Proceedings of the 42nd IEEE Conference on Decision and Control Maui, Hawaii USA, December 2003
FrP03-5
An Adaptive Approach to Control of Distributed Parameter Systems Belinda King Department of Mechanical Engineering Oregon State University Corvallis, OR 97331-6001
[email protected]
Naira Hovakimyan Department of Aerospace & Ocean Engineering Virginia Polytechnic Institute & State University Blacksburg, VA 24061
[email protected]
A BSTRACT A method is presented for augmenting a finite dimensional linear quadratic Gaussian (LQG) controller for a distributed parameter system with an adaptive output feedback element. The theory is discussed for a problem concerning control of vibrations in a nonlinear structure with bounded disturbance. I. I NTRODUCTION Control design for distributed parameter systems (DPS) is an area of research that has received much attention in the last fifteen years. Although a significant theory has been developed, in particular, for control of linear systems via methods such as linear quadratic Gaussian (LQG) or MinMax, this theory has not been widely implemented. A primary reason for this is that derivation of the DPS controllers requires approximation of the systems for computation of approximate controllers that are typically large scale. Since these controllers are large scale, their real-time implementation is not possible. To address this difficulty, model and control reduction techniques have been a topic of active research through the last decade. One of the challenges in reduction methods is ensuring that the resulting low order controller will control the DPS. At the first stage of approximating the PDE to compute a large-scale controller, theory exists for LQG and MinMax design to guarantee that this finite dimensional approximating controller converges to the PDE controller. However, even at this stage, nonlinearities and disturbances in the PDE system can present difficulties. For the available reduced order control methods, there is often little that can be said with respect to the low order controller’s relationship to the large-scale finite dimensional controller, much less its relationship to the PDE controller. To address these challenges, an approach to coupling an adaptive control with controllers designed for PDE systems is presented. Results reported on adaptive control of distributed parameter systems have been primarily for systems with linear dynamics [1]–[4] and only more recently for those with nonlinear dynamics [5]. In [1], the authors analyze adaptive control of infinite dimensional parabolic systems and include a stability proof ensuring parameter convergence. In [2], a discontinuous model reference adaptive control synthesis is developed for a certain hyperbolic distributed parameter systems. The synthesis is carried out in an infinite
0-7803-7924-1/03/$17.00 ©2003 IEEE
dimensional setting, the numerical approximation of which can be considered at the implementation stage. Here, we first discuss adaptive augmentation for largescale finite dimensional LQG designs, which are known to converge to the distributed parameter LQG controller. We show that the error dynamics could be written in a form for which the proof on ultimate boundedness presented in [6], [7] is valid. Next, using the approach in [7], we develop a reduced order controller design for the high order dynamics, by writing the original dynamics in its normal form and acknowledging the fact that the zeros are stable for the model problem considered in this paper. We then discuss adaptive augmentation of a reduced order controller obtained via balanced truncation for comparison. We characterize the conditions and assumptions on the reduced order model and discarded dynamics under which the adaptive output feedback approach, developed in [6], [7], can be applied for augmenting this reduced order controller. The ultimate goal is to compare the two approaches, one developed in the framework of nonlinear adaptive control, the other in the framework of distributed parameter systems using model reduction techniques. The remainder of this paper is organized as follows. In Section II, we give a motivating example that leads to the problem formulation. In Section III, we introduce the general PDE control framework and define the related infinite dimensional system dynamics along with its finite dimensional approximation. In Section IV we formulate the adaptive output feedback control approach specialized to our problem of interest. In Section V we discuss the framework of reduced order controllers and derive the associated adaptive controller. Section VI summarizes the paper.
II. A M OTIVATING E XAMPLE We consider the nonlinear structural dynamics problem discussed in [8] that models control of the vibrations of a cable mass system subject to disturbances. Specifically, this problem involves the vibrations of an elastic cable which is fixed at one end and attached to a mass at the other. The mass is suspended by a spring which has nonlinear stiffening terms and is forced by a sinusoidal disturbance. The equations
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describing this system are · ¸ ∂2 ∂ ∂ ∂2 ρ 2 w(t, s) = τ w(t, s) + γ w(t, s) ∂t ∂s ∂s ∂t∂s 0 < s < `, t > 0, (1) · ¸ ∂ ∂ ∂2 m 2 w(t, `) = − τ w(t, `) + γ w(t, `) ∂t ∂s ∂t∂s 3 −α1 w(t, `) − α3 [w(t, `)] + ϕ(t) + u(t), 2
with boundary condition w(t, 0) = 0. Initial conditions are chosen of the form ∂ w(0, s) = w1 (s). w(0, s) = w0 (s), ∂t
(2)
(3)
Here, w(t, s) represents the displacement of the cable at time t, position s, w(t, `) represents the position of the mass at time t, ρ and m are the densities of the cable and mass respectively, τ is Young’s modulus for the cable, and γ is a damping coefficient. The alphas are spring constants (linear and nonlinear) associated with the mass. The term ϕ(t) is viewed as a disturbance and u(t) is a control input. We assume that the only measurements of the system that are available are the mass position and velocity. This leads to the equation for the sensed measurement · ¸ ¸ · w(t, `) η1 (t) η(t) = . (4) = ∂ η2 (t) ∂t w(t, `) III. G ENERAL PDE C ONTROL D ESIGN F RAMEWORK Due to the special form of the physics of this problem, specifically, that the control and disturbance inputs, and the nonlinearity enter the system via forces on the mass, the infinite dimensional system dynamics can be presented in the following abstract form: w(t) ˙
= Aw(t) + B (u(t) + F (w(t)) + ϕ(t)) ,
w(0)
= w0
(5)
∂ ∂ (w(t, ·), w(t, `), ∂t w(t, ·), ∂t w(t, `)) = L2 (0, `) × R × L2 (0, `) × R and
where w(t) = is in the state space W u(t) and ϕ(t) are in R; F (w(t)) represents the nonlinearity in the system. The state measurement is given by η(t) = Cw(t)
(6)
where C : W → R2 . Notice that given (6), the vector relative degree of the measurement is r = [ 2 1 ] (for the definition of vector relative degree refer to [9]). Under standard assumptions, using the LQG theory for DPS (see for example, [10]), a feedback controller can be designed for the linear system obtained by linearizing (5) around an equilibrium. The resulting feedback control law is given by u(t) = Kwc (t) ,
(7)
where wc (t) is an estimate of the state obtained from the dynamical system: w˙ c (t) = Ac wc (t) + Lη(t),
wc (0) = wc0 .
(8)
The feedback gain operator K achieves the desired setpoint regulation. For this system, the equilibrium is the origin of the state space W , and this equilibrium is globally exponentially stable (see [8]). The operators K, Ac and L cannot, in general, be derived analytically, as they result from the solution of algebraic Riccati equations which, for PDE control problems, are functional differential equations. Therefore, to compute the control law in (7), (8), a finite dimensional approximation of the linearization of (5), (6) is derived using a convergent approximation scheme. Using, for example, a finite-element method as in [8], one can obtain an observable and controllable finite-dimensional approximation of (5), (6): w˙ N (t)
= AN wN (t) ¡ ¢ + B N u(t) + F N (wN (t)) + ϕ(t) , N
N
η(t) = C w (t),
N
w (0) =
w0N
(9) (10)
where N is the number of basis elements in the approximation scheme. We note that the vector relative degree is preserved in this approximation process. In [8], the disturbance, ϕ(t), was taken to be modeled by a cosine function. Here, we make the following more general assumption regarding ϕ(t). Assumption 3.1: Bounded disturbances ϕ(t) belong to a class of continuous time functions, describable by differential equations: w˙ ϕN (t) N
ϕ (t)
= fϕN (wϕN (t), wN (t)) =
(11)
gϕ (wϕN (t), wN (t)) .
The dynamics in (11) are input-to-state stable with wN (t) viewed as input [11], and in addition the overall system w˙ N (t) = AN wN (t) (12) ¢ ¡ N N N N N N + B u(t) + F (w (t)) + gϕ (wϕ (t), w (t))
w˙ ϕN (t) = fϕN (wϕN (t), wN (t)) N
(13)
N
η(t) = C w (t) is observable from its output η(t). Remark 3.1: Assumption 3.1 imposes only a mild restriction on the class of disturbances for which the adaptive control approach (developed below) is applicable. It also ensures that the framework of nonlinear control with its design tools can be applied for regulation of the dynamics in (9), see for example [12]. The approximations, introduced in (9), (10), are used to compute K N , LN , AN c , which can be shown to converge to the PDE controller given by K, L, Ac in (7), (8). Thus,
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the finite dimensional LQG controller for stabilization of the nominal linear system dynamics is given by uLQG (t) = −K N wcN (t) w˙ cN (t) N
N
=
N AN c wc (t)
N
N
+ L η(t), w (0) =
(14) wcN0
(15)
AN c
The K , L , are determined via the solution of two algebraic Riccati equations (see for example [8]). We assume that N is chosen large enough so that the matrices in (9) - (15) have converged to the respective operators in (5) (8). The finite dimensional approximation to the closed-loop LQG system without nonlinearity and disturbance (which henceforth will be referred to as full-order LQG system) is given by: ¸ · · N ¸ ¸· N AN −B N K N w˙ (t) w (t) = (16) w˙ cN (t) LN C N AN wcN (t) · N ¸ · cN ¸ w (0) w0 . (17) = wcN0 wcN (0) However, this LQG design does not account for the nonlinearities and disturbances in the system dynamics (9). In [8], extended Kalman filters were applied to address nonlinearities, in addition to MinMax controllers for disturbances. Here, for the regulation of the full order nonlinear dynamics in (9), consider the controller u(t) = uLQG (t) − uad (t) = −K N wcN (t) − uad (t) ,
(18)
where uad (t) ∈ R is an adaptive signal and will be designed to approximately cancel the nonlinearities and disturbances. With this addition, the full order closed-loop system will take the form: ¸ ¸· N ¸ · · N w (t) AN −B N K N w˙ (t) = wcN (t) LN C N AN w˙ N (t) c · Nc ¸ ¢ ¡ B − uad (t) − F N (wN (t)) − gϕN (wϕN (t), wN (t)) 0 w˙ ϕN (t) = fϕN (wϕN (t), wN (t)) N
where the definition of C¯ N implies that the estimates of states (controller states) are available for feedback and are added to the available measurements. Notice that the LQG design ensures that A¯N is Hurwitz. This implies that for arbitrary positive definite matrix QN > 0 there exists a unique positive definite symmetric matrix P N = (P N )T > 0 that solves the Lyapunov equation (A¯N )T P N + P N A¯N = −QN .
IV. A DAPTIVE O UTPUT F EEDBACK C ONTROL A PPROACH A. Neural Network Approximation of Nonlinearity Following the development in [13], given arbitrary ²∗ > 0 and a continuous function f (·), f : D ⊂ Rn → Rm , where D is compact, there exists a set of bounded constant weights W, V , such that the following representation holds for all x ∈ D: f (x) = W T σ(V T x) + ²(x) ,
(19)
¯ −B
¡ N
w ¯˙ N (t) = A¯N w ¯ N (t) ¢ uad (t) − F N (wN (t)) − gϕN (wϕN (t), wN (t)) w˙ ϕN (t) = fϕN (wϕN (t), wN (t)) η¯(t) = C¯ N w ¯ N (t)
= W T σ(V T µ(t)) + ²(µ(t)), k²k < ²∗ using the input vector: £ µ(t) = 1 η¯dT (t)
u ¯Td (t)
¤T
,
kµ(t)k ≤ µ∗ ,
(24)
(25)
where η¯d (t) , u ¯d (t) are vectors of difference quotients of the measurement η and the control variable u in (9), (10), respectively, and µ∗ is a known uniform bound on D. B. The Adaptive Control and Adaptation Laws We now design the adaptive element uad (t)
(21)
(23)
F N (wN (t)) + gϕN (wϕN (t), wN (t))
η(t) = C w (t)
and write the dynamics in (19) in the following compact form:
k²(x)k < ²∗ .
Here, the structure W T σ(V T x) is called a single hidden layer (SHL) neural network (NN), σ(·) is a vector of suitably chosen basis functions (its dimension being associated with th the number£ of nodes¤ in the £hidden ¤ layer), its i component T T ¯ i ), and ²(x) is the function defined as σ(V x ¯) i = σ( V x reconstruction error. In [14], [15], it was shown that for an observable system, such an approximation can be achieved using a finite sample of the output history. We recall the main theorem from [15] in the form of the following existence theorem. Theorem 4.1: Given ²∗ > 0, there exist bounded weights W, V and a positive time delay d > 0, such that the nonlinearity F (wN (t)) + gϕN (wϕN (t), wN (t)) in (21) can be approximated over a compact set D of the argument (wϕN , wN ) by a SHL NN
N
In the next section we introduce the adaptive output feedback control approach, developed in [6], and specialize it for the dynamics in (19). For the purposes of our further analysis, we introduce the following notation: ¸ · · N ¸ AN −B N K N w (t) N ¯ , A = w ¯ N (t) = , wcN (t) LN C N AN · N ¸ · N ¸c C 0 ¯N = B B , C¯ N = (20) 0 0 I
(22)
ˆ T (t)σ(Vˆ T (t)µ(t)) , = W
(26)
ˆ (t), Vˆ (t) are estimates of neural network weights, where W (see (24)), that will be adapted online. Using the representations in (24) and the definition of the adaptive control signal
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therefore any stabilizing controller for the dynamics in (29), will ensure boundedness of the z(t) states of the system w ¯˙ N (t) = A¯N w ¯ N (t) [11]. Thus, the original control problem formulation for the ¯ N (W ˆ T (t)σ(Vˆ T (t)µ(t)) − W T σ(V T µ(t)) − ²(µ(t))) −B dynamics in (9), (10) is reduced to defining a controller for N N ¯ η¯(t) = C w ¯ (t). (27) a second order system in (29). To this end, consider the approach outlined in [7], and The NN weight adaptation laws are similar to the ones rewrite the dynamics in (29)-(32) along with the dynamics proposed in [6]. for disturbances in (11), subject to Assumption 3.1, in the To this end, introduce the following linear observer for the following compact form: dynamics in (27) η˙ n (t) = An ηn (t) (33) w ˆ˙ N (t) = A¯N w ˆ N (t) + M N (¯ η (t) − ηˆ(t)) N N N N N + Bn (u(t) + F (w (t)) + gϕ (wϕ (t), w (t))) ηˆ(t) = C¯ N w ˆ N (t) (28) z(t) ˙ = Cz z(t) + Cη ηn (t) where M N is chosen to make A˜N , A¯N −M N C¯ N Hurwitz, w˙ ϕN (t) = fϕN (wϕN (t), wN (t)) and the following adaptive laws η(t) = Cn ηn (t). ˆ˙ (t) W A second order LQG controller can be designed for the ¯ N + kW ˆ (t)] = −F [(ˆ σ (t) − σ ˆ 0 (t)Vˆ T (t)µ(t))(w ˆ N (t))T P N B dynamics in (33) to stabilize the nominal linear dynamics ˙ ˆ Tσ ¯N W ˆ 0 (t) + k Vˆ (t)] Vˆ (t) = −G[µ(t)(w ˆ N (t))T P N B η˙ n (t) = An ηn (t) + Bn u(t) in which F, G are positive definite adaptation gain maη(t) = Cn ηn (t) trices, k > 0 is a constant, σ ˆ (t) , σ(Vˆ (t)µ(t)), and σ ˆ 0 (t) = σ 0 (Vˆ (t)µ(t)) is the Jacobian computed at the in the absence of nonlinearities and disturbances. Let the estimates Vˆ (t)µ(t). This choice of adaptive laws is based controller be defined by the following system on Lyapunov’s direct method and ensures bounded set point un (t) = −Kn ηc (t) (34) regulation for the dynamics in (21), as discussed in [6], [7], (35) η˙ c (t) = Anc ηc (t) + Ln η(t) [16]–[18]. For reasons of space the proof is not repeated here. C. Reduced Order Adaptive Output Feedback Control De- Following the approach presented previously for the full order controller, we augment this second order LQG controller sign with an adaptive element for stabilization of the dynamics in The adaptive output feedback control approach described (33). That is, above is presented in the context of the full-scale (high order) controller, and is therefore not practical for implementation u(t) = un (t) − uad (t) , (36) purposes. As an alternative idea, notice that η1 (t) has relative where uad is designed as in (26), degree 2, and following the approach presented in [7], ˆ T (t)σ(Vˆ T (t)µ(t)) , consider the dynamics in (9), (10) in its normal form [9]: uad (t) = W (37) ¸ · ¸ · ¸· η1 (t) η˙ 1 (t) 0 1 (29) and the adaptive laws take the form = η˙ 1 (t) η¨1 (t) 0 0 | {z } | {z } ˆ˙ (t) = −F [(ˆ ¯n + k W ˆ (t)] W σ (t) − σ ˆ 0 (t)Vˆ T (t)µ(t))ˆ η T (t)Pn B ηn An ¸ · ˙ ¯n W ˆ T (t)ˆ Vˆ (t) = −G[µ(t)ˆ η T (t)Pn B σ 0 (t) + k Vˆ (t)] . 0 + (u(t) + ∆1 (u(t), wN (t), wϕN (t))) b Here Pn solves the Lyapunov equation (22) for the Hurwitz | {z } Bn A¯n that defines the closed loop nominal linear dynamics z(t) ˙ = Cz z(t) + Cη ηn (t) (30) formed by using the LQG control (in the absence of nonN N N N w˙ ϕ (t) = fϕ (wϕ (t), w (t)) (31) linearities and disturbances): ¸ · ¸ · ¸· £ ¤ ηn (t) η˙ n (t) An −Bn Kn 1 0 ηn (t) , η1 (t) = (32) (38) = | {z } ηc (t) Ln Cn Anc η˙ c (t) Cn {z } | {z } | ¯n η(t) A where the constant matrices Cz , Cη are such that the dynam¯ η¯(t) = Cn η(t) , (39) ics z˙ = Cz z, representing the zero-dynamics of the system in (9), (10), are exponentially stable. Note that the exponential while ηˆ is defined using a linear observer for the closed loop stability of the zero dynamics follows easily from the global LQG dynamics in (38): exponential stability proven in [8]. Also notice that the zerodynamics are linear due to the structure of the problem, ηˆ˙ (t) = A¯n ηˆ(t) + Mn (C¯n η(t) − C¯n ηˆ(t)) (40) (26), the closed loop dynamics in (21) take the form
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As before, Mn is chosen to make A¯n − Mn C¯n Hurwitz. The stability proof, developed in [6], [7], [16]–[18], ensures bounded set point regulation for the ηn states in (33), guaranteeing boundedness of the internal z states. V. A DAPTIVE AUGMENTATION OF R EDUCED O RDER LQG C ONTROLLERS The above presented approach has been developed in the framework of nonlinear adaptive control, independently of the control theory for DPS. During the past several years various methodologies have been developed for design of reduced order controllers, specialized for DPS with the goal of obtaining reduced order models from high order finite dimensional approximations of PDE dynamics. Typically, stability proofs or convergence analysis for these low order models are lacking, and for nonlinear case remain still arguable. The purpose of this effort is to explore the potential of augmenting reduced order controllers with the adaptive output feedback architecture, proposed above, for cancelling the effect of the nonlinearities. Here we attempt to characterize the conditions under which the reduced order controllers are appropriate for the adaptive output feedback approach developed above. Specifically, we will augment a reduced order LQG controller with an adaptive output feedback element, as discussed in Section IV, and show that it achieves regulation of the states of the reduced order system, while guaranteeing boundedness of the states of the discarded dynamics. For the purposes of our discussion we choose balanced truncations, although any other model reduction algorithm might be considered, subject to the assumptions introduced below. Balanced realization and truncation is a common procedure that can be found in standard references on control, e.g., [19] for DPS and [20], [21] for finite dimensional systems, and we do not go into the details of the method here. It is based on the premise that a low order approximation to a given system with measurement as in (5), (6) could be obtained by eliminating any states that are difficult to control and to observe. To find these states, a balanced realization of the system is formed in which the states that are difficult to control coincide with the states that are difficult to observe. Then, after truncation, the resulting reduced system can be used for design of a low order controller that can be applied to the original dynamics. We point out that this is a technique for linear systems, so a linearization of any nonlinear system dynamics is used in the procedure. To this end, we apply the balanced realization method to the linearization of (9), (10) to obtain the balanced system. We write the balanced system as: w˙ bal (t) ηbal (t)
= Abal wbal (t) ¡ ¢ +Bbal u(t) + F (wN (t)) + ϕ(t) (41) = Cbal wbal (t) (42)
which can be partitioned as ¸ · ¸· ¸ · x(t) x(t) ˙ A11 A12 (43) = z(t) A21 A22 z(t) ˙ {z } | {z } | wbal A · bal¸ ¢ B1 ¡ + u(t) + F (wN (t)) + ϕ(t) B2 | {z } Bbal · · ¸ ¸ y1 (t) x(t) = [C1 C2 ] . (44) y2 (t) | {z } z(t) | {z } Cbal ηbal
Here, x(t), y1 (t) represent the state and measurement dynamics that will be retained, and z(t), y2 (t) those that will be discarded upon model reduction. Specifically, the dynamics for the reduced order model are given by x(t) ˙ = A11 x(t) + A12 z(t) ¡ ¢ + B1 u(t) + F (wN (t)) + ϕ(t)
y1 (t) = C1 x(t),
(45) (46)
while those for the discarded dynamics are given by ¡ ¢ z(t) ˙ = A21 x(t) + A22 z(t) + B2 u(t) + F (wN (t)) + ϕ(t) (47) Lemma 5.1: The origin of the system z(t) ˙ = A22 z(t) + B2 F (wN (t)) is globally exponentially stable. The proof follows from [8] wherein the proof is given that the full cable mass structure without control or disturbance input is globally exponentially stable. Lemma 5.1 ensures that given constant matrices A21 and B2 , any bounded input of x, u, ϕ will result in bounded response of z. Thus the dynamics in (47) are input-to-state stable [11]. To develop the adaptive output feedback approach for the dynamics in (45), notice that the term A12 z(t) introduces an unmatched uncertainty, as opposed to the case of (9). The following bound, though conservative, is easy to prove. Lemma 5.2: Let σmax (A12 ) be the maximum singular value of A12 . Then kA12 zk ≤ σmax (A12 )kzk (48) Notice that in [22], as well in other papers on balanced truncations, the reduced order models (45) are retained without the A12 z term. If to take that path, then the above developed adaptive output feedback approach will ensure bounded set-point regulation of x states in the absence of A12 z term. Following the philosophy of balanced truncations, which is based on discarding the least observable and least controllable modes, one may be able to argue that the above presented adaptive output feedback approach may still control the reduced order dynamics in (45), but the performance will be deteriorated within the bound of σmax (A12 )kzk. To be more rigorous, one needs to address the stability of the dynamics in (45), (47) with one Lyapunov function, as it is done in [23] (for reasons of space we do not repeat it here).
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The obtained bounds, however, will be overly conservative. A wiser approach will be to analyze in depth the properties of the retained and truncated dynamics, so that to come up with stronger claims about the truly achievable ultimate bounds. Investigation of these issues is in progress and will be reported in a forthcoming paper. VI. C ONCLUSIONS This paper presents the first attempt of applying an adaptive output feedback approach for control of distributed parameter systems. Using the potential of the approach and the properties of the model problem discussed in this paper, a second order adaptive output feedback controller is derived. As an alternative, reduced order models are discussed based on balanced truncations. A full comparison of adaptive augmentation of two LQG controllers, derived in separate frameworks of nonlinear control and control of DPS, will be provided in a forthcoming publication. ACKNOWLEDGEMENT Research of the second author is supported by AFOSR under Grant No. F4960-01-1-0024. VII. REFERENCES [1] K. S. Hong and J. Bentsman. Direct adaptive control of parabolic systems: algorithm synthesis and convergence and stability analysis. IEEE TRansactions on Automatic Control, 39:2018–2033, 1994. [2] Yu. Orlov. Model reference adaptive control of distributed parameter systems. Conference on Decision and Control, 1997. [3] M. Demetriou. Model reference adaptive control of slowly varying parabolic distributed parameter systems. Conference on Decision and Control, 1994. [4] Y. Miyasato. Model reference adaptive control for distributed parameter systems of parabolic type by finite dimensional controller. Conference on Decision and Control, 1990. [5] R. Padhi and S. N. Balakrishnan. Proper orthogonal decomposition based feedback optimal control synthesis of distributed parameter systems using neural networks. Proc. of the American Control Conference, 2002. [6] N. Hovakimyan, F. Nardi, A. Calise, and N. Kim. Adaptive Output Feedback Control of Uncertain Systems using Single Hidden Layer Neural Networks. IEEE Transactions on Neural Networks, 13(6), 2002. [7] N. Hovakimyan, B.-J. Yang, and A. J. Calise. An adaptive output feedback control methodology for nonminimum phase systems. Conference on Decision and Control 2002. [8] J. A. Burns and B. B. King. A reduced basis approach to the design of low order compensators for nonlinear partial differential equation systems. J. Vibr. & Contr., 4:297–323, 1998.
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