## 2 Observer-Based Estimation
### Kalman Filter  
The KF, along with its variants (extended Kalman filter and unscented Kalman filter), addresses the general problem of trying to estimate the state vector \(\mathbf{x}\in\Re^{n}\) of a discrete-time controlled process that is governed by the generic set of equations [126]:  
\[\mathbf{x}_{\mathbf{k}}=f(\mathbf{x}_{\mathbf{k-1}},\mathbf{u}_{\mathbf{k-1}},\mathbf{w}_{\mathbf{k-1}}) \tag{9}\] \[\mathbf{z}_{\mathbf{k}}=h(\mathbf{x}_{\mathbf{k}},\mathbf{v}_{\mathbf{k}})\]  
where \(\mathbf{u}\) is the system input, \(\mathbf{z}\) is the system output (i.e., the measured variables), and \(\mathbf{v}\) and \(\mathbf{w}\) are respectively the process and measurement noises. The process noise is introduced to account for the (unavoidable) difference between the dynamics of the actual system and the model used to represent it. Ideally, should a model be exactly representative of a system, \(\mathbf{w}\) would be zero. Similarly, measurement errors are accounted for in \(\mathbf{v}\), which depends on the accuracy of measurements (e.g., the quality of the sensors used). The noises are assumed to have a Gaussian distribution with zero mean and covariances \(\mathbf{Q}\) and \(\mathbf{R}\), respectively, for \(\mathbf{w}\) and \(\mathbf{v}\). The idea of a KF is that the state vector \(\mathbf{x}_{\mathbf{k}}\) could be estimated in two independent ways: (i) using the first equation in Formula (9), i.e., the dynamic evolution of the system; or (ii) using the second equation in Formula (9), inverting it to work out \(\mathbf{x}_{\mathbf{k}}\). However, both estimations would be affected by an error, depending on (i) the reliability of the model (parameters, unmodeled effects etc. ); and (ii) the reliability of the measures. These reliabilities depend respectively on \(\mathbf{Q}\) and \(\mathbf{R}\). The KF simply computes a weighted average between the two estimations of the state vector, the weight depending on \(\mathbf{Q}\) and \(\mathbf{R}\), which is calculated in order to minimise the covariance of the estimation error. The estimation error is the difference between the estimated state and its (unknown) actual value [126]. If the system is linear, then Formula (9) can be written in the form:  
\[\mathbf{x}_{\mathbf{k}}=A\mathbf{x}_{\mathbf{k-1}}+F\mathbf{u}_{\mathbf{k-1}}+\mathbf{w}_{\mathbf{k-1}} \tag{10}\] \[\mathbf{z}_{\mathbf{k}}=H\mathbf{x}_{\mathbf{k}}+\mathbf{v}_{\mathbf{k}}\]  
Denoting with \(\mathbf{\dot{x_{\mathbf{k}}}}^{-}\) the state estimate at step \(\mathbf{k}\) according to the dynamic evolution of the system (the first equation in Formula (10)), the KF estimation is:  
\[\mathbf{\dot{x_{\mathbf{k}}}}=\mathbf{\dot{x_{\mathbf{k}}}}^{-}+K(\mathbf{z}_{\mathbf{k}}-H\mathbf{\dot{x_ {\mathbf{k}}}}^{-}) \tag{11}\]  
One form of \(K\) is given in [126] by [PERSON] et al., and depends on \(\mathbf{Q}\) and \(\mathbf{R}\). For example, if \(\mathbf{R}\) approaches zero, i.e., in the hypothesis of an extremely reliable measurement, then \(K\approx H^{-1}\); hence, \(\mathbf{\dot{x_{\mathbf{k}}}}\approx H^{-1}\mathbf{z}_{\mathbf{k}}\). If the system is not linear, i.e., it can be described only in the general form (as in Formula (9)), it can be linearised and written in a form similar to Formula (10). Such approach is known as an extended Kalman filter (EKF). In [20], a comparison between EKF and SMO has been performed considering a non-linear heavy vehicle model, and the results showed that both methods were sufficiently accurate, but SMO has been preferred since it requires less input measurements. However, the KF is very simple to be used and implemented; moreover, its robustness, stability, and ability to deal with input and measurement noise make it the most used observer for VSA estimation [18]. This statement is also supported by [25], where after a comparison between the EKF, LO, and SMO concerning the VSA estimation using a non-linear single-track model, EKF achieved a smaller estimation error than the LO and SMO.