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This paper presents an integrated sensor/actuator device with multi-input and multi-output designed on the basis of a standard control representation called a distributed port-Hamiltonian system. The device is made from soft material called an ionic polymer-metal composite (IPMC). The IPMC consists of a base film of a polyelectrolyte gel and a double layer of plated metal electrodes. The electrodes of the experimental IPMC are sectioned, and it is implemented as a control system with four pairs of inputs/outputs. We stabilize the system, and detect changes in dynamics by using the control representation.

^{+} or tetraethyl-ammonium ions TEA^{+} [^{6} times [

Ionic polymer metal composite (IPMC).

IPMC consists of three physical systems, ^{+} or TEA^{+}) and water molecules in the gel are transferred to the side of the negative electrode by the potential between the electrodes. This side of the gel is swollen by the transfer. As a result, the swelling mechanically bends the whole film (^{+})–100 s (TEA^{+}) [

Mechanism for IPMC.

This paper presents an experimental example of a micro-mechanical construction incorporating controls. Micro-devices used in sub-centimeter environments are affected by the viscosity of the medium; therefore, they are subject to nonlinearities and must be treated as distributed parameter systems when control models are constructed for them. However, the design of such controls generally involves nonlinear systems and PDEs. An ordinary PDE control, for instance, assigns a state to the system domain. Thus, this method is based on analytical solutions to the systems.

On the other hand,

Moreover, DPH systems satisfy a power balance equation between the change in energy distributed throughout the system domain inside a boundary and the energy on the boundary. We can use this power balance for passivity-based boundary controls. This paper presents two experimental examples,

This section is devoted to explaining the experimental IPMC system and its control model.

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., [

Sectioned IPMC with Multiple Inputs and Multiple Outputs.

The spatial dimensions of the IPMC are 51 mm (length) × 19.2 mm (width) × 0.18 mm (thickness) (

Functions of Sectioned IPMC.

Experimental System.

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 [

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 [

On the other hand, film structures are modeled in terms of two-dimensional PDEs,

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

where

where we have defined the variables:

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 [

where we have defined the variables at the boundaries

The first term in Equation (4) is equivalent to the time variation of the Hamiltonian (with a minus sign),

We stabilize the IPMC and detect changes in its dynamics in terms of its power balance.

The pair

The input voltage to the first actuator and the sharer displacements

Experimental Results for Stabilization.

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,

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

Experimental Results on Detecting Dynamical Changes (obstacle).

Experimental Results on Detecting Dynamical Changes (water).

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

This paper presented experimental results on a multi-input/multi-output integrated ionic polymer metal composite (MIMO-IPMC) controlled by the passivity-based controls having a distributed port-Hamiltonian (DPH) system representation. The sectioned electrodes in the experimental IPMC had four inputs and four outputs. The IPMC was modeled with an Euler-Bernoulli equation, and the model was transformed into a DPH system with four pairs of collocated inputs and outputs for passivity-based controls. We applied two passivity-based controls to the experimental IPMC,

The DPH system representation can be applied to multi-physical and multi-scale systems including dissipative elements [

This work was supported by Grants-in-Aid for Young Scientists (B) (Nos. 19760298 and 19760169), and Scientific Research (C) (No. 19560435) from the Ministry of Education, Culture, Sports, Science and Technology and by the Japan-France Integrated Action Program (SAKURA) (No. 15994PF) of the JSPS and a Grant-in-Aid from the French Ministry of Foreign Affairs.

The authors would like to thank Kazuhiro Tanaka for his editing assistance.

Let us consider the Hamiltonian

where

where

The second term of the integrand in the second equality of (A2) can be transformed:

where we have used the Stokes theorem to change the spatial integrals into boundary terms, and we have applied integration by parts to the first term of each equation. Substituting (A3) into (A2), we obtain the relation:

The integral term of the first equality in (A4) must vanish, because it is the system equation itself.

Therefore, we obtain the power balance Equation (4) from