Fractional model parameter estimation in the frequency domain using set membership methods

Transcription

1 Fractional model parameter estimation in the frequency domain using set membership methods Firas KHEMANE, Rachid MALTI, Xavier MOREAU SWIM June 15th / 43

2 Objectives Fractional system modeling using uncertain frequency responses Extension of fractional derivatives and integrals to interval derivatives and integrals. Applying set membership approaches to estimate uncertain coefficients and uncertain derivative orders in fractional models Using tree set membership inclusion functions on frequency domain data based on rectangular, polar, and circular representations of complex intervals Merging all three solutions to obtain smaller intervals 2 / 43

3 Table of contents 1 From fractional derivative to interval derivative 2 Fractional systems 3 Set membership estimation using uncertain frequency response 4 Numerical example 5 Application : Thermal diffusion system 6 Conclusion

4 From fractional derivative to interval derivative 1 From fractional derivative to interval derivative 2 Fractional systems 3 Set membership estimation using uncertain frequency response 4 Numerical example 5 Application : Thermal diffusion system 6 Conclusion 4 / 43

5 From fractional derivative to interval derivative Fractional integration Cauchy formula I n f (t) = Riemann-Liouville integral Z t Z τ1 I ν f (t) = 1 Γ (ν)... Z t Z τn 1 f(τ n 2)dτ n 2... dτ 1 f (τ) (t τ) 1 ν dτ, ν R + Laplace transform L (I ν f(t)) = 1 s ν L (f(t)), 1 s ν Frequency characteristics of 8 1 >< Gain (db) : 2 log (jω) ν = 2ν log w >: 1 Phase (rad) : arg (jω) ν = ν π 2. Phase (deg) Magnitude (db) 2 1 1dB/dec Frequency (rad/s) Frequency (rad/s) 5 / 43

6 From fractional derivative to interval derivative Fractional derivative Grünwald definition D ν 1 f(t) = lim h h ν Laplace transform X k= ( 1) k L (D ν f(t)) = s ν L (f(t)).!! k f(t kh), ν R + ν 2 Frequency characteristics of s ν 8 < Gain (db) : 2 log (jω) ν = 2ν log(w), : Phase (rad) : arg ((jω) ν ) = ν π 2. Magnitude (db) Phase (deg) db/dec Frequency (rad/s) Frequency (rad/s) 6 / 43

7 From fractional derivative to interval derivative Interval integration Definition I [ν] f (t) = Example f(t) = n Z 1 t f (τ) o Γ (ν) (t τ) 1 ν dτ, ν [ν] ( t 2 if t, if t < Integration by real interval [ν] = [.5, 1.5] n I [ν] f (t) = 1 Γ (ν) Z t τ 2 o (t τ) dτ, ν [.5, 1.5] 1 ν I [ν] f(t) nu=.5 nu=.6 nu=.7 nu=.8 nu=.9 nu=1 nu=1.1 nu=1.2 nu=1.3 nu=1.4 nu= t 7 / 43

8 From fractional derivative to interval derivative Frequency characteristics Laplace transform n 1 L {I [ν] (f(t))} = s L {f(t)}, ν 1 s [ν] ν [ν] o Frequency characteristics of 8 < f G([ν]) = 1 1 = 2log, s db [ν] (jω) [ν] : f φ ([ν]) = arg = arg 1 s [ν] 1 (jω) [ν] Phase ( ) Gain (db) dB/dec 1dB/dec Pulsation (rad/s) Monotonicity Gain 8 < Phase : Pulsation (rad/s) 1 s [ν] db = [ 2ν log ω, 2ν log ω], ω ],1], 1 s [ν] db = [ 2ν log ω, 2ν log ω], ω [1, + [ f φ ([ν]) = [ ν π 2, ν π 2 ] 8 / 43

9 From fractional derivative to interval derivative interval derivative Grünwald n D [ν] f(t) = lim h 1 h ν X k= ( 1) k!! k o f(t kh), ν [ν]. ν Example ( t 2 if t, f(t) = if t <, D [ν] f(t) nu=.5 nu=.6 nu=.7 nu=.8 nu=.9 nu=1 nu=1.1 nu=1.2 nu=1.3 nu=1.4 nu= Derivative by real interval n D [ν] 1 X f(t) = lim h h ν k= ( 1) k t! k o!(t kh) 2, ν [.5, 1.5]. ν 9 / 43

10 From fractional derivative to interval derivative Frequency characteristics Laplace transform L {D [ν] (f(t))} = n s ν L {f(t)}, o ν [ν]. Gain (db) dB/dec 1dB/dec Frequency characteristics of s [ν] 8 < f G([ν]) = s[ν] db = 2log (jω) [ν], : f φ ([ν]) = arg s [ν] = arg (jω) [ν]. Phase ( ) Pulsation (rad/s) Pulsation (rad/s) Monotonicity Gain 8< = [2ν log ω,2ν log ω], when ω ] s[ν] db +,1 ], Phase : s[ν] db = [2ν log ω,2ν log ω], when ω [1 +,+ [ f φ ([ν]) = [ν π 2, ν π 2 ] 1 / 43

11 From fractional derivative to interval derivative Monotonicity of time domain response with respect to derivative order Fractional differential equation (FDE) D ν y(t) = f(y(t),u(t)) Uncertain fractional derivative equation (FDE) D [ν] y(t) = f(y(t),u(t)) Solution : Set of admissible trajectories Y(t) Objective : framing Y(t) by [y(t)] = [y(t),y(t)] D ν y(t) = f(y(t),u(t)), D ν y(t) = f(y(t),u(t)). It is a tough problem which cannot be solved in the general case 11 / 43

12 From fractional derivative to interval derivative Monotonicity of time domain response with respect to derivative order Example Differential equation D ν y(t) + λy(t) = u(t), λ = 1, with u(t) : Heaviside step function Transfer function Y (s) U(s) = 1 s(s ν + λ) Inverse Laplace transform of Y (s) U(s) X f ν(t) = ( 1) k+1 λ k 1 t νk Γ(νk + 1) k=1 Derivative of f ν(t) d dν fν(t) = X ( 1) k+1 λk 1 t νk k (log(t) Ψ(νk + 1)) Γ(νk + 1) k=1 12 / 43

13 From fractional derivative to interval derivative Monotonicity of time domain response with respect to derivative order Derivtive of f v(t) with ν =.9 Step response y(t) f ν (t) y(t) ν=.2 ν=.3 ν=.4 ν=.5 ν=.6 ν=.7 ν=.8 ν= Temps Temps Solving uncertain differential equation with uncertain differential orders still an open problem..! 13 / 43

14 Fractional systems 1 From fractional derivative to interval derivative 2 Fractional systems 3 Set membership estimation using uncertain frequency response 4 Numerical example 5 Application : Thermal diffusion system 6 Conclusion 14 / 43

15 Fractional systems Application Fractional systems are gaining more and more interest in the scientific and industrial communities Area of Control : CRONE controllers base on fractional calculus Fractional PID controllers Mechanics : Performance of CRONE suspension in vibration isolation Signal processing : Performance of fractional operators in modeling fractal noise Thermal domain : Performance of fractional operators in modeling flux diffusion 15 / 43

16 Fractional systems Fractional system representation Differential equation y(t) + a 1 D α 1y(t) + a 2 D α 2y(t) a N D α N y(t) = b D β u(t) + b 1 D β 1u(t) + b 2 D β 2u(t) b M D β M u(t), Commensurable order ν : Commensurable order ν is the biggest real number such that all derivative orders of the differential equation are its integer multiples α i β j ν N, i = 1,..., N and N, j =,..., M ν Fractional transfer function MP b j s β j G(s, θ) = j=, P 1 + N a i s α i i=1 θ = [b... b M, a 1... a N, β... β M, α 1... α N ] with 2(N + M + 1) parameters Commensurable transfer function MP b j s jν j= G(s, θ) =, P 1 + N a i s iν i=1 θ = [b... b M, a 1... a N, ν] with (N + M + 2) parameters 16 / 43

17 Fractional systems Fractional system representation State space representation j x (ν) (t) y(t) = Ax(t) + Bu(t) = Cx(t) + Du(t) Stability theorem Assume a fractional and commensurable transfer function and R ν = Q ν/p ν its rational form. F is stable iff : < ν < 2, and s k C, P ν(s k ) = such arg(s k ) > ν π 2. Im(s k) Im(s k) Im(s k) ν π 2 ν Re(s k) π 2Re(s k) Re(s k) ν π 2 a) ν < 1 b) ν = 1 c) ν > 1 17 / 43

18 Set membership estimation using uncertain frequency response 1 From fractional derivative to interval derivative 2 Fractional systems 3 Set membership estimation using uncertain frequency response 4 Numerical example 5 Application : Thermal diffusion system 6 Conclusion 18 / 43

19 Set membership estimation using uncertain frequency response Objective Having a set G n of N complex frequency responses, find the fractional models G consistent with all these frequency responses Frequency response : MP b j(jω) jν j= G(jω, θ) =. P 1 + N a i(jω) jν i=1 Imaginary vs Real part.5.1 G n.15.2 G p n.25 G r n G d n / 43

20 Set membership estimation using uncertain frequency response Set membership estimation using rectangular representation Rectangular inclusion function G r n : G r n = [Re(Gr n ) Re(Gr n )] + j[im(gr n ) Im(Gr n )] 8 Re(G r n ) = >< Re(G r n ) = Im(G r n) = >: Im(G r n) = min (Re(z 1), Re(z 2 ),..., Re(z I )), z i G n, i=1...i max (Re(z 1), Re(z 2 ),..., Re(z I )), z i G n, i=1...i min (Im(z 1), Im(z 2 ),..., Im(z I )), z i G n, i=1...i max (Im(z 1), Im(z 2 ),..., Im(z I )), z i G n. i=1...i CSP : Solution set S : 8 >< Re(G r n ) Re(G(jωn, θ)) Re(Gr n ), S = θ Θ f(ω n, θ) [y(ω n)], CSP : Im(G >: r n ) Im(G(jωn, θ)) Im(Gr n ), n {1,..., N}. θ Θ, 8! Re(G(jω n, θ)) f(ω n, θ) = >< Im(G(jω n, θ)) >: [y(ω n)] = [Re(G r n ), Re(Gr n )] [Im(G r n ), Im(Gr n )]! 2 / 43

21 Set membership estimation using uncertain frequency response Set membership estimation using polar representation Polar inclusion function G p n : G p n = ˆG n G n exp { j [ϕ n ϕ n] } 8 ϕ n >< ϕ n G n >: G n = min (arg(z 1), arg(z 2 ),...,arg(z I )), z i G n, i=1...i = max (arg(z 1), arg(z 2 ),...,arg(z I )), z i G n, i=1...i = min ( z 1, z 2,..., z I ), z i G n, z i G n, i=1...i = max ( z 1, z 2,..., z I ), z i G n. i=1...i CSP : 8 >< G n G(jω n, θ) G n, CSP : ϕ n ϕ(ω n, θ) ϕ n, >: θ Θ, 8 f(ω n, θ) = >< >: [y(ω n)] = G(jω n, θ) ϕ(ω n, θ) [ G n G n] [ϕ, ϕ]!! Solution set S : S = θ Θ f(ω n, θ) [y(ω n)], n {1,..., N}. 21 / 43

22 Set membership estimation using uncertain frequency response Set membership estimation using circular representation Circular inclusion function G d n : G d n = {c(g n), r(g n)}, CSP : CSP : ( G (jω n, θ) G d n θ Θ, Solution set S : S = θ Θ f(ω n, θ) [y(ω n)], n {1,..., N}. 8 >< f(ω n, θ) = X(c(G(jω n, θ)), r(g(jω n, θ))) >: [y(ω n)] = G d n 22 / 43

23 Set membership estimation using uncertain frequency response Set Inversion Via Interval Analysis (SIVIA) Jaulin and Walter, (1993) SIVIA allows to obtain an inner S and an outer S enclosures of the solution set S, based on partitioning the parameter set into three regions : feasible, indeterminate and unfeasible : S S S. Uses an inclusion test [t] which is a function allowing to prove if [θ] is unfeasible [θ] is ignored, feasible [θ] is added to the set S, undetermined [θ] is bisected and tested again, unless its size is less than a precision parameter η tuned by the user and which ensures that the algorithm terminates after a finite number of iterations. if w[θ] < η [θ] is added to the set S Algorithm SIVIA (in : [t],[θ], η ; out : S, S ) 1 If [t]([θ]) = [], return; 2 If [t]([θ]) = [1], then S := S [θ]; S := S [θ], return; 3 If w([θ]) η, S := S [θ]; Else bisect [θ] into [θ 1] and [θ 2]; 4 SIVIA (in : [t], [θ 1], η ; out : S, S) ; 5 SIVIA (in : [t], [θ 2], η ; out : S, S). 23 / 43

24 Numerical example 1 From fractional derivative to interval derivative 2 Fractional systems 3 Set membership estimation using uncertain frequency response 4 Numerical example 5 Application : Thermal diffusion system 6 Conclusion 24 / 43

25 Numerical example Fractional transfer function of the first kind Transfer function Uncertain linear model G 1(s, θ) = k s ν + b. G 1(s, θ i) = Uncertainties 8 >< k i = 3 + ρ (i) k, b i = 2 + ρ (i) b >:, ν i =.5 + ρ (i) ν k i s ν i + bi, i = 1,... I, 8 >< >: ρ (i) k ρ (i) ρ (i) = [.2,.2], b = [.2,.2], ν = [.5,.5] Vector of nominal parameters G 1(jω n, θ i) = k i (jω n) ν i + bi, ( i = 1,... 2, n = 1, / 43

26 Numerical example Uncertain frequency response Bode Nyquist-Circular form Nyquist-rectangular form 1 Phase ( ) Gain (db) Pulsation (rad/sec) Pulsation (rad/sec) Imaginary part Real part Imaginary part Real part / 43

27 Numerical example Set membership estimation using rectangular representation Complex frequency response k G 1(jω n, θ) = `b + cos(ν π )ωn ν + j `sin(ν, π 2 2 )ων n Real and imaginary part k `b + ωn ν cos `ν π 2 G 1(jω n, θ) = b 2 + 2bωn ν cos `ν π 2 + ω 2ν n Estimation using real and complex CSP kωn ν sin `ν π 2 + j b 2 + 2bωn ν cos `ν. π 2 + ω 2ν n Outer approximation S Parameters [k] [b] [ν] Real CSP [2.58, 3.52] [1.68, 2.4] [.46,.56] Complex CSP [2.56, 3.59] [1.66, 2.43] [.45,.56] Final rectangular inclusion [2.58, 3.52] [1.68, 2.4] [.46,.56] 27 / 43

28 Numerical example Results (a) Projection on (b, k) plan (b) Projection on (b, ν) plan 28 / 43

29 Numerical example Set membership estimation using polar representation Complex frequency response Gain and phase G 1(jω,θ) = G 1(jω, θ) exp jϕ(ω,θ) k G 1(jω,θ) = p (b + cos(ν π 2 )ων ) 2 + (sin(ν π 2 )ων ) 2 8 sin(ν π arctan 2 )ω ν, if Den(ω) >, Den(ω) >< ϕ(ω,θ) = «sin(ν >: π arctan π 2 )ω ν, if Den(ω n) < with Den(ω) = b + cos `ν π 2 ων. Den(ω) 29 / 43

30 Numerical example Set membership estimation using polar representation Estimation using real and complex CSP Outer approximation S Parameters [k] [b] [ν] Real CSP [2.56, 3.5] [1.69, 2.26] [.45,.56] Complex CSP [2.4, 3.74] [1.55, 2.49] [.44,.57] Final polar inclusion [2.56, 3.5] [1.69, 2.26] [.45,.56] 1 Gain (db) Pulsation (rad/sec) Phase ( ) Pulsation (rad/sec) Fig.: Projection on (b, k) plan Fig.: Projection on (b, ν) plan Fig.: Bode diagram 3 / 43

31 Numerical example Set membership estimation using circular representation Complex frequency response k G 1(jω n, θ) = `b + cos(ν π )ωn ν + j `sin(ν, π 2 2 )ων n Estimation using complex CSP Outer approximation S `[k],[b],[ν] = `[2.29, 3.94],[1.32, 2.88],[.43,.57], Fig.: Projection on (b, k) plan Fig.: Projection on (b, ν) plan Fig.: Nyquist diagram 31 / 43

32 Numerical example Merging all solution sets (a) Projection on (b, k) plan : rectangular (red line), polar (blue line) and circular (green line) (b) Projection on (b, ν) plan : rectangular (red line), polar (blue line) and circular (green line) Parameters [k] [b] [ν] Rectangular inclusion [2.58, 3.52] [1.68, 2.39] [.46,.56] Polar inclusion [2.56, 3.5] [1.69, 2.25] [.45,.56] Circular inclusion [2.29, 3.94] [1.32, 2.88] [.43,.56] Final inclusion [2.58, 3.5] [1.69, 2.25] [.46,.56] 32 / 43

33 Application : Thermal diffusion system 1 From fractional derivative to interval derivative 2 Fractional systems 3 Set membership estimation using uncertain frequency response 4 Numerical example 5 Application : Thermal diffusion system 6 Conclusion 33 / 43

34 Application : Thermal diffusion system Description Isolation thermique l(mm) Characteristics Aluminium rod l = 4cm, r = 1cm et S = 3.14cm 2 Heater 4.8Ω, λ = 237Wm 1K 1 et α = m 2 s 1 Input : Heat flux φ 34 / 43

35 Application : Thermal diffusion system Input-Output signals Flux step of 5kWm 2, l = 5mm PRBS of ±5kWm 2 Température( ) Flux(kW.m 2) 5 5 Température( ) Temps(s) Temps(s) Temps(s) 35 / 43

36 Application : Thermal diffusion system Non parametric identification of the frequency response Frequency response bg(jω) = b G yu(jω) bg uu(jω). 6 Gain (db) Pulsation (rad/s) 4 6 Phase ( ) Pulsation (rad/s) 36 / 43

37 Application : Thermal diffusion system Parametric estimation Criterion with P N n=1 J % (θ) = αj G + (1 α)j φ, b G(jω n) db b G(jωn) db «2 J G% = P N n=1 b G(jω n) db 2 1, and J φ% = P N n=1 «2 arg bg(jωn) arg( G(jωn)) b P N n=1 arg( G(jω b 2 1, n)) 37 / 43

38 Application : Thermal diffusion system Parametric estimation Model obtained k G(s) = 1 p 1 s 2ν + 1 p 2 s ν + 1 exp 2s with 8 k =.8 1 3, >< p 1 = 1 1 3, p 2 = >: 3, ν =.73 J =.3% Phase ( ) Gain (db) f (Hz) f (Hz) 38 / 43

39 Application : Thermal diffusion system Uncertainties Stochastic errors : Under assumption of normal distrubtion, errors due to time domain-frequency domain conversion are used to determine boundes on each frequency response. Standard deviation on gain : σ[ G ] b `1 γ 2 uy 1/2 G b γ, uy 2n d Standard deviation on phase : `1 γ 2 uy 1/2 σ[ bϕ] γ, uy 2n d Coherence function : γ 2 uy(ω) = b G uy(jω) 2 bg uu(jω) bg yy(jω), γ 2 uy(ω) 1 ω. 39 / 43

40 Application : Thermal diffusion system Uncertain frequency response Uncertain frequency response : Lower and upper boundes are obtained using confidance intervals with 3 standard deviation b G 3σ[ b G ] G b G + 3σ[ b G ] bϕ 3σ[ bϕ] ϕ bϕ + 3σ[ bϕ]. 6 Set membership approach is a bouded error approach Gain (db) Pulsation (rad/s) G n : >< G n : ϕ n : >: ϕ n : Lower bound on gain, Upper bound on gain, Lower bound on phase, Upper bound on phase Phase ( ) Pulsation (rad/s) 4 / 43

41 Application : Thermal diffusion system Set membership estimation Initial searching space ([k], [p 1],[p 2],[ν]) = `[., 5.], [., 5.], [., 5.], [., 2.]. Outer approximation ([k], [p 1], [p 2], [ν]) = `[4., 12] 1 4,[4.5, 15] 1 4, [.3, 2] 1 2,[.62,.97], 6 Gain (db) Pulsation(rad/sec) Phase ( ) 5 1 (c) Projection on (k, ν) plan (d) Projection on (p 1, p 2) plan Pulsation(rad/sec) 41 / 43

42 Conclusion 1 From fractional derivative to interval derivative 2 Fractional systems 3 Set membership estimation using uncertain frequency response 4 Numerical example 5 Application : Thermal diffusion system 6 Conclusion 42 / 43

43 Conclusion Conclusion Definitions of fractional integrals and derivatives are extended to interval integrals and derivatives. Set membership parameter estimation is applied to estimate all feasible parameters and differentiation orders of fractional models Three different inclusion functions of the complex frequency response are used in the set membership estimation and the final results are merged to obtain a smaller solution set Application to a thermal diffusion system and a set of feasible parameters is obtained 43 / 43