Control System for Multi-Input and Simple-Output Piezoelectric Beam Actuator Based on Macro Fiber Composite

Abstract

A control system for a piezoelectric beam actuator, which had one or two control inputs, was a subject of numerical and laboratory research. The actuator had a prismatic shape with a rectangular cross-section and consisted of one layer of carrying substrate made from PCB-FR4 and two patches of Macro Fiber Composite of P1 type. MFC patches were glued on both sides of the carrying substrate. An article presents a comparison of the control quality of piezoelectric actuator with one control signal (one-input actuator) and the control quality of piezoelectric actuator with two control signals (two-input actuator). An application of two-input actuator led to a reduction of control voltage compared to the control voltage of one-input actuator. The decrease in the maximum voltage was approximately from 30% to 39% in conducted laboratory experiments. An application of two-input actuator causes a reduction of an overshoot value compared to one-input actuator. An application of limit of maximum control voltage leads to a greater decrease of the control quality for one-input actuator compared to two-input actuator.

1. Introduction

A piezoelectric actuator is a device used to convert electric energy into a displacement of mechanical structure or into a force acting of selected structure. The basic part of such an actuator is made from piezoelectric material in which electric energy is converted into strain of this material. Mohith et al. [1] enumerated five groups of traditional structure of piezoelectric actuators: unimorph, bimorph, tube, multi-layer and amplified. The actuators from the first two groups are most often based on beam which is only fixed at one end. The actuator, which the mechanical structure is based on, has been called a piezoelectric cantilever actuator for several dozen years, e.g., in [2].

The piezoelectric cantilever actuator consists of a clamped-free beam, a voltage amplifier and a control system. The cantilever beams are most often constructed from one layer of non-piezoelectric substrate and one or two layers of piezoelectric material. Non-piezoelectric and piezoelectric materials are attached to each other by gluing. Initially, PZT ceramics [3] and PVDF polymers [4] were applied in the construction of this type of actuator. Since the beginning of the twentieth century, the rapid development of research on the control of cantilever actuators, built on the basis of Macro Fiber Composite (MFC), can be noticed. Some of the first research results were presented by Azzouz et al. [5] in this area. In the years that followed, many researchers pointed out advantages of piezoelectric actuators based on MFC in comparison to actuators based on piezoceramics, e.g., Nguyen et al. [6] noticed that the MFC have a better dynamic actuation than the bulk PZT type for range of high frequency.

Macro Fiber Composite (MFC), invented by NASA and commercialized by Smart Material Corp, is a material that consists of piezoceramic fibers with rectangular cross-section, non-piezoelectric epoxy filling the gaps between the fibers, copper electrodes adjacent to both sides of the fibers and polyamide kapton holding the structure together. Dimensions of fibers, epoxy layers, electrodes and polyamide kapton are known. P1 and P2 are two basic types of MFC: the first is dedicated to actuators and the second is used in the energy harvesting process. Structures of these types are varied in terms of the arrangement of the electrodes [7]. MFC P1 is equipped with interdigitated electrodes which gather both positive and negative charges on each side of piezoelectric fibers. A direction of polarization and a direction of stress in the beam are the same thanks to the application of such an electrodes arrangement [8]. The MFC P1 composite was modeled by many scientists. Williams et al. [9] presented for MFC patch the nonlinear tensile and shear stress–strain behavior and Poisson effects using various plastic deformation models. Deraemaeker et al. [10] proposed mixing rules for the determination of longitudinal and transverse piezoelectric coefficients of MFC patches. Zhang et al. [7] presented the FE model which is based on the Reissner–Mindlin hypothesis using linear piezoelectric constitutive equations. Steiger et al. [11] present a FE model, which was used for the effective computation of material constants of MFC. Emad et al. [12] proposed a model of MFC actuator based on the replacing of MFC actuator with an equivalent simple monolithic piezoceramic actuator using two electrodes only. Research is also conducted in the field of optimization of the localization of MFC in the actuator structure, e.g., Padoin et al. [13] proposed an optimum localization of the MFC in beam structure in order to maximize a controllability index. The cantilever actuator structures, containing both MFC and substrate, are most often modeled by the use of a linear motion equation, in which a tip displacement is only included, e.g., [14], or by the use of a linear matrix motion equation, e.g., [15]. The linear equations are expanded by descriptions of nonlinear phenomenon: hysteresis and creep, which appear in a control system of the actuator containing MFC. Hysteresis is modeled by the use of models applied for the actuators based on PZT, e.g., Yang et al. [14] proposed the Bouc-Wen approach and Xu et al. [16] applied the Prandtl-Ishlinskii model. The creep is also modeled by the use of approaches applied to describe piezoelectric ceramics, e.g., Schrock et al. [17] presented the application of the Prandtl-Ishlinskii model for the creep in actuator based on MFC.

Deflection control systems, which contain the cantilever beam actuator based on MFC, A/D board, a control algorithm implemented in a dedicated computer program, and a voltage amplifier, are also intensely researched. A selection of the A/D boards is not dependent on the kind of used piezoelectric material, so A/D boards are the same for the actuators based on PZT and actuators based on MFC. Control algorithms used for actuators based on PZT are also used for the actuators based on MFC: algorithms based on a linear function, e.g., PID [18], algorithms based on a state space, e.g., LQR [15], algorithms based on fuzzy functions, e.g., [19] and algorithms based on genetic methods, e.g., [20].

The significant difference in operation between the control system for actuator based on MFC and the control system for actuator based on PZT ceramics is apparent in the required control voltage values for these actuators. The control system for actuator based on MFC requires significantly higher control voltage in comparison to control system for actuator based on PZT ceramics. The initial depolarizing field for typical PZT ceramic (PZT-5A) is around 500 kV/m [21], which causes the maximum control voltage to be ±150 V for the thickness of the PZT layer equal to the thickness of the MFC layer (0.3 mm). A reducing the thickness of the PZT layer reduces the value of the maximum voltage, e.g., for 0.267 mm it will be ±134 V [22]. The maximum values of control voltage are significantly larger for MFC patch: maximum operational positive voltage is equal to +1500 V and maximum operational negative voltage is equal to −500 V [23]. Higher control voltage generally leads to higher control cost and lower economic efficiency [24]. In addition, it may lower the stability and reliability of the control system in practice because the piezoelectric actuator has the risk of being destroyed by too high control voltage [24]. The higher control cost mainly results from the necessity to use amplifiers that generate higher voltages. Nowadays, the amplifier generating maximum voltage of 1500 V is offered only by Smart Materials Corp. and is significantly more expensive than some of amplifiers of maximum voltage about ±200 V, which are on offer by other companies.

A deflection increase of the piezoelectric actuator (for a given carrying substrate) can only be achieved by an increase of control voltage [24]. Hence, the biggest deflection of cantilever actuator based on MFC may be obtained for +1500 V for each carrying substrate. The actuator containing two MFC patches can achieve a maximum deflection, whose value is the same as for control voltage equal to +1500V, for the use of lower control voltage than +1500 V. However, the application of two MFC patches causes a change of control system because the actuator with two MFC patches has two control inputs. Kumar et al. [25] presented an algorithm which generated one control signal for two PZT patches: for one patch this control signal was positive and for the second it was negative. A similar approach is presented by Jain et al. [26]. Kaci et al. [27] proposed another approach in which two control signals were generated, but one of these signals was a real part of the external force and the second was an imaginary part of this force. Control algorithms, presented in [25,26,27], enable effective control assuming that the cantilever beam is symmetrical so it can be treated as a SISO control object. A prismatic cantilever beam with two MFC patches is symmetrical if one MFC layer is glued on exactly the same as the second MFC layer and if both glued connections have the same properties, e.g., a contact area. In practice, meeting these conditions is difficult.

The novelty of this article is a control algorithm which generates two control signals which are independent of each other. In contrast to the literature approaches presented, the cantilever piezoelectric actuator containing two MFC patches was treated as a MISO (multi input and simple output) control object which allows for effective control regardless of whether both MFC layers are symmetrically glued on both sides of the carrying substrate layer. Obtaining the required control quality was realized by the use of two integrating action, which can be created independently for each control signal. A comparison of the control quality of piezoelectric actuator with one input and the control quality of piezoelectric actuator with two inputs was not presented before in the literature.

2. Synthesis of Control Algorithm

The subject of mathematical modeling is a cantilever bimorph beam and a control algorithm synthesis.

2.1. Simulation Model of a Cantilever Beam Containing Two MFC Patches

The finite element method (FEM) was used to establish the displacement of a tip of the composite cantilever beam. The cantilever beam motion can be written as [15]:

$$M_g\stackrel{••}{d}+C_g\stackrel{•}{d}+K_{g}d={{\rm E}}_1{\mathrm{V}}_1+E_2{\mathrm{V}}_2$$

(1)

where: M_g is the global mass matrix, C_g is the global damping matrix, K_g is the global stiffness matrix, E₁ and E₂ are the matrixes of distribution of the force generated by MFC patch no 1 and MFC patch no 2, V₁ and V₂ are the control voltages of MFC patch no 1 and MFC patch no 2, d is the vector of nodal displacements containing its vertical (w) and rotational displacements (φ). Mass and stiffness global matrixes were obtained by combining mass and stiffness local matrixes for all elements, into which the beam structure was divided. On the basis of an arrangement of the MFC patches in beam structure (Figure 1), five elements were defined. The beam structure, on which the active area of MFC patches was glued, was a one element (l_b2). The beam structure, on which the passive area of MFC patches was glued, was divided into two elements (l_b1 and l_b3). The beam structure without MFC patches was divided into next two elements (l_b4 and l_b5). A displacement of beam tip is measured in node no 4.

The local mass and stiffness matrixes were calculated as follows [28]:

$$\begin{array}{ccc}{M}_i^e=M_i^{eb}+2M_i^{eMFC}& & K_i^e=\end{array}{K}_i^{eb}+2K_i^{eMFC}$$

(2)

where: M_i^eb and K_i^eb are the mass and stiffness local matrixes of beam substrate for element no “i”, M_i^eMFC and K_i^eMFC are the mass and stiffness local matrixes of MFC patch for element no “i”.

$$\begin{array}{ccc}{\mathbf{M}}_{\mathbf{i}}^{\mathbf{eb}}=\frac{{\mathrm{\rho }}_{\mathrm{bi}}{\mathrm{A}}_{\mathrm{bi}}{\mathrm{l}}_{\mathrm{bi}}}{420}\left[\begin{array}{cccc}156& 22{\mathrm{l}}_{\mathrm{bi}}& 54& -13{\mathrm{l}}_{\mathrm{bi}}\\ 22{\mathrm{l}}_{\mathrm{bi}}& {4\mathrm{l}}_{\mathrm{bi}}^2& 13{\mathrm{l}}_{\mathrm{bi}}& -{3\mathrm{l}}_{\mathrm{bi}}^2\\ 54& 13{\mathrm{l}}_{\mathrm{bi}}& 156& -22{\mathrm{l}}_{\mathrm{bi}}\\ -13{\mathrm{l}}_{\mathrm{bi}}& -{3\mathrm{l}}_{\mathrm{bi}}^2& -22{\mathrm{l}}_{\mathrm{bi}}& {4\mathrm{l}}_{\mathrm{bi}}^2\end{array}\right]& & {\mathbf{K}}_{\mathbf{i}}^{\mathbf{eb}}=\frac{{\mathrm{E}}_{\mathrm{bi}}{\mathrm{I}}_{\mathrm{bi}}}{{\mathrm{l}}_{\mathrm{bi}}}\left[\begin{array}{cccc}12& 6{\mathrm{l}}_{\mathrm{bi}}& -12& 6{\mathrm{l}}_{\mathrm{bi}}\\ 6{\mathrm{l}}_{\mathrm{bi}}& {4\mathrm{l}}_{\mathrm{bi}}^2& -6{\mathrm{l}}_{\mathrm{bi}}& {2\mathrm{l}}_{\mathrm{bi}}^2\\ -12& -6{\mathrm{l}}_{\mathrm{bi}}& 12& -6{\mathrm{l}}_{\mathrm{bi}}\\ 6{\mathrm{l}}_{\mathrm{bi}}& {2\mathrm{l}}_{\mathrm{bi}}^2& -6{\mathrm{l}}_{\mathrm{bi}}& {4\mathrm{l}}_{\mathrm{bi}}^2\end{array}\right]\end{array}$$

(3)

$$\begin{array}{ccc}{\mathbf{M}}_{\mathbf{i}}^{\mathbf{eMFC}}=\frac{{\mathrm{\rho }}_{\mathrm{MFCi}}{\mathrm{A}}_{\mathrm{MFCi}}{\mathrm{l}}_{\mathrm{MFCi}}}{420}\left[\begin{array}{cccc}156& 22{\mathrm{l}}_{\mathrm{bi}}& 54& -13{\mathrm{l}}_{\mathrm{bi}}\\ 22{\mathrm{l}}_{\mathrm{bi}}& {4\mathrm{l}}_{\mathrm{bi}}^2& 13{\mathrm{l}}_{\mathrm{bi}}& -{3\mathrm{l}}_{\mathrm{bi}}^2\\ 54& 13{\mathrm{l}}_{\mathrm{bi}}& 156& -22{\mathrm{l}}_{\mathrm{bi}}\\ -13{\mathrm{l}}_{\mathrm{bi}}& -{3\mathrm{l}}_{\mathrm{bi}}^2& -22{\mathrm{l}}_{\mathrm{bi}}& {4\mathrm{l}}_{\mathrm{bi}}^2\end{array}\right]& & {\mathbf{K}}_{\mathbf{i}}^{\mathbf{eMFC}}=\frac{{\mathrm{E}}_{\mathrm{MFCi}}{\mathrm{I}}_{\mathrm{MFCi}}}{{\mathrm{l}}_{\mathrm{MFCi}}}\left[\begin{array}{cccc}12& 6{\mathrm{l}}_{\mathrm{bi}}& -12& 6{\mathrm{l}}_{\mathrm{bi}}\\ 6{\mathrm{l}}_{\mathrm{bi}}& {4\mathrm{l}}_{\mathrm{bi}}^2& -6{\mathrm{l}}_{\mathrm{bi}}& {2\mathrm{l}}_{\mathrm{bi}}^2\\ -12& -6{\mathrm{l}}_{\mathrm{bi}}& 12& -6{\mathrm{l}}_{\mathrm{bi}}\\ 6{\mathrm{l}}_{\mathrm{bi}}& {2\mathrm{l}}_{\mathrm{bi}}^2& -6{\mathrm{l}}_{\mathrm{bi}}& {4\mathrm{l}}_{\mathrm{bi}}^2\end{array}\right]\end{array}$$

(4)

where: ρ_bi is the density of substrate beam element no “i”, A_bi is the cross-section area beam element no “i”, l_bi is the length of beam element no “i”, E_bi is Young’s modulus of beam substrate, I_bi is the moment of inertia of beam substrate element no “i”, ρ_MFCi is the density of MFC patch for beam element no “i”, A_MFCi is the cross-section area of MFC patch for beam element no “i”, E_MFCi is Young’s modulus of MFC patch, I_{MFCi i} is the moment of inertia of MFC patch for element no “i”. The global damping matrixes can be calculated as proportional damping in the Rayleigh form [29]:

$$C_g=\mathsf{\alpha }{M}_g+\mathsf{\beta }{K}_g$$

(5)

where α and β are the dimensionless coefficients.

2.2. Model of Forces Generated by Two MFC Patches

Stress in MFC patches is induced by supply voltage. Dependence between the voltage and stress in piezoelectric material is determined from constitutive of piezoelectric materials [30]. Taking into account that a local polarization of piezoelectric fibers in MFC patch is along 3-axis and that compressive or tensile stress is also generated along 3-axis (Figure 2) [7], the constitutive equations are the following:

$$\begin{array}{l}{\mathrm{S}}_3={\mathrm{s}}_{33}^{\mathrm{E}}{\mathrm{T}}_3+{\mathrm{d}}_{33}{\mathrm{E}}_3\\ {\mathrm{D}}_3={\mathrm{d}}_{33}{\mathrm{T}}_3+{\mathsf{\epsilon }}_{33}^{\mathrm{T}}{\mathrm{E}}_3\end{array}$$

(6)

where: S₃ is the strain along 3-axis, T₃ is the strain along 3-axis, E₃ is the electric field intensity along 3-axis, D₃ is the electric induction along 3-axis, s^E₃₃ the compliance constant under constant electric field, d₃₃ is the piezoelectric charge constant, ε^T₃₃ is the permittivity under constant stress. Taking into account that strain along 3-axis is blocked; stress inducted by MFC patch can be calculated on the basis of the first equation of (9):

$${\mathrm{T}}_{3\mathrm{i}}=\frac{{\mathrm{d}}_{33}}{{\mathrm{s}}_{33}^{\mathrm{E}}}{\mathrm{E}}_{3\mathrm{i}}=\frac{{\mathrm{d}}_{33}}{{\mathrm{s}}_{33}^{\mathrm{E}}}\frac{{\mathrm{V}}_{\mathrm{i}}}{{\mathrm{t}}_{\mathrm{de}}}\begin{array}{ccc}& \mathrm{for}& \mathrm{i}=1,2\end{array}$$

(7)

where: t_de is the distance between electrode “+” and electrode “−”.

A bending moment per unit length produced in the composite beam by the actuator [31]:

$${\mathrm{M}}_{\mathrm{b}3\mathrm{i}}={\displaystyle \underset{\mathrm{A}}{\int }{\mathrm{T}}_{3\mathrm{i}}\mathrm{ydt}=}{\displaystyle \underset{\frac{1}{2}{\mathrm{t}}_{\mathrm{s}}}{\overset{\frac{1}{2}{\mathrm{t}}_{\mathrm{s}}+{\mathrm{t}}_{\mathrm{MFC}}}{\int }}\frac{{\mathrm{d}}_{33}}{{\mathrm{s}}_{33}^{\mathrm{E}}}\frac{{\mathrm{V}}_{\mathrm{i}}}{{\mathrm{t}}_{\mathrm{de}}}{\mathrm{w}}_{\mathrm{b}}\mathrm{y}}\mathrm{dy}=\frac{1}{2}\frac{{\mathrm{d}}_{33}}{{\mathrm{s}}_{33}^{\mathrm{E}}}\frac{{\mathrm{V}}_{\mathrm{i}}}{{\mathrm{t}}_{\mathrm{de}}}{\mathrm{w}}_{\mathrm{b}}\left({\mathrm{t}}_{\mathrm{s}}{\mathrm{t}}_{\mathrm{MFC}}+{\mathrm{t}}_{\mathrm{MFC}}^2\right)$$

(8)

where: t_MFC is the thickness of MFC patch. The force generated by MFC patch along 1-axis for tip of cantilever beam:

$${\mathrm{P}}_{1\mathrm{i}}=\frac{1}{{2\mathrm{l}}_{\mathrm{b}}}\frac{{\mathrm{d}}_{33}}{{\mathrm{s}}_{33}^{\mathrm{E}}}\frac{{\mathrm{V}}_{\mathrm{i}}}{{\mathrm{t}}_{\mathrm{de}}}{\mathrm{w}}_{\mathrm{b}}\left({\mathrm{t}}_{\mathrm{s}}{\mathrm{t}}_{\mathrm{MFC}}+{\mathrm{t}}_{\mathrm{MFC}}^2\right)$$

(9)

Four elements (from l_b1 to l_b4) were considered because the displacement measurement is difficult for node no 5 by the use of a laser sensor in practical application. Matrix of distribution of the force generated by MFC patch no 1:

$$E_1=\left(\frac{1}{{2\mathrm{l}}_{\mathrm{b}}}\frac{{\mathrm{d}}_{33}}{{\mathrm{s}}_{33}^{\mathrm{E}}}\frac{{\mathrm{V}}_1}{{\mathrm{t}}_{\mathrm{de}}}{\mathrm{w}}_{\mathrm{b}}\left({\mathrm{t}}_{\mathrm{s}}{\mathrm{t}}_{\mathrm{MFC}}+{\mathrm{t}}_{\mathrm{MFC}}^2\right)\right)\cdot {\left[\begin{array}{cccccccc}0& 0& 1& 0& 0& 0& 0& 0\end{array}\right]}^{\mathrm{T}}$$

(10)

Matrix of distribution of the force generated by MFC patch no 2:

$$E_2=\left(\frac{1}{{2\mathrm{l}}_{\mathrm{b}}}\frac{{\mathrm{d}}_{33}}{{\mathrm{s}}_{33}^{\mathrm{E}}}\frac{{\mathrm{V}}_2}{{\mathrm{t}}_{\mathrm{de}}}{\mathrm{w}}_{\mathrm{b}}\left({\mathrm{t}}_{\mathrm{s}}{\mathrm{t}}_{\mathrm{MFC}}+{\mathrm{t}}_{\mathrm{MFC}}^2\right)\right)\cdot {\left[\begin{array}{cccccccc}0& 0& 1& 0& 0& 0& 0& 0\end{array}\right]}^{\mathrm{T}}$$

(11)

Both forces act on the cantilever beam in node no 2.

2.3. Synthesis of Control Algorithm Generating Two Control Signals

LQR control algorithm was used for generating of control voltages. The piezoelectric cantilever actuator must be described by the use of state space model for LQR synthesis. The state space model has a well-known form [32]:

$$\begin{array}{l}\stackrel{•}{x}=Ax+Bu\\ y=Cx+Du\end{array}$$

(12)

where: x is the state vector containing vertical and rotational displacements and its derivatives for each nod (w_k, ${\dot{\mathrm{w}}}_{\mathrm{k}}$, φ_k, ${\dot{\varphi }}_{\mathrm{k}}$ for k = 1,…,8), u is the control vector, y is the output signals vector, A is the state matrix, B is the control matrix, C is the output matrix, D is the feed through matrix. The displacement of the node no 4 is an output signal.

$$\begin{array}{l}x={\left[\begin{array}{c}\mathrm{d}\\ \dot{\mathrm{d}}\end{array}\right]}_{16\times 1}\begin{array}{c}\\ \end{array}A={\left[\begin{array}{cc}{\mathrm{\Theta }}_{8\times 8}& I_8\\ -M_g^{-1}{K}_g& -M_g^{-1}{C}_g\end{array}\right]}_{16\times 16}\begin{array}{c}\\ \end{array}B={\left[\begin{array}{cc}{\mathrm{\Theta }}_{8\times 1}& {\mathrm{\Theta }}_{8\times 1}\\ M_g^{-1}{E}_1& M_g^{-1}{E}_2\end{array}\right]}_{16\times 2}\begin{array}{c}\\ \end{array}\\ C=\left[\begin{array}{cccccccccccccccc}0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]\begin{array}{c}\\ \end{array}D=\left[\begin{array}{cc}0& 0\end{array}\right]\end{array}$$

(13)

The control signals were expressed following:

$$\begin{array}{l}{\mathrm{u}}_1=-\left[\begin{array}{cc}{K}_1& {\mathrm{k}}_{\mathrm{n}+{1,\mathrm{u}}_1}^{}\end{array}\right]{\left[\begin{array}{cc}x& {\mathrm{x}}_{\mathrm{n}+1}\end{array}\right]}^{\mathrm{T}}+y_z\\ {\mathrm{u}}_2=-\left[\begin{array}{cc}{K}_2& {\mathrm{k}}_{\mathrm{n}+{1,\mathrm{u}}_2}^{}\end{array}\right]{\left[\begin{array}{cc}x& {\mathrm{x}}_{\mathrm{n}+1}\end{array}\right]}^{\mathrm{T}}+y_z\end{array}$$

(14)

where: n is the number of elements in the state vector, K₁ and K₂ are the gains matrixes corresponding to the control signals u₁ and u₂, k_n+1,u1 and k_n+1,u2 are the gains corresponding to the control signals u₁ and u₂ for additional state variable x_n+1, y_z is the reference values of the output. The state space model extended by the additional state variables is expressed following [33]:

$$\left[\begin{array}{c}\dot{x}\\ {\dot{\mathrm{x}}}_{\mathrm{n}+1}\end{array}\right]=\left[\begin{array}{cc}A& {\mathrm{\Theta }}_{16\times 1}\\ -C& 0\end{array}\right]\left[\begin{array}{c}x\\ {\mathrm{x}}_{\mathrm{n}+1}\end{array}\right]+\left[\begin{array}{c}B\\ 0\end{array}\right]u\begin{array}{cc}& y=\left[\begin{array}{cc}C& 0\end{array}\right]\left[\begin{array}{c}x\\ {\mathrm{x}}_{\mathrm{n}+1}\end{array}\right]+Du\end{array}$$

(15)

The gains K, k_n+1, were calculated by the minimization of the expanded quality index:

$$\mathrm{J}={\displaystyle \underset{0}{\overset{\infty }{\int }}\left({\left[\begin{array}{c}x\\ {\mathrm{x}}_{\mathrm{n}+1}\end{array}\right]}^{\mathrm{T}}Q\left[\begin{array}{c}x\\ {\mathrm{x}}_{\mathrm{n}+1}\end{array}\right]+u^{\mathrm{T}}Ru\right)\mathrm{dt}}$$

(16)

where: Q is the positive definite or semi definite weight matrix, R is the positive definite weight matrix.

The use of the LQR control algorithm is possible if the actuator is fully controllable. A matrix of modal controllability [34]:

$$P={\Phi }^{-1}B$$

(17)

where: Φ is the truncated matrix. The actuator is fully controllable if there is at least one non-zero element in each row of the matrix P.

3. Research Methods

3.1. Simulation Research

Simulation research was conducted in the MATLAB Simulink program (version 2019b), in which the FEM model, presented in the previous chapter, was implemented in matrix form. Solver ode23tb was used for the calculation of Equation (12). Algorithm ode23tb is an implementation of TR-BDF2, an implicit Runge–Kutta formula with a first stage that is a trapezoidal rule step and a second stage that is a backward differentiation formula of order two (the same iteration matrix is used in evaluating both stages) [35].Values of material properties, which were used in simulation research, are presented in

Table 1. Material and geometric properties used in simulation research.

Parameter	Symbol	Unit	Value
Young modulus of PCB FR-4 substrate [36]	Ys	Pa	18.6 × 109
Young modulus of MFC patch [23]	YMFC	Pa	30.336 × 109
Young modulus of passive area of MFC patch [10]	YMFCp	Pa	2.9 × 109
Density of PCB FR-4 substrate	ρs	kg/m3	1900
Density of active area of MFC patch [15]	ρMFCa	kg/m3	5400
Density of passive area of MFC patch [11]	ρMFCp	kg/m3	1420
Elastic compliance of MFC patch [11]	sE33	m2/N	18.8 × 10−12
Piezoelectric charge constant of MFC patch [10]	d33	C/N	436 × 10−12
Thickness of piezoelectric fibers in MFC patch [7]	tp	mm	0.18
Distance between electrodes in MFC patch [7]	tde	mm	0.5

.

Proportional damping in the Rayleigh form was used in simulation research. Coefficients α and β from Formula (5) were calculated on the basis [37]:

$$\left[\begin{array}{c}\mathsf{\alpha }\\ \mathsf{\beta }\end{array}\right]=\left[\begin{array}{cc}\frac{{2\mathsf{\omega }}_{\mathrm{i}}{\mathsf{\omega }}_{\mathrm{j}}^2}{{\mathsf{\omega }}_{\mathrm{j}}^2-{\mathsf{\omega }}_{\mathrm{i}}^2}& -\frac{{2\mathsf{\omega }}_{\mathrm{i}}^2{\mathsf{\omega }}_{\mathrm{j}}}{{\mathsf{\omega }}_{\mathrm{j}}^2-{\mathsf{\omega }}_{\mathrm{i}}^2}\\ -\frac{{2\mathsf{\omega }}_{\mathrm{i}}{\mathsf{\omega }}_{\mathrm{j}}}{{\mathsf{\omega }}_{\mathrm{j}}\left({\mathsf{\omega }}_{\mathrm{j}}^2-{\mathsf{\omega }}_{\mathrm{i}}^2\right)}& \frac{{2\mathsf{\omega }}_{\mathrm{i}}{\mathsf{\omega }}_{\mathrm{j}}}{{\mathsf{\omega }}_{\mathrm{i}}\left({\mathsf{\omega }}_{\mathrm{j}}^2-{\mathsf{\omega }}_{\mathrm{i}}^2\right)}\end{array}\right]\left[\begin{array}{c}{\mathsf{\xi }}_{\mathrm{i}}\\ {\mathsf{\xi }}_{\mathrm{j}}\end{array}\right]$$

(18)

where: ω_i and ω_j are the reference natural modes, ξ _i and ξ _j are the damping ratios of reference vibration mode. First and second natural modes were selected for a cantilever beam [38]:

$${\mathsf{\omega }}_1={\lambda }_1^2\sqrt{\frac{{\mathrm{Y}}_{\mathrm{b}}\mathrm{I}}{{\mathrm{m}}_{\mathrm{b}}{\left({\mathrm{l}}_{\mathrm{b}}\right)}^4}}\begin{array}{cc}& {\mathsf{\omega }}_2={\lambda }_2^2\sqrt{\frac{{\mathrm{Y}}_{\mathrm{b}}\mathrm{I}}{{\mathrm{m}}_{\mathrm{b}}{\left({\mathrm{l}}_{\mathrm{b}}\right)}^4}}\end{array}$$

(19)

where: λ₁ and λ₁ are the first two eigenvalues, Y_b is the Young’s modulus of whole cantilever beam, I is the inertia moment of whole cantilever beam, m_b is the mass of whole cantilever beam, l_b is the length of whole cantilever beam. The eigenvalues were determined as the product of a coefficient β and l_b. The coefficient β is determined by solving the equation [36]:

$$\frac{{\mathrm{d}}^4\mathrm{w}\left(\mathrm{x}\right)}{{\mathrm{dx}}^4}\begin{array}{cc}-{\mathsf{\beta }}^4\mathrm{w}\left(\mathrm{x}\right)=0& \mathrm{for}\end{array}\begin{array}{cc}& 0<\mathrm{x}<{\mathrm{l}}_{\mathrm{b}}\end{array}$$

(20)

For boundary conditions:

$$\begin{array}{cc}\mathrm{for}\begin{array}{c}\mathrm{x}=0\end{array}& \mathrm{w}\left(\mathrm{x}\right)=0,\frac{\mathrm{dw}\left(\mathrm{x}\right)}{\mathrm{dx}}=0\\ \mathrm{for}\begin{array}{c}\mathrm{x}={\mathrm{l}}_{\mathrm{b}}\end{array}& \frac{{\mathrm{d}}^2\mathrm{w}\left(\mathrm{x}\right)}{{\mathrm{dx}}^2}=0,\frac{{\mathrm{d}}^3\mathrm{w}\left(\mathrm{x}\right)}{{\mathrm{dx}}^3}=0\end{array}$$

(21)

The first two eigenvalues are following for each cantilever described by Equation (24) and (25): λ₁ = 1.8751, λ₂ = 4.6941.

Young’s modulus was calculated on the basis of Formula [36]:

$${\mathrm{Y}}_{\mathrm{b}}={\mathrm{Y}}_{\mathrm{s}}\left(\frac{{\mathrm{V}}_{\mathrm{b}}-{\mathrm{V}}_{\mathrm{MFC}}}{{\mathrm{V}}_{\mathrm{b}}}\right)+{\mathrm{Y}}_{\mathrm{MFC}}\left(\frac{{\mathrm{V}}_{\mathrm{MFC}}}{{\mathrm{V}}_{\mathrm{b}}}\right)$$

(22)

where: V_b is the volume whole cantilever beam, VMFC is the volume MFC patches. For values of damping ratios, which were assumed [36]: ξ1 = ξ2 = 0.005, the calculated values of coefficients were as follows: α = 5.652, β = 2.10961 × 10⁻⁶.

3.2. Laboratory Research

Experiments were carried out on a laboratory stand which is presented in Figure 3.

Tested beam actuator consisted of three layers: carrying substrate (2), MFC patch of P1 type (1) glued on bottom side of substrate and MFC patch of P1 type (3) glued on upper side of substrate. Whole beam structure was mounted in clamping assembly (4). Epoxy Adhesive DP490, produced by 3M company, was used to glue the MFC patches to the substrate. A schema of structure of the beam is presented in Figure 1 and Figure 2 and its geometric parameters are presented in

Table 2. Dimensions of the cantilever beam (values in mm).

Parameters	Symbol	Value
Distance between fixing and MFC active area	lb1	7.5
Length of MFC active area	lb2	85
Distance between MFC active area and end of MFC patch	lb3	7.5
Distance between end of MFC patch and measurement point	lb4	10
Width of beam	wb	20
Width of MFC patch	wMFC	20
Width of active area in MFC patch	wact	14
Thickness of PCB FR-4 substrate	ts	1
Thickness of MFC patch	tMFC	0.3

.

Control system consisted of LG5B65PI laser sensor (5) of displacement, RT_DAC/Zynq multifunctional board (6) and TD250-INV voltage amplifier (7). The laser sensor, produced by BANNER company, had a sensing range equal to 45–60 mm and a linearity ±60 μm. The detailed specification of this sensor can be found in [39]. The multifunctional board, produced by INTECO company, was dedicated to real-time data acquisition and control (contained FPGA chip) with the use of MATLAB Simulink program. The detailed specification of this card can be found in [40]. The voltage amplifier, produced by PIEZODRIVE, enabled the generation of three voltage signals with a value of ±500 V. The detailed specification of this amplifier can be found in [41]. Schema of control system for actuator is presented in Figure 4.

The implementation of LQR algorithm in practical applications requires a measurement of all of elements of the state vector. Taking into account that the number of elements was 16, such measurements would be practically impossible. The state vector should be estimated on the basis of measurements: beam tip displacement and control signals. Due to the fact that the measurement of the displacement was noisy, Kalman filter was used for the state vector estimation in the laboratory experiments. The following is a known model of state observer which was used in laboratory experiments [42]:

$$\begin{array}{l}\dot{\widehat{x}}=\left(A-HC\right)\widehat{x}+Bu+Hy\\ \widehat{y}=C\widehat{x}+Du\end{array}$$

(23)

where: $\hat{\mathrm{x}}$ is the estimated state vector, H is the matrix of observer gains, $\hat{\mathrm{y}}$ is the estimated output. Matrix H was determined with the use of MATLAB command on the basis of the assumed covariance matrix of state noise Q_c and a covariance matrix of measurement noise R_c.

The control vector was expressed as follows:

$$\begin{array}{l}{\mathrm{u}}_1=-\left[\begin{array}{cc}{K}_1& {\mathrm{k}}_{\mathrm{n}+{1,\mathrm{u}}_1}^{}\end{array}\right]{\left[\begin{array}{cc}\widehat{x}& {\mathrm{x}}_{\mathrm{n}+1}\end{array}\right]}^{\mathrm{T}}+y_z\\ {\mathrm{u}}_2=-\left[\begin{array}{cc}{K}_2& {\mathrm{k}}_{\mathrm{n}+{1,\mathrm{u}}_2}^{}\end{array}\right]{\left[\begin{array}{cc}\widehat{x}& {\mathrm{x}}_{\mathrm{n}+1}\end{array}\right]}^{\mathrm{T}}+y_z\end{array}$$

(24)

In order to reduce the computation cost, the model order reduction was presented in, e.g., [28]. The nodal displacement vector was transformed into the reduced vector:

$$d=\Phi \mathsf{\kappa }$$

(25)

where: κ is modal coordinate vector. Based on Equation (13), it can be seen that the truncated matrix consisted of eight eigenvectors. Hence, the truncated matrix was defined as the eight-modes shape matrix. Modal matrices were calculated:

$$K_m={\Phi }^T{K}_g\Phi \begin{array}{cc}& M_m={\Phi }^T{M}_g\Phi \end{array}\begin{array}{cc}& C_m={\Phi }^T{C}_g\Phi \end{array}\begin{array}{cc}& E_{1m}={\Phi }^T{E}_1\begin{array}{cc}& E_{2m}={\Phi }^T{E}_2\end{array}\end{array}$$

(26)

where: K_m is the modal stiffness matrix, M_m is the modal mass matrix, C_m is the modal damping matrix, E_1m and E_2m are the modal matrixes of force distribution. Based on the modal matrices, a second model in the state space was determined:

$$\begin{array}{cc}\hfill & {\mathbf{x}}_{\mathbf{m}}={\left[\begin{array}{c}\mathsf{\kappa }\\ \dot{\mathsf{\kappa }}\end{array}\right]}_{16\times 1}\begin{array}{c}\\ \end{array}{\mathbf{A}}_{\mathbf{m}}={\left[\begin{array}{cc}{\mathrm{\Theta }}_{\mathbf{8}\times \mathbf{8}}& {\mathbf{I}}_{\mathbf{8}}\\ -{\mathbf{M}}_{\mathbf{m}}^{-\mathbf{1}}{\mathbf{K}}_{\mathbf{m}}& -{\mathbf{M}}_{\mathbf{m}}^{-\mathbf{1}}{\mathbf{C}}_{\mathbf{m}}\end{array}\right]}_{16\times 16}\begin{array}{c}\\ \end{array}{\mathbf{B}}_{\mathbf{m}}={\left[\begin{array}{cc}{\mathrm{\Theta }}_{\mathbf{8}\times \mathbf{1}}& {\mathrm{\Theta }}_{\mathbf{8}\times \mathbf{1}}\\ {\mathbf{M}}_{\mathbf{m}}^{-\mathbf{1}}{\mathbf{E}}_{\mathbf{1}\mathbf{m}}& {\mathbf{M}}_{\mathbf{m}}^{-\mathbf{1}}{\mathbf{E}}_{\mathbf{2}\mathbf{m}}\end{array}\right]}_{16\times 2}\begin{array}{c}\\ \end{array}\hfill \\ \hfill & {\mathbf{C}}_{\mathbf{m}}=\left[\begin{array}{cc}\mathbf{\phi }& {\mathrm{\Theta }}_{\mathbf{8}\times \mathbf{8}}\end{array}\right]\begin{array}{c}\\ \end{array}{\mathbf{D}}_{\mathbf{m}}=\left[\begin{array}{cc}0& 0\end{array}\right]\hfill \end{array}$$

(27)

In the initial simulation tests, actuator displacement values for eight models differing in the number of modes were compared. Such a comparison for a step change in the value of the control signals (150 V) is shown in Figure 5.

The difference between the displacement value for eight-modes model and the displacement value for the first-mode model was equal to 0.003 mm, which was 0.6% of the output value. Similarly, small differences occurred for control steps of a different value in the range of 5 V–500 V. Therefore, in the laboratory tests, a model, containing only first mode, was adopted to determine the state vector estimate in the Kalman filter. Dimensions of matrixes and vector in this model are as follows:

$$\begin{array}{l}{x}_m={\left[\begin{array}{c}\mathsf{\kappa }\\ \dot{\mathsf{\kappa }}\end{array}\right]}_{2\times 1}\begin{array}{c}\\ \end{array}{A}_m={\left[\begin{array}{cc}{\mathsf{\Theta }}_{1\times 1}& {\mathrm{I}}_1\\ -M_{m1}^{-1}{K}_{m1}& -M_{m1}^{-1}{C}_{m1}\end{array}\right]}_{2\times 2}\begin{array}{c}\\ \end{array}{B}_m={\left[\begin{array}{cc}{\mathsf{\Theta }}_{1\times 1}& {\mathsf{\Theta }}_{1\times 1}\\ M_{m1}^{-1}{E}_{1m1}& M_{m1}^{-1}{E}_{2m1}\end{array}\right]}_{2\times 2}\begin{array}{c}\\ \end{array}\\ C_m=\left[\begin{array}{cc}{\varphi }_7 & {\mathrm{\Theta }}_{1\times 1}\end{array}\right]\begin{array}{c}\\ \end{array}{\mathrm{D}}_{\mathrm{m}}=\left[\begin{array}{cc}0& 0\end{array}\right]\end{array}$$

(28)

where: M_m1 is the modal stiffness matrix for first mode, M_m1 is the modal mass matrix for first mode, C_m1 is the modal damping matrix for first mode, E_1m1 and E_2m1 are the modal matrixes of force distribution for first mode, φ₇ is seventh element of truncated matrix.

The use of Kalman filter for state vector estimation is possible if the actuator is fully observable. A matrix of modal observability is as follows [34]:

$$R=C\Phi$$

(29)

The actuator is fully observable if there is non-zero element in each row of the matrix R.

A schema of the control system with the Kalman filter based on model (27) for beam actuator with two inputs is presented in Appendix A.

4. Results

4.1. Controllability and Observability of the Piezoelectric Beam Actuator

Matrix of modal controllability had a dimension 16 × 2 and it had all non-zero elements for tested actuator. All modes were controllable through both inputs, which were control voltages.

Matrix of modal observability had dimensions 1 × 16 and it had all non-zero elements for the tested actuator. All modes were observable in output, which was displacement of beam tip.

4.2. Weight and Covariance Matrixes

Simulation and laboratory tests were carried out for the experimentally determined weight matrixes.

Weight matrix Q, which relates to Equations (14) and (28), had a diagonal form:

$$Q=\left[\begin{array}{ccc}{\mathrm{q}}_{11}& 0& 0\\ 0& {\mathrm{q}}_{11}& 0\\ 0& 0& {\mathrm{q}}_{33}\end{array}\right]$$

(30)

The weights from q₁₁ and q₂₂ correspond to state variables from x_m1 to x_m2 that have no limitations. In case of no limitations, the corresponding weights based on Bryson’s rule should be equal to 0 [43]. The weights q₁₁ and q₂₂ were equal to 0 in all of experiments. Determination of the weight q₃₃ was based on trial-and-error procedure. First, low value for this weight was assumed and simulation runs were performed. In those simulations, stability was examined. Next, the weight q₃₃ was increased until instability appeared. If it was observed, weight q₃₃ was decreased by a few percent until stability was recovered. Such a procedure of weight selection is described in [44]. The weight q₃₃ was equal to 1 × 10¹² in all of the experiments.

Weight matrix R for two control inputs had the following elements:

$$R=\left[\begin{array}{cc}{\mathrm{r}}_{11}& 0\\ 0& {\mathrm{r}}_{22}\end{array}\right]$$

(31)

where: r₁₁ is the weight affecting the first control signal (u₁), r₂₂ is the weight affecting the second control signal (u₂). These weights were calculated on the basis of formulas [45]:

$${\mathrm{r}}_{11}=\frac{1}{{\mathrm{u}}_{1\max }^2}\begin{array}{cc}& {\mathrm{r}}_{22}=\frac{1}{{\mathrm{u}}_{2\max }^2}\end{array}$$

(32)

where: u_1max is the assumed maximum control voltage u₁, u_2max is the assumed maximum control voltage u₂.

Covariance matrixes Q_c and R_c were determined on the basis of trial-and-error procedure and had the following elements:

$$Q_c=\left[\begin{array}{cc}0.001& 0\\ 0& 0.001\end{array}\right]\begin{array}{cc}& {\mathrm{R}}_{\mathrm{c}}=1\times {10}^{-4}\end{array}$$

(33)

4.3. Impact of Weights in Matrix R on Control Quality in One-Input Actuator and Two-Input Actuator

Reference displacement of beam tip, which was used in all of simulation and laboratory experiments, is presented in Figure 6. Sample courses of the displacement of beam tip and the control signal or control signals for one-input actuator (with one MFC layer) and two-input actuator (with two MFC layer) are presented in Figure 7. Presented characteristics were obtained for weights in the R matrix, which were calculated assuming that the maximum control voltage was 150 V (based on Equation (32)).

Such characteristics were obtained for the next eight values of the assumed maximum control voltage u_imax; on the basis of which the weights r₁₁ and r₂₂ were calculated using Equation (32). A total of nine experiments were therefore carried out. The values of u_imax in experiments are as follows: 6.25 V, 12.5 V, 25 V, 50 V, 100 V, 150 V, 200 V, 250 V and 300 V. A maximum output value (y_max), a maximum value and a minimum value of the control signal (u_imax/u_imin) for both one-input and two-input actuators were determined on the basis of characteristics obtained in experiments. Values of these parameters are presented in

Table 3. Values of parameters in experiments.

Exper. No.	Assumed umax (V)	ymax (mm)		tr (s)		u1max/u1min (V)		u2max/u2min (V)
		One-Input	Two-Input	One-Input	Two-Input	One-Input	Two-Input	One-Input	Two-Input
1	6.25	0.516	0.515	>1	>1	337.8/−84.5	214.5/−64.8	X	64.88/−214.5
2	12.5	0.518	0.518	0.84	0.74	348.6/−89.5	219.4/−71.8	X	71.83/−219.5
3	25	0.534	0.545	0.43	0.37	374.4/−106.7	245.1/−79.4	X	79.66/−244.9
4	50	0.567	0.563	0.24	0.26	407.4/−143.3	249.7/−105.5	X	105.1/−249.7
5	100	0.607	0.594	0.26	0.27	460.8/−201.1	279.3/−128.9	X	128.6/−279.4
6	150	0.642	0.626	0.22	0.24	500.0/−241.7	307.2/−152.4	X	152.4/−307.2
7	200	0.638	0.618	0.20	0.22	500.0/−265.9	306.4/−166.9	X	166.9/−306.3
8	250	0.658	0.628	0.24	0.20	500.0/−302.2	311.7/−174.5	X	174.5/−311.7
9	300	0.678	0.663	0.23	0.26	500.0/−341.1	348.9/−195.1	X	195.1/−348.6

.

Settling time was measured from 1 s to the point where the output size was between 0.495 mm and 0.505 mm.

4.4. Impact of Voltage Limit on Control Quality in One-Input Actuator and Two-Input Actuator

Reference displacement of beam tip, which was used in all of simulation and laboratory experiments, is presented in Figure 6. Sample courses of the displacement of beam tip and the control signal or control signals for one-input actuator and two-input actuator are presented in Figure 8. Presented characteristics were obtained for weights in the R matrix, which were calculated assuming that the maximum control voltage was 150 V (based on Equation (32)). The presented characteristics were obtained in two experiments, in which the voltage limit for one-input actuator was ±440 V and for two-input actuator was ±240 V.

Such characteristics were also obtained for other values of the control voltage limit. Control voltage limits used in laboratory experiments for one-input actuator are as follows: 480 V, 460 V, 440 V, 420 V, 400 V, 380 V, 360 V, 350 V and 340 V. Control voltage limits used in laboratory experiments for two-input actuator are as follows: 250 V, 240 V, 230 V, 225 V and 220 V. A maximum output value (y_max) and settling time (t_r) for both one-input and two-input actuators were determined on the basis of characteristics obtained in experiments. Values of these parameters for one-input actuator are presented in

Table 4. Values of parameters in experiments with control voltage limits for one-input actuator.

Experiment No (--)	1	2	3	4	5	6	7	8	9
Voltage limit (V)	480	460	440	420	400	380	360	350	340
ymax (mm)	0.625	0.621	0.611	0.595	0.575	0.555	0.537	0.527	0.499
tr (s)	0.23	0.21	0.23	0.25	0.29	0.41	0.70	>1	>1

and for two-input-actuator in

Table 5. Values of parameters in experiments with control voltage limits for two-input actuator.

Experiment No (--)	1	2	3	4	5
Voltage limit (V)	250	240	230	225	220
ymax (mm)	0.570	0.553	0.539	0.535	0.523
tr (s)	0.29	0.32	0.57	0.87	>2

.

Settling time was measured from 1 s to the point where the output size was between 0.495 mm and 0.505 mm.

5. Discussion

5.1. Impact of Weights in Matrix R on Control Quality in One-Input Actuator and Two-Input Actuator

The courses of the output and control signals for one-input actuator and two-input actuator have been compared. The comparative analysis used the parameters presented in Table 3 and four additional parameters: an overshoot, a decrease of maximum control voltage for two-input actuator in comparison to one-input actuator, a decrease of minimum control voltage for two-input actuator in comparison to one-input actuator and two-input actuator and two-input actuator and calculated quality index:

$${\mathrm{I}}_{\mathrm{q}}={\displaystyle \underset{}{\overset{}{\int }}\left|{\mathrm{y}}_{\mathrm{z}}-\mathrm{y}\right|\mathrm{dt}}$$

(34)

The smaller the value of I_q index, the smaller the difference between the output value and the reference value. The index value was calculated for the time interval between 1 s and 2 s in each experiment. The values of these parameters are presented in

Table 6. Values of parameters in a comparison analysis.

Experiment No (--)		1	2	3	4	5	6	7	8	9
Assumed umax (V)		6.25	12.5	25	50	100	150	200	250	300
Weights: r11 and r22 (--)		0.0256	0.0064	0.0016	0.0004	0.0001	44 × 10−5	25 × 10−5	16 × 10−5	11 × 10−5
Overshoot (%)	One-input	103.12	103.56	106.86	113.42	121.32	128.38	127.60	131.52	135.50
	Two-input	102.94	103,50	108.92	112.58	118.80	125.26	123.50	125.58	132.66
Quality index Iq (--)	One-input	0.11300	0.06777	0.04509	0.03120	0.02567	0.02294	0.02195	0.02065	0.02018
	Two-input	0.08750	0.05595	0.03843	0.02946	0.02397	0.02192	0.02002	0.01884	0.01926
Decrease of maximum control voltagefor two-input (%)		−36.50	−37.06	−34.54	−38.71	−39.39	−38.56	−38.72	−37.66	−30.22
Decrease of minimum control voltage for two-input (%)		−23.22	−19.82	−25.51	−26.38	−35.90	−36.95	−37.23	−42.26	−42.80

.

On the basis of the presented results it can be noticed that the displacement of beam tip in actuator containing two MFC patches achieved its reference value at a lower control voltage compared to an actuator containing one MFC layer. However, the application of two MFC layers in the actuator does not lead to a two-fold reduction of the control voltage compared to the actuator with one MFC layer. The decrease in the maximum voltage was from 30.22% to 39.39%. The dependence between the maximum value of the control voltage and the weights in the R matrix is shown in Figure 9a.

It should be noted that the application of two control signals led to a reduction in overshoot in eight out of nine experiments; only for the weight 0.0016 was there an increase of 1.8% (Figure 9b). Two control signals did not significantly affect the settling time in seven out of nine experiments (Figure 9c). In the case of the weight 0.0064, the settling time decreased by 13% in comparison to one control signal and for the weight 0.0016 by 16%. The normalized I_q quality index, taking into account both the overshoot and the regulation time, indicated a higher quality of control for the system with two inputs in two experiments (for weight 0.0256 it was 10% and for weight 0.0064 it was 3%). In the case of the weight 0.0016, it indicated the same control quality. In the case of weights from 0.0004 to 11 × 10⁻⁵, it indicated a higher quality of control for a system with one input (on average for these six weights by about 9%)

5.2. Impact of Voltage Limit on Control Quality in One-Input Actuator and Two-Input Actuator

The impact of control voltage limit for one-input actuator and two-input actuator have been compared. The comparative analysis used the parameters presented in Table 4 and Table 5 and two additional parameters: an overshoot and quality index I_q (based on (34). The values of these parameters are presented in

Table 7. Parameters values of control system with voltage limits for one-input actuator.

Experiment No. (--)	1	2	3	4	5	6	7	8	9
Limit (V)	480	460	440	420	400	380	360	350	340
Overshoot (%)	125.08	124.26	122.2	118.9	115.08	111.02	107.36	x	x
Quality index Iq (--)	0.02268	0.02282	0.02331	0.02375	0.02507	0.02754	0.03296	x	x

and

Table 8. Parameters values of control system with voltage limits for two-input actuator.

Experiment No. (--)	1	2	3	4	5
Limit (V)	250	240	230	225	220
Overshoot (%)	114.04	110.5	107.82	106.98	x
Quality index Iq (--)	0.02397	0.02566	0.03008	0.03448	x

.

The application of a maximum control voltage limit led to the decrease of control quality for both the one-input actuator and the two-input actuator. However, the quality of the actuator control with one input had dropped more compared to the quality of the actuator control with two inputs. Figure 10 shows a comparison of the overshoot, settling time and normalized quality index.

The limit of maximum control voltage leads to a greater decrease of the control quality for one-input actuator compared to two-input actuator, as indicated by the comparison of quality indexes (Figure 10c). It was most influenced by the increase of overshoot, which was much greater for one-input actuator compared to two-input actuator (Figure 10a).

6. Conclusions

The control system of beam actuator, based on Macro Fiber Composite, was tested in laboratory and numerical research. The tested beam actuator contained PCB FR-4 substrate and two MFC patches.

An application of two control signals in the beam actuator led to a reduction of voltage, which controls this actuator, compared to the control voltage of actuator with one control signal. However, the application of two control signals does not lead to a two-fold reduction of the control voltage. The decrease in the maximum voltage was from 30.22% to 39.39% in conducted laboratory experiments.

An application of a control system with two control signals causes a reduction of the overshoot value compared to the system with one control signal. The decrease in overshoot occurred in eight out of nine experiments. The settling time was shortened in two experiments for two-input actuator in comparison to one-input actuator; in the remaining seven experiments, this time did not change significantly.

An application of limit of maximum control voltage leads to a greater decrease of the control quality for one-input actuator compared to two-input actuator. In the laboratory experiments, it was most influenced by the increase of overshoot, which was much greater for one-input actuator compared to two-input actuator.

The advantage of the presented control system is the possibility of generating two control signals, which are independent of each other. Thanks to this, it is possible to generate two control signals that are different from each other, which may be useful in the case of lack of symmetry in the structure of the actuator. The lack of symmetry may be caused by, e.g., unequal gluing of both MFC patches on a carrying substrate. The generation of two control signals that are different from each other will be the subject of future research.