Practical output tracking for a class of uncertain nonlinear time-delay systems via state feedback

In this paper, the problem of global practical output tracking is investigated by state feedback for a class of uncertain nonlinear time-delay systems. Under mild conditions on the system nonlinearities involving time delay, we construct a homogeneous state feedback controller with an adjustable scaling gain. By a homogeneous Lyapunov-Krasovskii functional, the scaling gain is adjusted to dominate the time-delay nonlinearities bounded by homogeneous growth conditions and render the tracking error can be made arbitrarily small while all the states of the closed-loop system remain to be bounded.


Introduction
Consider the following uncertain nonlinear time-delay system  ( ( ), , )) : x t x t R ,  u R , and ( )  y t R are the system state, control input and output, respectively.The constant 0  d is a given time-delay of the system, for 1,  i …, , n and the system initial condition is p and q are odd integers,  p q } ( 1, , 1)    i n are said to be the high orders of the system.
Global practical output tracking problem of nonlinear systems is one of the most important and challenging problems in the field of nonlinear control and has received a great deal of attention.By posed some conditions on system growth and power order, the practical output tracking problem of system (1) has been well-studied and a number of interesting results have been achieved over the past years, see [1][2][3][4][5][6][7][8], as well as the references therein.
However, the aforementioned results have not considered the time-delay effect.It is well known that time-delay phenomena exist in many practical systems such as electrical networks, microwave oscillator, and hydraulic systems, etc., due to the presence of time delay in systems, it often significant effect on system performance.Therefore, the study the problem of output tracking and stabilization of time-delay nonlinear systems has important practical significance and has received much attention in recent years.In recent years, by employing the Lyapunov-Krasovskii method to deal with the time-delay, control theory, and techniques for stabilization problem of time-delay nonlinear systems were greatly developed and advanced methods have been made; see, for instance, [9 -13] and reference therein.Compared with study the stabilization problem contain time-delay, the theory of output tracking control developed slower.In the case when the nonlinearities contain timedelay, for the output tracking problems, some interesting results have been obtained [14][15][16].However, in [14 -16] only considered special case for the system (1), i.e., 1  i p case.When the system under consideration is inherently time-delay non-linear, the problem becomes more complicated and difficult to solve.To the best of our knowlege, many interesting output tracking control problems for time delay inherently nonlinear systems unsolved yet.In this paper, we deal with such as the tracking problems via state feedback domination method in [17,18].

Mathematical preliminaries
We collect the definition of homogeneous function and several useful lemmas.
Definition1 ([19]).For a set of coordinates   For the simplicity, write Next, we introduce several technical lemmas which will play an important role and be frequently used in the later control design. Lemma1

 
 with respect to the same dilation Δ.
Lemma2 [19].Suppose : n V R R  is a homogeneous function of degree  with respect to the dilation weight  .Then, the following (i) and (ii) hold: (ii) There is a constant 0 Lemma3 [17].For all , x y R  and a constant 1 p  the following inequalities hold: Lemma4 [18].Let , c d be positive constants.Then, for any real-valued function ( , ) 0 x y   , the following inequality holds: This paper deals with the practical output tracking problem by state feedback for timedelay high-order nonlinear systems (1).Here, we give a precise definition of the problem.
The problem of global practical tracking by a state feedback: Consider system (1) and assume that the reference signal ( ) r y t is a time-varying 1  C -bounded function on [0, ). For any given 0   , design a state feedback controller having the following structure ( ) ( ( ), ( )),  r u t g x t y t (2) such that (i) All the state of the closed-loop system (1) with state controller (2) is well-defined and globally bounded on [0, )  .
(ii) For any initial condition, there is a finite time In order to solve the global practical output tracking problem, we made the following two assumptions: where Assumption2.The reference signal ( )

State feedback tracking control design
In this paper, we deals with the practical output tracking problem by delay-independent state feedback for high-order time-delay nonlinear systems (1) under Assumptions 1-2.To this end, we first introduce the following coordinate transformation: where 1 and 1 L  is a scaling gain to be determined later.Then, the system (1) can be described in the new coordinates i z as 1 1 ( , ( ), ( ), ), , where Now, using Assumption 1, Lemma 3, the fact that where In what follows, we will employ the homogeneous domination approach to construct a global state feedback controller for system (7).

Stability Analysis
First, we construct a homogeneous state feedback controller for the nominal nonlinear system without considering the non-linearity of ( ) Using SIMILAR the approach in [11,[17][18], we can design a homogeneous state feedback stabilizer for (8), which can be described in the following Theorem1.
Theorem1.For a real given number 0 


, there is a homogeneous state feedback controller of degree  such that the nonlinear systems (10) is globally asymptotically stable.
Proof.To prove the result, we use an inductive argument (recursive design method) to explicitly construct a homogeneous stabilizer for system (10).
, where 1 0 z   and From (10) where 2 z  the virtual controller and it is chosen as Step ( 2, , )   k k n .Suppose at the step k-1, there is a 1 C , positive definite and proper Lyapunov function We claim that ( 15) also holds at Step k, i.e., there is a 1 C , proper, positive definite Lyapunov function defined by (16) and virtual controller Since the prove of the claim ( 17) is very similar [4][5]14], so omitted here.Using the inductive argument above, we can conclude that at the -th step, there exists a state feedback controller of the form with the 1 C , proper and positive definite Lyapunov function, we arrive at  are positive constants.Thus, the closed-loop system (10) and ( 18) is globally asymptotically stable.

Tracking control design for the time-delay nonlinear system (1)
Now, we are ready to use the homogeneous domination approach to design a global tracking controller for the system (1), i.e., state the following main result in this paper.
Theorem 2. For the time-delay nonlinear system (1) under Assumptions 1-2, the global practical output tracking problem is solvable by the state feedback controller 7) and (18) Proof.From (18), we have .
Now, we define the compact notations Using the same notation ( 7) and ( 22), the closed-loop system ( 7) -( 18) can be written as the following compact form: ( ) where 1 0  m is constant.
From (35) it is not difficult to show that there is a finite time 0  T , such that from which it is clear that 1 z can be rendered smaller than any positive tolerance with a sufficiently large L .

Conclusion
In this paper, we have studied the practical output tracking problem for a class of uncertain nonlinear time-delay systems under a homogeneous condition.First, we design a homogeneous state feedback controllers have been constructed with adjustable scaling gains.Then, with the help of a homogeneous Lyapunov-Krasovskii functional, we've redesigned the homogeneous domination approach to tune the scaling gain for the overall the closed loop systems.It is shown that an appropriate choice of gain will enable us to globally track for a class of uncertain non-linear systems in finite time. ,

[19]. Denote
2 ( ) ( )V x V x is also homogeneous function with degree of 1 2  are homogeneous of degree 2    and 2 with respect to  , respectively.Therefore, by Lemma2, there are positive constants1 2, 