A Comparative Study of Nonsingular Terminal Sliding Mode and Backstepping Schemes for the Coupled Two-Tank System

.is paper presents an implementation of two radically different control schemes for a state-coupled two-tank liquid-level system. .is is due to the purpose of transferring theoretical studies to industrial systems. .e proposed schemes to be introduced and compared are the nonsingular terminal sliding mode control (NTSMC) and the backstepping control (BC). .e performances of the developed methods are experimentally tested on a particular class of second-order nonlinear systems..emain purpose of the considered control schemes is to achieve a tracking trajectory for a coupled-tank system. It is proved that the designed robust controllers guarantee the stability of the corresponding closed loop systems. .e obtained results are verified with the same setup test to ensure a suitable basis for their comparison. During the experiments, we resorted to adding an integrator to the backstepping control so that we improve the results, leading to the appearance of the integrator backstepping control (IBC). To focus on the adequacy and applicability of the suggested control layout, theoretical comparisons as well as experimental results are afforded and debated.


Introduction
Liquid-level control systems can be classified as an important process not only for draining but also in several major industries [1]. Various systems and devices have been developed to control the liquid level such as in food processing, water purification systems, filtration, pharmaceutical industries, decoration, boilers, beverage, and industrial chemical processing. Liquid level is so important and there are many parameters that should be mastered to measure it.
In the few past decades, some researchers have invented the design and the implementation of the liquid level of a coupled-tank system controller such as the Proportional-Integral-Derivative (PID) type controllers [2], the backstepping controller [3,4], the nonlinear constrained predictive algorithms based on the feedback linearization control [5], the second-order sliding mode control [6], Constrained Pole Assignment Control (CPAC) [7,8], and neurofuzzy sliding mode controller (NFSMC) [9]. erefore, industrial process control engineering has immensely benefited from the technology development brought by digital computers and their sophisticated software. us, these advanced technologies have allowed the ability of implementing advanced control algorithms that have been considered as quite complex in their implementation. In addition, the industry demands as well as the high precision required by modern systems encourage research in control engineering to develop and synthesize robust nonlinear control algorithms.
Despite practical control processes have been solved based on the above-mentioned approaches, these techniques require a measured state vector and a precise model.
As it is well known, the SMC is an efficient robust control for uncertain systems [26,27] and bounded external disturbances [27]. is approach consists of two steps. First, the system state path reaches a predefined surface according to the control objectives, called sliding surface. en, the designed control restricts the system trajectory to remain on this surface and to converge into its equilibrium state. Indeed, this task has been achieved by introducing a discontinuous term allowing a high frequency switching of system trajectories around the sliding surface. However, this procedure generally leads to the flawed "chattering phenomenon" which may generate some serious problems during the experimental applications. Some attention has been drawn to reducing the effect of this chattering phenomenon, which has led to other new scheme strategies based on the principle of SMC.
Among these scheme strategies, we can mention the terminal sliding mode control (TSMC) [28][29][30], the fast terminal sliding mode Control (FTSMC) [31], the integral terminal sliding mode control (ITSMC) [32,33], the nonsingular terminal sliding mode control (NTSMC) [34,35], and the fast nonsingular integral terminal sliding mode control (FNITSMC) [36]. In this framework, the TSMC has been designed to achieve the finite-time convergence of the system dynamics and it has been applied in many practical processes such as the rigid robotic manipulators [37,38], the PWM-based DC-DC [39], and robotic airships [40].
Compared to the conventional SMC, the TSMC which has a nonlinear sliding surface offers superior properties such as speed, convergence in finite time, and more accurate control [28]. However, it has two inconveniences which are, respectively, the singularity point problem and the requirement of the uncertainty limit problem. e first one can be overcome by the nonsingular approach in the NTSMC, and the second one can be solved through a well-designed uncertainty estimation [28,29].
To the best of authors' knowledge, there is no result in the literature applying TSMC to the two-tank system which motivates us to present this study.
is paper investigates a comparative study between NTSMC, BC, and IBC for the two-tank liquid-level system which has not been discussed yet. Indeed, more suitable control approaches have been considered to guarantee the desired performance for the liquid-level system. e main contributions of this paper are summarized as follows: e NTSMC, BC, and IB can be suggested among the most widespread and well-performing control approaches which have been simulated and implemented for a coupled two-tank system. e closed loop stability proofs of the NTSMC and BC schemes in the sense of Lyapunov have been developed. A theoretical and practical comparison for the interest of practicing engineers and researchers has been addressed.
is paper is structured as follows: In Section 2, the statecoupled two-tank system description and its mathematical model will be introduced. e NTSMC liquid-level approach to be considered is presented and its stability is studied in Section 3. Section 4 introduces the model based backstepping liquid-level controller and the study of its stability. A comparison of the control approaches, based on the proposed theoretical analysis, is presented in Section 5. Section 6 introduces the simulation and experimental results.
e conclusion and the future works are given in Section 7.

System Description
e model of a two degrees of freedom (DOF) state-coupled two-tank system is given in Figure 1. is system is composed of a liquid basin, a pump, and two equal-volumed tanks. ese tanks are equipped with "in" and "out" orifices and level sensors at their bottoms.
is system works as follows: First, the pump absorbs liquid from the basin to discharge it into tank 2. After that, the liquid exits from tank 2 to fill in tank 1. Finally, tank 1 liquid is emptied into the basin. e main characteristics and constraints of the dynamic system model are given as follows: e system input is the voltage u delivered by the pump which varies between 0 V and 12 V while the system output is the liquid level h 1 A pressure sensor is associated with both tank 1 and tank 2 for instantaneous measurement of the levels h 1 and h 2 e liquid levels h 1 and h 2 belong to the interval [4 cm, 30 cm] According to the already mentioned description, the dynamic equations of the liquid level in the two tanks are obtained as follows: e time change rate of liquid level in each tank is given by where h i (t), S i , F in i (t), and F out i (t) are the liquid level, crosssectional area, and inflow and outflow rates, respectively, for the i th tank. Next, the inflow rate into tank 2 is given by

Complexity
where K p is the pump constant (cm 3 /V) and u(t) is the voltage applied to the pump, such that, using Bernoulli's law for the flow through small orifices, the outflow velocity from the orifice at the bottom of each tank is v out en, the outflow rate for each tank is given in where g is the gravitational acceleration and s i denotes the cross-sectional area of the outflow orifice at the bottom of the i th tank. Finally, we note that for the two-tank liquid-level system: us, we obtain the following dynamic equations of the system: For convenience, in stating the main result of this section, we define the following constants: Consider that there is the same cross-sectional area for the two tanks: S 1 � S 2 .
In such conditions, (6) can be written as the following system: In what follows, we assume that the two sensors are available to measure the liquid levels in the two tanks. e level of h 1 will be controlled to follow a reference trajectory.

Typical Terminal Sliding Mode Control.
To summarize the basic principle of TSMC, we consider the following second-order system [28]: where x 1 and x 2 are the system states, f(x) and g(x) ≠ 0 are the known nonlinear functions, respectively, u is the control input, and d(x, t) is the disturbance such as |d(x, t)| ≤ L where L > 0. e sliding variable is selected as where β > 0, p, q, and (p > q) are positive odd numbers. e controller is designed as From (10), we have (q/p) − 1 < 0. When x 1 � 0 and x 2 ≠ 0 a singularity problem exists for the typical terminal controller.
us, a nonsingular terminal sliding mode control method is proposed by [31] to deal with the singularity problem. e nonsingular sliding variable is designed as where β > 0, p, q, and p > q are positive odd numbers. e nonsingular sliding mode controller is given as where 1 < (p/q) < 2 and η > 0.
In the next section, scheme (12) will be developed to deal with tracking problem position.

Nonsingular Terminal Sliding Mode Control for the
Coupled Two-Tank System. Many methods have been suggested to avoid the singularity problem in the typical TSMC. e first approach to be mentioned switches the sliding mode from TSMC to linear hyperplane-based sliding mode [28]. e second approach transfers the trajectory to a predefined open region where TSMC is not singular [29]. Complexity ese methods advocate indirect approaches to avoid singularity. In this work, a simple NTSMC, which is completely capable of overcoming this problem, is highly recommended. e suggested NTSMC model is interpreted as follows: e first step consists of transforming the model of the real system into an affine control model, described in (8). Starting from the model given in (7), we consider the following diffeomorphism: x � x 1 e time derivative of the new coordinates gives In the sequel, we apply the inverse of the diffeomorphism ϑ − 1 to recuperate the state in h for the controller. e inverse of the diffeomorphism is as follows: Figure 2 illustrates the proposed NTSM controller as follows.
e nonsingular adaptive sliding variable is designed by with where x 1 , x 2 are states of the system and r is a desired trajectory. β > 0, p, q, p > q, and 1 < (p/q) < 2 are positive odd numbers.

Stability Analysis of the NTSM Controller Design.
In what follows, the stability analysis of the corresponding closed loop system deduced from (15) will be addressed by using the NTSMC. We extend scheme (12) to overcome the tracking problem position. For this, we propose eorem 1 as follows.
Theorem 1. For system (13), the NTSMC law is designed as follows: where (19) will be reached in finite time and the errors e 1 and e 2 will converge to zero in finite time if there exist real parameters μ, L, p, q, and β such as μ > 0, L > 0, and 1 < (p/q) < 2.
Proof. Consider the following Lyapunov function for the resulting closed loop system inspired from (20) and (21): Its derivative along the closed loop system given in (13) and (15) is deduced as where From (22) and (15), we deduce According to (19), we have

Complexity
Finally, we obtain To complete this proof, it should be noticed that NTSMC law (19) is always nonsingular in the state space since 1 < (p/q) < 2. erefore, we assumed that 0 < (p/q) − 1 < 1.
In such condition, we guarantee that ( us, the Lyapunov stability of the considered system is checked.
is is the end of the proof of eorem 1.
□ Remark 1. e proposed NTSMC law in eorem 1 solves the problem of the control of liquid-level system that represents a special class of systems (n � 2). e method proposed can be extended to a class of n-order (n > 2) nonlinear dynamic systems that represent a broader class of problems: where . , x 2n ) T ∈ R n , f 1 and f 2 are smooth vector functions, g is a nonsingular matrix, and u � (u 1 , u 2 , . . . , u n ) T ∈ R n is the control vector. In fact, this considered approach can be applied to any system, which can be transformed to (27).

Backstepping Control
One of the benefits of the BC consists in stabilizing the nonlinear system without linearization. In fact, the presence of nonlinearization leads to multiple potential advantages. erefore, less effort is necessary for the system control. In addition, the corresponding scheme may depend on a less precise information model, which improves the robustness against the modelization errors and assures the global stability.
e BC aims to use the state as a virtual control. However, the system is then divided into united subsystems in a decreasing order. us, the corresponding scheme appears in the last step of the backstepping algorithm. During the stages intermediates, the instability of the nonlinear system is treated, and the order of the system is increased from one step to another. Global stability is guaranteed that it ensures continuity and regulation of nonlinear systems.

Backstepping Control for the Two-Tank System.
We consider the following class of triangular nonlinear SISO systems given in is considered control aims to stabilize system (28) with the reference trajectory r ∈ R.
In this work, according to the theoretical results in [41], a backstepping controller will be synthesized to ensure the tracking problem for the nonlinear system (28). In addition, the Lyapunov theory has been used to prove the existence of Tank 2   Tank 1 h2 Complexity 5 some conditions for the chosen feedback gains to guarantee the asymptotic stability of control system (28). e first step consists in rewriting the two-tank system model (7) in the triangular form as follows:

Stability Analysis of the Controlled
System. e proposed backstepping controller allows guaranteeing the global asymptotic stability of the system output respective to a variable reference signal.
is controller is formulated in eorem 2 as follows.
Theorem 2. We consider system (28) and let r be a reference signal with a bounded derivative and the following assumptions hold: e scheme will be synthesized as follows: where k ∈ R and ϑ: R ⟶ R makes y � r a globally asymptotically stable equilibrium. According to (30), we define an auxiliary input for system (29), which is given as follows: which is globally Lipchitz (r > 1) and k 2 > k 1 > 0. In such conditions, we can suggest a structure of BC with state feedback for the state-coupled two-tank system; it is given by Proof. Consider system (28) as well as the following change of variables: To find a stabilizing control law as function of h 1 given in (28), we denote by z 2 the command variable and we define the new variable as z des In the sequel, the variable control law with the Lyapunov control function will be defined as W(z 1 ) � (1/2)z 2 1 . e derivative of W with respect to time is obtained as en, (34) is defined negative if hypothesis A2 holds. Now, by introducing the residue z 2 � z 2 − z des 2 , system (28) can be presented as function of z 1 and z 2 such as en, we obtain From (36), we deduce Now, we select the following Lyapunov control function (LCF) for system (37): where F is a Lyapunov control function (CLF) of the subsystem z 1 . e derivative of the CLF given in (38) respective to the time is deduced as erefore, we have 6 Complexity Let us consider the following function: From (40) and (41), we get To reduce the complexity of the second term of equation (42), F is chosen such that the z 1 terms inside the braces neutralize each other.
is can be done by choosing F(z 1 ) such as By inserting this expression in (41), we obtain Consider the following control law: Equation (41) becomes If hypothesis A3 is verified, we have k > (zϑ(z 1 + r)/zz 1 ), and if A3 and A4 are checked, then U(z 1 ) > 0.
Finally, it is simple to see that _ V(z 1 , z 2 ) is negatively definite.
is is the end of this proof.

Comparative Study of the Theoretical Analysis
In this subsection, a comparative study for the proposed liquid-level schemes will be presented. Indeed, the comparison criteria are based on the controller design and implementation possibilities, the complexity of adjusting the controller parameters, and the stability properties of the uncertain parameters. Concerning the two schemes designs, the NTSMC expression depends on several parameters, which are very delicate to identify. It should be noted that the expression (L + μ)sign(S) will be substituted by (L + μ)sat(S) to minimize the chattering phenomenon with a particular choice of the parameters L and μ. erefore, the parameter β will be fixed after several simulations and practical tests. Indeed, we can deduce that this parameter has a lot of influences on the system performances and especially for its convergence speed. e BC method is simpler than the NTSMC in terms of the computational complexity and the mathematical development. en, the expression of the BC can be designed after a suitable choice of the gains k 1 and k 2 .
Note that the real system parameters affect the control law performance. In fact, the values of these parameters are mainly based on the calculation of the tank's sections and the piping connection between its components. However, this calculation is generally imprecise due mainly to the measurement errors. In such conditions, the considered controllers should be robust against uncertain parameters.

Simulation Results.
In this part, we present the obtained simulation results, respectively, to the NTSMC and the BC.
e numerical values of the parameters of the studied two-tank liquid-level system are given in Table 1.
Note that in the system model a bounded input voltage and bounded liquid levels have been assumed such as 0 V < u < 12 V and 4 cm < h i < 30 cm.
In what follows, two different disturbance types in tank 2 will be considered. Indeed, the disturbance scenarios are described as follows: (i) e external disturbance is equivalent to a shock stability moment of the system; in such situation, it could be applied at each constant level:10 cm, 16 cm, and 8 cm. In fact, in the simulation, this disturbance Complexity is equivalent to an impulse that will be applied at the instant t � 460 s during 20 s with an amplitude equal to 1. (ii) e parametric disturbance has been introduced in the simulation by considering the parameter c 2 � 0 from t � 430 s to t � 515 s. Figures 3 and 4 show the tracking results, respectively, to the NTSMC and the BC as follows.
From Figure 3, we can see that the system response is fast (the rise time for the NTSMC is t r (NTSM) � 50 s) which causes an overshoot at each climb level of the desired trajectory (10 cm, 16 cm).
us, 13% and 5.6% correspond, respectively, to the overshoot's levels (10 cm)and (16 cm). Despite the speed and the overshoot that degrade stability properties, the system maintains its stability under the two occurring disturbances.
On the other hand, from Figure 4, we can observe that the system response via the BC is slow (the rise time for the BC is t r (BC) � 100 s) which leads to static errors at each change of the desired liquid level. erefore, 27%, 20%, and 45% are the errors in percent relative, respectively, to the desired liquid level (10 cm, 16 cm, and 8 cm).
Finally, we can conclude that the tracking results from the different schemes are suitable and satisfactory. However, the NTSMC allows obtaining more performant results compared to the BC. Figure 5 represents the NTSMC and BC schemes signals in presence of external disturbances. Figure 5 shows that the obtained signals are regulated and their amplitude remains within a permissible and bounded limit. In addition, we can see that the BC is more economical than the NTSMC since the maximum and average values of the BC signal are smaller than those of the NTSMC.
According to the results presented in Figure 5, the motopump voltage takes its maximum value (12 V)to reach the liquid levels (10 cm and 16 cm). However, to go from 16 cm to 8 cm, the moto-pump stops or operates in the dead zone leaving tank 2 free to empty. ese results are similar for both controls. In addition, the voltage values remain equal to (7.1 V, 9 V, and 6.4 V) for the NTSMC and (6.7 V, 8.4 V, and 6 V) for the BC, respectively, to the regarding bearings. us, we can conclude that the BC is less aggressive than the NTSMC. Although all schemes have a positive reaction for disturbances rejection, the NTSMC makes 65 sto reject the external disturbances and the BC makes56 s for the same task with a small difference of 9 s.

Test Bench Description.
e practical two-tank liquidlevel system is presented in Figure 6. is system exists in the Laboratory Study of Industrial Systems and Renewable Energies "LAS2E" at the National Engineers School of Monastir, Tunisia. In fact, the electromechanical part of the system is made up of the tanks, the power card motor pumps, and conditioning cards for pressure sensors. e synthesized schemes will be implemented by using blocks in MATLAB/Simulink environment combined with the interface real time associated with the data acquisition board which is based on the Arduino Mega microcontroller.
Note that the schemes are made in the same environment and the same conditions. In addition, the initial liquid levels h 1 and h 2 are always equal to 4 cm and the conversion voltage/level of the two voltages measured from the two pressure sensors with their instrumentation amplifiers is made through multiplication by the coefficients a 1 � 9.3458 cm.V − 1 and a 2 � 8.5106 cm.V − 1 . Finally, the numerical parameters values are k 1 � 0.032 and k 2 � 0.11 for the BC and L � 1.1, μ � 1.3, β � 17, p � 5, and q � 3 for the NTSMC.

Remark 2.
e simulation tests allow obtaining approximate parameters for all schemes. However, the best values will be deduced from several experimental testings.
We suppose that the system meets two types of disturbances. Indeed, the disturbance scenarios are explained as follows: e external disturbance has been generated by suddenly adding at the instants 150 s, 500 s, and 800 s a water quantity in the tank 2.
e parametric disturbance will be affected by varying the value of c 2 . In such situation, we close the orifice related to c 2 for a few seconds at the same instants that have been chosen for the external disturbances.

Comparison of the Measurement Results.
e experimental results are illustrated in Figures 7-10 as follows which represent, respectively, the NTSMC tracking and its impact of the external disturbance, the NTSMC tracking and its impact on the parameter variation, the BC tracking and its impact on the external disturbance, and the BC tracking and its impact on the parameter variation.   From Figures 7 and 8, the following overshoot values 4%, 5%, and 10% are, respectively, obtained at levels 10 cm, 16 cm, and 8 cm. is result allows us to deduce that the corresponding system has satisfactory performance in terms of tracking, stability, fastness, and robustness when the NTSMC has been applied. erefore, the maximum overshoot percentage of NTSMC is equal to PO max (NTSMC) � 10% and the rise time for NTSMC is t r (NTSMC) � 30 s.
It can be shown from Figures 9 and 10 that there is a notable static error. e following overshoot values 10%, 12.5%, and 0.6% are, respectively, obtained at levels 10 cm, 16 cm, and 8 cm. en, the maximum overshoot percentage via BC equals PO max (BC) � 12.5% and the rise time for BC is t r (BC) � 45 s. In what follows, an integrator action will be added to the BC to eliminate the static error. In fact, in the recent years, the integrator backstepping control (IBC) has gained much focus since it provides a framework for attacking many electromechanical control troubles like the state-coupled two-tank system.
For systematic condition, requested control structure adjustments such as compensation for parametric uncertainty or eliminating state measurements can be one of the major benefits of the IBC family of control design tools.
is would be confirmed by the results given in Figures 11  and 12.
From Figures 11 and 12, the integral action avoids the static error. Despite the fact that the presence of this action degrades the stability performance, the system has a good tracking and a performed robustness in the presence of an external disturbance and parameters variation. Accordingly, the system response becomes rapid which leads to overshoots at each level change.
us, the IBC maximum overshoot percentage equals PO max (IBC) � 23% and the rise time of the IBC is t r (IBC) � 25 s. e control voltages in the presence of external disturbance and parameter variation for the NTSM, B, and IB controllers are given, respectively, in  e voltage provided by the NTSMC scheme is shown in Figure 13. Indeed, the voltage is equal to 10 V at the first desired slope. Once the level reaches10 cm, this voltage is kept at constant value which equals 5.5 V. At the second rise, the voltage is equal to 10.8 V. When the level reaches 16 cm, the voltage settles approximately at 7.5 V.
By some comparisons in terms of the energy consumption for three applied schemes, we defined the maximum and the minimum voltage values U max and U min for each controller such as U max (NTSMC) � 10.8 V and U min (NTSMC) � 0 V. Figures 14 and 15 present, receptively, the BC voltage with disturbances rejection and the IB voltage with disturbances rejection as follows.
It is seen from Figure 15 that the integral action slightly increases the moto-pump voltage so that h 1 liquid level reaches 10 and 16 cm desired values. For the change of level from 16 to 8 cm, the moto-pump ceased. is can be proved by the following values: For the IBC, the voltage is 10.3 V at the first desired slope. Once the level reaches10 cm, this voltage is kept at a constant value equal to5.8 V. At the second rise, the voltage is 11.3 V. When the level reaches 16 cm, the voltage settles approximately at 7.4 V.
At the third negative slope, the voltage is equal to 0 V which is explained by the free emptying of h 2 tank. en, the voltage rises to 11.6 V and it settles at 5.4 V when the level reaches 8 cm.
In such situation, the maximum and the minimum voltage values are U max (IBC) � 11.6 V and U min (IBC) � 0 V. As it is shown in Figure 14, the maximum and the minimum voltage values are U max (BC) � 11 V and U min (BC) � 0 V.
Concerning the external disturbance, we notice that the three schemes react to preserve the system stability and reject any negative interference, which proves their robustness. ese reactions are indicated in Figures 13-15.
e following values 35 s, 70 s and 50 s represent, respectively, the external disturbance rejection durations for the BC, IBC, and NTSMC.
In presence of the external disturbance, the pump interrupts instantly, which is explained by the increase of h 2 level due to the external liquid addition. en, after the disturbance fading, the pump signal resumes its previous voltage. By closing the orifice output in tank 2 for a few seconds, a variation of c 2 parameter occurs.
As a control response towards the parameter variation, the pump voltage increases which is explained by h 1 tank demand for liquid from h 2 tank. By cancelling this variation, the pump voltage decreases since h 2 level has increased. en, the pump signal resumes its previous voltage. By some conclusions, the NTSMC signal is less aggressive than those of BC and IBC ones. is means that NTSMC is more economical in terms of energy consumption.
On the other hand, the tracking error as a percent will be represented to master the tracking quality. Figure 16 shows the respective tracking errors of the NTSMC, BC, and IBC strategies. e tracking error is calculated according to (47).
To confirm the best tracking results extracted by the three schemes, we will compare the different tracking errors such as E(NTSM) < E(IBC) < E(BC).
We notice that the error related to NTSMC and IBC is less than that of BC. During disturbances, the tracking error increases to what is expected. e error and the overshoot expressions will be given, respectively, in the following equations: (48) Figure 16 shows the NTSMC, BC, and IBC liquid-level tracking errors in the presence of the external disturbance. e theoretical and practical obtained results are summarized in Table 2.

Remark 3.
e obtained results will be compared to those illustrated in [3]. In fact, in [3] the authors present some performance criteria for the backstepping scheme applied to the two-tank system for the tracking problem to the same desired trajectory as in the considered case.
Remark 4. Note that the SMC and BC have been combined to build a more efficient control named backstepping sliding mode control (BSMC) [42] that can be suggested to be applied later.

Conclusion and Future Works
In this paper, nonlinear controllers have been designed for the purpose of a precise liquid-level tracking in a statecoupled two-tank system by using the BC, the IBC, and the NTSMC techniques. Indeed, it has been proved that the corresponding closed loop process for all the considered schemes is stable.
To illustrate the enhanced performance of the proposed nonlinear controllers, we started with introducing various simulations and experimental results; then we proceeded with a detailed comparison of three different schememethods.
ese schemes are established based on their dynamic behavior, their stability, and their robustness properties. Particularly, this study has been compared to other related works presenting the practical results for the BC. e simulation results satisfy the performance and is proved by practice, which values the results given in this work.
Future research will extend the results of this work for MIMO systems. Future research will extend the results of this work for MIMO systems and implement an observer in cases when the state variables system is not all available. In addition, the estimation or the modeling of a physical model can be approximate by using type-2 fuzzy logic [43,44].

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare no conflicts of interest.