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A Nested Matrosov Theorem and Persistency of Excitation for Uniform Convergence in Stable Nonautonomous Systems Antonio Loría, Elena Panteley, Dobrivoje Popovic´, and Andrew R. Teel
Abstract—A new infinitesimal sufficient condition is given for uniform global asymptotic stability (UGAS) for time-varying nonlinear systems. It is used to show that a certain relaxed persistency of excitation condition, called uniform -persistency of excitation ( -PE), is sufficient for uniform global asymptotic stability in -PE of the right-hand side of a time-varying certain situations. differential equation is also shown to be necessary under a uniform Lipschitz condition. The infinitesimal sufficient condition for UGAS involves the inner products of the flow field with the gradients of a finite number of possibly sign-indefinite, locally Lipschitz Lyapunov-like functions. These inner products are supposed to be bounded by functions that have a certain nested, or triangular, negative semidefinite structure. This idea is reminiscent of a previous idea of Matrosov who supplemented a Lyapunov function having a negative semidefinite derivative with an additional function having a derivative that is “definitely nonzero” where the derivative of the Lyapunov function is zero. For this reason, we call the main result a nested Matrosov theorem. The utility of our results on stability analysis is illustrated through the well-known case-study of the nonholonomic integrator.
U
U
Index Terms—Matrosov theorem, nonholonomic systems, timevarying systems, uniform stability.
I. INTRODUCTION
I
N MANY interesting nonlinear control problems, the closed-loop control system can be modeled by the timevarying, not necessarily periodic, differential equation (1) When convergence of the trajectories of (1) to a given fixed point is required (this includes many trajectory tracking and adaptive control problems), perhaps the most appealing notion that includes such convergence is uniform asymptotic stability
Manuscript received July 29, 2003; revised July 30, 2004. Recommended by Associate Editor Z.-P. Zhang. This work was supported in part by a CNRS-NSF collaboration project and was done in part while the first two authors were visiting the Center for Control Engineering and Computation, the University of California, Santa Barbara. This work was also supported by the Air Force Office of Scientific Research under Grants F49620-00-1-0106 and F49620-03-10203 and by the National Science Foundation under Grants ECS-9988813, ECS0324679, and INT-9910030. A. Loría and E. Panteley are with the C.N.R.S, UMR 8506, Laboratoire de Signaux et Systèmes 91192 Gif s/Yvette, France (e-mail:
[email protected];
[email protected]). D. Popovic´ is with United Technologies Research Center, East Hartford, CT 06108 USA. A. R. Teel is with the Center for Control Engineering and Computation, Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA 93106-9560 USA. Digital Object Identifier 10.1109/TAC.2004.841939
(UAS) because of its inherent robustness (at least when has certain continuity properties that are uniform in ; see, e.g., [1, Lemma 5.4]) and its assertion that the convergence rate does not depend on the initial time in any significant way. See [2] for more detailed discussions on this topic and a list of related references. One can precisely characterize the property of UAS in many seemingly different, but in fact equivalent, ways. One way is estimates; that is, using a bounding function via so-called over the norm of the solutions that decreases uniformly with time and increases uniformly with the size of the initial states; see, e.g., [8]. Another common characterization is via the use of a smooth positive definite Lyapunov function with a uniformly negative definite total derivative; see, for instance, [1] and [3]. While these characterizations are very useful as intermediate steps in proving other properties (for instance, robust stability esof a perturbed system), they can be difficult to establish. timates are difficult to obtain because the solutions of (1) are not explicitly available. Lyapunov functions are appealing because explicit solutions to (1) are not needed. On the other hand, Lyapunov functions can be difficult to obtain because of the requirement that the derivative be uniformly negative definite. With regard to Lyapunov functions, in many model-based control applications it appears natural to use (when available) the closed-loop energy function as a candidate Lyapunov function. However, often the time derivative of this function is only negative semidefinite. For time-invariant problems, the typical analysis tool used to circumvent this problem is the Krasovskii–La Salle invariance principle. This principle, the use of which requires some information about the solutions of the system, asserts that the trajectories converge to the largest invariant set contained in the set of points where the derivative is zero intersected with a level set of the Lyapunov function [4]. In the special case where this invariant set is the origin, asymptotic stability can be asserted [5]. The Krasovskii–La Salle invariance principle is the key result that enables the so-called Jurdjevic´–Quinn control algorithm for open-loop stable nonlinear control systems [6]. When the closed-loop is time-varying, one tool that is often used when the derivative of a Lyapunov function is only negative semidefinite is Barb˘alat’s Lemma [7], [8]. This tool enables asserting that the derivative of the Lyapunov function converges to zero under certain regularity hypotheses. In adaptive control, Barb˘alat’s Lemma is frequently relied upon to establish convergence to zero of part of the state. Barb˘alat’s Lemma has also been used to establish convergence to the origin for a class of
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nonholonomic systems controlled by smooth time-varying feedback. However, even when Barb˘alat’s lemma can be used to show that the entire state converges to zero it typically cannot be used to conclude that the converge to zero is uniform in the starting time. It is well known that the invariance principle also applies to periodic time-varying systems (cf. [5] and [9]). Possibly motivated by these results, the study of stability for time-varying systems turned toward so-called asymptotically autonomous systems. Byproducts of this line of research are results based on the method of limiting equations.1 Among the most significant results based on the method of limiting equations, we cite [12] where necessary and sufficient conditions for UAS are established. Along the same line of research is the recent work of [13]–[16]. In these papers, the authors present a series of results among which we single out a generalization (in certain directions) of Krasovskii–La Salle’s invariance principle using the method of limiting equations. Roughly, in [16] UAS for a class of time-varying systems is concluded based on a reasoning a la La Salle on the system’s equations considered as the initial time approaches infinity. The originality of this work resides in the introduction of different notions of uniform detectability which aid to conclude UAS for the case when one has a Lyapunov function with a negative semidefinite derivative. See [11] for an early reference on “invariance principles” for non autonomous systems using the formalism of limiting equations. For time-varying systems, another tool that has been used, but more sparingly, is Matrosov’s theorem which first appeared in [17]; see also [18]. It pertains to the situation where one Lyapunov function that has a continuously differentiable establishes uniform stability and also a auxiliary function with appropriate properties. In particular, the auxiliary function should be bounded uniformly in time on bounded regions of the state space, and should have a “definitely nonzero” derivative on the set where a given, continuous, time-independent nonpositive upper bound on the Lyapunov function’s derivative vanishes. Roughly speaking such property on the second auxiliary function allows to conclude that the trajectories cannot remain trapped in the set where the first function’s derivative is zero but they necessarily converge to an invariant subset of the latter. Thus, Matrosov’s theorem can be regarded, to some extent, as an invariance principle for nonautonomous systems. Significantly, it does not require any explicit information about the solutions of the system. It is purely an infinitesimal condition. While Matrosov’s theorem relieves some of the burden from the first Lyapunov function, there are still no systematic methods for finding the second auxiliary function in Matrosov’s theorem. A simplified version of Matrosov’s theorem was used in [19] to establish one of the first results on uniform global asymptotic stability (UGAS) for robot manipulators in closed loop with a tracking controller. It also appears in the context of adaptive control in [20] and output feedback control, e.g., in [21]. There have been several interesting extensions of Matrosov’s theorem over the years, mostly found in the work of [10], [18], and the references therein. In [22], a vector auxiliary function 1Roughly speaking, dynamic equations describing the limiting behavior of the system when shifting time by t t t where ft g is an infinite unbounded sequence. For precise definitions and statements see [10, Chapter VIII, Section 5] and references therein as well as [11].
= +
is used while in [23] and [24] a family of auxiliary functions is considered. It is worth emphasizing that in the latter reference the family of auxiliary functions is possibly uncountable and extensions are given that pertain to stability of sets. Moreover, locally Lipschitz Lyapunov and auxiliary functions are allowed. In all of these cases, the behavior of the auxiliary functions is referenced to the set where the upper bound on the derivative of the Lyapunov function vanishes. In [19], a simplified condition is given for checking that the auxiliary function has a sign-definite derivative. Most recently, in [25] the authors have extended Matrosov’s theorem to pertain to differential inclusions, at the same time addressing stability of sets, using locally Lipschitz auxiliary functions and weakening the requirements on the upper bound of the derivative of the Lyapunov function. Despite all of the extensions of Matrosov’s theorem, it is difficult to give general constructive methods for constructing Matrosov functions in the same way that it is difficult to give general constructive methods for finding Lyapunov functions—notable recent exceptions are [26] and [63]. In order to meet the required conditions on the derivatives of Lyapunov (or auxiliary “Matrosov”) functions, one approach is to use observability-type arguments and/or exciteness conditions. Roughly speaking one tries to identify a converging output and then verifies whether all the modes of the zero-dynamics (i.e., the dynamics which is left by zeroing the output) are sufficiently excited so that all converge to zero. For linear systems such methods have been under investigation, based on [27], starting probably with [28], and followed by numerous works including [29]–[32]. In the nonlinear case we find for instance [2], [33], [34]. See also [15] which establishes sufficient conditions for uniform convergence in terms of a notion of (uniform) detectability and which is shown to be equivalent to different notions of persistency of excitation tailored for nonlinear systems and proposed therein as well as that presented in [2] (cf. Def. 4). The advantage of methods using observability-type arguments and related exciteness conditions is that for a certain class of systems one may infer the stability and convergence properties simply by “looking” at the dynamics of the system. For instance, for linear systems
with bounded , it was shown in [30] that it is necessary and be persissufficient for uniform asymptotic stability that and tently exciting (PE), namely that there exist such that for all unitary vectors (2) Equivalently, this system is uniformly (in ) completely (i.e., for all initial states) observable (see, e.g., [35]) from the output if and only if is PE. Notice that does not need to be full rank for any fixed . That PE is a necessary condition for uniform asymptotic (exponential) stability for linear systems has been well-known for many years now. In the case of general nonlinear systems, this was established for a generalized notion of persistency of excitation by Artstein in [12, Th. 6.2]. Relying on the notion of limiting equations it was proved in [12] that for the system (1) it
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is necessary for uniform convergence that for each and such that exist
there
(3) Sufficiency was also shown under other conditions involving Lyapunov functions and making use of a theorem establishing that uniform asymptotic stability of (1) is equivalent to asymptotic stability of all the limiting equations of (1). In view of the importance that persistency of excitation and related observability conditions have been proved to have in establishing convergence of linear and nonlinear systems, PE has been at the basis of the formulation of sufficient conditions for uniform asymptotic stability in many contexts. For example, see [12, Th. 6.3] and the results in [2], [15], and [27]–[34] among others. In the context of adaptive control of nonlinearly parameterized systems, other notions of persistency of excitation for nonlinear systems have been introduced recently in [36]. In [2] and [37], we introduced a sufficient condition for uniform attractivity for a certain class of nonlinear systems. In words, our condition is that a certain function evaluated along the trajectories of the system, be persistently exciting whenever the trajectories are bounded away from a -neighborhood of the origin (cf. Def. 4). Loosely speaking, this property, called uniform -persistency of excitation ( -PE), ensures that the zero-dynamics of the system (with respect to a converging output) is sufficiently excited. We wrap up this brief review with [26] where the author constructs Lyapunov functions for time-varying systems based on Lyapunov functions with negative semidefinite derivative and by smartly exploiting the persistency of excitation of certain functions. See also [63]. A. Contributions of This Paper The contribution of this paper is twofold. On one hand, we present a new definition of -PE which we show to be necessary for uniform attractivity of the origin of general nonlinear time-varying systems. On the other hand, we present a generalized Matrosov theorem and show that one useful application of it is in proving that under some additional assumptions, -PE is sufficient for uniform attractivity. The new definition of -PE that we present here is stated in a form that does not involve the state trajectories (see Def. 3) and, therefore, it is simpler to verify than its predecessors (cf. [2] and [37]). As a matter of fact, it is also a reformulation of the condition (3) and for the system (1), we will show both necessity and sufficiency for uniform convergence without relying on limiting equations theory. The former follows simply using a Lipschitz condition and Gronwall’s lemma. Our generalized Matrosov’s theorem distinguishes itself from its predecessors in that it uses a finite family of auxiliary functions (as opposed to only one) to establish uniform convergence. For simplicity of exposition, we will limit our discussion to differential equations, uniform asymptotic stability of the origin, and a means of checking a sign-definiteness condition that is similar to what was used in [19]. Nevertheless, the results that we present here extend to more general settings (stability
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characterizations of signof sets, differential inclusions, definiteness, etc.) Since we use a finite family of auxiliary functions, it is natural to wonder about the comparison to the vector auxiliary function used in [22] and the families of auxiliary functions used in [23] and [24]. The main difference is that, whereas in the previous references the behavior of all of the auxiliary functions and their derivatives is referenced to the set where the upper bound on the derivative of the Lyapunov function vanishes, our auxiliary functions are ordered and the behavior of an auxiliary function and its derivative is referenced to the set where the upper bounds on the derivatives of all of the preceding auxiliary functions vanish. We assume uniform stability, rather than assuming that we have a Lyapunov function which establishes it, and then none of our auxiliary functions is assumed to be sign definite. All of the auxiliary functions are assumed to be locally Lipschitz. We believe that the analysis tools that we present here may become an efficient tool to aid time-varying nonlinear control design, and may contribute to making the idea of using auxiliary functions for nonlinear systems, as introduced by Matrosov, a more versatile concept. A step forward in this direction is made in the early and extended versions of this paper [38], [39], where we solve some stabilization problems for fairly general nonlinear time-varying systems but having a clear impact in particular applications. See also [40] where necessary -PE and using our genand sufficient conditions in terms of eralized Matrosov theorem, are established for the stabilization of interconnected driftless systems. This setting generalizes the popular benchmark of nonholonomic chain-form systems. The rest of the paper is organized as follows. In Section II, we define our notation and give some basic definitions. In Section III, we present our nested Matrosov theorem. In Section IV, we present the definition of uniform -PE ( -PE) and related properties. In Section V, we give a short proof of necessity of -PE and a result on sufficiency of -PE which covers Artstein’s result. In Section VI, we illustrate, through the well-understood benchmark of nonholonomic systems, how to use our main results. We conclude with some remarks in Section VII. II. PRELIMINARIES Notation: Throughout this paper, stands for the Euclidean , where norm of vectors and induced norm of matrices, and , denotes the norm of time signals. In particular, , by we mean for a measurable function tity fine use is of class
for and . For two constants
denotes the quan, we de. We also will
. A continuous function if it is non decreasing. A continuous function is of class , if it is strictly increasing ; if in addition, as . and is of class if A continuous function for each fixed and as for each . Unless stated otherwise, throughout this paper we assume for that is locally bounded, the differential equation
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continuous almost everywhere, and that is continuous locally uniformly in . This property guarantees that soexist locally in time, but they are not lutions of necessarily unique. We will use to denote a solution . We will also use when we with initial condition wish to leave the initial condition implicit. By abuse of notation, for a locally Lipschitz function and for all points where is differentiable, which is almost everywhere, we define . This abuse of notation is justified here as follows: Since is locally Lipschitz and each solution of is absolutely continuous, the usual time derivative exists for almost all time where the solution is defined. Moreover, with the properties we have assumed for and assuming the same properties for it follows from Fubini’s theorem and the tools of nonsmooth analysis2 that if for almost all then, for each solution we also have that for almost all . As it has been motivated, e.g., in [2] and [44], for time-varying systems the most desirable forms of stability are those which are uniform in the initial time. Definition 1 (Uniform Global Stability): The origin of the system (1) is said to be uniformly globally stable (UGS) if there exists such that, for each each solution satisfies
Example 1: Let us consider the Lagrangian model of a rigidjoints robot (see, e.g., [45])
where the inertia matrix is positive definite for all , is skew-symmetric and are control torques. The control problem is to make the robot follow a smooth reference such that . For trajectory this, we apply the following control law, originally proposed in [19]
where , , . Now, we would like to analyze the stability of the closed-loop system, so we use the energybased Lyapunov function
to obtain that and, hence, that the origin of the system is uniformly globally stable (UGS). From this equality, it is also fairly standard to invoke Barb˘alat’s Lemma to (see, e.g., [46]). Alternatively, we can use conclude that the auxiliary function which is uniformly bounded on compact sets of the state and whose total time derivative satisfies3
(4) Definition 2 (Uniform Global Attractivity): The origin of (1) is said to be uniformly globally attractive if for each , there exists such that
We present in this section our main theorem for analysis of time-varying systems. In the succeeding section we will present a fairly general result for UGAS of the origin of (1) based on the property of -PE and which makes use of our generalized Matrosov’s theorem. To put our contribution in perspective we find it convenient to illustrate first how the original Matrosov’s theorem (see [17], but also [18, Th. 5.5] and [8, Th. 55.3]) works on tracking control and time-varying stabilization problems. To that end, let us first look at the example of tracking control of robots as addressed in [19].
Intuitively, since we know that we may think that so, loosely speaking, if we wait long enough (and we can do that because the system is UGS) we will have that for . In particular, is square-intelarge , grable so invoking again Barb˘alat’s Lemma one may accept that as well. it should hold that This idea can be made rigorous to actually prove UGAS via Matrosov’s theorem: one needs to observe that is sign-definite (it is actually negative for all nonzero values of the state) on that is, on . The argument the set at work here is, roughly speaking, that the sign-definiteness of and the fact that on the set imply that the system’s trajectories cannot remain trapped on the set unless they go to zero. See [8, p. 263] for a rigorous development on this idea and [19] for a rigorous analysis of this control system based on Matrosov’s theorem. One important characteristic of the control problem shown previously that helps to use [18, Th. 5.5] is that the system is of relative degree one with respect to the output . Another is that the system is stabilizable by static feedback. However, for systems not satisfying these conditions, such as the nonholonomic chain of integrators, the analysis is more involved. In particular, one seemingly needs several auxiliary functions with appropriate properties in order to conclude attractivity of the origin. Yet, the intuition a la Krasovskii–La Salle still holds as the following example illustrates. In Section VI, we establish precise
2For more precise statements, based on [41] and [42], see [25, proof of Prop. 1] and [43, item 5, p. 100].
3Using the facts that D (q ) [45] for further details.
(5) Furthermore, we say that the origin of the system is uniformly globally asymptotically stable (UGAS) if it is UGS and uniformly globally attractive. This property is equivalent to the exsuch that, for all and istence of (6) III. NESTED-MATROSOV THEOREM A. Motivational Examples
j
j
d
and
j
C (q; q_ )
j j _j. See, for instance, k
q
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results for nonholonomic systems using the Nested–Matrosov theorem (cf. Theorem 1). Example 2: Consider the system (7a) (7b) (7c) in closed loop with4 (8a) (8b)
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B. Main Result Theorem 1 (Nested Matrosov Theorem): Under the following assumptions the origin of (1) is UGAS.5 Assumption 1: The origin of the system (1) is UGS. and for each Assumption 2: There exist integers , there exist • a number ; • locally Lipschitz continuous functions , ; • a function ; , ; • continuous functions , and all such that, for almost all
where and has certain excitation properties (to be specified). The closed-loop system becomes (with yet to be defined) (9a) (9b) (9c) The general intuition to establish a proof of asymptotic stability of the origin of (9) can be explained in terms of Krasovskii–La Salle invariance principle. For this, let us restrict our attention to periodic feedbacks (as in [48]). First, taking the derivative of , we obtain that (10) From this inequality, we obtain that , hence, we may . In addition to this, also admit from (9c) that . This means that we have from (9b) that tends to a steady-state value which we denote by . Now, is periodic in it is reasonable to assume that it since . is also sufficiently rich (persistently exciting) for each If this is the case, then from the conjecture that necessarily the only constant value which may converge to is obtained from (9a) if is zero. Finally, the convergence of such that . we define The clear drawback of such an argument is that it cannot be made precise for general nonautonomous systems since it relies on Krasovskii–La Salle invariance principle. However, with the aid of auxiliary functions one can establish the right convergence properties for each variable. Roughly speaking, similarly to the previous example, one needs to find an auxiliary function (also for not necwhich allows to conclude that ) on the manifold defined by essarily periodic functions . Then, one more function is needed to conclude that, under appropriate exciteness conditions, . The appropriate exciteness conditions may be thought of as a condition of positivity, in an averaged sense, of the function uniformly in ; this is formalized in Section III-B, in terms of U -PE. The Nested–Matrosov theorem presented next formalizes the general intuition from the previous examples. The control problem of nonholonomic systems is addressed in Section VI in order to illustrate the use of use of Theorem 1. 4This control law was proposed first in [47]—see also [37] for a proof of UGAS—and is used here for the sake of illustration only.
(11) (12) Assumption 3: For each integer that6 A)
, we have , and all
implies that B) for all Assumption 4: We have that the statement A) , and all
.
implies that B) . Theorem 1 generalizes, in certain directions, [25, Pro. 2] (see also [49, Prop. 2]) which, as clearly shown in that reference is, in turn, an extension of the “classical” Matrosov theorem [17] which combines an auxiliary function with a Lyapunov function that establishes UGS. See also the more recent expositions of Matrosov’s theorem: [18, Th. 5.5, p. 58], [10, Th. 2.5, p. 62], and [8, Th. 55.3]. In particular, with respect to the formulation presented in [10, Th. 2.5, p. 62], it is worth remarking that: conditions i) and ii) imply UGS (assumed in Theorem 1), condition , condiiii) implies the first part of (11) for the case of tion v) (local uniform boundedness of ) is not imposed in Theorem 1 but we assume a local Lipschitz property. Besides the fact that the conclusion of Theorem 1 relies on finding, in general, more than one auxiliary function (besides the one to conclude UGS), the main difference with respect to [10, Th. 2.5, p. 62] is that we do not require that any of the functions to be sign-definite (see the cited reference for a precise definition) on the set where all the previous bounding functions, i.e., with , are zero. In particular, the bounds are allowed to depend on time through bounded (in ) functions as for instance in [19, Lemma in the Appendix]. Finally, we remark that Assumptions 3 and 4 imply Claim 1 (see further later) which in turn implies the sign-definiteness assumption in the original Matrosov’s theorem. See [25, Prop. 2 and Cor. 1] for precise statements. 5For clarity, we remind the reader that we do not assume that F is locally Lipschitz in x uniformly in t as is common use. See Section II. 6For the case k = 1 this assumption takes the form: “statement B) holds with k = 1.”
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It can be useful to note that when the is locally Lipschitz uniformly in then the uniform global stability assumption can be relaxed to uniform (local) stability and boundthat is uniform in . In paredness of trajectories for each ticular, the following hold. Theorem 2: If Assumption 1 in Theorem 1 is replaced by Assumption 5: 1) the origin is uniformly stable; such that 2) for each , there exists for all and ; 3) is locally Lipschitz uniformly in and Assumptions 2–4 hold then, the origin of (1) is UGAS. A sketch of proof is provided in the Appendix.
and and positive definiteness of . In this case, the result of Corollary 2 is closely related to the results of [53, Sec. III]. Proof of Corollary 2: UGS holds by hypothesis [cf. (18) and (19)]. For the application of the nested Matrosov theorem, and, for , and we take . We obtain that
C. Corollaries of Main Result
Observing that (19) implies the uniform boundedness, in , for all , we have that there exists of such that for all , and . Then, we may invoke Theorem 2 with all and the functions , for , , for and . Remark 2: Note that if we can write
A corollary of Theorem 2 generalizes well-known results for time-varying nonlinear cascades 7
(13) (14) Corollary 1: If, for (13) and (14), each initial condition produces trajectories that are bounded uniformly in the initial time, the functions and are locally Lipschitz uniformly in , and the origins of (14) and are UGAS then, the origin of (13) and (14) is UGAS. The following corollary, of Theorem 1, establishes that if one has a positive definite Lyapunov function with a negative semidefinite derivative bounded by a function that is “observable,” then UGAS follows. Observability is verified by differentiating the output as many times as needed so as to obtain a uniformly positive–definite function, if possible. More precisely, consider (1) with a -times continuously differentiable and define and for output , . Define also .. .
(15)
. Corollary 2: Consider the system (1) with output , and a locally LipsSuppose there exist: such that chitz function (16) (17)
(20) and (21)
(22a) (22b) (22c) .. .
(22d) (22e)
with the bounded for bounded , given by (15) redefined by (22), the result can be proved with and . This extension is useful if the same functions and are not differentiable. In other words, Corollary 2 . contains the special case where D. Proof of the Main Result (Theorem 1) To prove the theorem we first need to establish the following claims. , there exists such that Claim 1: Given A)
Then, the origin is UGAS. Remark 1: In the autonomous case, i.e., when and are independent of time, (18) and (19) are implied by continuity of
implies . B) Proof: We prove the claim by contradiction. Suppose that for each integer , there exist such that for all , and . By compactness of , the continuity of , and Assumption 3, the sequence has an accumulation point such that for all . By Assumption 4, this imwhich contradicts the fact that . plies that
7The result that is best known is for the autonomous case and is due to [50], [51]. The corresponding result for the time-varying case, under the assumption of uniform boundedness of trajectories in both t and x , first appeared in [52].
Claim 2: Let tion
and positive–definite functions
,
, and
such that (18) (19)
, and a continuous funcbe given. Then, Property 1 implies
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Property 2. Property 1: A)
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According to the conditions of the theorem and the previous , discussion, we have that, for almost all (27)
implies that B) Property 2: there exists A)
.
(28) such that and, using (11) together with (25) we obtain that for all
implies that . B) Proof: By Assumption 3 and , Property 2A implies that . Therefore, Property 2A implies
(29) and
Using Assumption 1, for each and such that
there exist (30)
(23) , then, due to Property 1, Property 2B Now, if whenever Property 2A holds. We claim holds for all such that Property 2B holds further that there exists whenever Property 2A holds and . Suppose not, i.e., for each integer there exists such that and
and (31) Let and generate and and then let and generate and through the aforementioned claims and definitions. Let (32)
(24) We claim that Then, by compactness of , continuity of , has an accumulation and Assumption 3, the sequence such that . However, then from point . By continuity of Property 1 we have that this contradicts (24) when is large and associated with a subsequence converging to the accumulation point. It now follows from the continuity of and compactness of that we can pick large enough to satisfy
(33) Suppose not. It follows from the previous discussion that for all . Owing to the remarks in Section II on the derivative of locally Lipschitz functions along trajectories, it then follows that for almost all
(34) Integrating and using (27), we have then Property 2A implies Property 2B. We now use these two claims to prove the theorem. According , to Claim 1, Property 1 of Claim 2 holds when and . An application of Claim 2 with these choices provides a value such that Property 1 of Claim 2 holds when , and . Continuing there exists with this iteration, it follows that for each and positive real numbers , such that, for all (25) Next, define the locally Lipschitz function as (26)
(35) which contradicts the choice of
in (32).
IV. UNIFORM -PERSISTENCY OF EXCITATION In this section, we present a new definition of -PE, a property originally introduced in [2] and [37]. The newly defined property is conceptually similar to each of the previous ones; however, it is technically different in the sense that: first, it is easier to verify since it is formulated as a property inherent to a nonlinear function instead of being directly related to the solutions of a differential equation. Second, the new property is also necessary for uniform attractivity of the system (1). We also stress that, in general, neither -PE as defined in [2] nor as defined later, implies the other. be partitioned as where Let and . Define the column vector function and the set .
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Definition 3: A function where is locally integrable, is said to be uniformly -persistently exciting there exist , ( -PE) with respect to if for each and s.t. (36)
If is -PE with respect to the whole state then we -PE.” This notation allows us to will simply say that “ is establish some results for nonlinear systems with state by imposing on a certain function the condition of -PE w.r.t. only part of the state. For the sake of comparison, we recall next the definition proposed in [2] which is stated as a property of a pair of functions where is the vector field in (1) and the matrix function is such that is locally inof (1). tegrable for each solution is called uniformly -perDefinition 4 [2]: The pair sistently exciting ( -PE) with respect to [along the trajecthere exist constants tories of (1)] if, for each and and , s.t. , all corresponding solutions satisfy
for all . We emphasize that Definition 4 is cited here only for the sake of comparison. Throughout this paper, when we say that a function is -PE we mean in the sense of Definition 3. From now on, we will refer to the property defined in Definition 4 as “ -PE along trajectories”. Even though in essence, the properties in both definitions are the same, as pointed out before, they are mathematically different. This is illustrated by the following example. Example 3: Consider the system
A. Characterizations of
-PE
In this section, we present some useful properties which are equivalent to Definition 3. The proofs of all statements are omitted for space constraints (cf. [38]). Our first charac-PE applies to the particular (but fairly wide) terization of class of uniformly continuous functions; we show that for such functions it is sufficient to verify the integral in (36) only for (i.e., for “large” states). each fixed such that is continuous uniformly in then Lemma 1: If is -PE with respect to if and only if • A) for each there exist and such that, for all (38) The following Lemma helps us to see that Definition 4 states is -PE with respect to in words that “a function if is PE in the usual sense8 whenever the states (or similarly, the trajectories ) are large.” This is important since it is the central idea to keep in mind when establishing sufficiency results based on the -PE property. This idea also establishes a relation with the original but also technically different definition given in [37]. It is also convenient to underline the similarity with the necessary condition (3), introduced in [12]. is -PE w.r.t. if and only Lemma 2: The function if • B) for each and there exist and such that, for all
(39) The last characterization is useful as a technical tool in the proof of convergence results as we show in Section V-B. is -PE w.r.t. if and only Lemma 3: The function if • C) for each there exist and continuous strictly decreasing such that, for all
(37) whose solutions with initial conditions take the form , function
and . Consider also the
Clearly, is locally integrable for each . One can see also that this function satisfies Definition 3 since but it does not satisfy Definition 4 with the initial conditions shown be. On the other hand, the function fore, since satisfies the trajectories-dependent property of Definition 4 but it does not satisfy Definition 3 for , . However, it is -PE in the sense of Definition 3 with respect to .
(40) Remark 3: We may summarize the previous characterizations as follows. • The following are equivalent: is -PE with respect to , statement B), and statement C). Also, each of these implies statement A. • For uniformly continuous functions, , it is sufficient to check statement A), which on occasions may be easier to verify. Consequently, B) and C) will also hold. 8That is, as defined for functions which depend only on time: that the function
A:
!
n is PE if there exist T > 0 and > 0 such that 2 we have that z A(s) A(s)zds .
,m for all unitary vectors z
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Remark 4: A bibliographical remark seems adequate at this point. • In [54] the authors pointed out that, for a nonlinear system where the time variations are due to an ex, the persistency of excitation as originally ternal input proposed for functions depending only on time, is neither sufficient nor necessary to assure uniform asymptotic stability. As a possible alternative, in [54, p. 157], the authors comment on the idea of defining a new persistency of ex, with respect to the citation condition for the vector . This has been done in differential equation [2], [15], and [37]. • In [12], a necessary condition, which is very close to -PE, for uniform (local) asymptotic stability is presented. It is important to mention that necessity for UAS is proved in that reference under more restrictive assumptions and in a less direct manner as we do in Section V-A. • Most recently and independently, in [15] were reported several definitions of persistency-of-excitation for nonlinear time-varying systems. In particular, the author proposed the notion of “output persistency of excitation ” where is an output (OPE) of the pair is as in (1). The interest of the results function and presented in [15] strives in different characterizations of a notion of uniform detectability previously introduced in [14] and which is shown to be necessary for UAS. It is also shown in [15] that detectability (in the sense defined in that reference) with respect to the output is equivalent to the property of -PE along trajectories, given in Definition 4. It can also be shown that Statement B in Lemma 2 is equivalent to “OPE of the pair with respect to the set .” See [15] for further details and definitions.
B. Properties of
-PE Functions
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Property 2 (Power of a -PE Function): If the scalar funcis -PE with respect to , with parameters , tion and then, for any the function is -PE with respect to , with parameters and , . It is also useful to remark that for functions linear in the state, -PE is equivalent to the usual PE property (when restricting to the set of nonnegative reals). where Property 3: Consider the function is locally integrable. Then, is -PE such that with respect to if and only if there exist and (43) The last property generalizes the well-known fact that for functions depending only on by which filtered (through strictly stable and proper transfer functions) regressors remain PE. This property is useful in control design. -PE Function): Consider the differProperty 4 (Filtered ential equation (44) where is -PE with respect to lowing. 1) is such that
(45) 2) There exist , such that with denoting the solution of (44), we have that (46) Then, defining tion V.
-PE funcWe present here some important properties of tions. Similar properties are well known for only-time dependent functions (cf. e.g., [32]) and some of which were proved which satisfy Definition to hold as well in [2], for pairs 4, i.e., along trajectories. The proofs are omitted due to space constraints and are provided in [38]. be -PE with respect to in the sense Let , , 2 be continof Definition 3 and let uous non decreasing functions. Assume that for all and almost all (41) (42) Property 1 (Multiplication of -PE Functions): If the funcis -PE then, necessarily each tion is -PE. The opposite is not necessarily true. The proof of the first part of Property 1 follows directly using the Cauchy–Schwartz inequality. A useful exception to the second part is the following.
and assume the fol-
is
and , the func-PE with respect to .
-PE IS NECESSARY AND SUFFICIENT FOR UGAS
Making use of the tools previously introduced we show, for a fairly general class of nonlinear time-varying systems, that -PE is necessary and sufficient for uniform the property of attractivity of the origin. Sufficiency is established by a corollary of Theorem 1. A. Necessity The following result, contained in [12, Th. 6.2], gives conditions under which -PE of the right-hand side of a differential equation is necessary for uniform asymptotic stability. The technical conditions that we use permit a relatively straightforward proof, based on Gronwall’s lemma, without recourse to the notion of limiting equations, as in [12, Th. 6.2]. : Assume that in (1) Theorem 3 is Lipschitz in uniformly in . If the origin of (1) is UGAS, then is -PE with respect to . Proof: The Lipschitz assumption on implies that such that there exists for each such that (47)
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. From the UGAS assumption on (1) it follows that s.t. (48)
Notice that without loss of generality, we may assume that for all . For the purposes of establishing a contradiction, assume that the statement of the theorem does not hold. More precisely, referring to the statement of Lemma 1, assume there exists such that for each and , there exist such that (49) . Let the Lipschitz asPick such that for sumption generate the constant and let . Let these values generate , and . By definition of consider the solution of (1) starting at solution (50)
In what follows, we use the following notation: , , , and . correspondingly, Theorem 4: Let Assumptions 1–3 hold. Suppose also the following. Assumption 6: We have that A) implies . B) Assumption 7: The function , is independent of , locally uniformly in , -PE w.r.t. and zero at the Lipschitz in origin. , we have Assumption 8: For all where is continuous and . Then, the origin of (1) is UGAS. Remark 5: In Assumption 6, there is no requirement that the size of matches the size of . Proof of Theorem 4: This result follows from Theorem 1 by using the additional function (54)
which also satisfies We claim that this function satisfies for almost all , (51) Now, setting
(55)
, we have from (49) that where (52)
so using the Gronwall–Bellman inequality, we obtain that
and (53)
represents a Lipschitz constant for . Then, the result follows defining
on the set
On the other hand, , hence We now prove that (55) holds. Let the -PE property of and Lemma 3 generate the functions and . Then, for any , we have that which contradicts (53). B. Sufficiency This result derives from the sufficient conditions for UGAS established in Theorem 1. Accordingly, the following sufficient conditions are expressed in terms of a finite number of auxiliary functions (cf. Assumption 2) having a certain nested property (cf. Assumption 3) and the property that when the bounds on the derivatives of these auxiliary functions are all zero, this implies that part of the state and a certain function are zero (cf. -PE with Assumption 6) and, moreover, that the function is respect to the rest of the state (cf. Assumption 7). The last technical assumption (cf. Assumption 8) bounds the derivative of the rest of the state in terms of the part of the state and the function that are zero in Assumption 6.
(56) Second, the partial derivatives of
The Lipschitz assumption on such that exists
are given by
implies that for each
there
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Hence, we obtain that
and for almost all
It follows that for almost all
So, (55) follows using the bound from (56) in the previous inin (56) is not defined equality. It is worth pointing out that [see statement C) in Lemma 3]. This motivates the at using the max of the two terms. definition of The following corollary covers [12, Th. 6.3]. Corollary 3: If the origin of (1) is UGS and the following assumptions hold then, the origin of (1) is also UGAS. Assumption 9: For each , there exist ; • a number ; • a locally Lipschitz continuous function ; • a continuous function such that, for almost all (57) (58) Assumption 10: We have that
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the property of robust stability with respect to disturbances uniformly bounded in time, in the same spirit as for instance that of [62]. In this section, we provide a new direct proof of UGAS of the origin and furthermore, we establish necessary conditions for UGAS. Moreover, the results presented here can be generalized to a framework of interconnected driftless systems (which includes nonholonomic integrators) through time-varying nonlinear functions of the state. These results are not presented here for space constraints. Invited readers are referred to [39] and [40]. The approach to stability analysis presented below encompasses, in particular, the case of periodic time-varying feedbacks as considered for instance in [48]. With regard to this reference it is also interesting to observe that the author used Krasovskii–La Salle invariance principle to obtain a direct proof of global asymptotic stability. Here, we use our generalization of Matrosov’s theorem which may be considered as the extension of the invariance principle, for general nonautonomous systems. For clarity of exposition, we address separately the cases of three and more states. A. Case of Three States Consider the problem of stabilizing the nonholonomic chained system as described in Example 2. Then, we have the following result. Proposition 1: Consider the system (7) in closed loop with (8). Let the following hold. Assumption 13: The map be such that , all its first and second partial derivatives are where is a nondecreasing uniformly bounded by function9 and, defining
Assumption 11: We have that
Assumption 12: The function Lipschitz in uniformly in and
(59) is locally -PE with respect to .
VI. NONHOLONOMIC INTEGRATOR: A CASE STUDY Motivated by Brockett’s celebrated paper [55], the problem of stabilization of nonholonomic systems of any dimension has been studied from numerous viewpoints. We suggest that interested readers see [56] for a tutorial with a very complete literature review up to 1995. Among recent works we cite [57], [58], and the seminal paper [59] which presents as a byproduct of the main results on controllability, universal controllers (i.e., for set-point and tracking) for practical stabilization of nonholonomic systems. In this regard, see also [60] where universal controllers achieving asymptotic stabilization for the (particular) case of underactuated (nonholonomic) ships are presented. The controllers used in this section are not original; we have sacrificed originality for clarity of exposition by choosing a well-understood benchmark to illustrate the use of the gen-PE eralized Matrosov’s Theorem 1 and the utility of the property in control design. The controllers that we study were originally proposed in [47] where global asymptotic stability was shown and restudied in [61] where UGAS was established. We stress that as a byproduct of UGAS, one recovers
assume that is -PE with respect to . Then, the origin of the closed-loop system is UGAS. Moreover, the origin with is -PE with respect is UGAS only if to . Remark 6: The necessary condition above is also sufficient. -PE condition imposed on As a matter of fact, the implies, via the “filtering property” (cf. Property 4), the -PE with with respect to . of B. Proof of Proposition 1 Necessity follows directly observing that UGAS implies, by Theorem 3, that is -PE, which implies in turn the statement of the proposition. The proof of sufficiency relies on Theorem 1 based on the intuition discussed in Example 2. First, UGS follows from the derivative of which yields (10). Integrating (10) from to we obtain that for all . Technically, this inequality is valid only on the interval of existence of the solutions. However, integrating on this window and using is bounded on the maximal interval of the fact that definition we proceed to integrate the -(9a) to obtain that 9For simplicity, we use (1) for a generic bound on any function which is uniformly bounded in t.
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for all and where is a generic bound on which exists due to Assumption 13. Therefore, the solutions exist for all and actually, the origin is UGS. Notice also that here, since . To prove attractivity we consider other differentiable functions which are bounded and have bounded derivatives on balls of the state–space. Our starting point in the pursuit of these additional functions to combine with is the observation that any terms in the derivative of subsequent auxiliary functions . that vanish with , can be ignored since we know that So, for example, we can take
yields (63) To obtain this inequality, we have used the smoothness of all functions, Assumption 13 (which in particular implies that is locally Lipschitz, uniformly in ), and the compactness . of Our fourth function is introduced to be used in combination -PE property of in order to infer that the with the may converge to is zero. This only constant value that function is
(60) and, defining
(64)
, we obtain that which satisfies for any
From smoothness of and Assumption 13 we obtain that its total derivative is bounded for bounded , uniformly in . In as a generic bound on the sequel, we will use the number continuous functions over compact sets. With this under consideration, we have that
(65) and we claim that from the previous equation and the -PE , it follows that there exists a continuous, assumption on such that and nondecreasing function (66)
For the sequel, we see that we can ignore terms in derivatives . As of auxiliary functions, that vanish with a matter of fact, from (7) and (8), we now see that we can igand . What is more, nore also terms that vanish with , if we were to consider the dynamics of the closed-loop system and is constant the -equation would define when the dynamics of a linear system with a time-varying input parameterized by a constant , i.e,
This follows by appealing to Property 4 of -PE functions and is defined by the differential equation observing that
Indeed, since is -PE with respect to so is . is also In other words, the “steady-state” value of when -PE. Then, appealing to Lemma 3 and using the inequality , the claim follows . with We proceed now to evaluate the time derivative of along the trajectories of the closed-loop system. To that end, we write 11
Differentiating on both sides and owing to the fact that , we obtain that satisfies the differential equation where is defined in (59), and whose solution is (61)
(67) is -PE. Based on these obAs we show farther down, servations, we introduce the next function with the aim at conand the cluding something about the difference between —in other words, between and steady-state solution its steady-state solution. Hence, we define our third auxiliary , where function as (62) along the trajec-
and observe that the time derivative of tories of10
(1)
(69) and observe that due to Assumption 13 all the partial derivatives in (67) and (68) which, are functions of , are uniformly . Since bounded in by a generic bound that we denote by also satisfies this boundedness property with we finally obtain, using (62), that for all and all
( ) const
10We emphasize that the choice of the function V 1; 1 is inspired by the g, however, we behavior of the system on the manifold fx ; x consider its total derivative along the trajectories of the closed-loop system, on B . In other words, we do not analyze the system’s dynamics only on the defined manifold.
0
(68)
11The use of the at zero.
min
f1g
function stems from the fact that
( ) is not defined 1
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and using (66) and the fact that for all and we finally obtain that
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C. Case of any Number of States We close Section VI with the extension of the previous result states, i.e., we now consider to the more general case of the set-point stabilization of the system
Thus, helps us to see that for the subsequent functions we can also ignore all the terms vanishing with . It is only left to find a function whose derivative is bounded by a negative term of and possibly positive , and . For this, we introduce terms involving whose total time derivative satisfies, for all and all
(75a) (75b) .. .
(75c) (75d) (75e)
We use the following smooth control laws which are the counterparts for states of the controller (8): Summarizing, we have that the functions (76a) (70) —see (62) and (61)
(71) (72) (73) (74)
satisfy where
for almost all
(76b) for all and . That is, the occurrence in alternates and the last term of is if is odd or, if is even. Interestingly, the equations of the closed-loop system has the following last form which reminds us of the controllability canonical form of linear systems. As shown first in [47] and also discussed in [61], one can write the equivalent closed-loop dynamics in the “skew symmetric” form (77a) (77b) where of
where
and , . So the result follows invoking the Nested–Matrosov’s Theorem 1. The verification of the assumptions is straightforward at this point. Remark 7: From a classical “Lyapunov perspective,” i.e., with a negwith aim at constructing a Lyapunov function is almost obvious ative–definite derivative, the choice of from the structure of the last two closed-loop equations (cf. we comments in Example 2). In particular, since and . The would be looking only for negative terms of comes almost naturally as a cross term that yields choice of in . Notice that this “nonpositive” terms of would be an obvious choice if were constant (i.e., if the subsystem were linear and strictly positive real). We also remark that, to some extent, this is the case of Example 1: a cross —with and —qualifies12 term of the type as auxiliary function. Since is not constant nor positive for all and , we look for a function that guarantees that converges to its steady-state value while another function is used to exploit -PE property via the characterization given by Lemma 3 the to obtain a negative term of . The choice of the last function is rather obvious. 12Notice
D
that since is full rank, in the case of one degree of freedom we would have that necessarily ( ) 0 for all which is tantamount to assuming constant in the current example.
u
dq >
q
.. .
..
.
..
.
..
.
..
.
..
.
..
.
.
..
.
..
.
..
.. . (78)
are specific linear combinations of ; see [61]. and Structurally, this system is a direct generalization of the closedloop equations for the case of three states. Notice in particular that one may easily conclude UGS of the origin. However, its analysis along the lines of [47] and [61] is much more involved than that of the previous case. In contrast to this, a direct proof of UGAS may be established via Theorem 1. To make a clear statement on the stability of (77), let us consider the system (1) with
(79) and, similar to (59), let us define
.. . Then, we have the following.
(80)
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Theorem 5: If the function defined above is -PE and the functions and are lo, the origin of (1), cally Lipschitz uniformly in and (79) is UGAS. If the origin is UGAS, then , with , is -PE with respect to . A direct consequence is the following. Proposition 2: Consider (75) in closed loop with (76). Let the following hold. be such that Assumption 14: The map , all its first and second partial derivatives be where is a nondecreasing uniformly bounded by function and let the function is -PE. Then, the closed-loop system (77) is UGAS. The proof of Theorem 5 is not presented here since it follows along the lines of the proof for the case of three states. However, for completeness we provide the guidelines for sufficiency. This follows by applying directly Theorem 1 with the following functions: (81) where
with for all , . In particular, this function allows to show UGS. The rest of the auxiliary functions are (82) (83) with
been proved to be necessary and sufficient for UGAS of general nonlinear time-varying systems. Sufficiency is established via our generalized Matrosov’s theorem. We have illustrated how to use our main results in the analysis of the popular benchmark of chained-form systems. APPENDIX SKETCH OF PROOF FOR THEOREM 2 The proof follows the same lines as the proof of Theorem 1. Up to (29), everything follows verbatim. From this point on, from Assumption 5. 2 we have that inequality (30) holds with and therefore, , and in (32) also depend on . Consequently, proceeding further as in the proof of Theorem 1, and there exists we obtain that for each such that (86) We use this fact together with ULS and the Lipschitz assumption to show uniform global boundedness. This together with ULS establishes UGS and so we recover all of the conditions of Theorem 1. We show uniform global boundedness by contradiction. Suppose that there exist a sequence of positive real numbers monotonically increasing to infinity, , and sequences with , and with such that (87) Let be an accumulation point for the sequence . Let generate according to Assumption 5. 2. In view of the ULS be such that assumption, let (88)
Defining for
Let and generate according to (86). Using Assumption 5.3, which implies continuity of solutions that is unifor all , we have that, for all form on the intervals sufficiently close to and all
,
(89) (84) so that using the latter, (86) and (88), we get that and, finally (90) (85) This contradicts (87) for sufficiently large. Note the similarity with the functions for the case of three states. In particular, note that here we need functions as defined functions as in (84) [see also (60) and (64)] in (82) and corresponds to the relative degree of (77b). where VII. CONCLUSION In this paper, we have presented a new tool for establishing uniform attractivity of the origin when the origin is uniformly stable. The tool involves the use of an arbitrary finite number of auxiliary functions whose derivatives are simultaneously zero only at the origin. This result generalizes Matrosov’s theorem on uniform asymptotic stability of the origin for nonlinear timevarying systems. We have also presented a new mathematical definition of the concept of -persistency of excitation which has
REFERENCES [1] H. Khalil, Nonlinear Systems, 2nd ed. New York: Macmillan, 1996. [2] E. Panteley, A. Loría, and A. Teel, “Relaxed persistency of excitation for uniform asymptotic stability,” IEEE Trans. Autom. Control, vol. 46, no. 12, pp. 1874–1886, Dec. 2001. [3] J. Hale, “Ordinary differential equations,” in Interscience. New York: Wiley, 1969. [4] J. P. La Salle and S. Lefschetz, Stability by Lyapunov’s Direct Method. New York: Academic, 1961. [5] N. N. Krasovskii, Problems of the Theory of Stability of Motion. Stanford, CA: Stanford Univ. Press, 1963. [6] V. Jurdjevic´ and J. P. Quinn, “Controllability and stability,” J. Diff. Equat., vol. 28, pp. 381–389, 1978. [7] I. Barb˘alat, “Systèmes d’équations différentielles d’oscillations nonlinéaires,” Revue de Mathématiques Pures et Appliquées, vol. 4, no. 2, pp. 267–270, 1959. [8] W. Hahn, Stability of Motion. New York: Springer-Verlag, 1967.
LORÍA et al.: NESTED MATROSOV THEOREM AND PERSISTENCY OF EXCITATION
[9] J. P. La Salle, An Invariance Principle in the Theory of Stability, J. K. Hale and J. P. La Salle, Eds. New York: Academic, 1967, ch. Differential Equations and Dynamical Systems. [10] N. Rouche, P. Habets, and M. Laloy, Stability Theory by Liapunov’s Direct Method. New York: Springer-Verlag, 1977, vol. 22. [11] Z. Artstein, Limiting Equations and Stability of Nonautonomous Ordinary Differential Equations, J. P. La Salle, Ed. Philadelphia, PA: SIAM, 1976, ch. Appendix A in Stability of Dynamical Systems. , “Uniform asymptotic stability via the limiting equations,” J. Diff. [12] Equat., vol. 27, pp. 172–189, 1978. [13] T. C. Lee, D.-C. Liaw, and B. S. Chen, “A general invariance principle for nonlinear time-varying systems,” IEEE Trans. Autom. Control, vol. 46, no. 12, pp. 1989–1993, Dec. 2001. [14] T. C. Lee and B. S. Chen, “A general stability criterion for time-varying systems using a modified detectability condition,” IEEE Trans. Autom. Control, vol. 47, no. 5, pp. 797–802, May 2002. [15] T. C. Lee, “On the equivalence relations of detectability and PE conditions with applications to stability analysis of time-varying systemss,” in Proc. Amer. Control Conf., Jun. 2003, pp. 1873–1878. , “Limit systems and attractivity for time-varying systems with ap[16] plications to nonholonomic systems,” presented at the 42nd. IEEE Conf. Decision Control, Maui, HI, 2003. [17] V. M. Matrosov, “On the stability of motion,” J. Appl. Math. Mech., vol. 26, pp. 1337–1353, 1962. [18] N. Rouche and J. Mawhin, Ordinary Differential Equations II: Stability and Periodical Solutions. London, U.K.: Pitman, 1980. [19] B. Paden and R. Panja, “Globally asymptotically stable PD+ controller for robot manipulators,” Int. J. Control, vol. 47, pp. 1697–1712, 1988. [20] R. Marino and P. Tomei, “Global adaptive output feedback control of nonlinear systems. Part I: Linear parameterization,” IEEE Trans. Autom. Control, vol. 38, no. 1, pp. 17–32, Jan. 1993. [21] S. Nicosia and P. Tomei, “On the control of flexible joint robots by dynamic output feedback,” in Proc. 4th. Symp. Robot Control, Capri, Italy, 1994, pp. 717–722. [22] N. Rouche, “Attractivity of certain sets proved by using several Liapunov functions,” in Symposia Matematica. New York: Academic, 1971, vol. 6, pp. 331–343. , “On the stability of motion,” Int. J. Nonlinear Mech., vol. 3, pp. [23] 295–306, 1968. [24] J. L. Corne and N. Rouche, “Attractivity of closed sets proved by using a family of Lyapunov functions,” J. Diff. Equat., vol. 13, pp. 231–246, 1973. [25] A. Teel, E. Panteley, and A. Loría, “Integral characterizations of uniform asymptotic and exponential stability with applications,” Math. Control Signals Syst., vol. 15, pp. 177–201, 2002. [26] F. Mazenc, “Strict Lyapunov functions for time-varying systems,” Automatica, vol. 39, pp. 349–353, 2003. [27] R. E. Kalman, “Contributions to the theory of optimal control,” Bol. Soc. Mat. Mexicana, vol. 5, pp. 102–119, 1960. [28] K. J. Åström and T. Bohlin, “Numerical identification of linear dynamic systems from normal operating records,” in Proc. 2nd IFAC Symp. Theory of Self-Adaptive Control Systems, P. H. Hammond, Ed., Teddington, U.K., 1965, pp. 96–111. [29] B. O. Anderson, “Exponential stability of linear equations arising in adaptive identification,” IEEE Trans. Autom. Control, vol. AC-22, no. 1, pp. 83–88, Jan. 1977. [30] A. P. Morgan and K. S. Narendra, “On the uniform asymptotic stability of certain linear nonautonomous differential equations,” SIAM J. Control Optim., vol. 15, no. 1, pp. 5–24, 1977. [31] S. Sastry and M. Bodson, Adaptive Control: Stability, Convergence and Robustness. Upper Saddle River, NJ: Prentice-Hall, 1989. [32] K. S. Narendra and A. M. Annaswamy, Stable Adaptive Systems. Upper Saddle River, NJ: Prentice-Hall, 1989. [33] D. Aeyels, R. Sepulchre, and J. Peuteman, “Asymptotic stability for time-variant systems and observability: Uniform and nonuniform criteria,” Math. Control Signals Syst., vol. 11, pp. 1–27, 1998. [34] Z. Artstein, “Stability, observability and invariance,” J. Diff. Equat., vol. 44, pp. 224–248, 1982. [35] B. D. O. Anderson, R. Bitmead, C. Johnson Jr, P. Kokotovic´ , R. Kosut, I. Mareels, L. Praly, and B. Riedle, Stability of Adaptive Systems. Cambridge, MA: MIT Press, 1986. [36] A. P. Loh, A. M. Annaswamy, and F. P. Skantze, “Adaptation in the prescence of a general nonlinear parameterization: An error model approach,” IEEE Trans. Autom. Control, vol. 44, no. 9, pp. 1634–1653, Sep. 1999.
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[37] A. Loría, E. Panteley, and A. Teel, “A new persistency-of-excitation condition for UGAS of NLTV systems: Application to stabilization of nonholonomic systems,” in Proc. 5th. Eur. Control Conf., 1999, Paper no. 500. [38] A. Loría, E. Panteley, D. Popovic´ , and A. R. Teel. (2003) Persistency of excitation for uniform convergence in nonlinear control systems. Univ. California, Davis, CA. [Online]. Available: http://front.math.ucdavis. edu/math.OC/0 301 335 [39] , (2003) Matrosov’s theorem using a family of auxiliary functions: An analysis tool to aid time-varying nonlinear control design. Univ. California, Davis, CA. [Online]. Available: http://front.math.ucdavis. edu/math.OC/0 304 255 [40] A. Teel, A. Loría, E. Panteley, and D. Popovic´ , “Control of port-interconnected driftless time-varying systems,” presented at the 7th. Eur. Control Conf., Cambridge, U.K., 2003, Paper 554. [41] F. H. Clarke, Y. S. Ledyaev, R. J. Stern, and P. R. Wolenski, Nonsmooth Analysis and Control Theory. New York: Springer-Verlag, 1998. [42] F. H. Clarke, Optimization and Nonsmooth Analysis. Philadelphia, PA: SIAM, 1990. [43] A. R. Teel and L. Praly, “On assigning the derivative of a disturbance attenuation control Lyapunov function,” Math. Control Signals Syst., vol. 13, pp. 95–124, 2000. [44] A. P. Morgan and K. S. Narendra, “On the stability of nonautonomous differential equations x_ = [A + B (t)]x with skew-symmetric matrix B (t),” SIAM J. Control Optim., vol. 15, no. 1, pp. 163–176, 1977. [45] M. Spong and M. Vidyasagar, Robot Dynamics and Control. New York: Wiley, 1989. [46] R. Ortega and M. Spong, “Adaptive motion control of rigid robots: A tutorial,” Automatica, vol. 25–6, pp. 877–888, 1989. [47] C. Samson, “Control of chained system: Application to path following and time-varying point stabilization of mobile robots,” IEEE Trans. Autom. Control, vol. 40, no. 1, pp. 64–77, Jan. 1995. [48] Z. P. Jiang, “Iterative design of time-varying stabilizers for multi-input systems in chained form,” Syst. Control Lett., vol. 28, pp. 255–262, 1996. [49] A. Teel, E. Panteley, and A. Loría, “Integral characterizations of set UGAS/UGES with applications to Matrosov’s theorem,” presented at the IFAC NOLCOS, St. Petersburg, Russia, July 2001, Paper 214. [50] P. Seibert and R. Suárez, “Global stabilization of nonlinear cascaded systems,” Syst. Control Lett., vol. 14, pp. 347–352, 1990. [51] E. D. Sontag, “Remarks on stabilization and input-to-state stability,” in Proc. 28th. IEEE Conf. Decision Control, Tampa, Fl, 1989, pp. 1376–1378. [52] E. Panteley and A. Loría, “Growth rate conditions for stability of cascaded time-varying systems,” Automatica, vol. 37, no. 3, pp. 453–460, 2001. [53] C. Byrnes, A. Isidori, and J. C. Willems, “Passivity, feedback equivalence, and the global stabilization of minimum phase nonlinear systems,” IEEE Trans. Autom. Control, vol. 36, no. 11, pp. 1228–1240, Nov. 1991. [54] K. Narendra and A. Annaswamy, “Persistent excitation in adaptive systems,” Int. J. Control, vol. 45, no. 1, pp. 127–160, 1987. [55] R. Brockett, “Asymptotic stability and feedback stabilization,” in Differential Geometric Control Theory, R. S. M. R. W. Brocket and H. J. Sussmann, Eds. Norwell, MA: Birkhäuser, 1983, pp. 181–191. [56] I. Kolmanovsky and H. McClamroch, “Developments in nonholonomic control problems,” Control Syst. Mag., pp. 20–36, Dec. 1995. [57] A. A. J. Lefeber, “Tracking control of nonlinear mechanical systems,” Ph.D. dissertation, Univ. Twente, Enschede, The Netherlands, 2000. [58] Z. P. Jiang and H. Nijmeijer, “A recursive technique for tracking of nonholonomic systems,” IEEE Trans. Autom. Control, vol. 44, no. 2, pp. 265–279, Feb. 1999. [59] P. Morin and C. Samson, “A characterization of the Lie algebra rank condition by transverse periodic functions,” SIAM J. Control Optim., vol. 40, no. 4, pp. 1227–1249, 2002. [60] K. D. Do, Z. P. Jiang, and J. Pan, “Universal controllers for stabilization and tracking of underactuated ships,” Syst. Control Lett., vol. 47, pp. 299–317, 2002. [61] A. Loría, E. Panteley, and K. Melhem, “UGAS of skew-symmetric timevarying systems: Application to stabilization of chained form systems,” Eur. J. Control, vol. 8, no. 1, pp. 33–43, 2002. [62] C. Prieur and A. Astolfi, “Robust stabilization of chained systems via hybrid control,” IEEE Trans. Autom. Control, vol. 48, no. 10, pp. 1768–1772, Oct. 2003. [63] F. Mazenc and D. Nesic, “Strong Lyapunov functions for systems satisfying the conditions of La Salle,” IEEE Trans. Autom. Control, vol. 49, no. 6, pp. 1026–1030, Jun. 2004.
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Antonio Loría was born in Mexico in 1969. He received the B.Sc. degree in electronic engineering from the ITESM, Monterrey, Mexico, in 1991, and the M.Sc. and Ph.D. degrees in control engineering from the UTC, France, in 1993 and 1996, respectively. From December 1996 to Dec. 1998, he was successively an Associate Researcher at the University of Twente, Enschede, The Netherlands; NTNU, Norway, and the CCEC of the University of California, Santa Barbara. He is currently “Charge de Recherche” at the French National Centre of Scientific Research (CNRS). He is an author/coauthor of more than 80 scientific articles and the books Passivity Based Control of Euler–Lagrange Systems ( New York: Springer Verlag, 1998) and Control of Robot Manipulators in Joint Space (New York: Springer Verlag, 2005). His research interests include modeling and control of Euler–Lagrange systems, stability analysis of nonlinear time-varying systems, biped locomotion, and output feedback stabilization. He is an Associate Editor of Systems and Control Letters. Detailed information and publications are available at http://www.lss.supelec.fr/~loria/.
Elena Panteley was born in Leningrad, U.S.S.R. She received the M.Sc. and Ph.D. degrees in applied mathematics from the State University of St. Petersburg, Russia. She holds a research position with the French National Centre of Scientific Research (CNRS), at Laboratoire de Signaux et Systèmes. From 1986 to 1998, she held a research position with the Institute for Problem of Mechanical Engineering of the Academy of Science of Russia, St. Petersburg. During 1998, she was an Associate Researcher at the Center for Control Engineering and Computation of the University of California, Santa Barbara. During 1999, she was with the INRIA Rhône Alpes, Monbonnot, France. She is a coauthor of more than 70 scientific articles and book chapters. Her research interests are stability of nonlinear time-varying systems, control of electromecanical systems, and nonlinear and robust control.
Dobrivoje Popovic´ received the B.S. degree in electrical engineering from the University of Belgrade, Belgrade, Serbia, in 1998, and the M.S. and Ph.D. degrees in electrical and computer engineering from the University of California, Santa Barbara, in 2000 and 2004, respectively. He is currently a Senior Engineer at the United Technologies Research Center, East Hartford, CT. His research interests include extremum seeking, nonlinear optimization, and multivariable control. He is a Member of the Society of Automotive Engineers.
Andrew R. Teel received his A.B. degree in engineering sciences from Dartmouth College, Hanover, NH, in 1987, and the M.S. and Ph.D. degrees in electrical engineering from the University of California, Berkeley, in 1989 and 1992, respectively. After receiving the Ph.D., he was a Postdoctoral Fellow at the Ecole des Mines de Paris, Fontainebleau, France. In September 1992, he joined the Faculty of the Electrical Engineering Department at the University of Minnesota, Minneapolis, where he was an Assistant Professor until September of 1997. In 1997, he joined the faculty of the Electrical and Computer Engineering Department at the University of California, Santa Barbara (UCSB), where he is currently a Professor. He is currently the Director of the Center for Control Engineering and Computation at UCSB. His research interests include nonlinear dynamical systems and control with application to aerospace and related systems. Dr. Teel has received the National Science Foundation Research Initiation and CAREER Awards, the 1998 IEEE Leon K. Kirchmayer Prize Paper Award, the 1998 George S. Axelby Outstanding Paper Award, and the first SIAM Control and Systems Theory Prize, in 1998. He was also the recipient of the 1999 Donald P. Eckman Award and the 2001 O. Hugo Schuck Best Paper Award, both given by the American Automatic Control Council.
