Control and Stability Analysis of Complex Dynamical Systems involved in Finance, Ecology and Engineering
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Dany Ivan Martinez, José de Jesús Rubio, Arturo Aguilar, Jaime Pacheco, Guadalupe Juliana Gutierrez, Victor Garcia, Tomas Miguel Vargas, Genaro Ochoa, David Ricardo Cruz, Cesar Felipe Juarez, "Stabilization of Two Electricity Generators", Complexity, vol. 2020, Article ID 8683521, 13 pages, 2020. https://doi.org/10.1155/2020/8683521
Stabilization of Two Electricity Generators
Abstract
In this research, a sliding mode regulator with sine mapping is suggested for the stabilization of electricity generators being affected by magnet interaction nonlinearities and generator nonlinearities. To reach this goal, our suggested regulator has the following contributions: (a) it starts from the sliding mode regulator with the modifications that the saturation mapping is used to reach a smoother performance instead of the signum mapping, and the sine mapping is applied to reach an upper bound in the proportional gain error, (b) it is used to reach some chosen constant behaviors for the angle position, angle speed, and current in the electricity generators, and (c) its stabilization is ensured based on the Lyapunov approach. We show the simulation of the suggested regulator in two electricity generators.
1. Introduction
The term alternative energy is utilized in the electricity generation from the environment by the utilization of renewable fuels, evading the necessity of a no renewable fuel. Some of the most relevant alternative energies can be classified in electricity generators based on wind turbines, magnets, or solar panels. The electricity generator of this research uses the interactions between static and dynamic magnets for the electricity generation [1, 2]. The motivation of this research is the stabilization of electricity generators. A regulator is one technique for the stabilization of electricity generators which let us to reach some constant behaviors for the angle position, angle speed, and current [3–8].
There are many kinds of nonlinearities which impede to reach the stabilization of electricity generators, and some examples of these nonlinearities are the arbitrary switching [9–13], the timedelays [14–18], the impulse perturbations [19, 20], or the unknown nonlinearities [21–24]. The major issue is that in most of the cases, the mentioned nonlinearities are unknown. The challenge of this research is the stabilization of electricity generators being affected by unknown nonlinearities.
The magnet interaction nonlinearities and generator nonlinearities are two unknown nonlinearities of the electricity generators which can severely affect their performance [21–24]. Hence, the research about regulators for the stabilization of electricity generators where the magnet interaction nonlinearities and generator nonlinearities are unknown, which is of interest, and two alternatives are the proportional derivation regulator of [3–5] and the sliding mode regulator of [6–8].
In this research, a sliding mode regulator with sine mapping is suggested for the stabilization of electricity generators being affected by magnet interaction nonlinearities and generator nonlinearities. To reach this goal, our suggested regulator has the following contributions:(a)The sliding model regulator starts from the proportional derivation regulator with one additional term of the signum mapping used to reach the stabilization. The sliding mode regulator with sine mapping starts from the sliding mode regulator with the modifications that the saturation mapping is used to reach a smoother performance instead of the signum mapping, and the sine mapping is applied to reach an upper bound in the proportional gain error.(b)Since the sliding mode technique is used in the suggested regulator instead of the magnet interaction nonlinearities and generator nonlinearities, the knowledge of these nonlinearities is not required in the electricity generators. Our suggested regulator is also used to reach some chosen constant behaviors for the angle position, angle speed, and current.(c)We ensure the stabilization of the regulator error in our suggested regulator based on the Lyapunov approach. This research is focused on the asymptotic stabilization.
We organize the paper as follows: in Section 2, we present the mathematical model of electricity generators. In Section 3, we present the sliding mode regulator with sine mapping for the stabilization of electricity generator. In Section 4, we simulate the suggested regulator for the stabilization of two electricity generators. In Section 5 we present the conclusions and future research.
2. Mathematical Model and Regulators for the Electricity Generators
In Appendix A of this research, the descriptions of the electricity generator symbols are shown in Table 1 and the parameter values for mathematical models are shown in Table 2. The descriptions of the mathematical model symbols are shown in Table 3, and the descriptions of the regulator symbols are shown in Table 4.




In this section, we address some concepts such as the mathematical model and regulators for the electricity generators.
We consider the electricity generator with two static magnets and two dynamic magnets of Figure 1 via the utilization of two static magnets and two dynamic magnets.
We express the electricity generator with two static magnets and two dynamic magnets as [1, 2]
Since all magnets are equal, .
We consider the electricity generator with four static magnets and four dynamic magnets of Figure 1 via the utilization of four static magnets and four dynamic magnets.
We express the electricity generator with four static magnets and four dynamic magnets as [1, 2]
Since all magnets are equal, .
From the electricity generator with two static magnets and two dynamic magnets of equation (1) and the electricity generator with four static magnets and four dynamic magnets of equation (2), we express the mathematical model for the electricity generators as
The fictitious input is affected by the generator nonlinearities, it is like a chattering movement presented in the electricity generator. We express the fictitious input as [6]
We see that the model of equation (1) can be expressed as the model of equations (3) and (4) with
We see that the model of equation (2) can be expressed as the model of equations (3) and (4) with
The generator nonlinearities of electricity generators are symmetric as in equation (4); consequently, we can express the fictitious input of equation (4) as
After some mathematical operations, the fictitious input of equation (7) is expressed as
We notice that the generator nonlinearities are bounded as
Remark 1. Even the mathematical model for the electricity generators of equations (3) and (4) seems to be simple, the complexity in this mathematical model is mainly focused on the magnets interaction nonlinearities , such that the design and validation of this mathematical model resulted in two publications [1, 2].
Remark 2. From equations (5) and (6), it is observed that the magnet interaction nonlinearities are the most complex part in the mathematical model of equations (3) and (4), while , , , and are scalar constants corresponding to the electricity generators. Thus, these magnet interaction nonlinearities are mainly focused on this research.
Now, we express the proportional derivation and sliding mode regulators due to they will be utilized for the comparisons in a future section.
We express the proportional derivation regulator as [3–5]We express the sliding mode regulator as [6–8]
3. Sliding Mode Regulator with Sine Mapping
In this section, we address the design and stabilization of the sliding mode regulator with sine mapping.
We notice that the magnet interaction nonlinearities of equation (3) are bounded as
Since we use the reference speed states as , we consider the stabilization case, and the reference position is a constant in this research. We express the sliding mode regulator with sine mapping aswhere is a constant such as , where is in equation (12) and is in equation (9). It is relevant to note that we do not know the behaviors of and , and we use their upper bounds and .
Remark 3. Since we consider the stabilization case, we do not require the application of the sine mapping to the term , i.e., since and , we obtain small values in . Since the state is dependent of the other states , , the stabilization of the states , yields the stabilization of the state and we do not require the state in the regulator.
Remark 4. The convergence speed of the suggested regulator can be affected by choosing the gains , , and . The complexity in the implementation of the suggested regulator is in the programming of the saturation mapping ; nevertheless, this complexity also is in all the regulators which use mappings.
In Figure 2, we show the sliding mode regulator with sine mapping of equations (13), (12), and (9) termed SMWS for the stabilization in the electricity generators of equations (3) and (4) termed EGM.
From Figure 3, it is shown that the proportional derivation regulator of equation (10) and [3–5] termed PD, the sliding model regulator of equations (11), (12,) and (9) and [6–8] termed SM, and the suggested sliding mode regulator with sine mapping of equations (13), (12), and (9) termed SMWS use the proportional gain and the derivation gain . Furthermore, the existence, similarities, and differences between PD, SM, and SMWS are detailed as follows: the sliding model regulator starts from the proportional derivation regulator with one additional term of the signum mapping used to reach the stabilization, the sliding mode regulator with sine mapping starts from the sliding mode regulator with the modifications that the saturation mapping is used to reach a smoother performance instead of the signum mapping, and the sine mapping is applied to reach an upper bound of the proportional gain error.
Now, we will discuss the stabilization of the regulator.
Theorem 1. The stabilization in the regulator error of the sliding mode regulator with sine mapping of equations (13), (12), and (9), and electricity generators of equations (3) and (4) are ensured, and the speed regulator error will converge towith as the final time, , the magnets interaction nonlinearities and generator nonlinearities are bounded as , , and .
Proof. We denote the candidate mapping aswith as the positive constant of equation (3) and as the positive constant of equation (13). We denote , we substitute equations (13) and (8) into equation (3), and we obtain the closedloop model asWe use the fact , and we obtain the derivation of equation (15) aswith and . We substitute last equation (16) into equation (17) asAfter some mathematical operations, of equation (18) is asEquation (19) can be expressed asForm equation (12), , and from equation (13), , we notice that there are three cases of the saturation mapping: (1) if , then and , and we substitute into equation (20) as(2) If , then and , and we substitute into equation (20) assince, in this case, , . (3) If , then and , and we substitute into equation (20) asFrom equations (21)–(23), the three cases have the same inequality expressed asFrom the results of [5, 6], the stabilization of the regulator error is ensured. We integrate equation (24) from the initial time to the final time asand we apply the to both sides of the last inequality of equation (25) asIf , then , and we comply with equation (14).
Remark 5. From equations (1) and (2), it can be seen that the angle speed behavior affects the angle position and current behaviors in the electricity generators. Then, the angle speed convergence of equation (14) is used to reach some chosen constant behaviors for the angle position and current in the electricity generators.
Remark 6. This research focused on electricity generators. Nevertheless, our suggested regulator could be applied to other plants with similar structure to equations (3) and (4), such as are the motors, machines, robots, pendulums, or cranes.
Remark 7. There are several stabilization results such as the exponential stabilization [14, 17, 18, 20], the asymptotic stabilization [5, 6, 16, 22, 23], the uniformly ultimately boundedness [9–11], the feedback stabilization [15, 24], the stochastic stabilization [12, 21], or the mean square stabilization [13]. This research focused on the asymptotic stabilization.
4. Simulations
In this section, we compare the sliding mode regulator with sine mapping of equations (13), (12), and (9) termed SMWS, the proportional derivation regulator of equations (10), (12), and (9), and [3–5] termed PD, and the sliding mode regulator of equations (11), (12), and (9), and [6–8] termed SM for the stabilization of the two electricity generators of equations (3) and (4). The existence, similarities, and differences between PD, SM, and SMWS are detailed in Figure 3. Our goal in the regulators is that the angle position, angle speed, and current in electricity generators must reach some chosen constant references for the angle position, angle speed, and current as fast as possible in presence of the magnet interaction nonlinearities and generator nonlinearities. The chosen constant references in the angle position are higher or equal to zero, while the chosen constant references in the angle speed and current are equal to zero. We utilize the root mean square error (RMSE) for the comparisons aswith for the states error or for the fictitious input error.
Electricity generator with two static magnets and two dynamic magnets.
We express the electricity generator with two static magnets and two dynamic magnets in equations (1)–(5).
PD of [3–5] is expressed by equations (10), (12), and (9), with parameters and .
SM of [6–8] is expressed by equations (11), (12), and (9), with parameters , , and .
SMWS is expressed by equations (13), (12), and (9), with parameters , , and .
We compare the fictitious input in Figure 4 and a zoom in Figure 5, we compare the states in Figure 6 and a zoom in Figure 7, and we show the RMSE of equation (27) in Figure 6 and Table 5 for the electricity generator with two static magnets and two dynamic magnets.

In Figures 6 and 7, since the angle position, angle speed, and current of SMWS reach better the constant references in the angle position, angle speed, and current than the angle position, angle speed, and current of PD and SM, we can see that SMWS has the best performance. In Figures 4 and 5, the fictitious input for SMWS is smaller than for PD and SM. And in Figure 8 and Table 5, since the RMSE for the SMWS is smaller than for PD and SM, we can show that SMWS has the best performance.
Electricity generator with four static magnets and four dynamic magnets.
We express the electricity generator with four static magnets and four dynamic magnets in equations (2)–(6).
PD of [3–5] is expressed by equations (10), (12), and (9), with parameters and .
SM of [6–8] is expressed by equations (11), (12), and (9), with parameters , , and . SMWS is expressed by equations (13), (12), and (9), with parameters , , and .
We compare the fictitious inputs in Figure 9 and a zoom in Figure 10, we compare the states in Figure 11 and a zoom in Figure 12, and we show the RMSE of equation (27) in Figure 13 and Table 6 for the electricity generator with four static magnets and four dynamic magnets.

In Figures 11 and 12, since the angle position, angle speed, and current of SMWS reach better the constant references in the angle position, angle speed, and current than the angle position, angle speed, and current of PD and SM, we can see that SMWS has the best performance. In Figures 9 and 10, the fictitious input for SMWS is smaller than for PD and SM. And in Figure 13 and Table 6, since the RMSE for the SMWS is smaller than for PD and SM, we can show that SMWS has the best performance.
Remark 8. The course of RMSE is not monotonically increasing in Figures 8 and 13 even that it is indirectly suggested in equation (27) based on the following two reasons: (a) the final time should increase for more simulation time, which is the denominator of equation (27), and (b) our goal is that the state error or the fictitious input error should decrease for more simulation time, which is the numerator of equation (27). The mentioned reasons result in the decreasing of RMSE.
Remark 9. Since Figures 4, 5, 9, and 10 describe the impulses used as the excitation to start moving these electricity generators and Figures 6, 7, 11, and 12 describe the angle position, angle speed, and current used as the responses of these electricity generators, the simulations could represent the real experiments.
5. Conclusions
In this research, we suggest a sliding mode regulator with sine mapping for the stabilization of two electricity generators being affected by unknown nonlinearities. In the simulations, we show that since angle position, angle speed, and current of our sliding mode regulator with sine mapping reach better the constant references of angle position, angle speed, and current than the angle position, angle speed, and current of the proportional derivation and sliding mode regulators, we can see that suggested regulator has the best performance in presence of the magnets interaction nonlinearities and generator nonlinearities. Our regulator illustrates the viability, efficiency, and potential of the approach especially relevant in electricity generators. Our suggested technique could also be applied to other mechatronic systems such as the electricity motors, machines, robots, pendulums, or cranes. The drawback of our regulator is that it is designed for models with structure of electricity generators, for models with other structures, the design should require modifications. In the future, we will apply our suggested regulator for the stabilization of robots or pendulums being affected by unknown nonlinearities.
Appendix
In this section, we express the tables used in this research.
Data Availability
The simulation data used to support this research are available from the second author upon request to the email rubio.josedejesus@gmail.com.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This research was financed by the Instituto Politécnico Nacional, Secretaría de Investigación y Posgrado, Comisión de Operación y Fomento de Actividades Académicas, and Consejo Nacional de Ciencia y Tecnología.
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Copyright
Copyright © 2020 Dany Ivan Martinez et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.