Control of an Energy-Harvesting System Using Sinha’s Theory for the Purpose of Energy Production

Abstract

The goal of this study was to develop a linear feedback control method to increase the energy produced for a parametrically excited energy-harvesting system. The chosen control design uses state feedback based on the Lyapunov–Floquet transformation to direct the system dynamics toward a previously chosen trajectory. The methodology of this work was based on the studies of Sinha and Butcher. They developed stability and control analysis techniques based on Floquet theory and the approximation of the state transition matrix of systems that have periodic coefficients in time. It was observed that the stability and instability conditions of the system were obtained from the choice of physical parameters related to the parametric excitation force. When the control technique was applied to the system, it was found that the suggested control effectively directed the system’s trajectory in the direction of the predetermined trajectory. The application of control significantly increased the energy generation compared with the uncontrolled system, which was seen when comparing the energy generated by the uncontrolled system and the net energy generated by the controlled system.

1. Introduction

Capturing ambient energy and turning it into a useable form is the process of energy harvesting. The most popular energy sources for energy-harvesting devices include solar energy, thermal gradients, and acoustic and mechanical vibrations [1]. When implementing new technologies, it is necessary to research and develop devices that are capable of producing the energy needed for these technologies to be self-sufficient, i.e., capable of producing enough energy for their own consumption. Mechanical vibrations are one way of obtaining energy from the environment. Vibration-energy-harvesting systems (VEHSs) are systems that employ vibrations as a source. Even though VEHSs produce less energy, they are nevertheless crucial because many gadgets need only modest quantities of energy to function. Vibration energy harvesting, known as supplemental power, can provide small amounts of energy for low-load applications or power and maintain remote devices or sensors that require small amounts of energy to operate. For example, hearing aids, pacemakers, spinal cord stimulators, and microelectromechanical systems. The cost of additional procedures to replace a pacemaker battery is high, and the patient must undergo a new treatment. A spinal cord stimulator must be constantly recharged, which is painful and time consuming. These gadgets would ideally be implanted with an energy-collecting system, enabling self-recharging and lifelong operation. The human body is the subject of research into the implementation of VEHSs, and early studies suggest that the contraction of blood vessels may provide up to 20 mW of energy per day [2]. In the literature, there are several studies on VEHSs, and among the most studied are the VEHSs that use piezoelectric materials as transducers, which are materials that convert mechanical energy into electrical energy. One of the most used electrical materials is Lead Zirconate Titanate (PZT) [3,4,5].

The amount of electrical power generated in this kind of system depends on several uncertain factors, including the amplitude, frequency, and physical parameters of the excitation force. Most research on the applied dynamics has not addressed the variations in the parameters of the system, meaning that no criterion has been established to verify which of them most affect its behavior. This issue can be tackled by a global sensitivity analysis [6], which is a technique with a low computational cost that uses the so-called Sobol indices, which are metrics based on the variance in plots of a polynomial expansion.

In the stability analysis, a method based on the Lyapunov–Floquet is adopted using the Chebyshev polynomial expansion to approximate the periodic terms. The state transition matrix is approximated by the Picard iterative method. This method allows for determining the Floquet multipliers and plotting the stability diagram of the dynamic system.

Both stability analysis and global sensitivity analysis are powerful complementary tools in the analysis of the dynamics of a system since the first one allows for analyzing the stability of the system by observing the terms of the system that make them periodic, and the second one allows for determining which of the physical parameters most cause the system to operate according to a predefined desired behavior. In this study, we were interested in solutions that were periodic (via the stability analysis) and that could produce the maximum possible energy output (via the global sensitivity analysis).

The two techniques mentioned above lead to relevant information that can be used to design more efficient controllers for the purposes of energy production. The control technique used in this study was based on the studies of Sinha and Joseph [7], David and Sinha [8], and Sinha and David [9]. This control technique is suitable for systems subject to a parametric excitation source, such as the system discussed in this study.

2. Energy-Harvesting System Model

The structures of vibrational-energy-harvesting systems are diverse, with most being common columns or overhanging plates partially or totally covered by a layer of PZT [3,5]. The structure analyzed in this study was a free-floating beam partially covered by a layer of piezoelectric material and parametrically excited, as illustrated by Daqaq et al. [10].

The piezoelectric material was connected to the beam and to a resistive load for the purpose of energy harvesting. The mathematical model representing the dynamics was described by Daqaq et al. [10], whose normalized equations are given by Equation (1), and the description of the parameters is given as per the following

Table 1. Parameters.

Parameters
R: resistor
${\mu }_1$: viscous damping
${\mu }_2$: quadratic damping (air resistance)
$\alpha$: constant
$\beta$: constant
$\theta$: electromechanical coupling
$C_p$: piezoelectric element capacitance
F: excitation amplitude
${\omega }_n$: natural frequency of the beam
$\omega$: excitation frequency
m: beam mass
$Lb$: beam length

:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\ddot{x}+2{\mu }_1\dot{x}+x+{\mu }_2\left|\dot{x}\right|\dot{x}+\alpha x^3+2\beta (x^2\ddot{x}+x{\dot{x}}^2)=xFcos(\Omega t)+{\displaystyle \frac{\theta }{K}}V,\hfill \\ \theta \dot{x}+C_p\dot{V}+{\displaystyle \frac{1}{R_{eq}}}V=0,\hfill \end{array}\right.\end{array}$$

(1)

where u: beam deflection in the x-direction, V: output voltage measured across resistor R, $x={\displaystyle \frac{u}{Lb}}$, $t=t\omega$, $V={\displaystyle \frac{V}{Lb}}$, ${\mu }_1={\displaystyle \frac{{\mu }_1}{{\omega }_n}}$, ${\mu }_2={\mu }_{2}Lb$, $\alpha =\alpha {\displaystyle \frac{{Lb}^2}{{\omega }_n^2}}$, $\beta =\beta {\left(Lb\right)}^2$, $F={\displaystyle \frac{F}{K}}$, $C_p=C_p$, $R_{eq}=R{\omega }_n$, $\Omega ={\displaystyle \frac{\omega }{{\omega }_n}}$, and $K=m{\omega }_n^2$.

The average output power associated with the system (Equation (1)) over a time interval of length h is given by

$${\displaystyle P={\displaystyle \frac{1}{h}}{\int }_{t_0}^{t_0+h}\lambda V{\left(t\right)}^{2}dt}$$

(2)

where $\lambda V{\left(t\right)}^2$ is the instantaneous power and $\lambda ={\displaystyle \frac{1}{R_{eq}{C}_p}}$ [11].

3. Material and Methods

The methodology used in this work was primarily based on techniques that complement each other for a more accurate analysis of the dynamics of periodic time-varying systems, namely, stability analysis via the Lyapunov–Floquer transformation and global sensitivity analysis. The stability analysis technique used in this work was proposed by Sinha et al. [12]. For the global sensitivity analysis, the technique based on the calculation of Sobol indices [6,13,14] was adopted. Both techniques can be interconnected, forming a type of ‘two-way street’ capable of providing a robust amount of information about the dynamics of the system, which, in turn, can be used to design a more efficient controller for the system. A schematic illustration of the methods applied in this work is shown in Figure 1. For the purpose of using the global sensitivity analysis results to assist in control design, we used the analysis results that were obtained by Cauz et al. [14].

3.1. Stability Analysis

In the analysis of the stability of the system (Equation (22)), we used the method developed by the French mathematician Floquet (1847–1920), together with a technique developed by Sinha et. al [12]. Floquet’s method is suitable for the study of the stability of closed trajectories if the analytical expressions of the corresponding solutions are known [15,16,17]. More precisely, Floquet showed, under certain conditions, that for a class of time-invariant linear systems, studying the solutions of a time-variant system is equivalent to studying the solutions of an invariant linear system. Unfortunately, the analytical study of these systems, as proposed by Floquet, has certain limitations.

The technique developed by Sinha and Butcher is based on Picard iterations and Chebyshev polynomial expansions [18], and aims to find approximate solutions for time-periodic systems. A detailed study on Sinha and Butcher’s technique can also be found in [12].

The main formulation and theorems on the Lyapunov–Floquet theory are presented below.

3.1.1. Some Results from Lyapunov–Floquet Theory

Consider a system of n first-order differential equations, which can be written in the form

$$\begin{array}{c}\hfill \dot{x}\left(t\right)=A\left(t\right)x\left(t\right)\end{array}$$

(3)

where $x\left(t\right)\in I{R}^n$ and $A\left(t\right)\in I{R}^{n\times n}$, where the elements $A_{ij}\left(t\right)$ of $A\left(t\right)$ are continuous and periodic functions in time of period T, and consequently, $A(t+T)=A\left(t\right)$ for all $t\ge 0$. Thus, this system (Equation (3)) is said to be periodic in time. Suppose that the system (Equation (3)) admits n linearly independent solutions $x^{\left(j\right)}\left(t\right)$. Such solutions form a fundamental set, in terms of which any other solution is written as their linear combination. This set can be expressed in the form of a matrix $\Phi \left(t\right)$ called the fundamental matrix of the system or the state transition matrix (STM):

$$\begin{array}{c}\hfill \Phi \left(t\right)=\left[\begin{array}{cccc}{x}^{\left(1\right)}\left(t\right)& x^{\left(2\right)}\left(t\right)& \cdots & x^{\left(n\right)}\left(t\right)\end{array}\right].\end{array}$$

In 1883, Floquet proposed the following result [19]:

Theorem 1.

If $\Phi \left(t\right)$ is an STM of the system described by Equation (3), where $A(t+T)=A\left(t\right)$ for all $t\ge 0$, then $\Phi (T+t)$ is also an STM of the system (Equation (3)). Moreover, for each STM $\Phi \left(t\right)$, there exists a non-singular periodic matrix $Q\left(t\right)$ of period T (with $Q\left(0\right)=I$) and a constant matrix R such that

$$\begin{array}{c}\hfill \Phi \left(t\right)=Q\left(t\right)e^{Rt}.\end{array}$$

(4)

More specifically, if R is a real (or complex) matrix of constant coefficients, then $Q\left(t\right)$ is $2T$-periodic (or T-periodic) [20]. Note that from Equation (4), one clearly sees that the decay of the system is determined by the term $e^{Rt}$ and, in the particular case when $t=T$, one can calculate the matrix R by

$$\begin{array}{c}\hfill \Phi \left(T\right)=e^{RT}\end{array}$$

where the matrix $\Phi \left(t\right)$ is calculated at the end of period T. The matrix $\Phi \left(T\right)$ is called the Floquet transition matrix (FTM) or the monodromy matrix. The eigenvalues ${\rho }_j$ of $\Phi \left(T\right)$ are called characteristic multipliers or Floquet multipliers [16]. The matrix R can be obtained from the FTM by means of the following identity [21]:

$$\begin{array}{c}\hfill R={\displaystyle \frac{1}{2T}}log\Phi \left(2T\right)={\displaystyle \frac{1}{2T}}log[\Phi {\left(T\right)]}^2.\end{array}$$

As a consequence of Theorem 1, when applied to a change in variables known as a Lyapunov–Floquet transformation or simply an L-F transformation, we have the following corollary:

Corollary 1.

The L-F T-periodic transformation

$$\begin{array}{c}\hfill x\left(t\right)=Q\left(t\right)z\left(t\right)\end{array}$$

with $Q(t+T)=Q\left(t\right)$ and $Q\left(0\right)=I_n$, where $I_n$ is the identity matrix of order n, reduces the time-variant system $\dot{x}\left(t\right)=A\left(t\right)x\left(t\right)$ to the time-invariant form

$$\begin{array}{c}\hfill \dot{z}\left(t\right)=Rz\left(t\right).\end{array}$$

The eigenvalues ${\lambda }_j$ of R are related to the characteristic multipliers through

$$\begin{array}{c}\hfill {\rho }_j=e^{{\lambda }_{j}T}.\end{array}$$

(5)

From Equation (5), it follows that the characteristic exponents are given by [22]

$$\begin{array}{c}\hfill {\lambda }_j={\displaystyle \frac{1}{T}}(log|{\rho }_j|+iarg{\rho }_j),j=1,2,\dots ,n.\end{array}$$

(6)

Remark 1.

Note from Corollary 1 of Theorem 1, a linear system with periodic coefficients (time invariant) can be transformed into a linear system with constant coefficients (time invariant).

The numbers ${\lambda }_j$ are called characteristic exponents or Floquet exponents. The stability of the system (Equation (3)) can be studied by analyzing the characteristic exponents, as can be seen from the following theorem [15]:

Theorem 2.

Regarding the stability of the system (Equation (3)), we have the following results:

$\left(i\right)$

If all characteristic exponents have a negative real part, then all solutions of the system (Equation (3)) are asymptotically stable.

$\left(ii\right)$

If at least one of the characteristic exponents has a positive real part, then the system (Equation (3)) is unstable.

$\left(iii\right)$

If all the characteristic exponents have a zero or negative real part, and those with a zero real part are simple roots of the characteristic polynomial of R, then the system is stable; if the roots with a zero real part are not simple, then the system is unstable.

As a consequence of Equation (6) and Theorem 2, the asymptotic stability of the system (Equation (3)) related to the Floquet multipliers is given in terms of the following corollary:

Corollary 2.

If $|{\rho }_j|<1$, for all $j=1,2,\dots ,n$, then the system Equation (3) is asymptotically stable. If $|{\rho }_j|>1$ for some j, the system is unstable.

3.1.2. Sinha and Butcher Technique for State Transition Matrix Approximation

The great difficulty in applying both Theorems 1 and 2 is in obtaining the state transition matrix (STM) $\Phi \left(t\right)$. This difficulty is overcome when the theorems are applied to commutative systems. However, in most cases, it is practically impossible to obtain the STM$\Phi \left(t\right)$. To circumvent this problem, Sinha et al. [12] developed a numerical-computational method to approximate the STM$\Phi \left(t\right)$.

Consider a nonlinear periodic dynamical system of dimension n of the form

$$\begin{array}{c}\hfill \dot{z}\left(t\right)=f(z\left(t\right),t,\alpha )=f(z\left(t\right),t+T,\alpha ),z\left(0\right)=z^0\end{array}$$

(7)

where $t\in I{\mathbb{R}}^{+}$ denotes the time, $z\in I{\mathbb{R}}^n$ is the state vector, $\alpha \in I{\mathbb{R}}^m$ is a vector of system parameters, and $f:I{\mathbb{R}}^n\times I{\mathbb{R}}^{+}\times I{\mathbb{R}}^m\to I{\mathbb{R}}^n$ is analytic in the z and $\alpha$ components and periodic with period T at time t. Suppose that $\overline{z}$ is an equilibrium or periodic solution of period $KT$ of system (7) and $x\left(t\right)=z\left(t\right)-\overline{z}\left(t\right)$ is a perturbation of this solution. Thus, by expanding as a Taylor series, system (7) around $z=\overline{z}$ becomes

$$\begin{array}{c}\hfill \begin{array}{c}\dot{x}=A(t,\alpha )x+f_2(x,t,\alpha )+f_3(x,t,\alpha )+\cdots +f_r(x,t,\alpha )+{O\left(\right|x|}^{r+1}),\hfill \\ x\left(0\right)=x^0=z^0-\overline{z}\left(0\right),\hfill \end{array}\end{array}$$

(8)

where $f_r$ represents the terms of order r ($r\ge 2$) of the Taylor series expansion of f, and $A(t,\alpha )$ and $f_r(x,t,\alpha )$ are periodic with period $KT$. Consider the linear part of Equation (8) given by

$$\begin{array}{c}\hfill \dot{x}(t,\alpha )=A(t,\alpha )x(t,\alpha ),x(0,\alpha )=x^0.\end{array}$$

(9)

Taking into account Equations (3) and (9) and Theorem 1, the local stability of Equation (8) is given by the Floquet multipliers ${\rho }_j$ of the Floquet transition matrix (FTM):

$$\begin{array}{c}\hfill \Phi (KT,\alpha ).\end{array}$$

To introduce the Sinha and Butcher technique, we change the variable $t=KT\tau$ to transform the linear system (Equation (9)) of period $KT$ into the following system of period 1:

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{d\tau }}x(\tau ,\alpha )=\overline{A}(\tau ,\alpha )x(\tau ,\alpha ),\overline{A}(\tau +1,\alpha )=\overline{A}(\tau ,\alpha ),x(0,\alpha )=x^0,\end{array}$$

(10)

where

$$\begin{array}{c}\hfill \begin{array}{c}\overline{A}(\tau ,\alpha )={\overline{A}}_1\left(\alpha \right)f_1\left(\tau \right)+{\overline{A}}_2\left(\alpha \right)f_2\left(\tau \right)+\cdots +{\overline{A}}_r\left(\alpha \right)f_r\left(\tau \right),\hfill \\ {\overline{A}}_i\left(\alpha \right)=T{A}_i\left(\alpha \right),i=1,2,\cdots ,r.\hfill \end{array}\end{array}$$

According to [12,23], the state transition matrix (STM) $\Phi (\tau ,\alpha )$ of the system (Equation (10)) can be approximated by

$$\begin{array}{c}\hfill {\Phi }^{(p,m)}(\tau ,\alpha )={\widehat{T}}^T\left(\tau \right)\left[\widehat{I}+\left(\sum _{k=1}^{p-1}{\left[L\left(\alpha \right)\right]}^{k-1}\right)P\left(\alpha \right)\right].\end{array}$$

where in the term $(p,m)$, p is the number of Picard iterations; m is the number of terms of modified Chebyshev polynomials $T_0^{*}\left(t\right),T_1^{*}\left(t\right),\dots T_{m-1}^{*}\left(t\right)$; ${\widehat{T}}^T$ is the Chebyshev polynomial matrix; $L\left(\alpha \right)={\widehat{G}}^T{\widehat{Q}}_D$; and $P\left(\alpha \right)={\widehat{G}}^{T}D\left(\alpha \right)$. The process of constructing the matrix $D\left(\alpha \right)$ and obtaining the operational matrices $\widehat{I}$, $\widehat{G}$, and ${\widehat{Q}}_D$ can be seen in [12].

3.2. Control of Periodic Systems Via Lyapunov–Floquet Transformation

Consider the time-invariant nonlinear system given by

$$\begin{array}{c}\hfill \dot{x}=f(x\left(t\right),t)+u_c\left(t\right),\end{array}$$

(11)

with $x\left(t\right)\in I{\mathbb{R}}^n$, and in which there is a chaotic attractor for a given set of parameters when $u_c\left(t\right)=0$. Let $y\left(t\right)$ be the desired trajectory and $x\left(t\right)$ be the trajectory of the controlled system. The objective of the control law $u_c$ is to drive the chaotic trajectory to the desired orbit. In this method, the control law $u_c\left(t\right)$ is defined in two parts:

$$\begin{array}{c}\hfill u_c\left(t\right)=u_f+u_t,u_f=\dot{y}-f(y\left(t\right),t),u_t=F\left(t\right)u\left(t\right).\end{array}$$

The $u_f$ part of the controller is called the feedfoward and the $u_t$ part is the feedback. The matrix $F\left(t\right)$ is called the gain matrix and is obtained next.

We can then rewrite the system (Equation (11)) in the form

$$\begin{array}{c}\hfill \dot{x}=f(x\left(t\right),t)+\dot{y}-f(y\left(t\right),t)+F\left(t\right)u\left(t\right).\end{array}$$

By defining $e\left(t\right)=x\left(t\right)-y\left(t\right)$ as the dynamic error between $x\left(t\right)$ and $y\left(t\right)$, it follows that

$$\begin{array}{c}\hfill \dot{e}=g(e\left(t\right),t)+F\left(t\right)u\left(t\right),\end{array}$$

(12)

where $g\left(e\right(t),t)=f\left(e\right(t)+y(t),t)-f\left(y\right(t),t)$ is a nonlinear function of class $C^1$.

If the following condition is satisfied:

$$\begin{array}{c}\hfill {\displaystyle \underset{\left|\right|e\left|\right|\to 0}{lim}\underset{t\ge 0}{sup}{\displaystyle \frac{\left|\right|g(e,t)-A\left(t\right)e+F\left(t\right)u\left(t\right)\left|\right|}{\left|\right|e\left|\right|}}=0,}\end{array}$$

where $A\left(t\right)\in I{\mathbb{R}}^{n\times n}$ and

$$\begin{array}{c}\hfill A\left(t\right)=\left[\begin{array}{c}{\displaystyle \frac{\partial g_j}{\partial e_i}}(0,t)\end{array}\right],\end{array}$$

then the linearization of (12) around $e=0$ can be written as

$$\begin{array}{c}\hfill \dot{e}=A\left(t\right)e\left(t\right)+F\left(t\right)u\left(t\right).\end{array}$$

(13)

Consider now that the system (13) has the general form

$$\begin{array}{c}\hfill \dot{z}=A\left(t\right)z\left(t\right)+B\left(t\right)u_t,\end{array}$$

(14)

where the matrices $A\left(t\right)$ and $B\left(t\right)$ have periodic coefficients with period T and the pair $[A,B]$ is controllable. Applying the Lyapunov–Floquet transformation [12]

$$\begin{array}{c}\hfill z\left(t\right)=Q\left(t\right)q\left(t\right),\end{array}$$

in Equation (14), there is

$$\begin{array}{c}\hfill \dot{q}=Rq\left(t\right)+Q^{-1}\left(t\right)B\left(t\right)u_t\end{array}$$

(15)

where $R={\displaystyle \frac{1}{2T}}ln\left({\Phi }^2\left(T\right)\right)$ and $\Phi \left(T\right)$ is the Floquet transition matrix.

Since the gain matrix of system (15), $Q^{-1}\left(t\right)B\left(t\right)$, is time invariant, one should construct an auxiliary system whose gain matrix is constant in time:

$$\begin{array}{c}\hfill \dot{\overline{q}}=R\overline{q}+B_{0}v\left(t\right),\end{array}$$

(16)

where $B_0$ is a constant matrix of full rank such that the pair $[R,B_0]$ is controllable. Consider the control law v given by

$$\begin{array}{c}\hfill v\left(t\right)=F_0\overline{q}\left(t\right),\end{array}$$

(17)

where $F_0$ is the gain matrix obtained from the controller $v\left(t\right)$, which can be obtained such that the system (Equation (17)) is asymptotically stable.

By defining the dynamic error $\epsilon \left(t\right)=q\left(t\right)-\overline{q}\left(t\right)$, using Equations (15)–(17) and adding and subtracting $B_0{F}_0\epsilon \left(t\right)$, we have

$$\begin{array}{c}\hfill \dot{\epsilon }\left(t\right)=(R+B_0{F}_0)\epsilon \left(t\right)+Q^{-1}\left(t\right)B\left(t\right)u_t\left(t\right)-B_0{F}_{0}q\left(t\right).\end{array}$$

(18)

With an appropriate choice of the matrix $F_0$, the stability matrix (18) is $(R+B_0{F}_0)$, and therefore, systems (15) and (16) can be considered equivalent if

$$\begin{array}{c}\hfill Q^{-1}\left(t\right)B\left(t\right)u_t\left(t\right)=B_0{F}_{0}q\left(t\right),forallt\ge 0.\end{array}$$

(19)

and therefore,

$$\begin{array}{c}\hfill u_t\left(t\right)=B^{*}\left(t\right)Q\left(t\right)B_0{F}_{0}q\left(t\right),where{B}^{*}={\left(B^{T}B\right)}^{-1}{B}^T.\end{array}$$

(20)

By applying the inverse Lyapunov–Floquet transformation $q\left(t\right)=Q^{-1}\left(t\right)z\left(t\right)$ to Equation (20), we obtain

$$\begin{array}{c}\hfill u_t\left(t\right)=B^{*}\left(t\right)Q\left(t\right)B_0{F}_0{Q}^{-1}\left(t\right)z\left(t\right),\end{array}$$

and therefore, we conclude that the time-varying gain matrix $F\left(t\right)$ is given by

$$\begin{array}{c}\hfill F\left(t\right)=B^{*}\left(t\right)Q\left(t\right)B_0{F}_0{Q}^{-1}\left(t\right).\end{array}$$

Therefore, the control law with state feedback can be written in the form

$$\begin{array}{c}\hfill u_t=-B^{*}\left(t\right)Q\left(t\right)B_0{F}_0{Q}^{-1}\left(t\right)z\left(t\right).\end{array}$$

(21)

4. Results and Discussions

In all the numerical results presented in this work, the following nominal values were adopted for the parameters: ${\mu }_1=0.01$, ${\mu }_2=0.01$, $\alpha =0.001$, $\beta =0.001$, $\theta =0.05$, $K=0.5$, and $R_{eq}=20$. The parameters F, $\Omega$, and $C_p$ were varied, corresponding to the amplitude and frequency of the parametric force and the capacitance of the piezoelectric element, respectively. MATLAB (version R2022a, MathWorks) was used to run all the simulations. The initial conditions were defined by the vector $(x_0,{\dot{x}}_0,V_0)=(0.1,0,0)$.

4.1. Stability Analysis

Initially, the numerical results for the stability analysis based on the Floquet method [15,16,17] and a state transition matrix approximation technique [12] are presented.

For the purpose of investigating the dynamic stability of the system and searching for possible static solutions, we rewrite the dimensionless dynamic equations of the system (Equation (1)) in the form of state equations. To do so, we introduce the changes in variables $x_1=x$, $x_2=\dot{x}$, and $x_3=V$ to obtain the state equations of the system in the following form:

$$\begin{array}{c}\hfill \left\{\begin{array}{ccc}\hfill {\dot{x}}_1& =& x_2\hfill \\ \hfill {\dot{x}}_2& =& {\displaystyle \frac{1}{1+2\beta x_1^2}}[-2{\mu }_1{x}_2-x_1-{\mu }_2\left|x_2\right|x_2-\alpha x_1^3-2\beta x_1{x}_2^2+x_{1}Fcos(\Omega t)\hfill \\ & +& {\displaystyle \frac{\theta }{K}}{x}_3]\hfill \\ \hfill {\dot{x}}_3& =& -{\displaystyle \frac{\theta }{C_p}}{x}_2-{\displaystyle \frac{1}{C_{p}Req}}{x}_3\hfill \end{array}\right.\end{array}$$

(22)

The system has three equilibrium points $(x_1^{*},x_2^{*},x_3^{*})$, namely, $(0,0,0)$, $(\sqrt{\frac{F-1}{\alpha }},0,0)$, and $(-\sqrt{\frac{F-1}{\alpha }},0,0)$. The last two equilibrium points only make sense when $\Omega =0$, i.e., if the system is not subject to periodic forcing. In this study, we dealt only with a system (1) subject to periodic forcing, and therefore, we focused only on the equilibrium point $(0,0,0)$.

To apply Sinha’s method [12], it is necessary to linearize the equations of states (Equation (22)) around the equilibrium point $(0,0,0)$. Thus, the equations of the system can be written in the form $\dot{x}(t,\mathcal{P})=A(t,\mathcal{P})x(t,\mathcal{P})$, where $\mathcal{P}$ represents its parameters and $A(t,\mathcal{P})$ is a periodic matrix of period $T=\frac{2\pi }{\Omega }$. By performing the transformation $t={\displaystyle \frac{2\pi }{\Omega }}\tau$, the linearized system is rewritten in the form ${\displaystyle \frac{dx}{d\tau }}=\overline{A}(\tau ,\mathcal{P})x(\tau ,\mathcal{P})$, where $\overline{A}(\tau ,\mathcal{P})=\overline{A_1}\left(\mathcal{P}\right)f_1\left(\tau \right)+\overline{A_2}\left(\mathcal{P}\right)f_2\left(\tau \right)$, $f_1\left(\tau \right)=1$, $f_2\left(\tau \right)=cos\left(2\pi \tau \right)$,

$$\begin{array}{c}\hfill {\overline{A}}_1\left(\mathcal{P}\right)={\displaystyle \frac{2\pi }{\Omega }}\left[\begin{array}{ccc}0& 1& 0\\ -1& -2{\mu }_1& {\displaystyle \frac{\theta }{k}}\\ 0& -{\displaystyle \frac{\theta }{C_p}}& -{\displaystyle \frac{1}{C_p{R}_{eq}}}\end{array}\right],{\overline{A}}_2\left(\mathcal{P}\right)={\displaystyle \frac{2\pi }{\Omega }}\left[\begin{array}{ccc}0& 0& 0\\ F& 0& 0\\ 0& 0& 0\end{array}\right].\end{array}$$

To analyze the structural stability, we investigated whether the modulus of each characteristic multiplier ${\rho }_j$ was greater than or equal to 1 for given values of the physical parameters. These multipliers could be calculated by the approximate fundamental matrix. In the approximation, the degree of the modified Chebyshev polynomial was $m=20$, and the number of Picard iterations was $p=40$. Figure 2 describes the stability of the system for different values of the parameters that characterize the parametric force. This was done according to the approach in Section 3.1.1 [12,23].

Figure 2a–d show the stability diagrams of the equilibrium point $(0,0,0)$ when the amplitude of the external force F was varied. Note in these figures that the dashed line is only a reference to observe the values of the moduli of the characteristic multipliers. The equilibrium point was asymptotically stable for values of F for which the moduli of the characteristic multipliers were below the dashed line. If the opposite occurred, the equilibrium was unstable. In Figure 2a, $\Omega =0.9$ was adopted and the amplitude F was varied in the interval $[0.01,1]$. There was a range of values of F for which the equilibrium point of the system was stable, i.e., the characteristic multipliers had a modulus less than 1, and the condition for instability in the range of values occurred when the multipliers had a modulus greater than 1. Figure 2b presents the stability diagram in the resonance region $1:1$$(\Omega =1).$ The stability change occurred for smaller values of the amplitude F when compared with what occurred for $\Omega =0.9$. In Figure 2c, the stability diagram is presented when the system was in the resonance regime $2:1$$(\Omega =2)$. In this condition, the stability change occurred for values of F that varied in a small range of values, i.e., $F\in [0.001,0.07]$. In Figure 2d, $\Omega =0.64$ and $0.01\le F\le 1$ were assumed. Note again that the system presented conditions for stability and for instability since there was a range for values of F for which the characteristic multipliers had a modulus smaller than 1 and a range for which the characteristic multipliers had a modulus larger than 1.

Figure 2e shows the stability (in blue) and instability (in green) regions of the system in the parameter space $\Omega F$. The blue region represents the values of $\Omega$ and F for which the characteristic multipliers had a modulus less than 1, while the green area represents the values of $\Omega$ and F for which the multipliers had a modulus greater than 1. Note that the stability (instability) conditions of Figure 2a–d can be observed in Figure 2e. In the stability region, the system always converged to the equilibrium point $(0,0,0)$. In particular, when $F=0.4$ and $\Omega =1.6$, $|{\rho }_1|=|{\rho }_2|=0.9615$ and $|{\rho }_3|=0.9902$, indicating stability, while if $F=0.056$ and $\Omega =2$, $|{\rho }_1|=1.0126$, $|{\rho }_2|=0.9274$, and $|{\rho }_3|=0.9922$, the system was unstable because $|{\rho }_1|>1$.

In order to observe the response of the system in some specific situations, three figures were generated that corresponded to the time history of the displacement of the beam (Figure 3). In Figure 3a,b, it is possible to see two different situations.

In Figure 3a, the parameters F and $\Omega$ were set to correspond to the stability region ($F=0.4$ and $\Omega =0.6$), which made the beam converge to the equilibrium position. In Figure 3b, $F=0.056$ and $\Omega =2$ represented an unstable condition, as previously observed by calculating the characteristic multipliers. In this configuration, the system first underwent a transition and then entered the periodic regime. In Figure 3c, we can see the behavior of the beam in the time interval $[7800,8000]$ and the Poincaré sections, which suggest that Figure 3b represents periodic behavior of the system.

4.2. Global Sensitivity Analysis

In Cauz et al. [14], it is shown that the system is more sensitive to the variation in parameters $\Omega$, F, and $C_P$ when the interest is to analyze the gain in electrical power generated by the system. Furthermore, it was observed that the average output power produces a maximum when $C_P=0.485$ in the $\Omega =2$ resonance region.

Taking into account the stability region (Figure 2e) and the sensitivity of the average output power of the system to the parameters $\Omega$ and $C_p$ in the neighborhood of $\Omega =2$ [14], the time histories of the displacement $x_1$ (Figure 4a) and the voltage $x_3$ (Figure 4b) are presented. Note that the system entered a periodic regime, as can be observed on the Poincaré section (Figure 5a) and the Lyapunov exponents (Figure 5b).

4.3. Control Design Via Lyapunov–Floquet Transformation

The purpose of this section is to design a controller that took the uncontrolled system to a hypothetically desired trajectory y so that the average net electrical power of the controlled system was greater than the average power of the uncontrolled system. For this purpose, we chose the following parameter values: $\Omega =2$, $F=0.062$, and $C_p=0.485$. The choice of $\Omega$ and F was made following the stability studies of Section 4.1 (Figure 2e). Under this condition, the equilibrium point $(0,0,0)$ of the system was unstable, but it entered the periodic regime (Figure 4 and Figure 5). The choice of the capacitance value $C_p$ was derived from the global sensitivity analysis (Section 4.2), for which the uncontrolled system presented a maximum in energy production.

Consider the nonlinear system (22) rewritten in the form

$$\begin{array}{c}\hfill \dot{x}=f(x,t)+u_c\left(t\right)\end{array}$$

(23)

where $u_c\left(t\right)$ is the control law, as described in Section 3.2; $x={(x_1,x_2,x_3)}^T$; and $f(x,t)={(f_1(x,t),f_2(x,t),f_3(x,t))}^T$, where

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill f_1(x,t)& =& x_2\hfill \\ \hfill f_2(x,t)& =& {\displaystyle \frac{1}{1+2\beta x_1^2}}\left[-2{\mu }_1{x}_2-x_1-{\mu }_2\left|x_2\right|x_2-\alpha x_1^3-2\beta x_1{x}_2^2+x_{1}Fcos(\Omega t)+{\displaystyle \frac{\theta }{K}}{x}_3\right]\hfill \\ \hfill f_3(x,t)& =& -{\displaystyle \frac{\theta }{C_p}}{x}_2-{\displaystyle \frac{1}{C_{p}Req}}{x}_3\hfill \end{array}\end{array}$$

(24)

For each instant in time t, consider the vector $e\left(t\right)$ given by $e\left(t\right)=x\left(t\right)-y\left(t\right)$, where $x\left(t\right)$ the trajectory of the controlled system (24) and $y\left(t\right)={(y_1,y_2,y_3)}^T$ is the desired trajectory. By applying Equation (24), we obtain

$$\begin{array}{c}\hfill \dot{e}=g(e\left(t\right),t)+B\left(t\right)u\left(t\right),suchthatg(e,t)={(g_1(e,t),g_2(e,t),g_3(e,t))}^T\end{array}$$

where

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill g_1(e,t)& =& e_2\hfill \\ \hfill g_2(e,t)& =& {\displaystyle \frac{1}{1+2\beta {(e_1+y_1)}^2}}\left[-2{\mu }_1(e_2+y_2)-(e_1+y_1)-{\mu }_2|e_2+y_2|(e_2+y_2)-\alpha {(e_1+y_1)}^3\right.\hfill \\ & & \left.-2\beta (e_1+y_1){(e_2+y_2)}^2+(e_1+y_1)Fcos(\Omega t)+{\displaystyle \frac{\theta }{K}}(e_3+y_3)\right]-{\displaystyle \frac{1}{1+2\beta y_1^2}}\left[-2{\mu }_1{y}_2\right.\hfill \\ & & \left.-y_1-{\mu }_2\left|y_2\right|y_2-\alpha y_1^3-2\beta y_1{y}_2^2+y_{1}Fcos(\Omega t)+{\displaystyle \frac{\theta }{K}}{y}_3\right]\hfill \\ \hfill g_3(e,t)& =& -{\displaystyle \frac{\theta }{C_p}}{e}_2-{\displaystyle \frac{1}{C_{p}Req}}{e}_3\hfill \end{array}\end{array}$$

By linearizing $g\left(e\right(t),t)$ around $e=0$ and by carefully selecting the trajectory y as $y_1=a+\epsilon cos(\Omega t)$, $y_2=b+\epsilon dsin(\Omega t)$, and $y_3=ccos(\Omega t)$ such that $a,b,c,d,\epsilon \in I\mathbb{R}$ are constants with $a>0$, $b>0$, $c>0$, $d\ne 0$, $b>\epsilon \left|d\right|$, and $0<\epsilon \ll 1$, we obtain

$$\begin{array}{c}\hfill \dot{e}=A\left(t\right)e\left(t\right)+B\left(t\right)u\left(t\right)+G(e\left(t\right),\epsilon )\end{array}$$

(25)

where $G\left(e\right(t),\epsilon )$ stands for the terms related to the powers of $\epsilon$,

$$\begin{array}{c}\hfill A\left(t\right)=\left[\begin{array}{ccc}{A}_{11}\left(t\right)& A_{12}\left(t\right)& A_{13}\left(t\right)\\ A_{21}\left(t\right)& A_{22}\left(t\right)& A_{23}\left(t\right)\\ A_{31}\left(t\right)& A_{32}\left(t\right)& A_{33}\left(t\right)\end{array}\right],A_{ij}\left(t\right)={\displaystyle \frac{\partial g_i}{\partial e_j}}(0,t),\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{c}{A}_{11}\left(t\right)=0,A_{12}\left(t\right)=1,A_{13}\left(t\right)=0\hfill \\ A_{21}\left(t\right)=-{\displaystyle \frac{4\beta a}{{(1+2\beta a^2)}^2}}\left[-2{\mu }_{1}b-a-{\mu }_2{b}^2-\alpha a^3-2\beta a{b}^2\right]+{\displaystyle \frac{1}{1+2\beta a^2}}\left[-1-3\alpha a^2-2\beta b^2\right]\hfill \\ -\left[{\displaystyle \frac{4\beta a}{{(1+2\beta a^2)}^2}}\left(aF+{\displaystyle \frac{\theta }{K}}c\right)-{\displaystyle \frac{F}{1+2\beta a^2}}\right]cos(\Omega t)\hfill \\ A_{22}\left(t\right)={\displaystyle \frac{1}{1+2\beta a^2}}\left[-2{\mu }_1-2{\mu }_{2}b-4\beta ab\right],A_{23}\left(t\right)={\displaystyle \frac{\theta }{K(1+2\beta a^2)}}\hfill \\ A_{31}\left(t\right)=0,A_{32}\left(t\right)=-{\displaystyle \frac{\theta }{Cp}},A_{33}\left(t\right)=-{\displaystyle \frac{1}{C_{p}Req}}\hfill \end{array}\end{array}$$

The system (Equation (25)) is rewritten by parameterizing the time $t=T\tau$ with $T={\displaystyle \frac{2\pi }{\Omega }}$ to give

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{d\tau }}e=\overline{A}\left(\tau \right)e\left(\tau \right)+B\left(\tau \right)u\left(\tau \right)+G(e\left(\tau \right),\epsilon )\end{array}$$

(26)

where $\overline{A}\left(\tau \right)=\overline{A}(\tau ,\Lambda )={\overline{A}}_1(\Lambda )f_1\left(\tau \right)+{\overline{A}}_2(\Lambda )f_2\left(\tau \right)$, $f_1\left(\tau \right)=1$, $f_2\left(\tau \right)=cos2\pi \tau$

$$\begin{array}{c}\hfill {\overline{A}}_1(\Lambda )=\left(\begin{array}{ccc}0& 1& 0\\ {\Theta }_{21}& {\Theta }_{22}& {\Theta }_{23}\\ 0& -{\displaystyle \frac{\theta }{C_p}}& -{\displaystyle \frac{1}{C_p{R}_{eq}}}\end{array}\right),{\overline{A}}_2(\Lambda )=\left(\begin{array}{ccc}0& 0& 0\\ \Gamma & 0& 0\\ 0& 0& 0\end{array}\right),\end{array}$$

${\Theta }_{21}=-{\displaystyle \frac{4\beta a}{{(1+2\beta a^2)}^2}}\left[-2{\mu }_{1}b-a-{\mu }_2{b}^2-\alpha a^3-2\beta a{b}^2\right]+{\displaystyle \frac{1}{1+2\beta a^2}}\left[-1-3\alpha a^2-2\beta b^2\right]$,

${\Theta }_{22}={\displaystyle \frac{1}{1+2\beta a^2}}\left[-2{\mu }_1-2{\mu }_{2}b-4\beta ab\right]$, ${\Theta }_{23}={\displaystyle \frac{\theta }{K(1+2\beta a^2)}}$ and $\Gamma =-\left[{\displaystyle \frac{4\beta a}{{(1+2\beta a^2)}^2}}\left(aF+{\displaystyle \frac{\theta }{K}}c\right)-{\displaystyle \frac{F}{1+2\beta a^2}}\right]$

By applying the Lyapunov–Floquet transformation $e=Q\left(t\right)q$, Sinha’s techniques for the state transition matrix approximation and using Equation (26), we can rewrite the system (25) in the form

$$\begin{array}{c}\hfill \dot{q}=Rq\left(t\right)+Q^{-1}\left(t\right)\overline{B}\left(t\right)u_t,\end{array}$$

(27)

where

$$\begin{array}{c}\hfill R=\left[\begin{array}{ccc}\hfill -0.0132& \hfill 0.0203& \hfill -0.0084\\ \hfill 0.0104& \hfill -0.0128& \hfill -0.0001\\ \hfill -0.0092& \hfill -0.0030& \hfill -0.1012\end{array}\right]\end{array}$$

The eigenvalues of the stability matrix of the system (27) were ${\lambda }_1=-0.0020$, ${\lambda }_2=-0.0272$, and ${\lambda }_3=-0.1020$. Since ${\lambda }_1=0.0020$, the time-invariant system (27) was unstable. Since the gain matrix of (27) was time invariant, there should be an equivalent time-invariant system (Equation (16)) and a matrix $B_0$ that satisfies the conditions given by Equations (16)–(19). $B_0$ was chosen to be equal to $I_3$, where $I_3$ is the identity matrix of order 3.

Following Section 3.2 and the procedures proposed by Sinha and Joseph [7], David and Sinha [8], and Sinha and David [9], one can design a linear controller using the pole allocation technique. Choosing $-1$, $-1$, and $-1$ as new poles, the time-invariant gain matrix $F_0$ obtained from the pole allocation is given by

$$\begin{array}{c}\hfill F_0=\left[\begin{array}{ccc}\hfill 0.9868& \hfill 0.0203& \hfill -0.0084\\ \hfill 0.0104& \hfill 0.9872& \hfill -0.0001\\ \hfill -0.0092& \hfill -0.0030& \hfill 0.8988\end{array}\right]\end{array}$$

Therefore, the control law of the nonlinear system (Equation (23)) is given in the form of Equation (21), where $Q\left(t\right)$ is given by Theorem 1.

Figure 6 shows the behaviors of the controlled and uncontrolled system as a function of time t. Note that the states of the controlled system converged quickly to the chosen orbit (Figure 6a–c). In addition, there was a significant increase in the oscillation amplitude of the $x_3$ voltage of the controlled system when compared with that of the uncontrolled system, which favored an increase in energy production (Figure 6c).

Figure 7a,b show the average power as a function of capacitance $C_p$ for $0.05\le C_p\le 1$. Figure 7a shows a comparison between the average power generated by the controlled and uncontrolled systems. Figure 7b shows a magnification of Figure 7a around the curve representing the average power of the uncontrolled system. Note from Figure 6c that the uncontrolled system produced small oscillations in the voltage-related $x_3$ coordinate. These oscillations were magnified in the controlled system. The increase in oscillation amplitude was reflected in the power gain, as shown in Figure 7.

4.4. Net Output Power

In 2005, Roundy and Zhang carried out one of the first studies to increase the efficiency of energy-harvesting systems using controllers. They presented the calculation of the controller’s actuation power and the system’s net power [24]. In 2010, Lallart et al. presented an improved actuation power equation by considering the average values and an actuation factor [25]. Briefly, in these studies, they compared the output powers of uncontrolled systems with that of controlled systems. In both cases, the power used for the control referred to the amount of change in the system parameters, that is, the energy cost to change the system behavior [26]. In general, the power used for control, also known as the actuation power, represents the amount of energy necessary to drive the uncontrolled system in a controlled state, which reflects the energy cost involved in changing the system’s behavior to achieve a result or desired state. This may include adjustments, requiring energy to make these modifications and keep the system running as designed. The greater the need for control and adjustment, the greater the power required to sustain these changes.

Based on the results, which were experimentally verified [24,25], the actuation power $P_{act}$ was determined using the fourth-order Runge–Kutta method to calculate the power generated by the controller feedback(Equation (21)). In the same way, the power generated by the controlled system was calculated (Equation (23)), and in this subsection, it is denoted by $P_c$. The values adopted for the parameters and the initial condition are given at the beginning of Section 4. The trajectory chosen for the system control was the same trajectory used in Section 4.3.

Figure 8 and Figure 9 present the net power $P_{net}$, actuation power $P_{act}$, power of the controlled system $P_c$, and power of the not-controlled system P. In both situations, the net power analysis was carried out by considering two situations chosen from the system stability diagram (Figure 2e).

Figure 8 and

Table 2. Output mean power values for not-controlled and controlled systems: $C_P=0.485$, $F=0.5$, and $0.6\le \Omega \le 1.3$.

Parameter	Output Mean Power
$\mathbf{\Omega }$	$\mathit{P}$	${\mathit{P}}_{\mathit{c}}$	${\mathit{P}}_{\mathit{act}}$	${\mathit{P}}_{\mathit{net}}$
$0.60$	$1.6651\times {10}^{-7}$	$0.051638$	$0.035705$	$0.015933$
$0.65$	$1.6866\times {10}^{-7}$	$0.051652$	$0.033895$	$0.017757$
$0.70$	$2.1682\times {10}^{-7}$	$0.051664$	$0.032137$	$0.019526$
$0.75$	$2.0655\times {10}^{-7}$	$0.051671$	$0.030446$	$0.021225$
$0.80$	$1.9426\times {10}^{-7}$	$0.051676$	$0.028777$	$0.022898$
$0.85$	$1.843\times {10}^{-7}$	$0.051674$	$0.027213$	$0.024462$
$0.90$	$1.7004\times {10}^{-7}$	$0.051666$	$0.025729$	$0.025937$
$0.95$	$1.5473\times {10}^{-7}$	$0.051655$	$0.024333$	$0.027322$
$1.00$	$3.143\times {10}^{-3}$	$0.051647$	$0.023018$	$0.028630$
$1.05$	$4.9621\times {10}^{-7}$	$0.051631$	$0.021786$	$0.029845$
$1.10$	$3.4989\times {10}^{-7}$	$0.051620$	$0.020632$	$0.030989$
$1.15$	$3.1814\times {10}^{-7}$	$0.051605$	$0.019549$	$0.032056$
$1.20$	$2.9985\times {10}^{-7}$	$0.051593$	$0.018533$	$0.033060$
$1.25$	$2.9405\times {10}^{-7}$	$0.051581$	$0.017582$	$0.033999$
$1.30$	$2.9371\times {10}^{-7}$	$0.051571$	$0.016693$	$0.034878$

were obtained when $F=0.5$ and $\Omega$ varied in the interval $[0.6,1.3]$ with a step size of $0.05$. Note that increasing the value of $\Omega$ reduced the energy used by the controller, which resulted in an increase in the net power. Note also that the uncontrolled system had an increase in electrical power generated when it passed through the resonance region ($\Omega =1$).

Figure 9 and

Table 3. Output mean power values for not-controlled and controlled systems: $C_P=0.485$, $F=0.1$, and $1.6\le \Omega \le 2.3$.

Parameter	Output Mean Power
$\mathbf{\Omega }$	$\mathit{P}$	${\mathit{P}}_{\mathit{c}}$	${\mathit{P}}_{\mathit{act}}$	${\mathit{P}}_{\mathit{net}}$
$1.60$	$3.4933\times {10}^{-7}$	$0.051573$	$0.012365$	$0.039207$
$1.65$	$3.4694\times {10}^{-7}$	$0.051580$	$0.011794$	$0.039785$
$1.70$	$3.4135\times {10}^{-7}$	$0.051592$	$0.011260$	$0.040331$
$1.75$	$3.3066\times {10}^{-7}$	$0.051593$	$0.010759$	$0.040834$
$1.80$	$3.1355\times {10}^{-7}$	$0.051599$	$0.010285$	$0.041314$
$1.85$	$2.8845\times {10}^{-7}$	$0.051604$	$0.009840$	$0.041764$
$1.90$	$2.5362\times {10}^{-7}$	$0.051606$	$0.009441$	$0.042184$
$1.95$	$2.4734\times {10}^{-7}$	$0.051615$	$0.009027$	$0.042588$
$2.00$	$1.8227\times {10}^{-3}$	$0.051613$	$0.008656$	$0.042957$
$2.05$	$7.9656\times {10}^{-7}$	$0.051612$	$0.008305$	$0.043307$
$2.10$	$9.3483\times {10}^{-7}$	$0.051609$	$0.007974$	$0.043636$
$2.15$	$6.5296\times {10}^{-7}$	$0.051604$	$0.007663$	$0.043941$
$2.20$	$5.7074\times {10}^{-7}$	$0.051598$	$0.007369$	$0.044228$
$2.25$	$5.3505\times {10}^{-7}$	$0.051588$	$0.007092$	$0.044497$
$2.30$	$5.1722\times {10}^{-7}$	$0.051580$	$0.006829$	$0.044752$

refer to the situations when $F=0.01$ and $1.6\le \Omega \le 2.3$ with a step size of $0.05$. With increasing nominal values of $\Omega$, there was a decrease in the energy used by the controller, and consequently, an increase in the net power generated by the system. Note also that the uncontrolled system had a significant increase in electrical power in the resonance region $2:1$$(\Omega =2)$.

Note that in the tested scenarios, the controlled system had a significant increase in the net power generated when compared with the original system (without control), which made the controlled system more efficient.

5. Conclusions

The stability analysis carried out in this study showed that the system presented conditions of stability (or instability) of its equilibrium point, depending on the choice of the nominal values of the parameters that characterized the parametric force. At the border of the region between stability and instability, it was possible to explore periodic regimes for the system.

The controller designed from the Lyapunov–Floquet transformation was shown to be effective in driving the trajectory of the system to a previously chosen one. Consequently, there was a significant increase in the production of energy generated by the system. This result indicates that the approach used contributed not only to the stability and direction of the system but also had a beneficial impact on the energy efficiency, which improved the system’s performance, and this was seen in the last part of this study, where in all the scenarios tested, the net mean power of the controlled system exceeded the mean power of the uncontrolled system.