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System Identification via the Proper Orthogonal Decomposition

Profile image of Christopher BeattieChristopher Beattie

Abstract

Although the finite element method is often applied to analyze the dynamics of structures, its application to large, complex structures can be time-consuming and errors in the modeling process may negatively affect the accuracy of analyses based on the model. System identification techniques attempt to circumvent these problems by using experimental response data to characterize or identify a system. However, identification of structures that are time-varying or nonlinear is problematic because the available methods generally require prior understanding about the equations of motion for the system. Nonlinear system identification techniques are generally only applicable to nonlinearities where the functional form of the nonlinearity is known and a general nonlinear system identification theory is not available as is the case with linear theory. Linear time-varying identification methods have been proposed for application to nonlinear systems, but methods for general time-varying systems where the form of the time variance is unknown have only been available for single-input single-output models. This dissertation presents several general linear time-varying methods for multiple-input multiple-output systems where the form of the time variance is entirely unknown. The methods use the proper orthogonal decomposition of measured response data combined with linear system theory to construct a model for predicting the response of an arbitrary linear or nonlinear system without any knowledge of the equations of motion. Separate methods are derived for predicting responses to initial displacements, initial velocities, and forcing functions. Some methods require only one data set but only promise accurate solutions for linear, time-invariant systems that are lightly damped and have a mass matrix proportional to the identity matrix. Other methods use multiple data sets and are valid for general time-varying systems. The proposed methods are applied to linear timeinvariant, time-varying, and nonlinear systems via numerical examples and experiments and the factors affecting the accuracy of the methods are discussed.

Figures (90)
Fig. 2.1 Nonlinear System Identification Process (from [3], used with author’s permission)
Fig. 2.1 Nonlinear System Identification Process (from [3], used with author’s permission)
Fig. 3.1 Example Mass-Spring System with Equal Masses >qual to the eigenmodes. Consider the nondimensional mass-spring system shown in Fig.
Fig. 3.1 Example Mass-Spring System with Equal Masses >qual to the eigenmodes. Consider the nondimensional mass-spring system shown in Fig.
This system is undamped and the mass matrix is proportional to (in fact, it is equal to) the identity matrix. The eigenmodes (scaled so that they are orthonormal) for the system are:
This system is undamped and the mass matrix is proportional to (in fact, it is equal to) the identity matrix. The eigenmodes (scaled so that they are orthonormal) for the system are:
3.2.2 POV Recalculation from a State-Space Perspective
3.2.2 POV Recalculation from a State-Space Perspective
After ®,,(t;) has been obtained from Eq. (3.27) at every time step, the POC histories representing a response to a new initial displacement profile may be calculated at each
After ®,,(t;) has been obtained from Eq. (3.27) at every time step, the POC histories representing a response to a new initial displacement profile may be calculated at each
Fig. 4.2 Cubic Spring Force for Nonlinear Beam Model
Fig. 4.2 Cubic Spring Force for Nonlinear Beam Model
Fig. 4.1 LTI (top), LTV (middle), and NL (bottom) Beam Models
Fig. 4.1 LTI (top), LTV (middle), and NL (bottom) Beam Models
Fig. 4.3 Time Variation of Tip Mass for LTI Beam Model
Fig. 4.3 Time Variation of Tip Mass for LTI Beam Model
Fig. 4.4 Applied Initial Displacement Profiles to form W . Thus the dimensions of W were (25 x 500) for both models.
Fig. 4.4 Applied Initial Displacement Profiles to form W . Thus the dimensions of W were (25 x 500) for both models.
Fig. 4.5 Original and Recalculated POVs for Linear Beam Model
Fig. 4.5 Original and Recalculated POVs for Linear Beam Model
Fig 4.6 Displacement Norms for LTI Beam Model in Response to w; mass results in a mass matrix with diagonal terms that are not close in value.
Fig 4.6 Displacement Norms for LTI Beam Model in Response to w; mass results in a mass matrix with diagonal terms that are not close in value.
Fig. 4.7 Percent Error of Displacement Norms for LTI Beam Model in Response to
Fig. 4.7 Percent Error of Displacement Norms for LTI Beam Model in Response to
Fig. 4.8 Displacement Norms for LTV Beam Model in Response to w;
Fig. 4.8 Displacement Norms for LTV Beam Model in Response to w;
Fig. 4.9 Percent Error of Displacement Norms for LTV Beam Model in Response to
Fig. 4.9 Percent Error of Displacement Norms for LTV Beam Model in Response to
Fig. 4.10 Displacement Norms for NL Beam Model in Response to w; they are dwarfed by the linearization error.
Fig. 4.10 Displacement Norms for NL Beam Model in Response to w; they are dwarfed by the linearization error.
Fig. 4.11 Percent Error of Displacement Norms for NL Beam Model in Response to
Fig. 4.11 Percent Error of Displacement Norms for NL Beam Model in Response to
Fig. 4.13 Displacement Norm for LTI Beam Model in Response to w,
Fig. 4.13 Displacement Norm for LTI Beam Model in Response to w,
Fig. 4.14 Percent Error of Displacement Norms for LTI Beam Model in Response to
Fig. 4.14 Percent Error of Displacement Norms for LTI Beam Model in Response to
Fig. 4.15 Displacement Norm for LTV Beam Model in Response to w , is attributed to projection error.
Fig. 4.15 Displacement Norm for LTV Beam Model in Response to w , is attributed to projection error.
Fig. 4.16 Percent Error of Displacement Norms for LTV Beam Model in Response to Wy,
Fig. 4.16 Percent Error of Displacement Norms for LTV Beam Model in Response to Wy,
Fig. 4.17 Displacement Norm for NL Beam Model in Response to w,
Fig. 4.17 Displacement Norm for NL Beam Model in Response to w,
Fig. 4.19 Pulse Load Applied to Beam Models models.
Fig. 4.19 Pulse Load Applied to Beam Models models.
Fig. 4.20 Five Locations of Pulse Load A pplication
Fig. 4.20 Five Locations of Pulse Load A pplication
Fig 4.21 Displacement Norms for LTI Beam Response to Load 5
Fig 4.21 Displacement Norms for LTI Beam Response to Load 5
Fig. 4.22 Percent Error of Displacement Norms for LTI Beam Response to Load 5
Fig. 4.22 Percent Error of Displacement Norms for LTI Beam Response to Load 5
Fig 4.23 Displacement Norms for LTV Beam Response to Load 5
Fig 4.23 Displacement Norms for LTV Beam Response to Load 5
Fig. 4.24 Percent Error of Displacement Norms for LTV Beam Response to Load 5
Fig. 4.24 Percent Error of Displacement Norms for LTV Beam Response to Load 5
Fig 4.25 Displacement Norms for NL Beam Response to Load 5
Fig 4.25 Displacement Norms for NL Beam Response to Load 5
Fig. 4.26 Percent Error of Displacement Norms for NL Beam Response to Load 5
Fig. 4.26 Percent Error of Displacement Norms for NL Beam Response to Load 5
Fig. 4.27 Initial Displacements in Excitation Sets a, b, and c dimensions of “W, ’W and ‘W were (25 x 500) for all beam models.
Fig. 4.27 Initial Displacements in Excitation Sets a, b, and c dimensions of “W, ’W and ‘W were (25 x 500) for all beam models.
Fig. 4.28 Initial Velocities in Excitation Sets a, b, and c Next, the POD was computed for each beam’s response to the first excitation set and the
Fig. 4.28 Initial Velocities in Excitation Sets a, b, and c Next, the POD was computed for each beam’s response to the first excitation set and the
Fig. 4.31 New Load A pplied Vertically at Beam Tip
Fig. 4.31 New Load A pplied Vertically at Beam Tip
Fig. 4.32 Displacement Norms for LTI Beam in Response to W
Fig. 4.32 Displacement Norms for LTI Beam in Response to W
Fig. 4.36 Displacement Norms for LTI Beam in Response to F (t)
Fig. 4.36 Displacement Norms for LTI Beam in Response to F (t)
Fig. 4.37 Percent Error of Displacement Norms for LTI Beam in Response to F (t)
Fig. 4.37 Percent Error of Displacement Norms for LTI Beam in Response to F (t)
Fig. 4.38 Displacement Norms for LTV Beam in Response to W, attributed to the fact that the tip mass causes high diagonalization error.
Fig. 4.38 Displacement Norms for LTV Beam in Response to W, attributed to the fact that the tip mass causes high diagonalization error.
Fig. 4.42 Displacement Norms for LTV Beam in Response to F (t)
Fig. 4.42 Displacement Norms for LTV Beam in Response to F (t)
Fig. 4.43 Percent Error of Displacement Norms for LTV Beam in Response to F (t)
Fig. 4.43 Percent Error of Displacement Norms for LTV Beam in Response to F (t)
Fig. 4.46 Displacement Norms for NL Beam in Response to W,
Fig. 4.46 Displacement Norms for NL Beam in Response to W,
Fig. 4.48 Displacement Norms for NL Beam in Response to F (¢)
Fig. 4.48 Displacement Norms for NL Beam in Response to F (¢)
Table 4.1 Truss Member Geometry and Material Properties
Table 4.1 Truss Member Geometry and Material Properties
Fig. 4.50 Cross Section and Schematic View of a Single Truss Element [C ourtesy Armaghan Salehian] members all have hollow circular cross sections and are joined by idealized pin joints.
Fig. 4.50 Cross Section and Schematic View of a Single Truss Element [C ourtesy Armaghan Salehian] members all have hollow circular cross sections and are joined by idealized pin joints.
Fig. 4.51 Coordinates for Truss Element [C ourtesy Armaghan Salehian] rotation around each pinned end. The structure is therefore a linear time-varying system.
Fig. 4.51 Coordinates for Truss Element [C ourtesy Armaghan Salehian] rotation around each pinned end. The structure is therefore a linear time-varying system.
Table 4.2 Static Loads A pplied to Generate Initial Displacements for Truss
Table 4.2 Static Loads A pplied to Generate Initial Displacements for Truss
Fig. 4.52 Displacement Norms for Truss Model in Response to w¢
Fig. 4.52 Displacement Norms for Truss Model in Response to w¢
Fig. 4.53 Percent Error of Displacement Norms for Truss Model in Response to w,
Fig. 4.53 Percent Error of Displacement Norms for Truss Model in Response to w,
Table 4.3 Impulsive Loads A pplied to Generate Initial Velocities for Truss
Table 4.3 Impulsive Loads A pplied to Generate Initial Velocities for Truss
Fig. 4.54 Displacement Norms for Truss Model in Response to Wg
Fig. 4.54 Displacement Norms for Truss Model in Response to Wg
Fig. 4.56 Pulse Load A pplied to Satellite Truss M odel the left boundary).
Fig. 4.56 Pulse Load A pplied to Satellite Truss M odel the left boundary).
Fig. 4.57 Displacement Norms for Truss Model in Response to Load method’s response prediction.
Fig. 4.57 Displacement Norms for Truss Model in Response to Load method’s response prediction.
valid for any applied load and the extra hits do not interfere as long as they are measured.
valid for any applied load and the extra hits do not interfere as long as they are measured.
Fig. 5.2 Application of Impulsive L oad via Shaker Strike For each load case, a Polytec OFV-303 laser vibrometer was set up to measure the
Fig. 5.2 Application of Impulsive L oad via Shaker Strike For each load case, a Polytec OFV-303 laser vibrometer was set up to measure the
Fig. 5.3 Load Cell Output With and Without Beam Impact
Fig. 5.3 Load Cell Output With and Without Beam Impact
Fig. 5.4 Forces Applied to Various Beam L ocations
Fig. 5.4 Forces Applied to Various Beam L ocations
Fig. 5.5 Two Most Dominant POMs for Linear Beam the SL and ML methods compared with the measured data.
Fig. 5.5 Two Most Dominant POMs for Linear Beam the SL and ML methods compared with the measured data.
Fig. 5.6 POVs for Experimental Linear Beam
Fig. 5.6 POVs for Experimental Linear Beam
Fig. 5.8 Percent Error of Displacement Norms for Linear Experimental Beam Response
Fig. 5.8 Percent Error of Displacement Norms for Linear Experimental Beam Response
Fig. 5.9 Rubber Band Acting as Linear Spring beam tip deflects toward the rubber band attachment point (the rubber band buckles as
Fig. 5.9 Rubber Band Acting as Linear Spring beam tip deflects toward the rubber band attachment point (the rubber band buckles as
Fig. 5.10 Rubber Band Providing no Stiffness
Fig. 5.10 Rubber Band Providing no Stiffness
Fig. 5.13 Phase Lag for Various Impulse M agnitudes excitation magnitude, indicating that the nonlinearity is active in the beam response.
Fig. 5.13 Phase Lag for Various Impulse M agnitudes excitation magnitude, indicating that the nonlinearity is active in the beam response.
Fig. 5.14 Forced A pplied to Locations on Nonlinear Beam initial impact.
Fig. 5.14 Forced A pplied to Locations on Nonlinear Beam initial impact.
Fig. 5.15 Displacement Norms for NL Experimental Beam Response to Load at 4” Location as in the numerical example.
Fig. 5.15 Displacement Norms for NL Experimental Beam Response to Load at 4” Location as in the numerical example.
Fig. 5.16 Percent Error of Displacement Norms for NL Experimental Beam Response
Fig. 5.16 Percent Error of Displacement Norms for NL Experimental Beam Response

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