# Survey on Modelling and Techniques for Friction Estimation in Automotive Brakes
## 2 Techniques for the Estimation of BLCF in Automotive Disc Brakes
### State Observer for BLCF Estimation
#### 2.5.1 Linear Kalman Filter  
The present work does not dwell on the mathematical subtleties of the Kalman filter theory. For a better understanding, the writer refers the reader to specific textbooks. Here, it is worth noticing that the KF is the minimum variance state estimator for linear dynamic systems with Gaussian noise. It is constituted of two stages: (i) the prediction step, based on the process model; (ii) the update step, based on sensor measurements. The noise on the input variables makes the model of the process less accurate; the noise of the sensors deteriorates the correction provided during the update step. The Kalman filter combines information coming from models and sensors, in the presence of uncertainty, in order to estimate the state variables. Hence, it is possible to define an associated state mean vector \(\hat{x}_{k}\) and covariance matrix \(P_{k}\). The general equation of a linear Kalman filter for the prediction phase can be expressed as:  
\[\hat{x}_{k}=F_{k}\hat{x}_{k-1}+B_{k}u_{k} \tag{8}\]  
\[P_{k}=F_{k}P_{k-1}F_{k}^{T}+Q_{k}. \tag{9}\]  
In the discrete state space formulation of Equations (8) and (9), \(\hat{x}_{k}\) represents the observed state vector at time step \(k\), obtained from the previous best estimate; \(u_{k}\) is simply the input vector. \(F_{k}\) represents the state matrix ruling the dependence between the states at increasing time step, \(B_{k}\) is the input matrix that distributes the input among the states; \(P_{k}\) is the covariance matrix of the state updated with the best previous estimate; \(Q_{k}\) represents the uncertainty that is attributed to the model (prediction part). Equations (8) and (9) are responsible for the information content of the \"model predicted part\" of the Kalman filter; in other words, the model that simulates the process must be transcribed in the illustrated state space formulation. The estimation of the state vector and its covariance matrix are performed by using the available information at the time step \((k-1)\) to feed the predictive model.  
Instead, the general equations for the measurement update are:  
\[K^{\prime}=P_{k}H_{k}^{T}\Big{(}H_{k}P_{k}H_{k}^{T}+R_{k}\Big{)}^{-1} \tag{10}\]  
\[\hat{x}_{k}^{\prime}=\hat{x}_{k}+K^{\prime}(z_{k}-H_{k}\hat{x}_{k}) \tag{11}\]  
\[P_{k}^{\prime}=P_{k}-K^{\prime}H_{k}P_{k}, \tag{12}\]  
where the superscript means that both the state vector and its covariance matrix are updated with the sensor information. \(K^{\prime}\) represents the Kalman gain; \(H_{k}\) is the output matrix that ties the state vector to the outputs \(z_{k}\); and \(R_{k}\) represents the uncertainty that is attributed to the sensors, generally linked to the white noise. Hence, the update step consists of comparing the real output \(z_{k}\) with the estimated output \(H_{k}\hat{x}_{k}\) in order to correct the state vector. It is worth remarking that the matrixes \(R_{k}\) and \(Q_{k}\) are the filter tuning parameters. The latter gives more importance to the prediction part or to the update part and must be tuned in accordance with the quality of the model and measurements. For example, the higher \(Q_{k}\) is, the more the sensors' output will lead the results because, as can be seen from the previous equations, the more the Kalman filter relies on the measurements.