An approach for the identification of dynamic single-input single-output bilinear
discrete-time fractional order system models within the errors-in-variables framework for the
case of white input and output noise sequences is presented. A criterion is obtained that allows
obtaining highly consistent estimates of the parameters of the system.The authors gratefully acknowledge the contributions of prof. O.A. Katsyuba in improving the article

Chertykovtseva, N.V.

Ivanov, D.V.

Sandler, I.L.

Samara University

Математическое моделирование физико-технических процессов и систем IV Международная конференция и молодёжная школа «Информационные технологии и нанотехнологии» (ИТНТ-2018) Identification of dynamic errors-in-variables bilinear systems of fractional order  D.V. Ivanov1, I.L. Sandler2, N.V. Chertykovtseva2 1Samara National Research University, Moskovskoe Shosse 34А, Samara, Russia, 443086 2Samara State University of transport, Svobody street 2 В, Samara, Russia, 443066   Abstract. An approach for the identification of dynamic single-input single-output bilinear discrete-time fractional order system models within the errors-in-variables framework for the case of white input and output noise sequences is presented. A criterion is obtained that allows obtaining highly consistent estimates of the parameters of the system. Keywords: bilinear dynamical systems, a difference of fractional order, least square, strongly consistents.  1. Introduction In order to describe processes of different nature, equations with derived differences of fractional order are increasingly used. Although there is no simple interpretation, which is possessed by derivatives, integrals and differences of integers, models described by equations of fractional order, used in physics and engineering [1-4], the branch of the theory of management related to the synthesis of regulators of fractional order is actively developing.   In connection with the active development and application of equations with differences and derivatives of fractional order for modeling and forecasting, methods of identification of systems described by equations and differences of fractional order are being actively developed. Most of the papers are devoted to the parametric identification of differential equations of fractional order with noise in the equation or the output signal [5-8].   The problem of identification with the presence of noises in the input and output signals is much more complicated, for example in [9] a method based on the cumulant, involving significant restrictions on signal and noise is proposed.  Articles [10-12] are devoted to identification of systems described by equations, with differences of fractional order.    In the field of physical system modelling the bilinear system (BS) models have been exploited extensively in various scientific areas, such as nuclear fission, electric networks, heat transfer, fluid flow, chemical kinetics etc., see e.g. [13-14]. Their popularity and broad applicability stems from the fact that BS models are able to satisfactorily approximate many nonlinear processes.  Furthermore, building on the theory of linear systems, the state dependent steady-state and dynamic properties of BS models are well understood. Therefore, the BS models are commonly utilised as a stepping stone when analyzing systems exhibiting nonlinear behavior. Consequently, a need to extend Математическое моделирование физико-технических процессов и систем D.V. Ivanov et al.IV Международная конференция и молодёжная школа «Информационные технологии и нанотехнологии» (ИТНТ-2018)   1840 the EIV system description to encompass BS models is prompted. This allows a combination of both, i.e. the increased applicability offered by the BS models and the generalized system setup afforded bythe EIV framework.In this paper an approach for the identification of single-input-single-output (SISO) discrete-time fractional EIV BS models is proposed considering the case when the input and output noise sequences are white. 2. Problem statementConsider the class of the discrete-time input-output SISO (single-input single-output) bilinear systemsthat can be represented by the following nonlinear auto-regressive with exogenous input process, i.e.( )( )( )( ) ( ) ( ) ( )( )( )0 01 1c31 21 2 300 1γα β− − − −= =∆ = ∆ + ∆∑ ∑ ∑∑mm mm m m krr rr mk(m) (m) mki f i f i f i fm= m= m kb z a x x z      (1) ξi i iy = z + , ζi i iw = x + ,  where     (1) (r)0 α α…< < , 110 β β… (r )( )< < ,  (k)310 γ γ… (r k)( k)< < , 21,r ,k =( )(m) (m) (m)(m)01 ,αα− − − ∆ −   ∑iji f i j fj=z = zj ( )(m)(m) (m)1 1(m)01 ,ββ− − − ∆ −   ∑iji f i j fj=x = xj ( ) ( )( ) ( ) (m) (k)2 3 2 3(mk)01 ,γγ− − − − − − ∆ −   ∑m ki(mk) ji f i f i j f i j fj=x z = x yj(m) (m)(m)1 ,1 1α αα  Γ  Γ Γ − ( + )=(j + ) ( j + )j (m) (m)(m)1 ,1 1β ββ  Γ  Γ Γ − ( + )=(j + ) ( j + )j (mk) (mk)(mk)1 ,1 1γ γγ  Γ  Γ Γ − ( + )=(j + ) ( j + )j10,αα∞− −Γ ∫ t( )= e t dt(m) (m) (m) (k)1 2 3, , ,f f f f are non-negative values of delays, i iz , y are noise-free output and input, i ix ,w  are the measured output and input, i i,ξ ζ  are noise of observation in the output and the input signals,   The following assumptions are introduced A1.The discrete-time bilinear system is asymptotically stable, observable and controllable. A2. The system structure, i.e.  (k)1 2 3, , ,r r r r(m) (m) (m) (k)1 2 3, , , ,f f f f(m) (m) (mk), ,α β γ  is known a priori. A3. The true input signal ix  is a random process persistently exciting of sufficiently high order with  ( ) 0,Ε ix = ( )2 2σΕ = ≤ ∞i xx : ( )( ) ( )11lim ,ϕ ϕ→∞==∑N Ti ix xN iHNa.s.,( )( ) ( )11liim ,ϕ ϕ→∞==∑N Ti ix xN iHN ( )(r )(1) 1(1) (r )11 1( ) , , ,β βϕ− −= ∆ ∆Tix i f i fx x   matrix H  is restricted positive definite. A4. Input/output noise sequences { }iξ  and { }iζ  are zero mean, ergodic, white signals with known variances, denoted 2xσ   and 2σ y . A5. Input/output noise sequences { }iξ  and { }iζ  mutually uncorrelated and uncorrelated with the noise-free signals{ }iz ,{ }ix . The system parameter vector is defined as Математическое моделирование физико-технических процессов и систем D.V. Ivanov et al.IV Международная конференция и молодёжная школа «Информационные технологии и нанотехнологии» (ИТНТ-2018)   1841 ( )0 0 0 0 ,θ =TT T Tb a c ( )(2) ( )0 0 01 ...=Trb b b , ( )1( )(1)0 0 0... ,=Tra a a( )( )(1) (1) 223 3 32(1 ) (2 ) ( )( 1)(11) (21)0 0 0 0 0 0 0 .=     r Tr r r rrc c c c c c cThe regressor vector for the measured signals is defined as ( ) ( ) ( )( )( ) ( ) ( )) ,ϕ = ϕ ϕ ϕ TT T Ti i ii y w wy  ( )(1) (r)(1) (r)( ) ,α α− −ϕ = ∆ ∆ Tiy i f i fy y( )(r )(1) 1(1) (r )11 1( ) , , ,β β− −ϕ = ∆ ∆Tiw i f i fw w ( )(m)3( )( ) ( 1) ( 2) ,ϕ = ϕ ϕ ϕ Tiri i iwy wy wy wy  2( ) ( )(1) ( )23 32 2( )(1 )( ) .k krr kkikwy i f i fi f i fw y w yγ γ− −− −  ϕ = ∆ ∆  Given N samples of the measured signals, i.e. { }iy ,{ }iw , determine the vector θ̂ . 3. Criteria for identificationThis system can be reformulated in equation error form as( ) ( ) ( )0 0 0 ,ϕ − ϕ − ϕ = εT i T i T iy w wy ia b cwhere the equation error or residual is given by ( )( ) ( ) ( ) ( ) ( )0 0 0 ,ξ ζ ξ ζ ζξε = ϕ − ϕ − ϕ + ϕ + ϕT i T i T i i ii x zb a c ( )(1) (r)(1) (r)( ) , , ,α αξ − −ϕ = ∆ ξ ∆ ξTii f i f  ( )(r )(1) 1(1) (r )11 1( ) , , ,β βζ − −ϕ = ∆ ζ ∆ ζTii f i f ( )(m)3( )( ) ( 1) ( 2) ,ξ ξ ξ ξϕ = ϕ ϕ ϕTiri i ix x x x2(1) ( ) ( ) ( )22 3 32( )(1 )( ) ,γ γξ − − −− ϕ = ∆ ξ ∆ ξ  k r kr kkikx i f i f i fi fx x ( )(m)3( )( ) ( 1) ( 2) ,ζ ζ ζ ζϕ = ϕ ϕ ϕ Tiri i iz z z z2(1) ( ) ( ) ( )22 3 32( )(1 )( ) ,γ γζ − − −− ϕ = ∆ ζ ∆ ζ  k r kr kkikz i f i f i fi fz z ( )(m)3( )( ) ( 1) ( 2) .ζξ ζξ ζξ ζξϕ = ϕ ϕ ϕ Tiri i i  From requirement A4, A5  it follows that  generalized error εi  has zero mean. We obtain that variance of generalized error equal to 2 2 2 20 0 0 0 0 0ε ς ξ α ζ β γσ = σ + σ + σ +T T Tb H b a H a с H c( )(11) (21) (1 )(21) (22) (2 )( ) ( ) 21(r1) ( 2) ( ),α α αα α αα ξ ξ ξ=α α α     = ϕ ϕ = σ        ∑   rrN Ti iir rrh h hh h hH Eh h h ( )( ) ( )( ) ( )(m) (k)1( )(m) (k)01lim ,−α αα − − →∞=   α α −= ∆ ξ ⋅∆ ξ = ⋅    − −   ∑m km kNmki f i f N jN jh EN Nj f j f 1,m r= , 1, ,k r=  ( )111 1 1 1(1 )(11) (12)(1 )(21) (22)( ) ( ) 21( 1) ( 2) ( ),rrN Ti iir r r rh h hh h hH Eh h hβ β ββ β ββ ζ ζ ζ=β β β     = ϕ ϕ = σ        ∑   ( )( ) ( )( ) ( )1 1(m) (k)1( )(m) (k)0 1 11lim ,m km kNmki f i f N jN jh EN Nj f j f−β ββ − − →∞=   β β −= ∆ ζ ⋅∆ ζ = ⋅    − −   ∑  11,m r= , 11, ,k r=Математическое моделирование физико-технических процессов и систем D.V. Ivanov et al.IV Международная конференция и молодёжная школа «Информационные технологии и нанотехнологии» (ИТНТ-2018)   1842 ( ) ( ) ( )( )222 2 2(1 )(11) (12)(2 )(21) (22)( ) ( ) ( ) ( ) ( ) ( )1( 2)( 1) ( ),crrN T T Ti i i i i iy y w wirr r rH H HH H HH EH H Hγ γγ γ γγ ζ ζ ξ ξ ζξ ζξ=γ γ γ     = ϕ ϕ + ϕ ϕ −ϕ ϕ =        ∑    ( )( )( ) ( ) ( ) ( )333 3 3 3(1r kk )(11kk ) (12kk )(2r kk )(21kk ) (22kk )(mm kk ) 2(r 1kk ) (r 2kk ) (r r kk ),kkk k k kh h hh h hHh h h′′′′′ ′γ γ γ′′ ′′ ′ γ γ γγ ξ′ ′ ′γ γ γ    = σ        ( ) ( ) ( ) ( )2 3 2 3( ) ( )( )m k m kmk m kmm kky i f i f i f i fh E y y′ ′′ ′γ γ′ ′ζ − − − − = ∆ ζ ⋅∆ ζ =  ( ) (k ) ( ) (k )3 3 2 2(mk) (m k )12( ) (k )0 2 21lim ,k kN f f f fykN jhN j f j f′ ′′ ′− − − −ζ ′→∞=   γ γ= σ     − −   ∑   ( ) (k ) ( ) (k ) ( ) (k ) ( ) (m )3 3 2 2 3 3 2 2 ,k k k mf f f f f f f fy yN jh hN′ ′ ′ ′− − + − − + −=  01 ,limNmy i i mi ih y yN +→∞ == ∑( ) ( ) ( ) ( )2 3 2 3( ) ( )( )m k m kmk m kmm kkw i f i f i f i fh E w w ′ ′′ ′γ γ′ ′ξ − − − − = ∆ ξ ⋅∆ ξ =  ( ) (k ) ( ) (k )3 3 2 2(mk) (m k )12( ) (k )0 3 31lim ,k kN f f f fwkN jhN g f j f′ ′′ ′− − − −ξ ′→∞=   γ γ= σ     − −   ∑ ( ) (k ) ( ) (k ) ( ) (k ) ( ) (m )3 3 2 2 3 3 2 2 ,k k k mf f f f f f f fw wN jh hN′ ′ ′ ′− − + − − + −=  01 ,limNmw i i mi ih w wN +→∞ == ∑( ) ( ) ( ) ( )2 3 2 3(mk) (m k )1( ) ( )( ) 2 2( ) (k )0 2 21limm k m kNmk m kmm kkki f i f i f i f N jN jh EN Nj f j f′ ′′ ′−′ ′γ γ′ ′ζξ ζ ξ ′− − − − →∞=   γ γ − = ∆ ζ ξ ⋅∆ ζ ξ = σ σ       − −     ∑With the known structure of the model, which means that orders r , 1r , 2r , 3(k)r , (m)f ,  1 2 3(m) (m) (k)f , f , f ,  (m)α , (m)β , (mk)γ  are determined, the following criterion can be applied to get strongly consistent estimates of parameters: ( )22 2 21.T ( i ) T ( i ) T ( i )Ny w wyT T Ta ,b,c i=a b cb H b a H a с H cmin ς ξ α ζ β γϕ − ϕ − ϕσ + σ + σ +∑       (2) Theorem. Suppose that a random process  { }, 1,0,1− iy i = is described by equation (1) with zero initial conditions and the assumptions 1-5 are met. Then estimate ( )ˆ Nθ , determined by expression (2) with probability 1 when N →∞ , exists and is a unique and strongly consistent estimate, i.e.  0Nâ a ,→∞→   a.s. 0Nb̂ b ,→∞→  a.s. 0Nĉ c .→∞→  a.s. Математическое моделирование физико-технических процессов и систем D.V. Ivanov et al.IV Международная конференция и молодёжная школа «Информационные технологии и нанотехнологии» (ИТНТ-2018)   1843 4. ConclusionsThe algorithm (2) compared to Least Square (LS). The dynamic system is described by the equations0.9 1.7 0.7 1.4 0.21 1 10.5 0.2 0.2 0.3 .i i i i i i iz z z x x x z− − −= ∆ − ∆ + ∆ − ∆ + ∆  ,i i iy z= + ξ ,i i iw x= + ζ                 (3) The coefficients of the bilinear system were given by ( )0 0.7 0.4 0.3 0.7 0.2 0.2 .Tθ = −  The noise-free  inputs ix is modelled as 1 2 1 20.8 0.6 1.7 0.5 ,i i i i i ix x x− − − −= − ⋅ − ⋅ + ζ + ⋅ζ + ⋅ζ  where iζ  - is are zero mean, white signal.  The estimates of the parameter vector are given in Table 1, as well as the normalized root mean square error, defined as   2 2 000 0ˆ 100 .δθ θ θ θ= − ⋅   The results are based on 50 independent Monte-Carlo simulations. The number of data points N in each simulation was 2000. For each method we have given the sample mean and sample standard deviation denoted by SD. Table 1. Monte-Carlo simulation results. Noise-signal ratio Mean square error ± SD zσσξ zσσζ δθ ,% LSδθ ,% 0.25 0.25 2.94±2.27 11.57±2.37 0.50 0.50 8.49±5.14 25.81±2.20 0.75 0.75 16.58±10.85 38.31±2.58 5. ConclusionsAn approach for the identification of dynamic bilinear discrete-time errors-in-variables system modelshas been developed. A Monte-Carlo simulation study compares two realizations of the proposedapproach with least square technique. The results obtained demonstrate the relatively high accuracyand the robustness against noise of the algorithms proposed.Future work could encompass potential extensions to handle the case of the coloured output noise together with the recursive implementation of the algorithm. 6. AcknowledgmentsThe authors gratefully acknowledge the contributions of prof. O.A. Katsyuba in improving the article.7. References[1] Stiassnie, M. On the application of fractional calculus for the formulation of viscoelastic models/ M. Stiassnie // Applied Mathematical Modelling. – 1979. – Vol. 3. – P. 300-302.[2] Bagley, R.L Fractional calculus – a different approach to the analysis of viscoelastically dampedstructures / R.L. Bagley // AIAA J. – 1983. – Vol. 21. – P. 741-748.[3] Reyes-Melo, M.E. Application of fractional calculus to modeling of relaxation phenomena oforganic dielectric materials / M.E. Reyes-Melo, J.J. Martinez-Vega, C.A. Guerrero-Salazar, U.Ortiz-Mendez // Proc Int Conf Solid Dielectrics. – 2004. – Vol. 2. – P. 530-533.[4] Vinagre, B.M.  Modeling and control of dynamic system using fractional calculus: Applicationto electrochemical processes and flexible structures / B.M. Vinagre, V. Feliu // Proc 41-st IEEEConf Decision Control, Las Vegas, NV. – 2002. – P. 214-239.[5] Mathieu, B. Identification of non integer order systems in the time domain / B. Mathieu, L.L.Lay, A. Oustaloup // IEEE SMC/IMACS Symp Control, Optimization and Supervision. 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Identification of dynamic errors-in-variables bilinear systems of fractional order

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Identification of dynamic errors-in-variables bilinear systems of fractional order

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