Global Stabilization of High-Order Time-Delay Nonlinear Systems under a Weaker Condition

and Applied Analysis 3 3. State Feedback Controller Design 3.1. Assumption. The following assumption is imposed on system (1) in this paper. Assumption 6. For i = 1, . . . , n, there are constants a > 0 and τ > −1/∑ n l=1 p 1 ⋅ ⋅ ⋅ p l−1 such that 󵄨 󵄨 󵄨 󵄨 f i (x i ( t) , x1 (t − d 1 ( t)) , . . . , xi (t − d i ( t))) 󵄨 󵄨 󵄨 󵄨


Introduction
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 [1].Therefore, the control design and stability analysis of time-delay systems has been an active research area within the automation and control community.In recent years, by employing the Lyapunov-Krasovskii method or Lyapunov-Razumikhin method to deal with the time delay, control theory, and techniques for time-delay linear systems were greatly developed and many advanced methods have been made; see, for instance, [2][3][4][5][6][7][8][9] and reference therein.However, due to no unified method being applicable to nonlinear control design, many important and interesting control problems for timedelay nonlinear systems remain unsolved.
System (1) represents an important class of systems which can model many frequently met practical systems, such as the underactuated, weakly coupled, unstable mechanical system, and the cascade chemical system.However, the global stabilization of (1) has been widely recognized as difficulty because its Jacobian linearization, being neither controllable nor feedback linearizable, leads to the traditional design tools hardly applicable to such system.Mainly, thanks to the method of adding a power integrator, when   () = 0, the 2 Abstract and Applied Analysis state feedback stabilization of (1) has been well studied and a number of interesting results have been achieved over the last decades; for example, one can see [10][11][12][13][14][15][16][17][18] and the references therein.
However, when   () ̸ = 0, the global stabilization of ( 1) is much more challenging because trade-off of time-delay effect and identification of time-delay restriction.In this regard, some results were also reported.In [19], high-order time-delay nonlinear systems were first investigated, and, by imposed some restriction on the system growth, a continuous state feedback controller was given.Later, the authors in [20][21][22][23] further relaxed conditions placed on the system growth and addressed state or output feedback stabilization problem for high-order nonlinear systems with multiple time delays, respectively.
Motivated by the continuous control ideas in [11,18], this paper continues the investigations in [19][20][21][22][23] and further addresses the stabilizing control design for high-order timedelay nonlinear systems (1).The main contributions of this paper are twofolds.(i) By comparison with the existing results in [19][20][21]23], the nonlinear growth condition is largely relaxed and a much weaker sufficient condition is given.(ii) By successfully overcoming some essential difficulties such as the weaker assumption on the system growth and the construction of a  1 , proper, and positive definite Lyapunov function, a new method to global stabilization of high-order time-delay nonlinear systems by state feedback is given and leads to much more general results than the previous ones.
The remainder of this paper is organized as follows.Section 2 presents some necessary notations, definition, and preliminary results.Section 3 formulates the control problem, presents the design scheme to the controller, and gives the main contributions of this paper.Section 4 gives a simulation example to demonstrate the effectiveness of the theoretical results.Section 5 addresses some concluding remarks.The paper ends with an appendix.

Mathematical Preliminaries
The following notations, definition, and lemmas will be used throughout the paper.
Notations. + denotes the set of all nonnegative real numbers and   denotes the real -dimensional space. + odd := {/ |  and  are positive odd integers} and  ≥1 odd := {/ |  and  are positive odd integers, and  ≥ }.For a given vector ,   denotes its transpose, and || denotes its Euclidean norm.  denotes the set of all functions with continuous th partial derivatives.K denotes the set of all functions:  + →  + , which are continuous, strictly increasing, and vanishing at zero; K ∞ denotes the set of all functions which are of class K and unbounded.For any  ∈  + and  ∈ , the function []  is defined as []  = sgn()||  .Besides, let ∑ 1 =1  1  0 = 1 and the arguments of the functions (or the functionals) will be omitted or simplified, whenever no confusion can arise from the context.For instance, we sometimes denote a function (()) by simply (), (⋅), or .Definition 1 (see [24]).Weighted homogeneity: for fixed coordinates ( 1 , . . .,   ) ∈   and real numbers   > 0,  = 1, . . ., , consider the following.
For simplicity, it is assumed that  = −/ with  being any even integer and  being any odd integer, under which and the definition of   in Assumption 6, we know that   ∈  + odd .
When  ∈ [0, +∞), it is equivalent to those in [20,21,23].Moreover, it is worth pointing out this assumption cannot be covered by Assumption 1 in [22], which can be represented as       (  () , where  ∈ [0, +∞) is a constant.For example, the simple system ẋ 1 =  2 +  3/5 1 ( − ), ẋ 2 =  cannot be globally stabilized using the design method presented in [22] because of the presence of low-order term  3/5 1 ( − ), but it is easy to verify that Assumption 6 in this paper is satisfied with  = 1 and  = −2/5 ∈ (−1/2, +∞).This means that, to some extent, the system studied in this paper is less restrictive and allows for a much broader class of systems.
The objective of this paper is to design a state feedback controller for system (1) under Assumption 6 such that the closed-loop system is globally asymptotically stable.
Then, under the new coordinates   's, system (1) is transformed into Remark 8. We need to emphasize that the gain  creates an extra freedom in control design.As a matter of fact, in the proof of Theorem 10, complex uncertainties will inevitably be produced in the amplification of nonlinearities.Hence, the gain  can be used to effectively dominate all the possible uncertainties.

State Feedback Controller Design for Nominal Nonlinear
System.We first construct a state feedback controller for the nominal nonlinear system of (8): Step 1.
, where  ≥ max 1≤≤ {1,  +   } is a positive number, and choose the Lyapunov function ) with  * 1 = 0. From (9), it follows that where the virtual controller is chosen as Step i ( = 2, . . ., ).In this step, we can obtain the following property, whose proof is given in the appendix.

Stability Analysis.
We state the main results in this paper.Theorem 10.For the time-delay nonlinear system (1) under Assumption 6, the state feedback controller    =    +1    in (7) and (17) renders that the equilibrium at the origin of the closed-loop system is globally asymptotically stable.
Proof.We prove Theorem 10 by four steps.
Step 1.We first prove that    preserves the equilibrium at the origin.
From (17) and  +1   =   + , we have By which and the definitions of   's and , we easily see that    =    +1    is a continuous function of  and    () = 0 for  = 0.This together with Assumption 6 implies that the solutions of -system is defined on a time interval [−,   ), where   > 0 may be a finite constant or +∞, and    preserves the equilibrium at the origin.
Step 2. We construct a Lyapunov-Krasovskii functional where  is a positive parameter to be determined later.
Because   (()) is  1 , positive definite radially unbounded and by Lemma 4.3 in [27], there exist two class K ∞ functions  1 and  21 such that According to the homogeneous theory, there are positive constants  and  such that where (()) is a positive definite function, whose homogeneous degree is 2.Therefore, the following inequality holds: with two class K ∞ functions   Defining  2 =  21 +  22 , from ( 21), (22), and (25), it follows that Step 3.Because   () and () are homogeneous of degree 2 −  and  with respect to Δ, by Lemmas 2 and 3, there is a constant  1 such that By (7), Assumption 6 and  > 1, we can find constants   > 0 and 0 < ]  ≤ 1 such that since it can be seen that by definition   =   +1/( 1 ⋅ ⋅ ⋅  −1 ), so Noting that for  = 1, .

Extension.
In this subsection, we can extend the results developed above to high-order time-delay nonlinear system in nontriangular form: under the following assumption.
It is obvious that Assumption 6 is a special case of Assumption 11, and, under this, some more general results are given.Theorem 12.For the time-delay nonlinear system (35) under Assumption 11, the state feedback controller    =    +1    in (7) and (17) renders that the equilibrium at the origin of the closed-loop system is globally asymptotically stable.
Proof.Similar to (28), Assumption 11 will directly lead to (30).The rest of the proof is similar to that of Theorem 10 and hence is omitted here.
It is worth pointing out that system (37) cannot be globally stabilized even by state feedback, using the design methods presented in [19][20][21][22][23] because of the presence of low-order term (1/4) ) satisfy Assumption 6 with  = 1/4.Moreover, noting that 0 ≤  1 () ≤ 1/5 and ḋ1 () = 1/5 cos  ≤ 1/5 < 1, the controller proposed in this paper is applicable.Thus, in terms of the design steps developed in Section 3, a continuous controller of system (37) can be given to ensure that the closed-loop system meets the conclusions of Theorem 10.
Let  = 1 and the initial states be  1 () = 0.5,  2 () = −0.3, and  ∈ [−0.5, 0].Using MATLAB, Figures 1 and 2 are obtained to exhibit the the trajectories of the closed-loop system states and the control input.From these figures, it can be seen that  1 and  2 are asymptotically regulated to zero, which demonstrates the effectiveness of the control method proposed in this paper.

Conclusion
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 [28][29][30][31][32][33][34][35][36], and so forth; an important problem is whether the results in this paper can be extended to stochastic high-order nonlinear systems.

Figure 1 :Figure 2 :
Figure 1: The trajectories of system states.