IPMC acts as an actuator when we apply a voltage to electrodes and as a sensor when we measure the counteraction voltage between electrodes without the voltage input. The sensor mode has been studied in, e.g., [12,13,14,15,16,17,18,19,20]. The experimental IPMC was implemented as a control system with four distributed inputs and four distributed outputs. The pattern of inputs/outputs for the electrode was made by laser cutting (Figure 3).

The spatial dimensions of the IPMC are 51 mm (length) × 19.2 mm (width) × 0.18 mm (thickness) (Figure 4). The sectioned electrodes consist of four actuator areas and four sensor areas, and they are connected to one side of the IPMC that is used as an interface connector to the control and measurement system. The control and measurement system is composed of a personal computer, an A/D convertor, a D/A convertor, amplifiers, and a laser measurement device (Figure 5). The inputs and outputs are processed by the personal computer. The signals for the inputs/outputs are transmitted through the convertors and the amplifiers. The interface side of the IPMC is fixed at in Figure 4. We positioned the IPMC horizontally with respect to the surface of the ground. We measured the share displacement vertically at each position for 1 with a laser measurement device to verify the sensor output of the IPMC. We assumed that the spatial difference in velocity for would be in proportion to the output voltage of the -th sensor from the left, i.e., , where is a constant and . Accordingly, we can approximate the spatial partial derivatives of the share displacements from the spatial differences.

Flexible structures are modeled in terms of PDEs. The mechanical characteristics of IPMC can be modeled, e.g., by using the Euler-Bernoulli beam model, the Timoshenko beam model, and the model with large deformations [21]. These beam models can be expressed as DPH systems [9]. The overall motion of IPMC is nonlinear; therefore, the large deformation beam would the best choice for the motion analysis. However, the large deformation beam is expressed by three physical variables, i.e., the axial displacement , shear displacement , and rotation , and we cannot directly obtain the output with respect to the rotation from the experimental IPMC. Thus, we decided to employ the Euler-Bernoulli beam model instead as a real-time control model. The Euler-Bernoulli beam model is actually a reduced large deformation beam made by assuming and making a simplification [9], wherein the subscript means the derivative with respect to the spatial coordinate . However, we must measure (higher order) spatial derivatives of instead of .

The Euler-Bernoulli beam model includes up to fourth-order spatial derivatives. Such higher-order PDEs are modeled in terms of higher-order DPH systems [22]. In the case of IPMC, the DPH system is only second order. The higher-order DPH systems have non-trivial higher-order boundary ports that are derived from the integration by parts formula (see Appendice A.1 and Appendice A.2). These special inputs/outputs pairs never appear in the standard (first order) DPH systems; therefore, they offer the possibility of applying passivity-based controls to IPMC. We verified this possibility by means of the experimentation described in Section 3.

On the other hand, film structures are modeled in terms of two-dimensional PDEs, i.e., flexible plate equations. The film model is used for, e.g., controlling the shape of the film. However, one-dimensional PDEs are appropriate control models in the case of the experimental IPMC, because, the control degree of the IPMC is unidirectional regardless of the dimension. Hence, we chose the DPH representation having a one dimensional Euler-Bernoulli beam equation as the control model.

Let us consider the one-dimensional Euler-Bernoulli beam model on the interval :

(1)

where is the length of the beam, is the mass per unit length, is the flexural stiffness, is the share displacement, is the time coordinate, and is the spatial coordinate. Here, the subscript of means partial derivatives with respect to or . Equation (1) can be transformed into a second order DPH system:

(2)

where we have defined the variables:

(3)

Note that the first row in Equation (2) is equivalent to that in Equation (1), and the second row in Equation (2) is an identity [7]. The advantage of this system representation is that it always satisfies the power balance equation at an instant in time:

(4)

where we have defined the variables at the boundaries ,

and (5)

(6)

The first term in Equation (4) is equivalent to the time variation of the Hamiltonian (with a minus sign), i.e., the total energy change in the system (see Appendix A.1), and the second and third terms in Equation (4) mean energies flowing across the boundaries, which are derived in terms of Stokes theorem and integration by parts with respect to the first term in Equation (4) (see Appendix A.2). Thus, we can detect the total energy change by calculating the boundary energy flows instead of integrating the first term in Equation (4) over the domain. This is the basic strategy of passivity-based boundary controls of DPH systems.

The pair for and in Equations (5) and (6) means a collocated pair of boundary inputs and outputs for passivity-based controls. Let us consider the pair in Equation (5) for the IPMC. The third-order derivative of the share displacement at can be approximated as , where is the share displacement at virtual position in Figure 4. We regard as the output in Equation (5). Hence, we send the feedback input to the first actuator distributed on the interval , where the input voltage of the actuator is determined by for a constant , and is feedback gain. We applied a band-pass filter for Hz to the output voltages, because the above assumption is valid around that frequency range.

The input voltage to the first actuator and the sharer displacements and are shown in Figure 6. The control input is added to the first actuator after. Two impact disturbances are applied to the tip of the IPMC at and . We can see that the residual vibration in the controlled IPMC after decreases more rapidly than in the uncontrolled IPMC before this time. This means the total energy of the first term in Equation (4) is dissipated through the second and third terms in Equation (4), because the negative feedback applied to boundary variables and acts as a dissipative element.

This section explains how we detect the change in the dynamics of the IPMC in terms of the power balance Equation (4). Time-independent Hamiltonian systems are conservative, i.e., time invariant with respect to energy. However, actual systems have dissipative elements, e.g., viscosity damping or distributed registers. Indeed, the swelling of IPMC relaxes by dissipation, but this effect has not been modeled in Equation (1). If the total energy stored in the system is dissipated inside the system domain, the observed boundary energy flow calculated with the second and third term in Equation (4) will be less than the total energy. Moreover, the change in the dissipation rate can be detected from the difference in observed energies.

We conducted experimental tests in two dissipative situations. The first situation was where one side of the overall motion of the IPMC was blocked around by placing an external object after . The second situation was where the tip of the IPMC was soaked in water and the water level was raised after . We believed these environmental changes might increase the energy dissipated by the IPMC.

Figure 7 and Figure 8 plot the responses of the IPMC to a sinusoidal input of 3 Hz. The responses were detected by the laser measurement device placed at (laser output), and the total boundary energy flows in the first and second sensor areas (total pow1 and total pow2) were calculated from the time integral of the product of input and output in Equation (5). The ranges of the figures have been normalized to be dimensionless. The estimated responses before the time of the change in dissipations are also plotted (estimates).

Figure 7 indicates that dissipation increased when the object came into contact after , and the change in the dissipation rate can be seen as a change in the slopes of the second and third graphs. We can see from Figure 8 that a change in dissipation is detected, because dissipation constantly increases because of the viscous drag of the water after .

It is difficult to detect the above changes solely from the laser outputs, which are the closest data to the real motion of the IPMC, because the amplitudes of the laser outputs are not so stable. We guess that the IPMC flexibly bends because of the obstruction. However, we can clearly detect the changes from the power balance from the change in the slopes calculated from the control model.

The amount of energy dissipated in Figure 7, i.e., the difference in the slopes of the total measured power and the estimate, is larger than in Figure 8. The movement of the IPMC is completely blocked by the obstacle in the first situation. However, the viscous drag of the water in the second situation does not stop it from moving. Therefore, we can guess that the dissipated energy is much larger in the first situation than in the second situation.