Robust switching adaptive control of multi-input ... - Semantic Scholar
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Robust Switching Adaptive Control of Multi-Input Nonlinear Systems Elias B. Kosmatopoulos and Petros A. Ioannou, Fellow, IEEE
Abstract—During the last decade a considerable progress has been made in the design of stabilizing controllers for nonlinear systems with known and unknown constant parameters. New design tools such as adaptive feedback linearization, adaptive backstepping, control Lyapunov functions (CLFs) and robust control Lyapunov functions (RCLFs), nonlinear damping and switching adaptive control have been introduced. Most of the results developed are applicable to single-input feedback-linearizable systems and parametric-strict-feedback systems. These results, however, cannot be applied to multi-input feedback-linearizable systems, parametric-pure-feedback systems and systems that admit a linear-in-the-parameters CLF. In this paper, we develop a general procedure for designing robust adaptive controllers for a large class of multi-input nonlinear systems. This class of nonlinear systems includes as a special case multi-input feedback-linearizable systems, parametric-pure-feedback systems and systems that admit a linear-in-the-parameters CLF. The proposed approach uses tools from the theory of RCLF and the switching adaptive controllers proposed by the authors for overcoming the problem of computing the feedback control law when the estimation model becomes uncontrollable. The proposed control approach has also been shown to be robust with respect to exogenous bounded input disturbances.
systems is still very much unexplored. For example, there exists no procedure for designing a globally stable feedback control system for multi-input feedback linearizable systems of the form (1.1) , denote the state and control input where are convectors of the system, respectively, , , stant unknown matrices and , are continuous matrix funcand is nonsingular for all tions satisfying . The existing adaptive control designs guarantee [1], [5], [19] closed-loop stability only for the case where the conand are known; an exception is the case stant matrices (i.e., the system (1.1) is single-input) and the pair where is in a special canonical form [9]. Another example is the system of the form (parametric-purefeedback system)
(1.2)
Index Terms—Feedback linearizable systems, robust adaptive control, switching control.
is a vector of unknown constant parameters, denotes the state vector of the system are continuous functions. The procedures proand , , posed in [6], [11], [20] are applicable to this system if both and , , where denotes the estimate of ; moreover, these procedures guarantee global stability only in the case where the input vector-field is independent of , i.e., in the case where and the . functions are independent of In this paper, we develop a general procedure for designing robust adaptive controllers for a large class of multi-input nonlinear systems with exogenous bounded input disturbances. The class of systems for which the proposed approach is applicable deis characterized by the assumption that the function pends linearly on unknown constant parameters, where denotes the input vector field, is a CLF (RCLF) for the system denotes the Lie derivative of with respect to . This and class of nonlinear systems includes as a special case the systems (1.1) and (1.2). The proposed approach combines the theory of CLF (RCLF) and the switching adaptive controller proposed by the authors [9] for overcoming the problem of computing the control law in the case where the estimation model becomes uncontrollable. Contrary to the classical adaptive approach where the control law depends on estimates of the system vector-fields, in our case, the control law depends on estimates of the “RCLF term” where
I. INTRODUCTION
D
URING the last decade a considerable progress has been made in the design of stabilizing controllers for nonlinear systems with known and unknown constant parameters. New design tools such as adaptive feedback linearization [1], [5], [19], adaptive backstepping [6], [12], [20], control Lyapunov functions (CLFs) and robust control Lyapunov functions (RCLFs) [2], [13], [21], [22], nonlinear damping and swapping [11], [12] and switching adaptive control [8], [9] have been introduced. Using these new design tools, globally stabilizing controllers have been constructed for various classes of nonlinear systems such as single-input feedback-linearizable systems [1], [5], [9] and parametric-strict-feedback systems [6], [12], [20]. Despite the success of the aforementioned design tools to resolve a variety of adaptive control problems for nonlinear systems, the problem of adaptive control of nonlinear Manuscript received July 13, 1998; revised January 20, 2000 and July 10, 2001. Recommended by Asociate Editor M. Krstic. This work was supported in part by the National Aeronautics and Space Administration (NASA) under Grant Number NAGW-4103, and in part by the National Science Foundation under Grant Number ECS 9877193. E. B. Kosmatopoulos is with the Department of Production Engineering and Management, Technical University of Crete, Chania 73100, Greece. P. A. Ioannou is with the Department of Electrical Engineering-Systems, University of Southern California, Los Angeles, CA 90089-2563 USA. Publisher Item Identifier S 0018-9286(02)03739-X.
0018-9286/02$17.00 © 2002 IEEE
KOSMATOPOULOS AND IOANNOU: ROBUST SWITCHING ADAPTIVE CONTROL OF MULTI-INPUT NONLINEAR SYSTEMS
which depends both on the system vector-fields and the RCLF function . The advantage of such an approach is that the Lyapunov inequalities relating the parameter estimation errors and the time-derivative of the RCLF are easy to handle. The disadvantage is that the controllers that are designed based on . the RCLF theory critically depend on the knowledge of Adaptive versions of such controllers may fail due to the fact that may have different sign, at certain times, the estimate of . Even worse, we may have the than the sign of the actual is close to zero and the actual case where the estimate of is far from zero, which implies that the estimavalue of tion model becomes uncontrollable while the actual model is not. This problem is overcome by using a switching adaptive control law. This control law is a modified version of the one originally proposed by the authors for overcoming the problem of computing the control law in the case where the estimation model becomes uncontrollable for the case of single-input feedback-linearizable systems in canonical form [9]. Such a control law appropriately switches between two adaptive controllers which have the following properties: (i) both controllers behave approximately the same in the nonadaptive case (i.e., in the case of known system parameters), and (ii) when the one of these controllers becomes nonimplementable, the other one is implementable. The significance of the proposed approach is that it is the first to overcome the problem of constructing globally stabilizing controllers for systems of the form (1.1) and (1.2). In addition, the proposed approach can be used to solve control problems for a wider class of plants than those described by (1.1) and (1.2). The paper is organized as follows. In Section II, we present the problem formulation and some results from the theory of Robust Control Lyapunov Functions. In Section III, we present and theoretically analyze the proposed approach. In Section IV, we present various classes of systems for which the proposed approach is applicable, and, finally, Section V is the conclusion section. We close this section by mentioning that in [23] an adaptive controller for muliple-input–multiple-output (MIMO) nonlinear systems is proposed where, similar to this paper, instead of linearly parameterizing the state equations the controller design is based on a linearly parameterized positive function. The class of systems dealt in [23] is not as broad as the ones dealt in this paper and more restrictive assumptions on the system dynamics are imposed. Also, in [3], a switching controller is proposed for nonlinear systems with unknown parameters that obey a CLF that may also depend on the unknown parameters. The assumptions made in [3] are that the vector of unknown parameters belong to a finite set and that the controller for the case where the system parameters are known satisfies some robustness conditions. A. Notation and Preliminaries denotes the Euclidean norm. In this paper, If is a vector, we use the following version of the signum function: if if We say a function tinuous, strictly increasing, and
is of class when is con. We say is of class
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when is of class and satisfies as . is of class when We say a function is of class for each fixed and decreases for each fixed . We say that a continto zero as when it has continuous first derivauous function belongs to tive, and is smooth when it has continuous derivatives of any order. If is a positive constant, then the set -Ball is defined . If is a subset of then to be the set . Finally, if is a -dimensional denotes the -dimensional vector square matrix, then whose first entries are the entries of the first column of , the next entries are the ones of the second column of , etc. II. PROBLEM FORMULATION In this paper, we consider nonlinear systems of the form (2.1) , , denote the vectors of where are system states, control inputs and disturbances and , , vector-fields of appropriate dimensions. We assume that the where is disturbance vector is bounded, i.e., a positive constant. The control objective is to find the control input as a function of such that all closed-loop signals are as . Since and is asbounded and sumed to be any general unknown bounded continuous function of time, in many cases, the best that we can hope is closed-loop signal boundedness and convergence of to a residual bounded set whose size is of the order of . be a compact subset of Definition 1: [2], [14] Let such that . Moreover, suppose that the control input is chosen as , where is an appropriate feedback. Then, the solutions of the closed-loop system are robustly globally uniformly asymptotically stable (RGUAS) with respect to (RGUAS- ) when there exists a class function such and any admissible disthat for any initial condition , all solutions of the closed-loop turbance and satisfy system starting from exist for all for all . The solutions of the closed-loop system are RGUAS when they are RGUAS-{0}. Definition 2: [2], [14] The system (2.1) is robustly asymptotically stabilizable (RAS) when there exists a control law such that the closed-loop system solutions are RGUAS. The system (2.1) is robustly practically stabilizable (RPS) when there exists a control law and a comfor every satisfying -Ball such that the sopact set lutions of the closed-loop system are RGUAS- . The system is Robustly Stabilizable (RS) when there exists a control law and a compact set satisfying such that the solutions of the closed-loop system are RGUAS- . be a function. We say that Let if , for all and there exists a function such that for all . class denote the Lie derivative of Let with respect to , and be defined as follows:
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where is a nonnegative constant. Note that since , is compact for all . Let us now recall the then definition of RCLF as defined in [2], [13] as well as the relation between the RCLF and the robust stabilization problem of system (2.1). function is an RCLF for the system Definition 3: A and, moreover, there exists a positive constant (2.1) if such that satisfies
where and is a compact subset. will be defined explicitly later. Moreover, let positive design constant satisfying
and be a
(3.3) Now, let
,
denote the following sets:
if (2.2) We let denote the smallest value of for which (2.2) is satisbe defined as follows: if , and fied. Let also is an arbitrarily small positive constant if . Finally, let be defined as
The next theorem [2] states that the existence of an RCLF for system (2.1) is a necessary and sufficient condition for robust stabilization of system (2.1). Theorem 1: If system (2.1) is RAS or RPS or RS via a locally , then there is a smooth RCLF Lipschitz control law for system (2.1). On the other hand, if there exists a RCLF for , then (2.1) system (2.1), then (2.1) is RS. If furthermore is RPS. The proof of Theorem 1 can be found in [2].
Obviously, following Lemma holds. Lemma 1: There exists a scalar compact set ) such that for all
for all
. The
(that depends on the the following holds:
Proof: Since and are continuous, we have that is continuous, too. Let be the set . Also, for each let be the largest for all number such that and let be defined as
From the continuity of and the fact that is compact we (a continuous function is uniformly continuous have that on a compact set, [18, Th. 4.19]). Then, is given by
III. THE ADAPTIVE CONTROLLER Our first assumption for system (2.1) is the following. A1) System (2.1) is RS (or RPS). Since system (2.1) is RS or RPS, from Theorem 1 we have that there exists a smooth RCLF for system (2.1). The time-derivative of this RCLF is given by
where
(3.1) can be written as Let us assume that the function linear combination of known functions and unknown constant parameters. That is, we assume the following. satisfies A2) The function
where denotes the boundary of . From the above definitions is continuous, it is obvious that and the fact that and that implies that , which concludes the proof. We are now ready to present the proposed controller. The control input is chosen as follows:
(3.2)
(3.4)
is a constant unknown matrix and the where is a known nonlinear vector regressor vector function. In Section IV, we give examples of classes of systems that satisfy Assumption A2). Let us define the sets
where • (3.5) • (3.6)
KOSMATOPOULOS AND IOANNOU: ROBUST SWITCHING ADAPTIVE CONTROL OF MULTI-INPUT NONLINEAR SYSTEMS
•
denotes the estimate of the -th entry , generated as vector
of the
Note that (3.14)
(3.7) is the estimate of [ denotes the th where column of the matrix defined in (3.2)]; is a design function chosen to satisfy •
Moreover, define
The following statements hold: a)
where
(3.8) •
is a continuous-switching signal which is used to conto control , and vice tinuously switch from control versa if if if
(3.15) b)
(3.9) Proof: The proof can be found in Appendix I. Let
are updated using the following The parameter estimates desmooth projection update law [17] (here, note the parameter estimation errors)
(3.10) is a symmetric positive–definite design matrix and where is defined as [17] (3.11), shown at the bottom of the page, where
where
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positive design constant. The variable is a hysteresis-switching signal to be defined is a discontinuous explicitly later in this section. Finally, function defined as follows:
(3.16) From Lemma A.1 part a) (see Appendix I), we have that, for , and therefore the term can be made arbitrarily close to 1 by increasing the design constant . Therefore, from Lemma 2 we have that for
is
if otherwise
(3.12)
is a positive design constant. From the definition of where , we have that the adaptive law (3.10) becomes inactive is smaller than . The role of will be made clear when during the proof of the main result of this paper. Let us analyze the proposed control law. More precisely, consider the following Lemma. Lemma 2: Consider the variables (3.13)
(3.17) (note that ). Using now the facts that is close is bounded from above by we can see to one and and from the above inequality that both control laws have approximately the same effect on the term . More precisely, the both have the effect of replacing the unknown term by the negative definite term , plus a term that . depends on the estimation error was absent in the right-hand side If the term (RHS) of (3.17) then, using standard arguments in adaptive control, we could show that the use of the parameter projection law takes care of the effect of the estimation error (3.10) with where
if if
and (3.11)
otherwise
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term . However, although the term has , it may have destabilizing efsmaller magnitude than fects to the closed-loop system due to the fact that it depends on the unknown function . In order to overcome the problem where the term may have a destabilizing effect, we proceed as follows: As it will be shown in the proof of Theorem 2, the following term appears in the RHS of the time-derivative of an appropriately defined Lyapunov-like function
The term appears in the Lyapunov equation due to the projection law (3.10). Define (3.18) is a continuous funcwhere . Ob-
and
where tion, satisfying
and for all . , are continuous functions for all , where is a convex set which is defined explicitly in Theorem 2 later in this section. Therefore, using standard arguments from the theory of approximation of nonlinear functions (see, e.g., [7], and the references therein) , there exist1 two convex sets we can show that for any , two constant vectors , and such that a nonlinear function2 viously
(3.19) and (3.20) It is no loss of generality to assume that the entries of , as well as the entries of all vectors or are3 nonnegative. Also, since we assume and that for all such that (again this is true provided we choose that appropriately; for instance the entries of could be chosen so ). Using (3.19) and (3.20) their magnitude is bounded by we obtain and the fact that
and
Using (3.20) and (3.21), we have that for
(3.22) denotes the estimate of and denotes the parameter estimation error. The first two terms in the RHS of inequality (3.22) can be taken care of by using adaptive update laws. Thus we have to choose in such a way that the rest three terms remain bounded by a small design constant. We will use the following hysteresis-switching adaptive scheme for : where
if otherwise (3.23) , , where . The problem now is in the design of the parameter estimation . There are two issues regarding the laws for design of such estimation laws. The first is the issue of keeping the parameter estimates bounded; such a problem can be easily addressed using projection laws similar to (3.10) and (3.11). The second issue has to do with the problem when, is very close to zero at certain time-instants, is large positive. In such a case, the term while may not be bounded by a small design constant. To round this problem, we proceed as follows. First, observe , , we have that4 that from the definitions of where
(3.24) and
Therefore, using the properties , we have that
(3.25) where
is the convex set defined as
(3.21) where
!
!1
that, in general, " 0 as n . notational simplicity we assume that both approximators have the same regressor terms. 3Such an assumption holds if we augment the regressor vector so that it contains both the entries ( ) and ( ). 1Note 2For
1
0
1
(3.26) 4The
integer j ? k is defined as j ? k =
n
(k
0 1) + j .
KOSMATOPOULOS AND IOANNOU: ROBUST SWITCHING ADAPTIVE CONTROL OF MULTI-INPUT NONLINEAR SYSTEMS
where is a large positive constant such that . The first two inequalities define the constraint that the entries of are positive, the third one defines the constraint that and the last inequality defines the constraint is bounded from above by . that Relation (3.25) motivates us to propose the following adap: tive laws for (3.27) where keeps
,
is a positive constant, , and is the projection law which and is defined as follows:
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the existence and uniqueness of solutions in the sense of Caratheodory [16] (the proof of [16] can be easily revised to include the case where the projection law is inactive when ). Therefore, using similar arguments as those in [15] (see also [24, Lemma A.1]) we can establish that there of maximal length on which the exists an interval hysteresis-switching closed-loop system possesses a unique and, moreover, solution with ’s piecewise constant on can that on each strictly proper subinterval , are switch at most finite times. Note also that bounded due to the projection update laws (see Lemma A.1, part (c) in Appendix I). denote the smallest positive constant satisfying Now, let
Also, let
denote the set
(3.28) if or and and , otherwise. Similarly, if or and and , otherwise. We are now ready to prove closed-loop stability under the the control law (3.4)–(3.11), (3.23), and (3.27). Theorem 2: Consider the unknown system (2.1) and the control law (3.4)–(3.11), (3.23) and (3.27). Assume that A1) and are vector-fields, and the disturbance A2) hold, that , is continuous and bounded. Moreover, assume that the following hold. satisfies (3.8) and , where is defined C1) in Lemma 1. where C2) is large enough so that
where
where is a positive constant. From the above definitions, we have that
Since , we have that there exists a positive real such that for all . Let us analyze the . Define the Lyapunov-like closed-loop system for function
In Appendix I, we show that for is large enough so that and where is a positive design constant. and for any positive conThen, for any compact set the following holds: there exist positive constant , , (dependent on ) and such that, for any stants , the control law (3.4)–(3.11) with initial state guarantees that all the closed-loop signals are bounded and converges to the residual set
(3.29)
Moreover,
where the constant satisfies where when . is continuous and bounded, is Proof: Since smooth with respect to its arguments [see Lemma A.1, part is a continuous signal, we have that the source of (c)], discontinuities in the closed-loop system dynamics are the and the parameter projection hysteresis-switching variables laws (3.10) and (3.27). Parameter projection laws guarantee
are positive terms that can be made arbitrarily where , (or, equivalently, decreasing ). small by increasing if and Using now the fact that , the definition of and the definition of , it is straightforward to show that, there exist positive constants and a positive–definite function such that for (3.30) provided that set
remains on
where
is the compact
In (3.30), is a positive constant. Such a constant exists, proare large enough. and defined in vided that Theorem 2 are related as follows: The variable is a positive . For the time being let us assume constant such that . Using now the that is the smallest constant for which
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definition of the function , part (c) of Lemma A.1, and the fact is decreasing as long as does not enter , we that remains in the interior of for all can see that provided that
(3.31) and independent of we have that Given any such that the above inequality is valid for all there exists (note that the term depends on the dimension of the nonlinear approximators and thus it may be increasing increases; note also that may as becomes larger, i.e., as by increasing ). become arbitrarily close to remains in the interior of for all , Since all the way to . On the other we have that we can extend is decreasing outside together with hand, the fact that due to the projection update the boundedness of . Finally, since outside laws implies that for all implies that enters in finite time . converges to , that is, there is a time We will show that such that . Let us define instant , i.e., . While is , the parameter adjustment laws become inactive, which in is constant while . in turn implies that the term enters at time-instant Consider the following situation: , enters at , exits at and exits at . Since outside , we have that will enter again at time-inat , where . When stant and the parameter adjustment laws are inactive, and thus . Moreover, from the definition of we have . that is decreasing for and . Finally, . This is since: Using the above, we can see that and (b) is decreasing at and (a) and thus is decreasing in . Thereat the time enters , is strictly smaller than its fore, entered previously. It can be seen that, by value when appropriately defining , the following also holds: . If the above claim is not true, we can always increase the value of to make5 it true. Therefore, each time enters the term is smaller than the one during the previous visit which in turn, implies that eventually stays in of in foreover. This claim can be proven by contradiction: if would not stay in then we would have that converges to converges to , and zero, which in turn implies that would be—at the limit—decreasing outside . thus,
2( ) 2( ) 2
5This can be done as follows: suppose that t t which in turn implies that t increased at t t ;t by an amount that is larger than t t . In other words, we have the case where has changed more during the trip of x t from the boundary of to the boundary of than it has changed while x t was outside . In that case, we can increase c (i.e., increase the size of ) so that x t never left . Note that the amount by which c has to be increased should be very small.
2( ) 2( ) 0 2( ) () ()
()
2[
)
Remark 1: As it is seen from (3.31) the reason we choose is to make , and thus , independent of the diof the nonlinear approximators. mension Remark 2: In the case where does not satisfy the linearin-the-parameters assumption A2), the proposed approach is can be still applicable, by assuming that the function approximated by linear-in-the-weights nonlinear functions. In [10], we show how we can apply the proposed strategy in the does not satisfy assumption A2) or, for the case where case where the system dynamics are completely unknown. IV. APPLICATION TO VARIOUS CLASSES OF SYSTEMS In this section, we present some examples of classes of nonlinear systems which satisfy assumption A2). For simplicity, we . The consider the case where the external disturbance results can be easily extended to the case where external input disturbances are present. A. Multi-Input Feedback-Linearizable Systems One class of systems that satisfy assumption A2) is the class of multi-input feedback-linearizable systems. For this class there exists no general methodology for designing adaptive controllers that guarantee global stability for the closed-loop system. Let us consider the class of multi-input feedback linearizable systems whose dynamics can be described as follows: (4.1) are unknown constant matrices, are where known nonlinear continuous vector functions and, moreover, the is stabilizable and the functions are matrix pair and is nonsingular for all . such that . Then, a Let be a stabilizing gain matrix for the pair where RCLF for system (4.1) is the function [2] is the symmetric positive–definite solution of the Lyapunov equation (4.2) is a symmetric positive–definite matrix. Then, the where and are given by functions (4.3) and (4.4) It is not difficult to see that if is a vector whose entries are the and , is a vector whose entries are the elements , is a vector whose entries are the elements of elements and and is a matrix whose entries are the elements , then, and can be written in the form (3.2). In of other words, systems of the form (4.1) satisfy assumption A2). In a similar way, we can show that the feedback linearizable systems of the form (4.5) are unknown satisfy assumption A2). Here, are known nonlinear continuous constant matrices,
KOSMATOPOULOS AND IOANNOU: ROBUST SWITCHING ADAPTIVE CONTROL OF MULTI-INPUT NONLINEAR SYSTEMS
vector functions and, moreover, the unknown matrix pair is stabilizable and the continuous functions are such that and is nonsingular for . For the case of the system (4.5) the functions all are given as follows:
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B. Linear-in-the-Parameters Nonlinear Systems With Linear-in-the-Parameters RCLF Consider now the class of nonlinear systems of the form (2.1) whose vector-fields and are linear combinations of known functions and unknown constant parameters, i.e.,
(4.6) and
and (4.7)
is defined in (4.2). Next we show how to chose where the controller parameters: First, let us examine the equality is valid. Since we have that which, due to the nonsingularity of , implies that and implies . On the therefore other hand, from the Lyapunov equality (4.2) it can be easily implies that seen that , and therefore, implies that . we conclude that where is defined in Definition 3 Thus, we have that can be any arbitrarily small constant. Once and therefore is chosen then can be chosen as follows: First note that is nonsingular, we have that since for some positive number . Assuming that a lower bound of is known we have that implies (4.8)
where
are known nonlinear vector functions and are unknown constant matrices (vectors). Moreover, assume that an RCLF for the system can be written as a linear combination of known functions and unknown constant parameters, i.e.,
where is a known nonlinear vector function and is an unknown constant vector. Then, it can be easily seen that the above system satisfies assumption A2) where the entries of are the elements of , the entries of are the elements , the entries of are the elements of , of are the elements of . and the entries of C. Parametric-Pure-Feedback Systems Let us now try to apply the results of Section III to nonlinear systems that take the form
and By adding and subtracting the terms in the RHS of (4.6) and using (4.2) we obtain after some algebraic manipulations (4.9) Using (4.8), we have that
implies
where is a known compact set satisfying . From the above inequality, we have that could be set equal to the largest that satisfies . Note that an upper is needed in order to solve the above problem. bound on can be then set equal to where The constant is the minimum value of that satisfies , where is defined in the proof of Theorem 2. can be chosen Finally, the proposed controller function as follows:
where
are positive constants satisfying . It is worth noticing that the existing adaptive control designs guarantee global stability only in the case where the matrix for the case of system (4.1) or the matrices and for the case of system (4.5) are known.
, are smooth known functions and where is the vector of constant but unknown system parameters. Let us rewrite (4.9) as
(4.10) . Systems of the form (4.10) are where called parametric-pure-feedback (PPF) systems [6], [12], [20]. Although the problem of constructing globally stable adaptive controllers for the simplest case of parametric-strict-feedback (PSF) systems 6 has been completely solved [6], [12], [20], the problem of constructing globally stable adaptive controllers for PPF systems remains an open problem and is the subject of this subsection. Although the results developed in Sections I–III are for state stabilization, for the particular case of PPF systems, they can be 6The class of PSF systems refers to the subclass of PPF systems (4.10) which satisfy f (z ; . . . ; z ) = 0; g (z ; . . . ; z ) = 0.
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readily extended to the case of asymptotic tracking as well. The control objective is to force the system output to asymptotically track a reference signal . We assume that the first time derivatives of are known. Also it is assumed that as time derivatives are bounded and smooth well as its first signals. Before we design the feedback law, we will transform the system (4.9) into a suitable form. The procedure we follow is that of [6] and [12], and it is based on the backstepping integrator principle [22]. . Let also be positive Step 0): Let constants to be chosen later. Step 1): Using the “chain of integrators” method, we see that, if was the control input in the -part of (4.10) and was known, then the “control law” (4.11) would result in a globally asymptotically stable tracking, since such a control law would transform the -part of (4.10) as follows
Step 3): Using the above definitions of that after some algebraic manipulations
(4.17) is a vector that consists of all elements that are where or of the form or of the form either of the form . In the system (4.17), we will think of as our control input. Therefore, as in Step 2), we define the new state as
(4.18) Substituting (4.18) into (4.17) yields (4.19) : Using the definitions of Step and working as in the previous steps we may express the derivative of as
However, the state is not the control. Therefore, we deto be the difference between the actual and its fine desired expression (4.11): (4.12) Using the above definition of , the definition of the -part of (4.10), we found that
and
, we have
(4.20) where the vector
contains all the terms of the form with . Defining now as
follows:
(4.21) (4.13) Step 2): Using the above definitions of
we obtain that
, we have that
(4.22) and working Step : Using the definitions of as in the previous steps we may express the derivative of as follows:
(4.23)
(4.14)
where the vector form
contains all the terms of the with , and
is a -dimensional vector that consists of where or of the form all elements that are either of the form where by we denote the -th entry of the vector . In the system (4.14), we will think of as our control input. Therefore, as in Step 1), we define the new state as
is given by
(4.24) (4.15)
, and by rearranging Using the definitions of terms, we may rewrite (4.23) as follows:
(4.16)
(4.25)
Substituting (4.15) into (4.14) yields
KOSMATOPOULOS AND IOANNOU: ROBUST SWITCHING ADAPTIVE CONTROL OF MULTI-INPUT NONLINEAR SYSTEMS
Therefore, using the aforementioned methodology, we have transformed system (4.10) into the following:
.. .
.. .
..
.
..
.
.. .
where and (note that
619
are appropriately defined known functions ) Therefore, we have that
.. .
and .. .
(4.26)
or (4.27) where of the previous system as
. Let us define the output
where . Obviously, the above system is feedback-linearizable, if the following assumption holds. for all . A1’) are not available for the Note that the variables control design since they depend on the unknown vector . We will show that, if the constants are chosen so that for all , then the function
From the above two equations, it can be easily seen that the and satisfy assumption A2), by defining the functions vectors and as the vectors whose entries are the elements and defining the functions and appropriately. The proposed controller parameters can be chosen as follows: , the constant can be chosen to since the constant be any arbitrarily small positive number. Working similar to the case of feedback linearizable systems in Section IV-A we can show that can be set equal to the largest that satisfies where is defined as follows
where
is an RCLF for system (4.9) and, moreover, that the resulting and satisfy assumption A2). By differentifunctions ating with respect to time, we obtain that
is a positive constant satisfying and is a known compact set satisfying . must be chosen so that and The constant where is defined in the proof of Theorem 2 . , one possible way Regarding the design of the function as follows: is to choose
(4.28) and, therefore (4.29) and (4.30) From the assumption A1’), it can be easily seen that . The quantity is for all , and thus, is negative definite provided that . Note now that an RCLF for system (4.9) and moreover using the definitions of we can rewrite ’s as follows:
where is a positive constant satisfying . Remark 3: Theorem 2 is not directly applicable to the class and in (4.29) of PPF systems (4.9), since the terms and (4.30) are explicit functions of . However, Theorem 2 can be easily modified to be applicable to such a case. The only modas well as the regressor ification needed is that the function terms in the adaptive laws should be explicit functions of (thus, for instance the regressor terms of the nonlinear approximators should be replaced by . If we carry out the same analysis as in the proof of Theorem 2, by incorporating the above modifications, it can be seen that results of Theorem 2 are applicable to the class of systems (4.9).
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V. CONCLUSION
and therefore, from (3.3) we obtain that, for all
In this paper, we proposed a switching adaptive controller for multi-input nonlinear systems whose dynamics are nonlinearly affected by external input disturbances. The proposed approach is applicable to nonlinear systems for which the function can be written as a linear combination of unknown constants and known nonlinear functions. By making use of the notion of RCLF [2] and a modified version of the switching adaptive controller of [9] we showed that the proposed controller guarantees bounded closed-loop signals and convergence of the state to a residual set. The proposed control approach is used to design stabilizing controllers for multi-input feedback-linearizable systems, PPF systems and linear-in-the-parameters nonlinear systems which admit a linearly parameterized RCLF. APPENDIX Lemma A.1: Assume that the conditions of Theorem 2 hold. Then the following statements are true. we have that a) For all
and
b)
, . is smooth with respect to its arguments. Moreover,
c)
b) c) d) e)
and thus, is bounded from above by for all . Using now the fact that is bounded from above for all and (3.16), it is straightforward to by verify the second inequality of part a). The proof is straightforward. The proofs are similar to those of [17] and [4, Th. 4.4.1]. The proof is straightforward. Since from part c) of this Lemma , we have that and with such therefore, there exists . Therefore that
We have that (since and the design assumption ), which concludes the proof. Proof of Lemma 2: a) From (3.5), we have that
and
where is a finite positive constant, independent of Similarly, for
.
Therefore d) e)
and Proof: we have that . Therea) Since fore, by taking into account (3.3) we conclude that for , i.e., the denominator of never becomes all . Since , we have that zero provided that
for all
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(A.1) b) Using (3.6), we obtain
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(A.2)
Proof of (3.29): Differentiating with respect to time and using (3.4)–(3.11) and (3.18)–(3.27), Lemma 2, Lemma we obtain that A.1, and the fact that , the equation at the bottom of the for page holds true,
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where
From (3.3), it can be seen that for any such that . Therefore
there exists an
and, thus, we finally obtain that
(A.3)
REFERENCES [1] G. Cambion and G. Bastin, “Indirect adaptive state feedback control of linearly parameterized nonlinear systems,” Int. J. Adapt. Control Signal Processing, vol. 4, pp. 345–358, Sept. 1990. [2] R. A. Freeman and P. V. Kokotovic, “Inverse optimality in robust stabilization,” SIAM J. Control Optim., vol. 34, no. 4, pp. 1365–1391, July 1996. [3] H. Hespana and A. S. Morse, “Supervision of families of nonlinear controllers,” in Proc. 35th IEEE Conf. Decision Control, vol. 4, Dec. 1996, pp. 3771–3773.
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[4] P. A. Ioannou and J. Sun, Stable and Robust Adaptive Control. Upper Saddle River, NJ: Prentice-Hall, 1996. [5] I. Kanellakopoulos, P. V. Kokotovic, and R. Marino, “An extended direct scheme for robust adaptive nonlinear control,” Automatica, vol. 27, pp. 247–255, Mar. 1991. [6] I. Kanellakopoulos, P. V. Kokotovic, and A. S. Morse, “Systematic design of adaptive controllers for feedback linearizable systems,” IEEE Trans. Automat. Contr., vol. 36, pp. 1241–1253, November 1991. [7] E. B. Kosmatopoulos and M. A. Christodoulou, “Techniques and applications of recurrent high order neural networks in the identification of dynamical systems,” in Neural Network Systems Techniques and Applications, C. Leondes, Ed. New York: Academic, 1999. [8] E. B. Kosmatopoulos, “Universal stabilization using control Lyapunov functions, adaptive derivative feedback and neural network approximators,” IEEE Trans. Syst., Man, Cybern. B, vol. 28B, pp. 472–477, June 1998. [9] E. B. Kosmatopoulos and P. A. Ioannou, “A switching adaptive controller for feedback linearizable systems,” IEEE Trans. Automat. Contr., vol. 44, pp. 742–750, Apr. 1999. [10] E. B. Kosmatopoulos, “Robust neural stabilizers for unknown systems,” IEEE Trans. Neural Networks, submitted for publication. [11] M. Krstic and P. V. Kokotovic, “Adaptive nonlinear design with controller-identifier separation and swapping,” IEEE Trans. Automat. Contr., vol. 40, pp. 426–440, Mar. 1995. [12] M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic, Nonlinear and Adaptive Control Design. New York: Wiley, 1995. [13] Y. Lin and E. D. Sontag, “Control Lyapunov universal formulae for restricted inputs,” Control: Theory Adv. Technol., vol. 10, pp. 1981–2004, 1995. [14] Y. Lin, E. D. Sontag, and Y. Wang, “A smooth converse Lyapunov theorem for robust stability,” SIAM J. Control Optim., vol. 34, pp. 124–160, 1996. [15] A. S. Morse, D. Q. Mayne, and G. C. Goodwin, “Applications of hysteresis switching in parameter adaptive control,” IEEE Trans. Automat. Contr., vol. 34, pp. 1343–1354, Sept. 1992. [16] M. M. Polycarpou and P. A. Ioannou, “On the existence and uniqueness of solutions in adaptive control systems,” IEEE Trans. Automat. Contr., vol. 38, pp. 474–480, Mar. 1993. [17] J.-B. Pomet and L. Praly, “Adaptive nonlinear regulation: Estimation from the Lyapunov equation,” IEEE Trans. Automat. Contr., vol. 37, pp. 729–740, 1992. [18] W. Rudin, Principles of Mathematical Analysis. New York: McGrawHill, 1984. [19] S. S. Sastry and A. Isidori, “Adaptive control of linearizable systems,” IEEE Trans. Automat. Contr., vol. 34, pp. 405–412, Apr. 1989. [20] D. Seto, A. M. Annaswamy, and J. Baillieul, “Adaptive control of nonlinear systems with a triangular structure,” IEEE Trans. Automat. Contr., vol. 39, pp. 1411–1428, July 1994. [21] E. D. Sontag, “A universal construction of Arstein’s theorem on nonlinear stabilization,” Syst. Control Lett., vol. 13, no. 2, pp. 117–123, 1989. [22] J. Tsinias, “Sufficient Lyapunov-like conditions for stabilization,” Math. Control, Signals, Syst., vol. 2, no. 4, pp. 343–357, 1989. [23] B. Yao and M. Tomizuka, “Adaptive robust control of a class of multivariable nonlinear systems,” IFAC World Congress, vol. F, pp. 335–340, 1996. [24] S. R. Weller and G. C. Goodwin, “Hysteresis switching adaptive control of linear multivariable systems,” IEEE Trans. Automat. Contr., vol. 39, pp. 1360–1375, July 1994.
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Elias B. Kosmatopoulos received the Diploma in production and management engineering, and the M.Sc. and Ph.D. degrees in electronics and computer engineering, all from the Technical University of Crete (TUC), Greece, in 1990, 1993, and 1995, respectively. He is currently a tenure-track Lecturer in the Department of Production and Management Engineering, and Deputy Director of the Dynamic Systems and Simulation Laboratory, at TUC. Prior to joining TUC, he was a Research Assistant Professor with the Department of Electrical Engineering-Systems, University of Southern California (USC), Los Angeles, and a Postdoctoral Fellow with the Department of Electrical and Computer Engineering, University of Victoria, Vistoria, BC, Canada. He is the author or coauthor of more than 20 journal articles and book chapters, and more than 40 conference publications in the areas of neural networks, adaptive and neural control, fuzzy systems, and intelligent transportation systems. He has been involved in various research projects funded by the European Community, the National Sciences of Engineering Research Council (NSERC) of Canada, the National Aeromautics and Space Administration (NASA), the Air Force, and the Department of Transportation involving virtual reality, fault detection and identification, manufacturing systems, robotics, fuzzy controllers, telecommunications, design and control of flexible and space structures, active isolation techniques for civil structures, control of hypersonic vehicles, automated highway systems, agile port technologies, and intelligent transportation systems. He has served as a reviewer for various journals and conferences, and has served as the session chairman or cochairman for various international conferences.
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Petros A. Ioannou (S’80–M’83–SM’89–F’94) received the B.Sc. degree (with First Class Honors) from University College, London, U.K., and the M.S. and Ph.D. degrees from the University of Illinois, Urbana, in 1978, 1980, and 1982, respectively. From 1975 to 1978, he held a Commonwealth Scolarship from the Association of Commonwealth Universities, London, U.K. From 1979 to 1982, he was a Research Assistant at the Coordinated Science Laboratory at the University of Illinois. In 1982, he joined the Department of Electrical Engineering-Systems, University of Southern California, Los Angeles, where he is currently a Professor in the same Department and the Director of the Center of Advanced Transportation Technologies. His research interests are in the areas of adaptive control, neural networks, vehicle dynamics, and control and intelligent vehicle and highway systems. He has been an Associate Editor for The International Journal of Control and Automatica. He has published five books and over 100 technical papers. He is a member of the AVCS Committee of Intelligent Transportation Systems (ITS) America, and a Control Systems Society member of the IEEE Technical Activities Board Committee on Intelligent Transportation Systems. Dr. Ioannou was awarded several prizes, including the Goldsmid Prize and the A.P. Head prize from University College, London, U.K. In 1984, he was a recipient of the Outstanding Transactions Paper Award for his paper “An Asymptotic Error Analysis of Identifiers and Adaptive Observers in the Presence of Parasitics, ” which appeared in the IEEE TRANSACTIONS ON AUTOMATIC CONTROL in August 1982. He was also the recipient of a 1985 Presidential Young Investigator Award for his research in Adaptive Control. He has been an Associate Editor for the IEEE TRANSACTIONS ON AUTOMATIC CONTROL
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Robust Switching Adaptive Control of Multi-Input Nonlinear Systems Elias B. Kosmatopoulos and Petros A. Ioannou, Fellow, IEEE
Abstract—During the last decade a considerable progress has been made in the design of stabilizing controllers for nonlinear systems with known and unknown constant parameters. New design tools such as adaptive feedback linearization, adaptive backstepping, control Lyapunov functions (CLFs) and robust control Lyapunov functions (RCLFs), nonlinear damping and switching adaptive control have been introduced. Most of the results developed are applicable to single-input feedback-linearizable systems and parametric-strict-feedback systems. These results, however, cannot be applied to multi-input feedback-linearizable systems, parametric-pure-feedback systems and systems that admit a linear-in-the-parameters CLF. In this paper, we develop a general procedure for designing robust adaptive controllers for a large class of multi-input nonlinear systems. This class of nonlinear systems includes as a special case multi-input feedback-linearizable systems, parametric-pure-feedback systems and systems that admit a linear-in-the-parameters CLF. The proposed approach uses tools from the theory of RCLF and the switching adaptive controllers proposed by the authors for overcoming the problem of computing the feedback control law when the estimation model becomes uncontrollable. The proposed control approach has also been shown to be robust with respect to exogenous bounded input disturbances.
systems is still very much unexplored. For example, there exists no procedure for designing a globally stable feedback control system for multi-input feedback linearizable systems of the form (1.1) , denote the state and control input where are convectors of the system, respectively, , , stant unknown matrices and , are continuous matrix funcand is nonsingular for all tions satisfying . The existing adaptive control designs guarantee [1], [5], [19] closed-loop stability only for the case where the conand are known; an exception is the case stant matrices (i.e., the system (1.1) is single-input) and the pair where is in a special canonical form [9]. Another example is the system of the form (parametric-purefeedback system)
(1.2)
Index Terms—Feedback linearizable systems, robust adaptive control, switching control.
is a vector of unknown constant parameters, denotes the state vector of the system are continuous functions. The procedures proand , , posed in [6], [11], [20] are applicable to this system if both and , , where denotes the estimate of ; moreover, these procedures guarantee global stability only in the case where the input vector-field is independent of , i.e., in the case where and the . functions are independent of In this paper, we develop a general procedure for designing robust adaptive controllers for a large class of multi-input nonlinear systems with exogenous bounded input disturbances. The class of systems for which the proposed approach is applicable deis characterized by the assumption that the function pends linearly on unknown constant parameters, where denotes the input vector field, is a CLF (RCLF) for the system denotes the Lie derivative of with respect to . This and class of nonlinear systems includes as a special case the systems (1.1) and (1.2). The proposed approach combines the theory of CLF (RCLF) and the switching adaptive controller proposed by the authors [9] for overcoming the problem of computing the control law in the case where the estimation model becomes uncontrollable. Contrary to the classical adaptive approach where the control law depends on estimates of the system vector-fields, in our case, the control law depends on estimates of the “RCLF term” where
I. INTRODUCTION
D
URING the last decade a considerable progress has been made in the design of stabilizing controllers for nonlinear systems with known and unknown constant parameters. New design tools such as adaptive feedback linearization [1], [5], [19], adaptive backstepping [6], [12], [20], control Lyapunov functions (CLFs) and robust control Lyapunov functions (RCLFs) [2], [13], [21], [22], nonlinear damping and swapping [11], [12] and switching adaptive control [8], [9] have been introduced. Using these new design tools, globally stabilizing controllers have been constructed for various classes of nonlinear systems such as single-input feedback-linearizable systems [1], [5], [9] and parametric-strict-feedback systems [6], [12], [20]. Despite the success of the aforementioned design tools to resolve a variety of adaptive control problems for nonlinear systems, the problem of adaptive control of nonlinear Manuscript received July 13, 1998; revised January 20, 2000 and July 10, 2001. Recommended by Asociate Editor M. Krstic. This work was supported in part by the National Aeronautics and Space Administration (NASA) under Grant Number NAGW-4103, and in part by the National Science Foundation under Grant Number ECS 9877193. E. B. Kosmatopoulos is with the Department of Production Engineering and Management, Technical University of Crete, Chania 73100, Greece. P. A. Ioannou is with the Department of Electrical Engineering-Systems, University of Southern California, Los Angeles, CA 90089-2563 USA. Publisher Item Identifier S 0018-9286(02)03739-X.
0018-9286/02$17.00 © 2002 IEEE
KOSMATOPOULOS AND IOANNOU: ROBUST SWITCHING ADAPTIVE CONTROL OF MULTI-INPUT NONLINEAR SYSTEMS
which depends both on the system vector-fields and the RCLF function . The advantage of such an approach is that the Lyapunov inequalities relating the parameter estimation errors and the time-derivative of the RCLF are easy to handle. The disadvantage is that the controllers that are designed based on . the RCLF theory critically depend on the knowledge of Adaptive versions of such controllers may fail due to the fact that may have different sign, at certain times, the estimate of . Even worse, we may have the than the sign of the actual is close to zero and the actual case where the estimate of is far from zero, which implies that the estimavalue of tion model becomes uncontrollable while the actual model is not. This problem is overcome by using a switching adaptive control law. This control law is a modified version of the one originally proposed by the authors for overcoming the problem of computing the control law in the case where the estimation model becomes uncontrollable for the case of single-input feedback-linearizable systems in canonical form [9]. Such a control law appropriately switches between two adaptive controllers which have the following properties: (i) both controllers behave approximately the same in the nonadaptive case (i.e., in the case of known system parameters), and (ii) when the one of these controllers becomes nonimplementable, the other one is implementable. The significance of the proposed approach is that it is the first to overcome the problem of constructing globally stabilizing controllers for systems of the form (1.1) and (1.2). In addition, the proposed approach can be used to solve control problems for a wider class of plants than those described by (1.1) and (1.2). The paper is organized as follows. In Section II, we present the problem formulation and some results from the theory of Robust Control Lyapunov Functions. In Section III, we present and theoretically analyze the proposed approach. In Section IV, we present various classes of systems for which the proposed approach is applicable, and, finally, Section V is the conclusion section. We close this section by mentioning that in [23] an adaptive controller for muliple-input–multiple-output (MIMO) nonlinear systems is proposed where, similar to this paper, instead of linearly parameterizing the state equations the controller design is based on a linearly parameterized positive function. The class of systems dealt in [23] is not as broad as the ones dealt in this paper and more restrictive assumptions on the system dynamics are imposed. Also, in [3], a switching controller is proposed for nonlinear systems with unknown parameters that obey a CLF that may also depend on the unknown parameters. The assumptions made in [3] are that the vector of unknown parameters belong to a finite set and that the controller for the case where the system parameters are known satisfies some robustness conditions. A. Notation and Preliminaries denotes the Euclidean norm. In this paper, If is a vector, we use the following version of the signum function: if if We say a function tinuous, strictly increasing, and
is of class when is con. We say is of class
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when is of class and satisfies as . is of class when We say a function is of class for each fixed and decreases for each fixed . We say that a continto zero as when it has continuous first derivauous function belongs to tive, and is smooth when it has continuous derivatives of any order. If is a positive constant, then the set -Ball is defined . If is a subset of then to be the set . Finally, if is a -dimensional denotes the -dimensional vector square matrix, then whose first entries are the entries of the first column of , the next entries are the ones of the second column of , etc. II. PROBLEM FORMULATION In this paper, we consider nonlinear systems of the form (2.1) , , denote the vectors of where are system states, control inputs and disturbances and , , vector-fields of appropriate dimensions. We assume that the where is disturbance vector is bounded, i.e., a positive constant. The control objective is to find the control input as a function of such that all closed-loop signals are as . Since and is asbounded and sumed to be any general unknown bounded continuous function of time, in many cases, the best that we can hope is closed-loop signal boundedness and convergence of to a residual bounded set whose size is of the order of . be a compact subset of Definition 1: [2], [14] Let such that . Moreover, suppose that the control input is chosen as , where is an appropriate feedback. Then, the solutions of the closed-loop system are robustly globally uniformly asymptotically stable (RGUAS) with respect to (RGUAS- ) when there exists a class function such and any admissible disthat for any initial condition , all solutions of the closed-loop turbance and satisfy system starting from exist for all for all . The solutions of the closed-loop system are RGUAS when they are RGUAS-{0}. Definition 2: [2], [14] The system (2.1) is robustly asymptotically stabilizable (RAS) when there exists a control law such that the closed-loop system solutions are RGUAS. The system (2.1) is robustly practically stabilizable (RPS) when there exists a control law and a comfor every satisfying -Ball such that the sopact set lutions of the closed-loop system are RGUAS- . The system is Robustly Stabilizable (RS) when there exists a control law and a compact set satisfying such that the solutions of the closed-loop system are RGUAS- . be a function. We say that Let if , for all and there exists a function such that for all . class denote the Lie derivative of Let with respect to , and be defined as follows:
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where is a nonnegative constant. Note that since , is compact for all . Let us now recall the then definition of RCLF as defined in [2], [13] as well as the relation between the RCLF and the robust stabilization problem of system (2.1). function is an RCLF for the system Definition 3: A and, moreover, there exists a positive constant (2.1) if such that satisfies
where and is a compact subset. will be defined explicitly later. Moreover, let positive design constant satisfying
and be a
(3.3) Now, let
,
denote the following sets:
if (2.2) We let denote the smallest value of for which (2.2) is satisbe defined as follows: if , and fied. Let also is an arbitrarily small positive constant if . Finally, let be defined as
The next theorem [2] states that the existence of an RCLF for system (2.1) is a necessary and sufficient condition for robust stabilization of system (2.1). Theorem 1: If system (2.1) is RAS or RPS or RS via a locally , then there is a smooth RCLF Lipschitz control law for system (2.1). On the other hand, if there exists a RCLF for , then (2.1) system (2.1), then (2.1) is RS. If furthermore is RPS. The proof of Theorem 1 can be found in [2].
Obviously, following Lemma holds. Lemma 1: There exists a scalar compact set ) such that for all
for all
. The
(that depends on the the following holds:
Proof: Since and are continuous, we have that is continuous, too. Let be the set . Also, for each let be the largest for all number such that and let be defined as
From the continuity of and the fact that is compact we (a continuous function is uniformly continuous have that on a compact set, [18, Th. 4.19]). Then, is given by
III. THE ADAPTIVE CONTROLLER Our first assumption for system (2.1) is the following. A1) System (2.1) is RS (or RPS). Since system (2.1) is RS or RPS, from Theorem 1 we have that there exists a smooth RCLF for system (2.1). The time-derivative of this RCLF is given by
where
(3.1) can be written as Let us assume that the function linear combination of known functions and unknown constant parameters. That is, we assume the following. satisfies A2) The function
where denotes the boundary of . From the above definitions is continuous, it is obvious that and the fact that and that implies that , which concludes the proof. We are now ready to present the proposed controller. The control input is chosen as follows:
(3.2)
(3.4)
is a constant unknown matrix and the where is a known nonlinear vector regressor vector function. In Section IV, we give examples of classes of systems that satisfy Assumption A2). Let us define the sets
where • (3.5) • (3.6)
KOSMATOPOULOS AND IOANNOU: ROBUST SWITCHING ADAPTIVE CONTROL OF MULTI-INPUT NONLINEAR SYSTEMS
•
denotes the estimate of the -th entry , generated as vector
of the
Note that (3.14)
(3.7) is the estimate of [ denotes the th where column of the matrix defined in (3.2)]; is a design function chosen to satisfy •
Moreover, define
The following statements hold: a)
where
(3.8) •
is a continuous-switching signal which is used to conto control , and vice tinuously switch from control versa if if if
(3.15) b)
(3.9) Proof: The proof can be found in Appendix I. Let
are updated using the following The parameter estimates desmooth projection update law [17] (here, note the parameter estimation errors)
(3.10) is a symmetric positive–definite design matrix and where is defined as [17] (3.11), shown at the bottom of the page, where
where
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positive design constant. The variable is a hysteresis-switching signal to be defined is a discontinuous explicitly later in this section. Finally, function defined as follows:
(3.16) From Lemma A.1 part a) (see Appendix I), we have that, for , and therefore the term can be made arbitrarily close to 1 by increasing the design constant . Therefore, from Lemma 2 we have that for
is
if otherwise
(3.12)
is a positive design constant. From the definition of where , we have that the adaptive law (3.10) becomes inactive is smaller than . The role of will be made clear when during the proof of the main result of this paper. Let us analyze the proposed control law. More precisely, consider the following Lemma. Lemma 2: Consider the variables (3.13)
(3.17) (note that ). Using now the facts that is close is bounded from above by we can see to one and and from the above inequality that both control laws have approximately the same effect on the term . More precisely, the both have the effect of replacing the unknown term by the negative definite term , plus a term that . depends on the estimation error was absent in the right-hand side If the term (RHS) of (3.17) then, using standard arguments in adaptive control, we could show that the use of the parameter projection law takes care of the effect of the estimation error (3.10) with where
if if
and (3.11)
otherwise
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term . However, although the term has , it may have destabilizing efsmaller magnitude than fects to the closed-loop system due to the fact that it depends on the unknown function . In order to overcome the problem where the term may have a destabilizing effect, we proceed as follows: As it will be shown in the proof of Theorem 2, the following term appears in the RHS of the time-derivative of an appropriately defined Lyapunov-like function
The term appears in the Lyapunov equation due to the projection law (3.10). Define (3.18) is a continuous funcwhere . Ob-
and
where tion, satisfying
and for all . , are continuous functions for all , where is a convex set which is defined explicitly in Theorem 2 later in this section. Therefore, using standard arguments from the theory of approximation of nonlinear functions (see, e.g., [7], and the references therein) , there exist1 two convex sets we can show that for any , two constant vectors , and such that a nonlinear function2 viously
(3.19) and (3.20) It is no loss of generality to assume that the entries of , as well as the entries of all vectors or are3 nonnegative. Also, since we assume and that for all such that (again this is true provided we choose that appropriately; for instance the entries of could be chosen so ). Using (3.19) and (3.20) their magnitude is bounded by we obtain and the fact that
and
Using (3.20) and (3.21), we have that for
(3.22) denotes the estimate of and denotes the parameter estimation error. The first two terms in the RHS of inequality (3.22) can be taken care of by using adaptive update laws. Thus we have to choose in such a way that the rest three terms remain bounded by a small design constant. We will use the following hysteresis-switching adaptive scheme for : where
if otherwise (3.23) , , where . The problem now is in the design of the parameter estimation . There are two issues regarding the laws for design of such estimation laws. The first is the issue of keeping the parameter estimates bounded; such a problem can be easily addressed using projection laws similar to (3.10) and (3.11). The second issue has to do with the problem when, is very close to zero at certain time-instants, is large positive. In such a case, the term while may not be bounded by a small design constant. To round this problem, we proceed as follows. First, observe , , we have that4 that from the definitions of where
(3.24) and
Therefore, using the properties , we have that
(3.25) where
is the convex set defined as
(3.21) where
!
!1
that, in general, " 0 as n . notational simplicity we assume that both approximators have the same regressor terms. 3Such an assumption holds if we augment the regressor vector so that it contains both the entries ( ) and ( ). 1Note 2For
1
0
1
(3.26) 4The
integer j ? k is defined as j ? k =
n
(k
0 1) + j .
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where is a large positive constant such that . The first two inequalities define the constraint that the entries of are positive, the third one defines the constraint that and the last inequality defines the constraint is bounded from above by . that Relation (3.25) motivates us to propose the following adap: tive laws for (3.27) where keeps
,
is a positive constant, , and is the projection law which and is defined as follows:
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the existence and uniqueness of solutions in the sense of Caratheodory [16] (the proof of [16] can be easily revised to include the case where the projection law is inactive when ). Therefore, using similar arguments as those in [15] (see also [24, Lemma A.1]) we can establish that there of maximal length on which the exists an interval hysteresis-switching closed-loop system possesses a unique and, moreover, solution with ’s piecewise constant on can that on each strictly proper subinterval , are switch at most finite times. Note also that bounded due to the projection update laws (see Lemma A.1, part (c) in Appendix I). denote the smallest positive constant satisfying Now, let
Also, let
denote the set
(3.28) if or and and , otherwise. Similarly, if or and and , otherwise. We are now ready to prove closed-loop stability under the the control law (3.4)–(3.11), (3.23), and (3.27). Theorem 2: Consider the unknown system (2.1) and the control law (3.4)–(3.11), (3.23) and (3.27). Assume that A1) and are vector-fields, and the disturbance A2) hold, that , is continuous and bounded. Moreover, assume that the following hold. satisfies (3.8) and , where is defined C1) in Lemma 1. where C2) is large enough so that
where
where is a positive constant. From the above definitions, we have that
Since , we have that there exists a positive real such that for all . Let us analyze the . Define the Lyapunov-like closed-loop system for function
In Appendix I, we show that for is large enough so that and where is a positive design constant. and for any positive conThen, for any compact set the following holds: there exist positive constant , , (dependent on ) and such that, for any stants , the control law (3.4)–(3.11) with initial state guarantees that all the closed-loop signals are bounded and converges to the residual set
(3.29)
Moreover,
where the constant satisfies where when . is continuous and bounded, is Proof: Since smooth with respect to its arguments [see Lemma A.1, part is a continuous signal, we have that the source of (c)], discontinuities in the closed-loop system dynamics are the and the parameter projection hysteresis-switching variables laws (3.10) and (3.27). Parameter projection laws guarantee
are positive terms that can be made arbitrarily where , (or, equivalently, decreasing ). small by increasing if and Using now the fact that , the definition of and the definition of , it is straightforward to show that, there exist positive constants and a positive–definite function such that for (3.30) provided that set
remains on
where
is the compact
In (3.30), is a positive constant. Such a constant exists, proare large enough. and defined in vided that Theorem 2 are related as follows: The variable is a positive . For the time being let us assume constant such that . Using now the that is the smallest constant for which
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definition of the function , part (c) of Lemma A.1, and the fact is decreasing as long as does not enter , we that remains in the interior of for all can see that provided that
(3.31) and independent of we have that Given any such that the above inequality is valid for all there exists (note that the term depends on the dimension of the nonlinear approximators and thus it may be increasing increases; note also that may as becomes larger, i.e., as by increasing ). become arbitrarily close to remains in the interior of for all , Since all the way to . On the other we have that we can extend is decreasing outside together with hand, the fact that due to the projection update the boundedness of . Finally, since outside laws implies that for all implies that enters in finite time . converges to , that is, there is a time We will show that such that . Let us define instant , i.e., . While is , the parameter adjustment laws become inactive, which in is constant while . in turn implies that the term enters at time-instant Consider the following situation: , enters at , exits at and exits at . Since outside , we have that will enter again at time-inat , where . When stant and the parameter adjustment laws are inactive, and thus . Moreover, from the definition of we have . that is decreasing for and . Finally, . This is since: Using the above, we can see that and (b) is decreasing at and (a) and thus is decreasing in . Thereat the time enters , is strictly smaller than its fore, entered previously. It can be seen that, by value when appropriately defining , the following also holds: . If the above claim is not true, we can always increase the value of to make5 it true. Therefore, each time enters the term is smaller than the one during the previous visit which in turn, implies that eventually stays in of in foreover. This claim can be proven by contradiction: if would not stay in then we would have that converges to converges to , and zero, which in turn implies that would be—at the limit—decreasing outside . thus,
2( ) 2( ) 2
5This can be done as follows: suppose that t t which in turn implies that t increased at t t ;t by an amount that is larger than t t . In other words, we have the case where has changed more during the trip of x t from the boundary of to the boundary of than it has changed while x t was outside . In that case, we can increase c (i.e., increase the size of ) so that x t never left . Note that the amount by which c has to be increased should be very small.
2( ) 2( ) 0 2( ) () ()
()
2[
)
Remark 1: As it is seen from (3.31) the reason we choose is to make , and thus , independent of the diof the nonlinear approximators. mension Remark 2: In the case where does not satisfy the linearin-the-parameters assumption A2), the proposed approach is can be still applicable, by assuming that the function approximated by linear-in-the-weights nonlinear functions. In [10], we show how we can apply the proposed strategy in the does not satisfy assumption A2) or, for the case where case where the system dynamics are completely unknown. IV. APPLICATION TO VARIOUS CLASSES OF SYSTEMS In this section, we present some examples of classes of nonlinear systems which satisfy assumption A2). For simplicity, we . The consider the case where the external disturbance results can be easily extended to the case where external input disturbances are present. A. Multi-Input Feedback-Linearizable Systems One class of systems that satisfy assumption A2) is the class of multi-input feedback-linearizable systems. For this class there exists no general methodology for designing adaptive controllers that guarantee global stability for the closed-loop system. Let us consider the class of multi-input feedback linearizable systems whose dynamics can be described as follows: (4.1) are unknown constant matrices, are where known nonlinear continuous vector functions and, moreover, the is stabilizable and the functions are matrix pair and is nonsingular for all . such that . Then, a Let be a stabilizing gain matrix for the pair where RCLF for system (4.1) is the function [2] is the symmetric positive–definite solution of the Lyapunov equation (4.2) is a symmetric positive–definite matrix. Then, the where and are given by functions (4.3) and (4.4) It is not difficult to see that if is a vector whose entries are the and , is a vector whose entries are the elements , is a vector whose entries are the elements of elements and and is a matrix whose entries are the elements , then, and can be written in the form (3.2). In of other words, systems of the form (4.1) satisfy assumption A2). In a similar way, we can show that the feedback linearizable systems of the form (4.5) are unknown satisfy assumption A2). Here, are known nonlinear continuous constant matrices,
KOSMATOPOULOS AND IOANNOU: ROBUST SWITCHING ADAPTIVE CONTROL OF MULTI-INPUT NONLINEAR SYSTEMS
vector functions and, moreover, the unknown matrix pair is stabilizable and the continuous functions are such that and is nonsingular for . For the case of the system (4.5) the functions all are given as follows:
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B. Linear-in-the-Parameters Nonlinear Systems With Linear-in-the-Parameters RCLF Consider now the class of nonlinear systems of the form (2.1) whose vector-fields and are linear combinations of known functions and unknown constant parameters, i.e.,
(4.6) and
and (4.7)
is defined in (4.2). Next we show how to chose where the controller parameters: First, let us examine the equality is valid. Since we have that which, due to the nonsingularity of , implies that and implies . On the therefore other hand, from the Lyapunov equality (4.2) it can be easily implies that seen that , and therefore, implies that . we conclude that where is defined in Definition 3 Thus, we have that can be any arbitrarily small constant. Once and therefore is chosen then can be chosen as follows: First note that is nonsingular, we have that since for some positive number . Assuming that a lower bound of is known we have that implies (4.8)
where
are known nonlinear vector functions and are unknown constant matrices (vectors). Moreover, assume that an RCLF for the system can be written as a linear combination of known functions and unknown constant parameters, i.e.,
where is a known nonlinear vector function and is an unknown constant vector. Then, it can be easily seen that the above system satisfies assumption A2) where the entries of are the elements of , the entries of are the elements , the entries of are the elements of , of are the elements of . and the entries of C. Parametric-Pure-Feedback Systems Let us now try to apply the results of Section III to nonlinear systems that take the form
and By adding and subtracting the terms in the RHS of (4.6) and using (4.2) we obtain after some algebraic manipulations (4.9) Using (4.8), we have that
implies
where is a known compact set satisfying . From the above inequality, we have that could be set equal to the largest that satisfies . Note that an upper is needed in order to solve the above problem. bound on can be then set equal to where The constant is the minimum value of that satisfies , where is defined in the proof of Theorem 2. can be chosen Finally, the proposed controller function as follows:
where
are positive constants satisfying . It is worth noticing that the existing adaptive control designs guarantee global stability only in the case where the matrix for the case of system (4.1) or the matrices and for the case of system (4.5) are known.
, are smooth known functions and where is the vector of constant but unknown system parameters. Let us rewrite (4.9) as
(4.10) . Systems of the form (4.10) are where called parametric-pure-feedback (PPF) systems [6], [12], [20]. Although the problem of constructing globally stable adaptive controllers for the simplest case of parametric-strict-feedback (PSF) systems 6 has been completely solved [6], [12], [20], the problem of constructing globally stable adaptive controllers for PPF systems remains an open problem and is the subject of this subsection. Although the results developed in Sections I–III are for state stabilization, for the particular case of PPF systems, they can be 6The class of PSF systems refers to the subclass of PPF systems (4.10) which satisfy f (z ; . . . ; z ) = 0; g (z ; . . . ; z ) = 0.
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readily extended to the case of asymptotic tracking as well. The control objective is to force the system output to asymptotically track a reference signal . We assume that the first time derivatives of are known. Also it is assumed that as time derivatives are bounded and smooth well as its first signals. Before we design the feedback law, we will transform the system (4.9) into a suitable form. The procedure we follow is that of [6] and [12], and it is based on the backstepping integrator principle [22]. . Let also be positive Step 0): Let constants to be chosen later. Step 1): Using the “chain of integrators” method, we see that, if was the control input in the -part of (4.10) and was known, then the “control law” (4.11) would result in a globally asymptotically stable tracking, since such a control law would transform the -part of (4.10) as follows
Step 3): Using the above definitions of that after some algebraic manipulations
(4.17) is a vector that consists of all elements that are where or of the form or of the form either of the form . In the system (4.17), we will think of as our control input. Therefore, as in Step 2), we define the new state as
(4.18) Substituting (4.18) into (4.17) yields (4.19) : Using the definitions of Step and working as in the previous steps we may express the derivative of as
However, the state is not the control. Therefore, we deto be the difference between the actual and its fine desired expression (4.11): (4.12) Using the above definition of , the definition of the -part of (4.10), we found that
and
, we have
(4.20) where the vector
contains all the terms of the form with . Defining now as
follows:
(4.21) (4.13) Step 2): Using the above definitions of
we obtain that
, we have that
(4.22) and working Step : Using the definitions of as in the previous steps we may express the derivative of as follows:
(4.23)
(4.14)
where the vector form
contains all the terms of the with , and
is a -dimensional vector that consists of where or of the form all elements that are either of the form where by we denote the -th entry of the vector . In the system (4.14), we will think of as our control input. Therefore, as in Step 1), we define the new state as
is given by
(4.24) (4.15)
, and by rearranging Using the definitions of terms, we may rewrite (4.23) as follows:
(4.16)
(4.25)
Substituting (4.15) into (4.14) yields
KOSMATOPOULOS AND IOANNOU: ROBUST SWITCHING ADAPTIVE CONTROL OF MULTI-INPUT NONLINEAR SYSTEMS
Therefore, using the aforementioned methodology, we have transformed system (4.10) into the following:
.. .
.. .
..
.
..
.
.. .
where and (note that
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are appropriately defined known functions ) Therefore, we have that
.. .
and .. .
(4.26)
or (4.27) where of the previous system as
. Let us define the output
where . Obviously, the above system is feedback-linearizable, if the following assumption holds. for all . A1’) are not available for the Note that the variables control design since they depend on the unknown vector . We will show that, if the constants are chosen so that for all , then the function
From the above two equations, it can be easily seen that the and satisfy assumption A2), by defining the functions vectors and as the vectors whose entries are the elements and defining the functions and appropriately. The proposed controller parameters can be chosen as follows: , the constant can be chosen to since the constant be any arbitrarily small positive number. Working similar to the case of feedback linearizable systems in Section IV-A we can show that can be set equal to the largest that satisfies where is defined as follows
where
is an RCLF for system (4.9) and, moreover, that the resulting and satisfy assumption A2). By differentifunctions ating with respect to time, we obtain that
is a positive constant satisfying and is a known compact set satisfying . must be chosen so that and The constant where is defined in the proof of Theorem 2 . , one possible way Regarding the design of the function as follows: is to choose
(4.28) and, therefore (4.29) and (4.30) From the assumption A1’), it can be easily seen that . The quantity is for all , and thus, is negative definite provided that . Note now that an RCLF for system (4.9) and moreover using the definitions of we can rewrite ’s as follows:
where is a positive constant satisfying . Remark 3: Theorem 2 is not directly applicable to the class and in (4.29) of PPF systems (4.9), since the terms and (4.30) are explicit functions of . However, Theorem 2 can be easily modified to be applicable to such a case. The only modas well as the regressor ification needed is that the function terms in the adaptive laws should be explicit functions of (thus, for instance the regressor terms of the nonlinear approximators should be replaced by . If we carry out the same analysis as in the proof of Theorem 2, by incorporating the above modifications, it can be seen that results of Theorem 2 are applicable to the class of systems (4.9).
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V. CONCLUSION
and therefore, from (3.3) we obtain that, for all
In this paper, we proposed a switching adaptive controller for multi-input nonlinear systems whose dynamics are nonlinearly affected by external input disturbances. The proposed approach is applicable to nonlinear systems for which the function can be written as a linear combination of unknown constants and known nonlinear functions. By making use of the notion of RCLF [2] and a modified version of the switching adaptive controller of [9] we showed that the proposed controller guarantees bounded closed-loop signals and convergence of the state to a residual set. The proposed control approach is used to design stabilizing controllers for multi-input feedback-linearizable systems, PPF systems and linear-in-the-parameters nonlinear systems which admit a linearly parameterized RCLF. APPENDIX Lemma A.1: Assume that the conditions of Theorem 2 hold. Then the following statements are true. we have that a) For all
and
b)
, . is smooth with respect to its arguments. Moreover,
c)
b) c) d) e)
and thus, is bounded from above by for all . Using now the fact that is bounded from above for all and (3.16), it is straightforward to by verify the second inequality of part a). The proof is straightforward. The proofs are similar to those of [17] and [4, Th. 4.4.1]. The proof is straightforward. Since from part c) of this Lemma , we have that and with such therefore, there exists . Therefore that
We have that (since and the design assumption ), which concludes the proof. Proof of Lemma 2: a) From (3.5), we have that
and
where is a finite positive constant, independent of Similarly, for
.
Therefore d) e)
and Proof: we have that . Therea) Since fore, by taking into account (3.3) we conclude that for , i.e., the denominator of never becomes all . Since , we have that zero provided that
for all
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(A.1) b) Using (3.6), we obtain
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(A.2)
Proof of (3.29): Differentiating with respect to time and using (3.4)–(3.11) and (3.18)–(3.27), Lemma 2, Lemma we obtain that A.1, and the fact that , the equation at the bottom of the for page holds true,
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where
From (3.3), it can be seen that for any such that . Therefore
there exists an
and, thus, we finally obtain that
(A.3)
REFERENCES [1] G. Cambion and G. Bastin, “Indirect adaptive state feedback control of linearly parameterized nonlinear systems,” Int. J. Adapt. Control Signal Processing, vol. 4, pp. 345–358, Sept. 1990. [2] R. A. Freeman and P. V. Kokotovic, “Inverse optimality in robust stabilization,” SIAM J. Control Optim., vol. 34, no. 4, pp. 1365–1391, July 1996. [3] H. Hespana and A. S. Morse, “Supervision of families of nonlinear controllers,” in Proc. 35th IEEE Conf. Decision Control, vol. 4, Dec. 1996, pp. 3771–3773.
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[4] P. A. Ioannou and J. Sun, Stable and Robust Adaptive Control. Upper Saddle River, NJ: Prentice-Hall, 1996. [5] I. Kanellakopoulos, P. V. Kokotovic, and R. Marino, “An extended direct scheme for robust adaptive nonlinear control,” Automatica, vol. 27, pp. 247–255, Mar. 1991. [6] I. Kanellakopoulos, P. V. Kokotovic, and A. S. Morse, “Systematic design of adaptive controllers for feedback linearizable systems,” IEEE Trans. Automat. Contr., vol. 36, pp. 1241–1253, November 1991. [7] E. B. Kosmatopoulos and M. A. Christodoulou, “Techniques and applications of recurrent high order neural networks in the identification of dynamical systems,” in Neural Network Systems Techniques and Applications, C. Leondes, Ed. New York: Academic, 1999. [8] E. B. Kosmatopoulos, “Universal stabilization using control Lyapunov functions, adaptive derivative feedback and neural network approximators,” IEEE Trans. Syst., Man, Cybern. B, vol. 28B, pp. 472–477, June 1998. [9] E. B. Kosmatopoulos and P. A. Ioannou, “A switching adaptive controller for feedback linearizable systems,” IEEE Trans. Automat. Contr., vol. 44, pp. 742–750, Apr. 1999. [10] E. B. Kosmatopoulos, “Robust neural stabilizers for unknown systems,” IEEE Trans. Neural Networks, submitted for publication. [11] M. Krstic and P. V. Kokotovic, “Adaptive nonlinear design with controller-identifier separation and swapping,” IEEE Trans. Automat. Contr., vol. 40, pp. 426–440, Mar. 1995. [12] M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic, Nonlinear and Adaptive Control Design. New York: Wiley, 1995. [13] Y. Lin and E. D. Sontag, “Control Lyapunov universal formulae for restricted inputs,” Control: Theory Adv. Technol., vol. 10, pp. 1981–2004, 1995. [14] Y. Lin, E. D. Sontag, and Y. Wang, “A smooth converse Lyapunov theorem for robust stability,” SIAM J. Control Optim., vol. 34, pp. 124–160, 1996. [15] A. S. Morse, D. Q. Mayne, and G. C. Goodwin, “Applications of hysteresis switching in parameter adaptive control,” IEEE Trans. Automat. Contr., vol. 34, pp. 1343–1354, Sept. 1992. [16] M. M. Polycarpou and P. A. Ioannou, “On the existence and uniqueness of solutions in adaptive control systems,” IEEE Trans. Automat. Contr., vol. 38, pp. 474–480, Mar. 1993. [17] J.-B. Pomet and L. Praly, “Adaptive nonlinear regulation: Estimation from the Lyapunov equation,” IEEE Trans. Automat. Contr., vol. 37, pp. 729–740, 1992. [18] W. Rudin, Principles of Mathematical Analysis. New York: McGrawHill, 1984. [19] S. S. Sastry and A. Isidori, “Adaptive control of linearizable systems,” IEEE Trans. Automat. Contr., vol. 34, pp. 405–412, Apr. 1989. [20] D. Seto, A. M. Annaswamy, and J. Baillieul, “Adaptive control of nonlinear systems with a triangular structure,” IEEE Trans. Automat. Contr., vol. 39, pp. 1411–1428, July 1994. [21] E. D. Sontag, “A universal construction of Arstein’s theorem on nonlinear stabilization,” Syst. Control Lett., vol. 13, no. 2, pp. 117–123, 1989. [22] J. Tsinias, “Sufficient Lyapunov-like conditions for stabilization,” Math. Control, Signals, Syst., vol. 2, no. 4, pp. 343–357, 1989. [23] B. Yao and M. Tomizuka, “Adaptive robust control of a class of multivariable nonlinear systems,” IFAC World Congress, vol. F, pp. 335–340, 1996. [24] S. R. Weller and G. C. Goodwin, “Hysteresis switching adaptive control of linear multivariable systems,” IEEE Trans. Automat. Contr., vol. 39, pp. 1360–1375, July 1994.
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Elias B. Kosmatopoulos received the Diploma in production and management engineering, and the M.Sc. and Ph.D. degrees in electronics and computer engineering, all from the Technical University of Crete (TUC), Greece, in 1990, 1993, and 1995, respectively. He is currently a tenure-track Lecturer in the Department of Production and Management Engineering, and Deputy Director of the Dynamic Systems and Simulation Laboratory, at TUC. Prior to joining TUC, he was a Research Assistant Professor with the Department of Electrical Engineering-Systems, University of Southern California (USC), Los Angeles, and a Postdoctoral Fellow with the Department of Electrical and Computer Engineering, University of Victoria, Vistoria, BC, Canada. He is the author or coauthor of more than 20 journal articles and book chapters, and more than 40 conference publications in the areas of neural networks, adaptive and neural control, fuzzy systems, and intelligent transportation systems. He has been involved in various research projects funded by the European Community, the National Sciences of Engineering Research Council (NSERC) of Canada, the National Aeromautics and Space Administration (NASA), the Air Force, and the Department of Transportation involving virtual reality, fault detection and identification, manufacturing systems, robotics, fuzzy controllers, telecommunications, design and control of flexible and space structures, active isolation techniques for civil structures, control of hypersonic vehicles, automated highway systems, agile port technologies, and intelligent transportation systems. He has served as a reviewer for various journals and conferences, and has served as the session chairman or cochairman for various international conferences.
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Petros A. Ioannou (S’80–M’83–SM’89–F’94) received the B.Sc. degree (with First Class Honors) from University College, London, U.K., and the M.S. and Ph.D. degrees from the University of Illinois, Urbana, in 1978, 1980, and 1982, respectively. From 1975 to 1978, he held a Commonwealth Scolarship from the Association of Commonwealth Universities, London, U.K. From 1979 to 1982, he was a Research Assistant at the Coordinated Science Laboratory at the University of Illinois. In 1982, he joined the Department of Electrical Engineering-Systems, University of Southern California, Los Angeles, where he is currently a Professor in the same Department and the Director of the Center of Advanced Transportation Technologies. His research interests are in the areas of adaptive control, neural networks, vehicle dynamics, and control and intelligent vehicle and highway systems. He has been an Associate Editor for The International Journal of Control and Automatica. He has published five books and over 100 technical papers. He is a member of the AVCS Committee of Intelligent Transportation Systems (ITS) America, and a Control Systems Society member of the IEEE Technical Activities Board Committee on Intelligent Transportation Systems. Dr. Ioannou was awarded several prizes, including the Goldsmid Prize and the A.P. Head prize from University College, London, U.K. In 1984, he was a recipient of the Outstanding Transactions Paper Award for his paper “An Asymptotic Error Analysis of Identifiers and Adaptive Observers in the Presence of Parasitics, ” which appeared in the IEEE TRANSACTIONS ON AUTOMATIC CONTROL in August 1982. He was also the recipient of a 1985 Presidential Young Investigator Award for his research in Adaptive Control. He has been an Associate Editor for the IEEE TRANSACTIONS ON AUTOMATIC CONTROL
