Modeling Hierarchy of Self-Sensing Actuators

As shown in Sect. 6.9.2 in small-signal range the operators Ts and Fa can be approximated by the linear system equations 

Here the coefficients 7e, 7s = 7a, and 7m correspond to the small-signal capacitance C, the effective piezoelectric charge constant dp and the inverse of the small-signal stiffness cp for a piezoelectric transducer and to the small-signal inductance L, the effective magnetostrictive constant du and the inverse of the small-signal stiffness cm for a magnetostrictive transducer, respectively. 

In this linear case the inverse operator (6.58) can be derived analytically by an evaluation of (6.66) and (6.67). Then the linear reconstruction model 

All these applications have confirmed the principle of a self-sensing actuator, they have also shown that the linear reconstruction model (6.68) and (6.69) is restricted for small amplitudes of the voltage and force. Furthermore the bridge circuit is strongly affected by external disturbances e. g. from temperature leading to a wrong evaluation of the sensory information. 

A further step to extend the validity of the model beyond the small-signal range is a description of the operators Fs and Fa in (6.55) and (6.56) with smooth nonlinear multidimensional characteristics 

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