# Stabilization of Two Electricity Generators

Authors: 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
Journal: Complexity
 (2020)

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## 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.


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## Body

## 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 time-delays [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 Section2, 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 Table1 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.Table 1
Description of the electricity generator symbols.
SymbolDescriptionmeThe mass of the link 1aThe width of the link 1cThe length of the link 1miThe mass of the dynamic magnetriThe radius of the surface in the magnetmThe massrgThe radius of the body rotationB11The magnet flux density produced by dynamic magnet 1 and static magnet 1B12The magnet flux density produced by dynamic magnet 1 and static magnet 2B21The magnet flux density produced by dynamic magnet 2 and static magnet 1B22The magnet flux density produced by dynamic magnet 2 and static magnet 2μThe magnet permeability constant in magnetshThe distance of a dynamic magnet and a static magnet on the axiszKmThe electromotive force constantRThe resistanceiThe currentLThe inductanceReThe load resistanceTable 2
Parameters of electricity generators.
ParameterValueParameterValuemi9.2×10–2 kgRe30 Ωμ9.42×10–5 Hm−1rg7.5 × 10–2 mB8.4×10–1 Tri2.9 × 10–2 mL6.03×10–1 Ha1.5 × 10–1 mKm45×10–2 Vsrad−1c1.5 × 10–2 mb1×10–1 kgm2 rads−1nr0.5me5×10–3 kgnl0.5h4×10–2 mwr0.5R6.96 Ωwl−0.5Table 3
Description of the mathematical model symbols.
SymbolDescriptionz1∈ℜThe angle positionz2∈ℜThe angle speedz3∈ℜThe currentv∈ℜThe fictitious input of torque produced by the magnets interactionOz1∈ℜThe magnets interaction nonlinearitiesz1, z2, z3The statesQ, J, X, YScalar constantsuThe input of the generator nonlinearitiesnr, nl, wr, wlConstant terms for the generator nonlinearitiesPuThe generator nonlinearitiesTable 4
Description of the regulators symbols.
SymbolDescriptionz˜1=z1−z1d∈ℜThe position regulator errorz1d∈ℜThe constant reference positionz˜2=z2∈ℜThe speed regulator errorK1∈ℜThe proportional gainK2∈ℜThe derivation gainsign⋅The signum mappingsat⋅The saturation mappingsin⋅The sine mappingcos⋅The cosine mappingKThe sliding mode gainz˜12=z1−z1d2The position regulator errorz˜22=z22The speed regulator errorz˜32=z32The current regulator errorIn 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 Figure1 via the utilization of two static magnets and two dynamic magnets.Figure 1
Electricity generator with static magnets and dynamic magnets.We express the electricity generator with two static magnets and two dynamic magnets as [1, 2](1)z•1=z2,z•2=1me1/24a2+1/24c2+rg2+miri2+2rg2v−πB2ri2rg2sinz1μh2+2rg21−cosz10.5+πB2ri2rg2sinz1μh2+2rg21+cosz10.5−bz2,z•3=−R+ReLz3+KmLz2.Since all magnets are equal,B11=B12=B.We consider the electricity generator with four static magnets and four dynamic magnets of Figure1 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](2)z•1=z2,z•2=1me1/24a2+1/24c2+rg2+miri2+2rg2v−πB2ri2rg2sinz1μh2+2rg21−cosz10.5−πB2ri2rg2cosz1μh2+2rg21+sinz10.5+πB2ri2rg2sinz1μh2+2rg21+cosz10.5+πB2ri2rg2cosz1μh2+2rg21−sinz10.5−πB2ri2rg2sinz1−π/4μh2+2rg21−cosz1−π/40.5−πB2ri2rg2cosz1−π/4μh2+2rg21+sinz1−π/40.5+πB2ri2rg2sinz1−π/4μh2+2rg21+cosz1−π/40.5+πB2ri2rg2cosz1−π/4μh2+2rg21−sinz1−π/40.5−bz2,z•3=−R+ReLz3+KmLz2.Since all magnets are equal,B11=B12=B21=B22=B.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(3)z•1=z2,Qz•2+Jz2+Oz1=v,z3•+Xz3+Yz1=0.The fictitious inputv is affected by the generator nonlinearities, it is like a chattering movement presented in the electricity generator. We express the fictitious input v as [6](4)v=nru−wr,u≥wr,0,wl<u<wr,nlu−wl,u≤wl.We see that the model of equation (1) can be expressed as the model of equations (3) and (4) with(5)Q=me124a2+124c2+rg2+miri2+2rg2,J=b,X=R+ReL,Y=−KmL,Oz1=πB2ri2rg2sinz1μh2+2rg21−cosz10.5−πB2ri2rg2sinz1μh2+2rg21+cosz10.5.We see that the model of equation (2) can be expressed as the model of equations (3) and (4) with(6)Q=me124a2+124c2+rg2+miri2+2rg2,J=b,X=R+ReL,Y=−KmL,Oz1=πB2ri2rg2sinz1μh2+2rg21−cosz10.5+πB2ri2rg2cosz1μh2+2rg21+sinz10.5−πB2ri2rg2sinz1μh2+2rg21+cosz10.5−πB2ri2rg2cosz1μh2+2rg21−sinz10.5+πB2ri2rg2sinz1−π/4μh2+2rg21−cosz1−π/40.5+πB2ri2rg2cosz1−π/4μh2+2rg21+sinz1−π/40.5−πB2ri2rg2sinz1−π/4μh2+2rg21+cosz1−π/40.5−πB2ri2rg2cosz1−π/4μh2+2rg21−sinz1−π/40.5.The generator nonlinearities of electricity generators are symmetric asnr=nl in equation (4); consequently, we can express the fictitious input v of equation (4) as(7)v=nlu,u≥wr,nlu,wl<u<wr,nlu,u≤wl,+−nlwr,u≥wr,−nlu,wl<u<wr.−nlwl,u≤wl,After some mathematical operations, the fictitious inputv of equation (7) is expressed as(8)v=nlu−Pu,Pu=nlwr,u≥wr,nlu,wl<u<wr,nlwl,u≤wl.We notice that the generator nonlinearitiesPu are bounded as(9)Pu≤P¯.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 Oz1∈ℜ, 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 Oz1∈ℜ are the most complex part in the mathematical model of equations (3) and (4), while Q, J, X, and Y are scalar constants corresponding to the electricity generators. Thus, these magnet interaction nonlinearities Oz1∈ℜ 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](10)u=1nl−K1z˜1−K2z˜2.
We express the sliding mode regulator as [6–8](11)u=1nl−K1z˜1−K2z˜2−Ksignz˜2,signz˜2=1,z˜2>0,0,z˜2=0,−1,z˜2<0.


## 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 nonlinearitiesOz1 of equation (3) are bounded as(12)Oz1≤O¯.Since we use the reference speed states asz2d=0, we consider the stabilization case, and the z1d∈ℜ reference position is a constant in this research. We express the sliding mode regulator with sine mapping u as(13)u=1nl−cosz˜1TK1sinz˜1−K2z˜2−Ksatz˜2,satz˜2=1,z˜2>1,z˜2,z˜2≤1,−1,z˜2<−1,where K is a constant such as O¯+P¯≤K, where O¯ is in equation (12) and P¯ is in equation (9). It is relevant to note that we do not know the behaviors of Oz1 and Pu, and we use their upper bounds O¯ and P¯.Remark 3.
Since we consider the stabilization case, we do not require the application of the sine mapping to the termK2z˜2, i.e., since z2d=0 and z˜2=z2, we obtain small values in z˜2. Since the state z3 is dependent of the other states z1, z2, the stabilization of the states z1, z2 yields the stabilization of the state z3 and we do not require the state z3 in the regulator.Remark 4.
The convergence speed of the suggested regulator can be affected by choosing the gainsK1, K2, and K. The complexity in the implementation of the suggested regulator is in the programming of the saturation mapping sat⋅; nevertheless, this complexity also is in all the regulators which use mappings.
In Figure2, 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 Figure3, 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 K1 and the derivation gain K2. 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.Figure 2
Sliding mode regulator with sine mapping.Figure 3
Relation between PD, SM, and SMWS.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 z˜2 will converge to(14)limsupT⟶∞z˜22=0,with T as the final time, z˜2=z2, the magnets interaction nonlinearities and generator nonlinearities are bounded as Oz1≤O¯, Pu≤P¯, and O¯+P¯≤K.Proof.
We denote the candidate mapping as(15)V1=12z˜2TQz˜2+12sinz˜1TK1sinz˜1,with Q as the positive constant of equation (3) and K1 as the positive constant of equation (13). We denote z˜2=z2, we substitute equations (13) and (8) into equation (3), and we obtain the closed-loop model as(16)Qz•2+Jz2+Oz1=v=nlu−Pu=nl1nl−cosz˜1TK1sinz˜1−K2z˜2−Ksatz˜2−Pu,⇒Qz˜•2=−cosz˜1TK1sinz˜1−K2z˜2−Jz˜2−Oz1−Pu−Ksatz˜2.
We use the factz˜2=z2, and we obtain the derivation of equation (15) as(17)V•1=z˜2TQz˜•2+z˜2Tcosz˜1TK1sinz˜1,with z˜•1=z•1−z•1d=z2−z2d=z2=z˜2 and z˜•2=z•2. We substitute last equation (16) into equation (17) as(18)V•1=z˜2TQz˜•2+z˜2Tcosz˜1TK1sinz˜1,⇒V•1=z˜2T−cosz˜1TK1sinz˜1−K2z˜2−Jz˜2−Oz1−Pu−Ksatz˜2z˜2Tcosz˜1TK1sinz˜1,⇒V•1=−z˜2TK2z2−z˜2TJz˜2−z˜2TOz1−z˜2TPu−z˜2TKsatz˜2−z˜2Tcosz˜1TK1sinz˜1+z˜2Tcosz˜1TK1sinz˜1.
After some mathematical operations,V•1 of equation (18) is as(19)V•1=−z˜2TK2z˜2−z˜2TJz˜2−z˜2TOz1−z˜2TPu−z˜2TKsatz˜2.
Equation (19) can be expressed as(20)V•1=−z˜2TK2+Jz˜2−z˜2TOz1−z˜2TPu−z˜2TKsatz˜2.
Form equation (12), Oz1≤Oz1+Pu≤O¯+P¯≤K, and from equation (13), satz˜2=1z˜2>1z˜2z˜2≤1−1z˜2<−1, we notice that there are three cases of the saturation mapping: (1) if z˜2>1, then satz˜2=1 and z˜2=z˜2, and we substitute into equation (20) as(21)V•1≤−z˜2TK2+Jz˜2+z˜2TO¯+z˜2TP¯−z˜2TK,⇒V•1≤−z˜2TK2+Jz˜2.
(2) Ifz˜2≤1, then satz˜2=z˜2 and z˜2Tz˜2=z˜2Tz˜2, and we substitute into equation (20) as(22)V•1=−z˜2TK2+Jz˜2+z˜2TO¯+z˜2TP¯−z˜2Tz˜2K,⇒V•1=−z˜2TK2+Jz˜2+z˜2TO¯+z˜2TP¯−z˜2Tz˜2K,⇒V•1=−z˜2TK2+Jz˜2−z˜2Tz˜2K−O¯−P¯,⇒V•1=−z˜2TK2+Jz˜2,since, in this case, z˜2≤1, z˜2K−O¯−P¯≥0⇒O¯+P¯≤z˜2K≤K. (3) If z˜2<−1, then satz˜2=−1 and z˜2=−z˜2, and we substitute into equation (20) as(23)V•1=−z˜2TK2+Jz˜2−−z˜2TOz1−−z˜2TPu−−z˜2TK−1,⇒V•1≤−z˜2TK2+Jz˜2+z˜2TO¯+z˜2TP¯−z˜2TK,⇒V•1≤−z˜2TK2z˜2.
From equations (21)–(23), the three cases have the same inequality expressed as(24)V•1≤−z˜2TK2+Jz˜2.
From the results of [5, 6], the stabilization of the regulator error is ensured. We integrate equation (24) from the initial time 0 to the final time T as(25)∫0Tz˜2TK2+Jz˜2dt≤V1,0−V1,T≤V1,0,⇒K2+JT∫0Tz˜22dt≤1T∫0Tz˜2TK2+Jz˜2dt≤1TV1,0,and we apply the limsupT⟶∞ to both sides of the last inequality of equation (25) as(26)limsupT⟶∞1T∫0Tz˜22dt≤V1,0K2+JlimsupT⟶∞1T=0.
IfT⟶∞, then z˜22=0, 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 as(27)RMSE=1T∫0Tz˜2dt1/2,with z˜2=z˜12+z˜22+z˜32 for the states error or z˜2=v2 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 K1=100 and K2=0.1.SM of [6–8] is expressed by equations (11), (12), and (9), with parameters K1=100, K2=0.1, and K=0.5.SMWS is expressed by equations (13), (12), and (9), with parameters K1=100, K2=0.1, and K=0.5.We compare the fictitious input in Figure4 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.Figure 4
Fictitious input of the first electricity generator.Figure 5
A zoom in the fictitious input of the first electricity generator.Figure 6
States of the first electricity generator.Figure 7
A zoom in the states of the first electricity generator.Table 5
Comparisons for the first electricity generator.
RMSE forz˜RMSE forvPD2.01540.7386SM1.82260.7463SMWS1.78350.6656In Figures6 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.Figure 8
RMSE of the first electricity generator.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 K1=100 and K2=0.1.SM of [6–8] is expressed by equations (11), (12), and (9), with parameters K1=100, K2=0.1, and K=0.5. SMWS is expressed by equations (13), (12), and (9), with parameters K1=100, K2=0.1, and K=0.5.We compare the fictitious inputs in Figure9 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.Figure 9
Fictitious input of the second electricity generator.Figure 10
A zoom in the fictitious input of the second electricity generator.Figure 11
States of the second electricity generator.Figure 12
A zoom in the states of the second electricity generator.Figure 13
RMSE of the second electricity generator.Table 6
Comparisons for the second electricity generator.
RMSE forz˜RMSE forvPD2.01080.6995SM1.82300.7098SMWS1.78210.6224In Figures11 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 Figures8 and 13 even that it is indirectly suggested in equation (27) based on the following two reasons: (a) the final time T should increase for more simulation time, which is the denominator of equation (27), and (b) our goal is that the state error z˜2=z˜12+z˜22+z˜32 or the fictitious input error z˜2=v2 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 Figures4, 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.



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