The beauty of the proportional-integral-derivative (PID) algorithm for feedback control is its simplicity and efficiency. Those are the main reasons why PID controller is the most common form of feedback. PID combines the three natural ways of taking into account the error: the actual (proportional), the accumulated (integral), and the predicted (derivative) values; the three gains depend on the magnitude of the error, the time required to eliminate the accumulated error, and the prediction horizon of the error. This paper explores the new meaning of integral and derivative actions, and gains, derived by the consideration of non-integer integration and differentiation orders, i.e., for fractional order PID controllers. The integral term responds with selective memory to the error because of its non-integer order λ , and corresponds to the area of the projection of the error curve onto a plane (it is not the classical area under the error curve). Moreover, for a fractional proportional-integral (PI) controller scheme with automatic reset, both the velocity and the shape of reset can be modified with λ . For its part, the derivative action refers to the predicted future values of the error, but based on different prediction horizons (actually, linear and non-linear extrapolations) depending on the value of the differentiation order, μ . Likewise, in case of a proportional-derivative (PD) structure with a noise filter, the value of μ allows different filtering effects on the error signal to be attained. Similarities and differences between classical and fractional PIDs, as well as illustrative control examples, are given for a best understanding of new possibilities of control with the latter. Examples are given for illustration purposes.
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Ionic polymer-metal composites (IPMCs) are electrically driven materials that undergo bending deformations in the presence of relatively low external voltages, exhibiting a great potential as actuators in applications in soft robotics, microrobotics, and bioengineering, among others. This paper presents an artificial eukaryotic flagellum (AEF) swimming robot made up of IPMC segments for the study of planar wave generation for robot propulsion by single and distributed actuation, i.e., considering the first flagellum link as an actuator or all of them, respectively. The robot comprises three independent and electrically isolated actuators, manufactured over the same 10 mm long IPMC sheet. For control purposes, a dynamic model of the robot is firstly obtained through its frequency response, acquired by experimentally measuring the flagellum tip deflection thanks to an optical laser meter. In particular, two structures are considered for such a model, consisting of a non-integer order integrator in series with a resonant system of both non-integer and integer order. Secondly, the identified models are analyzed and it is concluded that the tip displacement of each actuator or any IPMC point is characterized by the same dynamics, which remains unchanged through the link with mere variations of the gain for low-frequency applications. Based on these results, a controller robust to gain variations is tuned to control link deflection regardless of link length and enabling the implementation of a distributed actuation with the same controller design. Finally, the deflection of each link is analyzed to determine whether an AEF swimming robot based on IPMC is capable of generating a planar wave motion by distributed actuation.
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