## 2 The Interacting Multiple Model Unscented Kalman Filter  
The unscented Kalman filtering deals with the case governed by the nonlinear stochastic difference equations:  
\[\mathbf{x}_{k+1}=f(\mathbf{x}_{k},k)+\mathbf{w}_{k} \tag{1a}\] \[\mathbf{z}_{k}=h(\mathbf{x}_{k},k)+\mathbf{v}_{k} \tag{1b}\]  
where the state vector \(\mathbf{x}_{k}\in\Re^{n}\), process noise vector \(\mathbf{w}_{k}\in\Re^{n}\), measurement vector \(\mathbf{z}_{k}\in\Re^{m}\), and measurement noise vector \(\mathbf{v}_{k}\in\Re^{m}\). The vectors \(\mathbf{w}_{k}\) and \(\mathbf{v}_{k}\) are zero mean Gaussian white sequences having zero cross-correlation with each other:  
\[E[\mathbf{w}_{k}\mathbf{w}_{i}^{\mathrm{T}}]=\mathbf{Q}_{k}\delta_{ik}\ ;\ E[\mathbf{v}_{k} \mathbf{v}_{i}^{\mathrm{T}}]=\mathbf{R}_{k}\delta_{ik}\ ;\ E[\mathbf{w}_{k} \mathbf{v}_{i}^{\mathrm{T}}]=\mathbf{0}\qquad\text{for all $i$ and $k$}\]  
where \(E[\cdot]\) represents expectation, and superscript \"T\" denotes matrix transpose, \(\mathbf{Q}_{k}\) is the process noise covariance matrix and \(\mathbf{R}_{k}\) is the measurement noise covariance matrix. The symbol \(\delta_{ik}\) stands for the Kronecker delta function:  
\[\delta_{ik}= \begin{cases}1,&i=k\\ 0,&i\
eq k\end{cases} \]  
The IMM approach takes into account a set of models to represent the system behaviour patterns or system model. The estimator carries out a'soft switching' among various models by the model probability. The overall estimates is obtained by a combination of the estimates from the filters running in parallel based on the individual models that match the system modes. The measurements could be obtained from one or more sensors, and the model-matched filters could be linear or nonlinear. The algorithm of IMM-nonlinear filters is introduced to deal with the noise uncertainty and system nonlinearity simultaneously. Let a general system for multiple models in discrete time be described by:  
\[\mathbf{x}_{k+1}=f(\mathbf{x}_{k},k,M_{k})+\mathbf{w}(\mathbf{x}_{k},M_{k}) \tag{2a}\] \[\mathbf{z}_{k}=h(\mathbf{x}_{k},k,M_{k})+\mathbf{v}_{k}(\mathbf{x}_{k},M_{k}) \tag{2b}\]  
where \(f(\cdot)\) and \(g(\cdot)\) are the parameterized state transition and measurement functions, \(\mathbf{x}_{k}\) and \(\mathbf{z}_{k}\) are the dynamical state and measurement of the system in model \(M_{k}\), and the system itself is a Markov chain, \(expansion for any nonlinear system. One of the remarkable merits is that the overall computational complexity of the UKF is the same as that of the EKF [8]. The first step in the UKF is to sample the prior state distribution, _i.e._, generate the sigma points through the UT, which is a method for calculating the statistics of a random variable which undergoes a nonlinear transformation. The basic premise is that to approximate a probability distribution is easier than to approximate an arbitrary nonlinear transformation. The samples are propagated through true nonlinear equations, and the linearization of the model is not required. Suppose the mean \(\mathbf{\bar{x}}\) and covariance \(\mathbf{P}\) of vector \(\mathbf{x}\) are known, a set of deterministic vector called sigma points can then be found. The ensemble mean and covariance of the sigma points are equal to \(\mathbf{\bar{x}}\) and \(\mathbf{P}\). The nonlinear function \(\mathbf{y}=f(\mathbf{x})\) is applied to each deterministic vector to obtain transformed vectors. The ensemble mean and covariance of the transformed vectors will give a good estimate of the true mean and covariance of \(\mathbf{y}\), which is the key to the unscented transformation. The UKF approach estimates are expected to be closer to the true values than the EKF approach.