Page - 49 - in Integration of Advanced Driver Assistance Systems on Full-Vehicle Level - Parametrization of an Adaptive Cruise Control System Based on Test Drives

4.2. Side Slip Angle Estimation According to [DB06], the estimated state variable xˆ can be determined using the expres- sion ˙ˆx=A xˆ+b δ+L   y−cT xˆ︸ ︷︷ ︸ vωz−vωˆz   , (4.7) where A is the system matrix and b is the input vector of the system, both of which are described in eq. (B.8). The variable δ is the measured steering angle of the front wheel. The measurement of the vehicle yaw rate is described in eq. (4.2). The vector L is the observer gain vector that has to be found. For a robust observer, the real part of the poles of the characteristic equation of the observer must be negative. The characteristic equation reads det ( λ∗ I−(A−L cT))= 0, (4.8) where the vectorλ∗ consists of the desired Eigenvalues of the observer, and I is the iden- tity matrix. According to [KD97], the poles are placed atλ∗= [ −200 −2.4 fcy+rcym vvx ]T , where fcy and rcy are the lateral stiffnesses of one front and one rear wheel including influences of the suspension,m is the vehicle mass, and vvx is the longitudinal vehicle speed. The vector L can be determined using the method of Ackermann, [DB06]. The equation reads L= ( p0 I+p1 A+ · ··+pn−1 An−1 +An ) Q−1B      0 ... 0 1      , (4.9) where pi are the coefficients of the desired characteristic polynomial reading p0 +p1λ+ · ··+pn−1λn−1 +λn= n∏ i=1 (λ−λ∗i). (4.10) For the given system, the observer gain vector results in L= [ 1 a21 ( a211 +a12a21−a11(λ∗1 +λ∗2)+λ∗1λ∗2 ) a11 +a22−λ∗1−λ∗2 ] . (4.11) For a good observer performance, the parameters have to be tuned to achieve a good correlation between the measured and simulated data. In the described algorithm for the side slipangleobserver, the tuningparametersare the twoconstants in thedefinition of λ∗ (200 and 2.4), and the lateral tyre stiffnesses fcy and rcy. The test vehicle has the same tyres mounted on the front and rear axles. Therefore, the pure tyre stiffness has to be the same for the front and the rear tyres, assuming equal wheel loads. In general, the kinematics and elasto-kinematics of the vehicle suspension make the front tyres steer out of the curve while the rear tyres steer into the curve, thereby producing anundersteeringbehaviourduringcornering, [MW04]. This results ina smaller absolute 49