Hk is the observation model matrix, which maps the state vector space into the measurements vector space. x^k? is the a priori state estimate vector, resulting from the prediction stage. x^k+ is the a posteriori state estimate vector, derived from the measurements update stage. zk is the measurements vector obtained from the system sensors. Kk is the optimal Kalman gain matrix, which weights the importance of the innovation that introduces the measurements vector zk in the update stage. Pk? is the a priori state covariance matrix, which provides the a priori estimation error covariance after the prediction stage. Pk+ is the a posteriori state covariance matrix, containing the a posteriori estimation error covariance, given after the update stage.
Q is the process noise covariance matrix of the prediction stage noise, which somehow ponders the weight of the process estimates. R is the observation noise covariance matrix of the update stage noise, which in a way ponders the degree of confidence in each one of the measurements. The relative weights become
The vibration of the bridge when it is mounted in a violin is a dynamic contact vibration with two interfaces: strings-bridge, and bridge feet-top plate. According to the Hertzian contact vibration theory, the contact stiffness changing can cause the bridge resonance frequency shift and the resonance amplitude changing for each vibration mode. The influence of the dynamic contact stiffness on the bridge mobility has not been studied comprehensively up to now.The dynamic contact vibration of a rigid punch on an elastic medium has been studied by many researchers [9�C12].
Most researchers considered the contact to be equivalent to an elastic-spring support and adopted the Hertzian static-contact stiffness. Different dynamic Hertzian contact models based on a nonlinear mass-spring-damping system have been presented to investigate its nonlinear vibration theoretically and experimentally. In [11], an analytical model based on a linear-elastic theory for dynamic contact stiffness of a vibrating rigid sphere contacting a semi-infinite viscoelastic solid was proposed. The dynamic contact-pressure distribution at the interface between the rigid sphere and the viscoelastic solid was deduced first. Then, the dynamic contact stiffness at the interface was deduced from the approximate dynamic contact boundary conditions for displacements.
In [12], experimental results showed that the contact stiffness Brefeldin_A not only affects the resonance frequency position, but also the amplitude of the resonance.When a bridge is fitted in a violin, the contact stiffness in the two contact interfaces of strings-bridge and bridge feet-top plate is affected by a variety of factors such as the force generated by the strings, the materials and the surface roughness of both the bridge and the top plate, and the area of the contact surfaces.