# Nonsingular Fast Terminal Sliding Mode Tracking Control for a Class of Uncertain Nonlinear Systems

Authors: Siyi Chen; Wei Liu; Huixian Huang
Journal: Journal of Control Science and Engineering
 (2019)

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## Abstract

Aiming at the tracking control problem of a class of uncertain nonlinear systems, a nonsingular fast terminal sliding mode control scheme combining RBF network and disturbance observer is proposed. The sliding mode controller is designed by using nonsingular fast terminal sliding mode and second power reaching law to solve the problem of singularity and slow convergence in traditional terminal sliding mode control. By using the universal approximation of RBF network, the unknown nonlinear function of the system is approximated, and the disturbance observer is designed by using the hyperbolic tangent nonlinear tracking differentiator (TANH-NTD) to estimate the interference of the system and enhance the robustness of the system. The stability of the system is proved by the Lyapunov principle. The numerical simulation results show that the method can shorten the system arrival time, improve the tracking accuracy, and suppress the chattering phenomenon.


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## Body

## 1. Introduction

Sliding mode variable structure control is essentially a nonlinear control that the structure changes over time. Its significant advantage is its strong robustness to uncertain parameters and external disturbances. Therefore, it has been widely used in aerospace, robot control, and chemical control [1–8]. However, the traditional sliding mode control takes the linear sliding mode as the “sliding mode” of the system. The deviation between the system state and the given trajectory converges exponentially but cannot converge to zero in a finite time. Therefore, the nonlinear term is introduced in the design of terminal sliding mode (TSM) control, and the tracking error on the sliding mode surface can converge to zero in a limited time, which makes it widely used in various control systems [9–12]. Reference [13] proposed a terminal sliding mode control design scheme for uncertain dynamic systems with pure feedback form. In Reference [14], a new terminal sliding mode control design is proposed for the n-link rigid manipulator. Reference [15] proposed a new sliding mode control method for robot terminals. Reference [16] discussed the design of terminal sliding mode variable structure control for multi-input uncertain linear system. However, TSM has singularity [17–19]. Nonsingular terminal sliding mode (NTSM) control evolved on the basis of avoiding the TSM singularity problem. It avoids the control singular regions directly in the sliding mode design and preserves the finite time convergence characteristics of TSM [20–25]. In recent years, NTSM has developed rapidly. In [26], a continuous nonsingular terminal sliding mode control method was proposed for the suppression of mismatch interference. In [27], in order to realize the finite time tracking control of the axial position of the nonlinear thrust active magnetic bearing rotor, a robust nonsingular terminal sliding mode control system was proposed. In order to further improve the convergence speed of the sliding mode, a nonsingular fast terminal sliding mode (NFTSM) surface was designed. Reference [28] studied the fast finite time control of terminal sliding mode with nonlinear dynamics. In [29], a state-based nonsingular fast-terminal sliding mode controller was designed using direct instantaneous torque control. However, the system was often affected by the uncertainty of the model and the amount of interference, resulting in system chattering and even instability. In order to solve this problem, Reference [30] proposed a continuous nonsingular fast terminal sliding mode control scheme with extended state observer and tracking differentiator for second-order uncertain SISO nonlinear systems. In [31], the stability and attitude control of a class of quadrotor systems with uncertainties and unknown disturbances were studied. A nonsingular fast terminal sliding mode attitude control scheme for tracking differentiators and extended state observers was proposed. However, the extended state observer has an initial differential peak problem. Document [32] proposes an NDO design method for underactuated robot arm control. In [33], the NDO is combined with the dynamic surface to design the moving wheel inverted pendulum controller. They have a good effect on the estimation of the disturbance, but both of them have the disadvantage of relying on the prior knowledge of the disturbance. However, it is difficult to obtain prior knowledge of disturbance in practice. Literature [34] proposed a nonlinear disturbance observer based on tracking differentiator, which overcomes the shortcomings of the literature [32, 33] need to know the prior knowledge of disturbance. At the same time, it has the advantages of simple structure, excellent estimation effect on disturbance, and suppression of measurement noise.In this paper, to realize fast and stable tracking control for a class of second-order uncertain nonlinear systems, a nonsingular fast terminal sliding mode control strategy combining RBF network and disturbance observer is proposed. The contributions of this paper are as follows.(1) The sliding mode controller is designed by using nonsingular fast terminal sliding mode and second power reaching law, so that the system can converge to zero smoothly in a short time.(2) RBF neural networks have strong nonlinear fitting ability to map arbitrarily complex nonlinear relationships. At the same time, it has the advantages of simple learning rules and easy computer implementation. Using the universal approximation principle of RBF network, the unknown nonlinear function is approximated to solve the influence of unknown nonlinear function on the robustness of the system.(3) The nonlinear disturbance observer based on tracking differentiator has the advantages of simple structure, good disturbance estimation effect, and suppressing measurement noise. A hyperbolic tangent nonlinear disturbance observer is designed to estimate the external disturbance and unknown part of the model and compensate the controller. At the same time, an augmented nonlinear tracking differentiator designed in Reference [35] is used to filter the given signal, eliminating the influence of the given noise on the system.The numerical simulation results show that the designed control method can effectively shorten the convergence time, eliminate the noise of the given signal and the chattering phenomenon in the controller, and improve the control tracking accuracy.


## 2. Problem Description

Consider the following second-order uncertain nonlinear system:(1)x˙1=x2x˙2=fx+bxu+dty=x1wherex=[x1,x2]T∈R2 is the state variable; f(x) and b(x) are the unknown nonlinear functions; u∈R is the control input; y∈R is the system output; d(t) is the slow time-varying interference, and d(t)≤D, constant D>0.


## 3. Main Results

The control objective of the system is to design a robust controller that enables accurate and fast tracking of the desired input signal even in the presence of model uncertainties and external disturbances. In order to reach the target, an NFTSM control scheme combining RBF network and disturbance observer is designed. The structure diagram of the system controller is shown in Figure1.Figure 1
Structure diagram of the system controller.


### 3.1. Nonsingular Fast Terminal Sliding Mode Control

In order to solve the problem of singularity and slow convergence of traditional terminal sliding mode control, a novel nonsingular fast terminal sliding mode control method is proposed.


#### 3.1.1. Sliding Mode Design

x d is set the given signal and is second derivable. The position and velocity tracking errors of the system are defined as e1=x1-xd and e2=x2-x˙d, respectively. The designed nonsingular fast terminal sliding mode is(2)s=e1+α1e1γsgn⁡e1+α2e2q/psgn⁡e2whereα1, α2 are positive real numbers; q, p are odd integers, and 1<q/p<2, γ>q/p. e1 and e2 do not have negative exponential terms, ensuring that the sliding mode s based controller does not have singularity. When the system error state reaches the sliding mode (s=0), the following equation can be obtained:(3)e1+α1e1γsgn⁡e1+α2e2q/psgn⁡e2=0Substitutee˙1=e2 into (3), the following equation can be obtained:(4)e˙1=-1α2p/qe1+α1e1γsgn⁡e1p/q=-1α2p/qe1p/q1+α1e1γ-1p/qAssume the time taken frome1(tr)≠0 to e1(tr+ts)=0 is ts, and the following equation can be obtained through the time integrated on both sides of (4):(5)∫e1tre1tr+ts1e1p/qde1=-∫trtr+ts1α2p/q1+α1e1γ-1p/qdτ≤-∫trtr+ts1α2p/qdτSimplification of Inequality (5) (6) t s ≤ α 2 p / q q q - p e 1 t r 1 - p / qTherefore, the system error can converge to zero for a limited time.Remark 1.
When the system error state is far from the equilibrium point, the dominant role ofe1 in the sliding mode s causes the system trajectory to converge quickly. When the system error state approaches the equilibrium point, the dominant role of α1e1γsgn⁡e1 in the sliding mode s causes the system trajectory to converge rapidly. Therefore, NFTSM can achieve fast convergence of the whole state trajectory.


#### 3.1.2. Reaching Law Design

According to the sliding mode variable structure principle, the sliding mode reachability condition only ensures that the moving point at any position in the state space can reach the switching surface within a limited time, and there is no restriction on the specific trajectory of the reaching motion. The reaching law can improve the dynamic quality of reaching movement. The following second power reaching law is adopted in this paper.(7)s˙=-k3sc2sgn⁡s-k4sc3sgn⁡swherek3>0, k4>0, 0<c2<1, c3>1. When the system state is far from the sliding mode (s>1), -k4sc3sgn⁡s plays a leading role; when the system approaches the sliding mode (s<1), -k3sc2sgn⁡(s) dominates. When s=0, s˙=0, the speed of the state reaching the sliding mode is reduced to zero, and the smooth transition of the sliding mode is realized, which greatly weakens the system chattering phenomenon. Compared with the traditional power reaching law, exponential reaching law, and fast power reaching law, the reaching law (7) has faster convergence speed and better motion quality.


#### 3.1.3. Controller Design and Stability Analysis

Lemma 2 (see [36]).
Setx∈M⊂Rn, x˙=f(x), f:Rn→R as the continuous functions defined in the equilibrium point region M. Assume that a continuous function V:M→R satisfies the following conditions:
(1)V is positive definite.
(2)V˙ is negative definite except for the equilibrium point.
(3) Real numberk>0, α>0 and region N⊂M make V˙+kVα≤0, and the function x˙=f(x) converges at balanced zero point within finite time.Theorem 3.
For the system shown in (1), design the sliding mode according to (2), using the NFTSM controller shown in (9). The system state s converges to the following areas in a limited time:(8)s≤min⁡Dk31/c2,Dk41/c3s˙≤min⁡D,k3Dk4c1/c2+min⁡k4Dk3c2/c1,D
Sliding mode controller is designed as(9)u=-bx-1fx+1α2pqe22-q/psgn⁡e21+α1γe1γ-1+k3sc2sgn⁡s+k4sc3sgn⁡s-x¨d
whereα1, α2, k3, k4 are normal numbers, 0<c2<1, c3>1; q, p are odd integers, and 1<q/p<2, γ>q/p.Proof.
Define the following Lyapunov function:(10)V=12s2
Find the time derivative for (10):(11)V˙=se˙1+α1γe1γ-1sgn⁡e1e˙1+α2qpe2q/p-1sgn⁡e2e˙2=se2+α1γe1γ-1sgn⁡e1e2+α2qpe2q/p-1sgn⁡e2fx+bxu+dt-x¨d
Substitute (9) into the above equation:(12)V˙=se˙1+α1γe1γ-1e˙1+α2qpe2q/p-1e˙2=se2+α1γe1γ-1e2+α2qpe2q/p-1fx+bx-bx-1fx+1α2pqe22-q/psgn⁡e21+α1γe1γ-1+k3sc2sgn⁡s+k4sc3sgn⁡s-x¨d+dt-x¨d=sα2qpe2q/p-1-k3sc2sgns-k4sc3sgns+dt=-α2qpe2q/p-1k3sc2+1+k4sc3+1-sdt≤-α2qpe2q/p-1k3sc2+1+k4sc3+1-Ds
Deform (12):(13)V˙≤-α2qpe2q/p-1k3sc2+1-α2qpe2q/p-1k4sc3+1+α2qpe2q/p-1Ds=-α2qpe2q/p-1k3sc2+1-sα2qpe2q/p-1k4sc3-D
Becauseα2 is a normal number, c3>1; q, p are odd integers, and 1<q/p<2, so α2q/pe2q/p-1≥0. If k4sc3-D≥0, then (13) can be further changed to(14)V˙≤-α2qpe2q/p-1k3sc2+1=-α2qpe2q/p-1k3Vc2+1/2
Also, becausek3>0, 0<c2<1, that is, α2q/pe2q/p-1k3>0, c2+1/2>0, (14) satisfies Lemma 2, and the system converges on the finite time of the equilibrium zero and can ensure the convergence of the following regions in a finite time.(15)s≤Dk41/c3
Similarly, equation (12) can be transformed into(16)V˙≤-α2qpe2q/p-1k4sc3+1-sα2qpe2q/p-1k3sc2-D
Becauseα2q/pe2q/p-1≥0, if k3sc2-D≥0, then (16) can be further changed to(17)V˙≤-α2qpe2q/p-1k4sc3+1=-α2qpe2q/p-1k4Vc3+1/2
Also, becauseα2q/pe2q/p-1k3>0, c3>1, c3+1/2>0, (17) satisfies Lemma 2, and the system converges on the finite time to the equilibrium zero and can guarantee the convergence of the following regions in a finite time.(18)s≤Dk31/c2
Combining equations (15) and (18), the system state s converges to the following areas within a limited time:(19)s≤min⁡Dk31/c2,Dk41/c3Substituting the previous equation into (7), we can get(20)s˙≤k3sc2sgn⁡s+k4sc3sgn⁡s≤k3sc2+k4sc3≤k3·min⁡Dk31/c2,Dk41/c3c2+k4·min⁡Dk31/c2,Dk41/c3c3=min⁡D,k3Dk4c1/c2+min⁡k4Dk3c2/c1,D
Theorem3 is proved.


### 3.2. RBF Network Approximationf(x) and b(x)

The nonlinear functionsf(x) and b(x) are required according to the control law (9), but the actual control needs to be obtained based on empirical knowledge, and sometimes it is not available. Therefore, it is approximated by the approximation principle. Because the RBF neural network has strong nonlinear fitting ability, it can map arbitrarily complex nonlinear relationships. At the same time, it has the advantages of simple learning rules and easy computer implementation. Therefore, using the universal approximation principle of RBF network, the unknown nonlinear function is approximated to solve the problem of the acquisition of unknown nonlinear function in the control law and the influence of unknown nonlinear function on the robustness of the system.


#### 3.2.1. RBF Neural Network Structure

The RBF network is a 3-layer forward network with a simple structure and is suitable for real-time control. The structure of the RBF network with multiple inputs and single output is shown in Figure2.Figure 2
The RBF network with multiple inputs and single output.In the RBF network,x=x1x2⋯xnT is the network input, and hj is the Gaussian function, which represents the output of the jth neuron in the hidden layer; namely,(21)hj=exp⁡-x-cj22gj2,j=1,2,···,mwherecj=[cj1,cj2,···,cjm] is the center of the Gaussian function of the node j; gj is the width of the Gaussian function of the node j. The weight of the network is taken as θ=[θ1,θ2,···,θm]T.Output of RBF network is(22)ymt=θ1h1+θ2h2+···+θmhm


#### 3.2.2. Adaptive Approximation of RBF Network

For the unknown nonlinear functionsf(x) and b(x) in (9), the RBF network adaptive approximation is adopted. The RBF network input and output algorithm is(23)hj=exp⁡-x-cj22gj2,j=1,2,···,mfx=W∗Thfx+εf,bx=V∗Thbx+εbwherex is the network input; hj is the Gaussian function; W∗ and V∗ are the ideal network weight vectors of f(x) and b(x), respectively; εf and εb are the network approximation errors, εf≤εMf, εb≤εMb. hf(x), hb(x) are vector functions, and the elements of them are all Gaussian function. We can set x=x1x2T, and then the output of the RBF network is(24)f^x=W^Thfx,b^x=V^ThbxDesign control law is(25)u=-b^x-1f^x+1α2pqe22-q/psgn⁡e21+α1γe1γ-1+k3sc2sgn⁡s+k4sc3sgn⁡s-x¨dwhereα1, α2, k3, k4 are normal numbers, and 0<c2<1, c3>1. q, p are odd integers, and 1<q/p<2, γ>q/p. f^(x) and b^(x) are the RBF network output values.Find the time derivative for (2):(26)s˙=e˙1+α1γe1γ-1e2+α2qpe2q/p-1e˙2=e2+α1γe1γ-1e2+α2qpe2q/p-1fx+bxu+dt-x¨d=e2+α1γe1γ-1e2+α2qpe2q/p-1fx+b^xu+bx-b^xu+dt-x¨dSubstitute (25) into (26):(27)s˙=e2+α1γe1γ-1e2+α2qpe2q/p-1fx+b^x-b^x-1f^x+1α2pqe22-q/psgn⁡e21+α1γe1γ-1+k3sc2sgn⁡s+k4sc3sgn⁡s-x¨d+bx-b^xu+dt-x¨d=α2qpe2q/p-1fx-f^x-k3sc2sgn⁡s-k4sc3sgn⁡s+bx-b^xu+dt=α2qpe2q/p-1f~x-k3sc2sgn⁡s-k4sc3sgn⁡s+b~xu+dt=α2qpe2q/p-1W~Thfx-εf-k3sc2sgn⁡s-k4sc3sgn⁡s+V~Thbx-εbu+dtwhereW~=W^-W∗, V~=V^-V∗, and(28)f~=f-f^=W∗Thfh-W^Thfx-εf=W~Thfx-εfb~=b-b^=V∗Thbx-V^Thbx-εb=V~Thbx-εbDesign the Lyapunov function as(29)L=12s2+12r1W~TW~+12r2V~TV~wherer1>0, r2>0.Find the time derivative for (29) and substitute it into (27):(30)L˙=sα2qpe2q/p-1W~Thfx-εf-k3sc2sgn⁡s-k4sc3sgn⁡s+V~Thbx-εbu+dt+1r1W~TW^˙+1r2V~TV^˙=W~Tα2qpe2q/p-1shfx+1r1W^˙+V~Tα2qpe2q/p-1shbxu+1r2V^˙+α2qpe2q/p-1s-εf-k3sc2sgn⁡s-k4sc3sgn⁡s-εbu+dtTake the adaptive law as(31)W^˙=-r1α2qpe2q/p-1shfxV^˙=-r2α2qpe2q/p-1shbxuSubstitute (31) into (30):(32)L˙=α2qpe2q/p-1s-εf-k3sc2sgn⁡s-k4sc3sgn⁡s-εbu+dt=-α2qpe2q/p-1k3sc2+1+k4sc3+1+sεf+εbu-dt≤-α2qpe2q/p-1k3sc2+1+k4sc3+1+sεf+εbu-DSinceεf and εb, the RBF network approximation errors, are very small real numbers, δ is selected as δ≥εf+εbu-D. According to Theorem 3, the system state s converges to the following region within a finite time:(33)s≤min⁡δk31/c2,δk41/c3s˙≤min⁡δ,k3δk4c1/c2+min⁡k4δk3c2/c1,δ


### 3.3. Hyperbolic Tangent Nonlinear Disturbance Observer

For system (1), the interference of the system is not only the model unknown interference generated by the unknown nonlinear function, but also the external disturbance. However, from (33), an important condition for the system state s to reach the convergence region in a finite time is that the disturbance amount d(t) in (1) is d(t)≤D and D is affected by the disturbance amount d(t). When d(t) is larger, D becomes larger, which affects the stability of the system. In order to further eliminate the influence of the interference amount on the control system, the nonlinear disturbance observer using the tracking-differentiator has the advantages of simple structure, good interference estimation effect, suppression of measurement noise, etc., and a hyperbolic tangent nonlinear disturbance observer is designed. The interference in (1) is estimated and feedforward compensation is performed to improve the anti-interference ability of the system.Lemma 4 (see [37]).
Consider the following systemΣ1:(34)z˙1=z2tz˙2=-a1tanh⁡b1z1t-a2tanh⁡b2z2t
Ifa1, a2, b1, b2 are all positive real numbers, the system Σ1 is progressively stable at the origin (0,0). It satisfies(35)limt→∞⁡z1t=0limt→∞⁡z2t=0
An improved disturbance observer is designed by using the hyperbolic tangent tracking differentiator proposed in [37].(36)x^˙2=fx+bxu+d^d^˙=-R2a1tanh⁡b1x^2-x2+a2tanh⁡b2d^R
In the formula,R, a1, a2, b1, b2 are all positive real numbers; x^ and d^ are, respectively, estimates of x and d. If T>0, R>0, there are(37)limR→∞⁡∫0Tx^2-x2dt=0
WhenR→∞, it can be obtained that |d^˙|=-R2a1tanh⁡b1x^2-x2+a2tanh⁡b2d^/R gets closer to infinity, so d^ changes faster than f(x)+b(x)u; that is, limR→∞⁡dfx+bxu+d^/dt=d^˙. Therefore, (36) satisfy Lemma 4, and the designed disturbance observer is progressively stable.
For (1), the following control laws are used:(38)u=-b^x-1f^x+kd^t+1α2pqe22-q/psgn⁡e21+α1γe1γ-1+k3sc2sgn⁡s+k4sc3sgn⁡s-x¨d
whereα1, α2, k3, k4, k are normal numbers, 0<c2<1, c3>1. q and p are odd integers, and 1<q/p<2, γ>q/p. f^(x) and b^(x) are the output values of the RBF network. d^(t) is the output value of the hyperbolic tangent nonlinear disturbance observer. The system state s will converge to a smaller region for a limited time.
Take a derivative of function (29) according to the Lyapunov, combining with (26), (31), and (38):(39)L˙=se2+α1γe1γ-1e2+α2qpe2q/p-1fx+b^x-b^x-1f^x+kd^t+1α2pqe22-q/psgn⁡e21+α1γe1γ-1+k3sc2sgn⁡s+k4sc3sgn⁡s-x¨d+bx-b^xu+dt-x¨d+1r1W~TW^˙+1r2V~TV^˙=sα2qpe2q/p-1W~Thfx-εf-kd^t-k3sc2sgn⁡s-k4sc3sgn⁡s+V~Thbx-εbu+dt+1r1W~TW^˙+1r2V~TV^˙=W~Tα2qpe2q/p-1shfx+1r1W^˙+V~Tα2qpe2q/p-1shbxu+1r2V^˙+α2qpe2q/p-1s-εf-k3sc2sgn⁡s-k4sc3sgn⁡s-εbu-kd^t+dt=α2qpe2q/p-1s-εf-k3sc2sgn⁡s-k4sc3sgn⁡s-εbu-kd^t+dt=-α2qpe2q/p-1k3sc2+1+k4sc3+1+sεf+εbu+kd^t-dt≤-α2qpe2q/p-1k3sc2+1+k4sc3+1+sεf+εbu+kd^t-D
Settingδ′≥εf+εbu+kd^(t)-D, according to Theorem 3, the system state s converges to the following region within a finite time:(40)s≤min⁡δ′k31/c2,δ′k41/c3s˙≤min⁡δ′,k3δ′k4c1/c2+min⁡k4δ′k3c2/c1,δ′Theorem 5.
For the system shown in (1), design the sliding mode according to (2), using the NFTSM controller shown in (38). The system state converges to the following areas in a limited time:(41)e1≤min⁡δ′k3α1c21/γc2,δ′k4α1c31/γc3e2≤min⁡δ′k3α2c2p/qc2,δ′k4α2c3p/qc3Proof.
According to Theorem3 and (40), system state enters s≤min⁡δ′/k31/c2,δ′/k41/c3 region, and we can get(42)e1+α1e1γsgn⁡e1+α2e2q/psgn⁡e2=σwhere σ∈Ωs=s∣s≤min⁡δ′/k31/c2,δ′/k41/c3.
Equation (42) can be changed to(43)e1+α1e1γsgn⁡e1+α2-σe2q/psgn⁡e2e2q/psgn⁡e2=0
Whenα2-σ/e2q/psgn⁡(e2)>0, (43) satisfies the NFTSM form described in (2), and the system trajectory will converge to(44)α2≤σe2q/psgn⁡e2
Therefore, statee2 can converge rapidly to the region within a finite time ts:(45)e2≤σα2p/q≤min⁡δ′/k31/c2,δ′/k41/c3α2p/q≤min⁡δ′k3α2c2p/qc2,δ′k4α2c3p/qc3(46)ts≤α2-σ/e2q/psgn⁡e2p/qqq-pe1tr1-p/q
Using the same approach, (42) can be changed to(47)e1+α1-σe1γsgn⁡e1e1γsgn⁡e1+α2e2q/psgn⁡e2=0
Whenα1-σ/e1γsgn⁡e1>0, (47) satisfies the NFTSM form described in (2), and the system trajectory will converge to(48)α1≤σe1γsgn⁡e1
Therefore, statee1 can converge rapidly to the region within a finite time ts:(49)e1≤σα1γ≤min⁡δ′/k31/c2,δ′/k41/c3α1γ≤min⁡δ′k3α1c21/γc2,δ′k4α1c31/γc3
Theorem5 is proved.Remark 6.
As the amount of disturbanced(t) in (1) becomes larger, D and d^(t) become larger. It can be ensured that the system can still converge with a smaller region. Owing to the fact that the control rate is nonsingular and continuous and there is no switching term in the general control rate, the chattering caused by frequent switching is avoided, so that the proposed control rate u can get lower chattering.


## 3.1. Nonsingular Fast Terminal Sliding Mode Control

In order to solve the problem of singularity and slow convergence of traditional terminal sliding mode control, a novel nonsingular fast terminal sliding mode control method is proposed.


### 3.1.1. Sliding Mode Design

x d is set the given signal and is second derivable. The position and velocity tracking errors of the system are defined as e1=x1-xd and e2=x2-x˙d, respectively. The designed nonsingular fast terminal sliding mode is(2)s=e1+α1e1γsgn⁡e1+α2e2q/psgn⁡e2whereα1, α2 are positive real numbers; q, p are odd integers, and 1<q/p<2, γ>q/p. e1 and e2 do not have negative exponential terms, ensuring that the sliding mode s based controller does not have singularity. When the system error state reaches the sliding mode (s=0), the following equation can be obtained:(3)e1+α1e1γsgn⁡e1+α2e2q/psgn⁡e2=0Substitutee˙1=e2 into (3), the following equation can be obtained:(4)e˙1=-1α2p/qe1+α1e1γsgn⁡e1p/q=-1α2p/qe1p/q1+α1e1γ-1p/qAssume the time taken frome1(tr)≠0 to e1(tr+ts)=0 is ts, and the following equation can be obtained through the time integrated on both sides of (4):(5)∫e1tre1tr+ts1e1p/qde1=-∫trtr+ts1α2p/q1+α1e1γ-1p/qdτ≤-∫trtr+ts1α2p/qdτSimplification of Inequality (5) (6) t s ≤ α 2 p / q q q - p e 1 t r 1 - p / qTherefore, the system error can converge to zero for a limited time.Remark 1.
When the system error state is far from the equilibrium point, the dominant role ofe1 in the sliding mode s causes the system trajectory to converge quickly. When the system error state approaches the equilibrium point, the dominant role of α1e1γsgn⁡e1 in the sliding mode s causes the system trajectory to converge rapidly. Therefore, NFTSM can achieve fast convergence of the whole state trajectory.


### 3.1.2. Reaching Law Design

According to the sliding mode variable structure principle, the sliding mode reachability condition only ensures that the moving point at any position in the state space can reach the switching surface within a limited time, and there is no restriction on the specific trajectory of the reaching motion. The reaching law can improve the dynamic quality of reaching movement. The following second power reaching law is adopted in this paper.(7)s˙=-k3sc2sgn⁡s-k4sc3sgn⁡swherek3>0, k4>0, 0<c2<1, c3>1. When the system state is far from the sliding mode (s>1), -k4sc3sgn⁡s plays a leading role; when the system approaches the sliding mode (s<1), -k3sc2sgn⁡(s) dominates. When s=0, s˙=0, the speed of the state reaching the sliding mode is reduced to zero, and the smooth transition of the sliding mode is realized, which greatly weakens the system chattering phenomenon. Compared with the traditional power reaching law, exponential reaching law, and fast power reaching law, the reaching law (7) has faster convergence speed and better motion quality.


### 3.1.3. Controller Design and Stability Analysis

Lemma 2 (see [36]).
Setx∈M⊂Rn, x˙=f(x), f:Rn→R as the continuous functions defined in the equilibrium point region M. Assume that a continuous function V:M→R satisfies the following conditions:
(1)V is positive definite.
(2)V˙ is negative definite except for the equilibrium point.
(3) Real numberk>0, α>0 and region N⊂M make V˙+kVα≤0, and the function x˙=f(x) converges at balanced zero point within finite time.Theorem 3.
For the system shown in (1), design the sliding mode according to (2), using the NFTSM controller shown in (9). The system state s converges to the following areas in a limited time:(8)s≤min⁡Dk31/c2,Dk41/c3s˙≤min⁡D,k3Dk4c1/c2+min⁡k4Dk3c2/c1,D
Sliding mode controller is designed as(9)u=-bx-1fx+1α2pqe22-q/psgn⁡e21+α1γe1γ-1+k3sc2sgn⁡s+k4sc3sgn⁡s-x¨d
whereα1, α2, k3, k4 are normal numbers, 0<c2<1, c3>1; q, p are odd integers, and 1<q/p<2, γ>q/p.Proof.
Define the following Lyapunov function:(10)V=12s2
Find the time derivative for (10):(11)V˙=se˙1+α1γe1γ-1sgn⁡e1e˙1+α2qpe2q/p-1sgn⁡e2e˙2=se2+α1γe1γ-1sgn⁡e1e2+α2qpe2q/p-1sgn⁡e2fx+bxu+dt-x¨d
Substitute (9) into the above equation:(12)V˙=se˙1+α1γe1γ-1e˙1+α2qpe2q/p-1e˙2=se2+α1γe1γ-1e2+α2qpe2q/p-1fx+bx-bx-1fx+1α2pqe22-q/psgn⁡e21+α1γe1γ-1+k3sc2sgn⁡s+k4sc3sgn⁡s-x¨d+dt-x¨d=sα2qpe2q/p-1-k3sc2sgns-k4sc3sgns+dt=-α2qpe2q/p-1k3sc2+1+k4sc3+1-sdt≤-α2qpe2q/p-1k3sc2+1+k4sc3+1-Ds
Deform (12):(13)V˙≤-α2qpe2q/p-1k3sc2+1-α2qpe2q/p-1k4sc3+1+α2qpe2q/p-1Ds=-α2qpe2q/p-1k3sc2+1-sα2qpe2q/p-1k4sc3-D
Becauseα2 is a normal number, c3>1; q, p are odd integers, and 1<q/p<2, so α2q/pe2q/p-1≥0. If k4sc3-D≥0, then (13) can be further changed to(14)V˙≤-α2qpe2q/p-1k3sc2+1=-α2qpe2q/p-1k3Vc2+1/2
Also, becausek3>0, 0<c2<1, that is, α2q/pe2q/p-1k3>0, c2+1/2>0, (14) satisfies Lemma 2, and the system converges on the finite time of the equilibrium zero and can ensure the convergence of the following regions in a finite time.(15)s≤Dk41/c3
Similarly, equation (12) can be transformed into(16)V˙≤-α2qpe2q/p-1k4sc3+1-sα2qpe2q/p-1k3sc2-D
Becauseα2q/pe2q/p-1≥0, if k3sc2-D≥0, then (16) can be further changed to(17)V˙≤-α2qpe2q/p-1k4sc3+1=-α2qpe2q/p-1k4Vc3+1/2
Also, becauseα2q/pe2q/p-1k3>0, c3>1, c3+1/2>0, (17) satisfies Lemma 2, and the system converges on the finite time to the equilibrium zero and can guarantee the convergence of the following regions in a finite time.(18)s≤Dk31/c2
Combining equations (15) and (18), the system state s converges to the following areas within a limited time:(19)s≤min⁡Dk31/c2,Dk41/c3Substituting the previous equation into (7), we can get(20)s˙≤k3sc2sgn⁡s+k4sc3sgn⁡s≤k3sc2+k4sc3≤k3·min⁡Dk31/c2,Dk41/c3c2+k4·min⁡Dk31/c2,Dk41/c3c3=min⁡D,k3Dk4c1/c2+min⁡k4Dk3c2/c1,D
Theorem3 is proved.


## 3.1.1. Sliding Mode Design

x d is set the given signal and is second derivable. The position and velocity tracking errors of the system are defined as e1=x1-xd and e2=x2-x˙d, respectively. The designed nonsingular fast terminal sliding mode is(2)s=e1+α1e1γsgn⁡e1+α2e2q/psgn⁡e2whereα1, α2 are positive real numbers; q, p are odd integers, and 1<q/p<2, γ>q/p. e1 and e2 do not have negative exponential terms, ensuring that the sliding mode s based controller does not have singularity. When the system error state reaches the sliding mode (s=0), the following equation can be obtained:(3)e1+α1e1γsgn⁡e1+α2e2q/psgn⁡e2=0Substitutee˙1=e2 into (3), the following equation can be obtained:(4)e˙1=-1α2p/qe1+α1e1γsgn⁡e1p/q=-1α2p/qe1p/q1+α1e1γ-1p/qAssume the time taken frome1(tr)≠0 to e1(tr+ts)=0 is ts, and the following equation can be obtained through the time integrated on both sides of (4):(5)∫e1tre1tr+ts1e1p/qde1=-∫trtr+ts1α2p/q1+α1e1γ-1p/qdτ≤-∫trtr+ts1α2p/qdτSimplification of Inequality (5) (6) t s ≤ α 2 p / q q q - p e 1 t r 1 - p / qTherefore, the system error can converge to zero for a limited time.Remark 1.
When the system error state is far from the equilibrium point, the dominant role ofe1 in the sliding mode s causes the system trajectory to converge quickly. When the system error state approaches the equilibrium point, the dominant role of α1e1γsgn⁡e1 in the sliding mode s causes the system trajectory to converge rapidly. Therefore, NFTSM can achieve fast convergence of the whole state trajectory.


## 3.1.2. Reaching Law Design

According to the sliding mode variable structure principle, the sliding mode reachability condition only ensures that the moving point at any position in the state space can reach the switching surface within a limited time, and there is no restriction on the specific trajectory of the reaching motion. The reaching law can improve the dynamic quality of reaching movement. The following second power reaching law is adopted in this paper.(7)s˙=-k3sc2sgn⁡s-k4sc3sgn⁡swherek3>0, k4>0, 0<c2<1, c3>1. When the system state is far from the sliding mode (s>1), -k4sc3sgn⁡s plays a leading role; when the system approaches the sliding mode (s<1), -k3sc2sgn⁡(s) dominates. When s=0, s˙=0, the speed of the state reaching the sliding mode is reduced to zero, and the smooth transition of the sliding mode is realized, which greatly weakens the system chattering phenomenon. Compared with the traditional power reaching law, exponential reaching law, and fast power reaching law, the reaching law (7) has faster convergence speed and better motion quality.


## 3.1.3. Controller Design and Stability Analysis

Lemma 2 (see [36]).
Setx∈M⊂Rn, x˙=f(x), f:Rn→R as the continuous functions defined in the equilibrium point region M. Assume that a continuous function V:M→R satisfies the following conditions:
(1)V is positive definite.
(2)V˙ is negative definite except for the equilibrium point.
(3) Real numberk>0, α>0 and region N⊂M make V˙+kVα≤0, and the function x˙=f(x) converges at balanced zero point within finite time.Theorem 3.
For the system shown in (1), design the sliding mode according to (2), using the NFTSM controller shown in (9). The system state s converges to the following areas in a limited time:(8)s≤min⁡Dk31/c2,Dk41/c3s˙≤min⁡D,k3Dk4c1/c2+min⁡k4Dk3c2/c1,D
Sliding mode controller is designed as(9)u=-bx-1fx+1α2pqe22-q/psgn⁡e21+α1γe1γ-1+k3sc2sgn⁡s+k4sc3sgn⁡s-x¨d
whereα1, α2, k3, k4 are normal numbers, 0<c2<1, c3>1; q, p are odd integers, and 1<q/p<2, γ>q/p.Proof.
Define the following Lyapunov function:(10)V=12s2
Find the time derivative for (10):(11)V˙=se˙1+α1γe1γ-1sgn⁡e1e˙1+α2qpe2q/p-1sgn⁡e2e˙2=se2+α1γe1γ-1sgn⁡e1e2+α2qpe2q/p-1sgn⁡e2fx+bxu+dt-x¨d
Substitute (9) into the above equation:(12)V˙=se˙1+α1γe1γ-1e˙1+α2qpe2q/p-1e˙2=se2+α1γe1γ-1e2+α2qpe2q/p-1fx+bx-bx-1fx+1α2pqe22-q/psgn⁡e21+α1γe1γ-1+k3sc2sgn⁡s+k4sc3sgn⁡s-x¨d+dt-x¨d=sα2qpe2q/p-1-k3sc2sgns-k4sc3sgns+dt=-α2qpe2q/p-1k3sc2+1+k4sc3+1-sdt≤-α2qpe2q/p-1k3sc2+1+k4sc3+1-Ds
Deform (12):(13)V˙≤-α2qpe2q/p-1k3sc2+1-α2qpe2q/p-1k4sc3+1+α2qpe2q/p-1Ds=-α2qpe2q/p-1k3sc2+1-sα2qpe2q/p-1k4sc3-D
Becauseα2 is a normal number, c3>1; q, p are odd integers, and 1<q/p<2, so α2q/pe2q/p-1≥0. If k4sc3-D≥0, then (13) can be further changed to(14)V˙≤-α2qpe2q/p-1k3sc2+1=-α2qpe2q/p-1k3Vc2+1/2
Also, becausek3>0, 0<c2<1, that is, α2q/pe2q/p-1k3>0, c2+1/2>0, (14) satisfies Lemma 2, and the system converges on the finite time of the equilibrium zero and can ensure the convergence of the following regions in a finite time.(15)s≤Dk41/c3
Similarly, equation (12) can be transformed into(16)V˙≤-α2qpe2q/p-1k4sc3+1-sα2qpe2q/p-1k3sc2-D
Becauseα2q/pe2q/p-1≥0, if k3sc2-D≥0, then (16) can be further changed to(17)V˙≤-α2qpe2q/p-1k4sc3+1=-α2qpe2q/p-1k4Vc3+1/2
Also, becauseα2q/pe2q/p-1k3>0, c3>1, c3+1/2>0, (17) satisfies Lemma 2, and the system converges on the finite time to the equilibrium zero and can guarantee the convergence of the following regions in a finite time.(18)s≤Dk31/c2
Combining equations (15) and (18), the system state s converges to the following areas within a limited time:(19)s≤min⁡Dk31/c2,Dk41/c3Substituting the previous equation into (7), we can get(20)s˙≤k3sc2sgn⁡s+k4sc3sgn⁡s≤k3sc2+k4sc3≤k3·min⁡Dk31/c2,Dk41/c3c2+k4·min⁡Dk31/c2,Dk41/c3c3=min⁡D,k3Dk4c1/c2+min⁡k4Dk3c2/c1,D
Theorem3 is proved.


## 3.2. RBF Network Approximationf(x) and b(x)

The nonlinear functionsf(x) and b(x) are required according to the control law (9), but the actual control needs to be obtained based on empirical knowledge, and sometimes it is not available. Therefore, it is approximated by the approximation principle. Because the RBF neural network has strong nonlinear fitting ability, it can map arbitrarily complex nonlinear relationships. At the same time, it has the advantages of simple learning rules and easy computer implementation. Therefore, using the universal approximation principle of RBF network, the unknown nonlinear function is approximated to solve the problem of the acquisition of unknown nonlinear function in the control law and the influence of unknown nonlinear function on the robustness of the system.


### 3.2.1. RBF Neural Network Structure

The RBF network is a 3-layer forward network with a simple structure and is suitable for real-time control. The structure of the RBF network with multiple inputs and single output is shown in Figure2.Figure 2
The RBF network with multiple inputs and single output.In the RBF network,x=x1x2⋯xnT is the network input, and hj is the Gaussian function, which represents the output of the jth neuron in the hidden layer; namely,(21)hj=exp⁡-x-cj22gj2,j=1,2,···,mwherecj=[cj1,cj2,···,cjm] is the center of the Gaussian function of the node j; gj is the width of the Gaussian function of the node j. The weight of the network is taken as θ=[θ1,θ2,···,θm]T.Output of RBF network is(22)ymt=θ1h1+θ2h2+···+θmhm


### 3.2.2. Adaptive Approximation of RBF Network

For the unknown nonlinear functionsf(x) and b(x) in (9), the RBF network adaptive approximation is adopted. The RBF network input and output algorithm is(23)hj=exp⁡-x-cj22gj2,j=1,2,···,mfx=W∗Thfx+εf,bx=V∗Thbx+εbwherex is the network input; hj is the Gaussian function; W∗ and V∗ are the ideal network weight vectors of f(x) and b(x), respectively; εf and εb are the network approximation errors, εf≤εMf, εb≤εMb. hf(x), hb(x) are vector functions, and the elements of them are all Gaussian function. We can set x=x1x2T, and then the output of the RBF network is(24)f^x=W^Thfx,b^x=V^ThbxDesign control law is(25)u=-b^x-1f^x+1α2pqe22-q/psgn⁡e21+α1γe1γ-1+k3sc2sgn⁡s+k4sc3sgn⁡s-x¨dwhereα1, α2, k3, k4 are normal numbers, and 0<c2<1, c3>1. q, p are odd integers, and 1<q/p<2, γ>q/p. f^(x) and b^(x) are the RBF network output values.Find the time derivative for (2):(26)s˙=e˙1+α1γe1γ-1e2+α2qpe2q/p-1e˙2=e2+α1γe1γ-1e2+α2qpe2q/p-1fx+bxu+dt-x¨d=e2+α1γe1γ-1e2+α2qpe2q/p-1fx+b^xu+bx-b^xu+dt-x¨dSubstitute (25) into (26):(27)s˙=e2+α1γe1γ-1e2+α2qpe2q/p-1fx+b^x-b^x-1f^x+1α2pqe22-q/psgn⁡e21+α1γe1γ-1+k3sc2sgn⁡s+k4sc3sgn⁡s-x¨d+bx-b^xu+dt-x¨d=α2qpe2q/p-1fx-f^x-k3sc2sgn⁡s-k4sc3sgn⁡s+bx-b^xu+dt=α2qpe2q/p-1f~x-k3sc2sgn⁡s-k4sc3sgn⁡s+b~xu+dt=α2qpe2q/p-1W~Thfx-εf-k3sc2sgn⁡s-k4sc3sgn⁡s+V~Thbx-εbu+dtwhereW~=W^-W∗, V~=V^-V∗, and(28)f~=f-f^=W∗Thfh-W^Thfx-εf=W~Thfx-εfb~=b-b^=V∗Thbx-V^Thbx-εb=V~Thbx-εbDesign the Lyapunov function as(29)L=12s2+12r1W~TW~+12r2V~TV~wherer1>0, r2>0.Find the time derivative for (29) and substitute it into (27):(30)L˙=sα2qpe2q/p-1W~Thfx-εf-k3sc2sgn⁡s-k4sc3sgn⁡s+V~Thbx-εbu+dt+1r1W~TW^˙+1r2V~TV^˙=W~Tα2qpe2q/p-1shfx+1r1W^˙+V~Tα2qpe2q/p-1shbxu+1r2V^˙+α2qpe2q/p-1s-εf-k3sc2sgn⁡s-k4sc3sgn⁡s-εbu+dtTake the adaptive law as(31)W^˙=-r1α2qpe2q/p-1shfxV^˙=-r2α2qpe2q/p-1shbxuSubstitute (31) into (30):(32)L˙=α2qpe2q/p-1s-εf-k3sc2sgn⁡s-k4sc3sgn⁡s-εbu+dt=-α2qpe2q/p-1k3sc2+1+k4sc3+1+sεf+εbu-dt≤-α2qpe2q/p-1k3sc2+1+k4sc3+1+sεf+εbu-DSinceεf and εb, the RBF network approximation errors, are very small real numbers, δ is selected as δ≥εf+εbu-D. According to Theorem 3, the system state s converges to the following region within a finite time:(33)s≤min⁡δk31/c2,δk41/c3s˙≤min⁡δ,k3δk4c1/c2+min⁡k4δk3c2/c1,δ


## 3.2.1. RBF Neural Network Structure

The RBF network is a 3-layer forward network with a simple structure and is suitable for real-time control. The structure of the RBF network with multiple inputs and single output is shown in Figure2.Figure 2
The RBF network with multiple inputs and single output.In the RBF network,x=x1x2⋯xnT is the network input, and hj is the Gaussian function, which represents the output of the jth neuron in the hidden layer; namely,(21)hj=exp⁡-x-cj22gj2,j=1,2,···,mwherecj=[cj1,cj2,···,cjm] is the center of the Gaussian function of the node j; gj is the width of the Gaussian function of the node j. The weight of the network is taken as θ=[θ1,θ2,···,θm]T.Output of RBF network is(22)ymt=θ1h1+θ2h2+···+θmhm


## 3.2.2. Adaptive Approximation of RBF Network

For the unknown nonlinear functionsf(x) and b(x) in (9), the RBF network adaptive approximation is adopted. The RBF network input and output algorithm is(23)hj=exp⁡-x-cj22gj2,j=1,2,···,mfx=W∗Thfx+εf,bx=V∗Thbx+εbwherex is the network input; hj is the Gaussian function; W∗ and V∗ are the ideal network weight vectors of f(x) and b(x), respectively; εf and εb are the network approximation errors, εf≤εMf, εb≤εMb. hf(x), hb(x) are vector functions, and the elements of them are all Gaussian function. We can set x=x1x2T, and then the output of the RBF network is(24)f^x=W^Thfx,b^x=V^ThbxDesign control law is(25)u=-b^x-1f^x+1α2pqe22-q/psgn⁡e21+α1γe1γ-1+k3sc2sgn⁡s+k4sc3sgn⁡s-x¨dwhereα1, α2, k3, k4 are normal numbers, and 0<c2<1, c3>1. q, p are odd integers, and 1<q/p<2, γ>q/p. f^(x) and b^(x) are the RBF network output values.Find the time derivative for (2):(26)s˙=e˙1+α1γe1γ-1e2+α2qpe2q/p-1e˙2=e2+α1γe1γ-1e2+α2qpe2q/p-1fx+bxu+dt-x¨d=e2+α1γe1γ-1e2+α2qpe2q/p-1fx+b^xu+bx-b^xu+dt-x¨dSubstitute (25) into (26):(27)s˙=e2+α1γe1γ-1e2+α2qpe2q/p-1fx+b^x-b^x-1f^x+1α2pqe22-q/psgn⁡e21+α1γe1γ-1+k3sc2sgn⁡s+k4sc3sgn⁡s-x¨d+bx-b^xu+dt-x¨d=α2qpe2q/p-1fx-f^x-k3sc2sgn⁡s-k4sc3sgn⁡s+bx-b^xu+dt=α2qpe2q/p-1f~x-k3sc2sgn⁡s-k4sc3sgn⁡s+b~xu+dt=α2qpe2q/p-1W~Thfx-εf-k3sc2sgn⁡s-k4sc3sgn⁡s+V~Thbx-εbu+dtwhereW~=W^-W∗, V~=V^-V∗, and(28)f~=f-f^=W∗Thfh-W^Thfx-εf=W~Thfx-εfb~=b-b^=V∗Thbx-V^Thbx-εb=V~Thbx-εbDesign the Lyapunov function as(29)L=12s2+12r1W~TW~+12r2V~TV~wherer1>0, r2>0.Find the time derivative for (29) and substitute it into (27):(30)L˙=sα2qpe2q/p-1W~Thfx-εf-k3sc2sgn⁡s-k4sc3sgn⁡s+V~Thbx-εbu+dt+1r1W~TW^˙+1r2V~TV^˙=W~Tα2qpe2q/p-1shfx+1r1W^˙+V~Tα2qpe2q/p-1shbxu+1r2V^˙+α2qpe2q/p-1s-εf-k3sc2sgn⁡s-k4sc3sgn⁡s-εbu+dtTake the adaptive law as(31)W^˙=-r1α2qpe2q/p-1shfxV^˙=-r2α2qpe2q/p-1shbxuSubstitute (31) into (30):(32)L˙=α2qpe2q/p-1s-εf-k3sc2sgn⁡s-k4sc3sgn⁡s-εbu+dt=-α2qpe2q/p-1k3sc2+1+k4sc3+1+sεf+εbu-dt≤-α2qpe2q/p-1k3sc2+1+k4sc3+1+sεf+εbu-DSinceεf and εb, the RBF network approximation errors, are very small real numbers, δ is selected as δ≥εf+εbu-D. According to Theorem 3, the system state s converges to the following region within a finite time:(33)s≤min⁡δk31/c2,δk41/c3s˙≤min⁡δ,k3δk4c1/c2+min⁡k4δk3c2/c1,δ


## 3.3. Hyperbolic Tangent Nonlinear Disturbance Observer

For system (1), the interference of the system is not only the model unknown interference generated by the unknown nonlinear function, but also the external disturbance. However, from (33), an important condition for the system state s to reach the convergence region in a finite time is that the disturbance amount d(t) in (1) is d(t)≤D and D is affected by the disturbance amount d(t). When d(t) is larger, D becomes larger, which affects the stability of the system. In order to further eliminate the influence of the interference amount on the control system, the nonlinear disturbance observer using the tracking-differentiator has the advantages of simple structure, good interference estimation effect, suppression of measurement noise, etc., and a hyperbolic tangent nonlinear disturbance observer is designed. The interference in (1) is estimated and feedforward compensation is performed to improve the anti-interference ability of the system.Lemma 4 (see [37]).
Consider the following systemΣ1:(34)z˙1=z2tz˙2=-a1tanh⁡b1z1t-a2tanh⁡b2z2t
Ifa1, a2, b1, b2 are all positive real numbers, the system Σ1 is progressively stable at the origin (0,0). It satisfies(35)limt→∞⁡z1t=0limt→∞⁡z2t=0
An improved disturbance observer is designed by using the hyperbolic tangent tracking differentiator proposed in [37].(36)x^˙2=fx+bxu+d^d^˙=-R2a1tanh⁡b1x^2-x2+a2tanh⁡b2d^R
In the formula,R, a1, a2, b1, b2 are all positive real numbers; x^ and d^ are, respectively, estimates of x and d. If T>0, R>0, there are(37)limR→∞⁡∫0Tx^2-x2dt=0
WhenR→∞, it can be obtained that |d^˙|=-R2a1tanh⁡b1x^2-x2+a2tanh⁡b2d^/R gets closer to infinity, so d^ changes faster than f(x)+b(x)u; that is, limR→∞⁡dfx+bxu+d^/dt=d^˙. Therefore, (36) satisfy Lemma 4, and the designed disturbance observer is progressively stable.
For (1), the following control laws are used:(38)u=-b^x-1f^x+kd^t+1α2pqe22-q/psgn⁡e21+α1γe1γ-1+k3sc2sgn⁡s+k4sc3sgn⁡s-x¨d
whereα1, α2, k3, k4, k are normal numbers, 0<c2<1, c3>1. q and p are odd integers, and 1<q/p<2, γ>q/p. f^(x) and b^(x) are the output values of the RBF network. d^(t) is the output value of the hyperbolic tangent nonlinear disturbance observer. The system state s will converge to a smaller region for a limited time.
Take a derivative of function (29) according to the Lyapunov, combining with (26), (31), and (38):(39)L˙=se2+α1γe1γ-1e2+α2qpe2q/p-1fx+b^x-b^x-1f^x+kd^t+1α2pqe22-q/psgn⁡e21+α1γe1γ-1+k3sc2sgn⁡s+k4sc3sgn⁡s-x¨d+bx-b^xu+dt-x¨d+1r1W~TW^˙+1r2V~TV^˙=sα2qpe2q/p-1W~Thfx-εf-kd^t-k3sc2sgn⁡s-k4sc3sgn⁡s+V~Thbx-εbu+dt+1r1W~TW^˙+1r2V~TV^˙=W~Tα2qpe2q/p-1shfx+1r1W^˙+V~Tα2qpe2q/p-1shbxu+1r2V^˙+α2qpe2q/p-1s-εf-k3sc2sgn⁡s-k4sc3sgn⁡s-εbu-kd^t+dt=α2qpe2q/p-1s-εf-k3sc2sgn⁡s-k4sc3sgn⁡s-εbu-kd^t+dt=-α2qpe2q/p-1k3sc2+1+k4sc3+1+sεf+εbu+kd^t-dt≤-α2qpe2q/p-1k3sc2+1+k4sc3+1+sεf+εbu+kd^t-D
Settingδ′≥εf+εbu+kd^(t)-D, according to Theorem 3, the system state s converges to the following region within a finite time:(40)s≤min⁡δ′k31/c2,δ′k41/c3s˙≤min⁡δ′,k3δ′k4c1/c2+min⁡k4δ′k3c2/c1,δ′Theorem 5.
For the system shown in (1), design the sliding mode according to (2), using the NFTSM controller shown in (38). The system state converges to the following areas in a limited time:(41)e1≤min⁡δ′k3α1c21/γc2,δ′k4α1c31/γc3e2≤min⁡δ′k3α2c2p/qc2,δ′k4α2c3p/qc3Proof.
According to Theorem3 and (40), system state enters s≤min⁡δ′/k31/c2,δ′/k41/c3 region, and we can get(42)e1+α1e1γsgn⁡e1+α2e2q/psgn⁡e2=σwhere σ∈Ωs=s∣s≤min⁡δ′/k31/c2,δ′/k41/c3.
Equation (42) can be changed to(43)e1+α1e1γsgn⁡e1+α2-σe2q/psgn⁡e2e2q/psgn⁡e2=0
Whenα2-σ/e2q/psgn⁡(e2)>0, (43) satisfies the NFTSM form described in (2), and the system trajectory will converge to(44)α2≤σe2q/psgn⁡e2
Therefore, statee2 can converge rapidly to the region within a finite time ts:(45)e2≤σα2p/q≤min⁡δ′/k31/c2,δ′/k41/c3α2p/q≤min⁡δ′k3α2c2p/qc2,δ′k4α2c3p/qc3(46)ts≤α2-σ/e2q/psgn⁡e2p/qqq-pe1tr1-p/q
Using the same approach, (42) can be changed to(47)e1+α1-σe1γsgn⁡e1e1γsgn⁡e1+α2e2q/psgn⁡e2=0
Whenα1-σ/e1γsgn⁡e1>0, (47) satisfies the NFTSM form described in (2), and the system trajectory will converge to(48)α1≤σe1γsgn⁡e1
Therefore, statee1 can converge rapidly to the region within a finite time ts:(49)e1≤σα1γ≤min⁡δ′/k31/c2,δ′/k41/c3α1γ≤min⁡δ′k3α1c21/γc2,δ′k4α1c31/γc3
Theorem5 is proved.Remark 6.
As the amount of disturbanced(t) in (1) becomes larger, D and d^(t) become larger. It can be ensured that the system can still converge with a smaller region. Owing to the fact that the control rate is nonsingular and continuous and there is no switching term in the general control rate, the chattering caused by frequent switching is avoided, so that the proposed control rate u can get lower chattering.


## 4. Simulations

In order to verify the feasibility and effectiveness of the control method in this paper, the controlled object is taken as a single-stage inverted pendulum system, and its dynamic equation is(50)x˙1=x2x˙2=fx+bxuwhere f(x)=gsin⁡x1-mlx22cos⁡x1sin⁡x1/mc+mp/l4/3-mpcos2⁡x1/mc+mp, b(x)=cos⁡x1/(mc+mp)/l4/3-mpcos2⁡x1/mc+mp. x1 and x2 are, respectively, the swing angle and the swing speed, u is the control input, mc is the weight of the trolley, mp is the weight of the pendulum, l is the length of the pendulum, and g is the acceleration of gravity.The performance of system (50) contorted by the proposed method is compared with the NTSM control method [18] and the NFTSM control method [27] which are based on the following exponential reaching law.(1) The sliding mode, the reaching law, and the controller of the NTSM control method based on the exponential reaching law are as follows:(51)s=e1+1βe2q/p,s˙=-ws-ρsgn⁡s,u=-bx-1fx+βpqe22-q/p+ws+ρsgn⁡s-x¨d(2) The sliding mode, the reaching law, and the controller of the NFTSM control method based on the exponential reaching law are as follows:(52)s=e1+α1e1γsgn⁡e1+α2e2q/psgn⁡e2,s˙=-ws-ρsgn⁡s,u=-bx-1fx+1α2pqe22-q/psgn⁡e21+α1γe1γ-1+ws+ρsgn⁡s-x¨d(3) The sliding mode, the reaching law, and the controller of the NFTSM control method based on the second power reaching law (SPNFTSM) are as follows:(53)s=e1+α1e1γsgn⁡e1+α2e2q/psgn⁡e2,s˙=-k3sc2sgn⁡s-k4sc3sgn⁡s,u=-bx-1fx+1α2pqe22-q/psgn⁡e21+α1γe1γ-1+k3sc2sgn⁡s+k4sc3sgn⁡s-x¨d(4) The sliding mode, the reaching law, and the controller of the NFTSM control method based on the second power of RBF network (RBFSPNFTSM) are as follows:(54)s=e1+α1e1γsgn⁡e1+α2e2q/psgn⁡e2,s˙=-k3sc2sgn⁡s-k4sc3sgn⁡s,u=-b^x-1f^x+1α2pqe22-q/psgn⁡e21+α1γe1γ-1+k3sc2sgn⁡s+k4sc3sgn⁡s-x¨d(5) The sliding mode, the reaching law, and the controller of the NFTSM control method based on the second power of RBF network and disturbance observer (RBFDOSPNFTSM) are as follows:(55)s=e1+α1e1γsgn⁡e1+α2e2q/psgn⁡e2,s˙=-k3sc2sgn⁡s-k4sc3sgn⁡s,u=-b^x-1f^x+kd^t+1α2pqe22-q/psgn⁡e21+α1γe1γ-1+k3sc2sgn⁡s+k4sc3sgn⁡s-x¨d


### 4.1. Dynamic Performance Analysis

The initial state of the inverted pendulum system is set as[π/60,0], and the system model parameters and controller parameters are shown in Tables 1 and 2, respectively. Given signal xd=0, interference d(t)=sin⁡x1+5sin⁡(πt).Table 1
System model parameters.
the weight of the trolley (mc) the weight of the pendulum (mp) the length of the pendulum (l) the acceleration of gravity (g) 1kg 0.1kg 0.5m 9.8m/s2Table 2
Controller parameters.
parameter NTSM NFTSM SPNFTSM RBFSPNFTSM RBFDOSPNFTSM α 1 - 0.1 0.1 0.1 0.1 α 2 - 0.02 0.02 0.02 0.02 γ - 27/19 27/19 27/19 27/19 q p 21/19 21/19 21/19 21/19 21/19 β 10 - - - - w 1000 1000 - - - ρ 1 1 - - - k 3 - - 100 100 100 k 4 - - 1 1 0.1 c 2 - - 0.5 0.5 0.5 c 3 - - 1.5 1.5 1.5 k - - - - 0.291In order to better analyse the stability of the system, the following two performance evaluation indicators are adopted:(1) Integral of Error Squared Value (ISE) (56) I S E = ∫ 0 t x i 2 τ d τ(2) Integral of Absolute Value of Error (IAE) (57) Ι Α Ε = ∫ 0 t x i τ d τThe performance index values of the inverted pendulum system for the above two evaluation methods are shown in Table3. It can be seen from Table 3 that the performance evaluation index value of the state error e1 of the control method used in this paper is smaller than other control methods. Therefore, the system stability of RBFDOSPNFTSM is better than that of other control methods.Table 3
Performance index values of the inverted pendulum system.
ISE IAE e 1 e 1 NTSM 2.0317e-04 0.0229 NFTSM 1.6484e-04 0.0211 SPNFTSM 2.0334e-04 0.0154 RBFSPNFTSM 1.5577e-04 0.0062 RBFDOSPNFTSM 1.5375e-04 0.0050Figures3–6 show the response curves of the pendulum, swing speed, sliding mode, and system phase trajectory, respectively. It can be seen from them that the system state x1, x2 can converge to the equilibrium point at a faster speed under the control of the proposed scheme and can maintain a good stability state accuracy even if a large amount of interference is added. And the three-dimensional curve of the phase trajectory can intuitively show that the control process of the proposed method is smooth and jitter-free, and the system is more robust. Figure 7 shows the control input response curve. It can be seen from Figure 7 that the control method of the present invention controls the input signal to be smooth, which can reduce the wear of the motor. Figure 8 shows that the disturbance observer designed in this paper can estimate the interference value very well.Figure 3
Swing angle response curve.Figure 4
Swing speed response curve.Figure 5
Sliding modes response curve.Figure 6
System phase trajectory three-dimensional curve.Figure 7
Control input response curve.Figure 8
Disturbance observation result.


### 4.2. Signal Tracking Analysis after Adding Noise

If the initial states of the inverted pendulum system, system model parameters, controller parameters, and interference remain unchanged, consider the given signalxd=xd1+ηn(t) with the noise signal to analyze the tracking control performance of the system, where η=0.0001, n(t) is a random number between [-1, 1], and xd1 is shown as below:(58)xd1=0.5t-14πsin⁡2πt,0≤t<10.5,1≤t<2-0.5t+1.5+14πsin⁡2πt,2≤t<30,3≤t<4-0.5t+2+14πsin⁡2πt,4≤t<5-0.5,5≤t<60.5t-3.5-14πsin⁡2πt,6≤t<70,7≤t<8Figures9 and 10 show the swing angle tracking curve and the swing speed response curve of the inverted pendulum system with noise interference. It can be seen from Figures 9 and 10 that the control method of this paper has a poor suppression effect on noise and obvious chattering. Through the analysis of the control method of this paper, it is found that when the first and second derivatives are obtained for the given signal, a differential explosion phenomenon will be generated, and the noise signal in the given signal will be amplified. Therefore, the tracking differentiator needs to be designed to filter the given signal.Figure 9
Swing angle tracking curve (adding noise).Figure 10
Swing speed response curve (adding noise).The augmented nonlinear tracking differentiator (ATD) proposed in [34] is used to filter the given signal and suppress the influence of the noise signal on the control system. Its expression is as follows:(59)z˙0t=z1tz˙1t=z2tz˙2t=z3tz˙3t=-R4sigz0t-∫vτdτ;a0,b+sigz1tR;a1,b+sigz2tR2;a2,b+sigz3tR3;a3,bsigz;ai,b=ai1+e-bz-1-0.5wherea0>0, a1>0, a2>0, a3>0, b>0, R>0, and v(t) is an arbitrary bounded input signal. z1, z2, and z3 are estimated values, estimated values of first-order derivative, and estimated values of second-order derivative of v(t), respectively. Proof of stability has been given in [34]. The parameters of the augmented nonlinear tracking differentiator are shown in Table 4.Table 4
Parameters of the augmented nonlinear tracking differentiator.
a 0 a 1 a 2 a 3 b R 10 10 10 1 10 25Figures11–14 are the response curves of the swing angle, swing speed, sliding mode, and system phase trajectory of the inverted pendulum system with noise interference and filtered by the tracking differentiator. It can be seen from Figures 11–14 that the RBFDOSPNFTSM control method for filtering of the given signal by the augmented nonlinear tracking differentiator (ATDRBFDOSPNFTSM) has advantages in tracking accuracy, suppression of chattering, and the dynamic quality. Figure 15 shows the control input response curve. It can be seen from Figure 15 that the control method adding the ATDRBFDOSPNFTSM has basically no chattering phenomenon and has advantages in actual energy consumption and mechanical wear of operation.Figure 11
Swing angle tracking curve (adding noise).Figure 12
Swing speed response curve (adding noise).Figure 13
Sliding modes response curve (adding noise).Figure 14
System phase trajectory curve (adding noise).Figure 15
Control input response curve (adding noise).


## 4.1. Dynamic Performance Analysis

The initial state of the inverted pendulum system is set as[π/60,0], and the system model parameters and controller parameters are shown in Tables 1 and 2, respectively. Given signal xd=0, interference d(t)=sin⁡x1+5sin⁡(πt).Table 1
System model parameters.
the weight of the trolley (mc) the weight of the pendulum (mp) the length of the pendulum (l) the acceleration of gravity (g) 1kg 0.1kg 0.5m 9.8m/s2Table 2
Controller parameters.
parameter NTSM NFTSM SPNFTSM RBFSPNFTSM RBFDOSPNFTSM α 1 - 0.1 0.1 0.1 0.1 α 2 - 0.02 0.02 0.02 0.02 γ - 27/19 27/19 27/19 27/19 q p 21/19 21/19 21/19 21/19 21/19 β 10 - - - - w 1000 1000 - - - ρ 1 1 - - - k 3 - - 100 100 100 k 4 - - 1 1 0.1 c 2 - - 0.5 0.5 0.5 c 3 - - 1.5 1.5 1.5 k - - - - 0.291In order to better analyse the stability of the system, the following two performance evaluation indicators are adopted:(1) Integral of Error Squared Value (ISE) (56) I S E = ∫ 0 t x i 2 τ d τ(2) Integral of Absolute Value of Error (IAE) (57) Ι Α Ε = ∫ 0 t x i τ d τThe performance index values of the inverted pendulum system for the above two evaluation methods are shown in Table3. It can be seen from Table 3 that the performance evaluation index value of the state error e1 of the control method used in this paper is smaller than other control methods. Therefore, the system stability of RBFDOSPNFTSM is better than that of other control methods.Table 3
Performance index values of the inverted pendulum system.
ISE IAE e 1 e 1 NTSM 2.0317e-04 0.0229 NFTSM 1.6484e-04 0.0211 SPNFTSM 2.0334e-04 0.0154 RBFSPNFTSM 1.5577e-04 0.0062 RBFDOSPNFTSM 1.5375e-04 0.0050Figures3–6 show the response curves of the pendulum, swing speed, sliding mode, and system phase trajectory, respectively. It can be seen from them that the system state x1, x2 can converge to the equilibrium point at a faster speed under the control of the proposed scheme and can maintain a good stability state accuracy even if a large amount of interference is added. And the three-dimensional curve of the phase trajectory can intuitively show that the control process of the proposed method is smooth and jitter-free, and the system is more robust. Figure 7 shows the control input response curve. It can be seen from Figure 7 that the control method of the present invention controls the input signal to be smooth, which can reduce the wear of the motor. Figure 8 shows that the disturbance observer designed in this paper can estimate the interference value very well.Figure 3
Swing angle response curve.Figure 4
Swing speed response curve.Figure 5
Sliding modes response curve.Figure 6
System phase trajectory three-dimensional curve.Figure 7
Control input response curve.Figure 8
Disturbance observation result.


## 4.2. Signal Tracking Analysis after Adding Noise

If the initial states of the inverted pendulum system, system model parameters, controller parameters, and interference remain unchanged, consider the given signalxd=xd1+ηn(t) with the noise signal to analyze the tracking control performance of the system, where η=0.0001, n(t) is a random number between [-1, 1], and xd1 is shown as below:(58)xd1=0.5t-14πsin⁡2πt,0≤t<10.5,1≤t<2-0.5t+1.5+14πsin⁡2πt,2≤t<30,3≤t<4-0.5t+2+14πsin⁡2πt,4≤t<5-0.5,5≤t<60.5t-3.5-14πsin⁡2πt,6≤t<70,7≤t<8Figures9 and 10 show the swing angle tracking curve and the swing speed response curve of the inverted pendulum system with noise interference. It can be seen from Figures 9 and 10 that the control method of this paper has a poor suppression effect on noise and obvious chattering. Through the analysis of the control method of this paper, it is found that when the first and second derivatives are obtained for the given signal, a differential explosion phenomenon will be generated, and the noise signal in the given signal will be amplified. Therefore, the tracking differentiator needs to be designed to filter the given signal.Figure 9
Swing angle tracking curve (adding noise).Figure 10
Swing speed response curve (adding noise).The augmented nonlinear tracking differentiator (ATD) proposed in [34] is used to filter the given signal and suppress the influence of the noise signal on the control system. Its expression is as follows:(59)z˙0t=z1tz˙1t=z2tz˙2t=z3tz˙3t=-R4sigz0t-∫vτdτ;a0,b+sigz1tR;a1,b+sigz2tR2;a2,b+sigz3tR3;a3,bsigz;ai,b=ai1+e-bz-1-0.5wherea0>0, a1>0, a2>0, a3>0, b>0, R>0, and v(t) is an arbitrary bounded input signal. z1, z2, and z3 are estimated values, estimated values of first-order derivative, and estimated values of second-order derivative of v(t), respectively. Proof of stability has been given in [34]. The parameters of the augmented nonlinear tracking differentiator are shown in Table 4.Table 4
Parameters of the augmented nonlinear tracking differentiator.
a 0 a 1 a 2 a 3 b R 10 10 10 1 10 25Figures11–14 are the response curves of the swing angle, swing speed, sliding mode, and system phase trajectory of the inverted pendulum system with noise interference and filtered by the tracking differentiator. It can be seen from Figures 11–14 that the RBFDOSPNFTSM control method for filtering of the given signal by the augmented nonlinear tracking differentiator (ATDRBFDOSPNFTSM) has advantages in tracking accuracy, suppression of chattering, and the dynamic quality. Figure 15 shows the control input response curve. It can be seen from Figure 15 that the control method adding the ATDRBFDOSPNFTSM has basically no chattering phenomenon and has advantages in actual energy consumption and mechanical wear of operation.Figure 11
Swing angle tracking curve (adding noise).Figure 12
Swing speed response curve (adding noise).Figure 13
Sliding modes response curve (adding noise).Figure 14
System phase trajectory curve (adding noise).Figure 15
Control input response curve (adding noise).


## 5. Conclusions

This paper proposes a nonsingular fast terminal sliding mode control scheme combining RBF network and disturbance observer to solve the tracking control problem of uncertain nonlinear systems. Firstly, the sliding mode controller is designed by using nonsingular fast terminal sliding mode and second power reaching law to solve the problem of singularity and slow convergence of traditional terminal sliding mode control. Then, the unknown nonlinear function of the system is approximated by the RBF network, and the hyperbolic tangent disturbance observer is used to estimate the interference of the system and enhance the robustness of the system. Finally, the given signal is filtered by the augmented nonlinear tracking differentiator, eliminating the effects of noise in a given signal on the system. Through the numerical simulation results, the dynamic performance of the inverted pendulum system and the signal tracking of adding noise are analyzed. The control method can eliminate the noise pollution of a given signal, shorten the system arrival time, improve the tracking accuracy, and suppress the chattering phenomenon.



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*Source: 8146901-2019-05-30.xml*