Observer-based finite-time adaptive fuzzy back-stepping control for MIMO coupled nonlinear systems

: An attempt is made in this paper to devise a finite-time adaptive fuzzy back-stepping control scheme for a class of multi-input and multi-output (MIMO) coupled nonlinear systems with immeasurable states. In view of the uncertainty of the system, adaptive fuzzy logic systems (AFLSs) are used to approach the uncertainty of the system, and the unmeasured states of the system are estimated by the finite-time extend state observers (FT-ESOs), where the state of the observer is a sphere around the state of the system. The accuracy and efficiency of the control effect are ensured by combining the back-stepping and finite-time theory. It is proved that all the states of the closed-loop adaptive control system are semi-global practical finite-time stability (SGPFS) by the finite-time Lyapunov stability theorem, and the tracking errors of the system states converge to a tiny neighborhood of the origin in a finite time. The validity of this scheme is demonstrated by a simulation.


Introduction
In the past few decades, nonlinear system control methods have attracted the research interest of many scholars. Moreover, the systems with complexity, uncertainty and coupling are more challenging to control. In order to achieve accurate, stable and rapid control effect, many methods have been applied to nonlinear systems, such as adaptive control, fuzzy logic control, neural network control, nonlinear system with unmeasured states. Therefore, in this paper, a finite-time adaptive fuzzy backstepping control scheme for a class of MIMO coupled nonlinear systems with immeasurable states is devised. The solution of this issue is realized by the combination of fuzzy system, back-stepping and finite time theory.
The main advantages of this study can be summarized as follows: 1) The finite-time stability of MIMO nonlinear systems with uncertainty and coupling is studied by using back-stepping method, the systems are different from the non-coupled MIMO non-strict feedback system [35] and the SISO systems [36,37,39]. Since, it solves the problem that is difficult to control the non-strict feedback MIMO coupling nonlinear system with unmeasured states. 2) To solve the issue of the unpredictable system state, a FT-ESO is introduced to estimate the unmeasured state which fails to account for in the case where the system has unknown states [33,34,38]. Furthermore, the FT-ESO is different from the observers, where the output feedback is difficult to apply to MIMO coupling nonlinear systems [35,39].
This study not only guarantees that the observation errors converge to the tiny neighborhood of the origin in a finite time, but also ensures that the error of the system state and the reference quantity converges in a finite time. Through the finite time Lyapunov theory, it finally achieves SGPFS.

System description
Considering the following nonlinear MIMO non-strict feedback system: Where, is the vector of the state, = [ ⋯ ] , , and Y mean the input and output variables, and is bounded disturbances represented by . ( ) and ( ) are unknown nonlinear smooth functions with the following structure: Remark 1: The nonlinear function and of the controlled system (2.1) are completely unknown and FLSs are used to approach the uncertainties. Assumption 1: The total disturbance is bounded. Assumption 2: Considering the assigned reference signals, their first-order and second-order derivatives are bounded.

Fuzzy approximation theory
The FLS is a form of nonlinear function and can approximate all nonlinear functions with any precision, so it can be applied to all kinds of problems. FLSs are constructed by employing some specific inference, fuzzifier and de-fuzzifier strategies and form IF-THEN rules. Consequently, information from human experts in various fields can be integrated into the controller.
The design of a FLS is divided into four sections namely, the fuzzifier, the knowledge base, the fuzzy inference engine and the de-fuzzifier. Assuming that the FLS is composed of N fuzzy rules, the j th fuzzy rule can be denoted as: If is and ⋯ and is , then y is ( = 1,2, ⋯ , ).
Where, , are the fuzzy sets in R, related to the membership functions.
After adopting the product Inference Engine, the singleton fuzzifier and the center average defuzzification, the FLS's output is given as:

Observer-based finite-time fuzzy adaptive back-stepping design and stability analysis
This section introduces a FT-ESO to obtain the unmeasured states [42]. Then the observer-based finite time adaptive back-stepping controller is proposed. Finally, the system's stability is verified.

Finite-time extended fuzzy state observer design
Since the state of the system is unpredictable, the extended state observer needs to be designed in the following form to estimate immeasurable states: Where, the time-varying gains ( , ) are functions of a constant parameter T and the real-time t.
The convergence time is regulated by select parameter T later. The vector of estimation error is defined as: = − . The error models are generated as follows: In order to weaken the errors to a tiny area of 0 along with incline to the convergence time , the time-varying gains will be given. Considering nonlinear system (2.1) and the observer (3.1), which derive the error model (3.2), the extended state observer gains are presented as follows: Where, ̅ , for = 1,2, ⋯ , and = 2, ⋯ , + 1 are given as follows: Moreover, Where, ̅ , = 1 and for < one has ̅ , = 0 , ( , ) = . The scalar coefficients for = 1, ⋯ , are chosen such that the ( The error model (3.2) is stable along with the time T of convergence and the error model (3.2) is finite-time input-to-state stable (FT-ISS) for a positive constant > 1, Where, ( , ) = , and ( , ) = are time-varying functions. Moreover, gradually = [0 ⋯ 0 1] , m is an integer design parameter, is a lower triangular matrix. Moreover, ( ) is defined as .
The Lyapunov function is chosen as = ∑ , after differentiating, we get the following formula: ≤ − . Remark 2: The estimation error is equally bounded in finite time by the scheme without knowing the upper bound of the perturbation. Then, using the finite-time Lyapunov function described in the next part, a controller is built to maintain the closed-loop system's stability.

Finite-time adaptive back-stepping design and stability analysis
In this part, a finite-time adaptive fuzzy control approach is proposed by the adaptive backstepping technique and the stability analysis is shown through the finite-time Lyapunov function.
The system is converted as follows to apply back-stepping technique: Where, > . For the singularity problem in the subsequent derivation of , the method of obtaining the pseudo-inverse matrix is used to solve it. The differential expression is = . When =0, 0 < < 1 , ( ) •( ) has a singularity problem, so the pseudo inverse value needs to be obtained, and then ( ) •( ) =0. There are similar problems in ( ) •( ) , therefore, it will not be repeated. After the above analysis, the value at the singularity is = . For the first subsystem, considering the observer stability and position errors, define Lyapunov function as: When the estimation matrix is singular, the above operation is difficult to achieve. In order to overcome this disadvantage, we find the generalized inverse of . Then, It can be seen from the expression of , that c contains the model information of the above systems. In order to realize the control without model information, FLS is used to approximate .
Based on Lemma 1, we expect that the continuous function can be evaluated by the FLS as following Step 3: Stability analysis: , there is an optimal constant * that minimizes the approximation error. Define the reach time as Where, (ϛ(0)) is the initial value. Then, in the light of Lemma 5, This indicates that all the closed-loop state variables are SGPFS. Furthermore, the following inequality shows the tracking error gets into a tiny neighborhood around the origin after the .
Remark 3: The selection of the designed controller parameters is directly related to the capability of the dynamic response. Decreasing the neighborhood radius and increasing the convergence speed of the state vectors in system (2.1) is accomplished by increasing the parameter values. However, this leads to a greater control input that the designed parameters gain through trial-and-error method. We should gradually enlarge them from zero until the performance is contented. The structure of the above design approach is displayed in Figure 1.

Simulation study
In this section, the simulation example is considered to prove the validity of the presented control scheme. Dynamic model of the two-joint manipulator system is given as: Hurwitz, afterwards selecting the designed parameter independently to regulate the convergence time. The results of the simulation are displayed in Figures 2−8. In Figure 2, it is seen that the state follows the reference input , the estimated state follows and it exhibits the system state , the reference signal and the estimated state . Figure 3 indicates that the error for is bounded, the error f or -is bounded. Similarly, Figure 4 exhibits that the error for -is bounded, and the error for -is bounded. Figure 5 indicates that the estimated state follows the system state differential , the estimated state follows the system state differential . Figure 6 indicates the applied control input signal and . Figure 7 illustrates that the trajectories of adaptive rate ‖ ‖ . In order to compare the difference between the finite-time application results and the infinite-time application results, we carried out a comparative simulation. The premise is to adjust best, the parameters of the two controllers by the trial-and-error method. Figure 8 illustrates that the finite-time methods show better error convergence performance.   The results of the simulation clearly show that the finite-time fuzzy adaptive back-stepping control strategy based on FT-ESO can guarantee that all the state variables of the system are SGPFS, and the tracking error gets into a tiny neighborhood around the origin in a finite time.

Conclusions
This research proposed a fuzzy adaptive control method for a class of MIMO coupled nonlinear systems by combining the back-stepping technique, finite time theory and FT-ESO. The back-stepping technique is used by transforming the general form of the coupled nonlinear system. Then the virtual control input is introduced and the fuzzy adaptive control rate is designed for the new form of the system according to the back-stepping technique. The fuzzy approximation function is included in the actual control signal. Furthermore, due to the introduction of FT-ESO, the proposed control approach does not need the states of the control systems to be directly measured. The finite-time Lyapunov function can ensure the closed-loop systems stability. This method solves the problem of application of fuzzy adaptive control, which is difficult to apply to a class of coupled nonlinear systems with immeasurable states and uncertainties. The efficacy of the proposed strategy is verified by the simulations. Considering some practical situations, future research will focus on systems with the input saturations or dead zones.