IN THIS PAPER, we investigate the output feedback stabilization

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1 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 51, NO 9, SEPTEMBER Recursive Observer Design, Homogeneous Approximation, and Nonsmooth Output Feedback Stabilization of Nonlinear Systems Chunjiang Qian, Senior Member, IEEE, and Wei Lin, Senior Member, IEEE Abstract We present a nonsmooth output feedback framework for local and/or global stabilization of a class of nonlinear systems that are not smoothly stabilizable nor uniformly observable A systematic design method is presented for the construction of stabilizing, dynamic output compensators that are nonsmooth but Hölder continuous A new ingredient of the proposed output feedback control scheme is the introduction of a recursive observer design algorithm, making it possible to construct a reduced-order observer step-by-step, in a naturally augmented manner Such a nonsmooth design leads to a number of new results on output feedback stabilization of nonlinear systems One of them is the global stabilizability of a chain of odd power integrators by Hölder continuous output feedback The other one is the local stabilization using nonsmooth output feedback for a wide class of nonlinear systems in the Hessenberg form studied in a previous paper, where global stabilizability by nonsmooth state feedback was already proved to be possible Index Terms Homogeneous approximation, nonlinear systems, nonsmooth observers, nonsmooth stabilizability, nonuniform observability, output feedback stabilization I INTRODUCTION IN THIS PAPER, we investigate the output feedback stabilization problem for a class of nonlinear systems described by equations of the form (11) where and are the system state, output and control input, respectively For Manuscript received December 6, 2004; revised September 28, 2005 and January 16, 2006 Recommended by Associate Editor M-Q Xiao The work of C Qian was supported in part by the National Science Foundation under Grant ECS and by the University of Texas at San Antonio Faculty Research Award The work of W Lin was supported in part by the National Science Foundation under Grants DMS and ECS , and in part by the AFRL under Grant FA C-0110 C Qian is with the Department of Electrical and Computer Engineering, The University of Texas at San Antonio, San Antonio, TX USA ( chunjiangqian@utsaedu) W Lin is with the Department of Electrical Engineering and Computer Science, Case Western Reserve University, Cleveland, OH USA, and is affiliated with HIT Graduate School, Shenzhen, China ( linwei@caseedu) Digital Object Identifier /TAC is an odd positive integer, and,isa mapping with The function is and In [5] and [29], it was proved that every smooth affine system, ie, and (12) is feedback equivalent to the nonlinear system (11) with,bya local diffeomorphism and a smooth state feedback if, and only if, a set of necessary and sufficient conditions hold [5], [29] Moreover, it was pointed out [5] that the conditions are nothing but an extension of the exact feedback linearization conditions In the case when, the conditions of [5] reduce to the well-known necessary and sufficient conditions for affine systems to be exactly feedback linearizable Hence, (11) with a suitable form of is indeed a generalized normal form of affine systems when exact feedback linearization is not possible For system (11) in the Hessenberg form, ie,, the problem of global stabilization by state feedback was addressed in [26] Due to the presence of uncontrollable unstable linearization, system (11) may not be stabilized by any smooth state feedback, even locally [3] It is, however, globally stabilizable by Hölder continuous state feedback [26] By comparison, little progress has been made in the design of output feedback controllers for the nonlinear system (11) As a matter of fact, output feedback stabilization of (11) is a recognized challenging problem, because the lack of uniform observability and the unobservable linearization of (11) make the conventional output feedback design methods inapplicable For nonlinear systems with uncontrollable/unobservable linearization, there are very few results in the literature addressing difficult issues such as observer design [22], [23], [34], and output feedback stabilization Even in some relatively simple cases, for instance, the local case, a fundamental question of whether the nonlinear system (11) is locally stabilizable by nonsmooth output feedback remains unknown Over the past few years, attempt has been made to tackle this difficult problem and some preliminary results have been obtained towards the output feedback stabilization of lower-dimensional nonlinear systems (some elegant results on state feedback stabilization of two or three-dimensional systems can be /$ IEEE

2 1458 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 51, NO 9, SEPTEMBER 2006 found, for instance, in [1], [7], [8] [15]) The paper [27] considered a class of planar systems in the lower-triangular form to develop a nonsmooth feedback design approach for the explicit construction of a stabilizing, dynamic output compensator (14) (13) When, the first approximation of system (13) is not controllable nor observable As a result, the traditional Luenberger-type or high-gain observer proposed in [19], [9], [18], [17], [20] cannot be applied to the high-order system (13) To solve the global stabilization problem by smooth output feedback, we proposed in [27] a one-dimensional nonlinear observer that is constructed using a feedback domination design combined with the tool of adding a power integrator With the help of the reduced-order observer, a smooth output feedback stabilizer was designed for the planar system (13), under a high-order growth condition imposed on [27] The growth condition was relaxed later in [28], by employing nonsmooth rather than smooth output feedback Note that both papers considered only the output feedback stabilization for planar systems A major limitation of the papers [27], [28] is that the proposed output feedback control schemes are basically an ad-hoc design They are difficult to be extended to higher dimensional nonlinear systems with uncontrollable/unobservable linearization In the higher dimensional case, there are fewer results available in the literature, which address the question of how to stabilize non-uniformly observable systems via output feedback In [33], the problem of global stabilization by smooth output feedback was shown to be solvable for a chain of odd power integrators with same powers (ie, ) This was done by developing a new output feedback control scheme that allows one to design both high-order observers and controllers explicitly While the state feedback control law is constructed based on the standard tool of adding an integrator, the observer design was carried out by using a newly developed machinery which can be viewed as a dual of the adding a power integrator technique A novelty of the nonlinear observer design approach in [33] is that the observer gains can be assigned one-by-one, in an iterative manner Despite the aforementioned progress, many important output feedback control problems remain open and unsolved One of them, for instance, is whether a chain of odd power integrators with different powers can be stabilized by output feedback? The other is when the nonlinear system (11) in the Hessenberg form, which is impossible to be handled by smooth feedback, is locally or globally stabilizable by nonsmooth output feedback? These fundamental issues will be addressed in this paper, and some answers will be given to these questions In particular, a nonsmooth output feedback control scheme will be developed to tackle the output feedback stabilization problem of system (11) The objectives of this paper are twofold: To identify appropriate conditions under which a class of genuinely nonlinear systems (11) with arbitrarily odd integers is locally and/or globally stabilizable by nonsmooth output feedback, and 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 reducedorder observer for nonlinear systems that go substantially beyond the systems studied in [27], [28], [33] The proposed observer design approach possesses several unique features different from existing methods: i) The observer has a nonsmooth structure as a necessary tool to overcome the obstacle caused by the complex structure of (11), which seems to be hard to be handled by the smooth observer design [33]; ii) the reduced-order observer is constructed recursively, in an augmented manner by which the estimators of the unmeasurable states are built one by one, from top to bottom; (iii) the selection of the observer gains requires a more subtle procedure due to the use of nonsmooth observers Combining the new reduced-order observer design with the nonsmooth state feedback control method [26], we can achieve global stabilization by nonsmooth output feedback, for a number of nonlinear systems with unstabilizable/undetectable linearization In fact, the nonsmooth output feedback stabilization theory thus developed leads to several new and important conclusions One of them, among the other things, is that every chain of odd power integrators is globally stabilizable by Hölder continuous output feedback Another important result is that the local stabilization is possible by nonsmooth output feedback for nonlinear systems in the Hessenberg or -normal form [26], [5], where only global strong stabilization via nonsmooth state feedback was shown to be possible The rest of the paper is organized as follows Section II reviews some basic notions and results of homogeneous systems theory, which will be frequently used in the sequel In Section III, we first consider, for the sake of clarity, the problem of output feedback stabilization for a chain of odd power integrators with different powers For this simple yet genuinely nonlinear system, we explicitly construct a Hölder continuous output feedback controller In Section IV, we extend the result of Section III to a class of nonsmootly stabilizable systems with undetectable linearization The results on global output feedback stabilization are derived under restrictive growth conditions In the local case, we show how the growth conditions can be removed, and how new stabilization results can be derived by the theory of homogeneous systems [10], [7], [8], [14], [12], [15], [16], in particular, by the robust stability theorem of homogeneous systems [11][30] combined with the technique of homogeneous approximation Appendix collects some useful lemmas and all the proofs of the propositions to be used throughout this paper II HOMOGENEITY AND ROBUST STABILITY OF HOMOGENEOUS SYSTEMS Aside from aesthetic aspect, dynamic systems that exhibit homogeneity often possess some important and useful properties

3 QIAN AND LIN: RECURSIVE OBSERVER DESIGN, HOMOGENEOUS APPROXIMATION 1459 For instance, as in the case of linear systems, local asymptotic stability of a homogeneous system implies its global asymptotic stability Similarly, a nonlinear system is locally asymptotically stable if its homogeneous approximation is locally asymptotically stable These distinguished features make the stability analysis and synthesis of homogeneous systems much simpler and easier than general dynamic systems without homogeneity In this section, we review a number of basic definitions and concepts related to the notions of homogeneous vector fields, homogeneity with respect to a family dilations, homogeneous approximations 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 additional details A Standard Homogeneity The concept of homogeneity was introduced as a powerful tool for the stability analysis of nonlinear systems In the literature, the homogeneity was originally defined as follows: the vector field of the autonomous system with is said to be homogeneous if there is a constant (21) such that (22) With this definition, the following stability result was established for a perturbed system of (21), which is analog to the first stability theorem of Lyapunov Theorem 21: (see [35], [10]) Consider an autonomous system of the form (23) where satisfies (22) and If the homogeneous systems (21) is asymptotically stable, the perturbed system (21) is also locally asymptotically stable Due to the richness of the nonlinearities, the class of nonlinear systems satisfying (22) is rather limited To generalize Theorem 21 and to characterize a more general class of homogeneous systems, we review in the next subsection important concepts such as dilations, weighted homogeneity and homogeneous vector fields B Weighted Homogeneity The notion of weighted homogeneity was already discussed in the books [35], [2] as a natural extension of the standard homogeneity The concept and its applications were elaborated later in [12], [14] [16] This powerful notion, together with the idea of homogeneous approximation, has led to some important stability results for analysis and synthesis of general nonlinear control systems [11], [14], [16], [30] In what follows, we recall the notions such as dilations and homogeneous vector fields with weighted dilation, and Hermes theorem [11], [30] on robust stability of homogeneous systems to be used later on For a fixed coordinate and real numbers and : 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 homogeneous of degree if there is a real number such that for Remark 21: Notably, the standard homogeneity is a special case of the weighted one This conclusion follows immediately by setting and Throughout this paper, the weighted homogeneity will be used in the stability analysis of nonlinear systems The homogeneity and its properties reviewed so far for autonomous systems can be easily extended to the nonlinear control systems For instance, the vector field is homogeneous of degree if there exists a set of positive real number and such that The introduction of weighted homogeneity has led to a powerful way for the stability analysis of nonlinear systems Over the past few decades, the theory of homogeneous systems and its applications to feedback stabilization of nonlinear control systems have been developed extensively, and a number of important results have been obtained in the literature [35], [10], [11], [30], [16], [14] [8] Among many other things, a well-known result of homogeneous systems is the equivalence between local and global asymptotic stability Another important conclusion is that an asymptotically stable homogenous system admits a homogeneous Lyapunov function [35], [10], [11], [30] In the remainder of this section, we recall a robust stability theorem that is useful in proving asymptotic stability of nonlinear systems C Robust Stability of Homogeneous Systems With the aid of the concepts of dilation and homogeneous vector fields, the following stability result which is a natural extension of Theorem 21 can be established for general homogeneous systems Theorem 22: (See [11] and [30]) Suppose the following assumptions hold

4 1460 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 51, NO 9, SEPTEMBER ) is homogeneous of degree with respect to a family of dilations 2) The continuous vector field, with, satisfies, uniformly on Then, if the trivial solution of the homogeneous system (21) is locally asymptotically stable, the solution of the perturbed system is also locally asymptotically stable Remark 22: Clearly, applications of Theorem 22 depend crucially on how to identify a nilpotent approximating system which is homogeneous with respect to certain dilation This is an important but difficult issue that was discussed in [12] In this paper, we shall not address the issue of how to find a homogeneous approximation for general nonlinear systems Instead, we will focus on a class of nonlinear systems in the Hessenberg form [5] for which the homogeneous approximation can be 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 feedback stabilization for a relatively simple yet significant subclass of nonlinear systems (11) Specifically, we focus our attention on the nonlinear system with unstabilizable/undetectable linearization (31) known as a chain of odd power integrators When, global stabilization of system (31) has been shown to be solvable by smooth output feedback [33] The solution in [33] was derived based on a novel output feedback control scheme that enables one to construct recursively a 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 Jacobian linearization of high-order nonlinear systems, it is hard to be extended to the chain of power integrators (31) 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 compensators that globally stabilize system (31) A new ingredient will be the introduction of a recursive observer design algorithm, making it possible to construct a reduced-order observer step-by-step, in an augmented manner A combination of the new observer design with the nonsmooth state feedback control strategy [26] leads to a globally stabilizing, nonsmooth output feedback controller The main result of this section is the following theorem Theorem 31: There is a Hölder continuous output feedback controller of the form (14) globally stabilizing the nonlinear system (31) 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 technique [26] We then construct step-by-step, a nonsmooth reduced-order observer with a set of constant gains that will be determined in the last step of design Finally, we show that a careful selection of the observer gains guarantees global strong stability of the closed-loop system Nonsmooth State Feedback Design For a chain of odd power integrators (31), globally stabilizing nonsmooth state feedback controllers can be constructed using the method suggested in [26] The proposition below is a slight modification of the result in [26] It provides an explicit formula for the calculation of a global state stabilizer For the convenience of the reader, a sketch of the proof is given in the Appendix Proposition 31: There is a Hölder continuous state feedback controller of the form with being real constants, such that (32) (33) where is a positive definite and proper 1 Lyapunov function, whose form can be found in [26], and is a real constant Recursive Design of Nonlinear Observers In the state feedback case, it is easy to conclude from the Lyapunov inequality (33) that the nonsmooth controller globally stabilizes the chain of power integrators (31) In the case of output feedback, the state of (31) is not measurable and only is available for feedback design As a result, the controller (32) cannot be implemented directly To obtain an implementable controller, one must design an observer to estimate Motivated by the observer design methods in [28][33], we next develop a machinery that makes it possible to build a nonsmooth nonlinear observer step-by-step, in an augmented fashion This is the basic philosophy to be pursued below To see how a nonlinear observer can be recursively constructed, we first consider the case of system (31) with In this case, one can construct, similar to the design method in [28] with a suitable twist, the following one-dimensional nonlinear observer (34) 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 constructed for estimating the unmeasurable variables Of course, a desirable way for the recursive observer design is to keep the one-dimensional observer already built for unchanged, and being a part of the two-dimensional observer 1 A continuous function V : X! Y is said to be proper if for every compact A 2 Y, its inverse image V (A) is a compact set in X

5 QIAN AND LIN: RECURSIVE OBSERVER DESIGN, HOMOGENEOUS APPROXIMATION 1461 Fig 1 Block diagram of the nonsmooth observer With this idea in mind, we simply augment a one-dimensional observer of the form With this in mind, it is easy to see that (35) to the dynamic system (34), where is an estimate of the unmeasurable variable, and is a gain constant to be determined later In this way, we have obtained, in a naturally augmented manner, a two-dimensional observer consisting of (34) (35) for the chain of integrators (31) with There are three new ingredients in the construction of the observer (34) (35): i) no change of the structure is made to the observer for obtained in the previous step; ii) the augmented observer estimates the unmeasurable variable that is related to and, rather than a linear function of and as done in [33]; (iii) the observer (34) (35) is nonsmooth while the nonlinear observer designed in [33] is smooth Such an augmented design method enables us to construct a nonlinear observer recursively, going from lower dimensional systems to higher dimensional systems step-by-step, as shown in Fig 1 Indeed, for the -dimensional chain of odd power integrators (31), applying the augmented design algorithm repeatedly, we arrive at the following -dimensional observer: (36) where and is the estimate of the unmeasurable variable For the convenience of notations, we also denote and Let be the estimate errors Then, a direct calculation yields Now, consider the Lyapunov function which is positive definite and proper Clearly (310) (311) (312) where and In order to estimate the terms on the right-hand side of (312), we introduce two propositions whose proofs involve tedious calculations but nevertheless can be carried out using Lemmas A1 A3 The detailed proofs are included in the Appendix Proposition 32: For, given any, there is a constant such that (37) where and Proposition 33: There exist constants depending on the gain parameters, such that Note that (38) Thus (39) (314)

6 1462 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 51, NO 9, SEPTEMBER 2006 With the help of Propositions 32 and 33, the following inequality can be obtained by letting, and with Putting (315), (33) and (317) (318) together, we have (315) Nonsmooth Output Feedback Design Now, we apply the certainty equivalence principle to obtain an implementable output feedback controller Observe that the reduced-order observer (36) has provided an estimation for the unmeasurable states we replace Keeping this in mind, in the controller (32) by its estimate generated from the observer (36) Thus (319) The next proposition gives an estimation for one of the terms in (319) Its proof is given in Appendix Proposition 34: There exist constants depending on the gain parameters and a real constant independent of all the, such that (316) (320) Clearly, substituting the implementable controller (316) into inequality (33) results in a redundant term, where defined by (32) Using Lemmas A1 and A2, it is not difficult to prove that for the constant selected in (35), there is a constant such that By definition (32), it is easy to show that the choice of yields Substituting (320) and (319) into (319) yields (321) (317) where is defined by (32) On the other hand, the output feedback controller (316) can estimated as follows: From the previous inequality, it is clear that by choosing (318)

7 QIAN AND LIN: RECURSIVE OBSERVER DESIGN, HOMOGENEOUS APPROXIMATION 1463 we arrive at which is negative definite Therefore, the closed-loop system (31) (36) (316) is globally strongly stable in the sense of Kurzweil [24] 2 Remark 31: Theorem 31 has provided a global output feedback controller for a chain of odd power integrators with distinct powers s As a consequence, Theorem 31 recovers the previous result on global output feedback stabilization of the nonlinear system (31) when [33] While the controller obtained in [33] is smooth, the resulting output feedback stabilizer from Theorem 31 is continuous but nonsmooth, even if However, it is worth pointing out that the main advantage of Theorem 31 lies in the development of a systematic nonsmooth output feedback design method, which enables one to deal with not only the output feedback stabilization of (31) without requiring the same s, but also a number of nonsmoothly stabilizable systems with undetectable linearization, as shown in the next section IV OUTPUT FEEDBACK CONTROL OF NONSMOOTHLY STABILIZABLE SYSTEMS We now employ the output feedback design approach developed in the previous section to investigate the problem of output feedback stabilization for a class of triangular systems with undetectable linearization Although the systems under consideration cannot be dealt with by smooth feedback, we show that under appropriate conditions, they are globally stabilizable by nonsmooth output feedback We begin by introducing an important result that shows how Theorem 31 can be extended to a class of homogeneous systems A Output Feedback Stabilization of Homogeneous Systems In this subsection, we focus on the following class of homogeneous systems that are not smoothly stabilizable nor uniformly observable [9]: to show that global stabilization of (41) is also achievable by Hölder continuous output feedback Theorem 41: The homogeneous system (41) is globally stabilizable by non-smooth output feedback Proof: The proof is similar to that of Theorem 31, with an appropriate modification Due to the design of a nonsmooth nonlinear observer, the observer gains need to be chosen in a more subtle manner First of all, it is easy to verify that system (41) is homogeneous of degree with the dilation As a consequence, system (41) can also be globally stabilized by a nonsmooth state feedback controller of the form (32), with a different set of coefficients In other words, Proposition 31 holds and the Lyapunov inequality (33) is still true Next, we apply the recursive observer design method developed in the last section to construct a nonsmooth observer Since the homogeneous system (41) has a lower-triangular structure, a more rigorous observer needs to be designed In the case when, one can construct the following estimator for : (42) To build estimators for,define With the aid of this compact notation, a set of estimators can be constructed recursively as follows: (43) where and Letting, the error dynamics can be represented as (41) where is a constant and, are integers between 0 and, such that and Note that when In the previous work [4], [6], [26], [32], the state feedback stabilization of system (41) was, among the other things, extensively investigated In particular, [32] and [26] provided globally stabilizing, nonsmooth state feedback control laws for the homogeneous system (41) The objective of this subsection is 2 The notion of strong stability is nothing but a generalized notion of asymptotic stability for continuous systems with nonunique solutions [24][26] (44) Now, consider the Lyapunov function defined by (311) The time derivative of the Lyapunov function (311) along the trajectories of (44) satisfies (45)

8 1464 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 51, NO 9, SEPTEMBER 2006 Following the same line of the arguments in Theorem 41, one can prove that Proposition 32 can be modified to estimate the first term in (45) In fact, when Hence Since deduce from (48) and (49) that, it is not difficult to (410) (46) This inequality also holds for In this case, note that By the definition of and Lemma A2, we have where are constants depending on the gains Using (410), one can obtain, similar to inequality (414), the following estimation: (denoting ) which implies that (46) holds as well The relation (46), together with Lemma A2 (set and ), yields for some constants Substituting (47) and (411) into (45) results in (411) (47) Next, we prove that an estimation similar to Proposition 34 can be obtained for the last term in (45) To this end, observe that Applying (A10) to each term in (48) leads to (48) (412) which is of the form (315) The remaining part of the proof will be the use of the inequalities (33) and (412) to conclude global strong stability of the closed-loop system (41) (44) (316), which is almost identical to the proof of Theorem 31, and thus left to the reader as an exercise Theorem 41 is illustrated by the following example Example 41: Consider the homogeneous system (413) (49) Obviously, when This implies that For, since Similarly, and Hence, (413) is of the form (41) By Theorem 41, there is a nonsmooth output feedback controller globally stabilizing (413) Notably, system (413) is genuinely nonlinear and cannot be dealt with by smooth output feedback schemes including the one suggested in [33], because the linearized system

9 QIAN AND LIN: RECURSIVE OBSERVER DESIGN, HOMOGENEOUS APPROXIMATION 1465 of (413) at the origin is uncontrollable and unobservable, and moreover, the uncontrollable mode has an eigenvalue whose real part is positive In conclusion, global stabilization of (413) is only achievable by nonsmooth output feedback B Global Output Feedback Stabilization of Non-Homogeneous Systems It should be pointed out that the construction of the nonsmooth observer in Section IV-A relies heavily on the homogeneous property of the controlled plant Consequently, the nonsmooth output feedback control scheme developed so far can only be applied to homogeneous systems On the other hand, a careful examination of Theorem 41 suggests that with a suitable twist, it is possible to relax the homogeneous requirement and to establish a more general stabilization theorem for a class of non-homogeneous systems As a matter of fact, a global stabilization result can be proved for the nonhomogeneous system (11) under the following conditions Assumption 41: For, there are constants such that class of nonsmoothly stabilizable systems with undetectable linearization, which are not necessarily homogeneous For example, with the help of Theorem 42, we are able to solve the difficult problem of global output feedback stabilization for a benchmark example in [28], which is a simplified version of the underactuated unstable two degrees of freedom mechanical system studied in [31] Example 42: Consider the three-dimensional system (415) which is not smoothly stabilizable nor uniformly observable Due to the presence of, system (415) is not homogeneous and hence its output feedback stabilization problem is not solvable by Theorem 41 On the other hand, it is easy to verify that the conditions (414) (414) are fullfilled By Theorem 42, there does exist a nonsmooth output feedback controller globally stabilizing (415) Such an output dynamic compensator can be constructed as follows First, using the nonsmooth state feedback design method in [26], one can find the globally stabilizing, nonsmooth state feedback controller (see [26] for details) Assumption 42: For such that, there are constants (414) (416) Then, following the recursive observer design method proposed in this paper, a reduced-order nonsmooth observer can be constructed, ie, Under Assumptions 41 and 42, there is a nonsmooth output feedback controller of the form (14) globally stabilizing the nonhomogeneous system (11) Proof: Due to the nature of feedback domination design given in [26], it is easy to see that under the triangular growth condition (414), Proposition 31 still holds That is, the non-homogeneous system (11) is globally stabilizable by the nonsmooth state feedback controller (32) For the construction of a nonlinear observer, we use the same structure of the observer (43) but in the current case Consequently, the hypothesis (414) implies that This, together with (A10), leads to the relation (410) The rest of the proof is very similar to that of Theorem 41 and thus omitted here Since Assumptions 41 and 42 characterize a more general class of non-homogeneous systems than the homogeneous systems (41), Theorem 42 can be used to handle the problem of global stabilization by nonsmooth output feedback, for a (417) where and Now, substituting the estimated states and into (416) yields (418) Finally, a straightforward argument similar to the proof of Theorem 41 leads to the conclusion that by choosing the gain parameters and suitably, the nonsmooth output feedback controller (417) (418) globally stabilizes the nonhomogeneous system (415) The simulation results shown in Fig 2 illustrate the effectiveness of the nonsmooth output feedback controller (417) (418) In the simulation, the observer gains are selected as and V LOCAL STABILIZATION OF NONLINEAR SYSTEMS IN THE HESSENBERG FORM In the preceding sections, we identified a number of classes of nonsmoothly stabilizable systems (with undetectable linearization) for which global stabilization by nonsmooth output feedback has shown to be possible, under certain restrictive conditions such as Assumptions The purpose of this section

10 1466 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 51, NO 9, SEPTEMBER 2006 Fig 2 Transient esponses of (417) (418) 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 conditions imposed previously can be relaxed if a less ambitious control objective, namely, local rather than global output feedback stabilization, is sought It turns out that, perhaps not surprisingly, if one is only interested in a local result, the class of nonlinear systems that is stabilizable by output feedback can be significantly enlarged In particular, we will prove a general result on local output feedback stabilization of nonlinear systems That is, every nonlinear system in the so-called -normal form or Hessenberg form [26][5], which is neither uniformly observable nor smoothly stabilizable, is locally stabilizable by nonsmooth output feedback, without requiring any growth condition A Lower Triangular Systems To highlight the main idea, we first examine the case when the nonlinear system (11) is of a lower triangular structure, ie, In [25], it has been shown that this class of triangular systems is globally stabilizable by nonsmooth state feedback In what follows, we will show that without imposing any condition on, the triangular system (11) is locally stabilizable by nonsmooth output feedback Theorem 51: There is a Hölder continuous, dynamic output compensator of the form (43) (316) locally stabilizing system (11) when Proof: Since the vector field is and vanishes at the origin, by the Taylor expansion theorem (51) Clearly, there always exists an integer such that and Then, the homogeneous approximation of (51) under the dilation is (52) (53) Note that the remainder is a high-order term under this dilation As a matter of fact (54) because implies that, where From (53) and (54), it is clear that system (41) is a homogeneous approximation of (11) According to Theorem 41, there

11 QIAN AND LIN: RECURSIVE OBSERVER DESIGN, HOMOGENEOUS APPROXIMATION 1467 exists a Hölder continuous, dynamic output compensator composed of (43) and (316), globally stabilizing the homogeneous system (41) By construction, the resulted nonlinear observer that inequality, yields This, together with Young s is homogeneous of degree 0 under the dilation for for for (55) Moreover, it is easy to see that has a same degree of homogeneity as Therefore, the controller (316) is homogeneous under the composite dilations (52) (55) By Hermes robust stability theorem (ie, Theorem 22), it is concluded that the dynamic output compensator (43) (316) renders the triangular system (11) locally asymptotically stable B Nonlinear Systems in the Hessenberg Form In this subsection, we show how the result presented in Section V-A can be further extended to a more general class of nonlinear systems in the Hessenberg form or -normal form [26], [5] which is clearly a higher order term with respect to and, in the sense of homogeneity As a result, both the functions and have an exactly same homogeneous approximation, namely, the linear function Likewise, it is straightforward to prove that the homogeneous approximation of is identical to the homogeneous approximation of the function, which is equal to 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 Hessenberg form (56) has a homogeneously approximated system (41) that is stabilizable by the nonsmooth output feedback controller (43)-(316) Keeping this in mind and using the same arguments as in the proof of Theorem 51, we conclude immediately that the nonlinear system (56) is locally stabilizable by nonsmooth output feedback, for instance, by the output dynamic compensator (43) (316) We conclude this section by pointing out that Theorems 51 and 52 can be extended to a more general class of nonlinear systems, as summarized below Theorem 53: Suppose the nonlinear control system (56) where the functions, with, are It is worth noticing that global stabilization of (56) was already proved to be achievable by Hölder continuous state feedback [26] Theorem 52: There is a Hölder continuous output feedback controller rendering system (56) locally strong stable in the sense of Kurzweil [24][26] Proof: This result can be proved by using the idea of homogeneous approximation combined with Theorem 22 To begin with, observe that (57) has (41) as its homogeneous approximation with the dilation (52) and Then, there exists a Hölder continuous output feedback controller locally stabilizing system (57) The application of Theorem 53 can be demonstrated by the following example Example 51: Consider the nonlinear control system (58) By the Taylor expansion formula and the fact that, there is a function such which is neither in the lower triangular form nor in the -normal form (56) In addition, the linearized system is not detectable and and As a such, output feedback

12 1468 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 51, NO 9, SEPTEMBER 2006 Fig 3 State trajectories of (58) (510) with (x (0); x (0); x (0); ^z (0); ^z (0)) = (0:2; 0:2; 0:2; 0; 0) stabilization of system (58) is a difficult problem that cannot be addressed by existing methods in the literature However, a simple computation indicates that system (58) has a homogeneous approximation of the form (41) By Theorem 53, the local stabilization problem is solvable by nonsmooth output feedback Indeed, the homogeneous approximation of (58) is (59) with the degree and dilation Under this dilation, the remainders in (58) are actually highorder terms In fact, By Theorem 41, system (59) is stabilized by a homogenous output feedback controller of the form (43) (316), which is, in the present case, given by (510) According to Theorem 53, the same controller also renders system (58) locally strongly stable The simulation shown in Fig 3 confirms the conclusion The simulation was conducted with the parameters, and VI CONCLUSION In this paper, we have developed a nonsmooth output feedback framework that leads to a number of new results on local and global stabilization of nonlinear systems by output feedback The proposed output feedback control approach is nonsmooth in nature and couples effectively the tool of adding a power integrator [26] for the design of Hölder continuous state feedback controllers, and a novel recursive algorithm for the construction of nonsmooth nonlinear observers With the help of the new framework, we have identified appropriate conditions under which the problem of global stabilization is solvable by nonsmooth output feedback, for certain triangular systems that are not smoothly stabilizable, even locally, by any smooth state/output feedback (due to the presence of unobservable and uncontrollable unstable linearization) For a wider class of nonlinear systems beyond a triangular structure, such as systems in the -normal form or Hessenberg form [26][5], we proved that without imposing any growth condition, local stabilization by nonsmooth output feedback is possible This was done by means of the homogeneous systems theory The proof was constructive and carried out by explicitly designing output feedback stabilizers for the associated homogeneous systems The significance of our output feedback control schemes has been demonstrated by several examples whose local and global stabilization problems appear to be unsolvable by any existing method

13 QIAN AND LIN: RECURSIVE OBSERVER DESIGN, HOMOGENEOUS APPROXIMATION 1469 APPENDIX This section collects three useful lemmas that are frequently used throughout this paper It also includes proofs of the propositions introduced in Section III Lemma A1: Suppose is an odd integer Then, for any real numbers and (A1) Lemma A2: Suppose and are two positive real numbers, are continuous scalar-value functions Then, for any constant Lemma A3: Suppose a constant such that (A2) is an odd integer Then, there is (A3) The proofs of Lemmas A1 A3 are straightforward and hence left to the reader as an exercise The reader is also referred to [26] for details Proof of Proposition 31: As shown in [26], there are a Lyapunov function, which is positive definite and proper, and a set of virtual controllers, defined by Using the previous argument repeatedly, it is concluded that there is a constant satisfying (A8) Setting, (33) follows immediately from (A6) and (A8) Proof of Proposition 32: Given any, by Lemma A2 there is a constant such that (A9) Hence, (313) follows immediately from (A9) when When, a direct application of Young s inequality yields which, in turn, implies (313) Proof of Proposition 33: First of all, we prove that there is a constant such that (A10) This claim can be proved by an inductive argument Using (38) and Lemma A1, it is straightforward to show that (A10) is true for In fact When, from Lemma A1 it follows that (A4) with constants, and a Hölder continuous state feedback controller of the form such that (A5) Suppose at step,we have (A6) where is a positive real constant Putting (A5) and (A4) together, it is clear that there are appropriate constants such that (32) holds On the other hand, it follows from (A4) that Then, at step, there is a constant such that (A11) (A7)

14 1470 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 51, NO 9, SEPTEMBER 2006 Therefore, (A10) is true With the help of (A10) and Lemma A2, it can be shown that Now, we estimate each term in (A14) Using Lemma A3 yields This, together with Lemma A2, implies that for any, there are constants and so that for a constant Define Applying the relation (A11) to the last term in the previous inequality, we have Then, we arrive at (A15) (A12) In view of (A14), one can deduce easily from (A15) that (following the similar argument used in the proof of Proposition 33) from which (414) follows immediately, where the coefficients in (414) are given by (A13) Proof of Proposition 34: For any constant, it is clear from (39) that (A14) where and where the constants can be determined in a manner similar to (A13) and REFERENCES [1] A Bacciotti, Local Stabilizability of Nonlinear Control Systems Singapore: World Scientific, 1992 [2] A Bacciotti and L Rosier, Lyapunov Functions and Stability in Control Theory New York: Springer-Verlag, 2001, vol 267, Lecture Notes in Control and Info Sciences [3] R W Brockett, Asymptotic Stability and Feedback Stabilization, in Differential Geometric Control Theory, R W Brockett, R S Millman, and H J Sussmann, Eds Boston, MA: Birkäuser, 1983, pp [4] S Celikovsky and E Aranda-Bricaire, Constructive non-smooth stabilization of triangular systems, Syst Control Lett, vol 36, pp 21 37, 1983 [5] D Cheng and W Lin, On p-normal forms of nonlinear systems, IEEE Trans Autom Control, vol 48, no 7, pp , Jul 2003 [6] J M Coron and L Praly, Adding an integrator for the stabilization problem, Syst Control Lett, vol 17, pp , 1991 [7] W P Dayawansa, Recent advances in the stabilization problem for low dimensional systems, in Proc 2nd IFAC Nonlinear Control Systems Design Symp, Bordeaux, France, 1992, pp 1 8

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The Netherlands: Noordhoff, 1964 Chunjiang Qian (S 98 M 02 SM 04) received the BS and MS degrees in control theory from Fudan University, Shangai, China, in 1992 and 1994, respectively After working in industry for three years, he became a graduate student in the Department of Electrical Engineering and Computer Science, Case Western Reserve University, Cleveland, OH, where he received the PhD degree in 2001 Since August 2001, he has been with the Department of Electrical and Computer Engineering, The University of Texas at San Antonio, where he is now an Associate Professor His current research interests include robust and adaptive control, nonlinear systems and control, homogeneous systems theory, output feedback control, optimal control, and their applications to nonholonomic systems, underactuated mechanical systems, robotics, and biomechanical systems Dr Qian received the National Science Foundation CAREER Award in 2003 Currently, he is a Subject Editor of the International Journal of Robust and Nonlinear Control and a member of the IEEE CSS Conference Editorial Board Wei Lin (S 91 M 94 SM 99) received the DSc degrees in systems science and mathematics from Washington University, St Louis, MO, in 1993 He is currently a Professor in the Department of Electrical Engineering and Computer Science, Case Western Reserve University, Cleveland, OH He also held short-term visiting positions at a number of universities in the UK, Japan, Singapore, Hong Kong and China His research interests and publications can be found at: Dr Lin was a recipient of the NSF CAREER Award, the JSPS Fellow from Japan Society for the Promotion Science, and the Warren E Rupp Endowed Assistant Professorship of Science and Engineering He has served as an Associate Editor of the IEEE TRANSACTIONS ON AUTOMATIC CONTROL, a Guest Co-Editor of the Special Issue on New Directions in Nonlinear Control in the IEEE TRANSACTIONS ON AUTOMATIC CONTROL, and an Associate Editor of Automatica He was a Vice Program Chair for the 40th and 41st IEEE Conferences on Decision and Control, and a member of the Board of Governors of the IEEE Control Systems Society in Currently, he is a Subject Editor of the International Journal of Robust and Nonlinear Control, an Associate Editor of the Journal of Control Theory and Applications, and a member of the IFAC Technical Committee in Nonlinear Control