IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 5, MAY Bo Yang, Student Member, IEEE, and Wei Lin, Senior Member, IEEE (1.

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1 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 50, NO 5, MAY Robust Output Feedback Stabilization of Uncertain Nonlinear Systems With Uncontrollable and Unobservable Linearization Bo Yang, Student Member, IEEE, and Wei Lin, Senior Member, IEEE Abstract This paper investigates the problem of robust output feedback stabilization for a family of uncertain nonlinear systems with uncontrollable/unobservable linearization To achieve global robust stabilization via smooth output feedback, we introduce a rescaling transformation with an appropriate dilation, which turns out to be very effective in dealing with uncertainty of the system Using this rescaling technique combined with the nonseparation principle based design method, we develop a robust output feedback control scheme for uncertain nonlinear systems in the -normal form, under a homogeneous growth condition The construction of smooth state feedback controllers and homogeneous observers uses only the knowledge of the bounding homogeneous system rather than the uncertain system itself The robust output feedback design approach is then extended to a class of uncertain cascade systems beyond a strict-triangular structure Examples are provided to illustrate the results of the paper Index Terms Global robust stabilization, homogeneous observers, nonseparation principle design, rescaling transformation, smooth output feedback, uncertain nonlinear systems, uncontrollable/unobservable linearization I INTRODUCTION IN THIS paper, we consider the problem of global robust stabilization via a single smooth output feedback controller, for a family of uncertain systems of the form (11), and are the system input, state and output, respectively, and is an odd integer The mappings,, are functions that involve uncertainty and may not be precisely known It is worth to mention that a necessary and sufficient condition was first characterized in [3] and then extended in [21], for the Manuscript received April 12, 2004; revised December 6, 2004 Recommended by Associate Editor W Kang This work was supported in part by the National Science Foundation under Grants ECS and DMS , and in part by the Air Force Research Laboratory under Grant FA C-0110 The authors are with Department of Electrical Engineering and Computer Science, Case Western Reserve University, Cleveland, OH USA ( linwei@nonlinearcwruedu) Digital Object Identifier /TAC existence of a change of coordinates (diffeomorphism) and a state feedback control law transforming a smooth affine system and into the nonlinear system (11) with a suitable form of Therefore, (11) can be viewed as a generalized normal form of affine systems when exact feedback linearization is not possible For the class of affine systems that is topologically equivalent to (11), interesting stabilization results have been obtained over the years For example, local and global asymptotic stabilization of the system (11) with, and,, using smooth state feedback, were investigated in [4] and [2], respectively In the -dimensional case, a globally stabilizing smooth state feedback control law was explicitly designed in [17], for a class of nonlinear systems (11) under appropriate growth conditions that can be regarded as a high order version of feedback linearizable condition Much of the literature on stabilization of nonlinear systems has focused on the design of state feedback In the past two years, research efforts toward the development of output feedback control schemes for nonlinear systems with uncontrollable/unobservable linearization have gained momentum Reference [20] studied the global stabilization of the nonlinear system (11) in the plane via smooth output feedback Under suitable conditions imposed on,, a reduced-order nonlinear observer was designed in [20], resulting in a globally stabilizing, smooth dynamic output compensator Notably, the output feedback design in [20] does not rely on the separation principle Instead, it uses the idea of coupled controller observer construction In [7], Dayawansa proved the existence of a smooth output feedback stabilizer for the three-dimensional system (11) when and, His proof is based on the theory of homogeneous systems [9], [1] and some elegant design techniques from [5], [6], [12], [10], and [22] More recently, we have shown that for the -dimensional nonlinear system (11) with,, which is homogeneous, the problem of global stabilization is solvable by smooth output feedback [23] This was done by developing a new observe design technique for the construction of a homogeneous observer, combined with the method [17] for the design of a smooth state feedback controller When, and satisfy a global Lipschitz-like condition, we further showed that global stabilization of the nonhomogeneous /$ IEEE

2 620 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 50, NO 5, MAY 2005 system (11), which is not uniformly observable [8], can still be achieved via smooth output feedback [23] A key ingredient of the output feedback control strategy in [23] is the development a recursive algorithm for the design of homogeneous observers, which makes it possible to assign the gains of the homogeneous observer one-by-one, in a step-by-step manner Although such an observer design is substantially different from the Luenberger or high-gain observer design [13], [8], [14], [16], [11], [15], it still uses a copy of the original system and hence requires the precise information of the controlled plant In other words, the nonlinear functions,, in (11) must be independent of and involve no uncertainty As a result, the output feedback control scheme in [23] is not robust with respect to parametric or structural uncertainty Moreover, it cannot be applied to uncertain nonlinear systems such as (11) The main purpose of this paper is to address the robust issue discussed previously, and to develop a robust output feedback control scheme for a family of uncertain systems (11) under suitable growth conditions In the case of, such robust control problems have been studied, for instance, in [19] under a linear growth condition In view of the work [19], it appears to be natural to impose the homogeneous growth condition below, and to investigate the question whether global robust stabilization of the uncertain nonlinear system (11) is achievable by smooth output feedback Assumption 11: There exists a real constant such that (12) One of the objectives of this paper is to address this question and to provide an affirmative answer by constructing, under Assumption 11, a smooth dynamic output compensator (13) which globally robustly stabilizes the entire family of uncertain systems (11) Since Assumption 11 is weaker than the higher-order type of the global Lipschitz condition given in [23] (see Remark 35), the class of nonlinear systems considered in this paper is larger than those studied in [23] More significantly, because the design of the dynamic output compensator (13) uses only the knowledge of the upper bound of, ie, the condition (12) instead of itself, global output feedback stabilization will be achieved in a robust fashion, that is, in a manner which is not sensitive to perturbations and parametric uncertainty in the system This is one of the major differences between [23] and this paper The key for achieving robustness is the introduction of a rescaling technique with a subtle dilation, which transforms the original system (11) into a rescaled one for which a dynamic output compensator can be constructed using the output feedback design method in [23], with a suitable twist, in particular, by discarding the system uncertainty when designing homogeneous observers With the help of the rescaling technique, the uncertain nonlinearities in (11) can be dominated easily by tuning the rescaling factor In the case of uncertain systems with controllable/observable linearization (ie, ), the new design method provides not only a deeper insight but also an interesting alternative solution to the robust output feedback stabilization problem considered in [19] The other goal of the paper is to show how robust output feedback control strategies can be developed, under appropriate conditions, for a wider class of uncertain nonlinear systems with uncontrollable and unobservable linearization in the -normal form (31) and cascade form (412), which go beyond a triangular structure Several examples are given to demonstrate the applications of the robust output feedback design method proposed in this paper II ROBUST OUTPUT FEEDBACK DESIGN: THE CASE OF To better understand how global robust stabilization of the uncertain nonlinear system (11) with uncontrollable/unobservable linearization can be achieved by smooth output feedback under Assumption 11, we revisit a simple situation of (11), ie, the case when the first approximation of (11) is controllable and observable In this case, the uncertain system (11) can be rewritten as and Assumption 11 reduces to the linear growth condition (21) (22) In [19], we have shown that global robust stabilization of the uncertain system (21) satisfying (22) is solvable by a linear output dynamic compensator The proof was not based on the separation principle but instead, relied on a coupled controllerobserver design [19] Due to the linear nature of [19], it is, however, not easy to extend the output feedback design approach of [19] to a family of uncertain nonlinear systems such as (11) In this section, we explore an alternative output feedback control strategy that takes advantage of homogeneity of the system and, hence, might be naturally extended, in an intuitive and transparent manner, to the uncertain nonlinear system (11) with To this end, we introduce a rescaling transformation with a suitable dilation for the original system (21), which is motivated by[24] and turns out to be crucial for dominating the uncertainty of (21) To be precise, let (23) is a rescaling factor to be determined later Under the new coordinates s, the uncertain system (21) can be expressed as (24)

3 YANG AND LIN: ROBUST OUTPUT FEEDBACK STABILIZATION OF UNCERTAIN NONLINEAR SYSTEMS 621 By the hypothesis (22), the uncertain functions,, also satisfy the linear growth condition Consequently, the unmeasurable state of (24) can be estimated as follows: (211) By construction, the reduced-order observer (210) (211) is implementable Let,, be the estimate errors Then, the error dynamics is given by (25) For the rescaled uncertain system (24) with the constraint (25), it is straightforward to design recursively, in a fashion similar to the one in [19], a linear state feedback controller such that (26) (27) is a quadratic Lyapunov function,,, and with and being known constants independent of Next, we will design a linear observer for the rescaled system (24) Because is measurable and only unmeasurable states of (24) are, it is natural to design an -dimensional observer rather than a full-order observer Motivated by the reduced-order observer design for linear systems, we shall build an th-order linear observer to estimate, instead of the states, the unmeasurable variables defined by (212) Inspired by the certainty equivalence principle, we replace the unmeasurable state in the controller (26) by its estimate generated by the observer (210) (211) In this way, we get the following implementable controller: Substituting (213) into (27) yields is a constant independent of Now, consider the Lyapunov function (213) (214) for the closed-loop system (213) (212) (24) A simple calculation results in (28) are gain constants to be assigned later From (28), it is clear that (29) (215) Using the completion of square, together with the linear growth condition (25), it is not difficult to deduce from (215) (214) that In view of (29), we construct the th-order linear observer (regardless of the uncertain terms, ) (216) (210) is the estimates of the unmeasurable state are positive constants independent of, while and, are known constants independent of and all s

4 622 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 50, NO 5, MAY 2005 Choosing the gain parameters and one-by-one, in the order of as follows: we have (217) Hence, the uncertain system (21) is globally robustly stabilizable by the linear output feedback controller (210) (213) Remark 21: Notably, the reduced-order observer (210) is different from the one in [19] in two respects: 1) it involves gain parameters to be designed Only when and, (210) reduces to a traditional reduced-order high-gain observer; and 2) it contains a rescaling factor that is similar to the one introduced in [24] The rescaling factor creates an extra design freedom and turns out to be very effective in dealing with uncertainty of (21), under the linear growth condition (22) III ROBUST OUTPUT FEEDBACK DESIGN: THE -NORMAL FORM CASE Interestingly, the robust output feedback control scheme developed so far for the uncertain system (21) with controllable/observable linearization can be carried over, in a parallel manner, to its high order counterpart In this section, we show that in spite of the lack of controllability and observability in the first approximation, a robust output feedback control method can be developed for a family of uncertain systems such as (11) satisfying the growth condition (12) To this end, we first present a robust stabilization result for the following class of nonlinear systems: global robust stabilization of (31) by output feedback, we introduce the following hypothesis which is a natural generalization of the homogeneous growth condition (12) Assumption 31: There exists a constant such that (32) when Moreover, when With the aid of the growth condition (32), it is possible to establish the following output feedback stabilization theorem which is one of the main results of this paper Theorem 32: Under Assumption 31, there exists a smooth dynamic output compensator (13) making the uncertain system (31) globally asymptotically stable Proof: Similar to the philosophy of the previous section, the proof of this theorem is carried out by explicitly designing a robust smooth state feedback controller, and a homogeneous observer that does not require the knowledge of the system uncertainty, ie, The construction of the observer is substantially different from the one in [23] in the sense that here no copy of is used for the design of a robust observer, while the nonlinear observer in [23] did involve a copy of For this reason, in the work [23] must be known precisely Another new ingredient of our output feedback design is the development of a rescaling technique for handling the uncertain terms in the system (31) In particular, a higher order rescaling transformation with a suitable dilation is employed to deal with the system uncertainty effectively For the convenience of the reader, we break up the proof into three parts i) Rescaling of the -Normal Form: Observe that by the homogeneous systems theory (see, for instance, [9], [12], [10], and [22]), system (31) is homogeneous with dilation and degree when, Keeping this in mind and motivated by (23), we introduce the following rescaling transformation: (33) with dilation, is a rescaling factor to be assigned later In the rescaled coordinates s, the uncertain system (31) can be represented as (31) called -normal form [3],, and are the system input, state and output, respectively, and is an odd integer The mappings,, are, involve uncertainty and may be unknown As shown in [3] and [21], every smooth affine system is, under appropriate conditions, feedback equivalent to (31) To achieve (34)

5 YANG AND LIN: ROBUST OUTPUT FEEDBACK STABILIZATION OF UNCERTAIN NONLINEAR SYSTEMS 623 Using (32) and the fact that, it is easy to see that With the help of Lemma 65 and computation gives, a direct (39) is a constant independent of Substituting (39) into (38), we deduce from Lemmas 61 and 63 that (35) Applying Lemma 61 to (35), we obtain the following estimations ( ): (310) is a constant independent of Thus, the virtual controller, with being a constant independent of, is such that (36) is a known constant independent of In this way, a new parameter the rescaling factor is introduced for the design of dynamic output compensators It creates an extra freedom and plays an important role in dealing with the system uncertainty, ie,,, in (34) ii) State Feedback Design: For the rescaled system (34) satisfying the growth condition (36), one can construct a robust state feedback controller via the smooth feedback design method [17] Let with and choose the Lyapunov function Then, it is easy to deduce from (36) that (311) Using an inductive argument similar to the one in [17], one can find a set of virtual controllers, transformations, and Lyapunov functions and a smooth state feedback control law such that (312) (313) Keeping in mind, the virtual controller, with being a constant independent of, results in (37) Next, let and choose the Lyapunov function Using (34), (36), and (37), we have (314) all the constants and are known and independent of iii) Output Feedback Design: Since of the rescaled system (34) are unmeasurable but is measurable, we need only to design an -dimensional observer for (34) However, the reduced-order observer design method in [23] cannot be applied to the rescaled system (34) because it uses a copy of,, which are time-varying and not precisely known Motivated by the robust observer design in Section II, in what follows we shall construct an -dimensional robust homogeneous observer to estimate, instead of the states, the unmeasurable variables the parameters determined later From (315), it follows that (315) are gain constants to be (38) (316)

6 624 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 50, NO 5, MAY 2005 In view of (316), one can construct the homogeneous observer -dimensional To determine the observer gains change of coordinates, consider the (317) which does not involve the uncertain functions in (34) This is substantially different from the homogeneous observer proposed in [23] By construction, the reduced-order observer (317) is implementable Moreover, the estimates of s can be obtained based on the relationships (318) Let,, be the estimate errors Then, the error dynamics is given by (322) In the coordinates of and, (321) can be represented as (by Lemma 63) (323) are positive real constants independent of, and is a known constant independent of and all the s The error dynamics (319) in the coordinate can be rewritten as (319) By the certainty equivalence principle, the unmeasurable state in the controller (313) can be replaced by its estimate generated by the nonlinear observer (317) (318) In this way, one obtains the implementable feedback controller (320) Substituting (320) into (314) and using Lemmas 65 and 63, we have Now, consider the Lyapunov function A direct computation gives (324) (325) Similar to [23], it is not difficult to get the following estimations for each term on the right-hand side of (325): is a real constant related to s and independent of With the aid of (312) and Lemmas 63 and 61, the aforementioned inequality can be simplified as (321) is a constant independent of

7 YANG AND LIN: ROBUST OUTPUT FEEDBACK STABILIZATION OF UNCERTAIN NONLINEAR SYSTEMS 625, are positive constants independent of s and From (328), it is easy to conclude that if the gain parameters s and are assigned one-by-one, in the following manner: (329) (326) are positive constants independent of Substituting (326) into (325), a tedious but straightforward calculation leads to (327) are positive constants independent of, while and, are known constants independent of s and Finally, choose the Lyapunov function we have (330) This, in turn, implies that the uncertain high order system (31) is globally asymptotically stabilized by the dynamic output compensator (317) (320) As a consequence of Theorem 32, we have the following important result that provides a solution to the global stabilization problem of system (11) Corollary 33: For a family of uncertain systems (11) satisfying Assumption 11, there exists a smooth output feedback controller of the form (13), such that the closed-loop system (11) (13) is globally asymptotically stable at the equilibrium Proof: Corollary 33 follows immediately from Theorem 32 if one observes that system (11) is a special case of the uncertain system (31) and Assumption 31 reduces to Assumption 11 when for, The reader is referred to [18] for further details The application of Corollary 33 and the main features of the robust smooth output feedback control scheme developed so far can be illustrated by the following example Example 34: Consider the uncertain planar system for the closed-loop system in the coordinates follows from (323) and (327) that Then, it (331) (328) are positive constants independent of, while and, is a continuous time-varying function satisfying Clearly, global output feedback stabilization of the uncertain system (331) is a difficult problem for two reasons: 1) it requires the design of a single output feedback controller to stabilize a family of nonlinear systems, due to the presence of the time-varying parameter ; and 2) the output feedback control schemes proposed recently [23] cannot be applied to the uncertain system (331), because of the lack of effective design methods for the construction of robust observers and/or output compensators for uncertain nonlinear systems with uncontrollable/unobservable linearization

8 626 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 50, NO 5, MAY 2005 On the other hand, a simple calculation shows that the uncertain system (331) satisfies the homogeneous growth condition (12) Indeed By Corollary 33, a reduced-order dynamic output compensator can be designed as follows: is an odd integer The functions and,, are To tackle the problem of global robust stabilization by smooth output feedback for the cascade system (41), we make the following assumptions Assumption 41: There is a Lyapunov function, which is positive definite and proper, such that (332) which globally robustly stabilizes system (331) It should be pointed out that unlike in [23], the design of the output feedback controller (332) does not require the knowledge of the uncertain term This is substantially different from the work [23], a copy of the term must be used for the construction of a nonlinear observer As a result, the output feedback control scheme proposed in [23] is not robust and cannot be employed to control uncertain systems such as (331) Remark 35: It is worth pointing out that even in the absence of uncertainty, Corollary 33 is new Moreover, it has incorporated and generalized the output feedback stabilization results obtained previously in [23] Specifically, Corollary 33 can be applied to a larger class of nonlinear systems such as and because both of them satisfy the homogeneous growth condition (12) However, none of them satisfies the higher-order version of global Lipschitz-like condition, ie, [23, Ass 51] As a result, [23, Th 53] cannot be employed here to solve the output feedback stabilization problem for the two nonlinear systems shown previously IV OUTPUT FEEDBACK STABILIZATION OF UNCERTAIN CASCADE SYSTEMS The purpose of this section is to investigate how the robust output feedback stabilization result obtained in the previous section can be extended to a family of uncertain cascade systems of the form is a real constant Assumption 42: For and (42) (43) Clearly, Assumption 41 is a sort of ISS-like condition, while Assumption 42 is an extension of the homogeneous growth condition (32) With the help of these two conditions, we can prove the following result on global output feedback stabilization of the uncertain cascade system (41) Theorem 43: Under Assumptions 41 42, the uncertain cascade system (41) is globally asymptotically stabilizable by smooth output feedback Proof: The proof of this result is similar to that of Theorem 32 A key difference lies in the design of a partial-state rather than full-state feedback controller for the uncertain cascade system (41) For this reason, we give only a sketch of the proof highlighting the difference As done in the proof of Theorem 32, we first introduce a rescaling transformation that is composed of and (33) for the uncertain system (41) Such a transformation gives (44) In view of Assumption 42, it is straightforward to show that the uncertainty of system (44) satisfies the constraint (45) (41) and are the system input and output, respectively, and are the system states, and is a known constant independent of For the rescaled system (44) with the constraint (45), it can be proved that Assumptions 41 and 42 imply the existence of a globally stabilizing, partial-state feedback controller

9 YANG AND LIN: ROBUST OUTPUT FEEDBACK STABILIZATION OF UNCERTAIN NONLINEAR SYSTEMS 627 To see why this is the case, consider the Lyapunov function Then, it follows from (42) that (46) (47) Because the states are unmeasurable, the controller (49) cannot be directly implemented To obtain an implementable controller, we design an -dimensional observer for recovering of the rescaled system (44) Motivated by the robust observer design in the last section, we ignore the uncertain terms,, in system (44) and construct the dynamic output compensator Let and choose the Lynapunov function Using the fact and (47), one deduces from (44) (45) and Young s inequality that (411) are the observer gains to be assigned The remaining part of the proof is to determine the parameters as well as the rescaling factor, which is analogous to that of Theorem 32 and therefore left to the reader as an exercise In conclusion, one can prove that by suitably choosing the gain constants and one-byone, the closed-loop system (44) (411) can be rendered globally asymptotically stable at the origin Clearly, in the case when an uncertain system is of the form is a constant independent of Clearly, the virtual controller, with being a constant independent of, yields Assumption 42 reduces to the following Assumption 44: There exists a constant (412) such that Using a similar inductive argument we conclude at the th step that there exist a set of transformations (48) a Lyapunov function and a partial-state feedback control law of the form (49) such that (413) Then, we have the following useful corollary Corollary 45: Under Assumptions 41 and 44, the uncertain cascade system (412) is globally robustly stabilizable by smooth output feedback We conclude this section with an example that illustrates how Theorem 43 can be employed to solve the difficult problem of global robust stabilization by smooth output feedback, for uncertain cascade systems beyond a triangular structure Example 46: Consider the uncertain cascade system (410) all the parameters and are known constants independent of Note that inequality (410) reduces to (314) in the absence of -dynamics (414) is an unknown constant bounded by a known constant, for instance, by one Note that this nonlinear system has three key features that make global output feedback stabilization of (414) subtle First of all, system (414) is not in a lower triangular form due to

10 628 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 50, NO 5, MAY 2005 the second dynamic equation Second, the linearized system of (414) is given by both and are positive constants independent of Next, let with being a gain constant to be assigned later Since which is neither controllable nor observable While the latter makes the current output feedback design methods hard to be applied to system (414), the former prevents an application of the output feedback control scheme developed recently [23] to the cascade system (414) Finally, the presence of the unknown constant requires that a robust output feedback control scheme be used, and therefore the output feedback design method proposed in [23] cannot be applied to (414), due to the nature of the nonrobust design On the other hand, it is easy to see that the three-dimensional cascade system (414) is of the form (41) with and, and satisfies the growth condition (43) Moreover, the ISS-like inequality (42) holds for the -subsystem of (414) As a matter of fact, using the Lyapunov function and Lemma 61 yields we design the reduced-order observer (416) which is a copy of -dynamics without the uncertain term Using thus obtained and the certainty equivalence principle, we deduce from that (417) Finally, we show that the dynamic output compensator (416) (417) globally robustly stabilizes the uncertain cascade system (415) for all, if and are chosen suitably To this end, let be the estimate error The error dynamics is Moreover, the uncertain term Thus Choose the Lyapunov function Then That is, Assumption 42 holds By Theorem 43, there exists a dynamic output compensator of the form (13) such that the closed-loop system is globally asymptotically stable In what follows, a detailed design procedure is presented for the purpose of illustration First, we introduce the rescaling transformation,,, and, is a rescaling factor to be determined later Such a transformation results in Selecting yields (418) (415) Using the smooth state feedback design method in [17], one can find a Lyapunov function of the form is a constant independent of Now, consider the Lyapunov function for the closed-loop system (415) (416) (417) Using Lemmas 61 65, it is not difficult to prove that and a partial-state feedback controller, such that In view of the relationship (418) and, it is clear that the choices and result in

11 YANG AND LIN: ROBUST OUTPUT FEEDBACK STABILIZATION OF UNCERTAIN NONLINEAR SYSTEMS 629 Fig 1 Transient responses of the closed-loop system (414) (419) with =1and the initial condition (; ; ; ^z )=(1; 0:3;06;05) which implies global asymptotic stability of the closed-loop system (415) (417) for all The aforementioned design procedure leads to, for instance, the dynamic output compensator (419) that does the job The simulation shown in Fig 1 demonstrates the property of robust stability of the closed-loop system (414) (419) V CONCLUSION This paper has proved that under appropriate homogeneous growth conditions, global robust stabilization by smooth output feedback can be achieved for a family of uncertain nonlinear systems that are not uniformly observable [8] and have uncontrollable and unobservable linearization A robust output feedback design approach has been developed based on a rescaling technique and the idea of nonseparation principle design, enabling one to recursively construct a robust state feedback controller and a homogeneous observer that does not depend on the uncertainty of the system The main result of this paper has incorporated and generalized the robust output feedback stabilization theorem in [19], global robust stabilization was shown to be possible for a family of uncertain systems with controllable/observable linearization For high-order uncertain systems in a cascade form or in the so-called -normal form (which are beyond a strict-triangular structure), we have also identified suitable conditions for the problem of global robust stabilization to be solvable by smooth output feedback The applications of the proposed robust output feedback control schemes have been illustrated by several examples (also, see Remark 35) APPENDIX This section collects several useful lemmas that play a key role in deriving the main results of this paper Lemma 61: Given positive real numbers,,,,, and, the following inequality holds: Lemma 62: Given positive real numbers,,,,, and, the following inequality holds: Lemma 61 and 62 can be easily proved by Young s Inequality Lemma 63: Let be real numbers Then Lemma 64: Let and be any real numbers and be an odd integer Then, the following inequality holds: Lemma 65: For all and any odd positive integer, the following inequality holds:

12 630 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 50, NO 5, MAY 2005 The proofs of Lemmas are not difficult and, hence, left to the reader as an exercise REFERENCES [1] A Bacciotti, Local Stabilizability of Nonlinear Control Systems Singapore: World Scientific, 1992 [2] C I Byrnes and A Isidori, New results and examples in nonlinear feedback stabilization, Syst Control Lett, vol 12, pp , 1989 [3] D Cheng and W Lin, On p Normal forms of nonlinear systems, IEEE Trans Autom Control, vol 48, no 7, pp , Jul 2003 [4] P Crouch and M Irving, On sufficient conditions for local asymptotic stability of nonlinear systems whose linearization is uncontrollable, Univ Warwick, Warwick, UK, Control Theory Centre Rep, 1983 [5] W Dayawansa, Recent advances in the stabilization problem for low dimensional systems, in Proc 2nd IFAC Symp Nonlinear Control System Design, Bordeaux, France, 1992, pp 1 8 [6] W Dayawansa, C Martin, and G Knowles, Asymptotic stabilization of a class of smooth two dimensional 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IEEE Conf Decision Control, Maui, HI, 2003, pp [22] L Rosier, Homogeneous Lyapunov functions for homogeneous continuous vector field, Syst Control Lett, vol 19, pp , 1992 [23] B Yang and W Lin, Homogeneous observers, iterative design and global stabilization of high order nonlinear systems by smooth output feedback, IEEE Trans Autom Control, vol 49, no 7, pp , Jul 2004 [24], Further results on global stabilization of uncertain nonlinear systems by output feedback, in Proc 6th IFAC Symp Nonlinear Control Systems, Stuttgart, Germany, 2004, pp Also, a full version of the paper can be found in Int J Rob Nonlinear Control, vol 15, pp Bo Yang (S 01) was born in China in 1975 He received the BS degree in mathematics and MS degree in control theory, both from Fudan University, China, in 1996 and 1999, respectively After having worked in industry for two years, he is currently working toward the PhD degree in the Department of Electrical Engineering and Computer Science, Case Western Reserve University, Cleveland, OH His research interests include design of nonlinear observers, output feedback control of nonlinear systems with uncontrollable/unobservable linearization, robust and adaptive control, and homogeneous systems theory, with applications to nonholonomic, underactuated mechanical systems, robotics, and biologically inspired systems Wei Lin (S 91 M 94 SM 99) received the DSc degree in systems science and mathematics from Washington University, St Louis, MO, in 1993 During 1986 to 1989, he was a Lecturer in the Department of Mathematics, Fudan University, Shanghai, China From 1994 to 1995, he worked as a Postdoctoral Research Associate at Washington University Since Spring 1996, he has been a Faculty Member 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, Hong Kong, and China His research interests include nonlinear control, dynamic systems, homogeneous systems theory, nonlinear observer design, robust and adaptive control, and their applications to underactuated mechanical systems, nonholonomic systems, biologically inspired systems, and systems biology His recent research focus has been on the development of nonsmooth state/output feedback design methodologies for the control of nonlinear systems that cannot be dealt with by any linear or smooth feedback Dr Lin is a recipient of the National Science Foundation CAREER Award and of the Japan Society for the Promotion Science Fellow He was a Vice Program Chair (Short Papers) of the 2001 IEEE Conference on Decision and Control and a Vice Program Chair (Invited Sessions) of the 2002 IEEE Conference on Decision and Control He has served as an Associate Editor of the IEEE TRANSACTIONS ON AUTOMATIC CONTROL and a Guest Co-Editor of the Special Issue on New Directions in Nonlinear Control in the IEEE TRANSACTIONS ON AUTOMATIC CONTROL Currently, he is an Associate Editor of Automatica, a Subject Editor of Int J of Robust and Nonlinear Control, an Associate Editor of Journal of Control Theory and Applications, and a Member of the Board of Governors of the IEEE Control Systems Society