Decentralized Adaptive Control of Nonlinear Systems Using Radial Basis Neural Networks

Transcription

1 050 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO., NOVEMBER 999 Decentralized Adaptive Control of Nonlinear Systes Using Radial Basis Neural Networks Jeffrey T. Spooner and Kevin M. Passino Abstract Stable direct and indirect decentralized adaptive radial basis neural network controllers are presented for a class of interconnected nonlinear systes. The feedback and adaptation echaniss for each subsyste depend only upon local easureents to provide asyptotic tracking of a reference trajectory. Due to the functional approxiation capabilities of radial basis neural networks, the dynaics for each subsyste are not required to be linear in a set of unknown coefficients as is typically required in decentralized adaptive schees. In addition, each subsyste is able to adaptively copensate for disturbances and interconnections with unknown bounds. I. INTRODUCTION Decentralized control systes often arise fro either the physical inability for subsyste inforation exchange or the lack of coputing capabilities required for a single central controller. Furtherore, the difficulty of, and uncertainty in, easuring paraeter values within a large-scale syste ay call for adaptive techniques. Since these restrictions encopass a large group of applications, a variety of decentralized adaptive techniques have been developed. Model reference adaptive control (MRAC) based designs for decentralized systes have been studied in [] [3] for the continuous tie case and in [4] for the discrete tie case. These approaches, however, are liited to decentralized systes with linear subsystes and possibly nonlinear interconnections. Decentralized adaptive controllers for robotic anipulators were presented in [5] and [6], while a schee for nonlinear subsystes with a special class of interconnections was presented in [7]. It was shown in [8] that it is possible to provide stable tracking in decentralized systes which contain uncertainties which are bounded by polynoials with known order. These previous results consider subsystes which are linear in a set of unknown paraeters, or consider the uncertainties to be contained within the dynaics describing the subsyste interconnections which are bounded. On-line function approxiation approaches have been successfully applied to a wide variety of control probles, in particular in the area of nonlinear adaptive control of SISO systes (see [9] [3] for exaples). On-line function approxiation approaches adjust paraeters within a universal approxiator (such as a fuzzy syste or neural network) to estiate unknown nonlinearities which ay describe plant dynaics or a desired control law. Universal approxiators possess the property that, given an appropriate approxiator structure, it is possible to represent continuous nonlinearities over a copact space with arbitrary accuracy as described in [4] and [5]. This eans that entire classes of nonlinearities ay be represented with a single approxiator structure using different paraeter choices. In this correspondence, we exploit the function approxiation capabilities of radial basis functions to provide asyptotic tracking given a class of nonlinear subsystes with unknown interconnection Manuscript received July 5, 997; revised August 7, 998. Recoended by Guest Editors, M. Leon and A. Michel. J. T. Spooner is with the Control Subsystes Departent, Sandia National Laboratories, Albuquerque, NM USA (e-ail: jtspoon@sandia.gov). K. M. Passino is with the Departent of Electrical Engineering, The Ohio State University, Colubus, OH USA (e-ail: passino@ee.eng.ohio-state.edu). Publisher Ite Identifier S (99)08575-X. strengths. Using radial basis neural networks to approxiate unknown functions on-line allows us to extend the results in [6] to include the case of both paraetric and dynaic uncertainty. A direct adaptive approach approxiates unknown control laws required to stabilize each subsyste, while an indirect approach is provided which identifies the isolated subsyste dynaics to produce a stabilizing controller. Both approaches ensure asyptotic tracking using only local feedback signals. This correspondence is organized as follows. In Section II, an overview of radial basis neural networks is given. In Section III, the details of the proble stateent for the decentralized syste are presented. The adaptive algoriths for each subsyste using only local inforation are presented and coposite syste stability is established in Sections IV and V for the direct and indirect cases, respectively. Siulation exaples are given in Section VI, while concluding rearks are provided in Section VII. II. RADIAL BASIS NEURAL NETWORKS A radial basis neural network (RBNN) is ade up of a collection of parallel processing units called nodes. The output of the ith node is defined by a Gaussian function z i (x) =exp(0jx 0 c i j = i ), where x n is the input to the network, c i is the center of the ith node, and i is its size of influence. The output of a radial basis network, y = F(x; A), ay siply be calculated by either a weighted su so that F(x; A) = a i z i (x) () or by a weighted average a iz i(x) F(x; A) = p p z i (x) where A = [a; ;a p ] > is a vector of network weights. We notice that () and () ay be rewritten as F(x; A) = A > (X), where (X) is a set of radial basis functions defined by > (x) = [z(x); ;z p (x)] for the weighted su, (), and > p (x) =[z(x); ;z p(x)]= zi(x); for the weighted average (). Given a single RBNN, it is possible to approxiate a wide variety of functions siply by aking different choices for A. In particular, if there are a sufficient nuber of nodes within the network, then there exists soe A 3 such that sup xs jf (x; A 3 ) 0 f (x)j <W where S x is a copact set, and W> 0 is a finite constant provided f (x) is continuous [7]. This lets us express f (x) =F(x; A 3 )+ w(x) with jw(x)j < W when x S x. Notice that even though RBNN s are linear in a set of adjustable paraeters, we ay, e.g., approxiate a function f (x) =a + cos(bx > x) which is not linear in an independent set of paraeters [a; b] >. Thus we are using an approxiator which is linear in the paraeters to describe functions which are not necessarily linear in another set of paraeters. Even though we will be defining the control laws in ters of radial basis networks, it should be noted that any universal approxiator which is linear in the adjustable paraeters ay be considered. Other exaples are standard fuzzy systes with adjustable output () /99$ IEEE

2 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO., NOVEMBER ebership centers [9], Takagi Sugeno fuzzy systes [3], CMAC networks, aong others. III. A CLASS OF DECENTRALIZED SYSTEMS Here we consider each subsyste S i to be SISO such that _x i = f i (x ; ;x )+g u i (3) y i = h i (x ; ;x ) where x i n is the state vector, u i is the input, and y i is the output of S i. We assue that the functions f i (); g i () n and h i(), i =; ;are sooth. If each subsyste has strong relative degree d i, then the output dynaics ay be rewritten as y (d ) i = + u i + i (t; x ; ;x ) (4) where y (d ) i is the d i th tie derivative of y i [6]. Here we are assuing that the influence of the other subsystes S j ;j6= i, is represented by the i(t; x ;...;x ) ter in (4). We also assue that for soe 0 > 0, we have 0 so that the control gain is bounded away fro zero (for convenience we assue that > 0). Assuption Plant: The plant can be defined by (3) and transfored to (4) with input gain bounded by 0 < 0. The zero dynaics for each subsyste are exponentially attractive [8]. The ith subsyste input gain rate of change is bounded by j _ j B i where B i is a finite constant defined later in Theore. The tracking error for S i is defined by e i = r i 0 y i. Our objective is to design an adaptive control syste for each subsyste which will cause the output y i of a relative degree d i subsyste S i to track a desired output trajectory r i (i.e., e i! 0) in the presence of interconnections and unknown disturbances, using only local easureents. The desired output trajectory ay be defined by a signal external to the control syste so that the first d i derivatives of the ith subsyste s reference signal r i ay be easured, or by a reference odel, with relative degree greater than or equal to d i which characterizes the desired perforance. This requireent leads to the following assuption. Assuption Reference Model: The desired output trajectory and its derivatives r i ; ;r (d ) i for the ith subsyste S i are easurable and bounded. Let the scalars i; j quantify the strength of the interconnections and the output error vector for the ith subsyste be defined by e i = [e i ; ;e i ] >. It is assued that the interconnections satisfy i(x ; ;x ) = i(t) + i(x ; ;x ), where j i (x ; ;x )j j= i; jje j j and i (t) L. Let ~ 3 i = (sup( i (t)) 0 inf( i (t)))= be a easure of the variation of i (t) and i = (sup[ i(t)] + inf( i(t)))= be a easure of the center position of i (t). We ay thus let i (t) = i + ~ i (t) for soe ~ i (t), where j ~ i j ~ i 3, with ~ i 3 assued to be bounded. Also, if i is nonzero, we ay absorb i into within (4) as an unknown bias since it is not dependent upon the other subsyste states. The assuptions for the interconnections are suarized as follows. Assuption 3 Interconnections: The interconnections satisfy where i (x ; ;x )= i (t)+ i (x ; ;x ) j i (x ; ;x )j j= i; j je j j and i (t) L : This assuption on the interconnections can be satisfied by a variety of decentralized nonlinear systes. For instance, in [6] it is shown to be satisfied for an intervehicle spacing regulation proble in a platoon of an autoated highway syste. In this correspondence, we show that it is satisfied for the control of two inverted pendulus connected by a spring. It should be noted that if the interconnections satisfy j i ()j j= i; jjy j j (which is the case for any echanical systes), then j i()j i; j= j(je j j + jr j j ) where y i = [y i ; ;y (d ) i ] > and r i = [r i ; ;r (d ) i ] > which satisfies Assuption 3 provided each r i is bounded. IV. DIRECT ADAPTIVE CONTROL Because state inforation about the ith subsyste is only available for the ith controller, a standard feedback linearizing control law ay not be defined for the coposite syste, even if the plant dynaics are known. Ideally we ay, however, define a controller which copensates for the dynaics of each isolated subsyste. For the ith isolated subsyste, a feedback linearizing controller is defined by u 3 i (x 0 i(x i )+ i (t) i; i)= = u u (x i; i)+u k (5) where the signal i will be defined below, u u (x i; i) is the unknown portion of the control law that is sooth in its arguents, and u k (x i ) is a known part of the control which is assued to be well defined a priori. The ter u k is included to allow a priori control knowledge into the decentralized controller design. The ideal decentralized control function (5) ay be represented by an RBNN (or other approxiation structure), F u, such that u 3 i = F u x i; i;a 3 u + u k (x i)+w u (x i; i) (6) where the vector of ideal control paraeters is defined as A 3 u = arg in A sup x S ; S jf u (x i;;a u ) 0 u u (x i; i)j so that w u (x i; i) is the representation error which arises when u u (x i ; i ) is represented by an RBNN of finite size. Fro the universal approxiation property, we know that for a given approxiator structure, there exists A 3 u such that jw u jw u for soe finite W u > 0. The subspaces S x and S are defined as the copact sets through which the state trajectories for the ith subsyste and i ay travel. The subspace u is the convex copact set which contains feasible paraeter sets for A 3 u. The stability proof to follow will establish bounds for S x and S. The following assuption suarizes the controller requireents. Assuption 4 Control: If x i L n, then u k L. Also, assue that the representation error w u (x i ; i ) is bounded by soe W u > 0, i.e., jw u (x i; i)j W u. An adaptive algorith will be defined to estiate A 3 u with A u. These estiates are then used to define the control law as (7) u i = F u (x i ; i ;A u )+u k (x i ) (8) where F u (x i ; i ;A u ) is the RBNN used to approxiate an ideal controller for the ith subsyste. A paraeter error vector is defined as u = A u 0 A 3 u for each subsyste. It is desired that the output error of the ith subsyste follow e (d ) i + k i; d 0 e i + + k i; 0 e i = 0, where the coefficients are picked so that each ^L i (s) =s d + k i; d 0 s d k i; 0 has its roots in the open left-half plane (is Hurwitz). The error dynaics for the ith subsyste ay be expressed as e (d ) i = r (d ) i 0 0 u i 0 i (x ; ;x ): Adding and subtracting u 3 i and using the definition of u 3 i in (5), we obtain e (d ) i = r (d ) i 0 [u i 0 u 3 i ] 0 i (t) 0 i (x ; ;x ): (9)

3 05 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO., NOVEMBER 999 Let a (t) and a (t) be scalar tie functions and i = r (d ) i + k i; d 0e i + + k i; 0 e i + a (t) sgn e > i P ib i + a (t)e > i P ib i=: d d where P i is a positive definite atrix defined by a Lyapunov atrix equation and b i d is a vector. These will both be defined shortly. Using the definition of i, the error dynaics ay be expressed as where _e i =3 i e i + b i 0 [u i 0 u 3 i ] 0 a sgn e > i P i b i 3 i = 0 a e > i P i b i = 0 i (x ; ;x ) (0) k i; 0 0k i; 0k i; 0k i; d 0 and b i =[0; 0; ; 0; ] > d. In the analysis to follow, we will use the fact that u i 0 ui 3 = > u u 0 w u ; where > u u (x i ; i ;A u u with jw u jw u. Consider the following update laws: _ A u = u u e > i P ib i () _a = e > i P i b i () _a = e > i P i b i (3) where u > 0; > 0, and > 0; ; ;are adaptation gains. The update law () is used to estiate the dynaics of the subsyste under control, while the update laws () and (3) are used to stabilize the subsyste by estiating the effects of the interconnections. Both () and (3) increase onotonically and we require that a (0); a (0) 0 so a projection algorith ay be required to ensure that they do not becoe unnecessarily large. Theore : Given the decentralized syste with reference odels satisfying Assuption, subsystes satisfying Assuption, interconnections satisfying Assuption 3, and controllers satisfying Assuption 4, then the control law (8) with adaptation laws () (3) will ensure that for i =; ;if B i < ( 0 in(r i)= ax(p i)) then: ) the subsyste outputs and their derivatives, y i ; ;y i, are bounded; ) each control signal is bounded, i.e., u i + u k L ; 3) the agnitude of each output error, je i j, decreases asyptotically to zero, i.e., li t! je ij =0; 4) li t! j A _ u j =0; li t! j _a j =0, and li t! j _a j = 0; where R i is defined below. Proof: Consider the following Lyapunov-type function for the ith subsyste: v i = e> i P i e i + > u u + + (4) u where = a 0 3 i ; = a 0 3 i, and each P i d d is a positive definite and syetric atrix ( 3 i and 3 i will be defined shortly). Taking the tie derivative of v i yields _v i = e> i P i3 i +3 > i P i e i 0 e> i P i e i _ (x i ) i (x i) 0 e> i P ib i i(x ; ;x ) + > u _ u + e> i P i b i u 0 [u i 0 u 3 i ] 0 a sgn e > i P i b i 0 a e > i P i b i = + _ + _ : (5) Since each 3 i is negative definite, given soe positive definite R i, there exits a unique syetric positive definite P i satisfying P i 3 i +3 i > P i = 0R i, a Lyapunov atrix equation. Since _ u = A _ u ; _ =_a, and _ =_a, using the definition of the adaptive laws (5) ay be written as _v i = 0e> i R i e i 0 e> i P i e i _ (x i ) i (x i) 0 e> i P i b i i(x ; ;x ) + e > i P ib i + e> i P i b i w u 0 a sgn e > i P i b i 0 a e > i P i b i = + e > i P i b i : (6) Since = =, and a 0; a 0 we have _v i 0e> i R i e i 0 e> i P i b i i(x; ;x) 0 e> i P i e _ i (x i) +e > i P i b i w u 0 3 i e > i P i b i 0 3 i e > i P i b i : (7) If we choose 3 i = ~ 3 i = 0 + W u, then the inequality 0 e> i P i b i holds. Substituting (8) in (7) yields _v i 0e> i R ie i ~ i(t)+e > i P ib iw u 3 i e > i P ib i (8) 0 3 i e > i P i b i 0 e > i P ib i i(x ; ;x ) 0 e> i P i e _ i (x : (9) i) It is possible to set i 3 = 0 and use M-atrix techniques to deterine sufficient conditions for syste stability [9]. Due to the conservativeness of the M-atrix techniques, however, the resulting coposite syste stability results are very restrictive for systes with relative degree d i >. Here, we coplete the squares (in an analogous anner to []) to obtain _v i 0e> i R i e i + i (x ; ;x ) 3 i i (x i) + ax(p i )B i e > i e i i (x i) (0) where each i 3 > 0 and ax(p i) denotes the axiu eigenvalue of P i. Now consider the coposite syste Lyapunov candidate V = civi, where each ci > 0. Taking the derivative of V and using (0) and Assuption 3 gives _V c i 0 e> i R i e i + 3 i i (x i) + ax(p i )B i e > i e i i (x i) j= i;j je j j : ()

4 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO., NOVEMBER Since j= i; jje j j = > 0 i, where =[je j ; ; je j ] > and 0 i =[ i; ; ; i; ] >, () ay be written as _V c i + 0 in (R i )+ ax(pi)bi je ij 3 i i (x i) > 0 i0 > i () where in (R i ) is the real part of the eigenvalue of R i with the iniu agnitude. Require that B i < 0 in(r i)= ax(p i) so that 0 in (R i )+ ax (P i )B i = 0 < 0 i, where each i > 0 is soe finite constant. This gives us _V c i 0 i je i j + 3 i i (x i) > 0 i 0 > i : (3) Define K 3 = [ 3 ; ;]. 3 Let i 3 = 3 ;i = ; ;for soe 0 < 3, define D =diagfc = ; ;c = and M = c i 0 i 0 i > =0, so that V _ 0 > A, where A = D 0 (= 3 )M. Then for soe sufficiently large 3 > 0, the atrix A is positive definite. This diagonal doinance property ay be established using Gershgorin s Theore [0]. Now define K 3 = [ 3 ; ; 3 ] > as K 3 = arg in K 0< K 3 K 3 : A = D 0 3 M is positive definite : g (4) There exists sufficiently large 3 such that A, defined by (4), is positive definite, which iplies that V L, and thus jj L. Given bounded reference signals, Part is established. With exponentially attractive zero dynaics, the states for each subsyste are bounded. Boundedness of the Lyapunov function thus ensures that u i + u k L so Part holds. Also > A dt 0 _V dt+const (5) 0 so that jj L. Since all of the signals are well defined, we also have _e i L d so that d=dtje i j = e i > _e i =je i j j_ej L. Using Barbalat s Lea, we thus establish that li t! jj =0, thus we are guaranteed asyptotically stable tracking for each of the subsystes so Part 3 holds. In addition, since each of the plant and control signals is bounded and li t! je i j =0, convergence of the update law derivatives to zero is established by their definitions. Reark : The stability results are seiglobal in the sense that they hold for x i S x where S x is dependent upon the span of the RBNN. In the case that the RBNN s ay approxiate the feedback linearizing controller (5) globally, S x ay be defined as n and S as (rather than copact sets) so the results hold globally. Reark : The bound for the representation error, W u, does not need to be known for our choice of adaptation laws (all we needed to know earlier was that it existed and we are guaranteed this). In addition, the agnitude of the interconnections are estiated on-line to produce stable tracking. Reark 3: The direct adaptive control schee presented here does require that B i < 0 in (R i )= ax (P i ). That is, the rate at which the control gain changes ay influence the design of the update and control laws through the choice of P i given soe 3 i. For subsystes with a constant, then this requireent is always satisfied since B i = 0 is a valid choice. As the rate of change of the control gain increases, however, this bound becoes ore restrictive for the control design. 0 V. INDIRECT ADAPTIVE CONTROL The direct adaptive decentralized control law was defined using an RBNN with adjustable paraeters to approxiate u 3 i. For the indirect case, however, an identifier will be used to approxiate the isolated syste dynaics so that a feedback linearizing controller ay be defined based on the certainty equivalence principle. We will first represent the isolated syste dynaics (4) as = u + k (x i) and = u + k (x i ), where u and u represent the unknown dynaics, while k and k represent the known dynaics (which can be equal to zero and all the results to follow still hold). Representing the unknown dynaics with an RBNN, we see =F x i ;A 3 + k + w (6) =F x i;a 3 + k + w : (7) The paraeters for F (x i ;A 3 ) are defined by A 3 = arg in A sup x S jf (x i;a ) 0 u j (8) while the paraeters for F (x i;a 3 ) are defined by A 3 = arg in A sup x S jf (x i ;A ) 0 u j (9) where the representation errors for u and u are defined by w and w, respectively. The bounds on the representation errors are given by jw jw and jw jw for soe W > 0 and W > 0. The subspace S x is a copact set through which the state trajectory for the ith subsyste ay travel. The subspaces and are copact convex sets which contain the feasible paraeter sets for A 3 and A 3, respectively. The coponents of the isolated subsyste dynaics are approxiated by RBNN s so that ^ =F (x i;a (t)) + k (x i) (30) ^ =F (x i ;A (t)) + k (x i ) (3) where A (t) and A (t) will be updated on-line in an attept to identify the isolated syste dynaics. An adaptive algorith is used to estiate A 3 and A 3 with A and A, respectively. Paraeter error vectors are defined as = A 0 A 3 and = A 0 A 3. Using the current estiate for the ith subsyste with no interconnections, a certainty equivalence control ter for the ith subsyste is defined as u i = 0^ + i = ^ (3) assuing that ^ is bounded away fro zero (this ay be ensured using a projection algorith). Let i = r (d ) i + k i;d 0 e i + + k i; 0e i + a (t) sgn e > i P i b i + a (t)e > i P i b i = + a w (t) sgn e > i P i b i u i (33) where the adaptive paraeters a (t); a (t), and a w (t) are yet to be defined. The ter a sgn(e i > P i b i ) is used to reject unknown disturbances, while the ter a e i > P ib i= is used to copensate for unknown effects fro the interconnections. In addition, a sgn(e i > P i b i u i ) has been included to copensate for the representation error w. The control assuptions for the indirect adaptive controller are suarized as follows.

5 054 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO., NOVEMBER 999 Assuption 5 Control: If x i L n, then k ; k L. The representation errors w and w are bounded, i.e., jw jw and jw j W for soe W > 0 and W > 0. A projection algorith is used to ensure that ^ > 0. Proceeding in a siilar anner to the direct case, we have and e (d ) i =(^ 0 ) + ( ^ 0 )u i 0 i (x ; ;x ) 0 k i; d 0 e i 00k i; 0e i 0 a sgn e > i P ib i 0 a e > i P ib i= 0 a w sgn e > i P ib iu i (34) _e i =3 i e i + b i (^ 0 ) + ( ^ 0 )u i 0 i(x ; ;x ) 0 a sgn e > i P ib i 0 a e > i P ib i= 0 a w sgn e > i P ib iu i where 3 i and b i are as defined for the direct adaptive case. The identifier errors ay be expressed as ^ 0 = > 0 w (35) ^ 0 = > 0 w (36) where w and w are representation errors. At this point, we also define the total paraeter identification error for the isolated subsyste dynaics as =[ > ; > ] > with the total regressor as = [ > ; > u i ] >. The following update laws are now defined for the decentralized indirect adaptive controller: _A i = 0 e > i P i b i (37) _a w = w e > i P ib iu i (38) _a = e > i P ib i (39) _a = e > i P ib i (40) where ; w ;, and are fixed adaptive gains. The paraeter update law for the isolated syste identifier (37) is used to estiate the dynaics of the subsyste under control. The update law (38) is designed to copensate for the effects of the representation error w, while the update laws (39) and (40) are used to stabilize the subsyste by estiating the effects of the interconnections and w. The paraeter error associated with the representation error w is defined as w = a w 0 W, while and are as defined for the direct case. The adaptive paraeters are initialized such that a w (0) 0; a (0) 0, and a (0) 0 so that a w (t); a (t); a (t) reain positive. Theore : Given the decentralized syste with subsystes satisfying Assuption, interconnections satisfying Assuption 3, and controllers satisfying Assuption 5, then the control law (3) with adaptation laws (37) (40) will ensure that, for i =; ;: ) the subsyste output and its derivatives, y i ; ;y i, are bounded; ) each control signal is bounded, i.e., u i L; 3) the agnitude of each output error, je i j, decreases asyptotically to zero, i.e., li t! je i j =0; 4) li t! j _A j =0; li t! j _aw j =0; li t! j _a j =0, and li t! j _a j = 0. Proof: Consider the following Lyapunov-type function for the ith subsyste: v i = e > i P i e i + > w w (4) where each P i d d is a positive definite and syetric atrix. Fro the definition of ;, and w, we use a siilar approach to the direct case to obtain _v i = 0 e > i R ie i +e > i P ib i[0w 0 w u i 0 i(x ; ;x )] 0 3 i e > i P i b i 0 3 i e > i P i b i 0 W e > i P i b i u i : (4) Choosing 3 i = W + ~ 3 i ensures that e > i P i b i [w + ~ i (t)] 3 i je > i P ib ij. Also e > i P ib iw u i W je > i P ib iu ij so that _v i 0e > i R ie i 0 3 i e > i P ib i 0 e > i P ib i i(x ; ;x ): (43) We require each i 3 > 0, then coplete the squares to obtain _v i 0e i > R i e i +(=i 3 )i (x ; ;x ): Now consider the coposite syste Lyapunov candidate V = civi, where each ci > 0. Taking the derivate of V gives _V c i 0in(R i)je ij + 3 > 0 > i0 i (44) i where in (R i ) is the real part of the eigenvalue of R i with the iniu agnitude. Define K 3 =[ 3 ; ; 3 ]. Let D = diagfc in (R ); ;c in (R )g and M = c i0 i 0 i >,so that V _ 0 > A, where A = D0 M. Then for soe sufficiently large 3 > 0, the atrix A is positive definite. The reainder of the theore follows as for the direct adaptive case beginning with (4). Reark 4: The results obtained here are again seiglobal due to the definition of the RBNN s with global results obtained in the case where S x = n for i =; ;. Reark 5: Notice that for the indirect adaptive controller, we have u i a function of i and i a function of sgn(u i ) due to the a w (t) sgn(e i > P i b i u i ) ter in (33). For ipleentation purposes, it is possible to bias the controller output so that u i only takes on positive values so that i and thus u i ay be easily calculated. This is analogous to considering the controller output to be positive values passed out of an analog-to-digital board, while the scaling and biasing of the actuator are considered to be part of the plant dynaics. If, for a particular application and choice of the RBNN s, we ay set W =0, then it is possible to let a w (t) =0so that i = r (d ) i + k i;d 0 e i ++k i;0e i+a (t) sgn(e i > P ib i)+a (t)e i > P ib i=, thus eliinating any dependence upon u i. Reark 6: The indirect adaptive controller does require a projection algorith to ensure that the control signal is well defined for all tie. This ay be easily achieved for RBNN s with adjustable output weights using a weighted average radial basis calculation since the output of the RBNN is then no less than the value of the sallest weight. Reark 7: The indirect adaptive control routine does not ake any requireents upon the rate of change of the input gain for each subsyste. In addition, we did not need to know the interconnection strengths, representation errors, or bounds on i(t). Because of the functional approxiation properties of RBNN s, the functional for of the subsyste dynaics does not need to be known. VI. SIMULATIONS Within this section, we will present illustrative exaples for both the direct and indirect approaches. While the approach could be applied to intervehicle spacing regulation in a platoon of an

6 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO., NOVEMBER Fig.. Two inverted pendulus connected by a spring. Fig.. Control of the pendulus using the a proportional feedback controller (using u k ). autoated highway syste, since that syste fits the assuptions of our fraework [6], instead we study the control of two inverted pendulus connected by a spring as shown in Fig.. Each pendulu ay be positioned by a torque input u i applied by a servootor at its base. It is assued that both i and _ i (angular position and rate) are available to the ith controller for i =;. The equations which describe the otion of the pendulus are defined by _x ; = x ; (45) _x ; = gr J 0 kr 4J sin(x ; ) + kr J (l 0 b) + u J + kr 4J sin(x ; ) (46) _x ; = x ; (47) _x ; = gr J 0 kr 4J sin(x ; ) 0 kr J (l 0 b) + u J + kr 4J sin(x ; ) (48) where = x ; and = x ; are the angular displaceents of the pendulus fro vertical. The paraeters =kg and =:5 kg are the pendulu end asses, J =0:5 kg and J =0:65 kg are the oents of inertia, k = 00 N/ is the spring constant of the connecting spring, r =0:5 is the pendulu height, l =0:5 is the natural length of the spring, and g =9:8 /s is gravitational acceleration. The distance between the pendulu hinges is defined as b =0:4, where, in this exaple, b<lso that the pendulus

7 056 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO., NOVEMBER 999 Fig. 3. Control of the pendulus using the proposed direct adaptive decentralized technique. Fig. 4. Control of the pendulus using the proposed indirect adaptive decentralized technique. repel one another when both are in the upright position. It is easy to see that the pendulu equations of otion fit (4). Here we will attept to drive the angular positions to zero, so that ei = 0i [i.e., ri(t) = 0]. We will first deonstrate that siple decentralized proportional feedback controllers appear to stabilize the syste. Choosing ui =0ei for i =;, we find that the pendulus appear to be stabilized, but exhibit relatively large oscillatory behavior due to the lack of daping as shown in Fig.. Selecting uk =0ei, it is possible to augent the proportional controllers with the decentralized direct adaptive controllers to help regulate the syste. Here we choose to input i= and i=5 to the ith RBNN controller with centers evenly spaced between [0, ] for the i=5 input and [0, ] for the i= input with i =. The inputs were noralized because the RBNN definition considers

8 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO., NOVEMBER as a constant. Thus scaling the inputs ensures that the RBNN basis function s size of influence will be appropriate for each input. Using 0 centers for each input diension yields a total of 00 adjustable paraeters for each RBNN with a weighted average forulation. We additionally chose i = 000; =0; =0and each 3 i so that ^L(s) =s +4s +4has roots at (0, 0). The perforance of direct adaptive controller is shown in Fig. 3. Next, we apply the indirect adaptive schee using weighted average RBNN s for the inverted pendulu exaple. Rather than choosing soe u k, we ay now choose k and k if desired. Here we let k =0and k =. Since the control gains are siply constants for this exaple (i.e., the control inputs are ultiplied by constants =J and =J for u and u, respectively) we let W =0 for i = ;. This choice is valid since an RBNN ay exactly approxiate a constant. This iplies that we ay set a w (t) =0 to siplify the control law. In addition, a projection algorith was used to ensure that F + k 0:. This is done using a projection algorith such that each weight for F reains greater than 00:9 so that F > 00:9 when using a weighted average RBNN. We chose i = as the input for F and F with 0 centers evenly spaced between [0, ]. With i = 00; =, and =, the output trajectories for the inverted pendulus are shown in Fig. 4. VII. CONCLUDING REMARKS Within this paper, we presented direct and indirect adaptive control schees appropriate for a class of interconnected nonlinear systes using radial basis neural networks. Using an on-line approxiation approach, we have been able to relax the linear in the paraeter requireents of traditional nonlinear decentralized adaptive control without considering the dynaic uncertainty as part of the interconnections or disturbances. Seiglobal asyptotic stability results were obtained with global results achieved by placing additional assuptions upon the RBNN s. Although the adaptive schees presented here relax the linear in the paraeter requireents for subsystes, there are distinct disadvantages associated with the on-line approxiation approach to decentralized control. ) The schees presented here do not necessarily identify physically eaningful paraeters, which schees for linear in the paraeter subsystes ight do. Often, however, the paraeters identified using traditional approaches are a cobination of a nuber of physical paraeters so that it ay be difficult to extract useful inforation about the subsystes. ) Even though we have shown asyptotic stability for the tracking errors, we have ade no guarantee about convergence of the controller paraeters to their ideal values (e.g., A u ay not converge to A 3 u ). This ay not be a concern in cases where stable control is the priary objective as it is here. 3) The approaches here ay require a large nuber of adjustable paraeters for each RBNN due to the curse of diensionality associated with RBNN s which are linear in the paraeters. If using a decentralized approach was decided based upon coputational overhead, then there ay be circustances for which other less coputationally intensive approaches would be ore appropriate. REFERENCES [] P. A. Ioannou, Decentralized adaptive control of interconnected systes, IEEE Autoat. Trans. Contr., vol. 3, no. 4, pp. 9 98, Apr [] D. T. Gavel and D. D. Šiljak, Decentralized adaptive control: Structural conditions for stability, IEEE Trans. Autoat. Contr., vol. 34, pp , Apr [3] A. Datta, Perforance iproveent in decentralized adaptive control: A odified odel reference schee, IEEE Trans. Autoat. Contr., vol. 38, pp. 77 7, Nov [4] R. Ortega and A. 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