System Identification & Parameter Estimation

Transcription

1 System Identiication & Parameter Estimation Wb30: SIPE Lecture 9: Physical Modeling, Model and Parameter Accuracy Erwin de Vlugt, Dept. o Biomechanical Engineering BMechE, Fac. 3mE April 6 00 Delt University o Technology

2 Contents Parameter estimation in time domain resume previous lectures Overview o an experiment Basic steps in an ideal experiment Parameter estimation in requency domain: on-parametric models: requency response unction FRF Can be derived rom non-parametric time-domain models Models with physical parameters Transer unction o model as unction o Laplace operator s Model structure & model parameters: linear models Optimization algorithms: adapt model parameters or best it in requency domain xx

3 Parameter estimation in time-domain on-parametric models: ARMA, OE, Box-Jenkins, etc. Models with physical parameters Input-output data, simulation o model time domain Model structure & model parameters: linear and non-linear models Optimization algorithm: adapt model parameters or best it to simulation ote that ARX is a special case! ARX is linear in the parameters : no simulation/optimization required! 3 xx

4 System identiication & parameter estimation ut, yt Uω,Yω ARX ARMA Etc. non-parametric model non-parametric model Frequency Response Function FRF parametric model parametric model 4 xx

5 System identiication & parameter estimation Input signal Unknown system Output signal System identiication Input signal Unknown system Model + Parameter estimation - Output signal Predicted output 5 xx

6 Quantiication o validity Variance-Accounted-For VAF values: How much o the variance in the data can be explained by the model? VAF y t i i i y t i yˆ t i ˆ t y i yt i, u; t, ut i : recorded data 6 xx

7 Coherence and VAF High coherence, low VAF: Linear system, good SR, wrong model! High coherence, high VAF: Linear system, good SR, good model Low coherence, high VAF: on-linear system, good SR, good non-linear model Low coherence, low VAF: on-linear system or poor SR, poor model 7 xx

8 Parameter estimation o static and dynamic systems Measured data input signal xk output signal yk Model linear or non-linear Predicted output: ^ yk,uk : parameter vector > simulation o the model, depends on and uk error unction ek yk -,uk criterion unction least squares Jxk,yk, Σek ind which minimizes J iterative search requires many simulations! 8 xx

9 Accuracy o parameter it Single parameter: SEM: Standard Error o the Mean Multiple parameters: Covariance matrix Estimated rom Jacobian and residual error 9 xx

10 Standard error o the mean SEM How accurate can the parameters be estimated? Example: ormal distribution o data x : μ x, x Standard Error o the Mean: μ μ x x x x variance o the mean standard error deviation o the mean the more data samples to more accurate the estimation o the mean 0 xx

11 Co-variance matrix ˆ T T P P e e.[ J J] ˆ Cov P P : Parameter co-variance matrix or parameter vector Approximated by P limited number o data samples or estimation Cov : Variance o P diagcov xx

12 xx Covariance Matrix P Z : data vector with input vector u and output vector y V,Z : criterion value o : True, optimal parameter vector unknown! Expanding Taylor series st order around o :, arg min ˆ Z V 0,, ' Z V Z V o o o o o o o P Z V Z V Z V Z V. ˆ,. ], [ ˆ ˆ.,, 0 ˆ ' '' '' ' +

13 3 xx Derivation P e is white noise at o, and hence J T / e/ 0 T T V V V V P '' ' ' ''... ' ' ''. T T T V e V V J e V J V J J e +

14 Derivation P P becomes Where P V. V. V. V '' ' ' T '' T J J. J e. e. J. J J T T T λ. J J. J J. J J T T i i T i T λ. J J λ i i i T ee.. e. e 4 xx

15 5 xx Co-variance matrix And cov,, etc ˆ Cov M M M M M L M O M M L L

16 Matlab demo: parameter accuracy 6 xx

17 Basic steps in identiication scheme. Prepare experiment Choose sample requency, observation time, and number o repetitions Choose/design input signal. Perorm experiment Perorm experiments with care and prevent possible noise sources 3. Analyze results Check linearity! e.g. coherence Open-loop or closed-loop algorithms required? Do nonparametric analysis FRF or ARX/OE/ARMAX Fit parametric model onto data Check residue should be small and preerably white Check validity VAF and parameter uncertainty e.g. SEM 7 xx

18 . Prepare experiment Sample requency Should be high enough to see all relevant dynamics High sample requency will give more data storage! but will not necessarily give more inormation! Prevent aliasing Observation time Determines resolution in requency domain In general longer is better as long as system is time-invariant umber o repetitions Multiple observations > variations between observations Choose/design input signal persistently exciting > excite all relevant dynamics Prevent leakage 8 xx

19 . Perorm experiment Perorm experiments with care and prevent possible noise sources Electromagnetic intererence? Human subjects Unpredictable, to prevent anticipation clear instruction, no distractions, etc Oten data can not be ixed aterwards! 9 xx

20 3. Analyze results Check linearity Calculate coherence Open-loop or closed-loop algorithms required? Try to make a block scheme Do nonparametric analysis FRF or ARX/OE/ARMAX e.d. Bode diagram can give indication o system under investigation Fit parametric model onto data Do a irst check by inspecting the Bode diagram o data and model! Check residue should be small and preerably white What is not captured with itted model? Check validity o model VAF and parameter uncertainty e.g. SEM 0 xx

21 Options in parameter estimation Time-domain: direct it using derivatives HMC: inverse dynamics oise is ampliied by dierentiation direct it using simulation previous lecture Requires multiple model simulations: a lot o CPU power Can handle non-linear models! xx

22 Options in parameter estimation Frequency-domain:. FFT, estimation o FRF, estimation o parameters o prior assumptions are needed! Can easily cope with systems in closed-loop Estimates can be biased i very much noise is present E.g. depends on number requency bands used or averaging. OE/ARMAX it, estimation o parameters In general reasonable ast and accurate Order selection is needed requires a choice! Frequency reconstruction desired to estimate the model structure i system is unknown xx

23 Error unction in requency domain Simple approach: J e e Wrong approach Can give severely biased results H est H mod 3 xx

24 Model it in requency domain gain [-] Dynamic range: absolute errors vary between 0 vs requency [Hz] 4 xx

25 Model it in requency domain Reliability: coherence varies with requency coherence [-] requency [Hz] 5 xx

26 Model it in requency domain 0 0 Logarithmic requency axis: low emphasis on lower requencies 0 0 gain [-] requency [Hz] 6 xx

27 7 xx Log o Transer unction H.. ω ϕ ω ω ω ω j e A b j a H +. ln ln ln. ln ln ω ϕ ω ω ω ω ω ϕ ω ϕ j H e A e A H j j + + Logarithm eects the gain, not the phase!

28 Error unction in requency domain J e e. γ.ln H est ln H mod. γ.ln H est ln H mod + i. ϕ H est ϕ H mod ~ * γ *dierence in loggain + dierence in phase J. γ.ln H est ln H mod. γ H.ln H weighted by coherence to put more emphasis on reliable requencies weighted by /requency to compensate or ew data in low requency region est mod 8 xx

29 Error unction in requency domain Example: measurements on a mass-spring-damper system J e H H est mod e. γ.ln H S S uy uu M. s est + B. s + K ln H mod M.πj.. γ + B.πj. + K.ln H H est mod 9 xx

30 Assignment this week Estimate parameters in time and requency domain Compare the results between both approaches Goal: Show the dis-advantages and peculiarities o estimation in both time domain and requency domain and compare the two approaches 30 xx