Distributive vibration control in flexible multibody dynamics

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Compurers & Structures Vol. 61, No. 5, pp. 957-965, 1996 Ccmvrinht 82 19% Elsetier Science Ltd Printed’i;r &aiBritain. All d&ts reserved CO45-7949/96$15.00 + 0.00

VIBRATION CONTROL IN FLEXIBLE MULTIBODY DYNAMICS G. G. Yen

USAF Phillips Laboratory, Structures and Controls Division, Kirtland AFB, New Mexico 87117, U.S.A. (Received 24 February 1995)

Abstract-A distributive neural control system is advocated for flexible multibody structures. The proposed neural controller is designed to achieve trajectory slewing of a structural member as well as vibration suppression for precision pointing capability. The motivation to support such an innovation is to pursue a real-time implementation of a robust and fault tolerant structural controller. The proposed control architecture which takes advantage of the geometric distribution of piezoceramic sensors and

actuators has provided a tremendous freedom from computational complexity. In the spirit of model reference adaptive control, we utilize adaptive time-delay radial basis function networks as a building block to aliow the neural network to function as an indirect closed-loop controller. The horizon-of-one predictive controller cooperatively regulates the dynamics of the nonlinear structure to follow the prespecifiedreference models asymptotically. The proposed control strategy is validated in the experimental facility, called the planar articulating controls experiment which consists of a two-link flexible planar structure constrained to move over a granite table. This paper addresses the theoretical foundation of the architecture and demonstrates its applicability via a realistic structural test bed. Copyright 0 1996 Elsevier Science Ltd.

1. INTRODUCTION

Dynamic modeling, system identification, bration controls of a flexible multibody have active mission in the structures and controls of the USAF Phillips Laboratory for years.

Modern engineering technology is leading to increasingly complex space structures with ever more demanding performance criteria. Specifically, precision pointing devices (e.g. robotic manipulators and surveillance satellites) are often made of light-weight composites and equipped with piezoelectric and/ or piezoceramic sensors and actuators. These flexible multibody structures, which are likely to be highly nonlinear with time-varying structural parameters and poorly modeled dynamics have posed serious difficulties for all currently advocated control methodologies (e.g. robust, adaptive, and optimal controls) [l, 21. Furthermore, the ultimate pursuit of a higher degree of autonomous behavior, which calls for a highly sophisticated controller to ensure that demanding performance can be met, can be extremely difficult due to factors such as high dimensionality, multiple inputs and outputs, operational constraints, as well as the unexpected failures of sensors and actuators. Conventional control design approaches often depend upon the assumption of a high fidelity dynamic model containing identified system parameters. This hypothesis, which inevitably necessitates an iterative process of finite element analysis and system identification, is computationally expensive to validate. Consequently, design procedures to achieve the desired stability, robustness, and dynamic response for precision space structures with unknown parameters are incomplete.

and vibeen an division

At present, a critical need exists for the verification and comparison of various modeling and control theories based on an actual hardware experiment. To meet this need, Phillips Laboratory has constructed a flexible multibody structure which consists of two flexible beams connected in series with motors at both the hub and the elbow joint. Figure 1 shows the flexible multibody structure named the planar articulating controls experiment (PACE) [3,4]. The PACE arms are driven through the specified trajectory by d.c. motors. Piezoceramic actuators and sensors have been chosen for vibration suppression on PACE. Because of their stiffness, good linearity, relative temperature insensitivity, and ease of implementation, piezoceramics such as lead zirconate titanate (PZT) have been determined to be a good candidate as actuators for many structural control applications [!!I]. The mathematical formulation of a class of flexible multibodies based on equations of motion in terms of quasi-coordinates is derived for each substructure independently. The individual substructure is made to act as a single structure by means of a consistent kinematics synthesis. The resulting differential equations are nonlinear and hybrid, where the term “hybrid” implies that the equations for the rigidbody translations and rotations are ordinary differential equations and those for the elastic motions are 951

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partial differential equations [6]. The advantage of this approach is that it yields equations of motion in terms of body axes, which are the same axes used for the control forces and torques. In addition to modeling efforts, we propose to design and validate a distributive neural control system which is capable of withstanding structural failures, component deviation, and unpredictable perturbations. Neural networks which employ the well-known back-propagation learning algorithm are capable of approximating any continuous functions (e.g. nonlinear plant dynamics and complex control laws) with an arbitrary degree of accuracy [7]. Similarly, radial basis function networks [8] are also shown to be universal approximators [9]. These model-free neural network paradigms are more effective at memory usage in solving control problems than conventional learning control approaches. A typical example is the BOXES algorithm, a memory intensive approach, which partitions the control law in the form of a look-up table [lo]. Adaptive neural control system offers the capability of real-time adaptation and generalization while a look-up table approach would only provide discrete controller solutions in a lengthy and sequential search. Our goal is to approach structural autonomy by extending the control system’s operating envelope, which has traditionally required vast memory usage. Connectionist systems, on the other hand, deliver less memory intensive solutions to control problems and yet provide a sufficiently generalized solution space [1 11. In vibration suppression/trajectory following problems, we utilize the adaptive time-delay radial basis function network as a building block to allow the connectionist system to function as an indirect closed-loop controller. The decentralized nature of the control system provides a tremendous computation power to suppress the vibration modes that can be identified by the experimental modal testing or to follow a prespecified trajectory. Prior to training the compensator, a neural identifier based on an ARMA model is utilized to identify the open-loop system (see Fig. 2). The m horizon-of-one predictive controllers then cooperatively regulate the dynamics of the nonlinear structure to follow a prespecified reference system (in terms of m linearized systems for each mode interested) asymptotically as depicted in Fig. 3 (i.e. the model reference adaptive control

Fig. 1. PACE test article.

Yen

Fig. 2. System identification of flexible multibody. architecture) [12]. The m - 1 backup copies of ATDRBF neural identifier are created to facilitate the training of ATDRBF neural controllers. The reference models, which can be easily specified through an input-output relationship, described all desired features associated with the control task, e.g. a linear and highly damped model to suppress the vibration or a designate route to specify the desired trajectory. Each control subsystem, which were designed dedicatedly for one set of PZT actuator and sensor or for a d.c. motor, is utilized to suppress a specified vibration mode or to follow a designate trajectory, so that the computational load can be evenly distributed and executed on a real-time basis. The function of each ATDRBF neural controller is to map the system states into corresponding control actions in order to force the decomposed plant dynamics to match an output behavior, which is specified by a linearized reference model. However, we cannot apply the energy minimization procedure (e.g. gradient descent, conjugate gradient or Newton-Raphson method) to adjust the interconnection weights of the neural controllers because the desired outputs of the neural controllers are not available. To achieve the true gradient descent of the square of the error, we use dynamic back propagation [ 131 to accurately approximate the required partial derivatives. An adaptive time-delay radial basis function network is first trained to identify the open-loop system. The resulting neural identifier then serves as extended unmodifiable layers to train a set of neural controllers. If the structural dynamics are to change as a function of time, the neural identifier would require the learning algorithm to periodically update the network parameters accordingly (as well as the m - 1 backup copies). The proposed efforts address several issues to achieve a distributive fault tolerant control system in flexible multibody structures. In Section 2, the mathematical formulation and dynamic modeling of a class of flexible multibody are briefly discussed. In Section 3, adaptive time-delay radial basis function network is covered, providing an underlying issue pertaining to the learning algorithm. The proposed control strategy was validated on the PACE test article which consists of a two-link flexible planar structure constrained to move over a granite table in Section 4. The paper is concluded with a few pertinent observations in Section 5.

Distributive vibration control in flexible multibody dynamics

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Fig. 3. Decentralization model reference adaptive control. 2. DYNAMIC MODELING

OF FLEXIBLE MULTIBODY

A set of equations of motion suitable for the control task can be formulated by means of Lagrange’s equations for flexible bodies in terms of quasi-coordinates. The advantage of this approach is that it yields equations in terms of body axes, which are the same axes as those used to express control forces and torques. In using the approach of Ref. [ 141 to derive equations of motion for a chain of flexible multibody systems, it is convenient to adopt a kinematical procedure permitting the expression of the velocity vector of a nominal point in a typical body in terms of the velocity vector of the preceding body in the chain. The resulting differential equations are nonlinear and hybrid, where the term “hybrid” implies that the equations for the rigid-body translations and rotations are ordinary differential equations and those for the elastic motions are partial differential equations. Because maneuvering and control design in terms of hybrid equations is not feasible, the partial differential equations must be transformed into sets of ordinary differential Y

Fig. 4. Flexible

body.

equations by means of a discretization-in-space procedure, such as the finite element method [15] or a Rayleigh-Ritz based substructure synthesis [16]. The resulting formulation consists of a high-order set of nonlinear ordinary differential equations. A common approach to control design requires the solution of a two-point boundary value problem, which is not feasible for high-order systems, so that a different approach is advisable. The nonlinearity enters into the differential equations through the rigid-body motions. Indeed, the elastic motions tend to be small. In view of this, it appears natural to conceive of a perturbation approach whereby the rigid-body motions can be regarded as being of zero-order in magnitude and the elastic motions as being of first-order in magnitude. This approach permits dividing the problem into a low-dimensional set of nonlinear zero-order equations for the rigid-body motions and a highdimensional set of linear first-order equations for the elastic motions and the perturbations in the rigidbody motion, where the order is to be taken in a perturbation sense. Note that, because the zero-order solution enters into the first-order equations as a known function of time, the first-order equations represent a time-varying system. Moreover, the system is subjected to persistent disturbances. The perturbation approach just described was proposed in Ref. [ 171to maneuver and control flexible spacecraft. The kinematical synthesis [l&19] works quite well in the case in which the number of bodies in the chain is relatively small. When the number of bodies is larger than three, difficulties can be expected, so that a different approach is taken. In this paper, we consider a procedure whereby the equations of motion are derived first for each individual flexible body. Then, the sets of equations for the individual bodies are assembled into a global set by invoking the

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Distributive vibration control in flexible multibody dynamics kinematical relations. In the process, the redundant coordinates and velocities resulting from considering the individual bodies separately are eliminated. It is convenient to carry out the kinematical synthesis on the zero-order problem and first-order problem separately. Implementation of the kinematical synthesis is based on recursive relations that lend themselves to ready computer coding. The resulting zero- and first-order global sets of equations are particularly suited for maneuvering and control design, respectively. The zero-order nonlinear equations govern the maneuver as if the system consisted of articulated rigid bodies, where the maneuver amounts to driving the system from an initial state to a final state. The simplest approach is to carry out the maneuver by means of actuators that impart predetermined motions to the substructures relative to one another. The first-order equations govern the elastic vibrations and the perturbations in the rigid-body motions. They contain the zero-order solution as a known function of time. As a result, the system is time-varying. Moreover, it is subjected to persistent disturbances caused by the maneuver. The process can be likened to that in which the system must follow a reference state. In this case, the reference state is defined by the rigid-body maneuvering, which is characterized by zero elastic states. Then, the firstorder equations are simply the equations in terms of the difference between the actual states and the reference states, where this difference can be identified as perturbations in the state variables. The approach used in this paper is to derive equations of motion for the individual substructures separately and then impose kinematical relations of the type described earlier to obtain the system’s equations of motion. Although the approach is used for the case of a two-link flexible body system, the approach can be extended to the case of arbitrary N flexible multibody systems. Let us consider a typical flexible substructure moving on a horizontal surface (see Fig. 4) and introduce the inertial axes XY with the origin at 0 and a set of body axes x,y, with the origin at S and embedded in the undeformed substructure. Then, we can write the position vector of a typical point in the substructure with the spatial coordinates given symbolically as follows: W, = fi=(R, + CTus),

point relative to the body axes x,y,, respectively. Due space limitation, the detailed mathematical derivation can be found in Ref. [20]. The following equations represent the nonlinear ordinary differential equations for the single substructure:

- dfCT(LS, + N&q,)

-#fD~(LS,+N~b,q,)+I,~~+i5,ii,=T,,

c, =

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1’

(3b)

D, = K, represents the substructure

stiffness matrix, and T, is the torque applied to the origin of the body axis x,y, . Imposing the kinematical relations to the above equations to link two adjacent substructures is not an easy task. Thus, we propose the perturbation method to ease the kinematical synthesis and numerical calculations. Designing the slewing and control for articulated systems of substructures is very difficult, especially if the design is to be optimal in some fashion. The difficulty can be traced to the fact that the system is nonlinear and of high order. The nonlinearity can be attributed to the rigid-body motions and the high order to the elastic motions. The perturbation approach is based on the simple observation that rigidbody motions tend to be large compared to the elastic motions. Consistent with this, let us assume that the translations and rotation of the body can be divided into zero-order terms and first-order terms in magnitude. Elastic displacements are assumed to be small so that the generalized displacements associated with elastic motion can be regarded as a first-order term. Thus, we may write

(1)

[I Jl’ represents the column matrix consistunit vectors in X- and Y-directions correto inertial coordinates, C, = C(0,) is the the direction cosines which is given by cos 0, sin 0, - sin e6 cos &

(3a)

where m, = @,dx,, S, = jfi,x,dx,, I, = @,x:dx, in which Kri, is the mass density per unit length. In addition, a’, = jfi,cDSdx,, a, = @,x,@,dx, M, = jfi,@~@,dx,, in which @,(x,) is the vector of admissible functions and qs is the vector of generalized coordinates. Also L = [l O]‘, N = [0 llT,

Rs=Ra+R,,,

where A = ing of the sponding matrix of

- 28,D:Ntb,Q, = 0,

6,=0,+0,,,

T,=T,+T,,.

Inserting eqn (4) into trigonometric yields the following relations:

(4)

functions

cos es = cos 8, - sin e&e,,,

(2)

In addition, R, is the radius vector from I to S, and II, includes the radius vector from S to a typical point in s and the elastic displacement vector of the same

sin 19,= sin (Id)+ cos es0e,, ,

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which lead to C, = Cro - D,&, , D, = Dd + C&, . After mathematical manipulation, we fmally obtain the first-order equations of motion,

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Distributive vibration control in flexible multibody dynamics m#s so-D’$LSf? s so-t&CiLS

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as well as the first order equations of motion

- C&D; + tk,C,)N6),q,

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- SJLTDsoti,, + I,& + ib,ih - S,LTC,&&,

+ K,q, = - &,ib; - @fNTC,#,,.

A given adaptive time-delay radial basis function network can be completely described by its interconnecting topology, neuronic ~h~acte~sti~, temporal delays and learning rule. The individual processing unit performs its computations based only on local information. The basis function in the hidden layer produces a localized response to input stimulus, as do locally-tuned receptive fields in our nervous systems. The Gaussian function network, a realization of the RBF network using Gaussian kernels, is widely used in pattern classification and function approximation. The output of a Gaussian neuron in the hidden layer is defined by

(7~)

The main advantage of using the ~~urbation method is in that the zero-order equations can be solved independently of the first-order equations. The zero-order equations are nonlinear, but in low order. Once the zero-order equations are solved, the solution of the zero-order equations enters into the first-order equations and completes the numerical calculations. Due to space limitation, we are concerned with the single substructure. Please refer to Ref. [20] to see how to assemble the individual equations into global equations of motion by means of the kinematical synthesis. 3. ADAPTIVE TIME-DELAY RADIAL BASIS FUNCTION NETWORK

Biological studies have shown that variable timedelays do occur along axons due to different conduction time and different lengths of axonal fibers. In addition, temporal properties such as temporal decays and integration occur frequently at synapses. Inspired by this observation, the time-delay backpropagation network was proposed by Waibel et nl. for solving the phoneme recognition problem [21]. In this architecture, each neuron takes into account not only the current info~ation from all the neurons of the previous layer, but also a certain amount of past information from those neurons due to delay on the interconnections. However, a fixed amount of timedelay throughout the training process has limited the usage, mainly due to the mismatch of the temporal location in the input patterns. To overcome this limitation, Lin et al. have developed an adaptive time-delay back-propagation network to better accommodate the varying temporal sequences, and to provide more flexibility for optimization tasks [22].

(8)

where u,! is the output of the jth neuron in the hidden layer, x is the input vector, s$ denotes the weighting vector for the jth neuron in the hidden layer (i.e. the center of the jth Gaussian kernel), of is the normalization parameter of the jth neuron (i.e. the width of the jth Gaussian kernel), and N’ is the number of neurons in the hidden layer. Equation (4) produces a radially symmetric output with a unique maximum at the center dropping off rapidly to zero at large radii. That is, it produces a significant nonzero response only when the input falls within a small localized region of the input space. Inspired by the adaptive time-delay back-propagation network, the output equation of ATDRBF networks is described by yJ(r,)=

5

2

wg,,u;(t”-

ri,,), i = 1,. . I, P,

(9)

i=IL=l

where wk denotes the ~o~ection between the output of the ith neuron of the hidden layer and the input of the jth neuron of the output layer with an independent time-delay ri,,, uf (t, - r$,,) is the output vector from the hidden layer at time f, - 7.2, ~5,:denotes the number of delay connections from the ith neuron of the hidden layer to the jth neuron of the output neuron. Shared with generic radial basis function networks, adaptive time-delay Gaussian function networks have the property of undergoing local changes during training, unlike adaptive time-delay back-propagation networks which experience global weighting adjustments due to the characteristics of sigmoidal functions. The localized influence of each Gaussian neuron allows the learning system to refine its functional approximation in a successive and efficient manner. The hybrid learning algo~t~ [8] which employs the K-means clustering for the hidden layer and the least mean square (LMS) algorithm for the output layer further ensures a faster convergence and often leads to better ~~o~an~ and generalization. The combination of locality of representation and linearity of learning offers tremendous computational efficiency in real-time adaptive control. K-means

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algorithm is perhaps the most widely known clustering algorithm because of its simplicity and its ability to produce good results. The normalization parameters, uj, are obtained once the clustering algorithm is complete. They represent a measure of the spread of the data associated with each cluster. The cluster widths are then determined by the average distance between the cluster centers and the training samples,

CT;=&c J

xse,

IIx-wi’112,

where Oj is the set of training patterns belonging to jth cluster and Nj is the number of samples in Oj. This is followed by applying a LMS algorithm to adapt the time-delays and interconnecting weights in output layer. The training set consists of inp~t/output pairs, but now the input patterns are pre-processed by the hidden layer before being presented to the output layer. The adaptation of the output weights and time delays is derived based on error back-propagation to minimize the cost function,

(11) where d,(r,) indicates the desired value of the jth output neuron at time t.. The weights and timedelays are updated step by step proportional to the opposite direction of the error gradient, respectively,

(124

convex regions [23]. The control objective is to achieve trajectory slewing (i.e. defined by states 6,: absolute angular displacement of the upper arm; 8,: absolute angular velocity of the upper arm; &: relative angular displa~ment of the forearm; and p2: relative angular velocity of the forearm) as well as vibration suppression along the motion (i.e. defined by states r, : tip displacement of the upper arm; ii : tip velocity of the upper arm; r,: tip displacement of the forearm; and i,: tip velocity of the forearm) by applying the control forces (i.e. ul : torque applied to the shoulder and y: torque applied to the elbow). Our training strategy is to relax the vibration suppression task for the first few seconds. The r.m.s. errors are accumulated only based on the trajectory following states (i.e. 8,) 8,, f12,and &). After 1.5 s, we begin to suppress the structural vibration states (i.e. I~, i,, t2 and i,) while maintaining the progress of trajectory following. However, by confining the desired responses to strictly zeros will deteriorate the network performance, even affecting the trajectory following state components. The way we proposed in this study is to setup an implicit exponential delaying envelope, so the network will smoothly catch up the requirements. Whenever the output state falls within this exponential envelope, it indicates a zero error, meaning no weight adjustments will be taken. This process indeed speedup the training procedure significantly. Trajectory slewing/vibration suppression is performed by eight ATDRBF networks with 25 neurons each. The closed-loop controller regulates the dynamics of the PACE structure to follow the desired outputs as given below

B,(t)=G( 1 -cosy),

(13a)

(1% =V2tdjCfn)

-Yj(fn))w~,,U:'(tn

-*C&h

Wb)

where q, and Q are the learning rates for interconnecting weights and tapped delays, respectively. (13d) 4 PACE S~U~~ON

STUDY

The autonomous control of precision space structures requires a distributed computational architecture that provides the ability to perform on-line system ident~~tion and dynamic control after orbital deployment. The neural network based decentralized control system proposed in this paper provides an alternative way to reduce the need for a priori knowledge of structural qualitative behavior, although a minimum knowledge of modeling is assumed. The dynamics of the plant are assumed unknown. System identification is simulated by a single-layer ATDRBF network with 100 neurons to ensure the flexibility to approximate arbitrary non-

and tip displacements and their velocity are zeros, of course. Although the neural identifier learned to match the open-loop system in a reasonable time frame, the ~rn~n~tor took more than four days to converge to a reasonable accuracy, mean square error 0.000562 (with a 486-50 MHz computer). Figure 5 shows the open-loop performance with respect to all output variables, respectively, in response to random inputs, while Fig. 6 displays the closed-loop performance with respect to all output variables, respectively, in response to an impulse. The neural regulator learned to follow the specified trajectory and then damp out the structural vibration effectively.

Distributive vibration control in flexible multibody dynamics 5. CONCLUSIONS

The architecture proposed for distributed neural control system successfully demonstrates the feasibility and flexibility of our proposed solution for precision flexible multibodies. The salient features associated with the proposed control architecture are discussed. In a similar spirit, the proposed control structure can be extended to the dynamic control of aeropropulsion engines, underwater vehicles, chemical processes, power plants, and manufacturing scheduling. The applicability of the present methodology to various realistic CSI structural test beds will be pursued in our future research.

REFERENCES

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9. E. J. Hartman, J. D. Keeler and J. M. Kowalski. Layered neural networks with Gaussian hidden units as universal approximations. Neural Comput. 2, 210-215 (1990). -,,, D. Michie and R. A. Chambers, BOXES: an experiment ’ in adantive control. In: Machine InteIIigence (Edited by E. DaIe and D. Michie), pp. 137-152.~Springer, Berlin (1968). I’. G. G. Yen, Identification and control of large structures using neural networks. Comput. Struct. 52, 859-870 (1994). 12. G. G. Yen, Reconfigurable learning control in large space structures. IEEE Trans. Control Syst. Technol. 2, 362-370 (1994). 13. K. S. Narendra and K. Parthasarthy, Identification and control of dynamical systems using neural network. IEEE Trans. Neural Network 1. 4-27 (1990). 14. L. Meirovitch, Hybrid state equations for flexible bodies in terms of quasi-coordinates. AIAA J. Guidance, Control Dyn. 14, 1008-1013 (1991). 15. L. Meirovitch, Computational Methods in Structural Dynamics. Sijhoff and Noordhoff, Netherlands (1980). 16. L. Meirovitch and M. K. Kwak, A Rayleigh-Ritz based structure synthesis flexible multi-body systems. AIAA J. 29, 1709-1719 (1991). 17. L. Meirovitch and R. D. Quinn, Equations of motion for maneuvering flexible spacecraft. AIAA J. Guidance, Control Dyn. 10, 453-465 (1987). 18. L. Meirovitch and R. D. Quinn, Maneuvering and vibration control of flexible spacecraft. J. Astronaut. Sci. 35, 301-328 (1987). 19. L. Meirovitch and M. K. Kwak, Dynamics and control of a spacecraft with retargeting flexible antennas. AIAA J. Guidance, Control Dyn. 13, 241-248 (1990). 20. G. G. Yen and M. K. Kwak, The design of neural controller for flexible multibody systems. AIAA J. Guidance, Control Dyn. (submitted). 21 A. Waibel, T. Hanazawa, G. Hinton, K. Shikano and K. Lang, Phoneme recognition: neural networks versus hidden Markov models. In: Proc. IEEE Conf. on Acoustics, Speech and Signai Processing, pp. 107-I 10 (1988). 22. D. T. Lin, J. E. Dayhoff and P. A. Lignomenides, Adaptive time-delay neural network for temporal correlation and prediction. In: Proc. SPIE Conf. on Biological, Neural. Net and 3-D Methods, pp. 17&l 8 1 (1992). 23. G. G. Yen, Autonomous neural control in flexible space structures. Control Engng Practice 3, 471483 (1995).