle includes the product terms of the parameters of the static non-linear block and the dynamic linear block which induce redundancy in the estimation process. To
2019 IFAC MIM Berlin, Germany, August 28-30, 2019
Karima Hammar et al. / IFAC PapersOnLine 52-13 (2019) 206–210
Table 1. Simulation results for the noise free case key − term 0.100 0.300 0.200 −0.500 0.500 0.250 0.300 1.2e − 31
a1 a2 b1 b2 p2 p3 α J
over − parameterization 0.100 0.300 0.200 −0.500 0.500 0.250 0.300 5.8e − 30
209
Table 3. Simulation results for SN R = 26 dB key − term 0.116 0.3126 0.2052 −0.5315 0.4636 0.2213 0.2999 0.4540
a1 a2 b1 b2 p2 p3 α J
exact values 0.100 0.300 0.200 −0.500 0.500 0.250 0.300 −
over − parameterization 0.132 0.415 0.266 −0.613 0.491 0.236 0.198 0.498
exact values 0.100 0.300 0.200 −0.500 0.500 0.250 0.300 −
2 1.6
1.2
Over−parameterization principle δ
Key−term principle
1
δ
Key−term principle
1.5
1.4
Over−parameterization principle
1
0.8
0.5
0.6 0.4
0 0
0.2 0 0
20
40
60 Iteration
80
100
Fig. 2. Evolution of the error versus number of iterations for the noise free case Table 2. Simulation results for SN R = 50 dB key − term 0.101 0.300 0.200 −0.501 0.499 0.249 0.300 3.8e − 4
a1 a2 b1 b2 p2 p3 α J
over − parameterization 0.100 0.4157 0.202 −0.499 0.499 0.248 0.299 6.70e − 4
exact values 0.100 0.300 0.200 −0.500 0.500 0.250 0.300 −
Key−term principle
δ
Over−parameterization principle
0.5 0 0
20
40
60
80 100 Iteration
120
140
160
100
150 Iteration
200
250
300
Fig. 4. Evolution of the error versus number of iterations for SN R = 26dB enhance the efficiency of this algorithm when the level of noise is increased. REFERENCES
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180
Fig. 3. Evolution of the error versus number of iterations for SN R = 50dB overcome this drawback, the key-term principle can be used. This last allows the separation of the linear part and nonlinear part and at the same time, avoids the occurence of the cross-products coefficients. This study confirms the fact that LM algorithm can approximate any non linear system with a good accuracy for both principles and the key-term separation principle 214
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