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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 9, SEPTEMBER 2006 1457

Recursive Observer Design, HomogeneousApproximation, and Nonsmooth Output Feedback

Stabilization of Nonlinear SystemsChunjiang Qian, Senior Member, IEEE, and Wei Lin, Senior Member, IEEE

Abstract—We present a nonsmooth output feedback frameworkfor local and/or global stabilization of a class of nonlinear systemsthat are not smoothly stabilizable nor uniformly observable. Asystematic design method is presented for the construction ofstabilizing, dynamic output compensators that are nonsmoothbut Hölder continuous. A new ingredient of the proposed outputfeedback control scheme is the introduction of a recursive observerdesign algorithm, making it possible to construct a reduced-orderobserver step-by-step, in a naturally augmented manner. Sucha nonsmooth design leads to a number of new results on outputfeedback stabilization of nonlinear systems. One of them is theglobal stabilizability of a chain of odd power integrators byHölder continuous output feedback. The other one is the localstabilization using nonsmooth output feedback for a wide class ofnonlinear systems in the Hessenberg form studied in a previouspaper, where global stabilizability by nonsmooth state feedbackwas already proved to be possible.

Index Terms—Homogeneous approximation, nonlinear systems,nonsmooth observers, nonsmooth stabilizability, nonuniform ob-servability, output feedback stabilization.

I. INTRODUCTION

I N THIS PAPER, we investigate the output feedback stabi-lization problem for a class of nonlinear systems described

by equations of the form

...

(1.1)

where and arethe system state, output and control input, respectively. For

Manuscript received December 6, 2004; revised September 28, 2005 andJanuary 16, 2006. Recommended by Associate Editor M.-Q. Xiao. The workof C. Qian was supported in part by the National Science Foundation underGrant ECS-0239105 and by the University of Texas at San Antonio FacultyResearch Award. The work of W. Lin was supported in part by the NationalScience Foundation under Grants DMS-0203387 and ECS-0400413, and inpart by the AFRL under Grant FA8651-05-C-0110.

C. Qian is with the Department of Electrical and Computer Engineering,The University of Texas at San Antonio, San Antonio, TX 78249 USA (e-mail:[email protected]).

W. Lin is with the Department of Electrical Engineering and Computer Sci-ence, Case Western Reserve University, Cleveland, OH 44106 USA, and is affil-iated with HIT Graduate School, Shenzhen, China (e-mail: [email protected]).

Digital Object Identifier 10.1109/TAC.2006.880955

is an odd positive integer, and, is a mapping with . The function

is and .In [5] and [29], it was proved that every smooth affine system,

i.e.,

and (1.2)

is feedback equivalent to the nonlinear system (1.1) with, by a

local diffeomorphism and a smooth state feedbackif, and only if, a set of necessary and

sufficient conditions hold [5], [29]. Moreover, it was pointedout [5] that the conditions are nothing but an extension ofthe exact feedback linearization conditions. In the case when

, the conditions of [5] reduce tothe well-known necessary and sufficient conditions for affinesystems to be exactly feedback linearizable. Hence, (1.1) witha suitable form of is indeed a generalized normal formof affine systems when exact feedback linearization is notpossible.

For system (1.1) in the Hessenberg form, i.e.,, the

problem of global stabilization by state feedback was addressedin [26]. Due to the presence of uncontrollable unstablelinearization, system (1.1) may not be stabilized by anysmooth state feedback, even locally [3]. It is, however, globallystabilizable by Hölder continuous state feedback [26]. Bycomparison, little progress has been made in the design ofoutput feedback controllers for the nonlinear system (1.1).As a matter of fact, output feedback stabilization of (1.1) is arecognized challenging problem, because the lack of uniformobservability and the unobservable linearization of (1.1) makethe conventional output feedback design methods inapplicable.

For nonlinear systems with uncontrollable/unobservable lin-earization, there are very few results in the literature addressingdifficult issues such as observer design [22], [23], [34], andoutput feedback stabilization.

Even in some relatively simple cases, for instance, the localcase, a fundamental question of whether the nonlinear system(1.1) is locally stabilizable by nonsmooth output feedback re-mains unknown.

Over the past few years, attempt has been made to tacklethis difficult problem and some preliminary results have beenobtained towards the output feedback stabilization of lower-di-mensional nonlinear systems (some elegant results on state feed-back stabilization of two or three-dimensional systems can be

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1458 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 9, SEPTEMBER 2006

found, for instance, in [1], [7], [8]–[15]). The paper [27] con-sidered a class of planar systems in the lower-triangular form

(1.3)

When , the first approximation of system (1.3) is notcontrollable nor observable. As a result, the traditional “Luen-berger-type” or “high-gain” observer proposed in [19], [9], [18],[17], [20] cannot be applied to the high-order system (1.3). Tosolve the global stabilization problem by smooth output feed-back, we proposed in [27] a one-dimensional nonlinear observerthat is constructed using a feedback domination design com-bined with the tool of adding a power integrator. With the help ofthe reduced-order observer, a smooth output feedback stabilizerwas designed for the planar system (1.3), under a high-ordergrowth condition imposed on [27]. The growth conditionwas relaxed later in [28], by employing nonsmooth rather thansmooth output feedback. Note that both papers considered onlythe output feedback stabilization for planar systems. A majorlimitation of the papers [27], [28] is that the proposed outputfeedback control schemes are basically an ad-hoc design. Theyare difficult to be extended to higher dimensional nonlinear sys-tems with uncontrollable/unobservable linearization.

In the higher dimensional case, there are fewer results avail-able in the literature, which address the question of how to sta-bilize non-uniformly observable systems via output feedback.In [33], the problem of global stabilization by smooth outputfeedback was shown to be solvable for a chain of odd power in-tegrators with same powers (i.e., ). This wasdone by developing a new output feedback control scheme thatallows one to design both high-order observers and controllersexplicitly. While the state feedback control law is constructedbased on the standard tool of adding an integrator, the observerdesign was carried out by using a newly developed machinerywhich can be viewed as a dual of the adding a power integratortechnique. A novelty of the nonlinear observer design approachin [33] is that the observer gains can be assigned one-by-one, inan iterative manner.

Despite the aforementioned progress, many important outputfeedback control problems remain open and unsolved. One ofthem, for instance, is whether a chain of odd power integratorswith different powers can be stabilized by output feedback? Theother is when the nonlinear system (1.1) in the Hessenberg form,which is impossible to be handled by smooth feedback, is locallyor globally stabilizable by nonsmooth output feedback? Thesefundamental issues will be addressed in this paper, and someanswers will be given to these questions.

In particular, a nonsmooth output feedback control schemewill be developed to tackle the output feedback stabilizationproblem of system (1.1).

The objectives of this paper are twofold: To identify appro-priate conditions under which a class of genuinely nonlinearsystems (1.1) with arbitrarily odd integers is locallyand/or globally stabilizable by nonsmooth output feedback, and

to develop a nonsmooth feedback design approach for the ex-plicit construction of a stabilizing, dynamic output compensator

(1.4)

where is a but nonsmooth mapping.Inspired by the observer design approach in [28], we will

present in this work a new methodology to construct a reduced-order observer for nonlinear systems that go substantially be-yond the systems studied in [27], [28], [33]. The proposed ob-server design approach possesses several unique features dif-ferent from existing methods: i) The observer has a nonsmoothstructure as a necessary tool to overcome the obstacle caused bythe complex structure of (1.1), which seems to be hard to be han-dled by the smooth observer design [33]; ii) the reduced-orderobserver is constructed recursively, in an augmented manner bywhich the estimators of the unmeasurable statesare built one by one, from top to bottom; (iii) the selection of theobserver gains requires a more subtle procedure due to the useof nonsmooth observers. Combining the new reduced-order ob-server design with the nonsmooth state feedback control method[26], we can achieve global stabilization by nonsmooth outputfeedback, for a number of nonlinear systems with unstabiliz-able/undetectable linearization. In fact, the nonsmooth outputfeedback stabilization theory thus developed leads to severalnew and important conclusions. One of them, among the otherthings, is that every chain of odd power integrators is globallystabilizable by Hölder continuous output feedback. Another im-portant result is that the local stabilization is possible by non-smooth output feedback for nonlinear systems in the Hessen-berg or -normal form [26], [5], where only global strong stabi-lization via nonsmooth state feedback was shown to be possible.

The rest of the paper is organized as follows. Section IIreviews some basic notions and results of homogeneous sys-tems theory, which will be frequently used in the sequel. InSection III, we first consider, for the sake of clarity, the problemof output feedback stabilization for a chain of odd powerintegrators with different powers. For this simple yet genuinelynonlinear system, we explicitly construct a Hölder continuousoutput feedback controller. In Section IV, we extend the resultof Section III to a class of nonsmootly stabilizable systemswith undetectable linearization. The results on global outputfeedback stabilization are derived under restrictive growthconditions. In the local case, we show how the growth condi-tions can be removed, and how new stabilization results canbe derived by the theory of homogeneous systems [10], [7],[8], [14], [12], [15], [16], in particular, by the robust stabilitytheorem of homogeneous systems [11][30] combined with thetechnique of homogeneous approximation. Appendix collectssome useful lemmas and all the proofs of the propositions to beused throughout this paper.

II. HOMOGENEITY AND ROBUST STABILITY OF

HOMOGENEOUS SYSTEMS

Aside from aesthetic aspect, dynamic systems that exhibit ho-mogeneity often possess some important and useful properties.

QIAN AND LIN: RECURSIVE OBSERVER DESIGN, HOMOGENEOUS APPROXIMATION 1459

For instance, as in the case of linear systems, local asymptoticstability of a homogeneous system implies its global asymp-totic stability. Similarly, a nonlinear system is locally asymptoti-cally stable if its homogeneous approximation is locally asymp-totically stable. These distinguished features make the stabilityanalysis and synthesis of homogeneous systems much simplerand easier than general dynamic systems without homogeneity.

In this section, we review a number of basic definitions andconcepts related to the notions of homogeneous vector fields,homogeneity with respect to a family dilations, homogeneousapproximations and robust stability of homogeneous systems,which play a key role in proving the main results of this paper.The reader is referred to [11], [12], [14], [16], [7] and the books[2], [1], and [35], as well as the references therein for additionaldetails.

A. Standard Homogeneity

The concept of homogeneity was introduced as a powerfultool for the stability analysis of nonlinear systems. In the liter-ature, the homogeneity was originally defined as follows: thevector field of the autonomous system

with

(2.1)

is said to be homogeneous if there is a constant such that

(2.2)

With this definition, the following stability result was estab-lished for a perturbed system of (2.1), which is analog to thefirst stability theorem of Lyapunov.

Theorem 2.1: (see [35], [10]) Consider an autonomoussystem of the form

(2.3)

where satisfies (2.2) and . If the homo-geneous systems (2.1) is asymptotically stable, the perturbedsystem (2.1) is also locally asymptotically stable.

Due to the richness of the nonlinearities, the class of non-linear systems satisfying (2.2) is rather limited. To generalizeTheorem 2.1 and to characterize a more general class of homo-geneous systems, we review in the next subsection importantconcepts such as dilations, weighted homogeneity and homo-geneous vector fields.

B. Weighted Homogeneity

The notion of weighted homogeneity was already discussedin the books [35], [2] as a natural extension of the standard ho-mogeneity. The concept and its applications were elaboratedlater in [12], [14]–[16]. This powerful notion, together with theidea of homogeneous approximation, has led to some importantstability results for analysis and synthesis of general nonlinearcontrol systems [11], [14], [16], [30]. In what follows, we re-call the notions such as dilations and homogeneous vector fields

with weighted dilation, and Hermes’ theorem [11], [30] on ro-bust stability of homogeneous systems to be used later on.

For a fixed coordinate and real numbersand :

• a dilation is a mapping , defined by

where is the weighting of the th coordinate;• a function is said to be homogeneous of

degree if there is a real number such that

• a vector field is said to be homogeneousof degree if there is a real number such that for

Remark 2.1: Notably, the standard homogeneity is a specialcase of the weighted one. This conclusion follows immediatelyby setting and . Throughout thispaper, the weighted homogeneity will be used in the stabilityanalysis of nonlinear systems.

The homogeneity and its properties reviewed so far for au-tonomous systems can be easily extended to the nonlinear con-trol systems

For instance, the vector field ishomogeneous of degree if there exists a set of posi-tive real number and such that

The introduction ofweighted homogeneity has led to a powerful way for the sta-bility analysis of nonlinear systems. Over the past few decades,the theory of homogeneous systems and its applications tofeedback stabilization of nonlinear control systems have beendeveloped extensively, and a number of important results havebeen obtained in the literature [35], [10], [11], [30], [16],[14]–[8]. Among many other things, a well-known result ofhomogeneous systems is the equivalence between local andglobal asymptotic stability. Another important conclusion isthat an asymptotically stable homogenous system admits ahomogeneous Lyapunov function [35], [10], [11], [30]. In theremainder of this section, we recall a robust stability theoremthat is useful in proving asymptotic stability of nonlinearsystems.

C. Robust Stability of Homogeneous Systems

With the aid of the concepts of dilation and homogeneousvector fields, the following stability result which is a natural ex-tension of Theorem 2.1 can be established for general homoge-neous systems.

Theorem 2.2: (See [11] and [30]) Suppose the following as-sumptions hold.


1) is homogeneous of degree with respectto a family of dilations .

2) The continuous vector field , with ,satisfies

, uniformly on .Then, if the trivial solution of the homogeneous system(2.1) is locally asymptotically stable, the solution of theperturbed system is also locally asymptoticallystable.

Remark 2.2: Clearly, applications of Theorem 2.2 dependcrucially on how to identify a nilpotent approximating systemwhich is homogeneous with respect to certain dilation. This isan important but difficult issue that was discussed in [12]. Inthis paper, we shall not address the issue of how to find a homo-geneous approximation for general nonlinear systems. Instead,we will focus on a class of nonlinear systems in the Hessen-berg form [5] for which the homogeneous approximation canbe easily identified.

III. GLOBAL STABILIZATION OF A CHAIN OF POWER

INTEGRATORS BY NONSMOOTH OUTPUT FEEDBACK

In this section, we consider the problem of global output feed-back stabilization for a relatively simple yet significant subclassof nonlinear systems (1.1). Specifically, we focus our atten-tion on the nonlinear system with unstabilizable/undetectablelinearization

(3.1)

known as a chain of odd power integrators.When , global stabilization of system (3.1)

has been shown to be solvable by smooth output feedback [33].The solution in [33] was derived based on a novel output feed-back control scheme that enables one to construct recursivelya nonlinear observer and a smooth state feedback controller.Although the output feedback design method proposed in [33]overcomes the obstacle caused by unobservability of the Jaco-bian linearization of high-order nonlinear systems, it is hard tobe extended to the chain of power integrators (3.1) with different

, due to the nature of a smooth feedback design.In what follows, we present a nonsmooth output feedback

control scheme for the construction of dynamic output com-pensators that globally stabilize system (3.1). A new ingredientwill be the introduction of a recursive observer design algo-rithm, making it possible to construct a reduced-order observerstep-by-step, in an augmented manner. A combination of thenew observer design with the nonsmooth state feedback controlstrategy [26] leads to a globally stabilizing, nonsmooth outputfeedback controller. The main result of this section is the fol-lowing theorem.

Theorem 3.1: There is a Hölder continuous output feedbackcontroller of the form (1.4) globally stabilizing the nonlinearsystem (3.1).

Proof: We break up the proof into three parts.First, a nonsmooth but Hölder continuous state feedback

controller is designed via the adding a power integrator tech-nique [26]. We then construct step-by-step, a nonsmooth

reduced-order observer with a set of constant gains that will bedetermined in the last step of design. Finally, we show that acareful selection of the observer gains guarantees global strongstability of the closed-loop system.Nonsmooth State Feedback Design

For a chain of odd power integrators (3.1), globally stabi-lizing nonsmooth state feedback controllers can be constructedusing the method suggested in [26]. The proposition below is aslight modification of the result in [26]. It provides an explicitformula for the calculation of a global state stabilizer. For theconvenience of the reader, a sketch of the proof is given in theAppendix.

Proposition 3.1: There is a Hölder continuous state feedbackcontroller of the form

(3.2)

with being real constants, such that

(3.3)

where is a positive definite and proper1 Lyapunov func-tion, whose form can be found in [26], and is a realconstant.Recursive Design of Nonlinear Observers

In the state feedback case, it is easy to conclude from theLyapunov inequality (3.3) that the nonsmooth controller

globally stabilizes the chain of power integrators (3.1).In the case of output feedback, the state of (3.1)is not measurable and only is available for feedbackdesign. As a result, the controller (3.2) cannot be implementeddirectly. To obtain an implementable controller, one must de-sign an observer to estimate . Motivated by the ob-server design methods in [28][33], we next develop a machinerythat makes it possible to build a nonsmooth nonlinear observerstep-by-step, in an augmented fashion. This is the basic philos-ophy to be pursued below.

To see how a nonlinear observer can be recursively con-structed, we first consider the case of system (3.1) with .In this case, one can construct, similar to the design methodin [28] with a suitable twist, the following one-dimensionalnonlinear observer

(3.4)

In other words, we use a reduced-order observer to estimate,instead of the state , the unmeasurable variable

, where is a gain constant to be assigned later.When , a two-dimensional observer need to be con-

structed for estimating the unmeasurable variables .Of course, a desirable way for the recursive observer designis to keep the one-dimensional observer already built forunchanged, and being a part of the two-dimensional observer.

1A continuous function V : X ! Y is said to be proper if for every compactA 2 Y , its inverse image V (A) is a compact set in X .


Fig. 1. Block diagram of the nonsmooth observer.

With this idea in mind, we simply augment a one-dimensionalobserver of the form

(3.5)

to the dynamic system (3.4), where is an estimate of the un-measurable variable , and is a gain constantto be determined later.

In this way, we have obtained, in a naturally augmentedmanner, a two-dimensional observer consisting of (3.4)–(3.5)for the chain of integrators (3.1) with . There are threenew ingredients in the construction of the observer (3.4)–(3.5):i) no change of the structure is made to the observer forobtained in the previous step; ii) the augmented observer esti-mates the unmeasurable variable that is related to and

, rather than a linear function of and as done in [33];(iii) the observer (3.4)–(3.5) is nonsmooth while the nonlinearobserver designed in [33] is smooth.

Such an augmented design method enables us to construct anonlinear observer recursively, going from lower dimensionalsystems to higher dimensional systems step-by-step, as shownin Fig. 1.

Indeed, for the -dimensional chain of odd power integrators(3.1), applying the augmented design algorithm repeatedly, wearrive at the following -dimensional observer:

(3.6)

where and is the estimate of the unmeasurablevariable . For the convenience of notations,we also denote and .

Let be the estimateerrors. Then, a direct calculation yields

...

(3.7)

Note that

(3.8)

Thus

(3.9)

With this in mind, it is easy to see that

...

(3.10)

Now, consider the Lyapunov function

(311)

which is positive definite and proper. Clearly

(3.12)

where and .In order to estimate the terms on the right-hand side of (3.12),

we introduce two propositions whose proofs involve tediouscalculations but nevertheless can be carried out using LemmasA.1–A.3. The detailed proofs are included in the Appendix.

Proposition 3.2: For , given any , thereis a constant such that

where and .Proposition 3.3: There exist constants

depending on the gain parameters, such that

(3.14)


With the help of Propositions 3.2 and 3.3, the followinginequality can be obtained by letting

, and with

(3.15)

Nonsmooth Output Feedback DesignNow, we apply the certainty equivalence principle to obtain

an implementable output feedback controller. Observe that thereduced-order observer (3.6) has provided an estimation forthe unmeasurable states . Keeping this in mind,we replace in the controller (3.2) by its estimate

generated from the observer (3.6). Thus

(3.16)

Clearly, substituting the implementable controller(3.16) into inequality (3.3) results in a redundant term

, where defined by(3.2). Using Lemmas A.1 and A.2, it is not difficult to provethat for the constant selected in (3.5), there is a constant

such that

(3.17)

where is defined by (3.2).On the other hand, the output feedback controller (3.16) can

estimated as follows:

(3.18)

Putting (3.15), (3.3) and (3.17)–(3.18) together, we have

(3.19)

The next proposition gives an estimation for one of the terms in(3.19). Its proof is given in Appendix.

Proposition 3.4: There exist constantsdepending on the gain parametersand a real constant independent of all the , such that

(3.20)

By definition (3.2), it is easy to show that the choice ofyields

(3.21)

Substituting (3.20) and (3.19) into (3.19) yields

From the previous inequality, it is clear that by choosing

...


we arrive at

which is negative definite . Therefore,the closed-loop system (3.1)–(3.6)–(3.16) is globally stronglystable in the sense of Kurzweil [24]2.

Remark 3.1: Theorem 3.1 has provided a global output feed-back controller for a chain of odd power integrators with distinctpowers ’s. As a consequence, Theorem 3.1 recovers the pre-vious result on global output feedback stabilization of the non-linear system (3.1) when [33]. Whilethe controller obtained in [33] is smooth, the resulting outputfeedback stabilizer from Theorem 3.1 is continuous but non-smooth, even if . However, it is worthpointing out that the main advantage of Theorem 3.1 lies in thedevelopment of a systematic nonsmooth output feedback designmethod, which enables one to deal with not only the output feed-back stabilization of (3.1) without requiring the same ’s, butalso a number of nonsmoothly stabilizable systems with unde-tectable linearization, as shown in the next section.

IV. OUTPUT FEEDBACK CONTROL OF NONSMOOTHLY

STABILIZABLE SYSTEMS

We now employ the output feedback design approach devel-oped in the previous section to investigate the problem of outputfeedback stabilization for a class of triangular systems with un-detectable linearization. Although the systems under consider-ation cannot be dealt with by smooth feedback, we show thatunder appropriate conditions, they are globally stabilizable bynonsmooth output feedback. We begin by introducing an im-portant result that shows how Theorem 3.1 can be extended toa class of homogeneous systems.

A. Output Feedback Stabilization of Homogeneous Systems

In this subsection, we focus on the following class of homoge-neous systems that are not smoothly stabilizable nor uniformlyobservable [9]:

...

(4.1)

where is a constant and , are integers between0 and , such that and

. Note that when .In the previous work [4], [6], [26], [32], the state feedback

stabilization of system (4.1) was, among the other things, exten-sively investigated. In particular, [32] and [26] provided glob-ally stabilizing, nonsmooth state feedback control laws for thehomogeneous system (4.1). The objective of this subsection is

2The notion of strong stability is nothing but a generalized notion of asymp-totic stability for continuous systems with nonunique solutions [24][26].

to show that global stabilization of (4.1) is also achievable byHölder continuous output feedback.

Theorem 4.1: The homogeneous system (4.1) is globally sta-bilizable by non-smooth output feedback.

Proof: The proof is similar to that of Theorem 3.1, withan appropriate modification. Due to the design of a nonsmoothnonlinear observer, the observer gains need to be chosen in amore subtle manner.

First of all, it is easy to verify that system (4.1) is homoge-neous of degree with the dilation

As a consequence, system (4.1) can also be globally stabilizedby a nonsmooth state feedback controller of the form (3.2), witha different set of coefficients . In other words, Proposition 3.1holds and the Lyapunov inequality (3.3) is still true.

Next, we apply the recursive observer design method devel-oped in the last section to construct a nonsmooth observer. Sincethe homogeneous system (4.1) has a lower-triangular structure,a more rigorous observer needs to be designed. In the case when

, one can construct the following estimator for :

(4.2)

To build estimators for , defineWith the aid of this compact notation,

a set of estimators can be constructed recursively as follows:

(4.3)

where and Letting ,the error dynamics can be represented as

...

(4.4)

Now, consider the Lyapunov function defined by (3.11). Thetime derivative of the Lyapunov function (3.11) along the tra-jectories of (4.4) satisfies

(4.5)


Following the same line of the arguments in Theorem 4.1, onecan prove that Proposition 3.2 can be modified to estimate thefirst term in (4.5). In fact, when

. Hence

(4.6)

This inequality also holds for . In this case, note that. By the definition of and Lemma A.2, we have

which implies that (4.6) holds as well.The relation (4.6), together with Lemma A.2 (set

and ), yields

(4.7)

Next, we prove that an estimation similar to Proposition 3.4 canbe obtained for the last term in (4.5). To this end, observe that

(4.8)

Applying (A.10) to each term in (4.8) leads to

(4.9)

Since, it is not difficult to

deduce from (4.8) and (4.9) that

(4.10)

where are constants de-pending on the gains .

Using (4.10), one can obtain, similar to inequality (4.14), thefollowing estimation: (denoting )

(4.11)

for some constants .Substituting (4.7) and (4.11) into (4.5) results in

(4.12)

which is of the form (3.15).The remaining part of the proof will be the use of the inequal-

ities (3.3) and (4.12) to conclude global strong stability of theclosed-loop system (4.1)–(4.4)–(3.16), which is almost identicalto the proof of Theorem 3.1, and thus left to the reader as an ex-ercise.

Theorem 4.1 is illustrated by the following example.Example 4.1: Consider the homogeneous system

(4.13)

Obviously, when . This implies that .For , since . Similarly,and . Hence, (4.13) is of the form (4.1). By Theorem 4.1,there is a nonsmooth output feedback controller globally stabi-lizing (4.13). Notably, system (4.13) is genuinely nonlinear andcannot be dealt with by smooth output feedback schemes in-cluding the one suggested in [33], because the linearized system


of (4.13) at the origin is uncontrollable and unobservable, andmoreover, the uncontrollable mode has an eigenvalue whose realpart is positive. In conclusion, global stabilization of (4.13) isonly achievable by nonsmooth output feedback.

B. Global Output Feedback Stabilization of Non-HomogeneousSystems

It should be pointed out that the construction of the non-smooth observer in Section IV-A relies heavily on the homoge-neous property of the controlled plant. Consequently, the non-smooth output feedback control scheme developed so far canonly be applied to homogeneous systems.

On the other hand, a careful examination of Theorem 4.1 sug-gests that with a suitable twist, it is possible to relax the homoge-neous requirement and to establish a more general stabilizationtheorem for a class of non-homogeneous systems.

As a matter of fact, a global stabilization result can be provedfor the nonhomogeneous system (1.1) under the following con-ditions.

Assumption 4.1: For , there are constantssuch that

Assumption 4.2: For , there are constantssuch that

(4.14)

Under Assumptions 4.1 and 4.2, there is a nonsmooth outputfeedback controller of the form (1.4) globally stabilizing thenonhomogeneous system (1.1).

Proof: Due to the nature of feedback domination designgiven in [26], it is easy to see that under the triangular growthcondition (4.14), Proposition 3.1 still holds. That is, the non-ho-mogeneous system (1.1) is globally stabilizable by the non-smooth state feedback controller (3.2). For the construction ofa nonlinear observer, we use the same structure of the observer(4.3) but in the current case . Consequently, thehypothesis (4.14) implies that

This, together with (A.10), leads to the relation (4.10). The restof the proof is very similar to that of Theorem 4.1 and thusomitted here.

Since Assumptions 4.1 and 4.2 characterize a more generalclass of non-homogeneous systems than the homogeneoussystems (4.1), Theorem 4.2 can be used to handle the problemof global stabilization by nonsmooth output feedback, for a

class of nonsmoothly stabilizable systems with undetectablelinearization, which are not necessarily homogeneous. Forexample, with the help of Theorem 4.2, we are able to solvethe difficult problem of global output feedback stabilization fora benchmark example in [28], which is a simplified version ofthe underactuated unstable two degrees of freedom mechanicalsystem studied in [31].

Example 4.2: Consider the three-dimensional system

(4.15)

which is not smoothly stabilizable nor uniformly observable.Due to the presence of , system (4.15) isnot homogeneous and hence its output feedback stabilizationproblem is not solvable by Theorem 4.1.

On the other hand, it is easy to verify that the conditions(4.14)–(4.14) are fullfilled. By Theorem 4.2, there does exista nonsmooth output feedback controller globally stabilizing(4.15). Such an output dynamic compensator can be constructedas follows. First, using the nonsmooth state feedback designmethod in [26], one can find the globally stabilizing, nonsmoothstate feedback controller (see [26] for details)

(4.16)

Then, following the recursive observer design method pro-posed in this paper, a reduced-order nonsmooth observer can beconstructed, i.e.,

(4.17)

where and Now, substitutingthe estimated states and into (4.16) yields

(4.18)

Finally, a straightforward argument similar to the proof ofTheorem 4.1 leads to the conclusion that by choosing the gainparameters and suitably, the nonsmooth output feedbackcontroller (4.17)–(4.18) globally stabilizes the nonhomoge-neous system (4.15). The simulation results shown in Fig. 2illustrate the effectiveness of the nonsmooth output feedbackcontroller (4.17)–(4.18). In the simulation, the observer gainsare selected as and .

V. LOCAL STABILIZATION OF NONLINEAR SYSTEMS IN THE

HESSENBERG FORM

In the preceding sections, we identified a number of classes ofnonsmoothly stabilizable systems (with undetectable lineariza-tion) for which global stabilization by nonsmooth output feed-back has shown to be possible, under certain restrictive condi-tions such as Assumptions 4.1-4.2. The purpose of this section


Fig. 2 Transient esponses of (4.17)–(4.18) with (x (0); x (0); x (0); z (0); z (0)) = (2; 2; 2; 0; 0).

is to study, to what extent, the question that the restrictive con-ditions imposed previously can be relaxed if a less ambitiouscontrol objective, namely, local rather than global output feed-back stabilization, is sought.

It turns out that, perhaps not surprisingly, if one is onlyinterested in a local result, the class of nonlinear systems thatis stabilizable by output feedback can be significantly enlarged.In particular, we will prove a general result on local outputfeedback stabilization of nonlinear systems. That is, everynonlinear system in the so-called -normal form or Hessen-berg form [26][5], which is neither uniformly observable norsmoothly stabilizable, is locally stabilizable by nonsmoothoutput feedback, without requiring any growth condition.

A. Lower Triangular Systems

To highlight the main idea, we first examine the case whenthe nonlinear system (1.1) is of a lower triangular structure,i.e., In [25], it has beenshown that this class of triangular systems is globally stabiliz-able by nonsmooth state feedback.

In what follows, we will show that without imposing any con-dition on , the triangular system (1.1) is locally stabilizableby nonsmooth output feedback.

Theorem 5.1: There is a Hölder continuous, dynamic outputcompensator of the form (4.3)–(3.16) locally stabilizing system(1.1) when .

Proof: Since the vector field is and vanishes atthe origin, by the Taylor expansion theorem

(5.1)

Clearly, there always exists an integer such thatand . Then, the homogeneous

approximation of (5.1) under the dilation

(5.2)

is

(5.3)

Note that the remainderis a high-order term under this dilation. As a

matter of fact

(5.4)

because implies that, where

.From (5.3) and (5.4), it is clear that system (4.1) is a homoge-

neous approximation of (1.1). According to Theorem 4.1, there


exists a Hölder continuous, dynamic output compensator com-posed of (4.3) and (3.16), globally stabilizing the homogeneoussystem (4.1). By construction, the resulted nonlinear observer

...

is homogeneous of degree 0 under the dilation

for for

for (5.5)

Moreover, it is easy to see that has a same degree of homo-geneity as . Therefore, the controller (3.16) is homogeneousunder the composite dilations (5.2)–(5.5). By Hermes’ robuststability theorem (i.e., Theorem 2.2), it is concluded that thedynamic output compensator (4.3)–(3.16) renders the triangularsystem (1.1) locally asymptotically stable.

B. Nonlinear Systems in the Hessenberg Form

In this subsection, we show how the result presented in Sec-tion V-A can be further extended to a more general class ofnonlinear systems in the Hessenberg form or -normal form[26], [5]

(5.6)

where the functions, with , are .

It is worth noticing that global stabilization of (5.6) was al-ready proved to be achievable by Hölder continuous state feed-back [26].

Theorem 5.2: There is a Hölder continuous output feedbackcontroller rendering system (5.6) locally strong stable in thesense of Kurzweil [24][26].

Proof: This result can be proved by using the idea of homo-geneous approximation combined with Theorem 2.2. To beginwith, observe that

By the Taylor expansion formula and the fact that, there is a function such

that . This, together with Young’sinequality, yields

which is clearly a higher order term with respect to and ,in the sense of homogeneity.

As a result, both the functions andhave an exactly same homogeneous approximation,

namely, the linear function . Likewise, it is straight-forward to prove that the homogeneous approximation of

is identical to the homogeneousapproximation of the function , which is equalto with an integer satisfying

, and .In summary, similar to the case of a lower-triangular system

considered in Section V-A, the nonlinear system in the Hes-senberg form (5.6) has a homogeneously approximated system(4.1) that is stabilizable by the nonsmooth output feedback con-troller (4.3)-(3.16). Keeping this in mind and using the samearguments as in the proof of Theorem 5.1, we conclude imme-diately that the nonlinear system (5.6) is locally stabilizable bynonsmooth output feedback, for instance, by the output dynamiccompensator (4.3)–(3.16).

We conclude this section by pointing out that Theorems 5.1and 5.2 can be extended to a more general class of nonlinearsystems, as summarized below.

Theorem 5.3: Suppose the nonlinear control system

(5.7)

has (4.1) as its homogeneous approximation with the dilation(5.2) and . Then, there exists a Hölder continuousoutput feedback controller locally stabilizing system (5.7).

The application of Theorem 5.3 can be demonstrated by thefollowing example.

Example 5.1: Consider the nonlinear control system

(5.8)

which is neither in the lower triangular form nor in the -normalform (5.6). In addition, the linearized system is not detectableand and . As a such, output feedback


Fig. 3. State trajectories of (5.8)–(5.10) with (x (0); x (0); x (0); z (0); z (0)) = (0:2; 0:2;0:2;0; 0).

stabilization of system (5.8) is a difficult problem that cannot beaddressed by existing methods in the literature.

However, a simple computation indicates that system (5.8)has a homogeneous approximation of the form (4.1). By The-orem 5.3, the local stabilization problem is solvable by non-smooth output feedback. Indeed, the homogeneous approxima-tion of (5.8) is

(5.9)

with the degree and dilation .Under this dilation, the remainders in (5.8) are actually high-

order terms. In fact,

By Theorem 4.1, system (5.9) is stabilized by a homogenousoutput feedback controller of the form (4.3)–(3.16), which is, inthe present case, given by

(5.10)

According to Theorem 5.3, the same controller also renderssystem (5.8) locally strongly stable.

The simulation shown in Fig. 3 confirms the conclusion. Thesimulation was conducted with the parameters ,and .

VI. CONCLUSION

In this paper, we have developed a nonsmooth output feed-back framework that leads to a number of new results on localand global stabilization of nonlinear systems by output feed-back. The proposed output feedback control approach is non-smooth in nature and couples effectively the tool of adding apower integrator [26] for the design of Hölder continuous statefeedback controllers, and a novel recursive algorithm for theconstruction of nonsmooth nonlinear observers.

With the help of the new framework, we have identified ap-propriate conditions under which the problem of global stabi-lization is solvable by nonsmooth output feedback, for certaintriangular systems that are not smoothly stabilizable, even lo-cally, by any smooth state/output feedback (due to the presenceof unobservable and uncontrollable unstable linearization). Fora wider class of nonlinear systems beyond a triangular struc-ture, such as systems in the -normal form or Hessenberg form[26][5], we proved that without imposing any growth condi-tion, local stabilization by nonsmooth output feedback is pos-sible. This was done by means of the homogeneous systemstheory. The proof was constructive and carried out by explicitlydesigning output feedback stabilizers for the associated homo-geneous systems. The significance of our output feedback con-trol schemes has been demonstrated by several examples whoselocal and global stabilization problems appear to be unsolvableby any existing method.


APPENDIX

This section collects three useful lemmas that are frequentlyused throughout this paper. It also includes proofs of the propo-sitions introduced in Section III.

Lemma A.1: Suppose is an odd integer. Then, for anyreal numbers and

(A.1)

Lemma A.2: Suppose and are two positive real numbers,are continuous scalar-value functions. Then, for any

constant

(A.2)

Lemma A.3: Suppose is an odd integer. Then, there isa constant such that

(A.3)

The proofs of Lemmas A.1–A.3 are straightforward andhence left to the reader as an exercise. The reader is alsoreferred to [26] for details.

Proof of Proposition 3.1: As shown in [26], there are aLyapunov function , which is positive defi-

nite and proper, and a set of virtual controllers ,defined by

......

(A.4)with constants , and a Hölder continuousstate feedback controller of the form

(A.5)

such that

(A.6)

where is a positive real constant.Putting (A.5) and (A.4) together, it is clear that there are ap-

propriate constants such that (3.2) holds.On the other hand, it follows from (A.4) that

(A.7)

Using the previous argument repeatedly, it is concluded thatthere is a constant satisfying

(A.8)

Setting , (3.3) follows immediately from (A.6)and (A.8).

Proof of Proposition 3.2: Given any , by LemmaA.2 there is a constant such that

(A.9)

Hence, (3.13) follows immediately from (A.9) when .When , a direct application of Young’s inequality

yields

which, in turn, implies (3.13).Proof of Proposition 3.3: First of all, we prove that there

is a constant such that

(A.10)

This claim can be proved by an inductive argument.Using (3.8) and Lemma A.1, it is straightforward to show that

(A.10) is true for In fact

When , from Lemma A.1 it follows that

Suppose at step , we have

(A.11)

Then, at step , there is a constant such that


Therefore, (A.10) is true. With the help of (A.10) and LemmaA.2, it can be shown that

for a constant . Define

...

Then, we arrive at

(A.12)

from which (4.14) follows immediately, where the coefficientsin (4.14) are given by

...

(A.13)

Proof of Proposition 3.4: For any constant , it is clear from(3.9) that

(A.14)

where and .

Now, we estimate each term in (A.14). Using Lemma A.3yields

This, together with Lemma A.2, implies that for any ,there are constants and so that

Applying the relation (A.11) to the last term in the previousinequality, we have

(A.15)

In view of (A.14), one can deduce easily from (A.15) that(following the similar argument used in the proof of Proposi-tion 3.3)

where the constants andcan be determined in a manner similar to (A.13).

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Chunjiang Qian (S’98–M’02–SM’04) received theB.S. and M.S. degrees in control theory from FudanUniversity, Shangai, China, in 1992 and 1994, re-spectively. After working in industry for three years,he became a graduate student in the Department ofElectrical Engineering and Computer Science, CaseWestern Reserve University, Cleveland, OH, wherehe received the Ph.D. degree in 2001.

Since August 2001, he has been with the De-partment of Electrical and Computer Engineering,The University of Texas at San Antonio, where he

is now an Associate Professor. His current research interests include robustand adaptive control, nonlinear systems and control, homogeneous systemstheory, output feedback control, optimal control, and their applications tononholonomic systems, underactuated mechanical systems, robotics, andbiomechanical systems.

Dr. Qian received the National Science Foundation CAREER Award in 2003.Currently, he is a Subject Editor of the International Journal of Robust andNonlinear Control and a member of the IEEE CSS Conference Editorial Board.

Wei Lin (S’91–M’94–SM’99) received the D.Sc.degrees in systems science and mathematics fromWashington University, St. Louis, MO, in 1993.

He is currently a Professor in the Department ofElectrical Engineering and Computer Science, CaseWestern Reserve University, Cleveland, OH. He alsoheld short-term visiting positions at a number of uni-versities in the U.K., Japan, Singapore, Hong Kongand China. His research interests and publicationscan be found at: http://nonlinear.case.edu/~linwei/.

Dr. Lin was a recipient of the NSF CAREERAward, the JSPS Fellow from Japan Society for the Promotion Science,and the Warren E. Rupp Endowed Assistant Professorship of Science andEngineering. He has served as an Associate Editor of the IEEE TRANSACTIONS

ON AUTOMATIC CONTROL, a Guest Co-Editor of the Special Issue on ‘‘NewDirections in Nonlinear Control’’ in the IEEE TRANSACTIONS ON AUTOMATIC

CONTROL, and an Associate Editor of Automatica. He was a Vice ProgramChair for the 40th and 41st IEEE Conferences on Decision and Control, anda member of the Board of Governors of the IEEE Control Systems Society in2003–2005. Currently, he is a Subject Editor of the International Journal ofRobust and Nonlinear Control, an Associate Editor of the Journal of ControlTheory and Applications, and a member of the IFAC Technical Committee inNonlinear Control.

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