Non-convex sparse optimization-based impact force identification with limited vibration measurements

Abstract

Impact force identification is important for structure health monitoring especially in applications involving composite structures. Different from the traditional direct measurement method, the impact force identification technique is more cost effective and feasible because it only requires a few sensors to capture the system response and infer the information about the applied forces. This technique enables the acquisition of impact locations and time histories of forces, aiding in the rapid assessment of potentially damaged areas and the extent of the damage. As a typical inverse problem, impact force reconstruction and localization is a challenging task, which has led to the development of numerous methods aimed at obtaining stable solutions. The classical {\ell }_2 regularization method often struggles to generate sparse solutions. When solving the under-determined problem, {\ell }_2 regularization often identifies false forces in non-loaded regions, interfering with the accurate identification of the true impact locations. The popular {\ell }_1 sparse regularization, while promoting sparsity, underestimates the amplitude of impact forces, resulting in biased estimations. To alleviate such limitations, a novel non-convex sparse regularization method that uses the non-convex {\ell }_{1-2} penalty, which is the difference of the {\ell }_1 and {\ell }_2 norms, as a regularizer, is proposed in this paper. The principle of alternating direction method of multipliers (ADMM) is introduced to tackle the non-convex model by facilitating the decomposition of the complex original problem into easily solvable subproblems. The proposed method named {\ell }_{1-2}\text{-ADMM} is applied to solve the impact force identification problem with unknown force locations, which can realize simultaneous impact localization and time history reconstruction with an under-determined, sparse sensor configuration. Simulations and experiments are performed on a composite plate to verify the identification accuracy and robustness with respect to the noise of the {\ell }_{1-2}\text{-ADMM} method. Results indicate that compared with other existing regularization methods, the {\ell }_{1-2}\text{-ADMM} method can simultaneously reconstruct and localize impact forces more accurately, facilitating sparser solutions, and yielding more accurate results.

Graphical abstract

Keywords

impact force identification / inverse problem / sparse regularization / under-determined condition / alternating direction method of multipliers

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Nomenclature

Abbreviations
ADMM	Alternating direction method of multipliers
BVID	Barely visible impact damage
GRE	Global relative error
IRF	Impulse response function
LE	Localization error
LRE	Local relative error
MC-GSURE	Monte Carlo generalized stein unbiased risk estimate
PRE	Peak relative error
SISO	Single-input single-output
SNR	Signal-noise ratio
Variables
a(t)	Impulse response function
aij(t)	Impulse response function between the output position i and the input location j
aij(ω)	Frequency response function between the output position i and the input location j
A	Transfer matrix of the multiple-input multiple-output dynamic system
As	Transfer matrix of the single-input single-output dynamic system
B	Amplitude of the Gaussian-shaped impact force
c	All-ones vector
C	Damping matrix
e	Random noise in measurements
E	Elastic modulus
fi	Elements in the vector f
f(t)	Impact force excitation function
f	Force vector of the multiple-input multiple-output dynamic system
\hat{\mathit{f}}	Estimated vector of f
fp	Actual force vector at the impact position p
{\hat{\mathit{f}}}_p	Estimated force vector at the impact position p
{\hat{\mathit{f}}}_{\text{ML}}	Maximum likelihood estimation of the vector f
fs	Force vector of the single-input single-output dynamic system
g(f)	General representation function of penalty terms
G	Shear modulus
h	An additional vector for variable splitting
\hat{\mathit{h}}	Estimated vector of h
i	Looping variable within the summation operation
I	Identity matrix
k	Number of iterations
K	Stiffness matrix
m	Number of measurement responses
M	Mass matrix
n	Number of impact force excitations
nf	Total number of potential impact force locations
N	Data length of the discretized impulse response function
Nmax	Maximum number of iterations
O(nN)	Computational complexity of n × N
p	Serial number of the location subjected to impact force
q	Norm parameter defined in {\mathbb{R}}^{+\ast }
Q	Projection matrix
r	Gaussian white noise vector
\mathcal{R}(\mathit{f})	General expression for calculating the norm of vector f
s(t)	System response
s	Response vector of the multiple-input multiple-output dynamic system
\stackrel{\mathbf{~}}{\mathit{s}}	Noisy response vector
si	Response vector at a certain position i
ss	Response vector of the single-input single-output dynamic system
t	Time
t0	Occurrence time instant of the impact
Δt	Sampling interval
T	Impact duration
u	Sufficient statistic of the model Eq. (8)
wmax	Element with the largest absolute value in the vector·w
w	Intermediate vector defined as w = f (k+1) + z(k)
xλ(u)	Solution result of Eq. (8) when f = u
y(f)	Proximal operator
δ	Small positive parameter
δ	Lagrange multiplier vector
ε	Iteration termination threshold
λ	Regularization parameter
ρ	A positive penalty parameter
σ	Standard deviation of the vector \mathit{s}
σn	Standard deviation of noise in the measurements
τ	Time delayed operator
ν	Possion’s ratio
Γ	Threshold value defined as Γ = λ/ρ

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 52075414 and 52241502), and China Postdoctoral Science Foundation (Grant No. 2021M702595).

Conflict of Interest

The authors declare that they have no conflict of interest.

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