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

Authors: Hui Lv; Wei Xiang; Jun Zhu
Journal: Journal of Mathematics
 (2022)

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## Abstract

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.


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## Body

## 1. Introduction

Second-order nonlinear systems [1–6] 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 [6–14] 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 [12–18]. The 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. Therefore, 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 second-order nonlinear systems. The overall structure of this paper is as follows. In Section2, some preliminaries are presented. The 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.


## 2. System Descriptions and Problem Formulations

Consider the following uncertain second-order nonlinear system:(1)x¨t=ft,x+gt,xut,where xt∈ℝ is the system state, x=xt,x˙tT, ft,x is an unknown nonlinear function, and ut∈ℝ is the control input. Let xdt∈ℝ be a desired signal. The following assumptions are provided for later discussion.Assumption 1.
Statesxt,x˙t,xdt,x˙dt, and x¨dt are measurable.Assumption 2.
The nonlinear functionft,x is unknown but bounded.Assumption 3.
gt,x is known, and gt,x≠0 for all t≥0 and x.
Define the tracking erroret=xt−xdt; the aim of this paper is to limit the tracking error et within the preset boundary through the finite time prescribed performance control method.
In order to make the tracking erroret meet the steady-state performance and transient performance, the following constraint condition is designed:(2)−ρt<et<ρt,where ρt is a finite time performance function and defined as(3)ρt=a0sin3π2TT−t+a1,0≤t≤T,a1,T<t,where a0 and a1 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:(4)Γz=ez−e−zez+e−z.
Defineet=ρtΓz; if zt is bounded, then Γz satisfies −1<Γz<1, which implies that the tracking error et is limited within −ρt,ρt. Therefore, the following work focuses on the boundedness of zt.Remark 1.
Similar to performance function (3), we can also use the cosine function to design a finite time performance function as(5)ρt=a0cos3π2TT−t+a1,0≤t≤T,a1,T<t,where parameters α0 and α1 are the same as those in (3).Remark 2.
The design of the transformation function is also important for the PPC method, and transformation function (4) can also be changed as(6)z=etpt.
In order to ensurez<1, the construction of the barrier Lyapunov function needs to be considered.


## 3. Control Design and Stability Analysis

Now, the first derivative ofet can be obtained as(7)e˙t=ρ˙tΓz+ρt∂Γz∂zz˙t,which, together with (7), one gets(8)z˙t=Π1t+Λtx˙t−x˙dt,where Π1t=−ρ˙tΓz/ρt∂Γz/∂z,Λt=1/ρt∂Γz/∂z. According to (3) and (4), one has(9)0<a1≤ρt≤ρ0,0<∂Γz∂z=4ez−e−z2≤1.So,Λt satisfies the inequality Λt≥1/ρ0>0. From (7), one gets(10)e¨t=ρ¨tΓz+2ρ˙t∂Γz∂zz˙t+ρt∂2Γz∂z2z˙t2+ρt∂Γz∂zz¨t.Combining (1), (7), and (10), one obtains(11)z¨t=Π2t+Λtft,x+gt,xut−x¨dt,where Π2t=−ρ¨tΓz+2ρ˙t∂Γz/∂zz˙t+ρt∂2Γz/∂z2z˙t2/ρt∂Γz/∂z. In order to prove that zt is bounded, we introduce a new variable st as(12)st=z˙t+c1zt,where c1 is a positive design parameter.Sinceft,x is unknown, we will employ fuzzy logic systems to estimate ft,x in system (1). According to Lemma 2 in [19], there exists a fuzzy logic system (FLS) θ∗Twx such that(13)ft,x=θ∗Twx+εfx,where θ∗ is the ideal constant weight vector, wx is the basis function vector, and εfx is the bounded approximation error, i.e., there exists positive constant εf¯ such as εfx≤εf¯. Let f^x,θ^=θ^Twx, where θ^ is the estimation of θ∗ and θ˜=θ∗−θ^ is the estimation error. In this paper, the controller ut is designed as(14)ut=−Λtθ^Twx−Π3t−c2stΛtgt,x,where Π3t=Π2t−Λtx¨dt+c1Π1t+Λtx˙t−Λtx˙dt and c2 is a positive constant. And choose the adaptive law of θ^ as(15)θ^˙=γfΛtwxst−λfθ^,where γf>0andλf>0. Now, we give the main conclusions as follows.Theorem 1.
Under given Assumptions1–3, when initial conditions x0<ρ0 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 et is limited within constraint condition (2).Proof.
Consider the Lyapunov function as follows:(16)Vt=12s2t+1γfθ˜Tθ˜.
By taking the derivative ofVt, we can get(17)V˙t=sts˙t+1γfθ˜Tθ˜˙=stz¨t+c1z˙t+1γfθ˜Tθ˜˙=stΠ3t+Λtft,x+Λtgt,xut−1γfθ˜Tθ^=stΠ3t+Λtθ∗Twx+Λtεfx+Λtgt,xtut−1γfθ˜Tθ^.
Substituting (14) and (15) into V˙t yields(18)V˙t=stΛtθ˜Twx+Λtεfx−c2st−θ˜TΛtwxst−λfθ^=Λtstεfx−c2s2t+λfθ˜Tθ^.
Because the inequality described in the following holds,(19)Λtstεfx≤Λ2t4s2t+ε¯f2,λfθ˜Tθ^≤−λf2θ˜Tθ˜+λf2θ∗Tθ∗.
Substituting the above inequalities into (18), one gets(20)V˙t≤−c2−Λ2t4s2t−λf2θ˜Tθ˜+τ∗,where τ∗=ε¯f2+λf/2θ∗Tθ∗, and select positive constant c3 such that c3<min2c2−Λ2t/2,γfλf. So, one has(21)V˙t≤−c3Vt+τ∗,which implies that(22)Vt≤τ∗c3+V0−τ∗c˙3e−c3t≤2V0+τ∗c3.
Obviously, all signals in (16) are ultimately uniformly bounded. Assume that the bounded value of st is s¯. Let V¯t=1/2z2t; according to (8), one has(23)V¯˙t=ztz˙t=ztst−c1zt=ztst−c1z2t≤−c1−14z2t+s¯2.
Select parameterc1 so that c1>1/4, which means that zt is bounded. Therefore, we can conclude that et is limited within −ρt,ρt. This completes the proof.


## 4. Numerical Simulations

In this part, an uncertain gyroscope system [22] is used to show the availability of the proposed method in this paper. The gyroscope system is described as(24)x¨=ft,x+gt,xut,where x=x,x˙T, ft,x=−A121−cosx2/sin3x+A2sinx−A3x˙−A4x˙3+A5sinwtsinx, and gt,x=3+sinxsinx˙. Parameters A1,A2,A3,A4,A5, and w are selected as A1=10,A2=1,A3=0.5,A4=0.05,A5=35.7,andw=2. The initial values of system (24) are x0=−1,x˙0=1, and xdt=sint. The fuzzy sets are defined over [−5, 5] for x and x˙. The corresponding fuzzy membership functions are chosen as follows:(25)wξ=exp−12ξ+7.5−2.5l1.22,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:(26)et=xt−xdt,s¯t=e˙t+c1et,ut=1gt,x−θ^Twx+x¨dt−c1e˙t−c2s¯t,θ^˙=γfwxs¯t−λfθ^,where parameters c1,c2,γf,andλf are designed as c1=c2=5,γf=3,andλf=0.05. The control effect of traditional method (26) is shown in Figures 1–3.Figure 1
Time response trajectory of tracking errore by using method (26).Figure 2
Time response trajectories ofx and xd by using method (26).Figure 3
Time response trajectory of controlleru by using method (26).Obviously, the tracking erroret is not effectively controlled by using traditional method (26). With the above same parameters, the prescribed function ρt is designed as(27)ρ1t=1.97sin35−tπ10+0.05,t≤5,0.05,t>5.The control effect of controller (14) is shown in Figures 4–6. It can be seen that the control effect of et has been improved, and the tracking error et 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(28)ρ2t=1.97sin33−tπ6+0.02,t≤3,0.02,t>3.Figure 4
Time response trajectory of tracking errore by using method (14) under ρ1t.Figure 5
Time response trajectories ofx and xd by using method (14) under ρ1t.Figure 6
Time response trajectory of controlleru by using method (14) under ρ1t.The control effect is shown in Figures7–9. The above simulation results show that the proposed method (14) has better transient performance in practical applications.Figure 7
Time response trajectory of tracking errore by using method (14) under ρ2t.Figure 8
Time response trajectories ofx and xd by using method (14) under ρ2t.Figure 9
Time response trajectory of controlleru by using method (14) under ρ2t.


## 5. 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. Through 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.



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*Source: 7404460-2022-02-16.xml*