Robust Output Feedback Stabilization of Uncertain Nonlinear Systems ...
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 5, MAY 2005
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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
I
N 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
.. .
(1.1) , and are the system input, state and where is an odd integer. The mappings output, respectively, and , , 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-0400413 and DMS-0203387, and in part by the Air Force Research Laboratory under Grant FA8651-05-C-0110. The authors are with Department of Electrical Engineering and Computer Science, Case Western Reserve University, Cleveland, OH 44106 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2005.847084
existence of a change of coordinates (diffeomorphism) and a state feedback control law transforming a smooth affine system and into the nonlinear system (1.1) with a suitable form of . Therefore, (1.1) 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 (1.1), interesting stabilization results have been obtained over the years. For example, local and global asymptotic stabiliza, and , tion of the system (1.1) with , 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 (1.1) 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 (1.1) in the plane via smooth output feedback. Under , , a resuitable conditions imposed on duced-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 and , three-dimensional system (1.1) when . 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 (1.1) 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 de, sign of a smooth state feedback controller. When and satisfy a global Lipschitz-like condition, we further showed that global stabilization of the nonhomogeneous
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system (1.1), 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, , , in (1.1) must the nonlinear functions 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 (1.1). 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 (1.1) under , such robust suitable growth conditions. In the case of 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 (1.1) is achievable by smooth output feedback. Assumption 1.1: There exists a real constant such that (1.2) One of the objectives of this paper is to address this question and to provide an affirmative answer by constructing, under Assumption 1.1, a smooth dynamic output compensator (1.3) which globally robustly stabilizes the entire family of uncertain systems (1.1). Since Assumption 1.1 is weaker than the higher-order type of the global Lipschitz condition given in [23] (see Remark 3.5), 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 (1.3) , i.e., the uses only the knowledge of the upper bound of condition (1.2) 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 (1.1) 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 in (1.1) can be dominated easily uncertain nonlinearities by tuning the rescaling factor. In the case of uncertain systems ), the new with controllable/observable linearization (i.e., 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 (3.1) and cascade form (4.12), 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 (1.1) with uncontrollable/unobservable linearization can be achieved by smooth output feedback under Assumption 1.1, we revisit a simple situation of (1.1) where , i.e., the case when the first approximation of (1.1) is controllable and observable. In this case, the uncertain system (1.1) can be rewritten as .. .
(2.1) and Assumption 1.1 reduces to the linear growth condition (2.2) In [19], we have shown that global robust stabilization of the uncertain system (2.1) satisfying (2.2) 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 (1.1). 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 (1.1) with . To this end, we introduce a rescaling transformation with a suitable dilation for the original system (2.1), which is motivated by[24] and turns out to be crucial for dominating the uncertainty of (2.1). To be precise, let (2.3) is a rescaling factor to be determined later. where Under the new coordinates ’s, the uncertain system (2.1) can be expressed as .. .
(2.4)
YANG AND LIN: ROBUST OUTPUT FEEDBACK STABILIZATION OF UNCERTAIN NONLINEAR SYSTEMS
By the hypothesis (2.2), the uncertain functions , also satisfy the linear growth condition
,
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of (2.4)
Consequently, the unmeasurable state can be estimated as follows:
(2.11) By construction, the reduced-order observer (2.10)–(2.11) is implementable. , , be the Let estimate errors. Then, the error dynamics is given by
.. .
(2.5) For the rescaled uncertain system (2.4) with the constraint (2.5), it is straightforward to design recursively, in a fashion similar to the one in [19], a linear state feedback controller
.. .
(2.12)
(2.6) such that (2.7) where function,
,
Inspired by the certainty equivalence principle, we replace the unmeasurable state in the controller (2.6) by its generated by the observer (2.10)–(2.11). estimate In this way, we get the following implementable controller:
is a quadratic Lyapunov , and
(2.13) Substituting (2.13) into (2.7) yields
with and being known constants independent of . Next, we will design a linear observer for the rescaled system is measurable and only unmeasurable (2.4). Because states of (2.4) 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, th-order linear observer to estimate, we shall build an instead of the states , the unmeasurable variables defined by
(2.14) where is a constant independent of Now, consider the Lyapunov function
.
for the closed-loop system (2.13)–(2.12)–(2.4). A simple calculation results in
(2.8) are gain constants to be assigned later. where From (2.8), it is clear that (2.15) .. .
Using the completion of square, together with the linear growth condition (2.5), it is not difficult to deduce from (2.15)–(2.14) that (2.9)
In view of (2.9), we construct the (regardless of the uncertain terms
,
th-order linear observer )
.. .
(2.16) (2.10) is the estimates of the unmeasurable state
where .
where , while positive constants independent of , are known constants independent of all ’s.
are and and
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Choosing the gain parameters and as follows: order of
one-by-one, in the
global robust stabilization of (3.1) by output feedback, we introduce the following hypothesis which is a natural generalization of the homogeneous growth condition (1.2). such that Assumption 3.1: There exists a constant (3.2)
.. . (2.17) we have
Hence, the uncertain system (2.1) is globally robustly stabilizable by the linear output feedback controller (2.10)–(2.13). Remark 2.1: Notably, the reduced-order observer (2.10) is different from the one in [19] in two respects: 1) it involves gain parameters to be designed. Only when and , (2.10) reduces to a traditional reduced-order high-gain observer; and 2) it contains that is similar to the one introduced in a rescaling factor [24]. The rescaling factor creates an extra design freedom and turns out to be very effective in dealing with uncertainty of (2.1), under the linear growth condition (2.2). III. ROBUST OUTPUT FEEDBACK DESIGN: THE -NORMAL FORM CASE Interestingly, the robust output feedback control scheme developed so far for the uncertain system (2.1) 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 (1.1) satisfying the growth condition (1.2). To this end, we first present a robust stabilization result for the following class of nonlinear systems:
when . Moreover, where when . With the aid of the growth condition (3.2), it is possible to establish the following output feedback stabilization theorem which is one of the main results of this paper. Theorem 3.2: Under Assumption 3.1, there exists a smooth dynamic output compensator (1.3) making the uncertain system (3.1) 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 un. The construction of the observer is certainty, i.e., substantially different from the one in [23] in the sense that here is used for the design of a robust obno copy of server, while the nonlinear observer in [23] did involve a copy . For this reason, in the work [23] of 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 (3.1). 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 (3.1) is homogeneous with dilation and degree when , . Keeping this in mind and motivated by (2.3), we introduce the following rescaling transformation:
(3.3) with dilation , where is a rescaling factor to be assigned later. In the rescaled coordinates ’s, the uncertain system (3.1) can be represented as
.. .
.. . (3.1) called -normal form [3], where , 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 (3.1). To achieve
(3.4) where
YANG AND LIN: ROBUST OUTPUT FEEDBACK STABILIZATION OF UNCERTAIN NONLINEAR SYSTEMS
Using (3.2) and the fact that
, it is easy to see that
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, a direct
With the help of Lemma 6.5 and computation gives
(3.9) is a constant independent of . Substituting (3.9) where into (3.8), we deduce from Lemmas 6.1 and 6.3 that
(3.10) (3.5) Applying Lemma 6.1 to (3.5), we obtain the following estimations ( ):
(3.6) where 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 , , in (3.4). with the system uncertainty, i.e., ii) State Feedback Design: For the rescaled system (3.4) satisfying the growth condition (3.6), one can construct a robust state feedback controller via the smooth feedback design method [17]. with and choose the Lyapunov Let Then, it is easy to deduce from (3.6) function that
Keeping , with independent of , results in
in mind, the virtual controller being a constant
(3.7) Next, let function we have
and choose the Lyapunov Using (3.4), (3.6), and (3.7),
where is a constant independent of . , with Thus, the virtual controller being a constant independent of , is such that
(3.11) Using an inductive argument similar to the one in [17], one can find a set of virtual controllers, transformations, and Lyapunov functions
.. .
.. .
.. . (3.12)
and a smooth state feedback control law (3.13) such that
(3.14) and are known and where all the constants independent of . iii) Output Feedback Design: Since of the is measurrescaled system (3.4) are unmeasurable but -dimensional observer able, we need only to design an for (3.4). However, the reduced-order observer design method in [23] cannot be applied to the rescaled system (3.4) because , , which are time-varying it uses a copy of 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, , the unmeasurable variables instead of the states (3.15) where the parameters determined later. From (3.15), it follows that
are gain constants to be
.. . (3.8)
(3.16)
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In view of (3.16), one can construct the homogeneous observer
-dimensional
To determine the observer gains change of coordinates
, consider the
(3.22)
.. . In the coordinates of Lemma 6.3)
and , (3.21) can be represented as (by
(3.17) which does not involve the uncertain functions in (3.4). This is substantially different from the homogeneous observer proposed in [23]. By construction, the reduced-order observer (3.17) is implementable. Moreover, the estimates of ’s can be obtained based on the relationships (3.18) Let , Then, the error dynamics is given by
, be the estimate errors.
.. .
(3.23)
are where positive real constants independent of , and is a and all the ’s. known constant independent of The error dynamics (3.19) in the coordinate can be rewritten as
.. .
(3.19) By the certainty equivalence principle, the unmeasurable state in the controller (3.13) can be replaced by generated by the nonlinear observer its estimate (3.17)–(3.18). In this way, one obtains the implementable feedback controller (3.20)
(3.24) Now, consider the Lyapunov function
A direct computation gives
Substituting (3.20) into (3.14) and using Lemmas 6.5 and 6.3, we have
(3.25)
Similar to [23], it is not difficult to get the following estimations for each term on the right-hand side of (3.25): where is a real constant related to ’s and independent of . With the aid of (3.12) and Lemmas 6.3 and 6.1, the aforementioned inequality can be simplified as
(3.21)
where
is a constant independent of
.
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, are positive constants independent of ’s and . From (3.28), it is easy to conclude that if the gain parameters ’s and are assigned one-by-one, in the following manner:
.. .
(3.29) (3.26) are where positive constants independent of . Substituting (3.26) into (3.25), a tedious but straightforward calculation leads to
(3.27)
are where , while and positive constants independent of , are known constants independent of ’s and . Finally, choose the Lyapunov function
for the closed-loop system in the coordinates follows from (3.23) and (3.27) that
we have (3.30) This, in turn, implies that the uncertain high order system (3.1) is globally asymptotically stabilized by the dynamic output compensator (3.17)–(3.20). As a consequence of Theorem 3.2, we have the following important result that provides a solution to the global stabilization problem of system (1.1). Corollary 3.3: For a family of uncertain systems (1.1) satisfying Assumption 1.1, there exists a smooth output feedback controller of the form (1.3), such that the closed-loop system (1.1)–(1.3) is globally asymptotically stable at the equilibrium . Proof: Corollary 3.3 follows immediately from Theorem 3.2 if one observes that system (1.1) is a special case of the uncertain system (3.1) and Assumption 3.1 reduces to Assumption for , . 1.1 when The reader is referred to [18] for further details. The application of Corollary 3.3 and the main features of the robust smooth output feedback control scheme developed so far can be illustrated by the following example. Example 3.4: Consider the uncertain planar system
. Then, it (3.31)
(3.28)
where positive constants independent of
, while
are and
, is a continuous time-varying function satiswhere . fying Clearly, global output feedback stabilization of the uncertain system (3.31) 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 ; and 2) the output feedback contime-varying parameter trol schemes proposed recently [23] cannot be applied to the uncertain system (3.31), 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.
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On the other hand, a simple calculation shows that the uncertain system (3.31) satisfies the homogeneous growth condition (1.2). Indeed
By Corollary 3.3, 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 (4.1), we make the following assumptions. Lyapunov function , Assumption 4.1: There is a which is positive definite and proper, such that
(3.32)
(4.2)
which globally robustly stabilizes system (3.31). It should be pointed out that unlike in [23], the design of the output feedback controller (3.32) does not require the knowl. This edge of the uncertain term is substantially different from the work [23], where a copy of the must be used for the conterm struction 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 (3.31). Remark 3.5: It is worth pointing out that even in the absence of uncertainty, Corollary 3.3 is new. Moreover, it has incorporated and generalized the output feedback stabilization results obtained previously in [23]. Specifically, Corollary 3.3 can be applied to a larger class of nonlinear systems such as
and
because both of them satisfy the homogeneous growth condition (1.2). However, none of them satisfies the higher-order version of global Lipschitz-like condition, i.e., [23, Ass. 5.1]. As a result, [23, Th. 5.3] cannot be employed here to solve the output feedback stabilization problem for the two nonlinear systems shown previously.
is a real constant. where Assumption 4.2: For
(4.3) Clearly, Assumption 4.1 is a sort of ISS-like condition, while Assumption 4.2 is an extension of the homogeneous growth condition (3.2). With the help of these two conditions, we can prove the following result on global output feedback stabilization of the uncertain cascade system (4.1). Theorem 4.3: Under Assumptions 4.1–4.2, the uncertain cascade system (4.1) is globally asymptotically stabilizable by smooth output feedback. Proof: The proof of this result is similar to that of Theorem 3.2. A key difference lies in the design of a partial-state rather than full-state feedback controller for the uncertain cascade system (4.1). For this reason, we give only a sketch of the proof highlighting the difference. As done in the proof of Theorem 3.2, we first introduce a and (3.3) rescaling transformation that is composed of for the uncertain system (4.1). Such a transformation gives
.. .
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
and
(4.4) In view of Assumption 4.2, it is straightforward to show that the uncertainty of system (4.4) satisfies the constraint
.. .
.. .
(4.5) (4.1) where respectively,
and and
are the system input and output, are the system states, and
is a known constant independent of . where For the rescaled system (4.4) with the constraint (4.5), it can be proved that Assumptions 4.1 and 4.2 imply the existence of a globally stabilizing, partial-state feedback controller
YANG AND LIN: ROBUST OUTPUT FEEDBACK STABILIZATION OF UNCERTAIN NONLINEAR SYSTEMS
. To see why this is the case, consider the Lyapunov function (4.6) Then, it follows from (4.2) that (4.7) Let
and choose the Lynapunov function
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are unmeasurable, the conBecause the states troller (4.9) cannot be directly implemented. To obtain an imple-dimensional observer mentable controller, we design an for recovering of the rescaled system (4.4). Motivated by the robust observer design in the last section, we ignore , , in system (4.4) the uncertain terms and construct the dynamic output compensator
.. .
Using the fact and (4.7), one deduces from (4.4)–(4.5) and Young’s inequality that
(4.11) are the observer gains to be assigned. where The remaining part of the proof is to determine the parameas well as the rescaling factor , which is analters ogous to that of Theorem 3.2 and therefore left to the reader as an exercise. In conclusion, one can prove that by suitably and one-bychoosing the gain constants one, the closed-loop system (4.4)–(4.11) can be rendered globally asymptotically stable at the origin. Clearly, in the case when an uncertain system is of the form
where is a constant independent of . , with Clearly, the virtual controller being a constant independent of , yields
.. .
(4.12) Assumption 4.2 reduces to the following. Assumption 4.4: There exists a constant
Using a similar inductive argument we conclude at the th step that there exist a set of transformations (4.8) a Lyapunov function
and a partial-state feedback control law of the form (4.9)
such that
(4.13) Then, we have the following useful corollary. Corollary 4.5: Under Assumptions 4.1 and 4.4, the uncertain cascade system (4.12) is globally robustly stabilizable by smooth output feedback. We conclude this section with an example that illustrates how Theorem 4.3 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 4.6: Consider the uncertain cascade system
such that
(4.14) (4.10) and are where all the parameters known constants independent of . Note that inequality (4.10) reduces to (3.14) in the absence of -dynamics.
where 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 (4.14) subtle. First of all, system (4.14) is not in a lower triangular form due to
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the second dynamic equation. Second, the linearized system of (4.14) is given by
where both and are positive constants independent of . Next, let with being a gain constant to be assigned later. Since
we design the reduced-order observer which is neither controllable nor observable. While the latter makes the current output feedback design methods hard to be applied to system (4.14), the former prevents an application of the output feedback control scheme developed recently [23] to the cascade system (4.14). 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 (4.14), due to the nature of the nonrobust design. On the other hand, it is easy to see that the three-dimensional and , cascade system (4.14) is of the form (4.1) with and satisfies the growth condition (4.3). Moreover, the ISS-like inequality (4.2) holds for the -subsystem of (4.14). As a matter of fact, using the Lyapunov function and Lemma 6.1 yields
Moreover, the uncertain term Thus
.
That is, Assumption 4.2 holds. By Theorem 4.3, there exists a dynamic output compensator of the form (1.3) 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 , where is a rescaling factor to be determined later. Such a transformation results in
(4.16) which is a copy of -dynamics without the uncertain term . Using thus obtained and the certainty equivalence printhat ciple, we deduce from (4.17) Finally, we show that the dynamic output compensator (4.16)–(4.17) globally robustly stabilizes the uncertain cascade system (4.15) for all , if and are chosen suitably. To this end, let be the estimate error. The error dynamics is
Choose the Lyapunov function
. Then
Selecting (4.18) yields
Using the smooth state feedback design method in [17], one can find a Lyapunov function of the form
where is a constant independent of . Now, consider the Lyapunov function for the closed-loop system (4.15)–(4.16)–(4.17). Using Lemmas 6.1–6.5, it is not difficult to prove that
and a partial-state feedback controller , such that
In view of the relationship (4.18) and choices and
(4.15)
, it is clear that the result in
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0 0
Fig. 1. Transient responses of the closed-loop system (4.14)–(4.19) with = 1 and the initial condition (; ; ; z^ ) = (1; 0:3; 6; 5).
which implies global asymptotic stability of the closed-loop system (4.15)–(4.17) for all . The aforementioned design procedure leads to, for instance, the dynamic output compensator
(4.19) that does the job. The simulation shown in Fig. 1 demonstrates the property of robust stability of the closed-loop system (4.14)–(4.19). 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], where 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 3.5). APPENDIX This section collects several useful lemmas that play a key role in deriving the main results of this paper. Lemma 6.1: Given positive real numbers , , , , , and , the following inequality holds:
Lemma 6.2: Given positive real numbers , , , the following inequality holds:
, , , and
Lemma 6.1 and 6.2 can be easily proved by Young’s Inequality. be real numbers. Then Lemma 6.3: Let
Lemma 6.4: Let and be any real numbers and an odd integer. Then, the following inequality holds:
be
Lemma 6.5: For all and any odd positive integer , the following inequality holds:
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The proofs of Lemmas 6.3–6.5 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. 437–442, 1989. [3] D. Cheng and W. Lin, “On p—Normal forms of nonlinear systems,” IEEE Trans. Autom. Control, vol. 48, no. 7, pp. 1242–1248, 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, U.K., 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 systems,” SIAM J. Control Optim., vol. 28, pp. 1321–1349, 1990. [7] W. Dayawansa, private communication, 2002. [8] J. Gauthier, H. Hammouri, and S. Othman, “A simple observer for nonlinear systems, applications to bioreactors,” IEEE Trans. Automat. Control, vol. 37, no. 6, pp. 875–880, Jun. 1992. [9] W. Hahn, Stability of Motion. New York: Springer-Verlag, 1967. [10] H. Hermes, “Homogeneous coordinates and continuous asymptotically stabilizing feedback controls,” in Differential Equations Stability and Control, S. Elaydi, Ed. New York: Marcel Dekker, 1991, pp. 249–260. [11] A. Isidori, Nonlinear Control Systems. New York: Springer-Verlag, 1999, vol. II. [12] M. Kawski, “Stabilization of nonlinear systems in the plane,” Syst. Control Lett., vol. 12, pp. 169–175, 1989. [13] H. Khalil and A. Saberi, “Adaptive stabilization of a class of nonlinear systems using high-gain feedback,” IEEE Trans. Automat. Control, vol. AC-32, no. 11, pp. 1031–1035, Nov. 1987. [14] A. Krener and A. Isidori, “Linearization by output injection and nonlinear observer,” Syst. Control Lett., vol. 3, pp. 47–52, 1983. [15] A. Krener and W. Kang, “Locally convergent nonlinear observers,” SIAM J. Control Optim., vol. 42, pp. 155–177, 2003. [16] A. Krener and M. Xiao, “Observers for linearly unobservable nonlinear systems,” Syst. Control Lett., vol. 46, pp. 281–288, 2002. [17] W. Lin and C. Qian, “Adding one power integrator: A tool for global stabilization of high-order triangular systems,” Syst. Control Lett., vol. 39, pp. 339–351, 2000. [18] W. Lin and B. Yang, “Robust stabilization of uncertain high-order nonlinear systems by output feedback,” in Proc. 6th IFAC Symp. Nonlinear Control Systems, Stuttgart, Germany, 2004, pp. 67–74. [19] C. Qian and W. Lin, “Output feedback control of a class of nonlinear systems: A nonseparation principle diagram,” IEEE Trans. Autom. Control, vol. 47, no. 10, pp. 1710–1715, Oct. 2002. , “Smooth output feedback stabilization of planar systems without [20] controllable/observable linearization,” IEEE Trans. Autom. Control, vol. 47, no. 12, pp. 2068–2073, Dec. 2002. [21] W. Respondek, “Transforming a single-input system to a p-normal form via feedback,” in Proc. 42nd IEEE Conf. Decision Control, Maui, HI, 2003, pp. 1574–1579. [22] L. Rosier, “Homogeneous Lyapunov functions for homogeneous continuous vector field,” Syst. Control Lett., vol. 19, pp. 467–473, 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. 1069–1080, 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. 95–100. Also, a full version of the paper can be found in Int. J. Rob. Nonlinear Control, vol. 15, pp. 247–268.
Bo Yang (S’01) was born in China in 1975. He received the B.S. degree in mathematics and M.S. 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 Ph.D. 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 D.Sc. 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 U.K., 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.
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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
I
N 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
.. .
(1.1) , and are the system input, state and where is an odd integer. The mappings output, respectively, and , , 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-0400413 and DMS-0203387, and in part by the Air Force Research Laboratory under Grant FA8651-05-C-0110. The authors are with Department of Electrical Engineering and Computer Science, Case Western Reserve University, Cleveland, OH 44106 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2005.847084
existence of a change of coordinates (diffeomorphism) and a state feedback control law transforming a smooth affine system and into the nonlinear system (1.1) with a suitable form of . Therefore, (1.1) 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 (1.1), interesting stabilization results have been obtained over the years. For example, local and global asymptotic stabiliza, and , tion of the system (1.1) with , 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 (1.1) 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 (1.1) in the plane via smooth output feedback. Under , , a resuitable conditions imposed on duced-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 and , three-dimensional system (1.1) when . 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 (1.1) 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 de, sign of a smooth state feedback controller. When and satisfy a global Lipschitz-like condition, we further showed that global stabilization of the nonhomogeneous
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system (1.1), 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, , , in (1.1) must the nonlinear functions 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 (1.1). 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 (1.1) under , such robust suitable growth conditions. In the case of 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 (1.1) is achievable by smooth output feedback. Assumption 1.1: There exists a real constant such that (1.2) One of the objectives of this paper is to address this question and to provide an affirmative answer by constructing, under Assumption 1.1, a smooth dynamic output compensator (1.3) which globally robustly stabilizes the entire family of uncertain systems (1.1). Since Assumption 1.1 is weaker than the higher-order type of the global Lipschitz condition given in [23] (see Remark 3.5), 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 (1.3) , i.e., the uses only the knowledge of the upper bound of condition (1.2) 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 (1.1) 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 in (1.1) can be dominated easily uncertain nonlinearities by tuning the rescaling factor. In the case of uncertain systems ), the new with controllable/observable linearization (i.e., 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 (3.1) and cascade form (4.12), 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 (1.1) with uncontrollable/unobservable linearization can be achieved by smooth output feedback under Assumption 1.1, we revisit a simple situation of (1.1) where , i.e., the case when the first approximation of (1.1) is controllable and observable. In this case, the uncertain system (1.1) can be rewritten as .. .
(2.1) and Assumption 1.1 reduces to the linear growth condition (2.2) In [19], we have shown that global robust stabilization of the uncertain system (2.1) satisfying (2.2) 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 (1.1). 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 (1.1) with . To this end, we introduce a rescaling transformation with a suitable dilation for the original system (2.1), which is motivated by[24] and turns out to be crucial for dominating the uncertainty of (2.1). To be precise, let (2.3) is a rescaling factor to be determined later. where Under the new coordinates ’s, the uncertain system (2.1) can be expressed as .. .
(2.4)
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By the hypothesis (2.2), the uncertain functions , also satisfy the linear growth condition
,
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of (2.4)
Consequently, the unmeasurable state can be estimated as follows:
(2.11) By construction, the reduced-order observer (2.10)–(2.11) is implementable. , , be the Let estimate errors. Then, the error dynamics is given by
.. .
(2.5) For the rescaled uncertain system (2.4) with the constraint (2.5), it is straightforward to design recursively, in a fashion similar to the one in [19], a linear state feedback controller
.. .
(2.12)
(2.6) such that (2.7) where function,
,
Inspired by the certainty equivalence principle, we replace the unmeasurable state in the controller (2.6) by its generated by the observer (2.10)–(2.11). estimate In this way, we get the following implementable controller:
is a quadratic Lyapunov , and
(2.13) Substituting (2.13) into (2.7) yields
with and being known constants independent of . Next, we will design a linear observer for the rescaled system is measurable and only unmeasurable (2.4). Because states of (2.4) 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, th-order linear observer to estimate, we shall build an instead of the states , the unmeasurable variables defined by
(2.14) where is a constant independent of Now, consider the Lyapunov function
.
for the closed-loop system (2.13)–(2.12)–(2.4). A simple calculation results in
(2.8) are gain constants to be assigned later. where From (2.8), it is clear that (2.15) .. .
Using the completion of square, together with the linear growth condition (2.5), it is not difficult to deduce from (2.15)–(2.14) that (2.9)
In view of (2.9), we construct the (regardless of the uncertain terms
,
th-order linear observer )
.. .
(2.16) (2.10) is the estimates of the unmeasurable state
where .
where , while positive constants independent of , are known constants independent of all ’s.
are and and
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Choosing the gain parameters and as follows: order of
one-by-one, in the
global robust stabilization of (3.1) by output feedback, we introduce the following hypothesis which is a natural generalization of the homogeneous growth condition (1.2). such that Assumption 3.1: There exists a constant (3.2)
.. . (2.17) we have
Hence, the uncertain system (2.1) is globally robustly stabilizable by the linear output feedback controller (2.10)–(2.13). Remark 2.1: Notably, the reduced-order observer (2.10) is different from the one in [19] in two respects: 1) it involves gain parameters to be designed. Only when and , (2.10) reduces to a traditional reduced-order high-gain observer; and 2) it contains that is similar to the one introduced in a rescaling factor [24]. The rescaling factor creates an extra design freedom and turns out to be very effective in dealing with uncertainty of (2.1), under the linear growth condition (2.2). III. ROBUST OUTPUT FEEDBACK DESIGN: THE -NORMAL FORM CASE Interestingly, the robust output feedback control scheme developed so far for the uncertain system (2.1) 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 (1.1) satisfying the growth condition (1.2). To this end, we first present a robust stabilization result for the following class of nonlinear systems:
when . Moreover, where when . With the aid of the growth condition (3.2), it is possible to establish the following output feedback stabilization theorem which is one of the main results of this paper. Theorem 3.2: Under Assumption 3.1, there exists a smooth dynamic output compensator (1.3) making the uncertain system (3.1) 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 un. The construction of the observer is certainty, i.e., substantially different from the one in [23] in the sense that here is used for the design of a robust obno copy of server, while the nonlinear observer in [23] did involve a copy . For this reason, in the work [23] of 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 (3.1). 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 (3.1) is homogeneous with dilation and degree when , . Keeping this in mind and motivated by (2.3), we introduce the following rescaling transformation:
(3.3) with dilation , where is a rescaling factor to be assigned later. In the rescaled coordinates ’s, the uncertain system (3.1) can be represented as
.. .
.. . (3.1) called -normal form [3], where , 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 (3.1). To achieve
(3.4) where
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Using (3.2) and the fact that
, it is easy to see that
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, a direct
With the help of Lemma 6.5 and computation gives
(3.9) is a constant independent of . Substituting (3.9) where into (3.8), we deduce from Lemmas 6.1 and 6.3 that
(3.10) (3.5) Applying Lemma 6.1 to (3.5), we obtain the following estimations ( ):
(3.6) where 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 , , in (3.4). with the system uncertainty, i.e., ii) State Feedback Design: For the rescaled system (3.4) satisfying the growth condition (3.6), one can construct a robust state feedback controller via the smooth feedback design method [17]. with and choose the Lyapunov Let Then, it is easy to deduce from (3.6) function that
Keeping , with independent of , results in
in mind, the virtual controller being a constant
(3.7) Next, let function we have
and choose the Lyapunov Using (3.4), (3.6), and (3.7),
where is a constant independent of . , with Thus, the virtual controller being a constant independent of , is such that
(3.11) Using an inductive argument similar to the one in [17], one can find a set of virtual controllers, transformations, and Lyapunov functions
.. .
.. .
.. . (3.12)
and a smooth state feedback control law (3.13) such that
(3.14) and are known and where all the constants independent of . iii) Output Feedback Design: Since of the is measurrescaled system (3.4) are unmeasurable but -dimensional observer able, we need only to design an for (3.4). However, the reduced-order observer design method in [23] cannot be applied to the rescaled system (3.4) because , , which are time-varying it uses a copy of 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, , the unmeasurable variables instead of the states (3.15) where the parameters determined later. From (3.15), it follows that
are gain constants to be
.. . (3.8)
(3.16)
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In view of (3.16), one can construct the homogeneous observer
-dimensional
To determine the observer gains change of coordinates
, consider the
(3.22)
.. . In the coordinates of Lemma 6.3)
and , (3.21) can be represented as (by
(3.17) which does not involve the uncertain functions in (3.4). This is substantially different from the homogeneous observer proposed in [23]. By construction, the reduced-order observer (3.17) is implementable. Moreover, the estimates of ’s can be obtained based on the relationships (3.18) Let , Then, the error dynamics is given by
, be the estimate errors.
.. .
(3.23)
are where positive real constants independent of , and is a and all the ’s. known constant independent of The error dynamics (3.19) in the coordinate can be rewritten as
.. .
(3.19) By the certainty equivalence principle, the unmeasurable state in the controller (3.13) can be replaced by generated by the nonlinear observer its estimate (3.17)–(3.18). In this way, one obtains the implementable feedback controller (3.20)
(3.24) Now, consider the Lyapunov function
A direct computation gives
Substituting (3.20) into (3.14) and using Lemmas 6.5 and 6.3, we have
(3.25)
Similar to [23], it is not difficult to get the following estimations for each term on the right-hand side of (3.25): where is a real constant related to ’s and independent of . With the aid of (3.12) and Lemmas 6.3 and 6.1, the aforementioned inequality can be simplified as
(3.21)
where
is a constant independent of
.
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, are positive constants independent of ’s and . From (3.28), it is easy to conclude that if the gain parameters ’s and are assigned one-by-one, in the following manner:
.. .
(3.29) (3.26) are where positive constants independent of . Substituting (3.26) into (3.25), a tedious but straightforward calculation leads to
(3.27)
are where , while and positive constants independent of , are known constants independent of ’s and . Finally, choose the Lyapunov function
for the closed-loop system in the coordinates follows from (3.23) and (3.27) that
we have (3.30) This, in turn, implies that the uncertain high order system (3.1) is globally asymptotically stabilized by the dynamic output compensator (3.17)–(3.20). As a consequence of Theorem 3.2, we have the following important result that provides a solution to the global stabilization problem of system (1.1). Corollary 3.3: For a family of uncertain systems (1.1) satisfying Assumption 1.1, there exists a smooth output feedback controller of the form (1.3), such that the closed-loop system (1.1)–(1.3) is globally asymptotically stable at the equilibrium . Proof: Corollary 3.3 follows immediately from Theorem 3.2 if one observes that system (1.1) is a special case of the uncertain system (3.1) and Assumption 3.1 reduces to Assumption for , . 1.1 when The reader is referred to [18] for further details. The application of Corollary 3.3 and the main features of the robust smooth output feedback control scheme developed so far can be illustrated by the following example. Example 3.4: Consider the uncertain planar system
. Then, it (3.31)
(3.28)
where positive constants independent of
, while
are and
, is a continuous time-varying function satiswhere . fying Clearly, global output feedback stabilization of the uncertain system (3.31) 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 ; and 2) the output feedback contime-varying parameter trol schemes proposed recently [23] cannot be applied to the uncertain system (3.31), 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.
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On the other hand, a simple calculation shows that the uncertain system (3.31) satisfies the homogeneous growth condition (1.2). Indeed
By Corollary 3.3, 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 (4.1), we make the following assumptions. Lyapunov function , Assumption 4.1: There is a which is positive definite and proper, such that
(3.32)
(4.2)
which globally robustly stabilizes system (3.31). It should be pointed out that unlike in [23], the design of the output feedback controller (3.32) does not require the knowl. This edge of the uncertain term is substantially different from the work [23], where a copy of the must be used for the conterm struction 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 (3.31). Remark 3.5: It is worth pointing out that even in the absence of uncertainty, Corollary 3.3 is new. Moreover, it has incorporated and generalized the output feedback stabilization results obtained previously in [23]. Specifically, Corollary 3.3 can be applied to a larger class of nonlinear systems such as
and
because both of them satisfy the homogeneous growth condition (1.2). However, none of them satisfies the higher-order version of global Lipschitz-like condition, i.e., [23, Ass. 5.1]. As a result, [23, Th. 5.3] cannot be employed here to solve the output feedback stabilization problem for the two nonlinear systems shown previously.
is a real constant. where Assumption 4.2: For
(4.3) Clearly, Assumption 4.1 is a sort of ISS-like condition, while Assumption 4.2 is an extension of the homogeneous growth condition (3.2). With the help of these two conditions, we can prove the following result on global output feedback stabilization of the uncertain cascade system (4.1). Theorem 4.3: Under Assumptions 4.1–4.2, the uncertain cascade system (4.1) is globally asymptotically stabilizable by smooth output feedback. Proof: The proof of this result is similar to that of Theorem 3.2. A key difference lies in the design of a partial-state rather than full-state feedback controller for the uncertain cascade system (4.1). For this reason, we give only a sketch of the proof highlighting the difference. As done in the proof of Theorem 3.2, we first introduce a and (3.3) rescaling transformation that is composed of for the uncertain system (4.1). Such a transformation gives
.. .
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
and
(4.4) In view of Assumption 4.2, it is straightforward to show that the uncertainty of system (4.4) satisfies the constraint
.. .
.. .
(4.5) (4.1) where respectively,
and and
are the system input and output, are the system states, and
is a known constant independent of . where For the rescaled system (4.4) with the constraint (4.5), it can be proved that Assumptions 4.1 and 4.2 imply the existence of a globally stabilizing, partial-state feedback controller
YANG AND LIN: ROBUST OUTPUT FEEDBACK STABILIZATION OF UNCERTAIN NONLINEAR SYSTEMS
. To see why this is the case, consider the Lyapunov function (4.6) Then, it follows from (4.2) that (4.7) Let
and choose the Lynapunov function
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are unmeasurable, the conBecause the states troller (4.9) cannot be directly implemented. To obtain an imple-dimensional observer mentable controller, we design an for recovering of the rescaled system (4.4). Motivated by the robust observer design in the last section, we ignore , , in system (4.4) the uncertain terms and construct the dynamic output compensator
.. .
Using the fact and (4.7), one deduces from (4.4)–(4.5) and Young’s inequality that
(4.11) are the observer gains to be assigned. where The remaining part of the proof is to determine the parameas well as the rescaling factor , which is analters ogous to that of Theorem 3.2 and therefore left to the reader as an exercise. In conclusion, one can prove that by suitably and one-bychoosing the gain constants one, the closed-loop system (4.4)–(4.11) can be rendered globally asymptotically stable at the origin. Clearly, in the case when an uncertain system is of the form
where is a constant independent of . , with Clearly, the virtual controller being a constant independent of , yields
.. .
(4.12) Assumption 4.2 reduces to the following. Assumption 4.4: There exists a constant
Using a similar inductive argument we conclude at the th step that there exist a set of transformations (4.8) a Lyapunov function
and a partial-state feedback control law of the form (4.9)
such that
(4.13) Then, we have the following useful corollary. Corollary 4.5: Under Assumptions 4.1 and 4.4, the uncertain cascade system (4.12) is globally robustly stabilizable by smooth output feedback. We conclude this section with an example that illustrates how Theorem 4.3 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 4.6: Consider the uncertain cascade system
such that
(4.14) (4.10) and are where all the parameters known constants independent of . Note that inequality (4.10) reduces to (3.14) in the absence of -dynamics.
where 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 (4.14) subtle. First of all, system (4.14) is not in a lower triangular form due to
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the second dynamic equation. Second, the linearized system of (4.14) is given by
where both and are positive constants independent of . Next, let with being a gain constant to be assigned later. Since
we design the reduced-order observer which is neither controllable nor observable. While the latter makes the current output feedback design methods hard to be applied to system (4.14), the former prevents an application of the output feedback control scheme developed recently [23] to the cascade system (4.14). 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 (4.14), due to the nature of the nonrobust design. On the other hand, it is easy to see that the three-dimensional and , cascade system (4.14) is of the form (4.1) with and satisfies the growth condition (4.3). Moreover, the ISS-like inequality (4.2) holds for the -subsystem of (4.14). As a matter of fact, using the Lyapunov function and Lemma 6.1 yields
Moreover, the uncertain term Thus
.
That is, Assumption 4.2 holds. By Theorem 4.3, there exists a dynamic output compensator of the form (1.3) 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 , where is a rescaling factor to be determined later. Such a transformation results in
(4.16) which is a copy of -dynamics without the uncertain term . Using thus obtained and the certainty equivalence printhat ciple, we deduce from (4.17) Finally, we show that the dynamic output compensator (4.16)–(4.17) globally robustly stabilizes the uncertain cascade system (4.15) for all , if and are chosen suitably. To this end, let be the estimate error. The error dynamics is
Choose the Lyapunov function
. Then
Selecting (4.18) yields
Using the smooth state feedback design method in [17], one can find a Lyapunov function of the form
where is a constant independent of . Now, consider the Lyapunov function for the closed-loop system (4.15)–(4.16)–(4.17). Using Lemmas 6.1–6.5, it is not difficult to prove that
and a partial-state feedback controller , such that
In view of the relationship (4.18) and choices and
(4.15)
, it is clear that the result in
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0 0
Fig. 1. Transient responses of the closed-loop system (4.14)–(4.19) with = 1 and the initial condition (; ; ; z^ ) = (1; 0:3; 6; 5).
which implies global asymptotic stability of the closed-loop system (4.15)–(4.17) for all . The aforementioned design procedure leads to, for instance, the dynamic output compensator
(4.19) that does the job. The simulation shown in Fig. 1 demonstrates the property of robust stability of the closed-loop system (4.14)–(4.19). 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], where 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 3.5). APPENDIX This section collects several useful lemmas that play a key role in deriving the main results of this paper. Lemma 6.1: Given positive real numbers , , , , , and , the following inequality holds:
Lemma 6.2: Given positive real numbers , , , the following inequality holds:
, , , and
Lemma 6.1 and 6.2 can be easily proved by Young’s Inequality. be real numbers. Then Lemma 6.3: Let
Lemma 6.4: Let and be any real numbers and an odd integer. Then, the following inequality holds:
be
Lemma 6.5: For all and any odd positive integer , the following inequality holds:
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The proofs of Lemmas 6.3–6.5 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. 437–442, 1989. [3] D. Cheng and W. Lin, “On p—Normal forms of nonlinear systems,” IEEE Trans. Autom. Control, vol. 48, no. 7, pp. 1242–1248, 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, U.K., 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 systems,” SIAM J. Control Optim., vol. 28, pp. 1321–1349, 1990. [7] W. Dayawansa, private communication, 2002. [8] J. Gauthier, H. Hammouri, and S. Othman, “A simple observer for nonlinear systems, applications to bioreactors,” IEEE Trans. Automat. Control, vol. 37, no. 6, pp. 875–880, Jun. 1992. [9] W. Hahn, Stability of Motion. New York: Springer-Verlag, 1967. [10] H. Hermes, “Homogeneous coordinates and continuous asymptotically stabilizing feedback controls,” in Differential Equations Stability and Control, S. Elaydi, Ed. New York: Marcel Dekker, 1991, pp. 249–260. [11] A. Isidori, Nonlinear Control Systems. New York: Springer-Verlag, 1999, vol. II. [12] M. Kawski, “Stabilization of nonlinear systems in the plane,” Syst. Control Lett., vol. 12, pp. 169–175, 1989. [13] H. Khalil and A. Saberi, “Adaptive stabilization of a class of nonlinear systems using high-gain feedback,” IEEE Trans. Automat. Control, vol. AC-32, no. 11, pp. 1031–1035, Nov. 1987. [14] A. Krener and A. Isidori, “Linearization by output injection and nonlinear observer,” Syst. Control Lett., vol. 3, pp. 47–52, 1983. [15] A. Krener and W. Kang, “Locally convergent nonlinear observers,” SIAM J. Control Optim., vol. 42, pp. 155–177, 2003. [16] A. Krener and M. Xiao, “Observers for linearly unobservable nonlinear systems,” Syst. Control Lett., vol. 46, pp. 281–288, 2002. [17] W. Lin and C. Qian, “Adding one power integrator: A tool for global stabilization of high-order triangular systems,” Syst. Control Lett., vol. 39, pp. 339–351, 2000. [18] W. Lin and B. Yang, “Robust stabilization of uncertain high-order nonlinear systems by output feedback,” in Proc. 6th IFAC Symp. Nonlinear Control Systems, Stuttgart, Germany, 2004, pp. 67–74. [19] C. Qian and W. Lin, “Output feedback control of a class of nonlinear systems: A nonseparation principle diagram,” IEEE Trans. Autom. Control, vol. 47, no. 10, pp. 1710–1715, Oct. 2002. , “Smooth output feedback stabilization of planar systems without [20] controllable/observable linearization,” IEEE Trans. Autom. Control, vol. 47, no. 12, pp. 2068–2073, Dec. 2002. [21] W. Respondek, “Transforming a single-input system to a p-normal form via feedback,” in Proc. 42nd IEEE Conf. Decision Control, Maui, HI, 2003, pp. 1574–1579. [22] L. Rosier, “Homogeneous Lyapunov functions for homogeneous continuous vector field,” Syst. Control Lett., vol. 19, pp. 467–473, 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. 1069–1080, 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. 95–100. Also, a full version of the paper can be found in Int. J. Rob. Nonlinear Control, vol. 15, pp. 247–268.
Bo Yang (S’01) was born in China in 1975. He received the B.S. degree in mathematics and M.S. 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 Ph.D. 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 D.Sc. 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 U.K., 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.
