#### Abstract

This paper studies the problem of global output feedback stabilization for a class of nonlinear systems with a time-varying power and unknown output function. For nonlinear systems with a time-varying power and unknown continuous output function, by constructing a new nonlinear reduced-order observer together with adding a power integrator method, a new function to determine the maximal open sector of output function is given. As long as output function belongs to any closed sector included in , it is shown that the equilibrium point of the closed-loop system can be guaranteed globally uniformly asymptotically stable by an output feedback controller.

#### 1. Introduction

Consider nonlinear systems with the unknown output functionwhere , , and are the unmeasurable state, control input, and output, respectively. For a real constant , is defined as . The time-varying power is a continuous bounded function satisfying with two constants and . For , are continuous in the first argument and locally Lipschitz with respect to the rest variables with . Output function is an unknown continuous function with .

Over the past decade, with the help of adding a power integrator method, homogeneous domination method, and recursive observer design, there exist some interesting results on state/output feedback design of high-order nonlinear systems, whose powers are known constant ratios of odd integers; see [1â€“16] and the references therein.

In recent years, some interesting results have been achieved on output feedback design of nonlinear systems with known constant powers and unknown output function. For the nonlinear systems (1) with , when is a continuous differentiable function and its derivative with known upper and lower bounds, global output feedback stabilization and finite-time output feedback stabilization have been achieved in [17â€“19] and [20], respectively. When the derivative of is with unknown upper bound, [21] achieved semi-global output feedback control. Furthermore, when is extended to be only a continuous function, with the help of the given maximal sector region of output function, [22] achieved output feedback stabilization. Lately, in [23], a new design and analysis method for high-order nonlinear systems with unknown continuous output function was proposed based on adding a power integrator method and homogeneous domination method.

According to some practice [24â€“26], it is well-known that the timely deteriorated performance of system often results in different running data, which usually identify the powers of system. Therefore, the powers of system are usually not fixed and can be varying with a suitable bound even in the same working condition. For example, as a practical second-order dynamic model of reduced-order boiler-turbine unit, two typical different powers and have been identified in [25] and [26], respectively. For nonlinear systems with a time-varying power, [27, 28] achieved global state feedback stabilization based on interval homogeneous domination approach. As far as we know, [29] is the first paper to study the output feedback stabilization of nonlinear systems (1) with by the revamped method of adding a power integrator together with the recursive nonlinear observer design. Naturally, an interesting problem is put forward:* For more general nonlinear systems (1) with ** being an unknown continuous function, can we design an output feedback controller?*

In this paper, we make an attempt to handle this problem. Some essential technical difficulties in control design will be inevitably produced: (i) Compared with [29], since output function is unknown, we construct a new nonlinear reduced-order observer without using the unmeasurable state . (ii) Compared with [23], a new function to determine the maximal open sector of output function is given since the power of system is time-varying. As long as output function belongs to any closed sector included in , the equilibrium point of the closed-loop system is globally uniformly asymptotically stable under the constructed output feedback controller.

This paper is organized as follows. Section 2 gives some preliminaries. The design and analysis of output feedback controller are given in Section 3, following a simulation example in Section 4. Section 5 concludes this paper.

#### 2. Mathematical Preliminaries

Some notations, definitions, and lemmas are to be used throughout this paper.

In this paper, the argument of function will be omitted whenever no confusion can arise from the context. , , and denote the set of real numbers, the set of all nonnegative real numbers, and the real -dimensional space. For constant , let , and , , . For integers and constant , when .

*Definition 1 (see [30]). *A function is said to belong to the sector if , where and are constants with . If the inequality is strict, we write the sector as .

The following lemmas will serve as the basis for the development of output feedback controller. Lemmas 2â€“5 are used to enlarge inequalities. Lemmas 6 and 7 are Lyapunov stability theorem for the global uniformly asymptotically stable of the closed-loop system.

Lemma 2 (see [31]). *Let be a real-valued function of satisfying . For any , ,*

Lemma 3 (see [32]). *Let be positive real-valued functions of and be a positive real-valued function of . For any ,*

Lemma 4 (see [33]). *Let be a real-valued function of satisfying . For any ,where if and .*

Lemma 5 (see [33]). *Let be a real-valued function of satisfying . For any ,*

Lemma 6 (see [30]). *Let be a continuous positive definite and radially unbounded function defined on ; then there exist class functions and defined on such that for all .*

Lemma 7 (see [30]). *For the nonautonomous system , let be an equilibrium point of system and be a continuously differentiable function such that and hold for any and , where are continuous positive definite functions on and is radially unbounded. Then is globally uniformly asymptotically stable.*

#### 3. Output Feedback Controller Design and Stability Analysis

##### 3.1. Control Objective of This Paper

The objective of this paper is to construct an output feedback controller for system (1) such that the equilibrium point of the closed-loop system is globally uniformly asymptotically stable when the maximal open sector of output function is given.

*Assumption 8. *There is a known constant such that for ,for all , where is an arbitrary positive integer and the real-valued functions and satisfy the following relations:for all and , with and . By Lemma 3 and Assumption 8, there exists a constant such that

*Remark 9. *As far as we know, under Assumption 8, [29] is the first paper to study output feedback stabilization of nonlinear system (1) with . In this paper, we will consider system (1) with being an unknown continuous function.

##### 3.2. State Feedback Controller Design of System (1)

*Step 1. *Taking , , it follows from (1) and (8) thatChoose the virtual controller with , where is a constant to be designed. Due to , (9) becomes*Inductive Step.* Suppose that at step , there is a positive definite and radially unbounded Lyapunov function and a set of virtual controllers defined bysuch thatwhere , are positive constants, are constants to be designed, and are positive constants dependent on . In what follows, we will show that (12) still holds at step .

By (11), one has , which together with (1) lead toConstructing , it follows from (12) and (13) thatBy (8), (11), and Lemmas 2â€“4, we derivewhere , are positive constants, and and are positive constants dependent on and , respectively. Substituting (15) and (16) into (14) leads toChoose the virtual controller with , where is a constant to be designed. Due to , (17) becomesThis completes the inductive step.

*Step n. *The Lyapunov function and a positive constant givewhere is defined as in (11) for . Choose virtual controller with , where is a constant to be designed. Due to , (19) becomes

##### 3.3. Output Feedback Controller Design of System (1)

For system (1), since output function is unknown, the state is exactly unknown. We construct the output-driven nonlinear reduced-order observerwhere , , and the observer gains , are constants to be determined.

*Remark 10. *Compared with the observer in [29], since the unknown output function causes to be exactly unknown, a new output-driven nonlinear reduced-order observer (21) is designed without using but to rebuild unmeasured states. Moreover, not all but the first nonlinearities are used in the design of observer. For a second-order system Example 4.1 in [29], we can design an observer without using nonlinearity; see the simulation example in this paper for the details.

Based on , an output feedback controller is designed asDefine the errorBy (1), (21), and (23), one hasFor the Lyapunov function , it follows from (21), (23), and (24) thatBy (23), Lemma 5, and the fact that , we obtain for ,Next, we give the estimate of the others on the right-hand side of (25) by Propositions 11â€“14, whose proofs are included in the Appendix.

Proposition 11. *There is a positive constant , and positive constants , dependent on such that*

Proposition 12. *There is a positive constant , a positive constant dependent on , and positive constants , dependent on such that*

Proposition 13. *There is a positive constant , a positive constant dependent on , and positive constants , dependent on such that*

Proposition 14. *There is a positive constant , a positive constant dependent on , and positive constants , dependent on such that*

To estimate the term in (20), we give the following proposition whose proof is also included in the Appendix.

Proposition 15.

Define . By (20), (25), (26), and Propositions 11â€“15, one obtainswhere with , , it is easy to see that is a positive constant, and , are positive constants dependent on . For some positive constants , design such thatand for some positive constants , choose asBy (33) and (34), (32) becomes

##### 3.4. Stability and Convergence Analysis

Theorem 16. *If Assumption 8 holds for system (1), there is a maximal open sector with being a positive constant, as long as the unknown output function belongs to any closed sector included in , under the output feedback controller (21) and (22):**(i) the solutions of the closed-loop system (1), (21), (22) are well-defined on ,**(ii) the equilibrium point is globally uniformly asymptotically stable.*

*Proof. *(i) We firstly give the choice of , . At Step 1, for any given , by (33), we can choose ; then can be calculated. At step 2, for any given , by (33), we can choose ; then can be calculated when have been obtained by (15) and (16). One by one, at Step n, for any given , by (33), we can choose ; then can be calculated.

Secondly, we give the choice of . For the given , we can get . Then for any given , by (34), we can choose . Next, for the given , one get . Then for any given , can be chosen by (34). In turn, we can get .

Finally, we give the sector of output function. Denote , , , andIn fact, one can determine the supremum of ; that is, select constantas the supremum of . Hence, the maximal open sector of output function is . When output function in (1) belongs to any closed sector with , by Definition 1,from which one hasWith the help of , (36), and (37), we deriveFrom (39) and (40), (35) becomes Motivated by [29], with the help of , one obtainsfrom which (41) becomesIt is easy to see that is a continuous and positive function with respect to .

By the transformations (11), system (1) can be transformed into a -system:where is a continuous function with . Denote , by the existence and continuation of the solution, the solution of -system (24), (44) is defined on with .

Due to , it is obvious that is a continuously differentiable, positive definite, and radially unbounded function. By Lemma 6, there are class functions and such thatSince is a class function, then for any , one can always find a with such that . If , by (43) and (45), for all , which means that for all . Hence, is not an escape time; that is, is well-defined on , so is .

(ii) Since , from (43), (45), and Lemma 7, we know that the equilibrium point of -system (24), (44) is globally uniformly asymptotically stable. By the continuity of on and , it is easy to recursively prove that the equilibrium point of the closed-loop system (1), (21), (22) is globally uniformly asymptotically stable.

*Remark 17. *Since the power of system is time-varying, a new function (36) is given to acquire its supremum, which is used to determine the maximal open sector of output function. Moreover, a rigorous analysis is given to establish the stability of the closed-loop system.

#### 4. Simulation Example

Consider the second-order nonlinear system in [29]As discussed in [29], Assumption 8 holds and , . Following the design process as in Section 3 in this paper, we can design output feedback controller without using the nonlinearity ,Choose . By choosing , , we obtain ,