Sensitivity study of dynamic systems using polynomial chaos
Introduction
Very often, the equations of a model involve unknown parameters which must be estimated from experimental data. A number of parameters are estimated with more or less precision, which may lead to unacceptable uncertainty on the model output. Among all the parameters, however, only few have a small or insignificant influence on the model response and therefore do not need to be determined precisely. On the other hand, some parameters are decisive for the model response and thus influence its uncertainty significantly. These parameters may require additional measurement data in order to be estimated with relatively high accuracy. To prepare and plan the experiments, it is necessary to distinguish the parameters with an insignificant influence on the response uncertainty, so as to set them at their nominal value in their interval of variation, thanks to the sensitivity analysis. Numerous studies have focused on the sensitivity analysis for static non-linear models, for example [1], [2], [3], [4], [5], [6], [7]. The approaches may be local or global. Local approaches help to determine the impact of a small parameter variation around a nominal value [8]. Global approaches also allow the determination of the same impact but by varying the parameter in its entire range of variation. Global methods are often based on the analysis of the output variance and are known as ANOVA (ANalysis Of VAriance) techniques [9], [10], [7]. More recently, sensitivity moment-independent methods have been used, where emulation model is used to compute density-based sensitivity measure [11]. The emulator is the one of [12].
The model function is split into a sum of functions of increasing dimension [7]. This decomposition, known as High Dimensional Model Representation (HDMR), performs the separation of the effects of different input parameters, which are transmitted in the decomposition of the variance. The present study exclusively focuses on global approaches. In order to quantify the contribution of a parameter to the output variance, a sensitivity index is calculated, often analytically when the model function is known and relatively simple. However, some models may be complex with a high number of parameters so that analytical calculations of the sensitivity indices become time consuming or even impossible. It is therefore necessary to estimate them [3], [13], [14], [15], [7]. Very often, they are computed using Monte Carlo simulations, but for computationally demanding models, this can become intractable. To overcome this drawback, the model of interest is replaced by an analytical approximation, called metamodel, for example, by polynomials which are less expensive. The sensitivity indices are then obtained straightforwardly from the algebraic expression of the coefficients of the polynomial expansion. The polynomial chaos (PC) decomposition is an example of such metamodelling. The PC expansion appeared in the 1930s as an effective means to represent stochastic processes in mechanics [16]. It is based on a probabilistic framework and represents amounts from stochastic spectral expansions of orthogonal polynomials [17], [18]. It has recently been used in an original manner for sensitivity analysis purposes in [1], [19], [20], [21], [22], [23]. The great advantage of PC-based sensitivity approaches is that the full randomness of the response is contained in the set of the expansion coefficients.
On the other hand, the analysis of influential parameters is also important for dynamic models since most physical systems (biological, mechanical, electrical and so on) can be described by differential equations. Very few approaches have been proposed in the literature for the sensitivity analysis of dynamic models and the proposed ones are based on local derivatives or on one-at-a-time approaches [24], [25]. However, for some applications, mechanical or biological ones for instance, it can be of great importance to consider the entire uncertainty range of parameters since they can vary within large intervals depending on their meaning. Another advantage of global sensitivity analysis is that the sensitivity estimates of individual parameters are evaluated while all the other parameters are varied. In this way, the relative variability of each parameter is taken into account, thus revealing any existing interactions.
The global sensitivity analysis for dynamic models is addressed in this paper. In [26], [27], the PC expansion for stochastic differential equations has been studied to represent the model output and to get its statistic properties, but the parameter sensitivity has not been dealt with. Based on these studies, an original approach using the PC decomposition of the output is investigated here, to calculate the parameter sensitivity for dynamic models. At each time instant, the PC coefficients of the decomposition convey the parameter sensitivity and then a sensitivity function of each parameter can be obtained from the algebraic expression of the coefficients. The PC coefficients are determined either by regression or projection techniques which have the advantage of being non-intrusive methods. The proposed approach is illustrated with the well-known mass-spring-damper and DC motor systems.
The outline of this paper is as follows. The sensitivity functions for dynamic systems are presented in Section 2 and the PC expansion for the output of a differential equation in Section 3. Section 4 is focused on the determination of the PC coefficients. Section 5 proposes a PC-based approach to the estimation of the sensitivity functions, which approach is summed up in Section 6. Finally, Section 7 presents an analytical test case to show the convergence of the numerical results. Moreover, the provided approach is applied on some representative dynamic physical systems.
Section snippets
Global sensitivity analysis
Consider the following stochastic differential equation:where is a linear or non-linear differential operator and with , i=1, …, n, the n unknown parameters, considered as uniformly random and independent variables, defined on the unit cube K. The stochastic variable is used to indicate the randomness of the input variable p. For the sake of simplicity, will be omitted in the following and p stands for . The solution y=y(t,p),
PC expansion of the model output
The beginnings of the polynomial chaos can be traced back to Wiener [16]. He suggested that the spectral expansion of Hermite polynomials in terms of Gaussian random variables could be used to represent certain stochastic processes. It was then shown that the homogeneous chaos expansion could be used to approximate any function in the Hilbert space of square-integrable functions. As a result, any second-order random variable with finite variance can be represented by a spectral expansion
Determination of the PC coefficients
Once the structure of the PC expansion is obtained for the model output, as explained in the previous section, the deterministic coefficients must be computed. There exist two types of methods for this, the intrusive and the non-intrusive ones. Historically, the intrusive Stochastic Galerkin (SG) method was used [27], [28], [29]. A Galerkin projection helps to minimize the error of the truncated expansion, and all the resulting coupled equations are solved for the coefficients of the
PC-based sensitivity functions
Consider the PC expansion of the model output (17) with the coefficients computed with (19), (23). Due to the orthogonality of the basis, it can be shown that the mean and the variance of the output are, respectively, given byAs explained in [34], to compute the sensitivity function Si(t) (10) of parameter pi, it is necessary to reorder expansion (17) in order to separate the different contributions – single and collective – of each
Summary of the proposed approach
The computation of the sensitivity functions of the parameters for the dynamic model (2), requires the following steps:
- (a)
PC decomposition
- 1.
Polynomial type
Choose the polynomial type associated to the parameter distribution, see Table 1.
If necessary, map the parameters onto the required interval given in Table 1, in order to get an orthogonal basis for the chosen polynomials.
- 2.
PC degree
Set the required degree d for the PC.
- 3.
PC truncation
Compute the number M+1 of terms in expansion (17), according to (15).
- 1.
Application examples
A simple test case is provided in order to compute analytically the sensitivity indices and to show the convergence of the numerical results obtained using the PC decomposition. Then, two examples of dynamic system are given to illustrate the contribution of this work.
Conclusion
This paper has presented the problem of the global sensitivity of dynamic systems which proves important when parametric uncertainty falls within a wide range for several parameters and when the sensitivity changes significantly from one operation point to another.
The polynomial chaos expansion is used to calculate the sensitivity indices of the system. The expansion coefficients are calculated either through the technique of regression or projection, which are non-intrusive methods. As the
Acknowledgments
The authors wish to thank Michael Baudin (Consortium Scilab) for his valuable guidance in the use of software NISP.
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