Global Practical Output Tracking of Inherently Nonlinear Systems Using Continuously Differentiable Controllers

This paper considers the global practical output tracking problem by at least continuously differentiable (C) state feedback for a class of uncertain nonlinear systems whose linearization around the origin may contain uncontrollable modes. Based on utilizing the homogeneous domination approach, we not only propose conditions of constructing a global continuously differentiable (C) controller, but also provide explicit design schemes for such systems. Finally, a numerical example demonstrates the effectiveness of the result.


Introduction
The problem of global output tracking control of nonlinear systems is one of the most important and challenging problems in the field of nonlinear control.One of recent focuses in the nonlinear control research is the global practical output tracking problem for a class of inherently nonlinear systems, described by the following equations: where  = ( 1 , . . .,   ) T ∈   and  ∈  are the system state and the control input, respectively.For  = 1, . . ., ,   (, , ) are unknown continuous ( 0 ) nonlinear functions of the states and the control input and the power   ∈  + odd ( = 1, . . .,  − 1) is a positive odd integer or a positive ratio of odd integers with   := 1.
The uncertain system (1) represents a general class of nonlinear systems considered in the nonlinear control literature.When   = 1,  = 1, . . ., , system (1) reduces to the well-known feedback linearizable form, for which numerous design methodologies are developed; see [1][2][3][4][5] and the references therein.For the case that any one of the powers   ( = 1, 2, . . ., ) is greater than 1, system (1) is known as the power integrator system whose Jacobian linearization is uncontrollable.In recent years, the problem of global practical output tracking control of the power integrator systems in form (1) has been studied extensively with various restrictions on the integrator powers and the additive functions   (, , )'s, which directly influence the availability of smooth or nonsmooth controllers; see [6][7][8][9][10] and the references therein.For details, in [6], practical output tracking via state feedback for high-order (  ≥ 1,  = 1, . . ., ) nonlinear systems was considered.Further, in [8,9], the practical output feedback tracking problem was also investigated for a class of nonlinear systems with higherorder growing unmeasurable states, extending the results on stabilization in [11,12].
For the more general nonlinear systems with arbitrary   (> 0)'s, existing results toward the global output tracking problem for system (1) can be found in the literature.The global stabilization problem of system (1) for   > 0 (not restricted to be larger than or equal to one) has been studied for nonlinear systems in [13,14].In [13], a continuous controller under a certain nonlinear growth condition is studied.
The techniques from [11,14] were recently extended in [10,15] to the practical output tracking problem for nonlinear systems (1) by a continuous state feedback controller.However, from the practical point of view, the smoothness of the controllers is always desired because controllers at least  1 avoid the infinity controller gains around the origin and guarantee the uniqueness of the solution [16,17].Initial efforts were made in [18] to obtain  1 or smooth controllers by upgrading the unified homogeneous degree to a set of decreasing homogeneous degrees to solve the global stabilization problem of system (1) for   > 0. It was shown that these monotone degrees gave us much flexibility in the controller design, which will lead to some nicer features for the controlled system.
In this paper, we will further generalize the results in [18] to solve the practical output tracking problem.This work will develop a detailed recursive design method which constructs a series of integral Lyapunov functions as well as the explicit formula of the continuously differentiable controllers.
Throughout this study we use the following notations.
Notations. + denotes the set of all the nonnegative real numbers and   denotes the real -dimensional space.A function  :   →  is said to be   -function, if its partial derivatives exist and are continuous up to order , 1 ≤  < ∞.A  0 function means it is continuous.A  ∞ function means it is smooth; that is, it has continuous partial derivatives of any order.The arguments of functions (or functionals) are sometimes omitted or simplified; for instance, we sometimes denote a function (()) by (), (⋅), or .

Problem Statement and Preliminaries
The purpose of this paper is to solve the problem of global practical output tracking by state feedback.Let   () be a time-varying  1 -bounded on  ∈ [0, +∞) reference signal and, for any given tolerance  > 0, design a continuously differentiable state feedback controller of the form such that (i) all the states of the closed-loop system (1)-( 2) are welldefined on  ∈ [0, +∞) and globally bounded; (ii) for any initial condition (0) ∈   there is a finite time  > 0, such that      () −   ()     =      1 () −   ()     < , ∀ ≥  > 0. ( To construct a global practical output tracking  1 controller for nonlinear system (1), we introduce the following assumptions.
where   ( 1 , . . .,   ) > 0 are smooth functions and   's are defined as (ii) the   's defined by (i) satisfy the following condition: This section cites some definitions and technical lemmas which are used in the main body of this investigation.
Next, we will present several useful lemmas borrowed from [4,13,14,19], which will play an important role in our later controller design.Lemma 3.For all ,  ∈  and a constant  ≥ 1, the following inequalities hold: Lemma 5.For any positive real numbers ,  and  ≥ 1, the following inequality holds: Lemma 6.Let  1 , . . .,   ,  > 0 be real numbers.Then, the following inequality holds:

Continuously Differentiable State Feedback Controller Design
In this section, we will construct a continuously differentiable state feedback tracking controller which is addressed in a step-by-step manner for system (1).
Theorem 7.Under Assumptions 1-2, the global practical output tracking problem of system ( 1) can be solved by a continuously differentiable ( 1 ) state feedback controller of form (2).
such that Mathematical Problems in Engineering We claim that ( 18) also holds at step k.To prove this claim, consider the Lyapunov function . ( The function   ( 1 ,  2 , . . .,   ) can be shown to be  1 , proper, and positive definite with the following property: for  = 1, . . .,  − 1, and there is a known constant  > 0 such that Proofs of these properties proceed just in the same way as in the proofs for [20, Propositions 1 and 2] and [21], where the set of positive odd integers is considered instead of  odd which is used in this paper.With these properties, we obtain for a virtual controller  *   +1 to be determined later.In order to proceed further, a bounding estimate for each term in the right hand side of ( 22) is needed.The terms in (22) can be estimated using Propositions A.1-A.3 in the appendix.

Substituting the results of Propositions
with the where  = (2 −   )/2.Therefore, Inequality (31) will show that the state () of closed-loop system ( 11)-( 27) is well-defined on [0, +∞) and globally bounded.To prove this, first introduce the following set: and let () be the trajectory of ( 11) with an initial state (0).
Next, it will be shown that Therefore, for any  > 0, there is globally practical output tracking such that (36) holds.This completes the proof of Theorem 7.

An Illustrative Example
In this section, we give a simple numerical example to illustrate the correctness and effectiveness of the theoretical results by considering the following nonlinear system: where  1 = 7/3,  2 = 1 and  1 (),  2 () represent an unknown bounded time-varying function and parameter, respectively.Our objective is to design a practical continuously differentiable ( 1 ) output tracking controller such that the output of system (38) tracks a desired reference signal   , and all the states of system (38) are globally bounded.
(ii) When parameter  is increased to  = 0.0000225, then the tracking error reduces to about 0.025 as shown in Figures 2(a), 2(b), and 2(c).

Conclusion
This paper has developed a systematic approach to construct a continuously differentiable ( 1 ) practical output tracking controller for a class of inherently nonlinear systems, whose chained integrator part has the power of positive odd rational numbers.Such a controller guarantees that the states of the closed-loop system are globally bounded, while the tracking error can be bounded by any given positive number after a finite time.Further, a simple numerical example was performed to illustrate the effectiveness of the result obtained.

Appendix
Proposition A.