Under the weaker condition on the system growth, this paper further investigates the problem of global stabilization by state feedback for a class of high-order nonlinear systems with time-varying delays. By skillfully using the homogeneous domination approach, a continuous state feedback controller is successfully designed, which preserves the equilibrium at the origin and guarantees the global asymptotic stability of the resulting closed-loop system. A simulation example is given to demonstrate the effectiveness of the proposed design procedure.
Time-delay phenomena exist in many practical systems such as electrical networks, microwave oscillator, and hydraulic systems. It is well known that the existence of time delay often deteriorates the control performance of systems and even causes the instability of closed-loop systems [
In this paper, we consider a class of high-order time-delay nonlinear systems described by
System (
However, when
Motivated by the continuous control ideas in [
The remainder of this paper is organized as follows. Section
The following notations, definition, and lemmas will be used throughout the paper.
Weighted homogeneity: for fixed coordinates The dilation A function A vector field A homogeneous
Given a dilation weight
Suppose
there is a constant
For
Let
The following assumption is imposed on system (
For
For simplicity, it is assumed that
Assumption
The objective of this paper is to design a state feedback controller for system (
To this end, we first introduce the following coordinate transformation:
Then, under the new coordinates
We need to emphasize that the gain
We first construct a state feedback controller for the nominal nonlinear system of (
Let
In this step, we can obtain the following property, whose proof is given in the appendix.
Assume that at step
Hence at step
We state the main results in this paper.
For the time-delay nonlinear system (
We prove Theorem
From (
According to the homogeneous theory, there are positive constants
With the help of
By (
Noting that for
According to (
By (
In this subsection, we can extend the results developed above to high-order time-delay nonlinear system in nontriangular form:
For
It is obvious that Assumption
For the time-delay nonlinear system (
Similar to (
To illustrate the effectiveness of the proposed controller, we consider the following low-dimensional system:
It is worth pointing out that system (
Let
The trajectories of system states.
The trajectory of control input.
In this paper, a state feedback stabilization controller independent of time-delays is presented for a class of high-order nonlinear systems with time-varying delays under a weaker condition. The controller designed preserves the equilibrium at the origin and guarantees the globally asymptotic stability of the system. It should be noted that the proposed controller can only work well when the whole state vector is measurable. Therefore, a natural and more interesting problem is how to design output feedback stabilization controller for the systems studied in the paper if only partial state vector is measurable. In addition, in recent years, many results on stochastic nonlinear systems have been achieved [
We first prove that
Next, we prove that
Similarly, it can be shown that (
At last, we prove inequality (
Noting that
With the help of (
Choosing
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors thank the editor and the anonymous reviewers for their constructive comments and suggestions for improving the quality of the paper. This work is partially supported by National Nature Science Foundation of China under Grant 61073065 and the Key Program of Science Technology Research of Education Department of Henan Province under Grants 13A120016, 14A520003.