## Mathematical Theories and Applications for Nonlinear Control Systems

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# Global Output Feedback Stabilization for a Class of Nonlinear Cascade Systems

**Academic Editor:**Weihai Zhang

#### Abstract

This paper focuses on the problem of global output feedback stabilization for a class of nonlinear cascade systems with time-varying output function. By using double-domination approach, an output feedback controller is developed to guarantee the global asymptotic stability of closed-loop system. The novel control strategy successfully constructs a unified Lyapunov function, which is suitable for both upper-triangular and lower-triangular systems. Finally, two numerical examples are provided to illustrate the effectiveness of a control strategy.

#### 1. Introduction

It is well known that global output feedback stabilization is viewed as one of the most challenging fields of nonlinear control. Researchers have not yet found any unified way to handle the problem of global output feedback stabilization because the measure of states is difficult. Fortunately, with the help of nonseparation principle [1], homogeneous domination approach [2], and backstepping method, many interesting results such as [3â€“11] have been achieved.

It is worth pointing out that the structures of system output and nonlinear functions determine the possible forms of observer and controller. More specifically, the uncertainty of nonlinearities has led to the emergence of many kinds of observers, including high-gain observer, homogeneous observer, and time-varying observer. For example, [12] solved the problem of global output feedback stabilization based on linear high-gain observer for a class of uncertain nonlinear systems, where controller is independent of higher-order nonlinearities. Under uncertain linear growth condition in [13], a dynamic high-gain observer is proposed without requiring precise information of output function. References [14, 15] achieved system global stabilization by using time-varying observer, which uses the appropriate functions of time, rather than the dynamic compensator. Since some nonlinear functions satisfy neither the linear growth nor Lipschitz condition in practice, the existing approaches are not suitable. Therefore, [16â€“19] proposed homogeneous domination method to overcome this obstacle. Based on the existing results, some special observers are proposed, such as dual-observer [20] and reduced-observer [21]. In practice, complex systems are usually composed of simple subsystems. Therefore, cascade systems have become one of the most interesting topics of nonlinear systems. A great deal of research has been devoted to this subject over the last decades, as evidenced by the comprehensive books of [22, 23]. However, when zero-dynamics exist and obey mild conditions, the tracking problem cannot be solved by trivially extending the corresponding results without zero-dynamics; that is, there do not exist appropriate observers to tracking states of cascade systems. As further investigation, researchers now consider cascade connections in which the nonlinear systems are globally stable, but the input subsystem is more complex than just an integrator; for instance, [24â€“26] successfully investigated output feedback stabilization for uncertain cascade systems under growth condition. Regrettably, their approaches are only suitable for lower-triangular cascade systems. On the other hand, some literatures [27] achieved global output feedback stabilization when output function depends only on a state. References [28, 29] required that output function be continuous differentiability and initial value equals zero when output is unknown. The above conditions are restrictive; researchers turned to study time-varying output function. For instance, [30] further investigated the problem of global output feedback stabilization for a class of nonlinear systems with unknown measurement sensitivity. Meanwhile, a new method, namely, dual-domination approach, is proposed in [30].

In view of the above argument, an interesting question is proposed simultaneously:* Is it possible to find a new approach to solve the problem of global output feedback stabilization for nonlinear cascade systems with unknown time-varying output function, which is suitable for both upper-triangular and lower-triangular systems?* Based on above analysis and references, we will solve aforementioned question and provide satisfactory answer. There are three troublesome difficulties throughout the paper. The first is to find the appropriate Lyapunov function that is independent of the derivative of output function, since output function is unknown and does not satisfy differentiability. The second is to choose allowable sensitivity error, since it appears in the construction of controller. The third is to design rational observers to successfully track states, since nonlinearities and output function are unknown. A novel observer is proposed, which is different from the existing results [24, 25].

The main contributions of this paper are divided into three aspects: (i) double-domination approach is provided to handle time-varying sensitivity and uncertain nonlinearities, which is suitable for both upper-triangular and lower-triangular systems; (ii) linear observer does not rely on precise information of nonlinearities and output function; (iii) the construction of Lyapunov function avoids the use of the differentiability of output function.

#### 2. Preliminaries and Problem Formulation

##### 2.1. Preliminaries

We will adopt the following notations throughout this paper. denotes the set of real numbers, denotes the set of all nonnegative real numbers, and denotes Euclidean space with dimension . For any real vector , the norm is defined by , . For matrix , , , denotes the norm , where denotes maximum eigenvalue of square matrix . denotes the set of all functions with continuous th partial derivatives. A continuous function is said to belong to class if it is strictly increasing and . It is said to belong to class if and as . For a continuously differentiable function , it is positive definite if and if and only if ; it is radially unbounded if as . The arguments of functions are sometimes simplified; for instance, a function is denoted by or and is denoted by .

In the following, we list three lemmas that play an important role in proving the main results, and their proofs can be found in [31â€“33].

Lemma 1 (see [31]). *Let and be positive constants; given any positive real-valued function , the following inequality holds:*

Lemma 2 (see [32]). *For any , the following inequality holds: *

Lemma 3 (see [33]). *For and , is an integer, and the following inequality holds:*

##### 2.2. Problem Formulation

This paper investigates the nonlinear cascade system described bywhere and are system states with the initial values , , , being control input and output, respectively. is continuous function with and globally Lipschitz with respect to ; , , are continuous functions with . is a continuous function that represents time-varying sensitivity.

The following assumptions are needed.

*Assumption 4. *For the continuous function , there is a known positive parameter satisfying , where is an allowable sensitivity error and is the upper bound of the allowable sensitivity error.

*Assumption 5. *There exists a positive-definite and radially unbounded function such thatwhere , , and and are positive constants.

*Assumption 6. *There exists a constant such that

*Remark 7. *Since output function contains unknown parameter, it implies that the scope of this paper is more general than [1, 2, 12, 16] whose output function is equal to .

*Remark 8. *In terms of the appearance of , two obstacles will be encountered. The first is to find an appropriate observer, which does not use the information of output function. The second is to find the feasible range of , because the information of will be used in the design of controller.

#### 3. Main Results

##### 3.1. Output Feedback Controller Design for Upper-Triangular Case

We now summarise main results of this paper.

Theorem 9. *For system (4) under Assumptions 4â€“6, there exists an output feedback controller such that states of the closed-loop system are uniformly bounded over and .*

*Proof. *The proof is in four parts. At first, a linear observer with a domination gain is introduced to reconstruct all the states. Secondly, an output feedback controller composed of another domination gain is constructed to counteract the destabilized terms. Finally, a delicate selection of gains is provided and strict analysis is performed to guarantee that the closed-loop systems are globally asymptotically stable.*Part I: Design of an Observer.* We construct the following linear observer:where are coefficients of Hurwitz polynomial and the domination gain will be determined later. Define the estimation error as follows:Then, the error equation can be rewritten aswhereSince is Hurwitz, there exists a symmetric positive-definite matrix such that , where is an identity matrix of . Consider positive-definite and radially unbounded function ; a direct calculation givesAccording to Assumption 6 and Lemma 2, we obtainSubstituting (12) into (11), one hasIn addition, Lemma 1 shows thatandwhere , , and , are independent of a domination gain . Therefore, substituting (14) and (15) into (13) yields*Part II: Construction of a Controller*. Consider the following system:Define the following change of coordinates:where is a domination gain that will be determined later. Using above coordinate transform, (17) can be rewritten asDesign the output feedback control law as follows:where , , are coefficients of Hurwitz polynomial . Substituting (20) into (19), we havewhere is a Hurwitz matrix that shows that there exists a symmetric positive-definite matrix such that . Choose scalar function , which is positive-definite and radially bounded. Noting that , the time derivative of along the trajectories of (21) isFirstly, by virtue of Assumption 6 and Lemma 1, it holds thatwhere and . Furthermore, it is deduced from Lemma 1 thatand because of , one can conclude thatConsequently, one obtainswhere is independent of the domination gains and . *Part III: Determination of Domination Gains.* According to above arguments, it follows thatUsing Lemma 3, there isand putting (29) together with (27), we havewhere . Now, we choose the allowable sensitivity error asBecause of , one hasObviously, and (30) can be rewritten asSimilarly, substituting (29) into (16) yieldswhere Applying Assumption 5 and the property of Lipschitz function, one haswhere is a positive constant. By Lemma 1, it is easy to obtain that , and, according to Lemma 2, the following inequalities hold:Substituting (36) into (35), one hasAccording to (29) and the definition of norms, it is easy to deduce thatwhere ; therefore (37) can be further expressed asConsider positive-definite and radially unbounded function as follows:and substituting (33), (34), and (39) into (40), there isIn order to ensure that , we choose the domination gains and as follows. Firstly,implying thatand, with in hand, one deducesAs consequence, (41) can be rewritten asSecondly, we can select* L* to satisfy the following inequalities:and, in light of , it is straightforward to show thatand satisfiesUnder above choice of domination gains, it is clear that*Part IV: Stability Analysis.* Consider transformed systems (9), (20), and (21). By the existence and continuity of solution, the closed-loop system state composed of can be defined on a time interval , where may be a finite constant or infinity.*(i)â€‰â€‰For the Boundedness of . *Due to the fact that , , where and are functions; we obtainandwhere , , and are functions.

For any , noting that , one can always find a constant , such that . From (49), it follows that , . Then, if , it holds thatwhich implies that , ; that is, is bounded. Therefore, by virtue of estimation errors and coordinates transform, it is easy to get that the state is bounded on as well.*(ii)â€‰â€‰. *This claim can be shown by contradiction; if is finite, then would be a finite escape time; that is, the state would tend to as . However, the continuity of solution guarantees the boundedness of at , since is bounded on . This is clearly a contradiction. Therefore, the state of the closed-loop system is uniformly bounded over .*(iii)â€‰â€‰ and . *Above all, it is clear thatwhich implies that exists and is finite. On the other hand, the boundedness of over means that is uniformly bounded in over . Thus, is uniformly continuous in over and so is . Using well-known Barbalatâ€™s Lemma in [22], one obtains , , and , which shows that , , and . Finally, from (29), it is easy to see that . This completes the proof.

*Remark 10. *A double-domination method is proposed to handle the time-varying output function and nonlinearities in the proof of Theorem 9; that is, two domination gains and are used to dominate time-varying function and nonlinearities , respectively.

*Remark 11. *It should be noted from that the upper bound depends on coefficients for the Hurwitz polynomial . When are specified, the corresponding matrix as well as the upper bound can be computed. For example, , for ; , for .

##### 3.2. Extension to Lower-Triangular Case

Some subsystems do not satisfy upper-triangular structure in practical application, so we extend the subsystems to lower-triangular form and impose following assumptions on system (4).

*Assumption 12. *For , there exists a constant such that

Theorem 13. *For a class of nonlinear cascade system (4) under Assumptions 4, 5, and 12, there exists an output feedback controller, such that states of the closed-loop system are uniformly bounded over and .*

*Proof. *The proof is analogous to the proof of Theorem 9 with an obvious modification. To facilitate comparison, we select same notations as Theorem 9 and many similarities will be omitted.

Firstly, we construct the similar observer as (7)where and . With the help of the previous process, define the estimation errorand it is straightforward to show thatand since the definitions of the associated symbol are same as (9), we just give different symbol in the following paper: . Consider same scalar function , which is proper and radially unbounded. Evidently, following (16), we arrive atwhere , are independent of a domination gain .

Secondly, consider system described byThen, introduce the following transformations:where is a domination gain to be determined later. With the help of (60), it is easy to see that system (59) can be rewritten asIf the control law is designed as (20), then substituting (20) into (61) leads towhere the definitions of notations are the same as (21). Consider the same quadratic functionand, after calculations, we arrive at