Sequential deconvolution input reconstruction
Introduction
The reconstruction of forces from observed responses is a classic inverse problem in structural dynamics. In this paper the format examined is that where the observations are sampled values of m output signals and the r forces to be reconstructed have known positions and directions. There are, of course, other frequently inspected formats such as the estimation of moving loads, of interest in weight in motion applications [1], [2], [3] or the characterization of impact forces, in damage detection of composite plates [4], [5], to mention a couple. In this paper attention is restricted to the case where the system is linear and time invariant (LTI) and the input–output (I–O) maps are available either from a Finite Element model or from system identification.
Traditionally the input estimation problem has been solved (in batch mode) in the frequency domain [6]. The approach is commonly implemented with the periodic assumptions implied by the FFT but an “exact solution”, within the assumptions on the inter-sample behavior, is feasible using the integral transform if the loading is time limited and the output signals are fully observed. When the application is such that the lag in the input prediction is relevant a time domain algorithm is needed. Reconstruction in real time is theoretically feasible if the system has a full rank direct transmission but for general I–O arrangements the estimation has to be carried out with a delay respect to the measurements because transfer of information across space takes time. The inherent delay is a function of the wave propagation speeds and of the distances between inputs and outputs and it is clearly a lower bound for the prediction lag of any online reconstruction scheme.
This paper shows that the minimum realizable prediction lag can be significantly larger than the inherent delay and that it is typically determined by numerical stability considerations. The numerical stability constraint is derived and is found to be a function of: the controllability, the observability, the pseudo-inverse of the block Toeplitz matrix that maps inputs to outputs, the state transition matrix and the size of the window forward shift. Satisfying the stability constraint ensures against exponential error growth but does not guarantee that the solution is accurate. An expression that characterizes the error in the estimated inputs as a function of the noise and the error in the initial condition is derived and is used to decide how much regularization is needed, if any. Inspection of the Fisher information along the frequency line highlights the importance of formulating the inference model in such a way that the spectra of the inputs to be identified, and of the realized measurement noise, are within the system bandwidth.
In examining the input reconstruction problem it is essential to keep in mind that the Finite Dimensional (FD) models used to carry out analytical examinations have no dead time. This assertion can appear a bit surprising when encountered for the first time but it is a well-known result from system theory [7] and it is not difficult to rationalize if one notes that the response of FD models is made up of the sum of a series with finite terms. There are, of course, FD models where delays are “externally” added, so the assertion does not imply that delays cannot be incorporated into a FD model but rather that a finite element discretization of a continuum does not have impulse response functions with zero finite time segments. Needless to say, a well formulated FD model is accurate within the bandwidth that it is intended to function so the part of the impulse response functions that are zero in the continuum are very small in the model and have no consequence in the solution of forward problems. Very small, however, is infinitely large compared to zero and this finiteness, which does not exist in the real problem, creates the mathematical space for deriving schemes that offer input estimates with one time step delay, independently of spatial separation and independently of the time step, violating in general the physical constraints. In the rest of this paper references to delay in FD models should be understood to imply “quasi delay” not true dead time. A proof of the fact that FD models have no delay is presented in Appendix A.
Clarity in the subsequent discussions is enhanced by commenting briefly on the issue of conditioning and posedness. In the literature it is common to find statements indicating that the non-collocated problem is ill-conditioned and/or ill-posed with distinction between these two terms seldom addressed. The distinction can be important. A problem is poorly conditioned when the rate of change of the solution with respect to perturbations is high and, as shall be shown, in the input identification problem the matter depends to a large extent on the bandwidth of the disturbance relative to that of the model. Posedness, on the other hand, refers to uniqueness and points to whether the constraints from the physics are duly translated into the mathematical formulation and whether the questions asked are appropriate. Stating that deconvolution of none collocated inputs is ill-posed, when the question refers to inputs that are synchronous with the outputs is devoid of information since instantaneous information transfer across space is impossible. The relevant question is, of course, whether or not the I–O arrangement is such that knowledge of the outputs in [t0,t1], plus the state at t0, constrain the inputs over [t0, t1–d], for some d. It is shown in the paper that this question can be answered by inspecting the rank of a matrix that is computable from column partitions of the block Toeplitz that implements the discrete convolution, H. A related observation being that the identifiable part of the inputs (if any) is that for which the kernel of H is zero. The fact that the kernel makes transparent which part of the solution is affected by the delay has not been well appreciated and has led to several attempts to reformulate the equations with the objective of attaining full rank by time shifting, all proposals introducing approximation in the general case of m inputs and r outputs due to inevitable Markov Parameter truncation [8], [9], [10]. The reconstruction algorithm presented in this paper, designated as the segmented deconvolution reconstruction (SDR) scheme, takes advantage of the structure of the kernel and offers exact results.
The paper is organized as follows: Section 2 sets some notation. Section 3 presents the input output relations and derives the necessary and sufficient conditions for identifiability. Section 4 derives the stability condition of the sequential deconvolution and examines conditioning. Section 5 looks at conditioning along the frequency line and Section 6 summarizes the previous discussions as guidelines for a reconstruction scheme designated as SDR. Section 7 presents some observations on the likely effect of model error and Section 8 shows numerical results from simulations and from an experimental blind test where SDR was used to reconstruct an input applied to an aluminum cantilever beam.
Section snippets
Preliminaries
After spatial discretization the equations of dynamic equilibrium for a LTI representation, assuming viscous dissipation, can be written aswhere is the displacement vector, n is number of degrees of freedom (DOF), M, Cdam, are the symmetric mass, damping and stiffness matrices, is a vector describing the spatial distribution of the forces, are the unknown force histories and dots indicate differentiation. For simplicity we do not
Deconvolution
Accepting that the system and the inputs are finite dimensional the input-output relation can be obtained by following the recurrence in Eqs. (4), (5) and one getswhere
Stacking the inputs and outputs in columns gives
Or, with obvious notationwhere , is the observability block of order , total number of time steps and is the total number of time
Sequential deconvolution
Application of Eq. (19) to long duration sequences is computationally problematic because the dimensions of H grow with the number of time steps. The obvious solution to this issue is to implement the deconvolution in a window that slides along the time axis. In this instance two parameters have to be selected: the size of the window and the rate of advance,, where this “p” is the “p” in Eq. (19). Assume for now that λ and p are selected so the p identified inputs are not affected by the
Conditioning on the frequency line
A frequency examination provides a useful to guide to the formulation of the inference model. Here we obtain information on conditioning along the frequency line by inspecting the Cramer-Rao-Lower Bound (CRLB) of the attainable variance in the frequency domain. In doing this we take the simplest forward path by performing the analysis on the premise that the outputs are observed over all time. If the deterministic model for the observation is and the deviations are jointly
Segmented deconvolution reconstruction (SDR)
For convenience we synthesize the previous discussions in a computational approach labeled as the Segmented Deconvolution Reconstruction (SDR) scheme.
Off-Line
- 1)
Decide on the time step Δt and formulate the discrete time model. If the guidelines following Eq. (47) are followed the best conditioning that can be realized without regularization is attained.
- 2)
Select {p and } and possibly β and NSR (if the singular values of H are to be truncated) so stability is satisfied (Eq. 26). Typically p=1 and λ
Model error
As a final section prior to the numerical part of the paper we take a quick look at the likely effect of model error on the estimated inputs. Assuming zero initial conditions one haswhere the subscripts T and N are used to suggest “true” and “nominal” (or model). From Eq. (47) one gets (after neglecting higher order terms)
In the SISO scenario Eq. (49) states that the relative error in the estimated input is the product of the difference
Numerical illustrations
Let the system be a prismatic rod of length Lr area A, density ρ and modulus of elasticity E. We think of this rod as being in the horizontal plane with one end free and the other fixed. The wave propagation speed is and thus, if the distance from an input to the location of an output is the dead time is /c0. In some consistent set of units we take A=4, Lr=10, E=100 and , so c0=10.
Conclusions
The paper shows that the sequential deconvolution of inputs from measured outputs is computationally scalable and can be carried out in identical fashion independently of whether the I-O arrangement is collocated or not. A key issue is ensuring that the process is numerically stable and an expression to check the stability condition is given. The expression for the error in the inputs as a function of the noise sequence and the error in the initial condition provided contains all the
Acknowledgments
The research in this paper was carried out with support from NSF Grant CMMI-1000391, this support is gratefully acknowledged. Thanks are due to E. Hernandez and K. Erazo for the data and the model provided for the blind test and to the University of Trento in Italy, for the support provided to the second writer.
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