# Homeomorphism Mapping Based Neural Networks for Finite Time Constraint Control of a Class of Nonaffine Pure-Feedback Nonlinear Systems

Authors: Jianhua Zhang; Quanmin Zhu; Yang Li; Xueli Wu
Journal: Complexity
 (2019)

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## Abstract

This paper proposes a new scheme for solving finite time neural networks adaptive tracking control issue for the nonaffine pure-feedback nonlinear system. The procedure, based on homeomorphism mapping and backstepping, effectively deals with constraint control and design difficulty induced by pure-feedback structure. The most outstanding novelty is that finite time adaptive law is proposed for training weights of neural networks. Furthermore, by combining finite time adaptive law and Lyapunov-based arguments, a valid finite time adaptive neural networks controller design algorithm is presented to ensure that system is practical finite stable (PFS) rather than uniformly ultimately bounded (UUB). Because of using the finite time adaptive law to training weights of neural networks, the closed-loop error system signals are in assurance of bounded in finite time. Benchmark simulations have well demonstrated effectiveness and efficiency of the proposed approach.


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## Body

## 1. Introduction

During the past decades, lots of studies have been done to solve neural networks (NNs) control and improve the adaptive law for training neural networks weights [1–3]. Among these works, the core is NNs which are used as online approximation functions for the unknown nonlinearities, due to their inherent approximation capabilities [4, 5]. Almost all the neural adaptive control designs and stability analyses are Lyapunov uniformly ultimately bounded (UUB) results [5]; based on the Krasovskii–LaSalle invariance principle, it is challenging to establish a generalized powerful framework for neural control [6–8], even though it has been used to get sufficient conditions for smooth stabilization for closed-loop systems [9–11].Traditionally, Quadratic Lyapunov Functions are in common used to construct for analysis and control design of linear and nonlinear even time delay dynamic systems [12, 13]. There also exist some other formats of Lyapunov functions, such as integral Lyapunov function [14], barrier Lyapunov function [15, 16], and vector Lyapunov functions [17, 18]. Such attempts have enhanced Lyapunov function applications in control system design [19], for example, integral Lyapunov function for controller singularity problems and barrier Lyapunov function for output constraint problems. Nonlinear systems differ from linear system, because the complexity of the relation between states and input and therefore numerical techniques (open solutions) have played a significant role in controller design process [20–22]. Young’s Inequality as an important lemma has been widespread used to realize UUB (uniformly ultimately bounded) in neural network based adaptive control [23].In terms of complex of system structure such as pure-feedback and nonaffine, mean value theorem [24] is popular to use to eliminate. Due to its universal approximation capability for nonlinear systems, neural network has been extensively utilized to achieve modelling. Many elegant model structure design techniques, such as strict-feedback, pure-feedback, affine form, and nonaffine systems [3, 25, 26], have been obtained based on backstepping procedure and Lyapunov synthesis.Based on virtual control and iterative learning control [25], backstepping procedure provided an important methodology to control affine or nonaffine, strict-feedback, or pure-feedback form SISO nonlinear uncertain systems, which are split into several subsystems. In every subsystem, neural networks approximate the unknown nonlinear systems with adaptive law, based on Lyapunov stability theory, then the systems achieve the asymptotic stability or uniformly ultimately bounded stable. In order to better exploit the application of adaptive neural networks [27, 28], finite time stable results are more meaningful for uncertain system [29, 30], high order stochastic nonlinear system [31], and interconnected nonlinear system [32].It should be noted that the research on finite time neural networks control is still in a very beginning stage. Although there exist a few of adaptive finite control algorithms which have been developed for some special systems and bench tested with simulations and applications [33–35], it is very hard to extend the research results to the general systems or algorithms. The key issue is how to systematically obtain finite time adaptive law of neural networks [9, 36] weight from finite time convergence of close loop systems [37]. In regard to neural network based such adaptive control to the authors’ best knowledge, there are a few results about finite time adaptive neural networks control. That is because it is not easy to design the finite time neural networks adaptive controller and lack of relevant inequality skills to finish finite time stability analysis. There has been any reference to show finite time adaptive algorithm for weights of NNs having been expanded to solve the problems of finite control for pure-feedback nonaffine nonlinear systems.Motivated by the above critical analyses, we investigated neural backstepping control via finite time adaptive algorithm. At the first stage, the homeomorphism mapping is proposed in place of barrier Lyapunov functions and commonly utilized in traditional constraints control. Then with finite adaptive algorithm, adaptive neural networks backstepping control procedure is designed for nonaffine pure-feedback nonlinear systems. Sufficient condition is given via Lyapunov finite time stability theory to make sure all the signals in the closed-loop system are global uniformly bounded in finite time and the tracking error is remained in a bounded compact set. The main contributions of this paper can be summarized as follows:(1)
This paper is the first study to propose the finite time adaptive law based on Young’s inequality skill for adaptive neural networks control. The main advantages are that adaptive neural networks controls achieve finite time stabilization including weights of neural networks and closed-loop control system uniformly ultimately bounded stable in finite time.(2)
To keep full state constraints in a bounded compact set, the homeomorphism mapping is proposed in place of barrier Lyapunov functions and commonly utilized in traditional constraints control.(3)
New finite time adaptive neural networks techniques, including virtual control and finite adaptive law, are introduced for ensuring the successful design of the finite time adaptive neural networks backstepping control.The rest of this paper is organized as follows. Section2 gives problem formulation and preliminaries, including some definitions and necessary inequality with lemmas. Nonaffine pure-feedback nonlinear systems finite time adaptive neural networks control is considered in Section 3, and a new finite time adaptive law is developed for training neural networks. Lyapunov finite time stable theory is used to guarantee the closed-loop system signals with prescribed performance in finite time. Simulation study is presented in Section 4 to indicate efficiency and effectiveness of the procedure. The conclusion and future works are discussed in Section 5.


## 2. Problem Description and Preliminaries

In this section, problem description, the notation, basic definitions, and lemmas on the system stability are presented. Consider the following nonlinear system:(1)ξ˙t=φξt,ut,t,t≥0ξt=ϕt,t=0ζt=ψξt,ut,twhere ξ=ξ1ξ2⋯ξnT∈Rn, u∈R, ζ∈Rm, ϕt are state variable, control input, output, and initial condition, respectively, and φ·, ψ· are nonlinear smooth functions.The ideal output isζdt; therefore, the objective is to design an adaptive state feedback to ensure the output ζt tracking ζdt and controller in the form(2)ut=αξt,θ^tθ^˙t=βξt,θ^twhere θ^ indicate the estimated values of unknown parameters and the θ∗ indicate the ideal values. For system (1) with the homeomorphism mapping,(3)H:a_1a-1×a_2a-2×⋯×a_na-n→Rnpropose to transform the system to realize the constraints control, such as x=Hξ, and ξ∈a_1a-1×a_2a-2×⋯×a_na-n, x∈Rn, and then system (1) changes into(4)x˙t=fxt,ut,t,t≥0xt=ςt,t=0yt=gxt,ut,twhere x=x1x2⋯xnT∈Rn, u∈R, y∈R, ςt indicate the state, control, output, and initial condition, respectively, and f·,g· are nonlinear smooth functions. For the homeomorphism mapping, both x=Hξ and ξ=H-1x exist, and specially, y-yd=Hξ-ξd hold.The adaptive controller in the form(5)ut=αxt,θ^tθ^˙t=βxt,θ^tand the output of system (4) yt track the ydt.If controller (5) is designed based on system (4), then put the controller into system (1); the states ξ are constraint where ξ∈a_1a-1×a_2a-2×⋯×a_na-n, because the states ξ and x satisfy the of the homeomorphism mapping relation (3).Remark 1.
There are lots of researches about constraint control [15, 38, 39], which employ the barrier Lyapunov functions to realize constraints. The controller design is relatively complicated because of the complexity of derivation of barrier Lyapunov functions. The homeomorphism mapping skill is easy to achieve, as the mapping is used for transformation of states, which is independent for the controller design. After that, a new model emerged, then based on the new model, all control methods can be used, such as sliding model control [40], adaptive control [26, 41], and especially neural networks control which applied without model information. Such homeomorphism mapping is easy to choose as(6)ξ=2aπarctan⁡x,ξ=btanh⁡x,ξ=sgn⁡xc1-e-x21/2which satisfy x∈R and ξ<a,b,c, respectively.A continuous nonaffine pure-feedback nonlinear system can be described as follows:(7)ξ˙i=φiξ-i+1,1≤i≤n-1ξ˙n=φnξ-n,uζ=ξ1where ξ=ξ1ξ2⋯ξnT∈Rn, u∈R, ζ∈R, φt indicate the state, control, output, and initial condition, respectively, and ξ-i=ξ1,ξ2,…,ξiT∈Ri, φi· are smooth nonlinear functions.Assumption 2.
The nonlinear functionφ in system (1) is as follows:(8)∂φi∂ξi+1≠0,∂φn∂u≠0when 1≤i≤n-1.
Based on homeomorphism mapping (7), nonlinear pure-feedback nonaffine system can be described as follows:(9)x˙i=fix-i+1,1≤i≤n-1x˙n=fnx-n,uy=x1where x-i=x1,x2,…,xiT∈Ri, x=x1x2⋯xnT∈Rn, u∈R, y∈R, and φt indicate the state, control, output, and initial condition, respectively, and fi· are nonlinear smooth functions.Assumption 3.
Nonlinear functionfi in system (9) is as follows:(10)∂fi∂xi+1≠0,∂fn∂u≠0when 1≤i≤n-1.
Notation. Throughout this paper, W∈Rm×n represents the matrix, W=wijm×n, rational number η, and matrix, matrix Wη denote element-by-element powers, Wη=wijηm×n, W~ηT denote the transposition of matrix W~η, such as W~ηT=W~ηT. W∗,W^,W~ indicate ideal weight, estimated weight, and error between ideal and estimated weight.
In brief, the following radial basis function (RBF) NN is used to approximate the continuous functionFx:Rn→R over a compact set(11)FNNx,W=WTΨxwhere input x∈Ω⊂Rn, weight vector W=w1⋯wlT∈Rl, and node vector Ψx=ψ1x⋯ψlxT, with the element ψix being chosen as the commonly used Gaussian function as below(12)ψix=exp⁡-x-μiTx-μiηi2,i=1,2,…,l.where μi=μi1⋯μinT, the center of the receptive field, and ηi is the width of the Gaussian function.
It has been proven that RBF NN can approximate any continuous function over a compact setΩx⊂Rn as(13)Fx=W∗TΨx+εxwhere W∗ is the ideal NN weight and εx is the NN approximation error.(14)W∗=arg⁡minW∈Rl⁡sup⁡Fx-WTΨxDefinition 4.
Consider systemx˙=fx,u, where x is a state vector and u is the input vector. The solution is practical finite stable (PFS) if for all xt0=x0, there exist ε>0 and Tε,x0<∞, such that x<ε, for all t≥t0+T.Lemma 5.
Consider systemx˙=fx,u; suppose that there exist continuous function Vx and scalars λ>0,0<α<1, and 0<η<∞ such that(15)V˙x≤-λVαx+ηThen, the trajectory of system x˙=fx,u is PFS.Proof.
There exists a scalar0<θ≤1 such that inequality (15) can be expressed as(16)V˙x≤-θλVαx-1-θλVαx+ηClearly, V˙x≤-θλVαx if Vαx>η/λ1-θ. According to Lemma 5, the decrease of Vx in finite time drives the trajectories of the closed-loop system into Vαx≤η/λ1-θ. Therefore, the trajectories of the closed-loop system are bounded in finite time as(17)limθ→θ0⁡x∈Vαx≤ηλ1-θwhere 0<θ0<1. And the time needed to reach (17) is bounded as(18)T≤V1-αx0λθ01-αwhere Vx0 is the initial value of Vx.Lemma 6 ([42], Young’s inequality).
For any constanta,b∈R, the following inequality holds:(19)ab≤1pap+1qbqwhere p>1, q>1, and 1/p+1/q=1.Remark 7.
Based on Young’s inequality, then the following inequalities hold:(20)zε≤34z4/3+14ε4(21)-W~TW^1/3≤-12W~2/3TW~2/3+W∗2/3TW∗2/3There are two key inequalities in adaptive neural networks finite time control.


## 3. Control Design and Stability Analysis

Consider system (9), using the coordinate transform as(22)z1=x1-ydzi=xi-αi-1,i=2,3,…,nzn+1=u-αnwhere αi is the virtual control and the ideal virtual control are(23)α1∗=-k1z11/3+x2-f1+y˙dαi∗=-zi-1-kizi1/3+xi+1-fi+α˙i-1∗2≤i≤n-1αn∗=-zn-1-knzn1/3+u-fn+α˙n-1∗In the first step, consider system(24)z1=x1-ydThen it has(25)z˙1=f1x-2-y˙dBased on ideal virtual control law (23), choose the NN to approximate the nonlinear system f1, x-2∈Ω1⊂R2, and Ω1 is compact set(26)f1x-2-y˙d=W1T∗Ψ1x-2+ε1Therefore(27)z˙1=W1∗TΨ1x-2+ε1Choose the practical virtual control law(28)α1=-k1z11/3+x2-W^1TΨ1x-2Choose the adaptive law(29)W^˙1=Γ1z1Ψ1x-2t-σ1W^11/3where Γ1=Γ1T>0 and σ1>0 is positive constant design parameters.Then based on (27) and (28)(30)z˙1=W1∗TΨ1x-2+ε1-k1z11/3+x2-α1-W^1TΨ1x-2let (31)z2=x2-α1therefore(32)z˙1=-W~1TΨ1x-2-k1z11/3+z2+ε1where(33)W~1=W^1-W1∗Choose Lyapunov candidate function(34)V1=12z12+12W~1TΓ1-1W~1then(35)V˙1=z1z˙1+W~1TΓ1-1W~˙1=-k1z14/3+z1z2+z1ε1-σ1W~1TW^11/3Based on inequalities from Lemma 6 and Remark 7, the following hold:(36)z1ε1≤34z14/3+14ε14and(37)-σ1W~1TW^11/3≤-σ12W~12/3TW~12/3+σ1W1∗2/3TW1∗2/3Then based on (35), it gives(38)V˙1=-k1-34z14/3+z1z2-σ12W~12/3TW~12/3+14ε14+σ1W1∗2/3TW1∗2/3then it has(39)V˙1≤z1z2-η112z122/3-η112W~1TΓ1-1W~12/3+14ε14+σ1W1∗2/3TW1∗2/3≤z1z2-η1V12/3+δ1where(40)η1=min⁡41/3k1-34,σ121/3λmaxΓ12/3δ1=14ε14+σ1W1∗2/3TW1∗2/3The ith step 2≤i≤n-1(41)zi=xi-αi-1Choose the virtual control law (23), choose the NN to approximate the nonlinear system fix-i+1-α˙i-1, x-i+1∈Ωi⊂Ri+1, and Ωi is compact set(42)fix-i+1t-α˙i-1=Wi∗TΨix-i+1+εiwhere(43)α˙i-1=∑j=1i∂αi-1∂xjx˙j+∑j=1i-1∂αi-1∂W^jTW^˙jT+ydiBased on the system, (44)z˙i=Wi∗TΨix-i+1+εiChoose the practical virtual control law(45)αi=-zi-1-kizi1/3+xi+1-W^iTΨix-i+1Choose the adaptive law(46)W^˙i=ΓiziΨix-i+1-σiW^i1/3where Γi=ΓiT>0 and σi>0 is positive constant design parameters. Then(47)z˙i=-W~iΨix-i+1-zi-1-kizi1/3+zi+1+εiwhere(48)W~i=W^i-Wi∗Choose Lyapunov candidate function(49)Vi=12zi2+12W~iTΓi-1W~ithen(50)V˙i=ziz˙i+W~iΓi-1W~˙iT=-zi-1zi-kizi4/3+zizi+1+ziεi-σiW~iTW^i1/3Based on Lemma 6 and Remark 7, the following inequalities hold:(51)ziεi≤34zi4/3+14εi4and(52)-σiW~iTW^i1/3≤-σi2W~i2/3TW~i2/3+σiWi∗2/3TWi∗2/3Then based on (49), it yields(53)V˙i=ziz˙i+W~iTΓi-1W~˙i=-zi-1zi-ki-34zi4/3+zizi+1-σi2W~i2/3TW~i2/3+14εi4+σiWi∗2/3TWi∗2/3And(54)V˙i≤-zi-1zi+zizi+1-ηi12zi22/3-ηi12W~iTΓi-1W~i2/3+14εi4+σiWi∗2/3TWi∗2/3≤-zi-1zi+zizi+1-ηiVi2/3+δiwhere(55)ηi=min⁡41/3ki-34,σi21/3λmaxΓi2/3δi=14εi4+σiWi∗2/3TWi∗2/3The nth step is the most important step. Based on the system (56)zn=xn-αn-1Choose the NN to approximate the nonlinear system fnx-n,u-α˙n-1, x-n+1T≜x-nTu∈Ωn⊂Rn+1, and Ωn is compact set(57)fnx-n,u-α˙n-1=Wn∗TΨnx-n+1+εnwhere(58)α˙n-1=∑j=1n∂αn-1∂xjx˙j+∑j=1n-1∂αn-1∂W^jTW^˙jTBased on system (59)z˙n=Wn∗TΨnx-n+1+εnChoose practical virtual control law(60)αn=-zn-1-knzn1/3+u-W^nTΨnx-n+1Choose adaptive law(61)W^˙n=ΓnznΨnTx-n+1-σnW^n1/3where Γn=ΓnT>0 and σn>0 are positive constant design parameters, then(62)z˙n=-W~nTΨnx-nt-zn-1-knzn1/3+zn+1+εnwhere(63)W~n=W^n-Wn∗From the inequality(64)-σnW~nTW^n1/3≤-σn2W~n2/3TW~n2/3+σnWn∗2/3TWn∗2/3choose Lyapunov candidate function(65)Vn=12zn2+12W~nΓn-1W~nTThen(66)V˙n=znz˙n+W~nΓn-1W~˙nT=-zn-1zn-knzn4/3+znzn+1+znεn-σnW~nTW^n1/3Based on the lemma, the following inequalities hold:(67)znεn≤34zn4/3+14εn4and(68)-σnW~nTW^n1/3≤-σn2W~n2/3TW~n2/3+σnWn∗2/3TWn∗2/3Then, it gives(69)V˙n≤-zn-1zn-kn-34zn4/3+znzn+1-σn2W~n2/3TW~n2/3+14εn4+σnWn∗2/3TWn∗2/3then(70)V˙n≤-zn-1zn+znzn+1-ηn12zn22/3-ηn12W~nTΓn-1W~n2/3+δn≤-zn-1zn+znzn+1-ηnVn2/3+δnwhere(71)ηn=min⁡41/3kn-34,σn21/3λmaxΓn2/3δn=14εn4+σnWn∗2/3TWn∗2/3The n+1th step is the most important step. (72)zn+1=u-αnBased on system(73)z˙n+1=v-α˙nChoose the NN to approximate the nonlinear system -α˙nx-n+1, x-n+1T=x-nTu∈Ωn+1⊂Rn+1, and Ωn+1 is compact set(74)-α˙n=Wn+1∗TΨn+1x-n+1+εn+1where(75)α˙n=∑j=1n∂αn∂xjx˙j+∑j=1n∂αn∂W^jTW^˙jT+ydn+1Based on system (76)z˙n+1=v+Wn+1∗TΨn+1x-n+1t+εn+1Choose adaptive law(77)W^˙n+1=Γn+1zn+1Ψn+1x-n+1-σn+1W^n+11/3where Γn+1=Γn+1T>0 and σn+1>0 are positive constant design parameters.(78)v=-zn-kn+1zn+11/3-W^n+1TΨn+1x-n+1tthen(79)z˙n+1=-zn-kn+1zn+11/3-W~n+1TΨn+1x-n+1+εn+1where(80)W~n+1=W^n+1-Wn+1∗Choose Lyapunov candidate function(81)Vn+1=12zn+12+12W~n+1Γn+1-1W~n+1Tthen(82)V˙n+1=zn+1z˙n+1+W~n+1Γn+1-1W~˙n+1T=-znzn+1-kn+1zn+14/3+zn+1εn+1-σn+1W~n+1TW^n+11/3From the inequality(83)-σn+1W~n+1TW^n+11/3≤-σn+12W~n+12/3TW~n+12/3+σn+1Wn+1∗2/3TWn+1∗2/3and(84)zn+1εn+1≤34zn+14/3+14εn+14we have(85)V˙n+1≤-znzn+1-kn+1-34zn+14/3-σn+12W~n+12/3TW~n+12/3+14εn+14+σn+1Wn+1∗2/3TWn+1∗2/3Then (86)V˙n+1≤-znzn+1-ηn+112zn+122/3-ηn+112W~n+1TΓn+1-1W~n+12/3+δn+1≤-znzn+1-ηn+1Vn+12/3+δn+1where(87)ηn+1=min⁡41/3kn+1-34,σn+121/3λmaxΓn+12/3δn+1=14εn+14+σn+1Wn+1∗2/3TWn+1∗2/3Choose Lyapunov candidate functions(88)V=∑i=1n+1Vithen based on (39), (54), (70), and (86), it has(89)V˙≤-ηV2/3+δwhere(90)η=min⁡ηi,i=1,2,3,…,n+1δ=∑i=1n+1δiTheorem 8.
Under Assumptions2 and 3 and Lemmas 5 and 6, consider nonlinear system (9) for which the model dynamics is approximated by neural networks (26), (42), (57), and (74), the control law (78) with the virtual control (28), (45), and (60), and adaptive laws (29), (46), (61), and (77), then the following statements hold:(1)
All the signals of the closed-loop system, includingxi,αi,W^i, remain bounded all the time.(2)
The closed-loop signalz=z1z2⋯zn+1T∈Rn+1 converges to a compact set defined by(91)Ωs=z∣z2≤δηin finite time, whereδ,η are constants related to the design parameters.(3)
The finite timeT is given by(92)T≤3V1/3x0ηθ0whereη,θ0 are constants related to the design parameters and Vx0 is constant related to Lyapunov candidate functional.Proof.
Considering Lyapunov function candidate (88) and (89), then V is bounded in finite time. Therefore, it can conclude that, for all 1≤i≤n+1, the error signals zi, W~i are PFS.


## 4. Simulation Examples

In this section, simulation examples show the validity and feasibility of the proposed NNs adaptive control design approach.


### 4.1. Mathematical Example

Consider the following dynamic system:(93)x˙1=0.5sin⁡x1-x1+sin⁡x2+x2+5sin⁡t+5cos⁡tx˙2=-sin⁡x2-2x2+x2u+u+sin⁡uy=x1where x=x1x2T, u, y, and yd=5sin⁡t is state, control input, control output, and ideal output, respectively. The control objective of the proposed design method is to make the system output y follow the desired reference signal yd=5sin⁡t. Select the controller u˙ and virtual control α1,α2 as follows:(94)u˙=-z2-k3z31/3-W^3TΨ3x-2,u(95)α1=-k1z11/3+x2-W^1TΨ1x-2(96)α2=-z1-k2z21/3+u-W^2TΨ2x-2,uThe weight of NNs is updated by(97)W^˙i=ΓiziΨix-i+1-σiW^i1/3,i=1,2,3with z1=x1-yd, z2=x2-α1, and z3=u-α2; the initialization of the variables is selected by x10=4, x20=4, and u0=4.Applying finite time NNs control method, the simulation results are shown by Figures1–5. Figure 1 shows the trajectory of output and desired reference trajectory. Figure 2 shows the trajectory of controller. Figures 3 and 4 show the trajectory of NNs and function which approximated by NNs. Compared with [3], the method in this paper is more generally used, and both output and weight of NNs are convergence in finite time.Figure 1
Trajectories ofy,yd.Figure 2
Trajectory of control inputu.Figure 3
Trajectories off1, NN1.Figure 4
Trajectories off2-α˙1, NN2.Figure 5
Trajectories of error system.


### 4.2. Plate-Ball Example

Consider a Plate-Ball model [43] in form of(98)m+Ibrb2ξ¨x+mgux+fx=0m+Ibrb2ξ¨y+mguy+fy=0where m is ball mass, Ib is ball moment of inertia, rb is ball radius, ux,uy are plate inclination angle in the x or y direction, fx,fy are contain the centrifugal torque resulting from rotation and other modelling error, and yxd, yyd are ideal trajectory of ξx,ξy.Based on homeomorphism mapping technique, choose(99)ξx-yxd=rxtanh⁡xthen x is new state variable, and therefore ξx-yxd≤rx all the time. About new state variable, we have(100)ξ˙x=y˙xd+rx1-tanh2⁡xx˙and(101)ξ¨x=y¨xd-2rxtanh⁡xx˙2+rx1-tanh2⁡xx¨Let x1=x,x2=x˙, then the above equations can be changed as(102)x˙1=x2x˙2=fx-m+Ib/rb2y¨xd+2rxm+Ib/rb2tanh⁡x1x22-mguxrxm+Ib/rb21-tanh2⁡x1then control ux can be designed by regular control method based on Lyapunov theory, state variable x is not restricted, and ξx is restricted by reference signal yxd and bound rx.Choosefx as unknown function, reference signal as sinusoidal signal, and NNs control. The simulation result can be obtained by Figures 6 and 7 about ξy with cosine signal as reference signal. Figure 8 indicates trajectories of ball.Figure 6
Trajectories ofξx and sinusoidal signal.Figure 7
Trajectories ofξy and cosine signal.Figure 8
Trajectories of ball.


## 4.1. Mathematical Example

Consider the following dynamic system:(93)x˙1=0.5sin⁡x1-x1+sin⁡x2+x2+5sin⁡t+5cos⁡tx˙2=-sin⁡x2-2x2+x2u+u+sin⁡uy=x1where x=x1x2T, u, y, and yd=5sin⁡t is state, control input, control output, and ideal output, respectively. The control objective of the proposed design method is to make the system output y follow the desired reference signal yd=5sin⁡t. Select the controller u˙ and virtual control α1,α2 as follows:(94)u˙=-z2-k3z31/3-W^3TΨ3x-2,u(95)α1=-k1z11/3+x2-W^1TΨ1x-2(96)α2=-z1-k2z21/3+u-W^2TΨ2x-2,uThe weight of NNs is updated by(97)W^˙i=ΓiziΨix-i+1-σiW^i1/3,i=1,2,3with z1=x1-yd, z2=x2-α1, and z3=u-α2; the initialization of the variables is selected by x10=4, x20=4, and u0=4.Applying finite time NNs control method, the simulation results are shown by Figures1–5. Figure 1 shows the trajectory of output and desired reference trajectory. Figure 2 shows the trajectory of controller. Figures 3 and 4 show the trajectory of NNs and function which approximated by NNs. Compared with [3], the method in this paper is more generally used, and both output and weight of NNs are convergence in finite time.Figure 1
Trajectories ofy,yd.Figure 2
Trajectory of control inputu.Figure 3
Trajectories off1, NN1.Figure 4
Trajectories off2-α˙1, NN2.Figure 5
Trajectories of error system.


## 4.2. Plate-Ball Example

Consider a Plate-Ball model [43] in form of(98)m+Ibrb2ξ¨x+mgux+fx=0m+Ibrb2ξ¨y+mguy+fy=0where m is ball mass, Ib is ball moment of inertia, rb is ball radius, ux,uy are plate inclination angle in the x or y direction, fx,fy are contain the centrifugal torque resulting from rotation and other modelling error, and yxd, yyd are ideal trajectory of ξx,ξy.Based on homeomorphism mapping technique, choose(99)ξx-yxd=rxtanh⁡xthen x is new state variable, and therefore ξx-yxd≤rx all the time. About new state variable, we have(100)ξ˙x=y˙xd+rx1-tanh2⁡xx˙and(101)ξ¨x=y¨xd-2rxtanh⁡xx˙2+rx1-tanh2⁡xx¨Let x1=x,x2=x˙, then the above equations can be changed as(102)x˙1=x2x˙2=fx-m+Ib/rb2y¨xd+2rxm+Ib/rb2tanh⁡x1x22-mguxrxm+Ib/rb21-tanh2⁡x1then control ux can be designed by regular control method based on Lyapunov theory, state variable x is not restricted, and ξx is restricted by reference signal yxd and bound rx.Choosefx as unknown function, reference signal as sinusoidal signal, and NNs control. The simulation result can be obtained by Figures 6 and 7 about ξy with cosine signal as reference signal. Figure 8 indicates trajectories of ball.Figure 6
Trajectories ofξx and sinusoidal signal.Figure 7
Trajectories ofξy and cosine signal.Figure 8
Trajectories of ball.


## 5. Conclusions

For technique completeness, the neural networks enhanced finite time constraint control framework for a class of uncertain nonaffine pure-feedback nonlinear systems has developed analysis/proof, design, algorithms, and computational experiments. For reference of neurocontrol, this study attempts to strengthen neural network function to deal with complex nonlinear uncertain system control with different design constraints, which confirms again that the universal approximation property of neural networks feasibly effectively integrated with control system specifications. In line with further expansion of the study, some issues have been identified on the research card, such as how to design better homeomorphism mapping to realize the constraint control, how to extend the general order in finite time adaptive neural networks control, and fast computation algorithms for online computation.



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