Traveling wave ultrasonic motors are actuators relying on piezoelectric ceramics that combine many advantageous features, such as high stalling torque, fast response, compactness, and magnetic resonance compatibility. However, they suffer from nonlinear dynamics, loaddependent dead zones, and the difficulty to control low speeds. In this paper, we present a novel second-order model for traveling wave ultrasonic motors. It is based on a dry friction driving principle and features dead zone effects. Based on the model, a two-input sliding mode controller is designed. It controls both phase difference and frequency of the traveling wave, without the necessity of implementing a signum function. With this controller, the state-of-the-art is extended to the position control case, while at the same time using fine-grained phase difference control for low velocities. Moreover, we show global uniform asymptotic stability for bounded disturbances and that velocity jumps do not appear when the control domains of phase difference and frequency are switched. Finally, both the model and the controller are evaluated via simulations and experiments that include the response to a position step input under various opposing torques. Index Terms-Control engineering, piezoelectric resonators, switched systems.0278-0046

<div>The definition of the reactive power in nonsinusoidal conditions has been a controversial topic since the experiment published by C. P. Steinmetz in 1892. Where among Electrical Engineers there is no consensus on how to compute the reactive power in non-sinusoidal conditions. Therefore, in this paper it is used the mathematical concept of a Hilbert space, namely the L₂ and l₂ spaces, and the Least effort problem to obtain the definition of the generalized reactive power. Then, the obtained definition of the generalized reactive power is compared against some of the main power theories. Finally, it is used that definition to study the relation between the Tellegen’s theorem with the Least effort problem and a novel application is presented.</div>

<div>The definition of the reactive power in nonsinusoidal conditions has been a controversial topic since the experiment published by C. P. Steinmetz in 1892. Where among Electrical Engineers there is no consensus on how to compute the reactive power in non-sinusoidal conditions. Therefore, in this paper it is used the mathematical concept of a Hilbert space, namely the L₂ and l₂ spaces, and the Least effort problem to obtain the definition of the generalized reactive power. Then, the obtained definition of the generalized reactive power is compared against some of the main power theories. Finally, it is used that definition to study the relation between the Tellegen’s theorem with the Least effort problem and a novel application is presented.</div>

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