Instrumental Variable Identification of Hybrid Fractional Box-Jenkins Models

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1 Preprints o the 8th IFAC World Congress Milano (Italy) August 28 - September 2 2 Instrumental Variable Identiication o Hybrid Fractional Box-Jenkins Models Stéphane Victor Rachid Malti Pierre Melchior and Alain Oustaloup Université de Bordeaux IPB. IMS UMR 528 CNRS 35 cours de la Libération F 3345 Talence cedex France {stephane.victor rachid.malti pierre.melchior alain.oustaloup}@laps.ims-bordeaux.r Abstract: this paper deals with continuous-time system identiication using ractional dierentiation models in a colored noisy output context. An optimal instrumental variable method or identiying ractional models in a hybrid Box-Jenkins orm is described. The relationship between the measured input and the output is a ractional continuous-time transer unction and the noise is a discrete-time AR or ARMA process. Keywords: ractional continuous-time models; ractional dierentiation; instrumental variable; state variable ilter; system identiication; time-domain.. INTRODUCTION Instrumental Variable (iv) approaches to system identiication and parameter estimation have been developed since 97s as detailed in the survey paper by Young and Jakeman (98) and all reerences therein. The Simpliied Reined Instrumental V ariable (srivc) algorithm or ractional systems namely srivc has been developed in Malti et al. (28). In that paper the leastsquare (ls) method and the instrumental variable (iv) combined with state variable ilter (sv ) are recalled. The model structure is ixed and only the coeicients were identiied in a noisy context. An ative optimalinstrumental variable method or ractional continuous-time model has been implemented where the auxiliary model and the state variable ilters are updated at each ation. This optimal iv method presents the advantage o yielding asymptotically consistent estimates in the presence o additive noise and o minimum variance in the presence o white noise. However i the noise is not white the variance is no more minimum. In this paper the dimension o the true system and the commensurate order are assumed to be known. Although the iv aspects ensure that the parameter estimates are asymptotically unbiased even in noisy colored context the estimates are not statistically eicient in this situation because the sv are not suited in colored noise context. Recently Reined Instrumental Variables or Continuoustime system (Young et al. (26)) has been developed or identiying hybrid continuous-time Box-Jenkins models where the system model is continuous-time (CT) and the noise model is discrete-time (DT). Here the CT Box and Jenkins models are extended to ractional models: a ractional CT model links the input to the output and a DT ARMA process is used or noise modeling. All dierentiation orders are exactly divisible by the same number an integral number o times and the biggest number is always chosen. A ractional mathematical model is based on a ractional dierential equation: y + a p α y + + a ma p αm A y = b p β u + b p β u + + b mb p βm B u () where (a j b i ) R 2 p = d dt is the dierential operator the dierentiation orders < α < α 2 <... < α ma < β < β <... < β mb are allowed to be noninteger positive numbers. The concept o dierentiation to an arbitrary order (non-integer) p γ with γ R + (set o strictly positive real numbers) was deined in the 9 th century by Riemann and Liouville. The γ-orderractional derivative o y is deined as being an integer derivative o order γ + (. stands or the loor operator) o a non-integer integral ( o order γ γ Samko et al. (993): p γ y=p γ + I y) γ + γ = p γ + t y (τ)dτ (2) Γ ( γ + γ) (t τ) γ γ where I stands or the integral operator and the Euler Γ unction is deined in the set o real numbers except zero and negative whole numbers (x R \ N ) as: Γ(x) = e t t x dt. (3) The Laplace transorm is a more concise algebraic tool generally used to represent ractional systems Oldham and Spanier (974): { } L y (γ) = s γ X (s) i y = t. (4) This property allows to write the ractional dierential equation () in a transer unction orm: m B G(s) = B(s) b i s βi A(s) = i=. (5) + ma a j s αj j= Copyright by the International Federation o Automatic Control (IFAC) 434

2 Preprints o the 8th IFAC World Congress Milano (Italy) August 28 - September 2 2 Stability theorem (Matignon (998) extended): A commensurate transer unction G(s) = T(sγ ) R(s γ ) T and R being coprime polynomials is BIBO stable i and only i : < γ < 2 (6) and arg(s k ) > γ π 2 s k C / R(s k ) =. (7) Im(s k ) γ π 2 Stability axe Re (s k ) Stability axe Fig.. Stability region: a system is stable i its s γ -poles are inside the green area ( γ < ) The aim is to develop a method or time-domain system identiication workable with any ractional-model-timedomain simulation algorithm. Here the recursive approximation o a ractional dierentiation with poles and zeros developed by Oustaloup (995) is used or simulating the ractional operator. 2. SYSTEM IDENTIFICATION Most o the direct CT model identiication techniques do not consider the estimation o a noise model leading into statistically ineicient estimates (high estimate variance). The Reined Instrumental Variables or Continuous system (rivc: see Young and Jakeman (98)) method and its Simpliied version srivc (Young (22)) when the additive measurement noise is assumed to be white are the only iv methods that can be interpreted in optimal statistical terms. A logical extension o srivc or ractional models srivc has been developed in Malti et al. (28). This approach involves an atively-adaptive preiltering method based on a quasi-optimal statistical solution to the problem in this white noise case. This technique has the advantage that the preilters are automatically optimized in an ative algorithm. Optimal instrumental variables are obtained rom an atively updated auxiliary model. At the irst ation the auxiliary model is initialized using a least square estimate obtained with state variable ilters. The initial  (s) polynomial does not have to be particularly accurate provided the preilter is stable and does not seriously attenuate input/output signals within the system bandwidth. 2. Problem ormulation Consider K + regular samples o input u and noisy output y data collected rom t = to t K = KT s. In order to have good statistical estimation results it is necessary to consider stochastic errors that aect the output signal. When the additive measurement noise ξ is colored the model describing the system is o Box- Jenkins type: { y = G(p)u ξ = H(p)e (8) y = y + ξ where y is the hypothetical noise-ree deterministic system output the input/output model G(p) is described by (5) and the noise model H(p) supposed stable and invertible is described by an Auto-Regressive (AR) or Auto- Regressive Moving Average (ARMA) process: H(p) = C(p) D(p) (9) and where e is a zero-mean white additive gaussian noise. Since the data are sampled and the representation o a white noise is in discrete-time a more suited noise model is a discrete-time ARMA(r v) process: + v c i q i H(q ) = C(q ) D(q ) = i= + r () d j q j j= where q is the backward shit operator such that q l y(t k ) = y(t k l ). A hybrid model is then ormulated with y = G(p)u ξ(t k ) = H(q )e(t k ) y (t k ) = y(t k ) + ξ(t k ) () where e(t k ) is the white gaussian noise sampled at t k the system model is deined in continuous-time as in (5) and the noise model is discrete-time as in (). When the dierentiation orders are known the parameter vector θ = ρ η T (2) is composed o the m A +m B + coeicients o the system model ρ = b b...b mb a...a ma T and o the r + v + coeicients o the noise model η = c c... c v d... d r T. 2.2 Least square estimate In white gaussian additive noise (H(p) = ) the measured output is corrupted by an additive measurement noise e y = y + e and the equation error can be ormulated as: ε = y ϕ T θ (3) where u ϕ T (β ) u = (βm B ) y (α) y (αm A ) (4) and θ T = ρ T = b b... b mb a... a ma. (5) Minimizing the L 2 norm o ε with respect to θ leads to the ollowing least square estimates: t K tk ˆθ ls = ϕ T ϕ dt ϕ T y dt. (6) t 435

3 Preprints o the 8th IFAC World Congress Milano (Italy) August 28 - September 2 2 Ater a discretization bydeining Y as a column vectoro the system output and Φ as a regression matrix: Φ = ϕ (t ) ϕ (t )... ϕ (t K ) T (7) ˆθ ls can be approximated by: ( ˆθ ls = Φ T Φ ) Φ T Y. (8) Directractionaldierentiations onoisyoutput leadhowever to inaccurate results. Thus srivc is preerredto the direct ls estimate. 2.3 Simpliied reined instrumental variable or continuous-time ractional models (srivc) The srivc stochastic approach proposed by Young (22) includes an adaptive preiltering based on the optimal solution when the noise ξ is a white gaussian one (H(p) = and the parameter vector is resumed to θ = ρ). This method is a logical extension o the instrumental variable estimator with state variable ilters (Cois et al. (2)). This estimator extended to ractional models as srivc (srivc or ractional models) is recalled (see Malti et al. (28)). Following the Prediction Error Minimization (PEM) approach a suitable error unction ε in CT is given by the output error (OE): ε = y B (p) u (9) A(p) The minimization o the quadratic crion unction ε provides the basis or stochastic estimation. The error unction can be rewritten as: ε = A(p) A(p)y B (p)u ( ) ( ) = A(p) A(p) y B (p) A(p) u. (2) By considering F(p) = A(p) as the ilter y = F(p)y and u = F(p)u the error becomes ε = A(p)y B (p)u (2) or ε = y + a y (α ) + + a ma y (α ma ) b u (β) b u (β) b mb u (βm B ) (22) where { u βi = F(p)u i = m pβi B y α j = p αj F(p)y (23) j = m A. In practical cases A(p) is not known. The ilter F(p) is then updated by an ative procedure which estimates the denominator Â(p). This algorithm enables to optimize the instruments. Initialized by the ˆθ iv/sv estimates (see Malti et al. (28) or the details) as the irst estimation o the denominator is available the new derivatives o input and output signals are computed with the set o ilters: F (p) = Â(p) = (24) + ma â j p αj j= where = 2... stands or the ation number and â j the estimated a j coeicient at the ation. Instruments and input/output iltered derivatives are computed at each ation by: u (βi) = p βi F (p)u y(α j) = p αj F (p)y (25) y iv (α j) = p α j F (p)y iv where y iv = ˆB(p) u. The regression vectors Â(p) ϕ and are ormed as in (4): u (β) u (βm B ) y (α ) y ϕ T = (α ma ) u β T = u(βm B ) y iv (α ) y iv(α ma ). (26) srivc Finally the parameter vector ˆθ is computed as: ( ) ˆθ srivc T = Φ Φ ivt Y (27) with Φ iv Y = y (t ) y (t )...y (t K) T Φ = ϕ (t ) ϕ (t )... ϕ (t K) T Φ iv = (t ) (t )... (t K ) T. (28) As in the rational case the srivc estimator is asymptotically unbiased whatever additive zero-mean noise is present. When the additive noise is white srivc has minimum variance and the asymptotic covariance matrix o the estimation errors associated to the estimate o ˆθ srivc : T Φ iv Pˆθ = ˆσ 2 (Φ iv ) (29) where ˆσ 2 is the empirical estimation o noise variance. The associated estimation model can be written as y = ϕ T θ + ε(t k ) (3) where ε(t k ) is the equation error and ϕ T = u (β ) u u (β mb ) y (α ) y (α ma ). (3) Hence the estimation model (3) orms a basis or the deinition o alikelihood unction and maximum likelihood estimation. Summary o the srivc algorithm Step Use the iv method combined to ractional state variable ilter to generate an initial ( = ) estimate o the transer unction model parameter vector ˆθ (). Step 2 Iterative iv estimation with preilters. or = 2 to convergence (i) Generate the iv y iv rom the auxiliary model with the estimated polynomials based on the estimated srivc parameter vector ˆθ. (ii) Update the ilter F (s ˆθ) in (24) with the new estimated parameters. Then evaluate the preiltered derivatives o u y and y iv as in (25). (iii) Based on these preiltered data compute the new srivc estimates ˆθ as in (27). end or 436

4 Preprints o the 8th IFAC World Congress Milano (Italy) August 28 - September 2 2 Step 3 Compute the estimated parametric error covariance matrix with (29). It may happen when the bias is important that the estimated parameters rom Step lead to unstable models. In this case an empirical method 2 can be used to stabilize unstable poles. 2.4 Reined instrumental variable or continuous-time ractional Box-Jenkins models (rivc) I the noise ξ is colored (H(p) and the parameter vector θ is deined by ρ η T ) then the srivc method has no claim to optimality in statistical terms as the variance o the estimated parameters is not minimal. However thanks to the iv mechanism the estimates are consistent (asymptotically unbiased). The algorithm is very robust and normally produces estimates with reasonable statistical eiciency meaning that the variance is low but not minimum. The algorithm is extended to the colored noise case namely rivc (rivc or ractional systems see Garnier and Wang (28) or the rational case) method by using a hybrid approach: the system model is a ractional CT transer unction and the noise model is a DT AR or ARMA process. By ollowing the usual PEM approach in the hybrid case which corresponds to the maximum likelihood estimation given the gaussian hypothesis on e the error unction at t k is given by e(t k ) = D ( q ) C (q ) which can also be written as e(t k ) = D ( q ) C (q ) { y (t k ) B(p) } A(p) u(t k) { A(p) A(p)y (t k ) B(p)u(t k ) (32) } (33) where the DT preilter D(q ) C(q ) corresponds to the inverse o the noise model ARMA(r v). In the last two equations the DT and CT operators are used in order to indicate the hybrid nature o the estimation problem. Thus the operations such that B(p) A(p) u(t k) imply that the variable u(t k ) needs to be interpolated. The ls error crion minimisation on e(t k ) being at the heart o the stochastic estimation the equation (33) can be considered under the alternative orm e(t k ) = A(p)y (t k ) B(p)u (t k ) (34) where y (t k) and u (t k ) are the sampled outputs o the hybrid preiltering operations namely a CT iltering o y and u with the system model denominator: Fc opt (p) = A(p) (35) and a DT iltering o the error e with the inverse noise model: F opt d (q ) = D(q ) C(q ). (36) The system and noise models being unknown in practical cases ative procedure are suggested in the iv estima- 2 This method consists o replacing every unstable s γ -poles by its symmetrical one with respect to its closest stability axe (Fig. ) tion methods. The optimal preilters in (35) and in (36) are respectively replaced by F c (p) = (37) Â(p) and F d (q ) = ˆD(q ) Ĉ(q ) (38) computed atively where  Ĉ and ˆD correspond respectively to the estimation o A obtained at ation ( ) and to the estimations o C and D computed at the current ation (). Moreover the instrumental variable is generated rom the CT auxiliary model y iv = Â(p) u. (39) ˆB(p) The iltered derivatives o the input the output and the instrumental variable are computed at each ation by: u (βi) (t k ) = F d (q )u (βi) c (t k) i =...m B y(α j) (t k ) = F d (q )yc(α j) (t k ) j =...m A y iv (α j) (tk ) = F d (q )y iv (α j) c (tk ) j =...m A u (βi) c = F pβi c (p)u i =...m B yc(α j) = p αj F c (p)y j =...m A (α j) = p α j F c (p)y iv j =...m A. y iv The regression vector ϕ (t k) and the instrumental vector (t k) which depend on ˆρ and ˆη are computed: ϕ u (β) T = u (βm B ) T = y(α ) y(α ma ) u(βm B ) y iv (α ) y iv(α ma ). (4) u β Finally the parameter vector ˆρ rivc is calculated as: ˆρ rivc = Φ iv T Φ Φ ivt Y (4) with Y = y (t ) y (t )...y (t K) T Φ = ϕ (t ) ϕ (t )...ϕ (t K) T Φ iv = (t ) (t )... (t K ) T. (42) The parameter vector ˆη is estimated at each ation by applying any ARMA model estimation algorithm (Ljung (999); Söderström and Stoica (989)) on the output error assimilated to the colored noise at each ation: ˆξ(t k ) = y (t k ) y iv (t k ). (43) This algorithm based on linear regression is deined rom a quadratic error crionthat is convex. Thereore in all identiiable situations the rivc algorithm converges to the true parameters and the estimates will be consistent and asymptotically eicient. Asymptotic covariance matrix estimation P ρ associated to the estimates ˆρ rivc obtained at the convergence o ˆρ rivc is given by: 437

5 Preprints o the 8th IFAC World Congress Milano (Italy) August 28 - September 2 2 K Pˆρ rivc = ˆσ 2 (t k ) (t k ) T (44) k= where (t k) is the instrumental vector at convergence and ˆσ 2 is the empirical estimation o noise variance. power 2 2 Fig. a) Summary o the ull rivc Algorithm Step Initialization Apply the srivc algorithm in order to obtain an initial estimation at ation = o the parameter vector ˆρ this estimation being used to generate the CT initial preilter F c (p). ˆθ = ˆρ ˆη T where C/D =. Step 2 Iterative iv estimation do (i) = + Generate the instrumental vector y iv rom (39) and rom the estimated parameters ˆρ. (ii) Update the CT ilters F c (p) in (37) with the new estimated parameter vector ˆρ. Then evaluate the iltered derivatives o the input u output y and instruments y iv. (iii) Estimate the parameter vector o the noise model ˆη by using an ARMA estimation algorithm on the output error ˆξ(t k ) (43). (iv) Update the DT ilters F d (q ) in (38) with the estimated parameter vector ˆη. Then apply the DT ilter to the derivatives o signals calculated in (ii) namely the derivatives o u c (t k ) y c (t k ) and y iv c (t k ) in order to deine the regression vector ϕ (t k) and the instrumental vector (t k) as in (4). (v) From the iltered signals compute the new parameter vector ˆρ according to (4). The global rivc parameter vector ˆθ = ˆρ ˆη T is then obtained at ation. while max ˆθ jrivc jrivc j ˆθ > ǫ where corresponds to the j th element o the ˆθ jrivc rivc ˆθ parameter vector obtained at ation. Step 3 Parametric error estimation Compute the covariance matrix o the parametric error o the estimates ˆρ rivc with (44). 3. NUMERICAL EXAMPLE A simulation example illustrates the perormance o the srivc algorithm and o the proposed ull rivc method in case o colored output noise in system identiication using ractional models. The simulated model is given by: K ( Tp ν + ) G (p) = ( )( ( p ) 2ν + 2ζ ω ( p ) ν + ( p ) 2ν + 2ζ ω ω 2 ( p ) ν ) (45) + 2 ω 2 with ν =.5 K = T =.5 ω =.2 rad/s ζ =.4 ω 2 = rad/s and ζ 2 =.65. This ractional system has two resonant modes: one at ω =.2 rad/s with ζ =.4 and another one at ω 2 = rad/s with ζ 2 =.65 (or stable ractional systems a resonant peak may occur or negative pseudo-damping actors ζ 2 3 ). Furthermore the system has a positive s ν - zero at s ν = 2. 3 When ν ζ cannot be considered as a damping actor. outputs requency (rad/s) Fig. b) time (s) Fig. 2. Fig. a): Power spectral density o u Fig. b) Noisy output y and noise-ree output y The input signal u a pseudo random binary sequence o maximum length is used to excite the system (45) over its whole bandwidth. Its amplitude switches between 5 and 5 and the power spectral density is given on Fig. 2-a). The sampling interval is ixed to T s = 5. 2 s. Based on the CT process (45) the ollowing hybrid system is considered (see Fig. 2-b)) { y = G (p)u y (t k ) = y (t k ) + ξ (t k ) (46) where ξ (t k ) is a colored additive noise derived rom an ARMA(2 ) process /C(q ): C ( q ) D (q ) = +.92q.96q +.97q 2 (47) with a signal-to-noise ratio o db. The instrumental variables are computed according to the ative algorithm described in 2.4 in order to ripen the parametric estimation. The system model structure is assumed to be known and only the model parameters are estimated. The model is set to: b p ν + b G(p) = a 4 p 4ν + a 3 p 3ν + a p 2ν + a p ν +. (48) The commensurate order is set to the true value.5. However the commensurate order is not always known; when unknown the optimal commensurate order can be estimated. Moreover the noise model structure is supposed known: C ( q ) D (q ) = y y + c q + d q. (49) + d 2 q 2 A Monte-Carlo Simulation (MCS) o 2 runs is carried out with a white noise input to the ARMA model being selectedrandomlyateachrun. The MCSresults obtained rom the srivc and the ull rivc algorithms are shown in Table. From the srivc columns o Table it is shown that this algorithm delivers satisactory results however it is less accurate than the ull rivc method. Moreover the rivc algorithm clearlyimproves the statistical eiciency: in Table σ θj o rivc method are much lower than the ones in srivc. Furthermore an ative algorithm o the rivc method clearly improves the statistical estimation. 438

6 Preprints o the 8th IFAC World Congress Milano (Italy) August 28 - September 2 2 vrai srivc rivc (SR) rivc θ ˆθ σ ˆθ σ ˆθ σ a a a a b b c d d Table. Monte-Carlo simulation results: parameters o the models (48) and (49) evaluated with the srivc rivc (SR: Single Run) and rivc methods (ˆθ is the mean and σ is the standard deviation) with SNR = db Mag (db) Mag (db) Fig. a) Fig. b) 2 Frequency (rad/s) Fig. 3. Fig. a): Bode diagrams o the CT system ( ) and its estimated models or 2 runs o Monte-Carlo with the srivc (. ) and the rivc ( ) methods Fig. b): Bode diagrams o the DT system ( ) and its estimated models or 2 runs o Monte-Carlo with the rivc ( ) methods In Fig. 3 -a) the 2 ull rivc identiied models coincide with the true system; moreover the 2 srivc identiied models also converge to the true system with a higher parameter variance. The true noise model and its 2 ull rivc identiied noise models are plotted in the Bode diagram o Fig. 3 -b). As expected the true system and the identiied models coincide perectly. 4. CONCLUSION Ater a short introduction on ractional models and a synopsis o the srivc method the ull rivc algorithm or identiying hybrid Box-Jenkins ractional systems is presented when being in presence o colored noise. Even though the srivc method produces satisactory results it is shown through Monte-Carlo simulations that the ull rivc method improves clearly the statistical eiciency. When dierentiation orders are unknown as in practical physical systems they can be estimated using a gradientbased approach (see Victor et al. (29) or the white noise case). REFERENCES O. Cois A. Oustaloup T. Poinot and J.-L. Battaglia. Fractional state variable ilter or system identiication by ractional model. In 6th European Control Conerence ECC Porto Portugal 2. H. Garnier and L. Wang. Identiication o CTmodels rom sampled data. Springer-Verlag 28. L. Ljung. System identiication Theory or the user. Prentice-Hall Up. Sad. River N.J. USA 2 ed R. Malti S. Victor A. Oustaloup and H. Garnier. An optimalivmethodorct ractionalmodelidentiication. In The 7th IFAC World Congress (IFAC 8) Seoul Korea July 28. D. Matignon. Stabilityproperties or generalizedractional dierential systems. ESAIM proceedings - Systèmes Diérentiels Fractionnaires - Modèles Méthodes et Applications K.B. Oldham and J. Spanier. The ractionnal calculus - Theory and Applications o Dierentiation and Integration to Arbitrary Order. Academic Press N.-Y A. Oustaloup. La dérivation non-entière. Hermès - Paris 995. S.G. Samko A.A. Kilbas and O.I. Marichev. Fractional integrals and derivatives: theory and applications. Gordon and Breach Science 993. T. Söderström and P. Stoica. System Identiication. Series in Systems and Control Engineering. P. H S. Victor R. Malti and A. Oustaloup. IV method with optimal ractional dierentiation order or CT system identiication. In 5th IFAC Symposium on System Identiication (SYSID 29) Saint Malo France July 29. P.C. Young. Optimal IV identiication and estimation o CT TF models. In 5th World IFAC Congress Barcelona (Spain) 22. IFAC Elsevier. P.C. Young and A.J. Jakeman. Reined IV methods o time-series analysis: Part III extensions. International Journal o Control 3: P.C. Young H. Garnier and M. Gilson. An optimal IV approach or identiying hybrid CT Box-Jenkins models. In 4th IFAC Symposium on System Identiication (SYSID 26) pages Newcastle Australia