Fractional model parameter estimation in the frequency domain using - PowerPoint PPT Presentation

Fractional model parameter estimation in the frequency domain using set membership methods Firas KHEMANE, Rachid MALTI, Xavier MOREAU SWIM June 15th 2011 1 / 43

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

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

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

From fractional derivative to interval derivative Fractional integration Cauchy formula Z t Z τ 1 Z τ n − 1 I n f ( t ) = . . . f ( τ n − 2 ) dτ n − 2 . . . dτ 1 0 0 0 Riemann-Liouville integral Z t 1 f ( τ ) ∆ I ν f ( t ) ( t − τ ) 1 − ν dτ, ∀ ν ∈ R ∗ = + Γ ( ν ) 0 Laplace transform 20 − 10 dB/dec 10 Magnitude (dB) L ( I ν f ( t )) = 1 s ν L ( f ( t )) , 0 −10 −20 1 −2 −1 0 1 2 Frequency characteristics of 10 10 10 10 10 Frequency (rad/s) s ν 0 8 ˛ ˛ 1 Gain (dB) : 20 log ˛ = − 20 ν log w ˛ ˛ > (j ω ) ν > ˛ Phase (deg) < −45 “ ” > 1 = − ν π Phase (rad) : arg 2 . > : (j ω ) ν −90 −2 −1 0 1 2 10 10 10 10 10 Frequency (rad/s) 5 / 43

From fractional derivative to interval derivative Fractional derivative unwald definition Gr¨ ! ! ∞ X 1 k D ν f ( t ) = lim ( − 1) k f ( t − kh ) , ν ∈ R + h ν ν h → 0 k =0 Laplace transform L ( D ν f ( t )) = s ν L ( f ( t )) . 20 10 Magnitude (dB) 0 10 dB/dec Frequency characteristics of s ν −10 −20 8 20 log | (j ω ) ν | = 20 ν log( w ) , −2 −1 0 1 2 Gain (dB) : 10 10 10 10 10 Frequency (rad/s) < 90 arg ((j ω ) ν ) = ν π Phase (rad) : 2 . : Phase (deg) 45 0 −2 −1 0 1 2 10 10 10 10 10 Frequency (rad/s) 6 / 43

From fractional derivative to interval derivative Interval integration Definition Z t n o 1 f ( τ ) ∆ I [ ν ] f ( t ) = ( t − τ ) 1 − ν dτ, ν ∈ [ ν ] Γ ( ν ) 0 nu=0.5 500 nu=0.6 nu=0.7 nu=0.8 Example 400 nu=0.9 nu=1 nu=1.1 ( 300 nu=1.2 t 2 I [ ν ] f(t) if t ≥ 0 , nu=1.3 nu=1.4 f ( t ) = 200 nu=1.5 0 if t < 0 100 Integration by real interval 0 [ ν ] = [0 . 5 , 1 . 5] 1 2 3 4 5 6 7 8 9 10 t Z t n o τ 2 1 ∆ I [ ν ] f ( t ) = ( t − τ ) 1 − ν dτ, ν ∈ [0 . 5 , 1 . 5] Γ ( ν ) 0 7 / 43

From fractional derivative to interval derivative Frequency characteristics Laplace transform 40 − 14 dB/dec n 1 o 20 Gain (dB) L { I [ ν ] ( f ( t )) } = s ν L { f ( t ) } , ν ∈ [ ν ] 0 − 10 dB/dec −20 1 −40 Frequency characteristics of −2 −1 0 1 2 10 10 10 10 10 s [ ν ] Pulsation (rad/s) 8 ˛ ˛ 0 ˛ ˛ ˛ ˛ < ˛ 1 ˛ 1 f G ([ ν ]) = = 20 log ˛ ˛ , −20 s [ ν ] (j ω ) [ ν ] “ dB ” “ ” Phase ( ° ) −40 : 1 1 f φ ([ ν ]) = arg = arg −60 s [ ν ] (j ω ) [ ν ] −80 −2 −1 0 1 2 10 10 10 10 10 Pulsation (rad/s) Monotonicity ◮ Gain ˛ 1 ˛ 8 = [ − 20 ν log ω, − 20 ν log ω ] , ω ∈ ]0 , 1] , ˛ ˛ s [ ν ] dB < ˛ 1 ˛ = [ − 20 ν log ω, − 20 ν log ω ] , ω ∈ [1 , + ∞ [ : ˛ ˛ s [ ν ] dB ◮ Phase f φ ([ ν ]) = [ − ν π 2 , − ν π 2 ] 8 / 43

From fractional derivative to interval derivative interval derivative Gr¨ unwald ! ! n o ∞ X 1 k D [ ν ] f ( t ) = ( − 1) k lim f ( t − kh ) , ν ∈ [ ν ] . h ν ν h → 0 k =0 Example ( nu=0.5 80 t 2 nu=0.6 if t ≥ 0 , nu=0.7 f ( t ) = 70 nu=0.8 0 if t < 0 , nu=0.9 60 nu=1 nu=1.1 50 nu=1.2 D [ ν ] f(t) nu=1.3 40 nu=1.4 nu=1.5 30 20 10 0 1 2 3 4 5 6 7 8 9 10 t Derivative by real interval ! ! n o X ∞ 1 k D [ ν ] f ( t ) = ( − 1) k ( t − kh ) 2 lim , ν ∈ [0 . 5 , 1 . 5] . h ν ν h → 0 k =0 9 / 43

From fractional derivative to interval derivative Frequency characteristics Laplace transform 40 14 dB/dec n o 20 Gain (dB) L { D [ ν ] ( f ( t )) } = s ν L { f ( t ) } , ν ∈ [ ν ] . 0 10 dB/dec −20 −40 Frequency characteristics of s [ ν ] −2 −1 0 1 2 10 10 10 10 10 Pulsation (rad/s) 8 ˛ ˛ (j ω ) [ ν ] ˛ ˛ s [ ν ] ˛ ˛ ˛ ˛ < ˛ 80 f G ([ ν ]) = = 20 log ˛ , dB “ s [ ν ] ” “ (j ω ) [ ν ] ” 60 Phase ( ° ) : 40 f φ ([ ν ]) = arg = arg . 20 0 −2 −1 0 1 2 10 10 10 10 10 Pulsation (rad/s) Monotonicity ◮ Gain 8 ˛ ˛ s [ ν ] ˛ ω ∈ ]0 + , 1 − ] , = [20 ν log ω, 20 ν log ω ] , when ˛ dB < ˛ ˛ s [ ν ] ˛ ω ∈ [1 + , + ∞ [ = [20 ν log ω, 20 ν log ω ] , when : ˛ dB ◮ Phase f φ ([ ν ]) = [ ν π 2 , ν π 2 ] 10 / 43

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 ) [ y ( t )] = [ y ( t ) , y ( t )] Objective : framing Y ( t ) by 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

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 ) 1 U ( s ) = s ( s ν + λ ) ◮ Inverse Laplace transform of Y ( s ) U ( s ) ∞ t νk X ( − 1) k +1 λ k − 1 f ν ( t ) = Γ( νk + 1) k =1 ◮ Derivative of f ν ( t ) ∞ ( − 1) k +1 λ k − 1 t νk k d X dν f ν ( t ) = Γ( νk + 1) (log( t ) − Ψ( νk + 1)) k =1 12 / 43

From fractional derivative to interval derivative Monotonicity of time domain response with respect to derivative order Derivtive of f v ( t ) with ν = 0 . 9 Step response y ( t ) 0.5 0.9 0.4 0.8 0.3 0.7 0.2 0.6 0.1 ν =0.2 0.5 f ν (t)’ ν =0.3 y(t) 0 ν =0.4 0.4 −0.1 ν =0.5 ν =0.6 0.3 −0.2 ν =0.7 0.2 ν =0.8 −0.3 ν =0.9 0.1 −0.4 −0.5 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 1 2 3 4 5 6 Temps Temps Solving uncertain differential equation with uncertain differential orders still an open problem.. ! 13 / 43

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

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

Fractional systems Fractional system representation Differential equation y ( t ) + a 1 D α 1 y ( t ) + a 2 D α 2 y ( t ) + . . . + a N D α N y ( t ) = b 0 D β 0 u ( t ) + b 1 D β 1 u ( t ) + b 2 D β 2 u ( 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 ν ∈ N , j = 0 , . . . , M Fractional transfer function Commensurable transfer function M M b j s β j b j s jν P P j =0 j =0 G ( s, θ ) = , G ( s, θ ) = , N N P a i s α i P a i s iν 1 + 1 + i =1 i =1 θ = [ b 0 . . . b M , a 1 . . . a N , β 0 . . . β M , α 1 . . . α N ] θ = [ b 0 . . . b M , a 1 . . . a N , ν ] with 2( N + M + 1) parameters with ( N + M + 2) parameters 16 / 43

Fractional systems Fractional system representation State space representation  x ( ν ) ( t ) = Ax ( t ) + Bu ( t ) y ( 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 : 0 < ν < 2 , and | arg( s k ) | > ν π ∀ s k ∈ C , P ν ( s k ) = 0 such 2 . I m ( s k ) I m ( s k ) I m ( s k ) ν π ν π ν π 2 R e ( s k ) R e ( s k ) 2 R e ( s k ) 2 ◮ a) ν < 1 ◮ b) ν = 1 ◮ c) ν > 1 17 / 43

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