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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.

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 [

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 [

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.

Consider the following second-order uncertain nonlinear system:

where

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

Structure diagram of the system controller.

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.

where

Substitute

Assume the time taken from

Therefore, the system error can converge to zero for a limited time.

When the system error state is far from the equilibrium point, the dominant role of

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

Set

(1)

(2)

(3) Real number

For the system shown in (

Sliding mode controller is designed as

where

Define the following Lyapunov function:

Find the time derivative for (

Substitute (

Deform (

Because

Also, because

Similarly, equation (

Because

Also, because

Combining equations (

Theorem

The nonlinear functions

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 Figure

The RBF network with multiple inputs and single output.

In the RBF network,

where

Output of RBF network is

For the unknown nonlinear functions

where

Design control law is

where

Find the time derivative for (

Substitute (

where

Design the Lyapunov function as

where

Find the time derivative for (

Take the adaptive law as

Substitute (

Since

For system (

Consider the following system

If

An improved disturbance observer is designed by using the hyperbolic tangent tracking differentiator proposed in [

In the formula,

When

For (

where

Take a derivative of function (

Setting

For the system shown in (

According to Theorem

Equation (

When

Therefore, state

Using the same approach, (

When

Therefore, state

Theorem

As the amount of disturbance

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

The performance of system (

(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:

The initial state of the inverted pendulum system is set as

System model parameters.

the weight of the trolley ( | the weight of the pendulum ( | the length of the pendulum ( | the acceleration of gravity ( |
---|---|---|---|

1kg | 0.1kg | 0.5m | 9.8 |

Controller parameters.

parameter | NTSM | NFTSM | SPNFTSM | RBFSPNFTSM | RBFDOSPNFTSM |
---|---|---|---|---|---|

| - | 0.1 | 0.1 | 0.1 | 0.1 |

| - | 0.02 | 0.02 | 0.02 | 0.02 |

| - | 27/19 | 27/19 | 27/19 | 27/19 |

| 21/19 | 21/19 | 21/19 | 21/19 | 21/19 |

| 10 | - | - | - | - |

| 1000 | 1000 | - | - | - |

| 1 | 1 | - | - | - |

| - | - | 100 | 100 | 100 |

| - | - | 1 | 1 | 0.1 |

| - | - | 0.5 | 0.5 | 0.5 |

| - | - | 1.5 | 1.5 | 1.5 |

| - | - | - | - | 0.291 |

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

Performance index values of the inverted pendulum system.

ISE | IAE | |
---|---|---|

| | |

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 | 0.0050 |

Figures

Swing angle response curve.

Swing speed response curve.

Sliding mode

System phase trajectory three-dimensional curve.

Control input response curve.

Disturbance observation result.

If the initial states of the inverted pendulum system, system model parameters, controller parameters, and interference remain unchanged, consider the given signal

Figures

Swing angle tracking curve (adding noise).

Swing speed response curve (adding noise).

The augmented nonlinear tracking differentiator (ATD) proposed in [

where

Parameters of the augmented nonlinear tracking differentiator.

| | | | | |
---|---|---|---|---|---|

10 | 10 | 10 | 1 | 10 | 25 |

Figures

Swing angle tracking curve (adding noise).

Swing speed response curve (adding noise).

Sliding mode

System phase trajectory curve (adding noise).

Control input response curve (adding noise).

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.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.