Linear Kalman filter provides for optimal state estimates, in the minimum variance sense, under the conditions that the system state vector satisfies a linear continuous-time differential equation, or a discrete-time difference equation, driven by the Gaussian random excitation, while the observation vector is linearly dependent upon the system states, and is corrupted by the additive Gaussian noise (see Chap. 7 ). As mentioned in Sect. 8.3 , if the actual values of the system parameters and noise covariance are different from those used in the estimation, then the filter is suboptimal and, in some instances, may diverge. Therefore, state estimation could be improved by using an adaptive filtering technique. In general, adaptive filters prone to be nonlinear. Multiple model (MM) estimation, is an approach that allows for any possible values of system parameters and noise levels, and represents a type of adaptive and nonlinear filtering technique. This approach is presented in Sect. 9.1. In this sense, the extended Kalman filterKalman filterextended works well if the system nonlinearities are not too severe, because otherwise the Taylor series approximations fail, and if the various random vectors are approximately Gaussian distributed (see, Sect. 8.2 ). Moreover, recalculating the Jacobian matrices at every time is computationally expensive. Furthermore, it is too difficult to find the Jacobian matrices analytically in some applications, and numerical approximations of the Jacobian matrices are needed. Additionally, since the Kalman gain matrix depends on data, the stability of the extended Kalman filter is not guaranteed, while the theoretical analysisMatrixanalysis of the filter behaviors is difficult. In this sense, the extended Kalman filter does not guarantee unbiased state estimates. Finally, the calculated error covariances do not necessarily represent the true one, and the analysis of these effects is also very hard. Thus, improved state estimates could be obtained from the second or higher-degree filters that retain more terms in the Taylor series expansions than the extended Kalman filter. However, a statistically linearized Kalman filter generally performs as well or better than such a filter. Therefore, the statistical linearization method represents a viable alternative solution to the extended Kalman filter and the higher-order filters, since it generally produces a better approximation of the system nonlinearity than the Taylor series expansion method. Here the linear approximation of the system nonlinearity is used and, analogously to the estimation problem, the statistical mean-square error criterion is minimized to calculate the underlying coefficients. This, in turn, assumes the pdf of nonlinearity random argument to be known in advance, and the Gaussian pdf is adopted frequently. In addition, the statistical linear approximation can often be made for an adopted pdf in such a manner that the calculated coefficients provide for a more accurate result, in the statistical sense than the truncated Taylors series of a high order. Therefore, the statistical linearization method has the potential advantage for designing suboptimal nonlinear filters of approximate minimum variance type (see, Sect. 9.2). In addition, particle filtering is a new approach, which is generally applicable. It covers nonlinear, non-Gaussian continuous systems, but also discrete-time systems and hybrid, or mixed systems (see, Sect. 9.4). Moreover, a filter midway between the extended Kalman filter and particle one is the unscented Kalman filter. By analogy to the Kalman filter, assuming Gaussian densitiesDensityGaussian for the system states, the expectation and the covariance matrix may be represented by means of a number of samples, that are then used to calculate the effects of nonlinear system dynamics on these quantities. Unlike the particle filter, these samples are not randomly selected (see, Sect. 9.3). Analogously to the particle filter, transformed samples through the system nonlinear function are used further to reconstruct the underlying expectation and covariance matrix. However, such reconstruction is much more accurate than the approximation that is observed by means of the truncated Taylor series expansion.