A Generalization of Krasovskii–LaSalle Theorem for Nonlinear Time ...

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A Generalization of Krasovskii–LaSalle Theorem for Nonlinear Time-Varying Systems: Converse Results and Applications Ti-Chung Lee and Zhong-Ping Jiang, Senior Member, IEEE

Abstract—This paper presents a practically applicable characterization of uniform (global) asymptotic stability (UAS and UGAS) for general nonlinear time-varying systems, under certain output-dependent conditions in the spirit of the Krasovskii–LaSalle theorem. The celebrated Krasovskii–LaSalle theorem is extended from two directions. One is using the weak zero-state detectability property associated with reduced limiting systems of the system in question to generalize the condition that the maximal invariance set contained in the zero locus of the time-derivative of the Lyapunov function is the zero set. Another one is using an almost bounded output-energy condition to relax the assumption that the time derivative of the Lyapunov function is negative semi-definite. Then, the UAS and UGAS properties of the origin can be guaranteed by employing these two improved conditions related to certain output function for uniformly Lyapunov stable systems. The proposed conditions turn out to be also necessary under some mild assumptions and thus, give a new characterization of UGAS (and UAS). Through an equivalence relation, the proposed detectability condition can also be verified in terms of usual PE condition. To validate the proposed results, the obtained stability criteria are applied to a class of time-varying passive systems and to revisit a tracking control problem of nonholonomic chained systems. For the latter, under certain persistency of excitation conditions, the -exponential stability is achieved based on our approach. Index Terms—Detectability, nonholonomic systems, nonlinear time-varying systems, reduced limiting systems, uniform asymptotic stability.

I. INTRODUCTION

I

N THIS paper, we investigate uniform (global) asymptotic stability (for short, UAS and UGAS) for nonlinear time-varying, not necessarily periodic, systems. Our target is to present a new characterization of UGAS (and UAS) by extending the well-known Krasovskii–LaSalle theorem to general time-varying systems. The key concept employed throughout is Arstein’s limiting equations. The crucial observation of the paper is that, for a uniformly Lyapunov stable

Manuscript received November 24, 2003; revised July 28, 2004 and March 28, 2005. Recommended by Associate Editor M. Reyhanoglu. This work was supported in part by the NSC, Taiwan, R.O.C., under Contract NSC-91-2213-E159-004, and by the National Science Foundation Grants ECS-0093176 and INT-9987317. Part of this work was done when the first author was visiting the Polytechnic University, Brooklyn, New York. T. C. Lee is with the Department of Electrical Engineering, Ming Hsin University of Science and Technology, Hsinchu, Taiwan 304, R.O.C. (e-mail: [email protected]). Z. P. Jiang is with the Department of Electrical and Computer Engineering, Polytechnic University, Brooklyn, NY 11201 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2005.852567

system, uniform attractiveness can be verified by employing an output function and a detectability condition on the associated reduced limiting systems. Moreover, we demonstrate some interesting applications of the developed stability results in a new class of time-varying passive systems and in a revisited tracking problem for the popular class of nonholonomic chained systems. Stability is one of the most fundamental properties to be achieved in control systems design. In his classical work around the end of the 19th century [33], Lyapunov proposed a powerful analysis tool guaranteeing asymptotic stability for nonlinear time-varying dynamical systems. The theorem of Lyapunov was relaxed in [9] by admitting the use of Lyapunov functions having negative semidefinite, instead of negative definite, time derivatives for autonomous systems. A further extension to periodic systems can be found in [20]. The derived result also comes from the so-called LaSalle invariance principle [22]–[24] and is known as Krasovskii–LaSalle theorem in modern literature [46]. Its flexibility and popularity in controller design and analysis for control engineering applications has been abundant in past literature. The celebrated Krasovskii–LaSalle theorem gives two conditions to guarantee UAS for a uniformly Lyapunov stable system. One requires that the time derivative of certain Lyapunov function be negative semidefinite and another one, referred as an invariance property throughout this paper, asserts that the maximal invariant set constrained in the zero locus of the time derivative of the Lyapunov function is the zero set. To extend the Krasovskii–LaSalle theorem to more general (not necessarily periodic) dynamic systems, several possible generalizations have been proposed by improving the latter condition in past literature [6], [7], [41], [47]. In the early stage of development, many researchers focused on the extension of LaSalle invariance principle to nonperiodic equations [6], [41]. Since the -limit sets are not always invariant in general time-varying systems, the concept of limiting equations, that describe the limiting behavior of the original equation as initial time instants approach to infinity, was used to overcome this obstacle; see, for instance, the excellent survey [6] and the numerous references therein. Among these, Artstein established a very interesting pseudoinvariance property of the -limit set by employing limiting equations so that LaSalle’s invariance principle can be extended to a class of nonlinear time-varying systems [6]. In [41], a similar result was proposed for a large class consisting of asymptotically almost periodic (AAP) systems. Later on, Artstein also proposed a novel characterization of UAS in terms of (nonuniform) attrac-

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tivity of limiting equations [7]. By combining these results, UAS of the origin for the original system can then be analyzed by employing generalized LaSalle invariance principles proposed in [6] and [41] for every limiting equation. This generalizes classic Krasovskii–LaSalle theorem, but also leaves opening the problem of how to locate pseudo invariant sets for every limiting equation. In general, answering this question needs certain time-invariant properties related to Lyapunov functions [6], [7] and deserves further improvement. In present literature, there exist several different ways to guarantee the UAS property besides using the concept of limiting equations [1], [2], [8], [12], [18], [26], [39]. Most of these results used the concepts such as observability or detectability to verify attractivity. It is interesting to observe that a connection exists between observability and the invariant property stated in classical Krasovskii–LaSalle theorem. This with can be seen by defining a virtual output as denoting the time derivative of the Lyapunov function. Then, the invariance property is equivalent to saying that the system is zero-state observable (see [11] for a precise definition). Along this way, an interesting extension—named as integral invariance principle—was proposed for autonomous and periodic systems [12]. Recently, the concept of weak zero-state detectability (WZSD) was defined in [26] and used to guarantee UAS and UGAS for general nonlinear time-varying systems. It extends both classical Krasovskii–LaSalle theorem and [12, Th. 2.1] to nonperiodic systems. Besides using zero-state observability or detectability, another approach is employing observability Gramian to replace the invariance property. This can be traced back to the classical paper [3] where uniform complete observerbility (UCO) related to observability Gramian was defined and used to guarantee UGAS for linear time-varying (LTV) systems. Although the result was not necessary to follow classic Krasovskii–LaSalle theorem, the UCO condition can be used as an alternative description of the invariance property. For general nonlinear time-varying systems, the notion of uniform noticeability, often viewed as a nonlinear-version of UCO, was originally introduced by Artstein to guarantee UAS (and UGAS) [8]. Several interesting and related conditions were also proposed in [1], [18], and [39]; see, for instance, the nice survey in [2]. These results also generalize classic Krasovskii–LaSalle theorem to nonperiodic systems. It should be noticed that besides considering generalizations of classic Krasovskii–LaSalle theorem, there exist other approaches to guaranteeing UAS and UGAS for time-varying nonlinear systems in present literature. For example, an interesting criterion was proposed to guarantee UGAS by using a novel extension of conventional persistent excitation (PE) for a class of nonlinear time-varying systems [29], [30], [32], [40]. Based on Matrosov’s theorem [34], several further generalizations were also proposed in [31] and [44] quite recently. In this paper, we would like to continue this research line and propose an extension of the well-known Krasovskii–LaSalle theorem for nonlinear time-varying systems, followed by several examples illustrating its practical applicability. Our observation is that most generalizations available in present literature are not easily applicable to some physical systems with complex nonlinearities and structures. Indeed, they usually in-

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volve certain integral inequalities associated with observability Gramian (UCO-type conditions) or some integral equations related to the right-hand side of the differential equation (e.g., the WZSD condition). In particular, unlike the classical Krasovskii–LaSalle theorem in cases of time-invariant and periodic systems, it is difficult to perform a recursive reduction procedure for higher dimensional systems [11], [12]. This is the case for example for those applications arising in certain tracking control problems for higher order nonholonomic systems [16], [17], [28]. As a first contribution in this paper, we will employ the concept of limiting functions, proposed in [6], [7], and [41] to describe limiting equations, as a basic tool to improve the WZSD condition proposed in [26]. The limiting functions of output function and the nominal (i.e., unperturbed) function of the differential equation will be used simultaneously to define reduced limiting systems that describe the limiting behavior of the nominal system as initial time instants approach to infinity. Then, WZSD can be checked by employing a simpler “weak zero-state detectability” condition associated with reduced limiting systems. Roughly speaking, all solutions of reduced limiting systems constrained in the zero locus of the limiting output function describe “limiting zero-dynamics” of the original system. Thus, the proposed criterion can be used to simplify stability analysis by checking detectability for some “reduced systems” simpler than the original system. Recursively applying this reduction procedure, UAS and UGAS for higher dimensional time-varying systems can usually be achieved as done in [11] and [12]. This is reminiscent of the Jurdjevic–Quinn type argument used in nonlinear time-invariant systems. Several examples will be given to illustrate such a viewpoint. On the other hand, the proposed detectability condition can usually be checked by employing the nonzero property of limiting functions even though the exact form of reduced limiting systems is unknown. Then, by noticing an interesting equivalence relation between PE conditions and the nonzero property of limiting functions, it can also be verified in terms of usual PE conditions. In contrast with the results proposed in [7], we do not need to find a pseudoinvariant set. All that we need to do is checking a detectability condition for some known or chosen output function. From a systemic view-point, detectability is more suitable for those dynamic systems described by state models with inputs and outputs such as dissipative systems [11], [15], [45]. In comparison with the detectability condition proposed in [27], there are several different purposes and approaches in this paper. First, our target in this paper is guaranteeing UAS rather than (nonuniform) asymptotic stability (AS) that was the objective of [27]. In general, UAS enjoys a stronger (robustness) property than AS in time-varying systems [26], [31]. Furthermore, certain converse results can be achieved for UGAS (and UAS) in this paper, but, it is not trivial how to find a similar converse result for AS systems. Although the form of detectability seems similar, it is not clear how to directly apply the approach given in [27] that was based on Barbalat’s lemma in order to achieve the same result given in this paper. Besides extending the invariance property to time-varying systems, we would like to further improve another condition of Krasovskii–LaSalle theorem. That is, to relax the negative

LEE AND JIANG: GENERALIZATION OF KRASOVSKII–LASALLE THEOREM FOR NONLINEAR TIME-VARYING SYSTEMS

semi-definiteness of the time derivative of the Lyapunov function into “not always nonpositive.” Previous work along this line includes [12], [26] by using certain integrability conditions of the output function (also see [48] and [44] for other characterizations and, in particular, [31] and [32] where an interesting extension of Matrosov theorem is given.) Inspired by the integral lemma introduced in the interesting recent work [30], we will further improve the integrability condition proposed in [26] by employing an “almost bounded output-energy” condition. This way, the UAS and UGAS properties of the origin can be guaranteed by employing a “weak zero-state detectability” condition and an “almost bounded output-energy” condition for uniformly Lyapunov stable systems. This result clearly covers the classical Krasovskii–LaSalle theorem as well as the integral invariance principle in time-invariant and periodic systems, because any reduced limiting system is simply the original system up to a time-shifting operation. As a second contribution of this paper, we will show that the proposed conditions are not only sufficient but also necessary under some mild assumptions. It should be remarked that this was not yet achieved in [26] and [27]. Consequently, the proposed criterion provides a new characterization for the UGAS (and UAS) property. More interestingly, it will be seen that the proposed characterization reduces to a main theorem of Artstein given in [7] and the integral lemma given in [30] when output function is equal to the zero and the full state, respectively. As an illustration of our new stability results, we will study a class of time-varying passive systems, and revisit a tracking control problem for a class of nonholonomic chained systems that has received considerable attention in recent literature; see, for instance, [16], [17], [28], [37], [42], and [43]. For the former, a novel way of stability analysis will be presented. For the latter control application, it is shown that the tracking error system is feedback equivalent to a time-varying passive system. A new global tracking controller will then be proposed based on the passivity theory. The -exponential stability of the origin for the closed-loop systems will be established under certain PE conditions based on the developed framework. Parallel to the development for autonomous and periodic systems, these applications stress the importance of and steps in reducing the stability verification of the original system to the one of reduced limiting systems, a viewpoint advocated here for time-varying systems. II. PRELIMINARIES In this section, a stability criterion proposed in [26] will be reviewed and improved. It will serve as a preliminary step to derive the main results in this paper. One illustrative example is then given to motivate the necessity in replacing the integral equations by certain differential equations in order to apply the previously derived stability criterion to nonperiodic higherorder systems. We study a nonlinear time-varying system with an uncertain term described as (1) (2)

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with where is contained in an open subset of , and , and are all measurable functions defined on such that and for all . Assume that satisfies the Caratheodory condition so that the existence theorem and extension theorem of solutions of (1) are satisfied [14]. It should be noticed that a solution of (1) means an absolutely continuous function satisfying the following integral equation:

for some and . Moreover, the uniqueness of solutions is not necessary throughout this section. System (1)–(2) is further assumed in the output-injection form in the sense that , for any compact subset of and some continuous function with . This technical assumption is to relax the regto be an AAP ularity requirements (e.g., without requiring function) and simplify the form of reduced systems. (See [44] for other motivations.) Throughout this paper, when we assume that and are both uniformly bounded, we mean that, for any compact subset of , there exists a positive constant which dominates and over . It is said that a stateholds almost everywhere (a.e.) if the Lebesgue meament is false is zero [21]. Moresure of the set over, it is said that the solutions of (1) are globally uniformly there exists a such that bounded if, for any of (1) satisfying , for all solutions [18]. This condition is usually we have established using a positive definite proper Lyapunov function rather than derived from the other conditions made in this paper. In the following, we first recall the notion of limiting solutions used in [6] and [41], and an existence lemma from [26]. Roughly speaking, limiting solutions describe the limit behavior of solutions as initial time instants approach to infinity and, thus, can be used to study the asymptotic behavior of solutions. is called a limDefinition 1: A function iting solution of (1) with respect to an unbounded sequence in , if there exist a compact and a sequence of solutions of (1) such that the associated converges uniformly to on sequence . every compact subset of Lemma 1: Consider a system of the form (1)–(2) where and are both uniformly bounded. Then, for any compact and any sequence of solutions of (1), , there exist a limiting solution and a subsewith quence of such that the associated sequence converges uniformly to on every compact . subset of Remark 1: It is of interest to note that any limiting solution must be bounded and continuous. In fact, the boundedness of limiting solutions follows from the convergence in a compact set. Moreover, due to the uniform boundedness property of and , and the assumption of output-injection form, any sequence of solutions of (1) is equicontinuous. Thus, by the uniform convergence of solutions, it is not

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difficult to see that any limiting solution is continuous [21]. It is also possible to show that every limiting solution is locally Lipschitz and thus, absolutely continuous. Let us also recall the following two conditions imposed on the output function . C1) For each and each compact , there exists , such that, for each and all soluof (1) tions

uniformly in . Then, every limiting solution satisfying (4) is also a bounded solution of integral equation (5). Moreover, the and origin is UAS if H1) and C2) hold. In addition, if the solutions are globally uniformly bounded, then the origin is UGAS. In the following, one interesting example will be given to point out the limitation of Theorem 1. This example motivates the necessity for improving integral equation (5). Example 1: Consider the following system:

(3)

(7)

C2)

(Detectability in terms of limiting solutions): For any limiting solution with respect to an unbounded in , if the following equation sequence holds: (4)

then we have either for some or the origin is a -limit point of . Based on C1) and C2), the following result holds by combining [26, Th. 1] with [26, Prop. 1]. Proposition 1: Consider a system of the form (1)–(2) where and are both uniformly bounded. Suppose the origin is uniformly Lyapunov stable and is continuous in , uniformly in . Then, the origin is UAS when C1) and C2) hold. In addition, and the solutions are globally uniformly bounded, if then the origin becomes UGAS. Remark 2: Notice that under uniform Lyapunov stability, C2) is equivalent to a stronger conclusion that the origin is the . However, in unique -limit point of , i.e., practical applications, the statement in C2) seems to be more convenient. See Example 1. In the following, we would like to further improve Proposition 1. First, consider the following “integral-type” reduced limin iting equations for any unbounded sequence (5) The following hypothesis extends a condition from [26] and will be used to replace C1). It is inspired by the integral lemma in [30]; also see [40] and [44]. , with H1) There exists a continuous and for all such that, for and each compact , there exists each such that, for all and all solutions of (1)

where

is a bounded function, and is a skew-symmetric matrix such that is controllable. Define a virtual output map . . Then . This implies Let that H1) holds and that the origin is uniformly Lyapunov stable and the solutions are globally uniformly bounded [18] (also see , the proof of Theorem 2). Let for all and all . Since and for all , all and some positive constant , (7) is in the output injection form. Moreover, and are both uniformly bounded and are continuous in , uniformly in by their definitions. To guarantee UGAS of the origin, it remains to verify C2) in view of Theorem 1. By the definition of , every limiting solution satisfying integral equation (5) can be described by (8) A natural question is: How to compute the limiting solution by employing the integral equation above? Obviously, it is difficult to answer this question without additional conditions. has certain “continuous limiting Assume temporarily that , for all , functions,” say, with for some . Then, any limiting solution satisfying (8) is also a solution of the differential equation by taking the limit under the sign of integral [21]. Thus, a recursive procedure—like the one in the case of Krasovskii–LaSalle theorem for periodic systems—can be performed to show that C2) holds. Indeed, let be any limiting solution satisfying (4). Since is time-invariant, we . Differentiating this have equation, we have . Since and is continuous, there exists a nonempty open interval such that and thus . Continuing this process, it can be recursively shown that . Notice that

(6) The following theorem will be instrumental for the development of our main results. Its proof will be postponed to Appendix A, where one sees that H1) implies C1) with a different . Theorem 1: Consider a system of the form (1)–(2) with and both uniformly bounded. Suppose that the origin is uniformly Lyapunov stable and that and are continuous in ,

Thus, , by controllability of . This shows that C2) holds. As a result, the origin is UGAS when there exits a nonzero continuous limiting function of for any in . unbounded sequence Remark 3: From Example 1, it can be seen that the “continuous nonzero limiting function” plays a central role in verifying the detectability condition—C2). In next section, a definition for

LEE AND JIANG: GENERALIZATION OF KRASOVSKII–LASALLE THEOREM FOR NONLINEAR TIME-VARYING SYSTEMS

limiting functions will be recalled and several useful properties will also be proposed. Moreover, the nonzero property of limiting function will be connected to a PE condition. Thus, UGAS of (7) can also be guaranteed by employing a PE condition. Instead of using limiting functions, a similar result can be achieved from UCO-type argument and other methods, see for instance [4]. III. LIMITING FUNCTIONS, AAP FUNCTIONS, AND PE CONDITION From Example 1, one sees that it is necessary to improve Theorem 1 further to provide a time-varying stability criterion in the spirit of powerful Krasovskii–LaSalle theorem. In this section, we will pursue this target. Several interesting properties will also be given. Furthermore, an interesting equivalence relation between nonzero limiting functions and PE conditions will be given. It will then be used to define reduced limiting systems, improve Theorem 1 and to check the detectability in terms of usual PE condition. A. Definitions and Properties To make the paper self-contained, we recall definitions of limiting functions and AAP functions, and provide a simple criterion related to AAP functions. For the later development, the limiting function and the AAP function defined in [41] are reformulated as follows. We refer the interested reader to [6] and [7] for more discussions and extensions on limiting equations and limiting functions. Definition 2: Let be a continuous in is said function. An unbounded sequence to be an admissible sequence associated with if there exists a continuous function such that the converges associated sequence on every compact subset of . The uniformly to function is uniquely determined and called the limiting function of associated with . Definition 3: A continuous function is said to be an asymptotically almost periodic (AAP) function in there exists a if, for any unbounded sequence subsequence of so that is an admissible sequence associated with . A simple example of AAP functions is given here. Example 2: For any continuous time-independent function , every limiting function is equal to the original function . Notice that a single function may have many limiting functions. More precisely, consider the following example. be a continuous Example 3: Let is periodic for each fixed . Then, function such that each limiting function can be written as a time-shifting function of for some positive constant . Denote the set of all admissible sequences associated with an AAP function . Throughout this paper, it is said that a is equivalent to , denoted as , if subset of , there exists a subsequence of belonging for any to . In this case, since every subsequence of an admissible sequence yields the same limiting function by the uniqueness

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of the limit, it can be seen that every limiting function of can be determined by a sequence belonging to . Particularly, the following result can be established. Lemma 2: Let be an AAP function and be a subset of with . Then, every limiting function of is also . a limiting function associated with a sequence and be two AAP functions. It is not difficult Next, let to see that for any unbounded sequence in , there exists an admissible subsequence associated with both and by taking a suitable subsequence from admissible subsequences associated with . Thus, the following result is readable from the previous observation. be two AAP functions. Then, Lemma 3: Let and is also an AAP function and . The following lemma states a basic criterion to check when a function is an AAP function. Its proof is omitted because it is direct application of Arzela–Ascoli lemma [21]; see, e.g., [26]. is uniformly Lemma 4: Suppose for every compact subset continuous and bounded on of . Then, is an AAP function. Two further examples of AAP functions are given below. In each example, it is easy to see that is uniformly continuous and bounded on for every compact subset of . Thus, it is an AAP function according to Lemma 4. Moreover, most of the associated limiting functions can be explicitly given for these examples. for all Example 4: Consider a function in and in where is a continuous function defined and is bounded and uniformly on with and for each continuous. Then, for all in and in . We refer the reader to Lemma 7 for a detailed proof of this fact. Example 5: Let for all in and in where is defined as follows: if otherwise . It is straightforward to see that with is bounded. It can also be shown that is uniformly continuous. Thus, is an AAP function according to Lemma 4. However, it is not trivial to describe all limiting functions. We simply proand vide two interesting limiting functions. Let be two unbounded sequences in . Then, it can be directly checked that the associated limiting functions are and , and , respectively. Remark 4: In [5], there exists another definition for the terminology of AAP functions. In general, it is different from the one given in this paper. In fact, every function in the class defined in [5] is bounded and uniformly continuous. Thus, it must be an AAP function via satisLemma 4. Particularly, each function fying the definition of AAP functions given in [5] is also an AAP function by employing the definition in this paper. But, the converse is false. For example, consider the function given in Example 5. It does not satisfy the definition of AAP functions given and any in [5]. The reason is that it has a limiting function

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subsequence of the associated sequence of only converges uniformly to on every compact subset of rather than the whole infinite interval. Thus, does not satisfy the condition required in [5], according to Theorem 4.7.4 in that paper. B. Basic Properties of AAP Functions In this subsection, three basic properties of AAP functions will be given. They will be used throughout this paper. First, the following result can be proposed for AAP functions. be an AAP function. Lemma 5: Let Then, is continuous in , uniformly in . Proof: We prove this lemma by contradiction. If is not continuous in , uniformly in , there exist a positive constant , a sequence in and a sequence converging in such that , to a point for all in . Without loss of generality, we can assume that and are contained in a compact subset of . Since is continuous, it is also uniformly continuous on every com[21]. If is bounded, we have pact subset of , a contradiction. Therefore we may assume that is unbounded. From the definition of AAP functions, there exist a subseof and a continuous function such quence converges unithat the sequence on every compact subset of . This, toformly to gether with the continuity of , implies

Then, we reach a contradiction and hence is continuous in , uniformly in . It completes the proof. The following result states another property of AAP functions. be an AAP function. Lemma 6: Let Then, is uniformly bounded. Proof: By contradiction, suppose the lemma is false. of , a time seThen, there exist a compact subset quence and a sequence such that . If is bounded, is contained in a compact subset of and is bounded by the continuity of [21]. This implies that is unbounded by the fact . Due to the compactness of , there exists a subsequence of such that converges to a point in . of Since is an AAP function, there exists a subsequence such that converges uniformly to a continuous limiting function on every compact subset of . This results in . We reach a contradiction. This completes the proof of the lemma.

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Before closing this subsection, we propose the following result that states a basic property related to the composition of AAP functions. be an AAP function Lemma 7: Let be a continuous function. Then, the and is also an AAP function, composition function with and for any is the limiting function of associated with . be any admissible sequence assoProof: Let converges ciated with so that on every compact subset of uniformly to a limiting function . Notice that is uniformly bounded on in view of Lemma 6. Since is continuous, it is also uniformly [21]. Thus, it can continuous on every bounded subset of converges be concluded that uniformly to the function on every compact subset . This shows that is the limiting function of associated with and . Particularly, of is also an AAP function by the definition of AAP functions. Since is an AAP function, for any , there again by the defexists a subsequence of belonging to . It completes inition of AAP functions. Thus, the proof of the lemma. C. PE Conditions and Limiting Functions In this subsection, an interesting interpretation of PE condition given in adaptive systems theory will be proposed in terms of limiting functions. It will play a central role in checking detectabilty. First, let us recall a usual PE condition. Let be a bounded matrix-valued function. The following PE condition is well-known in the adaptive systems theory; see, e.g., [39].

some

and all

(9)

Consider a simple system with an output map defined as . Then, it is in the form (1)–(2) with . In this case, WZSD is reduced to C2) and, thus, C2) is equivalent to (PE) according to the main result in [25]. Particularly, the following result can be established. be a bounded matrixLemma 8: Let valued function. Then, satisfies (PE) if and only if for any and any unbounded sequence in , the following implication holds: (10) is an AAP When function, (PE) can be further interpreted as a nonzero property of limiting functions as follows. be a matrix-valued Proposition 2: Let be an function and AAP function. Then, satisfies (PE) if and only if for each and each is not admissible sequence a zero function.

LEE AND JIANG: GENERALIZATION OF KRASOVSKII–LASALLE THEOREM FOR NONLINEAR TIME-VARYING SYSTEMS

Proof: We will prove this proposition with the help of Lemma 8. First, let us prove the “if” part. Thus, we can asand each sume that for each admissible sequence is not a zero function. By Lemma 8, we only and need to show that implication (10) holds. Let be any unbounded sequence in . Since is an AAP function, there exists an admissible subsequence of . If a.e., we have a.e.. Thus, is a zero function according to the continuity of [21]. This by the assumption and therefore the “if” part folimplies lows from Lemma 8. Now, let us check the “only if” part. In this case, we can assume that satisfies (PE). This part will be proven by contradiction. Suppose there exist an admissible seand so that is the zero quence function. Then, . and we reach a conBy (PE) and Lemma 8, this results in tradiction. This completes the proofs of the necessity part and the proposition. . In In the following, let us consider the special case of this case, is a function taking as values in the space of row and vectors of dimension . Then, with according to Lemma 7. Moreover, for each admissible sequence for is not a zero function for all nonzero any in . Then, in if and only if is not a zero function. Particularly, the following result is readable from Proposition 2. is an AAP funcCorollary 1: Suppose and satisfies (PE) if and only if tion. Then, is not a zero for each admissible sequence function. Remark 5: In many practical applications, the nonzero property of limiting functions exactly implies (and often is implied by) detectability condition C2) as it was seen in Example 1.

AAP functions. For any system of (1)–(2) associated with

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, a reduced limiting is defined as (11) (12)

We impose the following simplified detectability hypothesis that, roughly speaking, describes a “weak zero-state detectability” [11] on reduced limiting system (11)–(12). H2) For any admissible sequence and any of reduced limiting bounded solution system (11)–(12) satisfying the equation , it holds that either the origin is a -limit for some . point of or In the following, C2) is further simplified by using reduced limiting systems as well as H2). Particularly, the following result can be derived from Theorem 1. Proposition 3: Consider a system of the form (1)–(2). Suppose that and are both AAP functions and that the origin is uniformly Lyapunov stable. Then, the origin is UAS if H1) and and the solutions are globally H2) hold. In addition, if uniformly bounded, then the origin is UGAS. and are both uniformly Proof: First, notice that bounded and are continuous in , uniformly in according to Lemmas 5 and 6. In view of Theorem 1, it remains to check C2). be any limiting solution w.r.t. To this end, let an unbounded sequence in such that (4) holds. By is also an AAP function and thus, there exists Lemma 3, of belonging to . a subsequence , a.e. This results in Since is a continuous function for all , it is a zero function [21]. According to Theorem 1, also satisfies integral equation (5). By taking the limit of functions under the integral sign, can be described by

IV. MAIN RESULTS In this section, reduced limiting systems will be defined by employing the notion of limiting functions. A simplified detectability hypothesis will be proposed to check condition C2) in Theorem 1. The derived criterion will be very similar to the forms of Krasovskii–LaSalle theorem and the integral invariance principle in periodic systems. We also present several converse results and a new characterization of UGAS that generalizes a theorem of Artstein’s in [7] as well as the integral lemma introduced in [30]. Two well-known examples from the adaptive systems theory are revisited to illustrate how the proposed criteria can be used to characterize UGAS by employing the nonzero property of limiting functions (or equivalently the PE condition based on Proposition 2) and to reduce the order of the original equation—in the spirit of Krasovskii–LaSalle theorem. A. Reduced Limiting Systems and Simplified Detectability Hypothesis Consider a system of the form (1)–(2) given in Section II. Throughout this subsection, we assume that and are both

where the uniform convergence property of limiting function was used. Applying the fundamental theorem of calculus, it can be concluded that is a bounded solution of the reduced lim. iting system that satisfies the equation Condition C2) follows from H2). This completes the proof of the proposition by applying Theorem 1. A direct application of Proposition 3 yields the following Lyapunov function based result that generalizes the classical Krasovskii–LaSalle theorem to nonperiodic systems. Theorem 2: Consider a system of the form (1)–(2). Let be a positive definite continuous function and be a continuously differentiable function such that (13) (14) for all

, all

where and are continuous positive definite functions . Suppose and are both AAP with functions. Then, the origin is UGAS when H2) holds.

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Proof: For any compact , let . From the inequality , the origin is uniformly Lyapunov stable and the solutions are globally uniformly bounded [18]. Integrating the previous inequality, it can be derived that the inequalities

(15) , any and any solution of (1). Thus, H1) is true. The proof of the theorem is thus completed in view of Proposition 3. Remark 6: For precise statements of the classic Krasovskii–LaSalle theorem and a generalization based on the integral invariance principle, we refer the interested readers to [46, p. 179] (also see [18, Cor. 3.1] and [12, Th. 2.1]), respectively. In the following, we would like to explain briefly why Theorem 2 as well as Proposition 3 can be viewed as a natural generalization of these results in nonperiodic systems. First, in the time-periodic case, every limiting function is a time-shifting of the original function [41]. Thus, every reduced limiting system is just a time-shifting of the . Define and original system when . It is easy to see that H2) holds provided that contains no the set nontrivial solution. Thus, Theorem 2 reduces to the well-known Krasovskii–LaSalle theorem for periodic systems. A similar argument can be applied to [12, Th. 2.1] by using Proposition 3. Remark 7: In [7], Artstein also proposed a generalization of Krasovskii–LaSalle theorem by employing the semi-quasi-invariance property established in [6] for every limiting equation (also see [41]). Particularly, UAS of the origin can be guaranteed for uniformly Lyapunov stable systems provided that we can find a set so that for each limiting equation, all of its solutions converge to and any semi-quasi invariant subset of is just the zero set. For periodic systems, can be chosen as and the criterion proposed in [7] reduces to classical Krasovskii–LaSalle theorem. This criterion also works well for asymptotically autonomous systems since all limiting equations are equal to the same autonomous system and simpler than the original system. However, it is not clear how to locate for general time-varying systems. For instance, even for the simple time-varying system in Example 6 below, how to locate is still a hard problem. In general, one needs certain time-invariant properties related to Lyapunov functions in order to locate the “time-invariant” set [6], [7]. In contrast with the generalized invariance principles proposed in [6], [7], and [41], we do not need to locate a pseudo invariant set that is often not an easy task. Within our framework, all that we need to do is checking a detectability condition for some a priori known or chosen auxiliary output function. Remark 8: The “weak zero-state detectability” hypothesis-H2) also captures a key idea behind the Krasovskii–LaSalle theorem, i.e., the order and the form of the original equation can be effectively reduced by considering only the “limiting hold for any

zero-dynamics” that consist of all solutions of limiting equation (11) constrained in the zero-locus of limiting output function (12). We will demonstrate how this reduction principle can be realized without knowing the exact form of limiting functions in Examples 6 and 7 and other examples in Section V. In fact, it will be seen that the only thing that we need to know is the nonzero property of limiting functions and in special but interesting situations, this turns to be equivalent to a PE condition according to the results proposed in Section III-C. Remark 9: In present literature, there exist other approaches leading to results like Theorem 2, for example, the UCO-type criteria proposed in [1], [2], [18], [38]. Roughly speaking, instead of using H2), these results employed the following integral inequality to guarantee UAS: (16) is any solution of (1) with where is a positive constant and is a continuous function with and for all [2]. Inequality (16) can be served to as a nonlinear-version of UCO. Some nice features of UCO—type criteria are the simplicity and the fact of requiring no extra assumptions such as AAP functions assumed implicitly in H2). But, a drawback is that such criteria need to evaluate or estimate (16) along all solutions of nonlinear time-varying system [38]. Even for LTV systems, it is not an easy job and in general, certain integral technique and inequalities are needed to conclude the UCO condition as done in [35] and [36]. Alternatively, a series of results related to limiting functions has been developed in this paper to provide different tools for the stability analysis of nonlinear time-varying systems. B. Converse Results and New Characterization for UGAS In this subsection, we show that the conditions of Theorem 1 and Proposition 3 are necessary under certain mild assumptions. As a by-product, a new characterization for the UGAS property of the origin is obtained. For simplicity of exposition, we only . The local case can be consider the global case, i.e., treated similarly. Our first result shows that hypothesis H1) is a necessary condition for UGAS. Some similar results and extensions can be found in [40] and [44]. For completeness, a detailed proof is given here. Proposition 4: Consider a system of the form (1)–(2) where , with , is uniformly bounded and is continuous in , uniformly in . Then, H1) holds provided that the origin is UGAS. be any continuous function Proof: Let with and for all . Such functions . are not vacuous, for instance, one can take is also Since is continuous, it is easy to see that uniformly bounded and is continuous at , uniformly in by the assumption of . Moreover, we have . In particular, for any positive constant , there exists a positive such that , for all and all constant . Since the origin is UGAS, for any compact subset of , there exist two positive constants and such

LEE AND JIANG: GENERALIZATION OF KRASOVSKII–LASALLE THEOREM FOR NONLINEAR TIME-VARYING SYSTEMS

that for any solution , we have Let

starting from , and

with . . Then and

This implies that H1) holds with . The proof of the proposition is therefore completed. In Theorem 1, it has been shown that every limiting solution satisfying (4) is a bounded solution of integral equation (5). In the following, we prove that the converse is also true under a mild assumption. Its proof is postponed to Appendix B. Proposition 5: Consider a system of the form (1)–(2) where and are both uniformly bounded. Suppose that the solutions are globally uniformly bounded and the functions and are locally Lipschitz, uniformly in . It is further assumed that , for any compact and some constant . Then, every continsubset of uous function satisfying (4)–(5) must be a limiting solution of (1). To state another main result of this paper, let us consider the following condition. C3) holds for any continuous function satisfying (4)–(5). Since any limiting solution of (1) satisfying (4) is also a bounded continuous solution of (5) in view of Theorem 1, it is straightforward to see that C3) is stronger than C2). We will show that C3) and thus C2) are necessary conditions of UGAS. Moreover, a new characterization for UGAS can be obtained based on Propositions 3–5 as follows. It also generalizes a theorem of Artstein [7] as well as the integral lemma in [30]. Theorem 3: Consider a system of the form (1)–(2) where and are both uniformly bounded with . Suppose that the solutions are globally uniformly bounded and the functions and are locally Lipschitz, uniformly in . As, sume further that for any compact subset of and some positive constant . Then, the following conditions are equivalent. 1) The origin is UGAS. 2) The origin is uniformly Lyapunov stable, and H1) and C3) hold. 3) The origin is uniformly Lyapunov stable, and H1) and C2) hold. If, in addition, and are both AAP functions, these conditions are all equivalent to the following condition. 4) The origin is uniformly Lyapunov stable, and H1) and H2) hold. Proof: This theorem will be proven by showing that , and provided that and are both AAP functions. In fact, the implication ” is trivial and the implication “ ” was “ just proven in Theorem 1. Let us show that the implication ” also holds in the following. Let be any “ continuous function satisfying (4)–(5). Then it is also a limiting

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solution of (1) in view of Proposition 5. Thus, there exist and a sequence a compact of solutions of (1), with , such that the associated sequence converges uniformly to on every compact subset of . If the origin is UGAS, . we have Particularly, C3) holds. Notice that H1) also holds by Propo” holds and hence, sition 4. Thus, the implication “ all three conditions (1)–(3) are equivalent. When and are both AAP functions, it was just proven that H2) implies C2) in Proposition 3. Thus, it remains to verify that the converse is also true, i.e., C2) implies H2). Let be any admissible sequence and be any bounded solution of . Then, we have (11) that satisfies the equation for all . Thus, satisfies (4). Moreover, the following equation also holds:

by the uniform convergence and taking the limit under the integral sign [21]. That is to say, also satisfies (5). Again by Proposition 5, is a limiting solution of (1). Thus, either the for some acorigin is a -limit point of or cording to C2). Therefore, H2) holds under C2). The proof of the theorem is then completed. Remark 10: In special cases, Theorem 3 reduces to certain . results in past literature. Consider for instance the case In this case, hypothesis H1) naturally holds. Based on Theorem 3, UGAS is equivalent to C3) when the solutions are globally uniformly bounded and the origin is uniformly Lyapunov stable. For the local case, it is possible to show that UAS is equivalent to C3) when the origin is uniformly Lyapunov stable along the line as in the Proof of Theorem 3. This is the result given in ([7, for any conTh. A]) since C3) just says that , our result states tinuous solution of (5). However, when a slightly different condition from the theorem of Artstein. That is, the reduced limiting systems are only required to have the detectability property and unlike the result given in [7], the reduced limiting systems may not be attractive. Consequently, it is possible to use merely the “weakly minimum phase” property for reduced limiting systems to establish uniform asymptotic stability for the original time-varying system as done in [11] for autonomous systems when we consider the “limiting zero dynamics.” See Examples 6–7 and Remark 11 for such situations. . In this case, the system is obAnother special case is servable and particularly, C2) holds. Then, Theorem 3 says that UGAS is equivalent to H1) provided that the solutions are globally uniformly bounded and the origin is uniformly Lyapunov stable. The sufficiency part, i.e., that H1) implies UGAS, was proposed in [30, Lemma 2] and is called as an integral lemma. We refer the interested readers to [40] and [44] for an interesting extension to and a complete characterization for the equilibria and the differential inclusion cases. As an illustrative example, consider the following well-known system in adaptive systems theory [39].

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Example 6: Consider a system of the form

Example 7: Consider a system of the form (17)

(18)

and is a bounded mawhere . trix-valued function. Define a virtual output map Let . Then . This implies that H1) holds, the origin is uniformly Lyapunov stable and the solutions are globally uniformly bounded using the same , and proof as in Theorem 2. Let for all and all . Then, for some positive constant , and system (17) is in the form (1)–(2). By definitions, the functions and are both uniformly bounded and locally Lipschitz, uniformly in , with . Thus, the origin is UGAS if and only if C2) holds in view of Theorem , every limiting solution is a solution of . 3. Since is also a solution of (5). If, Conversely, every solution of additionally, this solution satisfies (4), then it is a limiting solution of (17) according to Proposition 5. As a result, C2) holds for (17) if and only if it is true for the following reduced system:

where , with and are both bounded matrix-valued functions. Assume further that there exists a positive definite . We would like matrix so that to show that the origin is UGAS if and only if a PE condition holds by combining Theorem 3 with an equivalence relation and decompose the given in [25]. Indeed, let state into . Define a virtual . Then, direct computation shows that output map . Again, this implies that H1) holds, the origin is uniformly Lyapunov stable and the solutions are globally , uniformly bounded. Let and . Then, for some positive constant

and In other words, C2) is equivalent to (10) in Section III-C. Particularly, the origin is UGAS if and only if satisfies (PE) by Lemma 8. This is a classical result proven in [35]. To stress on the necessity of our conditions for UGAS, let us consider a special form of system (17) where , with being the function defined in Example 5. It is shown that the origin is not UGAS is an AAP function in this case. for this system. Indeed, with . Then, Let of by there exists a subsequence Lemma 3. The associated limiting function can be explicitly for some and determined as . Let be a nonzero vector. Then, we have all being a zero function. Thus, (PE) does not hold by Proposition 2 and hence, the origin of the system is not UGAS. A similar conclusion can also be achieved by employing H2). Although the origin is not UGAS, it is possible to show that the origin is globally asymptotically stable based on the results proposed in the paper [27]. Since this is not a primary concern in this paper, we omit the detailed proof. This example demonstrates that limiting functions can be used as a very useful tool in guaranteeing asymptotic stability. Moreover, in most cases, it does not need knowing the exact forms of all limiting functions to determine asymptotic stability. In next section, more examples will be proposed to illustrate such a viewpoint. Before closing this section, we would like to revisit another LTV system also well-known in the adaptive systems theory [39] to more clearly illustrate how the proposed criteria can really reduce the order of the original equation. The latter is a nice feature of Krasovskii–LaSalle theorem for autonomous and periodic systems. It should be noticed that it is possible to give a more general result for a class of nonlinear time-varying systems as done in [25, Th. 3] along similar lines.

, and system (18) is in the form (1)–(2). By definitions, the functions and are both uniformly bounded and . locally Lipschitz, uniformly in , with Thus, the origin is UGAS if and only if C2) holds in view of be any limiting solution Theorem 3. Let satisfying (4). According to Theorem 1, it also satisfies (5). and denote its first -dimensional component and Let is the last -dimensional component, respectively. Since time-invariant and continuous, (4) is reduced to . Moreover, (5) is equivalent to

and (19) in by the definition of for any unbounded sequence . Thus, every limiting solution satisfying (4) is in the form with satisfying (19). Conversely, any continuous solution satisfying (19) is also a limiting solution according to Proposition 5. Thus, C2) is reduced to the following implication: (20) This just is the “zero-state integral-detectability” condition defined in [25] for the following reduced-order system and By noticing an equivalence relation proposed in [25], we can conclude that C2) is also equivalent to the following SPE condition. SPE) Suppose there exist a and a such that for , there is a such that for any all unit vector , the following inequality holds: (21)

LEE AND JIANG: GENERALIZATION OF KRASOVSKII–LASALLE THEOREM FOR NONLINEAR TIME-VARYING SYSTEMS

Hence, the origin is UGAS if and only if (SPE) holds based on Theorem 3. This is a classical result proven in [36]. Remark 11: Notice that the reduced equation describes a limiting zero dynamics w.r.t. the output map . Moreover it only satisfies the weakly minimum phase condition other than the minimum phase condition [11]. As shown in Example 7 (or by the equivalence relation build in [25]), the SPE condition, that was used to guarantee the UGAS property for system (18), is equivalent to saying that the limiting together with a reduced output map zero dynamic is “zero-state integral-detectable.” Particularly, our results point out that usually, the UGAS property can be established from detectability of certain reduced-order systems. Such viewpoint is very closely related to the passivity theory developed in [11] for autonomous systems.

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for any given positive constant . Then, the closed-loop system . can be written into the form of (1)–(2) with Under C4), let us define an extended output as follows:

(25) Since is uniformly bounded, it is in a stronger output-injection form, namely , for any compact subset of and some positive . Define a function for all constant . According to C4), the following equations can be derived:

V. APPLICATIONS The purpose of this section is to validate the obtained stability results through their applications to time-varying passive systems and the tracking control of nonholonomic systems. In particular, a novel stability analysis is proposed for a class of time-varying systems and as a case study, a tracking control problem for the popular class of nonholonomic chained systems is revisited. A new scheme for global tracking controller design is developed and, with the help of our stability results, sufficient conditions for -exponentially stabilizing trackers are derived. The reader is referred to [43] for a definition of -exponential stability. A. A New Stability Criterion for Time-Varying Passive Systems Consider a class of time-varying systems as follows: (22) (23) is a state vector, is an input vector, where is an output vector; , and are all continuous functions such that for all , and defined on . We also assume that , and are all uniformly bounded. It is shown in [15] that a system of the form (22)–(23) is passive [11], [45] when the following basic condition holds. C4)

(KYP property): There exists a continuously differentiable, positive definite and proper (storage) function so that and

When a system is passive, a stabilizing output-feedback control law can be chosen as (24)

Since is a minimum value of , we have and hence . Moreover, it can also be seen that is an AAP function in view of Lemmas 3 and 7 when and are both AAP functions. Particularly, the following proposition is readable from Theorems 2 and 3. It will serve as a fundamental result in studying a class of nonlinear time-varying systems and a tracking control problem of nonholonomic chained systems in succeeding subsections. Proposition 6: Consider a system of the form (22)–(23) with are both AAP the output-feedback law (24), where and functions. Assume that C4) holds. If H2) holds with the new extended output in place of the original output function , then, the origin of the closed-loop system is UGAS. Conversely, UGAS implies H2) when the functions and are locally Lipschitz, uniformly in . B. Specification to a Class of Nonlinear Time-Varying Systems In this subsection, as a preliminary study for nonholonomic chained systems, Proposition 6 will be applied to nonlinear time-varying systems of the form (26) where

is a state vector, is an input vector, is a square matrix, and and are both continuous functions defined on with , is a bounded and uniformly continuous function. To verify the KYP property stated in C4) and simplify the discussion that follows, we assume that the following condition holds. and a positive–definite C5) There exist a matrix so that is a skew-symmetric matrix and , matrix, . for all To derive a stabilizer for (26), let us follow the passivity approach given in the previous subsection. Notice that (26) is in . Define a virthe form of (22) with . Let be a storage tual output map as

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function. In view of C5) and by direct computation, it is easy to check that

1) 2)

and 3)

This results in C4). According to output feedback law (24), the , for any given poscontroller can be chosen as itive constant . It can also be seen that for this system. By definitions, the functions and are locally Lipschitz, uniformly in . Since for each is bounded and uniformly continuous, the functions and are both AAP functions by Lemmas 4 and 7. According to Proposition 6, the origin of the closed-loop is UGAS has full-row rank, it is easy to see if and only if H2) holds. If when . This implies that H2) holds that and the origin of the closed-loop system is UGAS by Propois not full-row rank in the sition 6. Thus, we assume that sequel. To check H2), a technical condition is assumed as follows. is controllable and for each , each C6) and each , and implies . Notice that the functions and are in the form described in , the associated Example 4. Thus, for any reduced limiting system can be explicitly written as (27) (28) where

denotes the associated limiting function of . Define a set as

has full-row rank, the origin of the closed-loop If system is UGAS. and is not full-row rank, the origin When of the closed-loop system is UGAS if and only if satisfies (PE). and is not full-row rank, the origin When of the closed-loop system is UGAS if and only if satisfies (PE).

C. Tracking Control of Nonholonomic Chained Systems Revisited In this subsection, we revisit the tracking control problem of nonholonomic systems in chained form with the help of Proposition 7. It should be mentioned that both stabilization and tracking problems for this class of systems have been extensively studied in recent literature; see, for instance, [16], [17], [19], [28], [37], and the numerous references therein. Consider the following so-called chained form system: (30) (31) where and , with , are state variables; and are control variables; and is in the controllable canonical form (CCF) [13]. To study the tracking problem, suppose the reference trajectory is described by and

(32)

We assume that and are both bounded and uniformly continuous. It should be mentioned that unlike most previous and be differentiable with work, we do not require that and . Let . bounded derivatives Define the error variables and new control variables as follows:

(29) . The following lemma shows that H2) can be with checked in terms of PE conditions by dividing the cases into and . Its proof is postponed to Appendix C. is not full-row rank and C5)–C6) Lemma 9: Suppose holds. Then, the following results are true. , H2) holds if and only if 1) When satisfies (PE). , H2) holds if and only if satisfies 2) When (PE). Now, the following result is readable from Proposition 6, Lemma 9, and the previous discussions. It will be used in studying a tracking control problem of nonholonomic chained systems in next subsection. Proposition 7: Consider a system of the form (26). Suppose . C5) and C6) hold. Let for any given With the controller chosen as positive constant , the following results hold.

(33) Then, by direct computation, it can be seen that and for any , the equations shown at the bottom of the next page. Let . Then, the following error model for tracking system can be attained:

(34) where . Let contained in be any pre-assigned set of eigenvalues. such that the eigenvalues Then, there exists a matrix of are by controllability [13]. Then, it can be shown that there exists a unique positive–definite and symmetric matrix satisfying the following Ricatti equation:

(35)

LEE AND JIANG: GENERALIZATION OF KRASOVSKII–LASALLE THEOREM FOR NONLINEAR TIME-VARYING SYSTEMS

Define a Lyapunov function candidate . Then, the time derivative of let trajectory of (34) can be computed as follows:

and along the

This suggests that the following feedback transformation: and (36) can be used such that the original systems can be transformed into a passive system [11]. More precisely, under (36), the transformed system of (34) can be rewritten into the form of (26) , and the matrix , the where functions and are given as follows:

and (37) It is interesting to observe that the linearized system is also in the form of (26) where the matrix is unchanged, and the functions and are replaced in and

(38)

respectively. In the following, let us show that C5) and C6) hold for both the transformed system and its linearized system. Due to Ricatti equation (35), is a skew-symmetric machosen as in (38) and defining trix. With the matrix

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is controllable, is also consystems. Since is controllable by direct comtrollable. Consequently, , each and each putation. For each , it can be checked that and implies and, hence, in view of (37). A similar argument also holds if we replace in . Thus, C6) holds for both systems. Due to with , it is not full-row rank. Moreover, any vector in defined in (29) can be explicitly determined as and for both systems. Thus, if and only if is full-row rank. Since and are linearly independent, it can be concluded that if and only if for both systems. Furthermore, the controller is the same for both systems. Thus, Proposition 7 can be employed to guarantee the UGAS property for both the transformed system and its linearized system under the same conditions. Moreover, let us recall an interesting lemma that was given in [28]. Lemma 10: Consider the differential equation . The origin is globally -exponentially stable when the origin is UGAS and locally exponentially stable. It is well-known that the origin of a system is locally exponentially stable when the origin of its linearized system is UGAS [18]. In view of this fact and the previous discussions, it can be concluded that the origin of the closed-loop system is globally -exponentially stable based on Proposition 7 and Lemma 10 and satisfy (PE) for and , respecwhen tively. More explicitly, consider the following conditions (see Corollary 1). , there exist two positive constants and C7) For so that (40) C8)

For so that

, there exist two positive constants

and

(41) (39) it can be directly computed that is also skew-symmetric, and , for with . This implies that C5) holds for both all

Based on the previous discussions, the following result can be established. Theorem 4: Consider the nonholonomic chained system (30)–(31) with the reference model described by (32).

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Let contained in signed set of eigenvalues and are that the eigenvalues of Ricatti equation (35) and controllers as

be any preasbe a matrix such . Let be the solution of . Choose the

of (1) such that the associated sequence converges uniformly to on every compact subset of Furthermore, we have

.

and (A1)

(42) for any given positive constant where the error state variables and are defined as (33). Then, the origin of error system satisfies (34) is globally -exponentially stable when C7) for the case , and satisfies C8) for the case , respectively.

Due to the fact that the system is in the output injection form, it holds

VI. CONCLUSION A practically applicable generalization of the celebrated Krasovskii–LaSalle theorem has been obtained by employing H1) and H2), an almost bounded output-energy condition and a weak zero-state detectability condition related to reduced limiting systems, respectively. A new characterization of UGAS was also proposed and applied to studying a class of time-varying passive systems and the tracking control of nonholonomic chained systems. Based on the supplied examples and applications, it can be seen that the proposed approach is very suitable for analyzing uniform asymptotic stability of nonperiodic nonlinear time-varying systems. Interestingly, the verification process is rather simple and is indeed similar to the use of classic Krasovskii–LaSalle theorem in autonomous and periodic systems. Particularly, the proposed criterion can be used to replace the Barbalat lemma, that is often invoked to achieve the (nonuniform) asymptotic stability of the origin, in order to guarantee a stronger uniform convergence result for adaptive systems and many nonholonomic systems. The continuity assumptions as well as the definition of AAP functions given in this paper are provided just for simplifying the whole presentation and making the usefulness of limiting function clearer. It is possible to consider more general functions as done in [6]. It is our belief that the assumptions imposed within our framework are suitable for practical applications. Take the tracking control problem studied in Section V-C as an example. The assumptions are closely tied to reference (to-be-tracked) signals, that, as explained previously, are often (merely) bounded and uniformly continuous. Consequently, the reference signals are AAP functions and the proposed criterion can be used to guarantee UGAS of the closed-loop tracking systems. It is a topic of future study to examine the flexibility and practical efficiency of the presented framework in studying tracking problems for other classes of physical systems.

(A2) Since also have

and

are continuous in , uniformly in , we

and

where the latter inequality follows from (4), (A2), and the Lebesgue dominance theorem [21]. This implies that satisfies (5) in view of (A1). According to Proposition 1, it remains to check C1). Define a new output as . For any , let . By virtue of H1), for any compact , there exists a positive constant such that (6) holds. For any , choose a positive integer satisfying the inequality . Then, the following inequality can be derived:

By the choices of

and , we have

APPENDIX A PROOF OF THEOREM 1 Proof: Let be any limiting solution satisfying (4) w.r.t. in . Then, there exist a coman unbounded sequence and a sequence of solutions pact

Thus, C1) holds for the new output function with . It is straightforward to see that C2) also holds for new output

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when it is true for the old one The theorem follows thus from Proposition 1.

.

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is a sequence of solutions of (1), the following inequalities hold:

APPENDIX B PROOF OF PROPOSITION 4 Proof: Let be any continuous solution of (5) that satisin . Let fies (4) w.r.t. an unbounded sequence with be any sequence of solutions of (1). Since the solutions are globally uniformly bounded, is lying within a compact . Let be any given positive constant. Then, is bounded on and there is also a compact of containing such that . Thus, we have a subsequence of such that the associated sequence converges uniformly to a limiting solution in on every in view of Lemma 1. Since and are compact subset of locally Lipschitz, uniformly in , there exist three positive con, and such that stants and with . We claim that for any , the equation , can be deduced when the inequality holds, . It will be proven along similar lines in the proof of the uniqueness of solutions [14]. First, let us show that the following inequalities hold:

(A3) and

(A4)

by Fatou’s lemma [21]. This results in

by virtue of inequalities (A3)–(A4) with . From the Bellman–Gronwall inequality, one concludes [14], . Thus, the claim that by is true. Notice that . Thus, there exists a positive constant the definition of such that . Let . Then, and we have a sequence such and that . If , it can be seen that , by the claim. This implies that there exists a such that , again and . We reach a contradiction by the continuity of by the definition of . Thus, and one concludes that holds, , in view of the claim. Since is arbitrary, we also have . This shows that is a limiting solution and the proof is completed.

Inequality (A3) follows from APPENDIX C PROOF OF LEMMA 9

by employing the equation schitz continuity. Similarly, we have

for almost all

and the Lip-

in in view of (4), , and the Lipschitz continuity. Particuis a solution of (5) and larly, (A4) also holds. Since

Proof: : The “only if” part will be proven 1) The case of by contradiction as follows. If H2) holds and does not satisfy (PE), we have a such that the associated limiting function is a zero function by Corollary 1. Then, (27) is reduced , i.e., every constant function is a solution of to (27). Since we have assumed, in priori, that is not full-row rank, particularly there is a nonzero vector so that . Then, is a solution of (27) . This violates H2) and hence, and the “only if” part is true. Now, let us prove the “if” , part. We can assume that for any is not a zero the associated limiting function function by (PE) and Corollary 1. Let be a bounded solution of (27) satisfying . Differentiating this equation, the following equation can be derived: (A5)

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If there is a so that , we have in view of , (A5) and C6). Since , it can be concluded that and thus, H2) holds in this case. Otherwise, we can asis a zero function. In this case, (27) sume that and there exists a nonempty reduces to such that and, thus, open interval , in view of (A5). Repeatedly differentiating this equation and following the procedure given in Example 1, it can be recursively . shown that Notice that by the skew-symmetric property of

2)

. Thus, , by . This shows that H2) holds controllability of and completes the proof of part (1). : Similarly, the “only if” part will The case of be proven by contradiction as follows. Since and are both AAP functions, is also AAP function with by Lemma 3. If does not satisfy (PE), there exists a H2) holds and such that the associated limiting has the first component being function a zero function by employing Lemma 2 and Corol. Let lary 1. Then, (27) is reduced to be any nonzero vector in and define . This implies . Thus, is a solution of (27) and satisfies . This again violates H2) and hence, the “only if” part is true. Now, let us prove the “if” part. In this case, we can assume that for , the associated limiting funcany tion is not a zero function by employing Lemma 2, (PE) and Corollary 1. Let be a bounded solu. Diftion of (27) satisfying ferentiating this equation and along the line as in (1), (A5) can be derived. Based on (A5), we claim that . Indeed, for each with , we have again by (A5) and C6). Thus, the claim is true and (27) reduces to . Again by controllability of and noticing that is not a zero function, there exists a nonempty open interval such that by following the proof as in part (1). This shows that H2) holds and thus the proofs of part (2) and the lemma are completed.

ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their detailed and constructive comments that have helped us to improve the presentation and results of this paper. REFERENCES [1] D. Aeyels, “Asymptotic stability of nonautonomous systems by Liapunov’s direct method,” Syst. Control Lett., vol. 25, pp. 273–280, 1995.

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LEE AND JIANG: GENERALIZATION OF KRASOVSKII–LASALLE THEOREM FOR NONLINEAR TIME-VARYING SYSTEMS

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Ti-Chung Lee was born in Taiwan, R.O.C., in 1966. He received the M.S. degree in mathematics and the Ph.D. degree in electrical engineering from the National Tsing Hua University, Hsinchu, Taiwan, in 1990 and 1995, respectively. In August 1997, he joined the Ming Hsin University of Science and Technology at Hsinchu as an Assistant Professor of Electrical Engineering, and since 2005, he has been a Professor. His main research interests are stability theory, tracking control of nonholonomic systems, and robot control.

Zhong-Ping Jiang (M’94–SM’02) received the B.Sc. degree in mathematics from the University of Wuhan, Wuhan, China, in 1988, the M.Sc. degree in statistics from the Université de Paris-sud, France, in 1989, and the Ph.D. degree in automatic control and mathematics from the École des Mines de Paris, France, in 1993. From 1993 to 1998, he held visiting researcher positions in various institutions including INRIA (Sophia-Antipolis), France, the Australian National University, Canberra, the University of Sydney, Sydney, Australia, and the University of California. In January 1999, he joined the Polytechnic University, Brooklyn, NY, as an Assistant Professor of Electrical Engineering, and since 2002, he has been an Associate Professor. His main research interests include stability theory, the theory of robust and adaptive nonlinear control, and their applications to underactuated mechanical systems, congestion control, and wireless networks. In these areas, he has (co)authored four book chapters, 70 journal papers, and numerous conference papers. Dr. Jiang is a Subject Editor for the International Journal of Robust and Nonlinear Control, an Associate Editor for Systems and Control Letters, the IEEE TRANSACTIONS ON AUTOMATIC CONTROL, and European Journal of Control. He is a Recipient of the prestigious Queen Elizabeth II Fellowship Award from the Australian Research Council, the CAREER Award from the U.S. National Science Foundation, and the JSPS Invitation Fellowship from the Japan Society for the Promotion of Science.