Design of a Robust Interval Type-2 Fuzzy Adaptive Super Twisting Control for a Given Class of Disturbed MIMO Nonlinear Systems

This study deals with the tracking control problem of a large class of multi input multi output (MIMO) nonlinear systems with unknown dynamics and subject to unknown disturbances. First, two interval type-2 adaptive fuzzy systems (IT2-AFSs) are constructed to efficiently estimate the unknown nonlinear dynamics. Then, based on IT2-AFSs and super-twisting algorithm (STA), a new robust adaptive fuzzy-reaching STC law (AFRSTCL) has been added to the global control law to improve the robustness of the studied systems in the presence of approximation errors and unknown disturbances. In order to avoid the chattering phenomenon and guarantee simultaneously the best tracking performance, the gains of the designed AF-RSTCL are optimally online estimated. The adaptive parameters of the global synthesized control law are deduced from the stability analysis in the sense of Lyapunov. Finally, an example of simulation is used to confirm the effectiveness of the developed method in achieving the predetermined objectives of the tracking control.


INTRODUCTION
The design of robust control algorithms for disturbed multi input multi output (MIMO) uncertain systems with highly coupled nonlinearities is one of the biggest control problems encountered in the system control field. Therefore, in recent decades, several sophisticated control approaches have been developed for controlling such complex processes. Among them, those using fuzzy systems (FSs), adaptive control and sliding mode control (SMC) have been credited in many different real applications as effective and robust methods of control [1][2][3][4][5][6][7][8].
SMC is an efficient approach of robust control techniques widely utilized in many practical applications due to its capability to handle any uncertainties or disturbances influencing the system dynamics. However, the chattering phenomenon caused by the discontinuous term of the SMC law can be harmful to the actuators [9,10]. To eliminate or at least attenuate the chattering phenomenon, several control techniques have been employed in various technical and technological applications [11][12][13][14][15][16][17]. And in particular, those using super twisting algorithm (STA) algorithm, have recently been extensively used in the literature [18][19][20][21][22]. The STA has been introduced by Levant [23] to attenuate the chattering phenomenon while simultaneously ensuring a finite time convergence and the stability of the closed loop control system. However, the design of good values of the super-twisting control (STC) law gains remains one of the major disadvantages of such kind of algorithm. Indeed, the large gains values generate a big chattering in the control system and the small ones damage the system robustness, and can even lead to the instability of the closed loop control system. In [24], two terms are added to the conventional STC algorithm to guarantee the desired performance for uncertain nonlinear systems. And most recently, in [25], based on STC algorithm and a high order sliding mode observer, a robust controller-observer is proposed for an uncertain unmanned aerial system to ensure the desired tracking performance.
On the other hand, and due to the universal approximation property of FSs [26], fuzzy control methods have been largely studied and successfully implemented in control problem by many researchers [27][28][29][30][31][32]. However, type-1 FSs (T1-FSs), in the presence of linguistic uncertainties and measurement errors, are not able to process excellently such perturbations. Therefore, to overcome this constraint, type-2 FSs (T2-FSs), which constitute a generalization of T1-FSs, show appreciable and attractive features in taking into account such uncertainties [33][34][35][36]. The key feature is the so called footprint of uncertainty (FOU), which allows incorporating directly the uncertainties in the membership functions (MFs). In [35], to show the superiority of a reactive proposed control approach based on T2-FSs to drive a robot mobile towards a mobile target, it is compared against its counterpart using T1-FSs instead of T2-FSs. And, in [36], the proposed method, which is used to describe and control an underactuated mobile two wheeled inverted pendulum in the presence of modelling errors and external disturbances, shows that when it is based on T2-FSs rather than T1-FSs, it is more effective in achieving the desired performance of control.
Compared with the reported researches in the literature, the main contributions of this study are presented as follows: (1) A new effective tracking control design approach is developed for a general class of MIMO nonlinear systems characterized by the following features: • All dynamics are entirely unknown and suffer from time varying disturbances. • The disturbances that influence the studied systems dynamics are completely unknown, which means that no information of the upper bounds of these perturbations is available. • The physical parameters are unknown and present parametric variations.
(2) Two IT2-adaptive FSs (IT2-AFSs) are constructed to perfectly estimate the unknown nonlinear dynamics. (3) In order to avoid the chattering while simultaneously improving the tracking control performance in the presence of approximation errors and unknown disturbances, a new synthesized adaptive fuzzy-reaching STC law (AF-RSTCL) based on IT2-AFSs and STA has been added to the proposed control law. The AF-RSTCL gains are optimally online estimated in order to provide the good gains values that guarantee the system robustness and the best tracking performance with a smooth continuous control law. (4) The adaptation laws of both the IT2-AFSs and AF-RSTCL gains are deduced using the stability analysis theorem of Lyapunov. The closed loop stability and the tracking error convergence are guaranteed despite all the constraints discussed above. This paper is organized as follows. Section 2 describes the IT2-FSs. In section 3, the problem formulation is presented, and in section 4, we explain the proposed tracking control approach designed for a large class of disturbed MIMO nonlinear systems with unknown dynamics. Section 5 presents the simulation results of a two-link robot manipulator to confirm the effectiveness of the proposed tracking control algorithm. The conclusion is given in section 6.

INTERVAL TYPE-2 FUZZY SYSTEMS
IT2-FSs are used due to their excellent efficiency to handle directly measurement uncertainties and inaccurate linguistic information used to synthesize T2 fuzzy rules. Thus, the fuzzy sets for an IT2-FS are implemented in such a way that their associated MFs can easily incorporate the above discussed uncertainties through their FOU property.
The jth IT2 fuzzy rule of an IT2-FS, having q inputs and one output, can be formulated as follows [37]: R is the state vector; y ∈ R is the system (1) output; and M denotes the number of IT2 fuzzy rules.
For the system (1), if the product is used as an inference engine then the firing set associated to the jth IT2 fuzzy rule can be given as follows: represent respectively the left and right most values of the MFs associated to the IT2 fuzzy sets j i E . By using the center of sets technique, the output sets of the inference engine, which are IT2 fuzzy sets, are reduced to an IT2 fuzzy set called type reduced set. Then, based on the center of gravity method, the defuzzified output of the system (1) can be obtained as follows [38]: where l y and r y can be expressed as: are respectively the lower and upper vectors of fuzzy basis functions; are the lower and upper parameter vectors, respectively; j w is an element of the firing set l y and r y can be obtained easily using the Karnik-Mendel algorithm [39].

PROBLEM STATEMENT
Consider a general class of MIMO n th order nonlinear systems, having k inputs and p outputs, described by the following equation: R are the input and the output of the system (5), respectively, such that p k ≤ ; R is a vector of bounded unknown nonlinear continuous functions, and is a matrix of bounded unknown nonlinear continuous functions; ( 1) , ,..., is the first element of x , and q n p = × ; Assume that ( ) G x is a non null matrix, and

TRACKING CONTROL DESIGN METHOD
The control problem is to steer the state x of the system (5) to a desired state . Hence the first step in designing a robust tracking control despite unknown dynamics and unknown disturbances that affect the system (5) is to effectively estimate the unknown dynamics. Thus, in this paper, ( ) F x and ( ) G x are substituted by their IT2-AFSs.
Based on the IT2-FS defined in (1) and characterized by its output expressed in (3), the IT2-AFS outputs used to estimate the unknown nonlinear functions i f and , i j g can be given as: respectively the lower and upper vectors of fuzzy basis functions, as described by equation (4), and they are used to estimate ( ) F x ; likewise, the vectors of fuzzy basis functions used to estimate STC is an effective robust second order SMC capable to drive the system state trajectories with a continuous control law to the desired dynamics in finite time despite the existence of uncertainties and disturbances.
be the tracking error. Then, the sliding surface for the system (5) is designed as [41]: The time derivative of the sliding surface ( , ) s x t is given as: The main objective of the tracking control using STC is to ensure that the condition 0 The optimal parameters of ( ) , ( ) Then, The minimum approximation error of i f and , i j g can be expressed as: optimal approximations of i f and , i j g , respectively.
Based on STA and IT2-AFSs, the synthesized control law that ensures the predetermined objective of control is given as: is the reaching STC law (RSTCL), and it is given as [23]: are positive learning parameters. Theorem 1. Consider the general class of MIMO nonlinear systems expressed in (5), the IT2-AFSs presented in (6) and the adaptation laws given in (13). Then, the control law defined in (11), it guarantees the stability of the closed loop system and ensures the desired tracking performance despite approximation errors, unknown disturbances and unknown dynamics Proof 1. The following augmented Lyapunov function candidate is used to demonstrate the stability of the closed loop system: 1 11 ( ) ( ) ( , ) ( , ) The time derivative of v gives: From equation 11, one gets: Then, substituting the above expression in (15) gives: By subtracting and adding the optimal approximations 1 2 T into equation (17), one gets: ( ) By considering the adaptation laws defined in (13) and using equation (12), one gets: The gains i a and i b are chosen to satisfy the following condition: The inequality below guarantees that v becomes negative. And therefore, the proof 1 is completed.
The selection of good values of the gains i a and i b , which ensures the desired tracking performance while avoiding the chattering, remains among the most complicated tasks in real applications. The selection of large gains values can improve the tracking performance, but this cause much chattering in the control system, which may harm the system actuators. On the contrary, the small gains deteriorate the tracking performance and can even cause the system instability. In this study, to overcome this constraint of control, the optimal gains of the RSTCL i ST u are efficiently online estimated using IT2-AFSs, which guarantees both the desired tracking performance and a smooth continuous control input.

Interval Type-2 Adaptive Fuzzy Super Twisting Control Design: Step 2
Based on the IT2-FS defined in (1), which is characterized by its output expressed in (3) are substituted by their corresponding IT2-AFSs to excellently estimate the optimal gains that guarantee the desired tracking performance by ensuring the condition (20) while simultaneously avoiding the chattering.
The designed IT2-AFSs of ( ) a u i and ( ) b u i , are respectively given as: are the online tuned parameter vectors of the corresponding IT2-AFS defined by the above equation (21); N denotes the fuzzy rules number.
The optimal parameters of the IT2-AFSs ˆ( ) a u i and ˆ( ) b u i can be given by: The interval type-2 adaptive fuzzy super twisting control (IT2-AFSTC) law proposed in this paper is given as: The adaptation laws of the parameter vectors  (5), the IT2-AFSs described by equations (6) and (21), and the adaptation laws given by equations (13) and (24), then, the global control law presented by equation (23) ensures the stability of the closed loop system with the best tracking performance while avoiding the chattering, despite all the constraints that influence the control system, such as unknown dynamics, approximation errors and unknown disturbances.
Proof 2. To demonstrate the stability of the closed loop system, the following augmented Lyapunov function is used:  (14), (19) and (23), the time derivative of equation (25) can be given as: By considering equations (24) and ( Thus, the proof 2 is completed.

Dynamical Model of a Two Link Robot Manipulator
We present in this section the simulation results of a two link robot manipulator in order to confirm the effectiveness of the developed method of tracking control. A schematic representation of a two link robot manipulator is given in Figure 1. We assume that the masses 1 m and 2 m are unknown, and that they are subject to the time variation 1 m Δ and 2 m Δ , respectively. Hence, the system (31) can be reformulated as: where ( ) f x Δ and ( ) g x Δ represent the time varying disturbances that influence the system dynamics caused The dynamics ( ) f x and ( ) g x of the system (32) are assumed to be unknown and suffer respectively from the unknown time varying disturbances ( ) f x Δ and ( ) g x Δ . Therefore, we can apply the developed control law expressed by equation (23) to achieve the desired objective of control.

Simulation results
The physical parameters of the robot manipulator used for simulation are as follows: The unknown disturbances vector is represented as: We assume that 1 q and 2 q belong to 2 2 π π The designed IT2-AFSTC law defined by equation (23) can be expressed for the two-link robot manipulator system (32) as: The IT2-AFS F uses four inputs 1 q , 2 q , 1 q and 2 q to estimate F , whereas the IT2-AFS Ĝ uses two inputs 1 q and 2 q to estimate G . Three MFs are associated to each input of the IT2-AFS F , as shown in Figure 2. Likewise, three MFs are associated to each input of the IT2-AFS Ĝ , as depicted in Figure 3.
The AF-RSTCLˆS T u has the sliding surface as input vector, and uses three MFs for each of its inputs, as presented in Figure 4. In order to confirm the effectiveness of the designed tracking control method, a comparison was carried out with two approaches of tracking control, namely 1) fuzzy SMC (FSMC) approach, and 2) fuzzy STC (FSTC) approach. The first approach uses two type-1 adaptive fuzzy systems (T1-AFSs) to approximate F and G , and a conventional reaching SMC law to deal with the unknown disturbances and approximation errors; and the second approach uses two T1-AFSs to approximate F and G , and a STC law to reject the undesired effects caused by unknown disturbances and approximation errors.
The control laws used by the FSMC and FSTC approaches are given respectively by equations (34) and (35):   The constant parameters used by the proposed IT2-AFSTC, FSMC and FSTC approaches, are given in Table 1 below.  (1) 2800 The simulation results obtained from the comparison that was carried out between the three approaches of control are illustrated in  In comparison with the FSTC and FSMC approaches, the proposed IT2-AFSTC approach shows the best tracking performance with a smooth continuous generated control input vector.
This superiority of the IT2-AFSTC approach in ensuring the desired objective of control is due to its both features, namely 1) an optimal estimation of unknown dynamics, and 2) a great efficiency in rejecting all disturbances that influence the system robustness. On the other hand, the FSTC approach presents the better tracking performance in comparison with the FSMC.
In Figures 17 and 18 below, we notice that the more the gains of u ASFTC and v AFSMC become large, the more the tracking performance increases. However, this improvement in tracking performance comes at the expense of the smoothness of the applied control law. Indeed, as shown in Figures 19-22, the FSTC law presents big variations, and in the FSMC law, the chattering becomes more severe, which may be harmful to the system actuators.
And yet, even with this improvement in tracking control performance of both FSTC and FSMC approaches, which generate the chattering in their control inputs, the proposed IT2-AFSTC approach still shows better tracking performance while avoiding the chattering.

CONCLUSION
In this study, we designed a new rigorous tracking control approach for a general class of disturbed MIMO nonlinear systems with unknown dynamics. First, in order to effectively estimate the unknown dynamics, two IT2-AFSs have been constructed. Then, in the presence of approximation errors and unknown disturbances, a new robust synthesized AF-RSTCL has been introduced. Based on IT2-AFSs, the AF-RSTCL gains are optimally online estimated in order to guarantee the best tracking control performance while simultaneously avoiding the chattering. The adaptation laws are obtained using the stability analysis of Lyapunov. The simulation results confirm the mathematical proof of the proposed approach of control in ensuring the desired tracking performance with a smooth continuous control law, despite all the constraints that disturb the control system.