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A Unifying Framework for Global Regulation Via Nonlinear Output Feedback: From ISS to iISS Zhong-Ping Jiang, Senior Member, IEEE, Iven Mareels, Fellow, IEEE, David J. Hill, Fellow, IEEE, and Jie Huang, Senior Member, IEEE
Abstract—This paper presents a unifying framework for the problem of robust global regulation via output feedback for nonlinear systems with integral input-to-state stable inverse dynamics, subject to possibly unknown control direction. The contribution of the paper is two-fold. Firstly, we consider the problem of global regulation, instead of global asymptotic stabilization (GAS), for systems with generalized dynamic uncertainties. It is shown by an elementary example that GAS is not solvable using conventional smooth output feedback. Secondly, we reduce the stability requirements for the disturbance and demand relaxed assumptions for the system. Using our framework, most of the known classes of output feedback form systems are broadened in several directions: unmeasured states and unknown parameters can appear nonlinearly, restrictive matching and growth assumptions are removed, the dynamic uncertainty satisfies the weaker condition of Sontag’s integral input-to-state stability, and the sign of high-frequency gain may be unknown. A constructive strategy is proposed to design a dynamic output feedback control law, that drives the state to the origin while keeping all other closed-loop signals bounded. Index Terms—Input-to-state stability (ISS), integral ISS (iISS), nonlinear systems, output feedback, small-gain, universal adaptive control, unknown control direction.
I. INTRODUCTION A. Background
W
E investigate the problem of globally asymptotically stabilizing, via output feedback, single-input–single-output (SISO) nonlinear systems transformable into
.. . Manuscript received November 1, 2002; revised July 23, 2003. Recommended by Associate Editor R. Freeman. This work was supported in part by the Othmer Institute for Interdisciplinary Studies of Polytechnic University, the National Science Foundation under Grant ECS-0093176 and Grant INT-9987317, the Hong Kong Research Grant Council under Grant CUHK4168/03E, and the National Natural Science Foundation of China under Grant 60374038. Z.-P. Jiang is with the Department of Electrical and Computer Engineering, Polytechnic University, Brooklyn, NY 11201 USA (e-mail:
[email protected]). I. Mareels is with the Department of Electrical and Electronic Engineering, University of Melbourne, Vic. 3010, Australia. D. J. Hill is with the Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong, and also with the University of Sydney, Sydney, Australia. J. Huang is with the Department of Automation and Computer-Aided Engineering, Chinese University of Hong Kong, Shatin, Hong Kong. Digital Object Identifier 10.1109/TAC.2004.825663
(1) where is the control, is the output, and and are the states. It is further assumed are locally that the uncertain functions and Lipschitz and that the sign of the high-frequency gain is unknown. The states as well as the state of the -system, referred to as an “inverse system,” are not assumed to be measurable. Known as a special normal form, the class of systems of the form (1) is a subclass of the popular block strict-feedback systems, and includes as a particular case the class of output feedback form systems [26], [30] in which the control direchere—is known and the zero dytion—characterized by namics is globally exponentially stable. If the complete state information were available to the designer, the problem of global asymptotic stabilization would be easily solvable using various recursive designs proposed in the 1990’s literature of nonlinear control. See, for instance, [15], [18], [23], [24], [26], and [30], for many references for the case of known control direction, and [10], [11], [28], [29], [39], [52], and [53] for the case of unknown control direction. Achieving global stabilization using dynamic output feedback for the above class of systems is however an open challenge. The problem is not trivial because there is no generally accepted, low complexity, design methodology for constructing an observer for such a class of systems. Moreover, there is no separation principle, so observer design and feedback control design are inter-dependent. Finally, an internal state could exhibit finite escape which is not necessarily observable from the output. It is therefore of interest to delineate classes of systems for which dynamic output feedback (global) stabilization can be achieved. Indeed, much of the nonlinear control literature dealing with the problem of output feedback has taken this approach. Typical restrictions imposed on the system models are a combination of global Lipschitz continuity, output injection dynamics and linear disturbance dynamics. See [8], [26], [30], and [37], for earlier references and the survey paper [24] for an excellent account of recent work in observerbased global stabilization. In our earlier paper [37] (see [16] and [20] for the adaptive case), a novel solution to the nonadaptive, output feedback, global stabilization of system (1) was obtained without imposing that the unmeasured state occurs linearly in functions and . The essential assumption is that we impose the property of “input-to-state stability” (Sontag’s ISS [42]) from to for the subsystem . The key idea is to
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introduce an observer-like dynamic system of order or for the -part of the system, while the part is ignored as a disturbance. Then, one applies nonlinear small-gain theorems [21] for control design and stability analysis. See [48] for a review of recent results in nonlinear and adaptive control. A natural question arises, i.e., is ISS-stability for the disturbance dynamics ( to ) an essential ingredient in achieving global stability for systems of the form (1), or just a convenient vehicle to achieve the desired outcome? Some form of stability is obviously essential, as in the control design problem, is treated as an unmeasurable disturbance, affecting the -dynamics in an uncertain manner. In this paper, we show that the condition can be weakened, and to this end we exploit the notion of integral ISS (for short, iISS) proposed in [1] and [44]. In Section II, for completeness sake, we discuss ISS and iISS properties. In recent work [3], it was established that the cascade connection of an iISS system followed by a GAS system, may not possess a globally asymptotically stable trivial solution, despite the fact that each subsystem has this property. This does not augur well for exploiting the iISS property in a feedback situation, as is intended. An elementary example, which we adapt from [41], may shed some light on the difficulties that must be overcome. Example 1: Consider a nonlinear system of form (1), with known control direction
origin while maintaining the boundedness property for all other closed-loop signals. The control problem considered in this paper is precisely formulated as follows. Problem of global regulation via output feedback. Find a dynamic output feedback law of the form (3) in such a way that, for all initial conditions , the solutions of the closed-loop system (1), (3) are bounded. In and converge to zero as . particular, It is worth noting that we do not demand to achieve global asymptotic stability for the closed-loop system (1)–(3) at the origin. This is in sharp contrast to previous work carried out by a number of researchers [2], [24], [26], [30], [34], [37] in which the global stabilization problem includes the requirement of sending the state of (3) to the origin as well. This flexibility is exploited in this paper to overcome the impossibility of globally stabilizing systems (1) via conventional output feedback, like the one in Example 1. Our proposed control strategy is in the spirit of previous work in “universal” adaptive control for both linear and nonlinear systems; see, e.g, [7], [10], [28], [32], [39], [52], [53], [55], and the references there. Along the way, using the developed framework, a (nonsmooth) nonlinear output feedback controller is obtained to drive the state of system (2) to the origin. C. Organization of the Paper
(2) It can be verified that the subsystem to is iISS (see Section II for details) and that the subsystem to is ISS. Using the linear for , the subsystem has feedback law may not be bounded. a GAS trivial solution. Nevertheless, Indeed, given any particular feedback gain , there exist initial , and such that diverges. A formal conditions proof of this observation can be obtained using the Centre Manifold theorem, we refer the reader to [41, Ex. 4.16] for details. Essentially, the main reason is that the closed-loop signal converges so slowly that the integration of eventually destabilizes the -system which is only iISS, not ISS. This makes us that to conjecture that there is no smooth feedback law can achieve GAS. Nevertheless, the structure and the weak stability properties of the subsystems will allow us to construct a (nonsmooth) nonlinear controller that achieves regulation of whilst maintaining boundedness of —see Section V-B. The crucial trick in our solution is the exploitation of the roles of subsystems. It is worth different gain functions for the noting that a similar idea has already proven useful in global stabilization of nonlinear interconnected systems [21]. B. Problem Statement In this paper, we develop a new approach to the problem of output feedback global regulation (instead of stabilization). Taking a perspective, a new dynamic output feedback law will be proposed to globally drive the system states to the
The rest of the paper is organized as follows. To make our paper self-contained, Section II recalls Sontag’s notions of input-to-state stability (ISS) and integral input-to-state stability (iISS). Two preliminary technical results are also proposed there. Section III studies some simple situations for the main purpose of paving the way for a complete, though involved, answer to the above-stated control problem. Section IV first introduces an input-driven filter, and then presents a new output feedback design strategy to accommodate unmeasured states, unknown parameters and indefinite control direction. In Section V, we illustrate our methodology by means of examples, and show in particular how our unifying framework recovers previously published results. Section VI concludes the paper with some brief remarks and pointers for future work. II. MATHEMATICAL PRELIMINARIES A. Notions of ISS and iISS , and positive definite functions are Classes of , extensively used to introduce concepts of stability in the recent work of Sontag and his co-workers [1], [42], [44], [45], [47]. is For completeness, we recall that a function and if . A function positive definite if is of class if and is continuous if additionally it is and (strictly) increasing. It is of class unbounded. A function is of class if decreases to zero as . A function is if it is of class with respect to its first argument of class and is of class with respect to its second argument.
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Now, we recall the definitions of ISS [42], [45] and iISS [44], which play a key role in recent literature of constructive nonlinear control (cf. [24]). Consider a control system of the general form (4) where is locally Lipschitz. Definition 1: System (4) is input-to-state stable with respect -function and -function such that, for to if there exist each initial condition and each measurable, locally , the solution essentially bounded function exists for every and satisfies (5) is the truncated function of at , is the usual where is the -norm. Such a function in Euclidean norm, and (5) is often referred to as an ISS-gain for system (4). Definition 2: System (4) is integral input-to-state stable with satisfy respect to if, instead of (5), the solutions (6) where , and . Here, is referred to as an iISS-gain. As shown in [42], ISS implies that (4) is boundedand that its zero input–bounded-state stable when ) is GAS. However, the converse is not true solution (with in general. For the iISS property, Sontag et al. [1], [44] have shown that iISS is strictly weaker than ISS but stronger than forward completeness plus 0-input GAS. In view of [1], [47], both ISS and iISS properties can be equivalently characterized using Lyapunov functions. Proposition 1: The following statements hold for system (4): • System (4) is ISS if and only if there exists a positive definite and proper function , called an ISS Lyapunov function, such that
Let us use the systems involved in Example 1 to illustrate is iISS these notions. For example, , with an iISS Lyapunov function of the type while is ISS with an ISS Lyapunov function . Clearly, the -system is not ISS because is and (and more generally for unbounded when for all ). As another exall input functions such that ample, the ISS property also appears in other nonlinear systems such as the popular class of output feedback form systems [26], [30]. As is well-known in the past literature (see, e.g., [37]), the inverse dynamics of a minimum-phase nonlinear system in the output-feedback form is naturally ISS. See Section V for more details. B. Preliminary Results In this subsection, we gather two technical results that are used later in the paper. For notational simplicity, we use to mean that for some constant and all in a small neighborhood of the origin. Proposition 2: Consider an iISS system (4) with an iISSsatisfying (8). Take any smooth funcLyapunov function tion with the following property (10) Moreover in case tion holds
is bounded, the following additional condi-
(11) Then there always exist a positive-definite function function such that class-
and a
(12)
(7) where , . • System (4) is iISS if and only if there exists a positive definite and proper function , called an iISS Lyapunov function, such that (8) where is merely a positive definite continuous function -function. and is a Remark 1: It is of interest to note that the function in (8) can be seen as an iISS-gain for system (4); see [1]. However, the function in (7) is generally not the same as the ISS-gain function in (5). Indeed, as shown in [42] and [43], from the meeting (5) can be differential inequality (7), an ISS-gain , with and obtained as satisfying (9)
(13) Moreover, if the iISS-gain in (8) is such that , so is . Remark 2: It is worth noting that Proposition 2 is an extension of [28, Lemma 6.4.12] to nonlinear plants. As shown in [28], an integral inequality like (13) plays a crucial role in universal adaptive control for multivariable linear systems. Proof: Consider a function (14) where is a positive continuous function. By hypotheses, we have (15)
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Using the technique of changing supply rates [46] as in the proof of [19, Th. 1], it follows that we can find a suitable function in such a way that (16) where is a positive continuous function. Indeed, can be taken as a constant when both (10) and (11) hold. Notice that in (8) function when is unbounded. can be replaced by a classIntegrating both sides of (16) yields immediately (12) which, in turn, implies (13) by completing the squares. Therefore, the proof of Proposition 2 is completed. Borrowing the terminology from [27] and [39], the second preliminary result of this section is concerned with scaling-invariant Nussbaum functions. As shown in early literature of high-gain parameter adaptive control [28], [32], [52], Nussbaum-type functions are of paramount importance in the design of universal controllers for multivariable linear systems with reduced knowledge of high-frequency gain. Recall that share the following Nussbaum-type functions “Nussbaum properties:”
III. NONLINEAR SYSTEMS—SIMPLE CASES The main purpose of this section is to prepare the reader for the involved design scheme for the class of interconnected systems in the general form (1). Indeed, as it is shown in Section V, our algorithm can be adapted/modified for handling larger classes of systems than (1), with or without the a priori knowledge of control direction. Consider a nonlinear system with unknown control direction, again in a simplified form of (1):
(20) System (20) is of relative degree one when is considered as is not assumed to be constant (i.e., may the output. Here, be time-varying or state-dependent). It is only assumed that stays in a bounded interval of the type , with or . In order to achieve the control objective stated in Section I-B, we need the sufficient conditions: C1) The -subsystem of (20) is iISS with respect to in the sense that there exists an iISS-Lyapunov function so that (21)
(17) The following proposition can be directly proved using techniques from [27], [39], [54] and is an immediate extension of the fundamental lemmas used in these papers. Specifically, it covers [39, Prop., p. 755] in which is piecewise constant, and . We give a precise statement [54, pp. 931–932] in which of such a generalized result for the sake of robust redesign and reproduce an outlined proof for the sake of completeness. and for any Proposition 3: For any constant , with time-varying bounded function either or , of the following scaled Nussbaum function if otherwise
(18)
satisfies the Nussbaum properties (17). Proof: It is sufficient to examine the case where ; otherwise, consider . To prove that satisfies the Nussbaum properties (17), one needs to , with as for find two subsequences , 2, such that
with being a positive definite continuous function and a classfunction. C2) There exist two unknown positive constants and , and two known positive semidefinite, smooth functions and such that (22) Notice that C1) is equivalent to A1), while C2) is more general than A3) in the sense that significant parametric uncertainty may be allowed in the interconnection term . C2) was introduced in previous studies [16], [20]. The interested reader should consult [48] for a fairly detailed account of recent results in nonlinear adaptive control. the Nussbaum function As before, denote . Let us consider the dynamic output-feedback law of the following form (23) where , are design constants and is a positive design function. Proposition 4: Assume that the conditions C1) and C2) hold with the following properties (24) and, in case
is bounded,
(19) (25) Using the same techniques as in [39, Proof of Prop., pp. and 755–756], it is directly shown that will do the job.
in C1), then the solutions of (20)–(23) are If well-defined and bounded over for appropriately chosen
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positive smooth function . Furthermore, the convergence property holds (26) Remark 3: It is of interest to note that all conditions of Proposition 4 are trivially satisfied by the class of minimum-phase linear systems with relative degree one studied in, e.g., [28], [32], and [52]. As a result, Proposition 4 is a nonlinear version of the main results presented in [28], [32], and [52]. In addition, Proposition 4 lays a solid foundation for an immediate extension of [54, Th. 1] to the universal partial-state feedback control of nonlinear systems with stable zero-dynamics, instead of full-state feedback control without zero-dynamics [54]. Proof: Differentiating the function along the solutions of (20)–(23) yields
is bounded on . This property together with C1) implies ; see [44, Prop. 6]. Therefore, that remains bounded on and the first statement of Proposition 4 is proved. In order to establish the second statement on the convergence property (26), notice that the boundedness of together with is inte(23) yields that the smooth positive function . The fact that is uniformly continuous grable over follows from the boundedness of and . Using Barbalat’s and thus converge to zero lemma [23], it follows that . This, in turn, implies the convergence of to zero. as to zero follows from C1) The asymptotic convergence of and Proposition 1; also see [44, Prop. 6]. Indeed, together with , Remark 1, it follows that there exist functions , of class , such that and , of class
(27) where we have used C2) and (23). Under the stated assumptions we may appeal to Prop. 2 to conclude (28) where is of class and quadratic near the origin. Now, pick a smooth, positive function so that (29) always exists because of the fact that Such a function is smooth and vanishes at the origin and both and are quadratic near the origin. With these observations, from (27) it follows that (30) Upon integration of both sides of (30), using (28) and (29), we obtain
(31) where
and
is defined by
This together with the fact that is bounded and tends to zero implies that goes to . Finally, the proof of Proposition 4 is comzero as pleted. Remark 4: It is of interest to point out that the assumptions of Proposition 4 can be relaxed when the -system is composed of an ISS subsystem and an iISS subsystem. See Section V-B for the details. IV. INTERCONNECTED SYSTEMS—GENERAL CASES In this section, we now turn our attention to the general class of systems (1) and consider the general case where the relative degree is any arbitrary positive integer. The only price we pay is now asfor this generality is that the high-frequency gain sumed to be an unknown nonzero constant with indefinite sign. Throughout this section, the following hypotheses are made on system (1). H1) Repeat C1). , H2) Similar to C2). More precisely, for each and , and there exist two unknown positive constants and two known positive semidefinite, smooth functions such that (33)
Assume that the closed-loop solutions are defined on a right, with . We prove the boundmaximal interval over by contradiction. Assume that is edness of unbounded. Since , is increasing and tends to as . Dividing both sides of (31) by for sufficiently large , we have
A. Dynamic Feedback Design Our dynamic feedback strategy begins with an input-driven filter, followed by a recursive controller design procedure. An Input-Driven Filter: For notational convenience, denote (34)
(32) , the By Proposition 3 and the boundedness assumption of right side of (32) is unbounded from below, while the left side . This leads to of (32) converges to a finite number as . From a contradiction. Consequently, is bounded on . Using (29), (23) (31), it follows that is bounded on and the fact that is bounded, we conclude that
Introduce a -dimensional dynamic system
.. .. . .
.. . (35)
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’s are design parameters such that the matrix
Step 1) Begin with the -subsystem of (40) and consider as the virtual control input. Led by the analysis in the proof of Proposition 4, we choose the following virtual (dynamic) control law
.. .
(41)
is asymptotically stable, i.e., its eigenvalues are in the open left plane. Notice that, at the price of notational complexity, a -dimensional dynamic system can be used instead of (35) as in our previous work [16]. , define . We have For each
with , two design parameters and design function to be determined later on. Introduce a new intermediate variable
a
(42) Then, the new variable .. .
satisfies
.. .. . .
(43) (36)
which, in compact notation, is rewritten as (37) with
. Denote Step 2) Now, we consider the augmented system composed of the -subsystem and (43) for which we view (or equivalently ) as the virtual control input. To find a virtual control law for , consider the function
. It is clear that (37) inherits an ISS property from the input to . Due to the uncertainty in the gains, the ISS gain will be uncertain. In order to obtain a computable gain function for the universal control strategy, we scale the system (37) by defining
(44) Then, differentiating gives
along the solutions of (40)
(38) (45)
Clearly, (37) becomes By completing the squares, there holds
(39) Therefore, combining (1), (35) and (39), the controlled dynamical system for feedback design is taken as
.. .
.. .. . . (40)
At this point, let us recall that both as well as are available for control design. Controller Design Procedure: The adaptive scheme calls for “backstepping” to propagate the controller obtained in Section III from the relative-degree-one case to systems with higher relative-degree. Next, we provide details on this stepwise strategy. For simplicity, we will drop the arguments of functions whenever these are clear from the context.
Let
where is a small design parameter. be an (unknown) constant such that (46)
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and set
(47)
: For notational coherence, denote Step i . Using similar techniques as in Step 2, it follows from an induction argument that the time derivative of the augmented function (56)
After simple manipulations, (45) implies
(57)
satisfies (48)
Let be the estimate of the unknown parameter . To take into account the presence of parametric uncertainties, augment with a quadratic term of the parameter estimation error , such as
(58)
(49) where Set
is a design parameter.
(50)
In (57) and (58), for every . In (58), and are appropriate functions of and of , respectively. In order to and , we give derive explicit recursive relations for , a sketch of an induction-based proof of the above claim, which is reminiscent of the “tuning functions” backstepping design [26, Ch. 4]. See also the combined small-gain and backstepping design in [16]. satisfies similar properties to (57) and (58). Assume that Noticing that
(51) (52)
(59)
is a design parameter. where With the aid of (50)–(52), (42) and (48), it is directly verified satisfies that the time derivative of
(60)
there holds (53) In addition, the new variable
satisfies (54)
where, for notational simplicity,
is introduced as
(55)
(61)
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As in Step 2, by completing the squares
Using (64), the second last term in (65) can be merged into the last bracketed term so that it can be cancelled out by the vir). This is the essence of the tual control (or equivalently popular “tuning functions” adaptive backstepping method. With this observation in mind, we choose the following virtual conof the form trol law for
Now, define (66) (62) with
. Letting
This together with (46) and (61) implies
(67) from (65) and (66), the desired inequality (58) follows readily. Therefore, at Step where the real control input shows up, we obtain the smooth dynamic output feedback law of the form
(68) such that the time derivative of the function (69) (63) satisfies Introduce the notation (64) Substituting (64) into (63) yields
(70) Similar to [16], [20], the adaptive controller (68) uses one adapted parameter to cover all unknowns, instead of adapting all parameters. B. Main Result Now, we state the main theorem of this paper that gives a general, but constructive, solution to the problem of global regulation via output feedback posed in Section II. Theorem 1: Assume that the hypotheses H1) and H2) hold with the following properties (71) and, in case
(65)
is bounded, (72)
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for all . If in H1), then the solutions of for appropri(1), (68) are well-defined and bounded over ately chosen smooth function . Furthermore, the convergence property holds (73) Remark 5: Unlike earlier related work [7], [10], [53], we do not assume that the nonlinearities in system (1) satisfy some kind of polynomial bounds. No growth condition of any kind is is imposed on system nonlinearities for large signals, when unbounded, or equivalently, when -subsystem of (1) is ISS. Remark 6: In practice, deadzone or -tracker techniques should be inserted into the adaptive laws (68). This is necessary for preventing parameter drift instability in the presence of nonvanishing disturbances. A price paid for this modification is that the output is only driven to within a small neighborhood of the origin, as shown in [11], [53]. In order to maintain the boundedness property for the state of the iISS inverse systems, some kind of “bounded-input bounded-state” property is required but only for small inputs. Before proving Theorem 1, we give a technical lemma. Lemma 1: There exists an unknown positive constant , dependent on for , such that
By hypotheses, a direct application of Proposition 2 implies (79) is positive definite, and where quadratic near the origin. so that Take a smooth
is of class
and
(80) Such a function always exists because of the conditions of Theorem 1. Then, it follows from (78) that
(81) As in the proof of Proposition 4, integrating both sides of (81) yields
(74) (82) where
and
are constants given by
(75) Proof: With the help of the “completion of squares,” it , as in (47) and (62), follows directly from the definition of and the definition of as in (37) together with Assumption H2). Proof of Theorem 1: The proof follows along the lines of be the proof of Proposition 4. Toward this end, let the solution to the Lyapunov matrix equation (76) and consider the function (77) In view of (70) and Lemma 1, the time derivative of
satisfies
(78)
From (82), it follows by contradiction that must be bounded of the closed-loop soluover the right-maximal interval is bounded over tions. Again using (82), we conclude that . By definition of as in (77), it holds that the closed-loop signals , , , and are also bounded . Thanks to the choice of in (80) and the fact over that is bounded, is bounded on . This remains bounded property together with H1) implies that ; see [44, Prop. 6]. From the definitions of ’s and on ’s in our recursive control design, it is not hard to prove that , and are bounded on . Therefore, . Finally, the convergence property (73) can be established in the same vein as in the proof of Proposition 4. Indeed, on implies that the integral the boundedness of is bounded on . Thus, goes to zero. With the help of (39), (79) and (80), this fact together converges to zero as . with Lemma 1 yields that By the definition of and , it holds that tends to zero as . Again using Lemma 1, we have that is absolutely
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integrable on . As a result, it follows from (69) that , are square integrable. A direct the signals , application of Barbalat’s lemma [23] implies the asymptotic convergence of these signals to the origin. Going back to the and as in (50)–(52), (64), (66) recursive relations of , , , converge and (67), it is not hard to prove that . Therefore, by the definition of and , we to zero as conclude that and converge to zero. Again, applying similar arguments used in the proof of Proposition 4 to the iISS -system, the asymptotic convergence of to the origin follows readily. This completes the proof.
As it can be directly verified, we have (89)
(90) for each choice of
. This leads us to make the following so as to cancel out the last term in (90): (91)
V. EXAMPLES AND DISCUSSIONS We use some common examples studied in the work of others to illustrate the generality of the proposed framework. In particular, it is shown that the popular class of output feedback form systems (see, e.g., [7], [26], [30], [55], and the numerous references therein) can be viewed as a special member in the family of uncertain systems of the form (1). A. Specification to Output Feedback Form Systems
Letting as
, from (87),
can be rewritten
(92) is the (1, 1)th element of where the constant for each . satisfies Then, the time derivative of
Example 2: Consider the class of SISO nonlinear systems in the well-known “output feedback form” studied in [26], [30], and other references: (83) (84) where is the state, , are the input and output, respectively, is a vector of unknown parameters, and the pair is in observable canonical form, i.e., .. .
(93) Hence, for each , term in (93) is cancelled out:
is selected so that the last (94)
(85)
Using the recursive relations (94) and (91), the constants , and in (87), are completely with determined. In addition, the variable satisfies
(86)
(95)
It is assumed in [26], [30] that the sign of the nonzero high-freis known, while this assumption is not needed in quency gain more recent work [7], [11], [55]. We first show that any member in (83) and (84) can be transformed into a form (1). Then, a lower-order dynamic output feedback controller can be derived using the proposed recursive method. Toward these goals, for some constants ’s, introduce the following change of coordinates
where
is defined as
(96) Therefore, in the new -coordinates, the original system (83) in the output-feedback form is brought into a special case of (1), that is
.. .
.. .. . .
.. .
.. .. . .
(87) where by
,
and
.. .
is defined
(88)
(97)
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Notice that, in this case, the functions , , do not depend on the state of the inverse system. In addition, appears linearly in the function in (1). A common assumption in previous studies [7], [26], [30], [55] is that the system in question is minimum-phase, or equivis Huralently, the polynomial is stable. As a result, witz. Under this assumption, the matrix the -system (95) is ISS with respect to the input . Furthermore, it is rather easy to see that all the conditions of Theorem 1 are fulfilled. Thus, a direct application of the constructive controller design in Section IV-A results in a lower-order dynamic output feedback control law than those in earlier work [7], [55]. It should be mentioned that, similar to [16] and [20], the adaptive controller in (68) uses one adapted parameter to cover all unknowns, instead of adapting all parameters as in [26] and [30].
applying Proposition 2 to the tegral inequality of the form
-system yields the following in-
(100) Hence, it follows from the proof of Proposition 4 that the following Lipschitz continuous, dynamic, output feedback regulator achieves the desired control goal: (101) with , . Example 4: The class of partially linear composite systems with known control direction has been studied by several researchers; see, e.g., [40] and [41, Ch. 4]. It is shown that our unifying framework can cover a class of cascade systems with unknown control direction and output-only measurements:
B. Two More Examples Example 3: Consider the third-order nonlinear system
(102) Without going into the details, we illustrate the application of Theorem 1 to (102) by means of a partially linear cascade of third-order (98)
where is a positive integer. We take as the output. For this system, we are not aware of any existing algorithm solving the problem of global asymptotic regulation via output feedback. Nevertheless, there are plenty of semiglobally stabilizing output feedback controllers; see, for instance, [15], [22], [50], and quite a few references therein. and the Case 1 (Known Control Direction): When is known, say , system (98) high-frequency gain reduces to system (2) in Example 1. In this case, the use of Nussbaum function is not needed. Also note that the iISS inverse -system is composed of an ISS -subsystem and an iISS -subsystem. Additionally, the nonlinear interconnection term only depends on the state of the ISS subsystem. Taking advantage of this structural information, it is not hard to adapt the proof of Proposition 4 in order to yield a (discontinuous) global output feedback regulator
(103) where , are unknown constants and constant with indefinite sign, and satisfies
(104) Indeed, (103) is transformed into
(105) Then, a direct application of our proposed methodology yields a dynamic output-feedback controller of the form
(106)
(99) Indeed, it is sufficient to consider the Lyapunov function candi. The stability analysis date of the form for the closed-loop system can be done in the same way as in the proof of Proposition 4. and the Case 2 (Unknown Control Direction): When high-frequency gain is unknown, system (98) obviously belongs to the family of plants (1). Unfortunately, it does not fulfill is unbounded while the conditions of Proposition 4 because is only a bounded, continuous positive-definite function. (To see this, consider .) Therefore, (25) cannot hold. Nonetheless, it is shown that our design method can be adapted to come up with a global asymptotic regulator. Indeed,
is a nonzero
with , . Notice that the aforementioned controller is conand is discontinuous when . tinuous when For simulations in Fig. 1, we make the following choice of parameters and initial conditions: (107) (108) VI. SUMMARY AND FUTURE WORK In this paper, a unifying framework has been presented for global output-feedback regulation of a broad class of nonlinear
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Fig. 1. Plots of time histories of y (t), z (t), u(t) and k (t).
systems with unmeasured iISS inverse dynamics and unknown control direction. We motivated our work by an elementary example for which the global stabilization problem is not solvable by conventional smooth output feedback. The wide applicability and flexibility of our framework is justified by showing that the standard output feedback form, the partially linear cascade and nonlinear parametrization can be treated in our framework, and that earlier results on these topics are extendible to the case of unknown control direction. In addition, unlike earlier related work [7], [10], [53], no growth condition of any kind is imposed on system nonlinearities for large signals when -subsystem of (1) is ISS—see Remark 5. In particular, Propositions 2–4 can be seen as nonlinear versions of tools employed in early results in universal adaptive control for linear plants. As such, recent works in nonidentifier-based high-gain adaptive control for nonlinear plants are somehow covered as special cases [7], [39], [55]. Our results can be directly generalized to the problem of universal -tracking [11], [53] if the idea of dead-zone is applied. More detailed analysis is however required to make Remark 6 stick. It is also our belief that the proposed framework can be adapted to cover other classes of uncertain systems with more general interconnecting functions as described in [25], [36], and [38]. Our future research will be directed at extending the proposed framework to the practical situation where nonvanishing uncertainties occur as is the case for [6], [9], [14], [19], [33], and [49], and to practically important classes of large-scale systems with strong nonlinearities as is the case for [17] and [56]. Another topic of future research is to look at the extension of our framework to nonlinear systems described by functional differential
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[40] A. Saberi, P. V. Kokotovic´ , and H. J. Sussmann, “Global stabilization of partially linear composite systems,” SIAM J. Control Optim., vol. 28, pp. 1491–1503, 1990. [41] R. Sepulchre, M. Jankovic´ , and P. V. Kokotovic´ , Constructive Nonlinear Control. New York: Springer-Verlag, 1997. [42] E. D. Sontag, “Smooth stabilization implies coprime factorization,” IEEE Trans. Automat. Contr., vol. 34, pp. 435–443, Mar. 1989. [43] , “Further facts about input-to-state stabilization,” IEEE Trans. Automat. Contr., vol. 35, pp. 473–476, Apr. 1990. [44] , “Comments on integral variants of ISS,” Syst. Control Lett., vol. 34, pp. 93–100, 1998. [45] , “The ISS philosophy as a unifying framework for stability-like behavior,” in Nonlinear Control in the Year 2000, A. Isidori, F. Lamnabhi-Lagarrigue, and W. Respondek, Eds. Berlin, Germany: Springer-Verlag, 2000, vol. 2, pp. 443–468. [46] E. D. Sontag and A. R. Teel, “Changing supply rates in input-to-state stable systems,” IEEE Trans. Automat. Contr., vol. 40, pp. 1476–1478, Oct. 1995. [47] E. D. Sontag and Y. Wang, “On characterizations of the input-to-state stability property,” Syst. Control Lett., vol. 24, pp. 351–359, 1995. [48] J. T. Spooner, M. Maggiore, R. Ordonez, and K. M. Passino, Stable Adaptive Control and Estimation for Nonlinear Systems. New York: Wiley, 2002. [49] A. R. Teel and L. Praly, “On output feedback stabilization for systems with ISS inverse dynamics and uncertainties,” in Proc. 32nd IEEE Conf. Decision Control, 1993, pp. 1942–1947. , “Tools for semiglobal stabilization by partial-state and output [50] feedback,” SIAM J. Control Optim., vol. 33, no. 5, pp. 1443–1488, 1995. [51] M. Vidyasagar, Input-Output Analysis of Large-Scale Interconnected Systems. New York: Springer-Verlag, 1980. [52] J. C. Willems and C. I. Byrnes, Global Adaptive Stabilization in the Absence of Information on the Sign of High Frequency Gain. New York: Springer-Verlag, 1984, vol. 62, Lecture Notes in Control and Information Sciences, pp. 49–57. [53] Y. Xudong, “Universal -tracking for nonlinearly-perturbed systems without restrictions on the relative degree,” Automatica, vol. 35, pp. 109–119, 1999. , “Asymptotic regulation of time-varying uncertain nonlinear [54] systems with unknown control directions,” Automatica, vol. 35, pp. 929–935, 1999. [55] X. Ye, “Adaptive nonlinear output-feedback control with unknown high-frequency gain sign,” IEEE Trans. Automat. Contr., vol. 46, pp. 112–115, Jan. 2001. [56] X. D. Ye and J. Huang, “Decentralized adaptive output regulation for a class of large-scale nonlinear systems,” IEEE Trans. Automat. Contr., vol. 48, pp. 276–281, Feb. 2003. 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, Paris, France, in 1989, and the Ph.D. degree in automatic control and mathematics from the École des Mines de Paris, Paris, France, in 1993. From 1993 to 1998, he held visiting researcher positions in several institutions including INRIA (Sophia-Antipolis), France, the Department of Systems Engineering, the Australian National University, Canberra, and the Department of Electrical Engineering, the University of Sydney, Sydney, Australia. In 1998, he also visited several U.S. universities. In January 1999, he joined the Polytechnic University, Brooklyn, NY, as an Assistant Professor of Electrical Engineering, and since 2002, he has been a tenured Associate Professor. His main research interests include stability theory, optimization, and robust and adaptive nonlinear control, with special emphasis on applications to underactuated mechanical systems and communication networks. He has authored or coauthored over 120 refereed technical papers in these areas. He is a Subject Editor for the International Journal of Robust and Nonlinear Control and an Associate Editor for Systems and Control Letters. Dr. Jiang is an Associate Editor for the IEEE TRANSACTIONS ON AUTOMATIC CONTROL. He served as an Associate Editor for the IEEE Control Systems Society Conference Editorial Board from 2000 to 2002. He was the recipient of a prestigious Queen Elizabeth II Fellowship Award (1998) from the Australian Research Council and a CAREER Award (2000) from the U.S. National Science Foundation. He is appointed as an Othmer Junior Fellow at the Othmer Institute for Interdisciplinary Studies, Polytechnic University.
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Iven Mareels (S’86–M’86–SM’94–F’01) was born in Aalst, Belgium, in 1959. He received the B.S. degree in electromechanical engineering from Gent University, Ghent, Belgium in 1982, and the Ph.D. degree in systems engineering from the Australian National University, Canberra, in 1987. He is currently a Professor in the Department of Electrical and Electronic Engineering, the University of Melbourne, Melbourne, Australia, where he holds the Chair of Electrical and Electronic Engineering, a position he took in 1996. Previously, he was a Reader at the Australian National University (1990–1996) and a Lecturer at the University of Newcastle, Newcastle, Australia (1988–1990) and the University of Gent (1986–1988). He has extensive experience in consulting for both industry and government. He has taught a broad range of subjects in both mechanical and electrical engineering curricula. His research interests are in adaptive and learning systems, nonlinear control, and modelling. At present, his research interests are in modelling and controlling of large-scale systems. He is a co-Editor-in-Chief of the international journal Systems and Control Letters. Dr. Mareels was a recipient of the Vice-Chancellor’s Award for Excellence in Teaching in 1994. He is Fellow of the Academy of Technological Sciences and Engineering, Australia, a Member of the Society for Industrial and Applied Mathematics, a Fellow of the Institute of Engineers Australia, Vice-Chair of the Asian Control Professors Association, Chairman of the Education Committee of the latter, and the Chair of the Steering Committee for the Asian Control Conference. He is a Member of the Board of Governors of the IEEE Control Systems Society. He is registered with the Institute of Engineers Australia as a professional engineer.
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David J. Hill (M’76–SM’91–F’93) received the B.E. and B.Sc. degrees from the University of Queensland, Australia, in 1972 and 1974, respectively, and the Ph.D. degree in electrical engineering from the University of Newcastle, Australia. He is currently Chair Professor and Head of the Department of Electronic Engineering at City University of Hong Kong. Previous appointments include research positions in the Electronics Research Laboratory, University of California, Berkeley, from 1978 to 1980, and a Queen Elizabeth II Research Fellowship at the University of Newcastle, Newcastle, Australia, from 1980 to 1982. During 1986–1987, he was a Guest Professor in the Department of Automatic Control, Lund Institute of Technology, Lund, Sweden. From 1982 to 1993, he held various academic positions at the University of Newcastle. From 1996 to 1999, he was Head of the School of Electrical and Information Engineering at the University of Sydney, Sydney, Australia, where he now holds an honorary position. He also holds an honorary professorship at Huazhong University of Science and Technology, Wuhan, China. His basic research interests are mainly in nonlinear networks and systems, particularly their stability and control. His applied work has consisted of various projects in power systems and industrial control carried out in collaboration with utilities in Australia and Sweden. He is also a Fellow of the Institution of Engineers, Australia, and a Foreign Member of the Royal Swedish Academy of Engineering Sciences.
Jie Huang (S’90–M’91–SM’94) was born in Fuzhou, China. He studied at Fuzhou University, received the M.S. degree from Nanjing University of Science and Technology and studied, and the Ph.D. degree from the Johns Hopkins University, Baltimore, MD. He is currently a Professor at the Chinese University of Hong Kong and also holds a Professorship at South China University of Technology sponsored by the Cheung Kong Scholars Award Program. His research interests include nonlinear control, flight control, intelligent control, and internet-based control. Dr. Huang is an Associate Editor of the IEEE TRANSACTIONS ON AUTOMATIC CONTROL, a Member of the Editorial Board of Communications in Information and Systems, and an Executive Member of the Editorial Board of Control Theory and Applications. He was Associate Editor of the Asian Journal of Control from 1999 to 2001, and has been Guest Editor for the International Journal of Robust and Nonlinear Control and the Asian Journal of Control.