Optimal observability-based modelling, design and characterization of piezoelectric microactuators

This optimization tool is based on a multiobjective genetic algorithm approach, which searches the optimal distribution of available building blocks. The resulting structure is thus an assembly of elementary passive, active or sensitive compliant blocks chosen in a library (figure 2). Some blocks can act as pure mechanical stiffeners, whereas others integrate actuation functions and others sensing functions. The optimization problem appoints not only topological specifications, but also an optimal set of boundary conditions (fixed frame location, contacts, end-effectors, etc), dimensions and materials, depending on some optimization criteria selected by the designer.

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The library of building blocks is composed of 36 building block elements, made of beam assembly. These are sufficient to build a large variety of topologies. Moreover, the block feasibility related to fabrication process constraints can also be taken into account during the problem specification. The blocks present various topologies, as shown in figure 2. Their advantage is that they can furnish multiple coupled DOFs, thus generating more complex movements with only one building block than a method which would have consisted of optimizing directly the location of the elementary beams of the truss-like structure. Thus, the function of the building block method is to save computational time during the optimization process by reducing the complexity of the problem combinatorics. It should also be noted that the calculation of the different matrices of each valued block is done once only at the beginning of the optimal design problem (before running the genetic algorithm), which saves running time.

The modelling of active and sensitive piezoelectric building blocks is based on a 2D finite element method (FEM) as presented in this section.

2.2.1. FE formulation of the piezoelectric beam.

In the optimization procedure, the computation of different criteria requires the FE model of each block of the library. To obtain the FE formulation of the piezoelectric blocks, a model of a piezoelectric beam exploits direct and inverse piezoelectricity effects for sensing and actuation purposes respectively (figure 3). In the current version of FlexIn, a block is used either as sensor or as actuator but not for the two functions simultaneously. Nevertheless, it could change in future version of FlexIn.

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The matrices characterizing each block (mass, stiffness, electromechanical coupling and electric capacity) are computed at the beginning of the optimization algorithm, by the association of the corresponding beam matrices in the global coordinate system of the structure. The compliant mechanism is assumed to undergo structural planar deformation, mainly due to flexion, leading us to consider Navier–Bernoulli beam type FEs. Exploiting the transverse effect of piezoelectricity, longitudinal deformation along the L dimension is generated under the transverse electric field along the e_p thickness. Using the least action Hamilton principle for electromechanical systems [17], a piezoelectric beam model is established as a set of two equations, one for the actuation mode (inverse effect) and one for the sensing mode of working (direct effect),

$$\begin{eqnarray} &&{\mathbf{M}}_{\mathbf{b}}\ddot{{\eta }_{\mathbf{b}}}+{\mathbf{K}}_{\mathbf{b}}{\eta }_{\mathbf{b}}={\mathbf{G}}_{\mathbf{b}}{\boldsymbol{\Phi}}_{\mathbf{b}}+{\mathbf{Fr}}_{\mathbf{b}} \end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray} &&{\mathbf{q}}_{\mathbf{b}}={\mathbf{G}}_{\mathbf{b}}^{t}{\eta }_{\mathbf{ b}}+{\mathbf{C}}_{\mathbf{b}}^{t}{\boldsymbol{\Phi}}_{\mathbf{ b}} \end{eqnarray} \tag{ 2 }$$

where ${\eta }_{\mathbf{b}}=({u}_{A},{v}_{A},{\omega }_{A},{u}_{B},{v}_{B},{\omega }_{B})_{{R}_{\mathrm{p}}}^{t}$ is the nodal displacement vector in the beam's coordinate system R_p = (A,x_p,y_p,z_p) (figure 4). M_b,K_b,G_b and C_b are respectively the mass, stiffness, electromechanical coupling and electric capacitive beam matrices. Corresponding matrices are given in the appendix section. Φ_b = [φ₁,φ₂]^t and q_b = [q₁,q₂]^t are respectively the vectors of electric potentials and electric charges on the upper and lower boundaries of the piezoelectric beam.

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Matrix G_b in (1) induces piezoelectric loads, which make the actuator beam expand (or contract) proportionally to the external applied voltage φ₁ − φ₂. The force vector Fr_b, is due to external mechanical loads at the extremities of the beam ${\mathbf{Fr}}_{\mathbf{b}}=\left (\begin{array}{@{}cccccc@{}} \displaystyle {\mathbf{R}}_{\mathbf{A}}^{\mathbf{x}},&\displaystyle {\mathbf{R}}_{\mathbf{ A}}^{\mathbf{y}},&\displaystyle {\mathbf{H}}_{\mathbf{ A}}^{\mathbf{z}},&\displaystyle {\mathbf{R}}_{\mathbf{ B}}^{\mathbf{x}},&\displaystyle {\mathbf{R}}_{\mathbf{ B}}^{\mathbf{y}},&\displaystyle {\mathbf{H}}_{\mathbf{ B}}^{\mathbf{z}} \end{array}\right )_{{R}_{\mathrm{p}}}^{t}$(see figure 4). The displacement field is related to the corresponding node displacement η_b by means of the shape functions, computed according to Navier–Bernoulli kinematic assumptions. The relationship (2) is used to compute the electric charges induced on the sensing beam electrodes according to the direct effect of piezoelectricity. It must be noted that experimental measurement of electric charges requires an electronic charge–voltage converter using operational amplifiers. In ideal mode, the upper and lower electrodes of the sensor piezoelectric beam are then considered short-circuited, as would be the case when they are connected to the operational amplifier. A zero-voltage difference φ₁ − φ₂ can therefore be assumed in the following and relationship (2) becomes ${\mathbf{q}}_{\mathbf{b}}={\mathbf{G}}_{\mathbf{b}}^{t}{\eta }_{\mathbf{b}}$ for sensing blocks. This assumption ensures that the sensing and actuation functions of the piezoelectric beams are decoupled.

2.2.2. FE model of the FlexIn piezoelectric structures.

The mass, stiffness, electromechanical coupling and electric capacity matrices of each block are computed, considering every combination of the discrete values allowed for the structural optimization variables, i.e. the material and the size of the blocks. Each block results from the association of the corresponding beam matrices in the global coordinate system of the structure. Thus, the calculation of the different matrices of each valued block is done once only at the beginning of the optimal design problem (before running the genetic algorithm), which saves computing time.

Each flexible structure synthesized using blocks by FlexIn is defined as a finite-dimensional linear system modelled as,

$$\begin{eqnarray} &&\begin{array}{@{}c@{}} \displaystyle {\mathbf{M}}_{\mathbf{g}}\ddot{{\eta }_{\mathbf{g}}}+{\mathbf{K}}_{\mathbf{g}}{\eta }_{\mathbf{g}}={\mathbf{E}}_{\mathbf{g}}\mathbf{u}\\ \displaystyle {\mathbf{y}}_{\mathbf{co}}=\delta ={\mathbf{F}}_{\mathbf{g}}{\eta }_{\mathbf{g}}\\ \displaystyle {\mathbf{y}}_{\mathbf{ob}}=\mathbf{q}={\mathbf{L}}_{\mathbf{g}}{\eta }_{\mathbf{g}}. \end{array} \end{eqnarray} \tag{ 3 }$$

The foregoing second-order differential matrix equations represent the undamped dynamic behaviour of the device. M_g and K_g are respectively the structural mass and stiffness matrices, arising from the assembly of all block matrices which constitute the general structure.

Considering the integers p,s, and r, as respectively the numbers of DOFs of the structure, number of inputs (i.e. actuators), and number of tip displacement outputs, η_g is then the p × 1 nodal displacement vector and u is the s × 1 input vector. The p × s input matrix E_g reflects the location of the actuated DOFs, while y_co is the r × 1 controlled output vector representing the output tip displacement δ through the output displacement (r × p)-matrix F_g. The third equation expresses the electric charges (y_ob = q) obtained by the integrated sensing function from the direct piezoelectric effect. Note that L_g is the 1 × p single output matrix pointing out the placement of the piezoelectric sensor in the structure. Hence, it is important to note that there are two kinds of output: controlled output variable y_co, and observed output variables y_ob, that are different variables. This point does not correspond to the usual case of collocated control (output is the observed variable).

The harmonic solutions of the first equation in (3) give the eigenvector matrix Ψ and natural frequencies ω_i of the system. Details of the modal representation computation can be found in [13],

$$\begin{eqnarray} &&\begin{array}{@{}c@{}} \displaystyle \ddot{\mathbf{z}}+\mathbf{diag}(2{\xi }_{i}{\omega }_{i})\dot {\mathbf{z}}+\mathbf{diag}({\omega }_{i}^{2})\mathbf{z}={\boldsymbol{\Psi}}^{t}{\mathbf{E}}_{\mathbf{g}}\mathbf{u}\\ \displaystyle {\mathbf{y}}_{\mathbf{co}}=\delta ={\mathbf{F}}_{\mathbf{g}}\boldsymbol{\Psi}\mathbf{z}\\ \displaystyle {\mathbf{y}}_{\mathbf{ob}}=\mathbf{q}={\mathbf{L}}_{\mathbf{g}}\boldsymbol{\Psi}\mathbf{z} \end{array} \end{eqnarray} \tag{ 4 }$$

where z is the p × 1 modal displacement vector, and ξ_i is the ith modal damping ratio introduced using Basil's hypothesis. For a p-size degree of freedom mechanical system, a 2 × p-size state vector must be defined. A typically 2p × 1 state vector x (see [18] for advantages of this choice for flexible structures), consists of modal velocities and frequency-weighted modal displacements,

$$\begin{eqnarray} \mathbf{x}=\left (\begin{array}{@{}ccccc@{}} \displaystyle {\dot {z}}_{1}&\displaystyle {\omega }_{1}{z}_{1}&\displaystyle \cdots &\displaystyle {\dot {z}}_{\mathrm{p}}&\displaystyle {\omega }_{\mathrm{p}}{z}_{\mathrm{p}} \end{array}\right )^{t}. \end{eqnarray} \tag{ 5 }$$

Since controlled and observed output vectors are not the same, the modal state-space representation can be written as follows:

$$\begin{eqnarray} &&\begin{array}{@{}c@{}} \displaystyle \dot {\mathbf{x}}=\mathbf{Ax}+\mathbf{Bu},\\ \displaystyle {\mathbf{y}}_{\mathbf{co}}={\mathbf{C}}_{\mathbf{co}}\mathbf{x},\\ \displaystyle {\mathbf{y}}_{\mathbf{ob}}={\mathbf{C}}_{\mathbf{ob}}\mathbf{x} \end{array} \end{eqnarray} \tag{ 6 }$$

which leads to two matrices triplets (A,B,C_co) and (A,B,C_ob) related to control and observation state-space models. The matrices take the forms A = diag(A₁,...,A_p), $\mathbf{B}=({\mathbf{B}}_{\mathbf{1}}^{t},\ldots ,{\mathbf{B}}_{\mathbf{p}}^{t})^{t}$, C_co = (C_co1,...,C_cop), and C_ob = (C_ob1,...,C_obp), with, for i = 1,...,p,

$$\begin{eqnarray} {\mathbf{A}}_{\mathbf{i}}=\left [\begin{array}{@{}cc@{}} \displaystyle -2{\zeta }_{i}{\omega }_{i}&\displaystyle -{\omega }_{i}\\ \displaystyle {\omega }_{i}&\displaystyle 0 \end{array}\right ] \end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray} &&{\mathbf{B}}_{\mathbf{i}}=\left [\begin{array}{@{}c@{}} \displaystyle {\mathbf{b}}_{\mathbf{i}}\\ \displaystyle \mathbf{0} \end{array}\right ] \end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray} {\mathbf{C}}_{{\mathbf{co}}_{\mathbf{i}}}=\left [\begin{array}{@{}cc@{}} \displaystyle \mathbf{0}&\displaystyle \frac{{\mathbf{c}}_{{\mathbf{co}}_{\mathbf{i}}}}{{\omega }_{i}} \end{array}\right ] \end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray} {\mathbf{C}}_{{\mathbf{ob}}_{\mathbf{i}}}=\left [\begin{array}{@{}cc@{}} \displaystyle \mathbf{0}&\displaystyle \frac{{\mathbf{c}}_{{\mathbf{ob}}_{\mathbf{i}}}}{{\omega }_{i}} \end{array}\right ] \end{eqnarray} \tag{ 10 }$$

where b_i,c_coi, and c_obi are the ith components of Ψ^tE_g,F_gΨ and L_gΨ respectively. Matrix A is a function of the structural parameters (eigen frequencies and damping ratio), whereas matrix B depends only on the location of actuated DOF, matrix C_co on the location of desired displacement output, and matrix C_ob on the location of integrated piezoelectric sensors.

The observability Gramian (W_ob) between state x and measured output q is found to be convenient to characterize the mode observability by the mean of the measured electric charge q. Its energetic and geometric interpretations are emphasized in [5, 19]. For stable A matrix, W_ob is the algebraic solution of the following Lyapunov equation:

$$\begin{eqnarray} &&{\mathbf{A}}^{t}{\mathbf{W}}_{\mathbf{ ob}}+{\mathbf{W}}_{\mathbf{ob}}\mathbf{A}+{\mathbf{C}}_{\mathbf{ob}}^{t}{\mathbf{C}}_{\mathbf{ ob}}=\mathbf{0}. \end{eqnarray} \tag{ 11 }$$

Assuming that the damping ratios are infinitely small and the natural frequencies well spaced, which is widely accepted for flexible structures, the block diagonal forms of the (A,C_ob) couple can be exploited to give a closed-form analytical expression of the modal observability Gramian matrix [20]. The solution is then diagonal and equal to,

$$\begin{eqnarray} {\mathbf{W}}_{\mathbf{ob}}=\mathbf{diag}\left (\begin{array}{@{}ccc@{}} \displaystyle {\mathbf{W}}_{{\mathbf{ob}}_{\mathbf{11}}},&\displaystyle \ldots ,&\displaystyle {\mathbf{W}}_{{\mathbf{ob}}_{\mathbf{pp}}} \end{array}\right ) \end{eqnarray} \tag{ 12 }$$

with, for i = 1,...,p,

$$\begin{eqnarray} &&{\mathbf{W}}_{{\mathbf{ob}}_{\mathbf{ii}}}=\frac{{\gamma }_{{q}_{i i}}}{4{\xi }_{i}{\omega }_{i}^{3}}{\mathbf{I}}_{\mathbf{2}}={\alpha }_{i}{\mathbf{I}}_{\mathbf{2}} \end{eqnarray} \tag{ 13 }$$

where ${\gamma }_{{q}_{i i}}={\mathbf{c}}_{{\mathbf{ob}}_{\mathbf{i}}}^{t}{\mathbf{c}}_{{\mathbf{ob}}_{\mathbf{i}}}$, and I₂ is the 2 × 2 identity matrix. For a given mode (ξ_i,ω_i),γ_qii scalars represent the way the ith mode is seen through the piezoelectric sensor blocks.

The dominant vibrational mode contributions, which govern the dynamics of the tip deflection, have to be fully observable from the integrated sensing area of the device, in order to efficiently reconstruct and control the tip deflection. This will give the kind of transfer function sketched in figure 5: few dominant modes (high peaks) at low frequency in the transfer function between the tip position δ and the voltage input control u. Moreover, these few dominant modes have to be strongly coupled—located at the same frequencies—with the dominant modes of the transfer function between the electric charge q in the sensing areas and the voltage input control u.

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These dominant modes are symbolized by high Hankel singular values (HSVs), which defines the balanced Gramian matrix W_eδ of the (A,B,C_co) system as follows:

$$\begin{eqnarray} &&{\mathbf{W}}_{\mathbf{c}\delta }={\mathbf{W}}_{\mathbf{o}\delta }={\mathbf{W}}_{\mathbf{e}\delta }=\mathbf{diag}\left ({\sigma }_{i}\right ) \end{eqnarray} \tag{ 14 }$$

where σ_i are the HSVs of the (A,B,C_co) system. Balanced Gramian is therefore a useful tool to quantify the joint controllability and observability of the system. When damping ratios decrease to zero, it can be shown that modal state coordinates are approximately balanced, and the approximate ith HSV for a flexible structure in this asymptotic situation is given by,

$$\begin{eqnarray} &&{\sigma }_{i}=\frac{\sqrt{{\mathbf{c}}_{{\mathbf{co}}_{\mathbf{i}}}^{t}{\mathbf{c}}_{{\mathbf{co}}_{\mathbf{i}}}{\mathbf{b}}_{\mathbf{i}}{\mathbf{b}}_{\mathbf{i}}^{t}}}{4{\xi }_{i}{\omega }_{i}^{2}}. \end{eqnarray} \tag{ 15 }$$

The HSV describes the degree of the corresponding modal state's input–output energy flow through the system [20].

As emphasized previously, the design of active structures with both integrated actuation and sensing capabilities is non-intuitive and the designer is faced with two main issues, as follows.

• (i)  
  A reduced model of the structure must be synthesized, which includes the few dominant low-frequency modes, without destabilizing the system by rejecting the residual modes (i.e. high roll-off after the dominant modes as sketched in figure 5). Resonance peak amplitudes must be maximized in the frequency bandwidth to increase authority control on the dominant modes of the deflection transfer. In contrast, the amplitudes of resonance peaks after cutoff frequency must be minimized to increase gain margin and to limit mode destabilization in this area (spillover phenomenon).
• (ii)  
  If the dominant modes are not all well observed by the sensing area of the flexible structure, the reconstruction of the deflection δ will not be guaranteed in an optimal way by the observer.

Hence, we propose a new criterion which solves these issues. Forcing the optimal structure to have k first dominant modes in the δ/u frequency response function solves the first point. Guaranteeing a high level of observability of these dominant modes by means of the electric charges q solves the second point. The algorithm 1 expresses this procedure.

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In this algorithm, k is the number of the first dominant modes specified by the designer. ${\sigma }_{i=1\rightarrow k}^{\mathrm{min}}$ and ${\sigma }_{j=k+1\rightarrow p}^{\mathrm{max}}$ are respectively the level of the least dominant modes belonging to the first k modes and the level of the most dominant modes belonging to the residual modes. thv is a threshold value specified by the designer. As can be seen in figure 5, the condition $\frac{{\sigma }_{i=1\rightarrow k}^{\mathrm{min}}}{{\sigma }_{j=k+1\rightarrow p}^{\mathrm{max}}}\geq \mathit{thv}$ means that the k first modes dominate by at least thv times the residual modes. The threshold value is used as a preliminary test condition in order to avoid the complex computation of the numerical criterion J^k,n,thv for each structure. A penalty value is assigned to structures that do not meet the condition requirement. These would lead to structures with poor criterion value and these will not be considered in the following generation of synthesized structures. In contrast, when the threshold inequality condition is fulfilled, the way the k first modes dominate the residual modes is emphasized through the coefficient $\frac{{\sigma }_{i=1\rightarrow k}^{\mathrm{min}}}{{\sigma }_{j=k+1\rightarrow p}^{\mathrm{max}}}$ in the criterion formula. The threshold value can be chosen depending on the designer requirements for his own application. From a practical point of view, this tends to increase gain margin and to limit modal spillover phenomena. Based on the authors' experience, a compromise value of threshold of about 3 is recommended. Using a higher value for thv can possibly eliminate too many candidates in the population of the genetic algorithm. This would lead to a variety of poor designs. However, in contrast, choosing too low a value of thv can lead to residual modes with high authority.

${\alpha }_{i}=\frac{{\gamma }_{{q}_{i i}}}{4{\xi }_{i}{\omega }_{i}^{3}}$ corresponds to the coefficient of the ith observability modal Gramian W_obii and $({\sigma }_{i}/{\sigma }_{j}^{\mathrm{max}})^{n}$ is a weighing ratio ∈[0,1]. When this ratio is close to 1, the corresponding ith mode is a dominant mode belonging to the first k modes. The exponent n is a tuning parameter to accelerate the convergence towards ${\sigma }_{i}^{\mathrm{max}}$ (to emphasize the most dominant modes). Thus, the corresponding α_i is privileged compared to other mode observability. Maximizing this criterion favours flexible structures with vibrational modes where good observability of δ coincides with its dominant modes.

The monolithic structure is seen as the actuator of a left-arm gripper and is supposed to be made of a single piezoelectric material PIC151 from PI PiezoCeramic Technology [21]. The structure topology is considered to have a maximal size of 15 mm × 18 mm, and a constant thickness of 200 μm. This device includes either passive, active or sensitive blocks inside a 2 × 3 mesh (see figure 6). The number of active (resp. sensitive) blocks in the structure can vary between 1 and 4. These are optimally chosen in the library displayed in figure 2. When external voltages are applied to the block electrodes, the output node (tip of the structure) has to move along the x-axis. To evaluate static mechanical criteria, the electric voltage value applied between upper and lower electrodes of active blocks is 200 V. The size ratio of the blocks can vary as b_max/b_min∈〚1;2〛 and a_max/a_min∈〚1;2〛 (a and b geometric parameters are displayed in figure 6). The number of blocked nodes is between 1 and 3 among the locations permitted by the problem specifications, which are reported in figure 6. The boundary conditions of possibly 1, 2 or 3 fixed nodes are exclusively devoted to the nodes that are reported in the bottom line of the mesh structure. Thus, the number and choice of the fixed nodes are considered as optimization parameters.

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Finally, three numerical criteria to be maximized with FlexIn are considered, as follows:

• J¹: free mechanical displacement δ_x at the output node in x-direction when a constant voltage is applied to the system.
• J²: amplitude of the sensing electric charges induced on the sensing piezoelectric blocks.
• J^k,n,thv: modal observability of the mechanism output δ by the observed sensitive block charges q. In this example, we chose k = 2,n = 2 and thv = 3.

Note that J¹ and J² consider only the static behaviour of the structure, while J^k,n,thv considers its dynamic behaviour.

The optimization procedure was started according to the previous specifications. When the genetic algorithm does not find any new pseudo-optimum during 200 subsequent generations, the best compromises are kept and can be found on Pareto fronts. The designer can choose from among these solutions, after studying their static and dynamic behaviours. In our case, the structure chosen appears to have the best dynamic desired behaviour according to J³, while static criteria remain relatively good. This structure, displayed in figure 7, allows a tip displacement of 7 μm, and induces sensing charges around 1.9 × 10⁻⁹ C under a 200 V input voltage.

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Figure 8 reports the simulated dynamic behaviour (Bode diagrams) of the δ/u control transfer function and the q/u observation transfer function. According to design specifications, these results show that maximizing the J^2,2,3 criteria has led to the occurrence of two dominant resonant modes in the low-frequency spectrum of the deflection response, and these latter are perfectly observed by the electric charges.

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The microsystem prototype is clamped, and placed on x–y–z micropositioning linear stages, which are manually operated. Figure 9 shows a photograph of the experimental setup. The piezoelectric actuator requires high voltage (about ±100 V) to provide micrometric deflection. Thus, the device is supplied with a linear power amplifier (amplification ratio=  50). As seen in figure 10, the device is computer-controlled using Matlab–Simulink software and a NI Labview PXI board (sampling frequency f_e = 20 kHz). Output displacement of the tip of the piezoelectric structure is directly measured along the x-axis using a 0.01 μm-resolution Keyence laser sensor.

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An electronic charge–voltage converter is used to measure the electric charges induced in the sensing areas. The analog outputs of the laser sensor and charge amplifiers are directly wired to a 4th-order low-pass anti-aliasing filter. Corresponding double Sallen-Key electronic circuits are tuned to provide more than 75 dB attenuation at the $\frac{{f}_{e}}{2}=1 0~\mathrm{kHz}$ Shannon frequency. This permits us to filter accurately high frequency noise and high frequency unmodelled dynamics.

In ideal operating mode, the upper and lower electrodes of the sensing piezoelectric beams are considered short-circuited to respect the optimization design hypothesis. This assumption ensures that sensing piezoelectric beams are only related to the local deformation of the microactuator. The electronic signal conditioner is based on an operational amplifier in a current integrator operating mode (see figure 11). This analog conditioner for the charge measurement on the sensing electrode pattern uses a resistor R_cr in parallel to the feedback capacitance C. The role of the resistor is to discharge the capacitance C to avoid saturation effects at low frequencies. This ensures the proper functioning of the electric charge integration. The measured output voltage V_s is related to the electric charge q in s-domain as follows:

$$\begin{eqnarray} &&{V}_{s}(s)=-\frac{1}{C}\left (\frac{s}{s+\frac{1}{R C}}\right )Q(s). \end{eqnarray} \tag{ 16 }$$

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Micropower precision operational amplifiers (LF412ACN, National Semiconductor Corporation) are used because they present a very low voltage drift with respect to the temperature (max. =± 0.4 μV °C⁻¹), and a small offset voltage (max. =± 150 μV max). The performance of this model in terms of stability and accuracy and its high bandwidth (130 kHz) make it suitable as a signal conditioner. In this case, it is able to measure nC electric charges. The C feedback capacity is chosen equal to 2.2 nF, a value close to the experimental capacity of the proprioceptive sensing area of the structure. The value of the resistance R_cr = 10 MΩ must be large enough so as not to disrupt the integration function of the circuit, but not so large so as to avoid the discharge current passing through it. The choice also depends on the cutoff frequency of the system, which should stay low to avoid filtering of low frequencies.

The vibrational dynamics are recorded experimentally by applying to the piezoactuator a low-amplitude sine input U of increasing frequency. At low amplitude and relatively high frequencies, the creep and hysteretical behaviour of PZT can be neglected [14]. Using a spectrum analyser device (HP3562A), a Bode diagram is recorded and the corresponding transfer is directly identified in Laplace domain (see figure 12). As expected by FlexIn optimization, the first two resonance modes are dominant over the following vibrational modes: this is the first point of interest in considering the criterion J^2,2,3 for the design. Let us also note that the pole/zero alternate pattern is kept into this desired spectrum of interest. An identification process is thus performed considering only these first two modes in a reduced model. For identification, we consider a second-order modal transfer expansion,

$$\begin{eqnarray} &&{G}_{\mathrm{co}}(s)=\sum _{i=1}^{2}\frac{{k}_{{c}_{i}}}{1+\frac{2{\xi }_{i}}{{\omega }_{n i}}s+\frac{1}{{\omega }_{n i}^{2}}{s}^{2}}=\frac{{N}_{\mathrm{co}}(s)}{D(s)}. \end{eqnarray} \tag{ 17 }$$

This transfer function is related to the state-space model according to the following relationship:

$$\begin{eqnarray} &&{G}_{\mathrm{co}}(s)={C}_{\mathrm{co}}\left (s I-A\right )^{-1}B. \end{eqnarray} \tag{ 18 }$$

The damping ratios ξ₁ = 2.0% and ξ₂ = 2.5% have been calculated from the measured quality factor at −3 dB on the Bode diagram, according to ${Q}_{i}^{-3~\mathrm{dB}}\approx \frac{1}{2{\xi }_{i}}$, while values of the natural pulsations ω_n1 and ω_n2 are easily estimated owing to the measured resonance frequencies. Identified natural frequency values are reported in table 1.

Static gain k_c1 and k_c2 are computed to respect the frequency of the first antiresonance (between the two first resonances) and the whole system static gain. This identification procedure gives,

$$\begin{eqnarray} &&\begin{array}{@{}l@{}} \displaystyle \fl {N}_{\mathrm{co}}(s)=2.3 1 1\times 1{0}^{-9}{s}^{2}+2.2 2\times 1{0}^{-6}s+0.0 5 0 2\\ \displaystyle \fl D(s)=8.8 3 7\times 1{0}^{-1 8}{s}^{4}+1.5 5 3\times 1{0}^{-1 4}{s}^{3}\\ \displaystyle +~6.8 0 2\times 1{0}^{-9}{s}^{2}+4.9 3 4\times 1{0}^{-6}s+1. \end{array} \end{eqnarray} \tag{ 19 }$$

The identified response model G_co(s) is compared with the experimental behaviour in frequency (figure 12) and time (figure 13) domains.

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The experimental protocol used for the identification of the piezoelectric sensing dynamics behaviour is similar to the one used for the identification of the deflection dynamics: the spectrum analyser sends an excitation voltage to the actuation area of the smart structure, while the output of the electronic measurement circuit is recorded for the frequency response analysis. Measurements are reported in figure 14.

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Taking into account the criterion J^k,n,vs, the optimization design procedure of the system is effective: the modal observability by the charge transfer function G_ob(s) of the first two dominant modes of the deflection transfer function G_co(s) is important (see figure 15). The roll-off at higher frequencies allows us to bring the identification of the G_ob(s) transfer function to a model of order 4, with a vibrational dynamics D(s) similar to that corresponding to the actuation one,

$$\begin{eqnarray} &&{G}_{\mathrm{ob}}(s)=\sum _{i=1}^{2}\frac{{k}_{{o}_{i}}}{1+\frac{2{\xi }_{i}}{{\omega }_{n i}}s+\frac{1}{{\omega }_{n i}^{2}}{s}^{2}}=\frac{{N}_{\mathrm{ob}}(s)}{D(s)} \end{eqnarray} \tag{ 20 }$$

or equivalently,

$$\begin{eqnarray} &&{G}_{\mathrm{ob}}(s)={C}_{\mathrm{ob}}\left (s I-A\right )^{-1}B. \end{eqnarray} \tag{ 21 }$$

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The computation of the static gain values k_o1 and k_o2 are based on the identification of the zero (antiresonance) of the experimental charge transfer function and static gain, so that,

$$\begin{eqnarray} {N}_{\mathrm{ob}}(s)&=&-1.4 0 7\times 1{0}^{-1 1}{s}^{2}\nonumber\\ &&-1.5 0 6\times 1{0}^{-8}s-0.0 0 7. \end{eqnarray} \tag{ 22 }$$

With the same damping ratios ξ₁ and ξ₂ as for the deflection transfer function D(s) of G_co(s), the identified response model G_ob(s) well matches the experimental response, in frequency (figure 14), and time (figure 16) domains.

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The reduced model can then be used with confidence owing to the optimal criterion J^k,n,thv. The proposed optimization procedure is therefore relevant to the design of compliant structures that will be easily observable and controllable in an integrated way.