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Since existing results about fixed-time stabilization are only applied to strict feedback systems, this paper investigates the nonsingular fixed-time stabilization of more general high-order nonlinear systems. Based on a novel concept named coordinate mapping of time domain, a control method is first proposed to transform the nonsingular fixed-time convergence problem into the finite-time convergence problem of a transformed time-varying system. By extending the existing, adding a power integrator technique into the considered time-varying system, a periodic controller is constructed to stabilize the original system in fixed time. The results of simulations verify the effectiveness of the proposed method.

In recent years, more and more attention has been paid to the controller design of high-order nonlinear systems due to its wide application in modeling aerospace craft, rigid robotic systems, and machine systems with underactuation, weak coupling, and instability [

For high-order nonlinear system, the finite-time stabilization is studied from different perspectives in [

The concept of fixed-time stabilization is proposed in [

Based on the above, the nonsingular global fixed-time stabilization of high-order nonlinear systems is proposed. The main difficulty lies in the design complexity caused by various power terms. Particularly, this issue would be intensified if we adopt traditional double-power-term law. The obstacle is partially avoided in this work by using the time-domain mapping, with which the nonsingular fixed-time stabilization problem of the original system is transformed into the finite-time stabilization problem of the corresponding time-varying system; by using the power integration method, the finite-time stabilization problem of the time-varying system is realized. The main innovations are summarized as follows:

A control method based on time-domain mapping is proposed, which transforms the nonsingular fixed-time stabilization problem of the original system into the finite-time stabilization problem of the corresponding time-varying system and provides a new idea for the design of the fixed-time convergence control law. Compared with the traditional double-power-reaching law, the proposed method is designed with a single-power-reaching law, which is more effective and simpler.

In essence, the fixed-time stabilization can be regarded as the optimal control with fixed terminal time, and the design of its control law is easy to produce singularity, while the control law of the proposed method is nonsingular.

The existing method can only solve the problem of fixed-time stabilization for strict feedback systems with only matching uncertainties, while the proposed method can solve the fixed-time stabilization problem of high-order nonlinear systems with unmatched uncertainties. It is noted that the strict feedback systems are special cases of high-order nonlinear systems; the results of this paper greatly extend the research scope of fixed-time stabilization.

For the convenience of description, we define

Consider the following high-order nonlinear systems:

There exists a function

For

To accurately describe the concept of fixed-time stability, we give the following definitions.

If any solution

The goal of the paper is to design a control input for system (

The Lyapunov function

For any positive real number

When

For any positive real number

In this section, the concept of time-domain mapping is proposed for the first time. The problem of nonsingular fixed-time stabilization of system (

Assume that the upper bound of the convergence time of system (

The inverse transformation of it is as follows:

When

For the converted time-varying system (

The coordinate mapping from time domain

In this section, a finite-time state feedback controller is designed in time domain

Extending the power integral method to the time-varying system (

Calculating derivation of equation (^{1} function, which can be defined as

According to formula (^{1} functions. Substituting equation (

Suppose that, in step

Define virtual control law and error as follows:^{1} functions. Besides,

As ^{1} functions, ^{1} function.

Designing ^{0} virtual control law ^{1} positive Lyapunov function (equation (

According to equations (

Defining that

Let us estimate each term on the right side of equation (

According to Lemma ^{1} function

For the convenience of narration, the following lemmas are given to estimate the residual terms on the right side of equation (

There exits ^{1} function

There exists ^{1} function

There exists ^{1} function

It is noticed that the time after transformation is only explicitly included in

According to ^{1} function, it can be concluded that the partial derivative about ^{1} function.

Assuming that when ^{1} function

It should be noticed that ^{1} function and ^{1} function.

When ^{1} function.

Substituting formulas (

The ^{0} virtual control law in step ^{1} function. Substituting formula (

According to the above derivation process, the following ^{0} state feedback controller can be designed in step ^{1} positive Lyapunov function

Considering that moment

In the time domain

From formula (

Transform the mapping in (

Consider that we have already made controller (

Then, any solution trajectory of system (

For the time set

The control law can be designed as follows:

The solution trajectory of system (

In conclusion, the fixed-time stabilization control law in time domain

The proposed control scheme provides a novel perspective of fixed-time stabilization. Compared with the traditional method composed of high-power and low-power terms [

To verify the effectiveness of the proposed control law, a practical example simulation is used to compare the proposed control method with recent literature.

To the best of our knowledge, existing fixed-time results consider at most so-called normal form systems [

In the above formula, all physical quantities are in Si basic units, and the values of parameters in the simulation process are selected as

For the fixed-time controller (

Considering the value range of

Response of system (

Response of system (

It can be seen from Figures

The problem of fixed-time stabilization for high-order nonlinear systems is studied in the paper. A control method based on time-domain mapping is firstly introduced. Compared with the existing literature, the paper proposes a new idea to realize fixed-time stabilization based on time-domain mapping, which greatly expands the research scope of fixed-time stabilization. Note that all the states should be accessible in this work; an interesting problem is the observer design in the case of partially unknown state, which will be considered in our further work.

The data used to support the findings of this study are included within the article. The original data ca be obtained from the corresponding author upon request.

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

The authors would like to acknowledge the Major Science and Technology Projects in Anhui Province (no. 18030901058).