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

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 functionofthesystemisapproximated,andthedisturbanceobserverisdesignedbyusingthehyperbolictangentnonlineartracking differentiator(TANH-NTD)toestimatetheinterferenceofthesystemandenhancetherobustnessofthesystem.Thestabilityof thesystemisprovedbytheLyapunovprinciple.Thenumericalsimulationresultsshowthatthemethodcanshortenthesystem arrivaltime,improvethetrackingaccuracy,andsuppressthechatteringphenomenon.


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][2][3][4][5][6][7][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][10][11][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][18][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][21][22][23][24][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.

Problem Description
Consider the following second-order uncertain nonlinear system: where  = [ 1 ,  2 ]  ∈  2 is the state variable; () and () are the unknown nonlinear functions;  ∈  is the control input;  ∈  is the system output; () is the slow time-varying interference, and |()| ≤ , constant  > 0.

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 Figure 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.
. . .Sliding Mode Design.  is set the given signal and is second derivable.The position and velocity tracking errors of the system are defined as  1 =  1 −   and  2 =  2 − ẋ  , respectively.The designed nonsingular fast terminal sliding mode is where  1 ,  2 are positive real numbers; ,  are odd integers, and 1 < / < 2,  > /. 1 and  2 do not have negative exponential terms, ensuring that the sliding mode  based controller does not have singularity.When the system error state reaches the sliding mode ( = 0), the following equation can be obtained: Substitute ė 1 =  2 into (3), the following equation can be obtained: Assume the time taken from  1 (  ) ̸ = 0 to  1 (  +   ) = 0 is   , and the following equation can be obtained through the time integrated on both sides of (4):

Simplification of Inequality ( )
Therefore, the system error can converge to zero for a limited time.
Remark .When the system error state is far from the equilibrium point, the dominant role of  1 in the sliding mode  causes the system trajectory to converge quickly.When the system error state approaches the equilibrium point, the dominant role of  1 | 1 |  sgn( 1 ) in the sliding mode  causes the system trajectory to converge rapidly.Therefore, NFTSM can achieve fast convergence of the whole state trajectory.
. . .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. where When the system state is far from the sliding mode (|| > 1), − 4 ||  3 sgn() plays a leading role; when the system approaches the sliding mode (|| < 1), − 3 ||  2 sgn() dominates.When  = 0, ṡ = 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.

Proof. Define the following Lyapunov function:
Find the time derivative for (10): Substitute ( 9) into the above equation: Deform ( 12): Because  2 is a normal number,  3 > 1; ,  are odd integers, and 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.

|𝑠| ≤ (
4 ) Similarly, equation ( 12) can be transformed into Because 16) can be further changed to (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.

|𝑠| ≤ ( 𝐷 𝑘
3 ) Combining equations ( 15) and (18), the system state  converges to the following areas within a limited time: Substituting the previous equation into (7), we can get Theorem 3 is proved.
. .RBF Network Approximation () and ().The nonlinear functions () and () 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.
. . .RBF Neural Network Structure.The RBF network is a 3layer 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 Figure 2.
In the RBF network, is the network input, and ℎ  is the Gaussian function, which represents the output of the th neuron in the hidden layer; namely, where is the center of the Gaussian function of the node ;   is the width of the Gaussian function of the node .The weight of the network is taken as . . .Adaptive Approximation of RBF Network.For the unknown nonlinear functions () and () in ( 9), the RBF network adaptive approximation is adopted.The RBF network input and output algorithm is where  is the network input; ℎ  is the Gaussian function;  * and  * are the ideal network weight vectors of () and (), respectively;   and   are the network approximation errors, |  | ≤   , |  | ≤   .ℎ  (), ℎ  () are vector functions, and the elements of them are all Gaussian function.We can set  = [ 1  2 ] , and then the output of the RBF network is Design control law is where  1 ,  2 ,  3 ,  4 are normal numbers, and 0 <  2 < 1,  3 > 1. ,  are odd integers, and 1 < / < 2,  > /.f() and b() are the RBF network output values.
Find the time derivative for (29) and substitute it into (27): Take the adaptive law as Substitute ( 31) into (30): Since   and   , the RBF network approximation errors, are very small real numbers,  is selected as  ≥ |  +  −|.According to Theorem 3, the system state  converges to the following region within a finite time: . .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  to reach the convergence region in a finite time is that the disturbance amount () in ( 1) is |()| ≤  and  is affected by the disturbance amount ().When () is larger,  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 system Σ 1 : If  1 ,  2 ,  1 ,  2 are all positive real numbers, the system Σ 1 is progressively stable at the origin (0, 0).It satisfies An improved disturbance observer is designed by using the hyperbolic tangent tracking differentiator proposed in [ ].
In the formula, ,  For ( ), the following control laws are used: where  1 ,  2 ,  3 ,  4 ,  are normal numbers, 0 <  2 < 1,  3 > 1.  and  are odd integers, and 1 < / < 2,  > /.f() and b() are the output values of the RBF network.d() is the output value of the hyperbolic tangent nonlinear disturbance observer.e system state  will converge to a smaller region for a limited time.
Take a derivative of function ( ) according to the Lyapunov, combining with ( ), ( ), and ( ): Setting   ≥ |  +    +  d() − |, according to eorem , the system state  converges to the following region within a finite time: Theorem 5.For the system shown in ( ), design the sliding mode according to ( ), using the NFTSM controller shown in ( ). e system state converges to the following areas in a limited time: Proof.According to Theorem 3 and (40), system state enters || ≤ min((  / 3 ) 1/ 2 , (  / 4 ) 1/ 3 ) region, and we can get When  1 − /| 1 |  sgn( 1 ) > 0, (47) satisfies the NFTSM form described in (2), and the system trajectory will converge to Therefore, state  1 can converge rapidly to the region within a finite time   : Theorem 5 is proved.
Remark .As the amount of disturbance () in (1) becomes larger,  and d() 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  can get lower chattering.

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 where .  1 and  2 are, respectively, the swing angle and the swing speed,  is the control input,   is the weight of the trolley,   is the weight of the pendulum,  is the length of the pendulum, and  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: (2) The sliding mode, the reaching law, and the controller of the NFTSM control method based on the exponential reaching law are as follows: (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: (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: (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:   . .Dynamic Performance Analysis.The initial state of the inverted pendulum system is set as [/60, 0], and the system model parameters and controller parameters are shown in Tables 1 and 2, respectively.Given signal   = 0, interference () = sin( 1 ) + 5 sin().
In order to better analyse the stability of the system, the following two performance evaluation indicators are adopted: The performance index values of the inverted pendulum system for the above two evaluation methods are shown in Table 3.It can be seen from Table 3 that the performance evaluation index value of the state error  1 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.
Figures 3-6 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  1 ,  2 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.
. .Signal Tracking Analysis a er Adding Noise.If the initial states of the inverted pendulum system, system model parameters, controller parameters, and interference remain unchanged, consider the given signal   =  1 + () with the noise signal to analyze the tracking control performance of the system, where  = 0.0001, () is a random number between [-1, 1], and  1 is shown as below: Figures 9 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.
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: where  0 > 0,  1 > 0,  2 > 0,  3 > 0,  > 0,  > 0, and V() is an arbitrary bounded input signal. 1 ,  2 , and  3 are estimated values, estimated values of first-order derivative, and estimated values of second-order derivative of V(), respectively.Proof of stability has been given in [34].The parameters of the augmented nonlinear tracking differentiator are shown in Table 4.   Figures 11-14 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.

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.

Figure 2 :
Figure 2: The RBF network with multiple inputs and single output.

Table 1 :
System model parameters.

Table 3 :
Performance index values of the inverted pendulum system.

Table 4 :
Parameters of the augmented nonlinear tracking differentiator.