Tutorial on system identification using fractional differentiation models

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1 Tutorial on system identiication using ractional dierentiation models Rachid Malti, Mohamed Aoun, Jocelyn Sabatier, Alain Oustaloup To cite this version: Rachid Malti, Mohamed Aoun, Jocelyn Sabatier, Alain Oustaloup. Tutorial on system identiication using ractional dierentiation models. International Federation o Automatic Control IFAC. 14th IFAC SYmposium on System IDentiication, Mar 26, Newcastle, Australia. pp , 26. <hal > HAL Id: hal Submitted on 19 Oct 27 HAL is a multi-disciplinary open access archive or the deposit and dissemination o scientiic research documents, whether they are published or not. The documents may come rom teaching and research institutions in France or abroad, or rom public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diusion de documents scientiiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche rançais ou étrangers, des laboratoires publics ou privés.

2 14 th IFAC Symposium on System Identiication, Newcastle, Australia, 26 TUTORIAL ON SYSTEM IDENTIFICATION USING FRACTIONAL DIFFERENTIATION MODELS Rachid Malti, Mohamed Aoun Jocelyn Sabatier, and Alain Oustaloup LAPS - UMR 5131 CNRS Université Bordeaux I ENSEIRB, 351 cours de la Libération, 3345 Talence cedex, France. {FirstName.LastName}@laps.u-bordeaux1.r Abstract: This paper presents a tutorial on system identiication using ractional dierentiation models. The tutorial starts with some general aspects on time and requency-domain representations, time-domain simulation, and stability o ractional models. Then, an overview on system identiication methods using ractional models is presented. Both equation-error and output-error-based models are detailed. Copyright c 26 IFAC. Keywords: Fractional dierentiation, ractional integration, identiication, simulation, output error models, equation error models, state variable ilter. 1. INTRODUCTION 1.1 Mathematical background A ractional mathematical model is based on ractional dierential equation: Although ractional non integer operators remained or a long time purely a mathematical concept, the rise o digital computers oered an easy way or simulating numerically non integer integro-dierentiation o mathematical unctions. The last two decades have witnessed considerable development in the use o ractional dierentiation in various ields. Fractional dierentiation is now an important tool or the international scientiic and industrial communities. In that scope, the 1st IFAC Workshop on Fractional Dierentiation and its Applications FDA 4 was hold in 24 in France. The use o ractional dierentiation models in system identiication was initiated in the late nineties and the beginning o this century Le Lay, 1998; Lin, 21; Cois, 22; Aoun, 25. Fractional models are now enough mature and are widely used in representing some diusive phenomena thermal diusion, electrochemical diusion and in modeling viscoelastic materials. y t+b 1 D β 1 y t+ + b mb D βm B y t = a D α u t+a 1 D α 1 u t+ + a ma D αm A u t 1 where dierentiation orders, β 1 <β 2 <... < β mb, α < α 1 <... < α ma, are allowed to be non-integer positive numbers. The concept o dierentiation to an arbitrary order non-integer, D γ Δ = d dt γ γ R + was deined in the 19 th century by Riemann and Liouville. The γ ractional derivative o xt is deined as being an integer derivative o order m = γ +1. stands or the loor operator o a non-integer integral o order 1 m γsamko et al., 1993: D γ x t = D m I m γ xt Δ = m t 1 d x τ dτ 2 Γm γ dt t τ 1 m γ 66

3 where t>, γ R +, and the Euler s Γ unction is deined as: Γx = e t t x 1 dt x R \ { N } 3 Magnitude db logη logα logη γ 2dB logα logη A discrete-time deinition o ractional derivative was proposed by Grünwald 1867, γ R +: D γ 1 xt = lim h h γ 1 k γ xt kh 4 k k= where Newton s binomial γ k is generalized to non-integer orders by the use o Euler s Γ unction: γ Γγ +1 = 5 k Γk +1Γγ k +1 Equation 4 is generally used in time-domain simulations o ractional dierentiation. As Newton s binomial γ k does not converge rapidly to zero with k when γ is non integer, the computation o D γ xt depends on all values o xt between and t supposing that xt is relaxed at t =,i.e. xt = t <. Since ractional derivatives o a unction depend on whole its past, ractional operators are known to have long memory behavior. A more concise algebraic tool can be used to represent ractional systems: the Laplace transorm Oldham and Spanier, 1974: L {D γ x t} = s γ X s i xt = t < This property allows to write the ractional dierential equation 1, provided ut andyt are relaxed at t =, in a transer unction orm: m A a i s αi i= F s = 1+ mb 6 b j s βj j=1 where a i,b j R 2, α i,β j R 2 +, i =, 1,...,m A, j =1, 2,...,m B. Deinition 1. A transer unction F s is commensurate o order γ i it can be written as F s = Ss γ, where S = T R is a rational unction with T and R two co-prime polynomials. Moreover, the commensurate order γ is the biggest number satisying the aorementioned condition. In other words, the commensurate order γ is deined as the biggest real number such that all dierentiation orders are integer multiples o γ. A modal orm transer unction can then be obtained, provided 6 is strictly proper: N v k A k,l F s = s γ s k q, 7 k=1 q=1 where s k,k =1,,N are known as the s γ -poles o integer multiplicity q. Phase rad π/2 ω 1 ω 1 ω 2 ω 2 ω N 1 ω N ω N Frequency γ π/2 Fig. 1. Recursive approximation o a ractional dierentiator with poles and zeroes 1.2 Stability theorem Matignon, 1998, revisited A commensurate o order γ transer unction F s is BIBO stable i <γ<2 8 and or every s γ -pole, s k C o F s : arg s k >γ π Equivalence with rational models 9 Due to the consideration that real physical systems generally have bandlimited ractional behavior and due to the practical limitations o input and output signals Shannon s cut-o requency or the upper band and the spectrum o the input signal or the lower band, ractional operators are usually approximated by high order rational models. As a result, a ractional model and its rational approximation have the same dynamics within a limited requency band. The most commonly used approximation o s γ in the requency band [ω A,ω B ] is the recursive distribution o zeros and poles proposed by Oustaloup Trigeassou et al suggested to use an integrator outside the requency range [ω A,ω B ] instead o a gain: s γ s γ [ω A,ω B] = C s where ω i = αω i, 1+ s 1 γ ω A 1+ s ω B N C s k=1 ω i+1 = ηω i and 1+ s ω k 1+ s ω k 1 γ =1 log α 11 log αη α and η are real parameters which depend on the dierentiation order γ. The bigger N the better the approximation o the integrator s γ. 67

4 2. SYSTEM IDENTIFICATION Frequency-domain system identiication using ractional models was initiated by the Ph. D. thesis o Le Lay, Time-domain system identiication using ractional dierentiation models was initiated by the Ph.D. theses o Le Lay 1998, Lin 21, and Cois 22. Mainly two classes o models were developed: Equation-Error-based models and Output-Error-based models, both o which are presented in this section. Recently Malti et al. 25 synthesized ractional orthogonal bases generalizing Laguerre, Kautz and BOG bases to ractional dierentiation orders. 2.1 Equation-error models Equation-error-based models are linear in coeicients. The identiied system is assumed to be initially at rest, modeled by 1, and characterized by input/output coeicient s vector: θ =[a...a ma b 1...b mb ] T 12 Aprioriknowledge is generally used to ix the dierentiation orders α,...,α ma,β 1,...,β mb.usually a commensurate order γ is chosen and then all its multiples ixed up to a given order, say β mb, α ma is generally set to β mb γ or strictly proper systems. F s = 1+ αm A γ i= βm B γ j=1 a i s iγ b j s jγ 13 Consider observed data ut andy t =yt + pt, where pt is a perturbation signal, collected at regular samples: k T s, k +1T s,...,k + K 1T s. The most basic estimation method consists o computing ractional derivatives o input/output signals rom sampled data by applying 4. The output can be written in a regression orm: yt =φ tθ 14 where parameters and regression vectors are respectively given by 12 and: φ t = [ D α u t D αm A u t D β1 y t D βm B y t ] 15 The estimated parameters vector ˆθ o θ is obtained by minimizing the quadratic norm o the error: J ˆθ = E T E 16 and E = [ ε k T s ε k +1T s ε k + K 1 T s ε t =y t φ tˆθ ] T The minimum o J is given by the classical least squares: ˆθ opt = Φ T Φ 1 Φ T Y Φ = [ φ T k T s φ T k +1T s... φ T k + K 1T s ] T 17 As in the integer case, ractional dierentiation o noisy signals ampliies the noise. Hence, a linear transormation low-pass ilter can be applied to 14 so as to obtain a linear continuous regression o iltered input, u t, and output, y t, signals: y t =φ tθ 18 where φ t = [ D α u t D αm A u t D β1 y t D βm B y t ] 19 The ilter is generally chosen to be causal, stationary, and low-pass. Among the possible ilters, Cois et al. 21 extend the concept o State Variable Filters SVF to ractional dierentiation systems. They propose to use the ollowing ractional ilter: A H s = α + α 1 s γ α N 1s γn 1 + s γn γn is ilter s order. The design must respect the ollowing speciications: N > max β mb,α ma Coeicients α,α 1,...,α N 1 must be chosen such that Hs isstable. A particular choice o SVF, proposed by Cois et al. 21, is the ractional Poisson s ilter: 1 H s = n N = s ω +1 s nn + N 1 ω n sn N 1 ω nn.. N N 1 n N 1 ω s n + ω nn which is simply an extension o the rational Poisson s ilter to ractional dierentiation orders. Frequency ω is ixed by the user according to the requency characteristics o the system to be identiied close to the highest corner requency. The state vector, composed o ractional derivatives o iltered input or output signals, is deined by: [ ] D N x = 1γ z t, D N 2γ T z t,...,d γ z t,z t 2 where z denotes either u or y. The ractional state space representation o the ilter is given by: D γ x t =A x t+b z t 21 where A = 68

5 instrumental variables. The authors also suggest to optimize instruments by an iterative method. 2.2 Output-error models Fig. 2. Fractional state variable ilters N 1 ω γ N 2 ω 2γ N N 1 ω γn and B = [ ] T ω γn... ω γn Each state represents the derivative o a given order o input or output signals see igure 2. Fractional Poisson s ilters are simulated using 4 The estimated parameters vector ˆθ o θ is now obtained by minimizing the quadratic norm o the iltered equation error: J ˆθ = E T E 22 and φ E = [ ε k T s ε k +1T s ε k + K 1 T s ε t =y t φ tˆθ, being deined by 19. The solution is given by the classical least squares: 1 ˆθ = Φ T Φ T Φ Y 23 Φ = [ φ T k T s φ T k +1T s... ] T φ T k + K 1T s ] 24 As in the classical case, Cois et al. 21 showed that the least squares estimator 23 is biased in presence o noisy output. To eliminate the bias, they propose to use instrumental variable method. Parameters are estimated according to: ˆθ opt IV = Φ IV T 1 Φ IV T Φ Y where Φ IV is the regression matrix ormed o derivatives o iltered inputs and derivatives o Output-error-based models allow simultaneous estimation o dierentiation orders and model parameters. Mainly, two identiication methods were proposed in the literature. They dier in the way ractional derivatives are simulated and, at the same time, the way dierentiation orders are estimated. The irst approach is based on discretetime simulation o ractional models and assumes that the ractional behavior is present in the whole requency band, whereas the second one is based on continuous-time simulation o ractional models and assumes that the ractional behavior is present in a limited requency band Method based on discrete-time simulation o ractional models Here, the system to be identiied is assumed to be initially at rest, modeled by 6. It is now characterized by input/output vector ormed o coeicients and dierentiation orders θ =[a,...a ma,b 1,...b mb,α,...α ma, β 1,...β mb ]. When the number o parameters in 6 is high, optimization algorithms applied on θ are illconditioned due to the absence o constraints on dierentiation orders. One way or introducing a constrained optimization on dierentiation orders and, at the same time, limiting the number o parameters consists o optimizing the commensurate order γ instead o all dierentiation orders. In this case, the ractional transer unction 6 is rewritten in a commensurate orm as in 13. Numerator and denominator orders, respectively α ma and β mb both multiples o γ, are ixed as in classical rational models. Henceorth, the system is entirely characterized by coeicients vector: θ =[a,...,a ma,b 1,...,b mb,γ]. As ar as identiication o stable systems is concerned, the commensurate order can be constrained to ], 2[ see condition 8 o stability theorem. Considering observed data ut and y t = yt + pt, pt being an output white noise, the quadratic norm: k +K 1 J ˆθ = ε 2 kt s, ˆθ 25 k=k o output error: ε kt s, ˆθ = y kt s ŷ kt s, ˆθ 26 is now minimized. Model s output ŷkt s, ˆθ being non linear in ˆθ, gradient-based algorithms, such as 69

6 the Marquardt algorithm Marquardt, 1963, are used to estimate ˆθ iteratively: ˆθ i+1 = ˆθ { } i [J θθ + ξi] 1 J θ k +K 1 J θ = 2 ε kt s S kt s, ˆθ :gradient θ=ˆθ i 27 k=k k+k 1 J θθ 2 S kt s, ˆθ S T kt s, ˆθ : hessian 28 k=k S kt s, ˆθ = ŷ kt s, ˆθ : output sensitivity θ ξ : Marquardt parameter Output sensitivity unctions can be computed by dierentiating 13 with respect to a i, b i, γ. In doing so, one can notice the presence o lns, in Fs Fs γ.consequently, γ is computed numerically rather than analytically. The idea o optimizing the commensurate order instead o all dierentiation orders was irst introduced in Cois et al., 2 who chose to write the transer unction 13 in a modal orm as in 7. They however constrained all s γ -poles to be real and o multiplicity one v k =1, k. In general, s γ -poles can be real or complex conjugate, and o multiplicity greater or equal to one Method based on continuous-time simulation o ractional models Trigeassou et al take as a building block o ractional models a non integer integrator bounded in the requency band as shown in 1 and described in section 1.3. The estimation o dierentiation order is carried out by estimating the parameters α and η o the recursive distribution o poles and zeros 1. Once α and η known, the dierentiation order γ is deduced according to 11. For the sake o simplicity, consider the ollowing ractional dierential system: D γ yt+a yt =b ut 29 Identiication algorithm System is identiied in an output error context as deined in section In the case, the ractional behavior is believed to be naturally limited in a requency band, say [ω A,ω B ], then authors propose to estimate the parameter vector: θ T =[a,b,α,η] 3 In all cases the optimized criterion is deined as in 25 and 26. The coeicients are computed recursively according to 27 and 28. Sensitivity unctions are now obtained by computing partial derivatives o 1 with respect to each o the parameter o 3. Fig. 3. Insulated long aluminium rod heated by a resistor System output System input Time s Time s Fig. 4. Estimation data 3. EXAMPLE To illustrate the use o ractional models in system identiication, a semi-ininite dimensional thermal system is considered. It is constituted o a long aluminium rod heated by a resistor. To ensure unidirectional heat transer, the entire surace o the rod is insulated. The temperature o the rod is measured at a distance x =5mm rom the heated end igure 3. The thermal system is considered as a semiininite plane homogenous medium initially at ambient temperature. Losses on the surace where the thermal lux is applied are neglected. Cois et al. 2 have shown that the analytical model linking the lux density applied on the outgoing normal surace o the medium to the temperature measured at an abscissa x inside the medium has a commensurate order o.5. First o all, the system was identiied by applying equation-error model and more precisely the SVF method. Identiication data are plot on igure 4. The commensurate order was set to.5 andthe ollowing three-parameter model was obtained: H 1 s =.256s s s 31 Then, equation error model was applied and the commensurate order optimized. The ollowing 61

7 system H 1 H 2 H Time s Fig. 5. Validation data ive-parameter model was obtained:.319 H 2 s = s s The optimal commensurate order is close to.5 as in the analytic model Cois et al. 2. Next, or comparison purposes, a twelve parameters rational model was identiied: H 3 s =.1s s 4.7s s 2 +.3s +1 5 s s s s s s +.1 The normalized mean squared errors computed on validation data or both ractional models are close to each other: NMSEH 1 NMSEH ; whereas the normalized mean squared error o the rational model is: NMSEH As shown on validation data o igure 5, the identiied models give satisactory results. 4. CONCLUSION AND OUTLOOKS This paper presents a tutorial on system identiication using ractional dierentiation models. Mainly equation-error and output-error models were detailed. In the ormer dierentiation orders are ixed and only model s parameters are estimated. In the latter both dierentiation orders and model s coeicients are estimated. One way or limiting the number o parameters consists o estimating the commensurate order and ixing all its multiples. For the time being, only white additive noise was considered. All model classes including colored noise should be extended to ractional dierential orders. Moreover, system identiication using stochastic signals is worth consideration. Multiple other questions regarding ractional system identiication remain unanswered. One o the most challenging is how to take into account initial conditions? This question cannot be answered as easily as in the rational case because a noninteger derivative o a signal depends on its whole past. Lorenzo and Hartley 2 showed that the eect o the past can be considered by taking into account an initialization unction instead o a limited number o points. Can such a unction be identiied? REFERENCES Aoun, M. 25. Systèmes linéaires non entiers et identiication par bases orthogonales non entières. PhD thesis. Université Bordeaux 1. Talence, France. Cois, O. 22. Systèmes linéaires non entiers et identiication par modèle non entier : application en thermique. PhD thesis. Université Bordeaux 1. Talence, France. Cois, O., A. Oustaloup, E. Battaglia and Battaglia J.-L. 2. Non integer model rom modal decomposition or time domain system identiication. In: proceedings o 12th IFAC Symposium on System Identiication, SYSID. Santa Barbara, USA. Cois, O., A. Oustaloup, T. Poinot and J.-L. Battaglia 21. Fractional statevariable ilter or system identiication by ractional model. In: IEEE 6th European Control Conerence ECC 21. Porto, Portugal. Grünwald, A.K Ueber begrenzte derivationen und deren anwendung. Zeitschrit ür Mathematik und Physik pp Le Lay, L Identiication réquentielle et temporelle par modèle non entier. PhD thesis. Université Bordeaux I. Talence, France. Lin, J. 21. Modélisatoin et identiication de systèmes d ordre non entier. PhD thesis. Université de Poitiers. France. Lorenzo, C. and T. Hartley 2. Initialized ractional calculus. Int. J. o Applied Mathematics 33, Malti, R., M. Aoun, F. Levron and A. Oustaloup 25. Fractional Dierentiation and its Applications. Vol. 1 Mathematical tools, geometrical and physical aspects. Chap. Uniied construction o ractional generalized orthogonal bases, pp Marquardt, D.W An algorithm or leastsquares estimation o non-linear parameters. J. Soc. Industr. Appl. Math. 112, Matignon, D Stability properties or generalized ractional dierential systems. ESAIM proceedings - Systèmes Diérentiels Fractionnaires - Modèles, Méthodes et Applications. Oldham, K.B. and J. Spanier The ractionnal calculus. Academic Press, New-York and London. Oustaloup, A Systèmes asservis linéaires d ordre ractionnaire. Masson - Paris. Samko, S.G., A.A. Kilbas and O.I. Marichev Fractional integrals and derivatives: theory and applications. Gordon and Breach Science. Trigeassou, J.-C., T. Poinot, J. Lin, A. Oustaloup and F. Levron Modeling and identiication o a non integer order system. In: ECC. Karlsruhe, Germany. 611