Adaptive fast finite‐time control for nonlinear systems subject to output hysteresis by fuzzy approach

This article investigates an adaptive fast finite‐time tracking control problem for a class of uncertain nonlinear systems with output hysteresis. The idea of output hysteresis compensation is skillfully extended to adaptive design by employing a hysteresis inverse transformation and barrier Lyapunov function. The backstepping technique is adopted to establish the fast finite‐time control strategy, which can realize finite‐time convergence of the closed‐loop system. A rigorous theoretical analysis is performed to illustrate that the signals in the closed‐loop systems are bounded in finite time, and the tracking error can converge into a compact set with sufficient accuracy. Finally, a numerical simulation is given to confirm the effectiveness of the presented strategy.


INTRODUCTION
Over the last decade, adaptive control for nonlinear systems has raised widespread attention in many fields.Various control methods, including adaptive control, backstepping, and neural network or fuzzy techniques, have been presented by trailblazers in References 1-9.For the case of unmeasurable states, the observer-based control schemes were discussed in References 1 and 2. In Reference 3, the feedback domination approach for the explicit construction of a smooth adaptive controller was designed based on a new parameter separation technique and the tool of adding a power integrator.
The adaptive control research on parameterized nonlinear systems was given in Reference 4. Furthermore, the generalization of the concept of sliding mode control was addressed for arbitrary dynamic systems governed by a shift operator in Reference 5.Among these techniques, the backstepping control technology based on elementary Lyapunov theory has become one of the most popular control methods.The adaptive backstepping technique has promoted many remarkable works in nonlinear systems, see papers 10 and 11.The robust recursive design technique was proposed in Reference 10 for uncertain nonlinear plants in the vectorial strict feedback form to handle the output maneuvering problem based on the backstepping technique.In Reference 11, the semi-global tracking controller has been designed for strict-feedback nonlinear systems.Previous control methods for nonlinear systems are practically limited by the assumption that uncertainties can be parameterized.With the rapid development of smart technology, the adaptive neural network and fuzzy control problems have been extensively investigated within the backstepping design procedure, resulting in numerous significant presented in References 12-18.The fuzzy control strategy for nonlinear systems was proposed by constructing a high-gain observer for state estimation. 18However, these control schemes have not taken output constraints into account.
It is of great significance to solve the output constraint problem of nonlinear systems.In modern control processes, due to the impact of system performance requirements, physical limitations or security indicators, many practical systems are subjected to various forms of constraints.The violation of constraints will produce undesirable stability problems or performance degradation, such as the joint position of the robotic manipulator should be limited to a preset area to prevent possible structural damage.Therefore, it is essential to pay much attention to the research on systems with output constraints or state constraints, numerous meaningful works have been covered in References 19-23.Although the constrained control problems have made great progress in the field of adaptive control, there are some challenging problems yet.The non-smooth characteristics such as the hysteresis phenomenon widely occur in many industrial processes, which can easily become the source of system instability.In Reference 24, by using the backstepping technique, the event-triggered containment control scheme was developed for a class of stochastic nonlinear multiagent systems with known Bouc-Wen hysteresis.The nonlinear hysteresis exists in the devices or actuators and is reflected in the output of nonlinear systems as well.Due to the complicated processing of output hysteresis, there are few scholars have investigated the adaptive control issue for nonlinear systems subject to output hysteresis.By adopting the barrier Lyapunov function and robust filtering methods, the authors exploited a new adaptive neural algorithm for a class of nonlinear systems with unknown hysteresis output and unmeasurable states. 25In Reference 26, a novel adaptive consensus policy for the switched nonlinear system subject to output hysteresis and unmodeled dynamics has been proposed.However, it should be noted that the foregoing control works merely paid attention to the convergence of systems with infinite time.
Compared with asymptotic stability, it is more meaningful to guarantee the system achieves the desired control objective within a finite time.The theorem of fast finite-time stability, based on Lyapunov analysis, was first elaborated by the authors in Reference 27.Inspired by these pioneering works, researchers have made efforts in the field of finite time and put forward abundant research results on various nonlinear systems. 28-33The fast finite-time tracking control strategy was proposed for full-state constrained pure-feedback nonlinear systems in Reference 33.In Reference 34, the finite-time fuzzy control problem was considered for the first time in nonlinear systems with a pure feedback form.Meanwhile, the output feedback finite-time control problem was investigated for nonlinear systems by devising a state observer. 35 In addition, it should be pointed out that the problem of slow convergence will exist in designed finite-time controllers when system states are far from the equilibrium.This is primarily because when the initial condition is far from the origin, its convergence rate cannot prevail over the control Lyapunov function in linear dynamic equation form V(⋅) + bV(⋅) ≤ 0. To improve the convergence rate, a theory called fast finite-time stability (fast FTS) theory, which meets the relation V ≤ −c 1 V − c 2 V  + , was proposed to ensure the fast finite-time stability of nonlinear systems.In Reference 36, the authors considered the fast finite-time consensus tracking problem for first-order multi-agent systems with unmodeled dynamics.The fast finite-time control scheme for robotic manipulator systems was skillfully proposed based on the terminal sliding mode method. 37From a convergence rate point of view, the fast finite-time control indeed possesses an optimal property.Nevertheless, the fast finite-time stabilization will be unavailable whenever the nonlinear system is unknown.Coincidentally, by the distinct merits of fuzzy logic systems or neural networks, an innovative fast finite-time stabilization criterion had been constructed in Reference 38.Based on the prescribed performance function, Reference 39 studied the fast finite-time dynamic surface tracking control problem for a single-joint manipulator system.Unfortunately, the current published works with fast finite-time stabilization do not consider the output hysteresis of the system.
1.This article makes the first attempt to address the fast finite-time fuzzy control problem of nonlinear systems, in which the output hysteresis and external disturbance exist simultaneously.In contrast to the existing works, this article has tackled this problem within a unified framework.2. Compared with Reference 25, this article further considers the finite time control problem.We incorporate both linear and fractional terms into the controllers and adaptive laws, respectively.Based on the fast finite-time theorem, this method can ensure that all closed-loop system signals converge quickly within finite time without violating the output constraint.
The remainder of this article is arranged as follows.In Section 2, the key lemmas and problem formulation of output hysteresis control design are presented.In Section 3, the finite-time fuzzy controller is constructed with the help of approximation-based backstepping.Moreover, it gives the stability analysis which can achieve the control objective.In Section 4, a simulation example is presented, which is followed by Section 5 to conclude the article.

System description and problem statement
This article considers the uncertain nonlinear system described as where x i (t) = [x 1 (t), x 2 (t), … , x i (t)] T ∈ R i , x(t) ∈ R n denotes the state vector, u(t) ∈ R is control input, and d i (t) is a parameter vector denoting disturbance.f i (⋅) ∈ R is an unknown continuous function.y = H(x 1 ) is the system output which is defined later.The objective of this article is to design a continuous adaptive fuzzy controller such that all signals for closed-loop system are bounded, y(t) can track reference signal y d (t) in finite time, and the tracking error S i can reach zero as time approaches to infinity.To achieve the target of this article, some helpful preparations will be introduced in the following.

Necessary preparations
Assumption 1 (26).The external disturbance d i (t), and reference signal y d (t) are bounded, which satisfies that (26).There exists a strictly increasing smooth Γ i (.): Remark 1.Since x 1 (t) is unavailable, Assumption 2 is given to address the unmeasurable problem caused by output hysteresis.
Lemma 1 (40).For given positive constants  1 ,  2 , , and  ∈ (0, 1), assume that there exists a function V() holds.The trajectory of the system is in fast finite-time convergent while the convergence time satisfies T ≤ where Lemma 3 (41).

Bouc-Wen hysteresis
As pointed out in 45 , the system output of modified Bouc-Wen hysteresis is expressed as follows where  1 and  2 are unknown hysteresis parameters with the same sign, the variable  is specified as where The hysteresis variable  > || is the shape,  is the amplitude of the model.n ≥ 1 determines the smoothness from the initial slope to the asymptote's slope.The auxiliary variable  is bounded by a constant , that is, Remark 2. It is worth noting that the sign of  1 determines hysteresis direction.This article assumes that the hysteretic parameters  1 ,  2 , , , and n are unknown.However, x 1 (t) becomes unmeasurable due to the unknown hysteresis in the nonlinear systems output mechanism.Consequently, dealing with the unknown functions containing x 1 (t) in backstepping control design is challenging.To overcome this obstacle, the Nussbaum function and a variable transformation method are introduced.
The modified hysteresis inverse model can be rewritten as where ) ̇yf ) .
First of all, we introduce the new variable (t) where ) .

Fuzzy logic systems (FLSs)
Construct the fuzzy logic systems as follows and y ∈ R are the input and output of the fuzzy system, respectively.F l i and G l are fuzzy sets in R, N is the number of rules.By adopting a singleton fuzzifier, center average defuzzifier, and product inference, the FLSs can be formulated as where and where (x) = ( 1 (x),  2 (x), … ,  N (x)) T .The FLSs can be rewritten as Let f (x) be a continuous function defined on a compact set Θ, if the fuzzy membership function is selected as a Gaussian function, for ∀ > 0, there exists a FLSs (15) which satisfies

Design of FLSs-based adaptive controller
In this part, we will propose a novel fuzzy control strategy for system (1) by applying the backstepping method and variable separation technique.The FLS is utilized to approximate unknown packaged function f .To further facilitate the control design, we take the transformation as follows where S i is tracking error,  i−1 denotes the stabilizing function which can be established later.
To ensure that the output constraint is not violated, the following barrier Lyapunov function is chosen as where  d1 (t) and  d2 (t) are time-varying functions such that S 1 ∈ (− d1 (t),  d2 (t)).Define the time-varying output bounds as k c1 (t) < y(t) < k c2 (t),  d1 (t) and  d2 (t) are given as Define new variables where Step 1.According to ( 17) and ( 10), we have Let V 1 = V * , according to ( 20) and ( 21), it yields It is obvious that Eq. ( 23) is positive definite and continuously differentiable when q < 1.
= S 1 where , m 0 is an unknown constant.Since f 1 contains all the unknown functions, there exists a FLS Φ T 1  1 (x 1 ) such that where Based on Young's inequality, we can obtain where  1 = ||Φ 1 || 2 .Substituting ( 29) into (27) produces Design virtual controller  1 as where k 1 , k * 1 , a 1 , r 1 , and  are design parameters.Substituting ( 31) and ( 32) into (30) yields Step 2. The Lyapunov function candidate is chosen as Calculating the derivative of S 2 , one has where α1 = . With the definition of α1 , the term −S 2 α1 can be expressed as where Consider the term −S 2 ) x 2 , by using Young's inequality and Lemma 4, it is directly deduced that According to ||x|| ≤ ∑ k j=1 |x j |, together with Assumption 2, it holds that Substituting ( 39) and ( 40) into (38) yields where ) .
According to (28), for given  2 > 0, there is a FLS Φ T 2  2 (X 2 ) to approximate the unknown f 2 such that where X 2 = [y, y d , ̇yd , ÿd , ̇θ, ξd1 , ξd2 , ξd1 , ξd2 ] T .With the similar procedure as in (29), it follows that The virtual controller is constructed as Substituting ( 30), (43), and ( 44) into (41), it holds that ) Remark 3. As can be seen from the above process, the term S 2 S 3 generated in V 2 can be eliminated by the constructed virtual controller  3 in step 3.By analogy, the terms S n−1 S n generated in step (n − 1) can be eliminated by the actual controller  n in step n.
Step i The Lyapunov function candidate is designed as The time derivative of V i can be expressed as Consider the time derivative of where αi−1 Similar to (37) and (39), it holds that By using Young's inequality, we have Substituting ( 50) and ( 51) into (47) yields where Similarly, a FLS Φ T i (X i ) is utilized to approximate f i , we can get where , ̇θ, ξd1 , ξd2 , ξd1 , ξd2 ] T .Following the similar procedure as (43), we have Design the virtual controller as According to (45), the following inequality holds Based on the above design procedure, we select u =  n and note that S n S n+1 = 0 while i = n.Consequently, by iterating (56) from i = 3 to i = n, we can obtain the following inequality

Stability analysis
In this subsection, the main result is described by the following theorem.
1.The output y will remain in the bounded set, that is to say, the output constraint is never violated.2. The closed-loop system is finite time stable, all signals are bounded and converge to an arbitrary small area around the origin within a finite time.
Proof.The total Lyapunov function V o is constructed as where The adaptive laws are defined as ) where l i and  i1 ,  i2 are positive parameters.θi =  i − θi , θi represents the estimate of  i .

SIMULATION
To exemplify our proposed design approach, a specific simulation example to demonstrate our result.
Example: Consider the following nonlinear system where y is the output hysteresis with the parameters given as According to the design procedure in Section 3, the controllers and adaptive laws are as follows The design parameters are selected as The reference signal is given as y d = 0.8 sin(0.5t).
, it can be shown that the convergence interval is influenced by the design parameters k i ,  i1 , and .The convergence time T o is influenced by the parameters k * i ,  i2 , h, and .Increasing  will make  more smaller.By increasing the values of , k i , and  i1 , the approximation error and estimation error can be minimized.Similarly, by increasing k * i ,  i2 , and decreasing k * i ,  i2 , the settling time can be adjusted as small as possible.The simulation results are shown in Figures 1-5.As shown in Figure 1, system output y(t) can track ideal signal y d (t) in finite time under the proposed control scheme.while the output y does not violate the constraints.Figure 2 shows the system tracking error.Figure 3 depicts the state trajectory of the system with the proposed approach.As can be seen, the system state is bounded.We can see the system state tends to the zero equilibrium rapidly.Figure 4 gives the trajectory of the controller u(t).Figure 5 shows the adaptive parameters curves.The simulation results highlight that the proposed control algorithm could achieve the control objective.

CONCLUSION
This article has presented an adaptive fast finite-time tracking control strategy for a class of uncertain nonlinear systems with output hysteresis.A fast finite-time stability criterion is given to achieve adaptive finite-time stabilization with a fast convergence rate.The idea of output hysteresis compensation is skillfully extended to adaptive design by employing a hysteresis inverse transformation and barrier Lyapunov function.The backstepping technique is adopted to establish the fast finite-time control strategy, which ensures finite-time convergence of the closed-loop system.A rigorous theoretical analysis is performed to illustrate that the signals in the closed-loop systems are bounded in finite time, and the tracking error can converge into a compact set with sufficient accuracy.Furthermore, it would be a significant problem to address the event-triggered control design for nonlinear systems with output hysteresis.This article only investigates traditional time-triggered control, the event-triggered control can reduce both communication costs and transmission frequency.In our future work, we will focus on the event-triggered prescribed finite-time control problem for nonlinear systems with output hysteresis.

Remark 4 .
According to the definition of  1 , T o , and the region (

F I G U R E 1
The trajectories of y(t) and y d (t).

1 F I G U R E 2
The trajectory of tracking error S 1 .

2 F I G U R E 3 4
The trajectory of system state x 2 .The trajectory of controller u(t).

1 F I G U R E 5
The curves of adaptive laws θ1 and θ2 .