Finite Time Prescribed Performance Control for Uncertain Second-Order Nonlinear Systems

In this article, we discuss the ﬁnite time stability problem for second-order systems with an uncertain nonlinear function. A ﬁnite time performance function with the sinusoidal function is constructed, and the constrained problem of the original system is transformed into the stability problem of the equivalent system. Combining prescribed performance control and fuzzy logic systems, an eﬀective control method is proposed. The simulation results also prove that the method we adopted is eﬀective.


Introduction
Second-order nonlinear systems [1][2][3][4][5][6] have a variety of potentials in the real world, such as the robotic manipulator, horizontal platform system, navigation system, and gyro system. At present, many control methods [6][7][8][9][10][11][12][13][14] have been proposed for second-order nonlinear systems. For example, Zhao et al. [7] developed an output feedback control method for the second-order nonlinear system with uncertain parameters. For uncertain second-order nonlinear systems, a saturated controller was proposed in [8], which can overcome the system uncertainty and external disturbance. In [9], an adaptive fuzzy distributed control scheme was proposed, which realized the consensus tracking consistent of the second-order multiagent system. An adaptive PID control method was developed in [10], which makes the output error of the system tend to zero and ensures the stability of the system. In order to deal with the fault regulation problem of a class of second-order nonlinear systems, Van [11] proposed a high-order terminal sliding mode control method. It should be pointed out that the above control methods can only ensure that the tracking error enters a small neighborhood of zero, but the neighborhood and the time to reach the neighborhood cannot be set in advance.
Recently, many researchers have been studying the prescribed performance control (PPC) method [12][13][14][15][16][17][18]. e idea is to transform the restriction problem of an original system into the stability problem of an equivalent system by using the performance function and transformation function. However, the traditional PPC method cannot solve the presetting time problem. erefore, in recent years, different types of finite time performance functions were proposed. In [19], a finite time performance function with the exponential function was proposed, and the designed PPC method realized that the tracking error converges to the predefined zone in presetting time. Using the same finite time performance function in [19], Tran and Ho [20] studied the PPC strategy of uncertain horizontal platform system. In [21], a finite time performance function was constructed by polynomials, based on the partial persistent excitation condition, the tracking error of the strict feedback system can approach the predefined zone in presetting time and unknown functions can also be estimated accurately.
Inspired by the above work, this paper will construct a finite time performance function through the sinusoidal function to investigate the stability of uncertain secondorder nonlinear systems. e overall structure of this paper is as follows. In Section 2, some preliminaries are presented. e finite time PPC method and its stability analysis are investigated in Section 3. Section 4 gives an example for simulation. Finally, the conclusion is given in Section 5.

System Descriptions and Problem Formulations
Consider the following uncertain second-order nonlinear system: where x) is an unknown nonlinear function, and u(t) ∈ R is the control input. Let x d (t) ∈ R be a desired signal. e following assumptions are provided for later discussion.
e nonlinear function f(t, x) is unknown but bounded.
Define the tracking error e(t) � x(t) − x d (t); the aim of this paper is to limit the tracking error e(t) within the preset boundary through the finite time prescribed performance control method.
In order to make the tracking error e(t) meet the steadystate performance and transient performance, the following constraint condition is designed: where ρ(t) is a finite time performance function and defined as where a 0 and a 1 are preset positive parameters and T is the preset time. Usually, the tracking error e is transformed into an equivalent expression using the transformation function.
In this article, the transformation function is defined as follows: Define e(t) � ρ(t)Γ(z); if z(t) is bounded, then Γ(z) satisfies − 1 < Γ(z) < 1, which implies that the tracking error e(t) is limited within (− ρ(t), ρ(t)). erefore, the following work focuses on the boundedness of z(t). (3), we can also use the cosine function to design a finite time performance function as

Remark 1. Similar to performance function
where parameters α 0 and α 1 are the same as those in (3).

Remark 2.
e design of the transformation function is also important for the PPC method, and transformation function (4) can also be changed as In order to ensure |z| < 1, the construction of the barrier Lyapunov function needs to be considered.

Control Design and Stability Analysis
Now, the first derivative of e(t) can be obtained as which, together with (7), one gets where (3) and (4), one has So, Λ(t) satisfies the inequality Λ(t) ≥ (1/ρ(0)) > 0. From (7), one gets Combining (1), (7), and (10), one obtains where . In order to prove that z(t) is bounded, we introduce a new variable s(t) as where c 1 is a positive design parameter.
Since f(t, x) is unknown, we will employ fuzzy logic systems to estimate f(t, x) in system (1). According to Lemma 2 in [19], there exists a fuzzy logic system (FLS) where θ * is the ideal constant weight vector, w(x) is the basis function vector, and ε f (x) is the bounded approximation error, i.e., there exists positive constant ε f such as where θ is the estimation of θ * and θ � θ * − θ is the estimation error. In this paper, the controller u(t) is designed as where x d (t)) and c 2 is a positive constant. And choose the adaptive law of θ as where c f > 0 and λ f > 0. Now, we give the main conclusions as follows.

Theorem 1. Under given Assumptions 1-3, when initial
conditions |x(0)| < ρ(0) are satisfied, then controller (14) and the parameter adaptive law (15) can guarantee that all signals of the closed-loop system are bounded, which also means that the tracking error e(t) is limited within constraint condition (2).

Numerical Simulations
In this part, an uncertain gyroscope system [22] is used to show the availability of the proposed method in this paper. e gyroscope system is described as x. e corresponding fuzzy membership functions are chosen as follows: where ξ � x, _ x and l � 1, 2, 3, 4, 5. In order to compare with the proposed method in this paper, the traditional feedback method is designed as follows: where parameters c 1 , c 2 , c f , and λ f are designed as c 1 � c 2 � 5, c f � 3, and λ f � 0.05. e control effect of traditional method (26) is shown in Figures 1-3.    Obviously, the tracking error e(t) is not effectively controlled by using traditional method (26). With the above same parameters, the prescribed function ρ(t) is designed as e control effect of controller (14) is shown in Figures 4-6. It can be seen that the control effect of e(t) has been improved, and the tracking error e(t) has been limited within [− 0.05, 0.05] after 5 seconds. One strong point of the proposed method (14) is that it can change the control time and control interval, such as setting the prescribed performance function as e control effect is shown in Figures 7-9. e above simulation results show that the proposed method (14) has better transient performance in practical applications.      x, x d x by using the method (14) x d Figure 8: Time response trajectories of x and x d by using method (14) under ρ 2 (t).

Conclusion
We studied the finite time control problem of uncertain second-order nonlinear systems in this article. For the sake of the tracking error reaching the preset zone at the preset time, a finite time performance function was introduced. Meanwhile, the fuzzy logic system (FLS) was used to evaluate the uncertain function of the system. rough theoretical analysis, this paper proves that the proposed control method achieves the expected control effect. At the same time, this conclusion is also verified by simulation.
Data Availability e data used in this paper are reflected in the manuscript.

Conflicts of Interest
e authors declare no conflicts of interest.