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# Electrical Engineering Colloquium ECE 6800

Utah State University

GPA 3.86

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This 85 page Class Notes was uploaded by Seth Gibson on Wednesday October 28, 2015. The Class Notes belongs to ECE 6800 at Utah State University taught by Staff in Fall. Since its upload, it has received 9 views. For similar materials see /class/230399/ece-6800-utah-state-university in ELECTRICAL AND COMPUTER ENGINEERING at Utah State University.

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Date Created: 10/28/15

ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 1 Fractionalorder Calculus Fractionalorder FilterI land Fractionalorder Control An Overview I land Some Recent Developmentsl Speaker YangQuan Chen chhen ieeeorg eceusuedu Center for Self Organizing and Intelligent Systems CSOIS Dept of Electrical and Computer Engineering Utah State University March 25 2003 k Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 2 1 Fractional order Calculus A bit history de nitions examples 2 Fractional order ODE and Laplace Transform Fractional order Dynamic Filtering Fractional order Modeling and Control Implementation Techniques To Probe Further TQO lrPw Concluding Remarks inbetween thinking Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 3 Leibnitz introduced the notation dnydx In a letter to L Hospital in 1695 Leibniz raised the following question Can the meaning of derivatives with integer order d yxdxn be generalized to derivatives with nonintegral orders so that in general n E C The story goes that L Hospital was somewhat curious about that question and replied by another question to Leibniz What if n 12 Leibniz in a letter dated September 30 1695 replied It will lead to a paradoa from which one day useful consequences will be drawn Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 4 Some special functionsl Euler s Gamma function Special case When 16 n F771 17 b 221T1 71 1 2 Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 5 Some special functions ContinuedI MittagLef er function in two parameters k Ea z m agt0 gt0 3 It is a generalization of exponential function Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 6 MittagLef er function ContinuedI More particular cases E2 COShZ7 E12z E1212 ZerfC 2 6 Elw Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 7 A generalization of differential and integral operators dadto RM gt 0 an 1 Rm 0 7 fdTa Rm lt 0 There are two commonly used de nitions for the general fractional order differentiation and integral ie the GriinwaldLetnikov de nition and the RiemannLiouville de nition Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 8 I Gr nwaldLetnikov de nition I d ft ft h EN f t 53 f 8 d2 H f t f t h 11040 53 hm1ft ft hft h ft 2h h gtO h h ft 2ft hft 2h 113 h2 9 d3 M ft 3ft h3ft 2h ft 3h Eat tl1i h3 k 0 Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 9 Ingeneral 31 t 2 f t 53 h ijZJ m 3112 jhgt 11 n nn l n 2n 391 n j X j J Zan jquot 12 Df t lim i Z 1j0fft jh 13 Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 10 I RiemannLiouville de nition I fatat mfg2ft1dt1dt2dtn fd n fOld fornENngt0 1 t a1ft ant am PM t f 51 d7 16 for aaER alt0 a 01 t L aDt ft PM a dtn t Tan1d7 n 1lt oz Chem a and t are the limits of an t Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 11 Properties of CLDtO I 1 If fz is an analytic function of z the derivative aDg z is an analytic function of z and 04 2 The operation aDg gives the same result as the usual differentiation of integer order n 3 The operator of order or 0 is the identity operator 4 Fractional operators are linear aDZ afz bhz aaD fZ baDi hCZl 18 5 For fractional integrations of arbitrary order or gt O gt 0 Rem gt 0 Re gt 0 the additive index law semigroup property holds aDgaaDg z aDamfz 19 k Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 12 1 Fractional order Calculus A bit history de nitions examples 2 Fractionalorder ODE and Laplace Transform Fractional order Dynamic Filtering Fractional order Modeling and Control Implementation Techniques To Probe Further TQO lrPw Concluding Remarks inbetween thinking Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 13 Laplace Transformation of CLDtO I From the RiemannLiouville de nition oo n l e8t ODf t it SO F8 2 8k 0Dk1ftlt0 20 0 k0 for n 1 lt 04 g n Where F8 ft is the normal Laplace transformation From the GriinwaldLetnikov de nition 000 e ODf t it SO F8 21 k Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 14 Laplace Transform of Mittag Lef er function I Applying the above de nition for Laplace transformation 00 04 18 st ak 1 oz 104 0 e t Ea laiClt w R 8 gt CL 22 The particular case of 22 for 04 12 00 1 kl e sttkTEgmia mt 1 Res gt a2 23 0 2 2 g 3F 0J is useful for solving semidifferential equations Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 15 A typical n term linear FODE in time domain is give by anDt quotyta1DflytaonOyt 0 24 Where akk 0 1 n are constant coef cients of the FODE m k 0 1 2 n are real numbers Without loss of generality assume that n gt n1 gt gt l gt g 2 0 Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 16 Anmy a ThmeDmn n bk bnofFODEsI 1 0 1m yo 5 E mmmmwm m mkykkw2m k0207kn 220 TL 2 W H 15W n 1m n2302 n l jW 1 a z 0 Eon an l t n n 1 1671 1671 1gt16n20218n 1I6jkj an 7 Where EMLQZ is the Mittag Lef er function in two parameters 4 and E E E M y dyn MLCU 3831 Fj An LL7 En n012 Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 17 lGive me a break Too heavy math I Show me the pictures or some cartoons I httpwwwcsoisusueduilCcontrolhumor Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU An Overview 18 ECE6800 Seminar Fractional order Calculus in Signal Processing and Control I iguana mamas 43 5quot a causmwmm 2 3 a 4 mrmmarmu ngmm 0F H150quot THEdim 55 f w mm Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 19 lFractional derivatives of Heviside function Fractional derivatives of function yHt k W derivative order independent variable Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 20 Fractional derivatives of sine function Fractional derivatives of function ysint derivative order independent variable Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECESEOO Seminal anctionalrozdez Calculus in Signal oncessing and Contzol An Ovezview 21 Fractional derivatives of ramp function Dz YangQuan Chen Centez fox SelfrOiganizing and Intelligent Systems c3013 EOE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 22 lSolVing FODE Laplace methodI Consider a simple FODE Uni2flttgtaflttgt0 we infW 0 25 t0 Applying the Laplace transform we obtain C 12 my C OD f F8 130 and the inverse transform With a help of 23 gives the solution of 25 ft Ct12E az 26 Using series expansion 4 of Ea t it is easy to check that for a 1 solution 26 is identical to an alternative solution f t C 1 eterfct obtained by a more complicated procedure Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 23 Solving FODE Numerical SolutionI Basis Da t IE 5 K0 A H V A D I V t 3 SD M w 3 A m 9 V Kb Q E m 3 T3 89 For example Dayt byt qt tgt 0 n 1 lt 04 g n 29 y 0 k 0 1 k h azw yk j 53 qk 30 90 Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 24 We 111 1939 j 012 31 tk khv yk 90 07 Qk k071727 Algorithm yZ 0 212 n l 32 k yk bh0 yk1 ngaykj l hO qk k 7171 133 j1 Short Memory Principle Daft 2tL Daft tgt L 34 elttgtlDaflttgt tLDaflttgtl L991 35 ml cm Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 25 M1 tlte L t t12gtL2mE 36 Computation of coef cients Oz 04 1 04 105921 w 1 wf jl k 12 37 oz 00 k 05 00 oz 1 z 1 zk 2w bk 38 160 k0 With z 6 9 1 ejwa Zwlgaejkw 39 k0 Fourier Transform 111200 g 0 1 6 9 ejkwdw 40 Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 26 1 Fractional order Calculus A bit history de nitions examples 2 Fractional order ODE and Laplace Transform Fractionalorder Dynamic Filtering Fractional order Modeling and Control Implementation Techniques To Probe Further TQO liPw Concluding Remarks inbetween thinking Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 27 Dynamic Models de sI Continuous Models anDany l l aODO Oyt me mu l l bOD OuO 41 Commensurate order systems 1 aka kka7 157 akao y ikako u 43 k0 k0 Discrete Models anAgnya aOAgoya bmA mua bOAQOua 44 k Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 28 InputOutput Representations transfer functionI Y bm Gm bm Gm l b 60 6718 2 8 Z 8 18 08 45 ansan an1804n 1 0080 bm wk 1 bm 1wz 1 m 1w be wz 1 0 an ad z 1aquot l an1wz 1aquot1 a0 w 2 10 0 46 02 Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 29 l StateSpace Representation I Dakszx l Bu yCxDu Do xl bn1 bn2 b0 1 13 0 0 I I I l l l 1 l l D0231 0 0 10 y CLm am1 a1 CLO Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 30 lState Transition Matrixl xt 1Xs 1SO I A 1 BU3 3amp1 A1 x0 47 By de ning ltIgtt E 180 I A1 t Z 0 Xt tx0 ltIgtt gtllt But ltIgttx0 ltIgtt 739Bu739d739 0 48 Me a A2xlt0gt tm Akxm F1a F12a F1ka x05 2 1 00 Alstka a x0 Ea At X0 03740 Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 31 General Stability ConditionI 3M lGsl g M V8Res Z 0 49 For commensurate order systems largml gt a 50 Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 32 1 Fractional order Calculus A bit history de nitions examples 2 Fractional order ODE and Laplace Transform Fractional order Dynamic Filtering Fractionalorder Modeling and Control Implementation Techniques To Probe Further TQO liPw Concluding Remarks inbetween thinking Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 33 Modeling heat transferl 92211Eat 23yt W k 825 v 51 tgt0 0ltxltoo y0t Wt 52 yx70 0 53 xlirgoyw lt oo 54 Transfer function W k28YJc8 55 C2078 M8 56 1320 58l lt oo 57 Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 34 YJc s Asp WE Bsek 58 As 1408 M8 59 33 0 60 YJc s Mew WE 61 8 Yx8 e kamE G M8 62 think about transfer function e I Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 35 Fractional order speed control of DC motorI System transfer function G8 W J being the payload inertia Phase margin of controlled system chm arg momma 7r Controller Cs k1 k2 T giving a constant phase margin klk Pm arg CjwGjw 7T arg l Step response k J kkl kkl t 06 1 a a 1 saw Idan 1 2 Jay Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU Rachonalrozdex Calculus m ngnal Pxocesslng and Conuol An Ovexvxew 86 EcEsaoo Semmaz m 3 Note the isodamping similar overshoot F F T n ofUSU Dz YangQuan Chen ECESBOO Seminar Fractionalrorder Calculus in Signal Processing and Control An Overview 37 PID from point to plane ut mt mikes de et I AM 5A w Cfs KP My lei g 3 f I Dr YangQuan Chen Center for SelfrOrganizing end Intelligent Systems CSOIS EOE Dept ofUSU ECESEOO Seminaz Fractionalroxdex Calculus in Signal oncessing and Contzol An Ovezview 38 Control led system Temperature Control System 1 C8 m PD controller D1s 6447 12395 F ractional PD controller 1325 6447 489950 5 Dz YangQuan Chen Center for selfeotgenazang and Intelligent Systems 03013 EOE Dept of USU ECE6800 Seminar Hactionalrorder Calculus in Signal Processing and Control An Overview 39 time min Dr YangQuan Chen Center for SelfeOrgantztng and Intelligent Systems 05013 ECE Dept ofUSU ECE6800 Semmax Racuonalrozdex Calculus m ngnal Pxocessmg and Conuol An Ovemew 40 Performance comparison for 10 mins run Performance comparison DzYangQuan Chen w w n m of USU ECESEOO Seminar F ractionalrorder Calculus in Signal Processing and Control An Overview 41 lUSO5371670 on TID by B J Lurie 1994 3param tunable tiltintegral deriv controller77I Dr YangQuan Chen Center for SelfrOrganizing and Imemgem Systems CS 013 EOE Depc of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 42 lOustaloup s CRONE ControlI CRONE French abbreviation for Contr le Robuste d Ordre Non Entier non integer order robust control Since 1981 Based on concept of Fractal Robustness the isodamping and the vertical sliding form of frequency template in the Nichols chart Given plant 618 how to design 08 The ideal situation is to make G8C8 78 so that the characteristic equation is 1 78 0 Which is Fractal Robust Real life applications car suspension control exible transmission hydraulic actuator etc Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Semmax ButlenaLoxdex Calculus m Signal Pxocessmg and Conuol An Ovemew 43 Isodamping half stralght lines Dz YangQuan Chen ECESEOO Seminar F ractionalrorder Calculus in Signal Processing and Control An Overview 44 Robustness 1n chols chart Dr YangQuan Chen Center for SelfrOrganizing and Imemgem Systems CS 013 EOE Depc of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 45 So simply the controller is Cjw jwO G0jw G0jw is the nominal plant model Frequency speci cations of the open loop behavior for the nominal plant Will be given such as o the accuracy speci cations at low frequencies o the uertical template around unit gain frequency Lou o the input sensitiuity speci cations at high frequencies Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 46 For a stable minimum phase plant it turns out that the behavior thus de ned can be described by a transmittance based on the frequencylimited real n0n integer dt erenttator ie 8 Kbamp1nb lt 1wuWb218whgta Kh 1 1wuwh2 18wb 18wh 64 With 12 12 Kb 1 wbwu2 and Kh 1 tauwh2 65 Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 47 In the particular case Where transitional frequencies cab and wh are suf ciently distant from frequency tau around this frequency ie cab lt w lt wh 8 can be reduced to transmittance M8 Wu8 66 Which is the same as that described by the template The order 04 transmittance of relation 64 describes the frequency truncation of the template de ned by the transitional frequencies cab and ML This transmittance results from the substitution of the part raised at power 04 for the transmittance cabp Which is used in the description of the template between frequencies wA and wB Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 48 1 Fractional order Calculus A bit history de nitions examples 2 Fractional order ODE and Laplace Transform Fractional order Dynamic Filtering Fractional order Modeling and Control Implementation Techniques To Probe Further TQO liPw Concluding Remarks inbetween thinking Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Semmax chuonalrozdez Calculus m Signal Pxocessmg and Conuol An Ovemew 49 Analog 1 using op amp DzYangQuan Chen p pr 39 m n ECESSOO Seminar Fractionalrorder Calculus in Signal Processing and Control An Overview 50 Phase plot deg vs rad se Dr YangQuan Chen Center for SelfeOrganizing and Intelligent Systems 05015 EOE Dept of USU ECESSOO Seminar Fractionaleorder Calculus in Signal Processing and Control An Overview 51 Magnitude plot dB vs radsec Dr YangQuan Chen Center for SelfeOrganizing end Intelligent Systems 03013 EOE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 52 Digital Implementation indirect methodI Approximate in continuous time domain and then discretize it Oustaloup s method Given a frequency range of practical interest w E ca3 wH we can immediately get a rational transfer function of nite order which is a t to the given fractional order differentiator In practice we need set the transitional frequency range much larger than the ca3 mg for example 01wL 10mg Using this transitional frequency range and order of approximation 2N 1 the following formulae are the so called Oustaloup Recursive Approximation ORA Ali moo DN8 Ds 87quot 67 where N Dms 7 7quot H M 68 wH kN18Wk Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 53 and wu xwHwL 69 W 7quot w LHkNO5 05 2N17 70 60L w wk ULfkNO5O5r2N1t 71 Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 54 The 9th order Outaloup approximation can be obtained 3162s 3164s 6808s 1468s 3162s 6816s 1467s 03165s 006809s 001468 G s 9 s 6812s 1469s 316s 664s 2053s 0107s 003222s2 02818 2128 The Prewarp discretization of G9s at T 0001 sec can also be obtained as 121986z 099954z 09922z 09641z 08436z 04348z 02929 G9 2 z 099973z 09964z 09832z 09242z 06909z 05951z 008188 Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 55 Bode Diagram Magnitude dB 4 O N O Phase degrees 5 1o0 105 1o10 Bode plot of E 99 approximation Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU A o o ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 56 Nichols Chart 40 odB 3039 2 o25dB I 05dB a 10 gdg 3 cu 0 39C E 39g10 U 2 20 quot3039 3 I 39 39 333 25 3 i 39 I f f f 3 Phase degree 40 50 I 400 300 200 100 0 100 Nichols chart of 3 99 approximation Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 57 Digital Implementation direct method FIR Generating function 8 wz1 Backward difference rule wz1 1 z1T performing the power series expansion PSE of 1 z1ir gives the discretization formula for Griinwald Letnikov formula By using the short memory principle the discrete equivalent of the fractional order integro di erential operator wz1ir is given by LTl I j 39 wz 1ir TH E 1quotlt 70 ZWT 7 72 O J J where T is the sampling period L is the memory length is the i ooring operator and 1 7 7 are binomial coe icients J GET j 01 where 7quot 7 1 i T cg 1 cg 1 cgjl 73 k Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 58 Digital Implementation direct method IIR Recursive formula The trapezoidal Tustin rule as a generating l7 function wz1ir ltwz 1gtgt lt3gt 1 lt3 hm Aquotz 7quot 74 T 1z 1 T n gtoltgt Anz 1 r Where A0z1r 1 Anz1r An1z1r annAn1zr 75 and rn n is odd on 76 0 n is even Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 59 magnitude i i 250 300 freqency i i i i i i 0 50 100 150 200 350 400 450 500 J O l l 0 O l l phase deg M O l l O l l l l l l l l l l 0 50 100 150 200 250 300 350 400 450 500 freqency Bode plot of n21 approximation I Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 60 0391 0 l magnitude 8 O l i i i i i i i i 0 50 100 150 200 250 300 350 400 450 500 freqency 60 50 7 7 7 7 7 7 7 a 40 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 D E a 30 77 7 U 2 Q 20 7 r r r r r r r r r 7 10 7 7 7 7 7 7 7 0 l l l l l l l l l 0 50 100 150 200 250 300 350 400 450 500 freqency Bode plot of n23 approximation I Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 61 0391 0 l magnitude 0 O l i i i i i 250 300 350 400 450 500 freqency i i i 0 50 100 150 200 J O l 0 O l phase deg M O l O l l l l l l l l l 0 50 100 150 200 250 300 350 400 450 500 freqency Bode plot of n25 approximation I Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 62 magnitude 3 O l O O l i i i i i i i i 0 50 100 150 200 250 300 350 400 450 500 freqency 60 a i i D E w U m C Q l l l l l l l l l 50 100 150 200 250 300 350 400 450 500 freqency Bode plot of n27 approximation I Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 63 m magnitude 3 O l O O l i i i i i i i i 0 50 100 150 200 250 300 350 400 450 500 freqency 60 a i i D E w U m C Q l l l l l l l l l 50 100 150 200 250 300 350 400 450 500 freqency Bode plot of n29 approximation I Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 64 Direct Discretization Using Al Alaoui Operator Al Alaoui operator weighted sum of rectangular rule or Euler operator 025 and the trapezoidal rule 075 FIR and IIR above 8 1 51 77 1 7quot M2 Clearly 77 is an in nite order of rational discrete time transfer function To approximate it With a nite order rational one continued fraction expansion CFE is an ef cient way In general any function G can be represented by continued fractions in the form of 512 b2 m a2ltzgta3 ivhere the coef cients at and 97 are either rational functions of the Gz zaoz a1z Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 65 variable z or constants By truncation an approximate rational function can be obtained The resulting discrete transfer function approximating fractional order operators can be expressed as 7quot 1 7quot 8 1 z CFE 7T 1z17 paq Where CFEu denotes the continued fraction expansion of u 9 Diquot 22 and q are the orders of the approximation and P and Q are polynomials of degrees 19 and q Normally we can set 19 q n k Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 66 CFE MATLAB Scripts undocumented In MATLAB Symbolic Toolbox7 by the following script7 for a given n we can easily get the approximate direct discretization of fractional order derivative syms X rmaple withnumtheory aa 1X1X7 r n5 n22n maple cfe cfrac charaa Xn2 maple PoverQ nthconver cfe n2 maple P nthnumer cfe n2 maple Q nthdenom cfe n2 Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 67 m 7 7 503951Tustin CFE t0001s N1 1507 05 5 r r V 7 s AI Alaow CFE tS 0001s N 1 3 3 3 3 3 05 continuous Bode 3 3 3 3 2 10073 7 r 7 7 r 7 7 7 r r 17 rgrr r 7 37 50 r w gii i i i i i i i i 0 50 100 150 200 250 300 350 400 450 500 50 i 39 3 3 39 40739 w w r r r r i i i 307 r 7 r 3 20w quot 39 i 10 quot 0 l l l l l l l l l 0 50 100 150 200 250 300 350 400 450 500 Bode plot of pq1 approximation Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar 200 Fractional order Calculus in Signal Processing and Control An Overview 68 7 7 503951Tustin CFE ts0001sN3 i i i 150 7503951AI AlaouiCFEtSO001sN3 3 803952 continuous Bode 3 3 3 3 10073 3 r 3 3 r 3 3 3 r r 13 r 3313 7 i 450 500 i 400 i i i i i i 0 50 100 150 200 250 300 350 l l l l l l l l 0 50 100 150 200 250 300 350 400 450 500 Bode plot of pq3 approximation Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 69 m 7 7 503951Tustin CFE ts0001sN5 150 39 7503951AI AlaouiCFEts0001sN5 39 39 39 39 39 quot i i i i 05 continuous Bode 100 r n a r n r r 7 n i 74 e 507 t 7 477 i i 400 450 500 i 350 i i i i i 0 50 100 150 200 250 300 0 l l l l l l l l l 0 50 100 150 200 250 300 350 400 450 500 Bode plot of pq5 approximation Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 70 m 7 7 503951Tustin CFE ts0001s N7 150quot 7503951AI AlaouiCFEtS0001sN7 V 39 39 39 39 39 39 V quot 39 80395100ntinuous Bode 10077 n r r r n r r 7 e i i 400 450 500 i 350 i i i i i 0 50 100 150 200 250 300 50 4ora 307777 20 r w 7 107777 0 l l l l l l l l l 0 50 100 150 200 250 300 350 400 450 500 Bode plot of pq7 approximation Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 71 m 7 7 503951Tustin CFE t0001s N9 150739 05 s c r r r r r quot7quot 7 7 s 39 AI Alaow CFE ts0001s N9 39 05 continuous Bode i i i 2 10077 w a r r r r a w r r r 7 0 i i i i i i i i i O 50 100 150 200 250 300 350 400 450 500 50 4ora 307777 20 r w 7 107777 0 l l l l l l l l l 0 50 100 150 200 250 300 350 400 450 500 Bode plot of pq9 approximation Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU EcEsaoo Semmaz Rachonalrozdex Calculus in Signal Pxocessing and Conuol An Ovexview 72 a idem quotgr nfm f gumy Jx ggzgi gge Zm k Wcluvkrlmllinl mm AH wsmva bdamw u mnndmwmv We have a brilliant solutio lAll we have to do now is to nd the problem DzYangQuan Chen P w n m ofUSU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 73 lAll around sO where 04 E R I 0 its de nition in terms of fractional calculus o implications possible bene ts in ltering and control iso damping constant phase margin etc o realization techniques analog indirect digital direct digital FIR IIR etc o In brief we got a new tuning knob 80 in our engineering practice How to better use this knob is your business Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 74 1 Fractional order Calculus A bit history de nitions examples 2 Fractional order ODE and Laplace Transform Fractional order Dynamic Filtering Fractional order Modeling and Control Implementation Techniques To Probe Further TQO lrPw Concluding Remarks inbetween thinking Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 75 ITO Probe Furtherl 0 Books Podlubny I Fractional Di erential Equations Academic Press San Diego 1999 Fractional Di erential Systems Models Methods and Applications by D Matignon and G Montseny editors ESAIM Proceedings vol 5 December 1998 SMAI a conference proceedings online available Oustaloup A La Derivation non Entiere Hermes Paris1995 Samko S G Kilbas A A and Marichev O I Fractional Integrals and Derivatives Yverdon Switzerland Gordon and Breach 1993 K Nishimoto K Fractional Calculus New Haven CT Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 76 University of New Haven Press 1989 Oldham K B and Spanier J The Fractional Calculus Academic Press New York 1974 0 URLs httpmechatronicseceusuedufoc events news researchers tutorials MATLAB scripts ETEX BiBTeX library for FOC research etc httpXxxlanlgov some online papers with ETEX source on FOC httpwwwemathfrMathsProcVol5contentshtm an online proceedings on FOC some papers in French Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 77 1 Fractional order Calculus A bit history de nitions examples 2 Fractional order ODE and Laplace Transform Fractional order Dynamic Filtering Fractional order Modeling and Control Implementation Techniques To Probe Further TQO lrPw Concluding Remarks inbetween thinking Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 78 Concluding Remarks inbetween thinking I Examples of inbetween thinking Fuzzy logic inbetween thinking about binary logic 0 and 1 Fractional splines inbetween thinking about splines of integer orders see http bigwww epfl chdemofractsplinesindex html So With Fractional Order Calculus you may be able to extend a lot of things Good luck Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 79 Some Recent Developmentsl Refer to httpmechatronicsece usuedufoc Some highlighted developments 0 Special Issue on FOC Nonlinear Dynamics 2002 0 Special Issue on FOSP Signal Processing 2003 0 First IEEE CDC Tutorial Workshop on FOC Las Vegas 2002 0 First ASME Symposium on FOC Chicago 2003 0 First IFAC Symposium on FOC France 2004 0 First funding on FOC from National Research Council 2003 2005 Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 80 Acknowledgments I I d like to take this chance to thank Blas Vinagre and Ivo Petras for their great help and discussions Thanks also go to Sabatier Jocelyn A Oustaloup Igor Podlubny M A AlAlaoui Yoichi Hori and Dinh Nho Hao for many useful reference papers in either soft or hard copies Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 81 QA Session lYangQuan Chen Room EL 152 Tel 7970148I Email chhen ieeeorg eceusueduI FOC web httpmechatronicseceusuedufocI Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 82 backup slides 0 Some results from CSOIS Group 0 On Da type ILC results Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 83 Some results from CSOIS GroupI Done 0 Analytical stability bound for a class of delayed fractional order dynamic systems for IEEE Trans AC 0 Discretization Schemes for Fractional Order Differentiators and Integrators for IEEE Trans CASl 0 Direct Discretization of Fractional Order Derivative Using Al Alaoui Operator for 2001 IEEE CDC 0 On Da type Iterative Learning Control for 2001 IEEE CDC To do 0 make a TODO list At least have a look at QFT and Molli cation techniques Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 84 eb vs ILC iter no l ra0 G O eb deg 7Q 0906 Ooonmnnnnmnnmn 30 iteration number angular position tracking error VS ILC iter Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU ECE6800 Seminar Fractional order Calculus in Signal Processing and Control An Overview 85 eb vs ILC iter no 2 iteration number langular velocity tracking error VS ILC iter I Dr YangQuan Chen Center for Self Organizing and Intelligent Systems CSOlS ECE Dept of USU

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