Linear equivalent system

There is no closed-form solution for the auto-correlation function or the power spectral density function of this nonlinear system. Therefore, an equivalent linear system is utilized to obtain the approximated mean of the spectral density estimator. Multiplying Equation (3.76) with x(t - t) and taking expectation yields ... 

Figure 3.20 shows the response spectrum for 5/o = 0.08 s. Even though the theoretical mean spectrum of the equivalent linear system agrees well with the spectral density estimator computed using the measurement, the positions of their peaks are slightly different. Therefore, for nonlinear systems subjected to moderate to strong excitation, a more accurate computational method for the response spectrum is necessary. An alternative is to obtain the theoretical spectrum by simulation but this is a computationally demanding task. 

The modal frequencies of the building are also identified. They correspond to the equivalent linear system of the building since this structure may exhibit nonlinear behavior under this level of excitation. Figure 3.29 shows the variation of the modal frequencies with the associated plus and minus three standard derivations ( 3a) confidence intervals during the typhoon Kammuri. This interval includes a probability of 99.7% for the equivalent modal frequency falling in this range since the posterior PDF is approximately Gaussian. When the intensity of the excitation... 

The capability to calculate low-order equivalent linear systems from time domain simulations of SOFC models using the ARX algorithm has been established. After the SOFC model was created, it was reduced to transfer fimctions using the ARX algorithm thus, the transfer function (reduced-order model) exhibited the same dynamic response as the original SOFC model. 

The difference between the free energy change dM at an arbitrary state Xh and that at the stationary state X/" under the conditions of given Ai, Bi, dAi, dSj, and dXj is the excess work for box i in this equivalent linear system... 

In a non-linear system the driving force is the potential difference between state Xi and a reference state X, which is the stationary state of the equivalent linear system at the specified value of Xi. Thus from (5.8) and (5.9) we have... 

Next we analyse chemical reaction systems with autocatalytic steps in which the kinetic reaction terms may be non-linear functions of the concentrations of the intermediate species. We have in mind, once again, a reaction mechanism as shown in (4.11). We require that the number of X molecules changes by 1 or 0 in each elementary reaction step, and similarly for Y. For each set of [Xi, Yi) we can construct at each instance a thermodynamically and kinet-ically equivalent system the mapping from the non-linear to the equivalent linear system is unique. The linear equivalent system is chosen as shown in (5.26), but now the coefficients satisfy the relations given in Table 5.1. 

The differential excess work for the equivalent linear system is given in (5.29), and the instantaneous differential excess work for the non-linear system is... 

Since the non-linear system is indistinguishable from the instantaneously equivalent linear system, we have... 

There is, moreover, an extremely important point to stress here. We need only observe the response vector (y(t), t [0,T] and evaluate (3.3) with x(s) replaced by y(s). Thus, in the white noise case, the actual form of the non-linearity does not have to be known. Indeed, we can obtain our statistically equivalent linear system without any assumptions on the nature of the true non -linearity. All the information that is required will reside in the response, y(t), for the white noise excited nonlinear system. 

The reference concentrations x y ) approach the steady state concentrations for each stationary state of the system. Furthermore, they are stationary state concentrations in the thermodynamically and kinetically indistinguishable linear system corresponding to the nonlinear model. There is an unique transformation (mapping) from the nonlinear system for each of the values of (xj, t/j) to the instantaneously equivalent linear system [31, 35,36]. 

This heuristic local search method is the online version of the half-or-double optimization method in Yuen et al. (2007). If the updated noise parameter vectors of ten consecutive time steps remain unchanged, the training process will be terminated. However, this training process is enforced to be not shorter than one fundamental period of the underlying dynamical system (or its equivalent linear system). This one-period requirement ensures that sufficient information can be gained from the data for the preliminary estimation of the augmented state vector. 

Determination of the matrices Cg and Kg of the equivalent linear system in terms of the identification parameters of x. 

After finding cOeq and Seq, the RMS levels of response of the equivalent linear system can be obtained from... 

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