Rozwiązanie zagadnienia odwrotnego wyznaczania sił obciąŝających konstrukcje w czasie eksploatacji

Rozwiązanie zagadnienia odwrotnego wyznaczania sił obciąŝających konstrukcje w czasie eksploatacji Tadeusz Uhl Piotr Czop Krzysztof Mendrok Faculty of Mechanical Engineering and Robotics Department of Robotics and Mechatronics Kraków, 7..009

Contents Model Inversion System Identification Evaluation of Model Accuracy Case Study Simulation Case Study Laboratory Frame Case Study Semi-Active Shock Absorber Case Study Freight Car Case Study Power Plant Boiler Summary

Model Inversion The term inverse refers to the fact that the roles of the input and the output are exchanged comparing to the usual forward system structural dynamics problem. In this approach, the model is inverted to propagate measured signal to obtain the input physical load. The methodology introduced here deals with a time-domain parametric approach to model inversion, the so-called dynamic correction/estimation, adopted mainly from the control theory and the digital signal processing DSP theory. FORWARD INVERSE ut MODEL yt MODEL u * t INPUT INPUT * OUTPUT

Model Inversion Transfer Function Model nb nb na na s b s b b s a s a a s y s u s G s G K K = = = 0 0 0 0 nb na c nb nb na na r st s b s b b s a s a a s y s u s G s G s G = = = K K

Model Inversion State-Space Model

Model Inversion in Applications Case Study Laboratory Frame Reconstruction of operational load e.g. wind to which mechanical structure are subjected to, e.g. drilling platforms, buildings, bridges, masts, other structures Benefits: no possibilities for direct measurements Case Study Semi-Active Shock Absorber Reconstruction of operational displacement of a rod inside the pressure tube to provide a feedback to controller of a semi-active suspension Benefits: cost of an accelerometer or a small load cell is significantly lower than cost of an inductive displacement transducer

Model Inversion in Applications Case Study Freight Car Reconstruction of operational load and/or profile of a rail Benefits: cost of an accelerometer is significantly lower than cost of an optical laser device required for measurements of a rail profile Case Study Power Plant Boiler Reconstruction of frequency contents of operational acoustic pressure which is generated by a pipeline system installed in a boiler interior to detect healthy and abnormal i.e. steam leakage conditions. Benefits: avoidance of expensive microphones working in high temperatures, separation of structural properties of a boiler construction from operational load

System Identification Linear vs. Nonlinear Model Linear or nonlinear data-driven models can be considered in appplication to capture fordward or inverse dynamics of a system or a structure Linear models have to be first identified in the forward mode and then inversed using mathematical transformations, while nonlinear models can be directly identified in the inverse mode Both, linear and nonlinear models can be formulated in a form of a transfer function or state-space representation Linear model Non-linear model Linear System Identification Theory Any arbitrary nonlinear regressor mapping inputs into outputs neural networks neuro-fuzzy systems

System Identification The following three components are essential ingredients of the system approach to experimental data-driven modeling; these are i system isolation, ii inputs-outputs selection iii model economy.

System Identification Transfer Function Model Gz - and Hz - are discrete-time transfer functions containing adjustable coefficients and represent the input-to-output dynamics and the disturbance-to-output dynamics, respectively. i e z D z A z C i u z F z A z B i e z H i u z G i y = =,,...,,,...,,,...,,,...,,,..., 0 nf nf nd nd nc nc nb nb na na z f z f z f z F z d z d z d z D z c z c z c z C z b z b z b b z B z a z a z a z A = = = = =

System Identification Transfer Function Model

System Identification State-Space Model

System Identification Extraction Of Modal Properties For example, in case of Linear Time Invariant LTI systems, roots α k of the polynomial Az - are expressed as i [ ζ ω T ] α = exp ω k k i dk ln α = ζ ω iω k k k T dk T ln α * = ζ ω iω k k kt dk T ζ k = ln α ln α k ω kt * k ω = k ln α ln α k T * k

System Identification Extraction Of Modal Properties Modal properties can also be extracted from a nonlinear model e.g. neural network model by means of linearization In turn, a nonlinear model can be converted into an equivalent linear model at specific operating point within the model is valid

System Identification Estimation Methods polynomial equation error method output error method

System Identification overview of linear model structures

System Identification selected nonlinear model structures

Evaluation of Model Accuracy a Ljung measure b Pearson correlation coefficient R N N i= = N N y i y i= y i i r = N i= [ y i mean y ] [ yˆ i mean yˆ ] N std y std yˆ reference corrected; fit: 69.6% 00% a Ljung measure is very sensitive to any offset value e.g. linear trend b Corelation measure is very senitive to a signal lag

Evaluation of Model Accuracy Power Spectrum Density [db/hz] 0-0 -40-60 -80-00 reference corrected; fit: 3% 79.% -0 0 0 0 30 40 50 60 Frequency [Hz] Reference: raw data from the sensor Power Spectrum Density [N /Hz] 0.4 0.3 0. 0. reference corrected; fit: 3% 79.% 0 0 0 0 30 40 50 60 Frequency [Hz] Corrected: raw data from the sensor with removed mean value

Evaluation of Model Accuracy 40 30 reference corrected; fit: 69.6% 00% Amplitude[N] 0 0 0-0 -0 Reference: raw data from the sensor Corrected: raw data from the sensor with removed mean value -30 0 0 0 30 40 50 60 70 80 90 00 Time [s]

Case Study Simulation ut Ky Dy y M = && & [ ] = K Ds Ms Hs Dut Cxt yt But Axt xt = = & 3 3 = k k s d d m s k sd k sd k k s d d m s s H = 3 3 s M k k s d d m s s M k sd s M k sd s M k k s d d s m s H

Case Study Simulation Performance and efficiency of the inversion procedure was tested on data sets obtained from a two-degrees-of-freedom system representing a mechanical system with viscous damping. Sinusoidal and square excitation signals were used to identify the ARX MIMO model, to generate the response of the system, and then to filter the response via the inverse model and obtain the model input. = square sin t f t f t u π π,, ARX,, 4 4 4 4 ARX Incorrect model structure too low degrees of polynomials Correct model structure <<< Input load

Case Study Simulation An example of the incorrect model structure during identification of the forward model ARX,, ARX

Case Study Simulation An example of the correct model structure during identification of the forward model ARX,, 4 4 4 4 ARX

Case Study Laboratory Frame

Case Study Laboratory Frame SISO model BJ0,8,8,0,; up: 5z Power Spectrum Density [db/hz] 00 50 0-50 -00 60dB measured reconstructed; fit: 94% -50 0 50 00 50 00 50 Frequency [Hz]

Case Study Laboratory Frame SISO model Power Spectrum Density [db/hz] 00 50 0-50 -00 BJ0,8,8,0,; up: 5z 60dB measured reconstructed; fit: 94% -50 80 90 00 0 0 30 40 Frequency [Hz]

Case Study Laboratory Frame

Case Study Laboratory Frame MIMO model ch# Power Spectrum Density [db/hz] 00 50 0-50 -00 50dB measured reconstructed; fit: 99% -50 0 50 00 50 00 50 Frequency [Hz]

Case Study Laboratory Frame MIMO model ch# Power Spectrum Density [db/hz] 00 50 0-50 -00 measured reconstructed; fit: 99% -50 80 90 00 0 0 30 40 Frequency [Hz]

Case Study Laboratory Frame MIMO model ch# Power Spectrum Density [db/hz] 00 50 0-50 -00 50dB measured reconstructed; fit: 98% -50 0 50 00 50 00 50 Frequency [Hz]

Case Study Laboratory Frame MIMO model ch# Power Spectrum Density [db/hz] 00 50 0-50 -00 measured reconstructed; fit: 98% -50 80 90 00 0 0 30 40 Frequency [Hz]

Case Study Semi-Active Shock Absorber

Case Study Semi-Active Shock Absorber Input: displacement Output: force or acceleration, or internal pressure Test rig: Hydraulic dynamometer 50kN A semi-active shock is equipped with 6 pressure sensors: sensors built in the rod compression, rebound strokes, sensor in the reserve tube at the height of the servovalve and 3 pressure sensors in the third tube.

Case Study Semi-Active Shock Absorber road load data model: OE5,9, sinewave f=hz

Case Study Semi-Active Shock Absorber Data for estimation of parameters of forward model road load data Data for load reconstruction sinewave 4 y y 0 0 - - 0 0.5.5.5 3 Time u 4 0 - -4 0 0.5.5.5 3 Time - 0 0.5.5.5 3 Time u 0 - - 0 0.5.5.5 3 Time

Case Study Semi-Active Shock Absorber 0 Power Spectrum Density [db/hz] -0-40 -60-80 -00 measured reconstructed; fit: 97% 00% -0 0 0 0 30 40 50 60 Frequency [Hz]

Case Study Semi-Active Shock Absorber 0.35 0.3 measured reconstructed; fit: 97% 00% 0.5 Amplitude[.] 0. 0.5 0. 0.05 0 0 0 0 30 40 50 60 Frequency [Hz]

Case Study Semi-Active Shock Absorber.5 Amplitude[N] 0.5 0-0.5 measured reconstructed; fit: 90% 00% - -.5 0 0 40 60 80 00 0 40 Time [s]

Case Study Freight Car Input: vertical load Q Output: vertical acceleration of the bearing box z* ψ z Y L Q L y l L ω l R Y R Q R θ

Case Study Freight Car Selected model: linear model OE5,9, nb=5 nf=9 k= delay Model identified: dataset CMK Model validated: dataset CMK5 Applied linear model OE5,9,

Case Study Freight Car Sensitivity analysis of orders of model polynomials, i.e. nb and nf -60-40 -0 0 0 0 5 0 Fit of the model to data in % Values of AIC criterion 8 8 6 6 4 4 e r F ord n 0 e r F ord n 0 8 8 6 6 5 0 5 nb order 5 0 5 nb order

Case Study Freight Car Sensitivity analysis of orders of model polynomials, i.e. nb and nf 0 0 40-50 0 50 fit in the frequency domain fit in the time domain 4 4 0 0 e r F ord n 8 6 e r F ord n 8 6 4 4 5 0 5 nb order 5 0 5 nb order

Case Study Freight Car 35 30 measured reconstructed; fit: 3% 75% 5 e [.] d p l itu m A 0 5 0 5 0 0 4 6 8 0 4 6 8 0 Frequency [Hz]

Case Study Freight Car 30 z ] /H B y [d it D ens m tru S pec w e r P o 0 0 0-0 -0-30 -40 measured reconstructed; fit: 3% 75% -50 0 4 6 8 0 4 6 8 0 Frequency [Hz]

Case Study Freight Car 5 0 measured reconstructed; fit: -0% 4% e [-] ud p l it m A 5 0-5 -0 0 000 4000 6000 8000 0000 000 Time [s]

Case Study Freight Car 0 8 6 measured reconstructed; fit: -0% 4% e [-] d p l itu m A 4 0 - -4-6 -8 7300 7350 7400 7450 7500 7550 7600 7650 7700 7750 Time [s]

Case Study Freight Car 3 e [-] ud p l it m A 0 - - -3 measured reconstructed; fit: -0% 4% 500 600 700 800 900 000 00 00 Time [s]

Case Study Freight Car identified: CMK validated: CMK5 nonlinear model NARX0,3, linear element in the second layer 5 nonlinear elements in the first layer

Case Study Freight Car 5 0 measured reconstructed; fit: 40% 80% e [.] d p l itu m A 5 0 5 0 0 4 6 8 0 4 6 8 0 Frequency [Hz]

Case Study Freight Car 30 z ] /H B y [d it D ens m tru S pec w e r P o 0 0 0-0 -0-30 measured reconstructed; fit: 40% 80% -40 0 4 6 8 0 4 6 8 0 Frequency [Hz]

Case Study Freight Car 5 0 measured reconstructed; fit: -% 39% e [-] d p l itu m A 5 0-5 -0 0 000 4000 6000 8000 0000 000 Time [s]

Case Study Freight Car 5 4 measured reconstructed; fit: -% 39% 3 e [-] d p l itu m A 0 - - -3 600 700 800 900 000 00 Time [s]

Case Study Freight Car 0 8 measured reconstructed; fit: -% 39% 6 Amplitude[-] 4 0 - -4-6 7450 7500 7550 7600 7650 7700 7750 Time [s]

Case Study Power Plant Boiler Input: acoustic pressure inside the boiler Output: acoustic pressure outside the boiler

Case Study Power Plant Boiler OUTPUT INPUT

Case Study Power Plant Boiler Linear Model OE7,,0 0 8 6 4 measured reconstructed; fit: 34% 76% e [.] ud p l it m A 0 8 6 4 0 0 500 000 500 000 500 3000 Frequency [Hz]

Case Study Power Plant Boiler Nonlinear Model NARX,9,0 4 measured reconstructed; fit: 3% 74% 0 e [.] d p l itu m A 8 6 4 0 0 500 000 500 000 500 3000 Frequency [Hz]

Summary A procedure of inverting a model does not correspond to any physical phenomenon and so inverse models have always a tendency of being unphysical. As all physical systems have time delay as well as limited bandwidth, the exact inverse advances the signal as a result of a delay and amplifies high frequency noise without any bound if bandwidth is not restricted to upper frequency band of the inversing load. This phenomenon is the so-called ill-posedness of the inversion and requires regularization techniques to be used to correctly estimate the load up to a certain bandwidth, typically given by the frequency at which amplitudes of the signal and the noise are equal. A standard solution of the regularization problem is to use a lowpass noise filters, which leads to a model and its realized approximation to the prototype of inversion.

Summary This fact suggests that sampling rate of input and output signals has to be adapted to the maximal frequency of the reconstructed signal, although a very low sampling rate can result in severe aliasing. On the other hand, the sampling rate should allow the most important dynamics vibration modes of the structure or mechanical system to be captured correctly. An example of amplified noise in the reconstructed load

Summary Experimental validation tests confirm that the methodology proposed herein, i.e. parametric system identification and model inversion, is valid for both, the SISO and the MIMO, model structures. Results obtained from the study are sufficient to constitute foundations for implementing an inverse model as an inverse filter on a DSP platform. A development board with a fixed-point DSP processor TMS30LF407A

Q&A