Substructural neural network controller

Computers and Structures 78 (2000) 575±581

www.elsevier.com/locate/compstruc

Substructural neural network controller M. Sunar a,*, A.M.A. Gurain b, M. Mohandes b a

b

Department of Mechanical Engineering, King Fahd University of Petroleum and Minerals, P.O. Box 1205, Dhahran 31261, Saudi Arabia Department of Electrical Engineering, King Fahd University of Petroleum and Minerals, P.O. Box 1885, Dhahran 31261, Saudi Arabia Received 30 September 1998; accepted 18 February 2000

Abstract A substructure-based neural network is proposed for the active control of ¯exible structures. A ¯exible structure is divided into substructures. Subsequently, subcontrollers are designed for these substructures using the linear quadratic regulator (LQR) control method. These subcontrollers are assembled to obtain the central feedback controller for the whole structure. A radial basis function neural network is trained to emulate the behavior of this central controller designed from substructure levels. The training is based only on the outputs of sensors collocated with the actuators. Therefore, two distinct advantages of the proposed neural network controller are noted as its training being based on substructural LQR controller and collocated sensor data. The performance of the neural network controller is compared favorably with the complete structural LQR controller for various input forces acting on a large ¯exible structure. Ó 2000 Elsevier Science Ltd. All rights reserved. Keywords: Linear quadratic regulator; Substructural control; Radial basis function; Neural network

1. Introduction Due to lightly damped elements, vibration control of ¯exible structures is extremely important. Various optimal control techniques have been developed and applied to structural control for this purpose. Among many control techniques, linear quadratic regulator/linear quadratic Gaussian (LQR/LQG) and H1 techniques are frequently employed in structural control [1,2]. These techniques attempt to satisfy the structural control objectives despite the existence of various sources of noise. The design of low-order robust controllers for singleinput/single-output disturbance rejection for controlled structures was considered by Campbell and Crawley [3]. The controller design contained a blend of optimal and classical control concepts. Structural models with parametric uncertainties were used by Campbell and Crawley [4] for structural control. Hysteresis was used by Sain *

Corresponding author. E-mail address: [email protected] (M. Sunar).

et al. [5] as a nonlinear e€ect in structural model and control. The importance of developing robust control approaches in the presence of unreliable structural models was discovered by Skelton [6] with aerospace applications. The models produced by realistic ¯exible structures are usually complex and large, which makes the design of structural controllers very dicult. Hence, decentralized and substructural control techniques are often utilized for controller synthesis. These techniques o€er local damping or aid in assembling the centralized global controller from simpler substructure levels [7,8]. In the substructural control techniques presented by Sunar and Rao [9] and Su et al. [10], the concept of obtaining the global controller from subcontrollers of substructures was advocated. Neural networks have been employed recently in vastly di€erent areas. After a sucient amount of training, they can be used as a structural model predictor and controller. This is especially advantageous when the actual system is complex and large. The tracking control of mechanical systems using a proportional plus

0045-7949/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 5 - 7 9 4 9 ( 0 0 ) 0 0 0 3 9 - 0

576

M. Sunar et al. / Computers and Structures 78 (2000) 575±581

derivative controller and a feedforward neural network was presented by Efrati and Flashner [11]. Control of a system by state feedback using the systemÕs inverse dynamics was carried out by Szepesvari et al. [12] via neural networks. The neural network control of ¯exiblelinks and that of rigid robots were achieved by Yesildirek et al. [13], Kim and Lewis [14], and Jung and Hsia [15]. The tip position tracking and vibration suppression of a single-link ¯exible arm was studied by Jain et al. [16] with a genetic algorithm-trained neural network controller. Neural network control was used in a four-wheel anti-lock brake system by Yuan et al. [17]. Training of the neural network identi®cation and control was based on the Kalman ®lter update scheme. A neural network inverse dynamics controller with adjustable weights was compared by Nordgren and Meckl [18] with a computed-torque type adaptive controller. In this work, a large ¯exible structure is decomposed into substructures and subcontrollers designed for these substructures are assembled to obtain a central global controller for the entire structure. This substructural control scheme is based on the LQR control method [9]. A radial basis function neural network (RBFNN) is trained to emulate the behavior of this substructurebased global controller. The training is carried out using only outputs of displacement and velocity sensors collocated with actuators of the structure. This is contrary to the full sate feedback LQR control scheme, which assumes the availability of all the sensory data. Hence, the advantages of the proposed RBFNN controller are twofold, one originates from its substructure-based controller training and the other is the training data obtained from collocated sensors with actuators. The trained RBFNN controller is compared with the full model controller of the structure using various types of input forces. It is concluded that the proposed RBFNN controller can replace the full state feedback LQR controller and can operate through collocated sensors with actuators.

2. Control scheme The model of a ¯exible structure in con®guration space is given by M d ‡ Kd ˆ Du;

…1†

where M, K and D are the mass, sti€ness and input matrices, and d and u are the displacement and control input (force) vectors, respectively. The above model is written in state space as z_ ˆ Az ‡ Bu;

…2†

where the state vector z, the state matrix A, and the input matrix B are de®ned as

zˆ

  d ; d_

 Aˆ

0 ÿM ÿ1 K

 I ; 0

 Bˆ

0

ÿ1

M D

 ;

…3†

where I stands for the identity matrix. The LQR control scheme optimizes a performance index J of the form Z 1 Jˆ …zT Qz ‡ uT Ru† dt; …4† 0

where Q and R are positive semi-de®nite and positive de®nite weighting matrices, respectively. The control input vector u is de®ned as u ˆ ÿGz;

…5†

where G is a constant gain matrix found as G ˆ Rÿ1 BT P

…6†

and P is a unique positive semi-de®nite solution matrix to the following algebraic Riccati equation (ARE): AT P ‡ PA ÿ PBRÿ1 BT P ‡ Q ˆ 0:

…7†

3. Substructural control Assume that the ¯exible structure is decomposed into r substructures [9]. The model of a kth substructure in the con®guration space is given as Mk  dk ‡ Kk dk ˆ Dk uk :

…8†

Eq. (8) can be rearranged by grouping together the internal and boundary degrees of freedom (DOF) of the kth substructure. The resulting equations of the kth substructure are expressed as        MAk Ak MAk Bk KAk Ak KAk Bk dAk dAk ‡ B MBk Ak MBk Bk KBk Ak KBk Bk dBk d k    DAk Ak DAk Bk uAk ˆ ; …9† DBk Ak DBk Bk uBk where subscripts Ak and Bk stand for the internal and boundary DOF for the kth substructure, respectively. According to the Guyan condensation scheme [19], the relation between displacements corresponding to the internal and boundary DOF is given as dAk ˆ ÿKAÿ1k Ak KAk Bk dBk :

…10†

Hence, the transformation between these displacements is obtained as     dAk ÿKAÿ1 KAk Bk k Ak ˆ dBk ˆ Tk dBk : …11† dBk IBk By de®ning proper sets for the internal and boundary DOF, transformation matrices can be found between any kth substructure and all of its surrounding substructures.

M. Sunar et al. / Computers and Structures 78 (2000) 575±581

The substructural control scheme is iterative in nature. In each iteration, subcontrollers are designed for all the substructures with the inclusion of subcontroller forces from surrounding substructures. At the end of the iteration, subcontrollers are assembled to obtain the global controller and the closed-loop eigenvalues for the whole structure are computed. The iterations continue until the closed-loop eigenvalues cease changing considerably. The following is a summary of basic steps at ith iteration: The state space model of the kth substructure is given as z_ k ˆ ‰Ak ‡ FBk Šzk ‡ Bk uk ˆ Ak zk ‡ Bk uk ; where 2

3 dAk 6 dBk 7 7 zk ˆ 6 4 d_ Ak 5; d_ Bk

 Ak ˆ

0 ÿMkÿ1 Kk

 I ; 0

…12†

 Bk ˆ

0

Mkÿ1 Dk

 ;

…13† and FBk zk ˆ

s  X jˆ1

0

Mkÿ1 FBjk

 ;

…14†

where it is assumed that the kth substructure is surrounded by s number of substructures. In Eq. (14), FBjk is the subcontroller force vector generated by the jth surrounding substructure of the kth substructure in the (i)1)th iteration. The subcontroller for the kth substructure is designed using the LQR feedback control law of Eqs. (5) and (6) as T uk ˆ ÿGk zk ˆ ÿRÿ1 k Bk Pk zk ;

…15†

where  T Tk ˆ k 0

 0 : Tk

Recently, the RBFNN has become popular because of its structural simplicity and training eciency [20,21]. An RBFNN consists of two fully connected layers, namely hidden and output layers as shown in Fig. 1. The input nodes are directly connected to the hidden layer neurons. The output of the jth hidden neuron can be written as u X ÿ cj hj ˆ ; …20† rj where hj is the output of the jth neuron, u, the nonlinear radial basis transfer function, X, the input vector, cj , neuronÕs center and rj , the center spread parameter. The nonlinearity of the RBFNN is due to its transfer function u. The most commonly used type of radial basis function is the Gaussian given as " # X ÿ cj 2 hj ˆ exp ÿ : …21† r2j The neurons of the output layer have a linear transfer function. It is simply the weighted summation of outputs of all hidden neurons connected to that output neuron. For the kth neuron, the output Yk is Yk ˆ

m X Wkj hj ; jˆ1

…16†

The subcontroller force generated within the kth substructure, f k , is found as 8 9 dA > > >   < k> = f Ak d fk ˆ ˆ Dk uk ˆ ÿDk Gk zk ˆ ÿDk Gk _ Bk ; d > f Bk > > : _ Ak > ; dBk …17† where f Ak and f Bk are the subcontroller forces at the internal and boundary DOF of the kth substructure, respectively. Eq. (11) is used to condense this subcontroller force f k into the boundary DOF of the surrounding substructures of the kth substructure. Eq. (17) is written as   d f k ˆ ÿDk Gk Tk _ Bk ; …18† dBk

…19†

4. Radial basis function neural networks

where Pk satis®es the ARE, T AkT Pk ‡ Pk Ak ÿ Pk Bk Rÿ1 k Bk Pk ‡ Qk ˆ 0:

577

Fig. 1. The structure of RBFNN.

…22†

578

M. Sunar et al. / Computers and Structures 78 (2000) 575±581

where Wkj is the synaptic weight connecting the hidden neuron j to the output neuron k and m is the number of hidden layer neurons. 5. Case study A 45-bar truss, ¯exible structure shown in Fig. 2 is considered as a numerical example to implement the substructure-based RBFNN control scheme outlined above. The structure is made of aluminum with modulus of elasticity E ˆ 107 psi and mass density q ˆ 0:1 lbm/ in.3 . The cross-sectional areas of all members are assumed to be 2 in.2 . The structure is decomposed into six substructures for the substructural control. The members contained in these substructures are given in Table 1. Note that the members 8, 9, 19, 20 and 23 appear twice in the substructure list due to their being at the boundary of two substructures. The cross-sectional areas of these members are taken as 1 in.2 in each substructure. The actuators are placed at nodes 7, 8, 9, 10, 13, 14, 18 and 19. These actuators can generate forces in the x and y directions. The Q and R weighting matrices of the

Table 1 Substructure partitioning of 45-bar truss Substructures

Members

1

1, 2, 3, 14, 15, 16, 17, 18, 19, 40, 41, 42, 43, 44, 45 4, 13, 19, 20, 38, 39 5, 6, 11, 12, 20, 21, 23, 34, 35, 36, 37 7, 8, 22, 25, 28, 29 8, 9, 23, 26, 30, 31 9, 10, 24, 27, 32, 33

2 3 4 5 6

LQR control scheme are chosen as Q ˆ 1000 IQ and R ˆ IR , where IQ and IR represent the identity matrices with dimensions of 72  72 and 16  16, respectively, when designing the LQR controller using the complete structural model (complete structural control). For the substructural model (substructural control), the sizes of these matrices will depend on the size of the substructure under consideration. An RBFNN is trained to emulate the LQR controller of the structure based on the substructural control. An LQR controller for the whole structure is designed using the substructural control technique described before. This is numerically much advantageous due to great reduction in the sizes of matrices used in solving the AREs. In this problem, the largest size of state matrices of substructures is 24  24. This size is due to the substructure 1, which is much smaller than that of the whole structure. The controller obtained by the substructural control technique is then used in training the RBFNN controller. The LQR feedback controllers at nodes 7, 8, 9, 14 and 18 are trained utilizing only the displacement and velocity sensor outputs at corresponding nodes. This is also much useful due to the collocation of sensors and actuators. Piezoelectric materials can be used for this purpose, as the simultaneous sensing and actuation phenomenon allows them to be used as sensors and actuators at the same time [22]. Unit step and sinusoidal forces in the x and y directions at node 20 are taken as input forces to the structure during the training. Speci®c training data for feedback controllers are given in Table 2. Hence, inputs to the RBFNN controller (X ) are taken Table 2 The neural network training of feedback controllers

Fig. 2. 45-bar truss.

Feedback controller node

Input forces for the training

7 8 9 14 18

Unit Unit Unit Unit Unit

step force in the x direction step force in the y direction sinusoidal force in the y direction sinusoidal force in the x direction step force in the y direction

M. Sunar et al. / Computers and Structures 78 (2000) 575±581

as displacement and velocity sensor outputs from only corresponding nodes, and outputs (Y ) are the controller outputs (ux and uy ). The RBFNN used in this work has one hidden layer and one output layer in addition to input sensors (nodes). The network is fully connected with four inputs (x and y displacements, and velocities),

579

Fig. 3. Closed-loop structure with RBFNN.

two neurons in the output layer (ux and uy ), and seven neurons in the hidden layer. The spread constant is taken as 0.000151 for training. After the completion of training, the RBFNN controller is assumed to be replacing the actual full-state feedback LQR controller as shown in Fig. 3, where C is the sensor output matrix. After the replacement, it is compared with the actual controller. The testing is carried out for untrained feedback controllers at nodes 10, 13 and 19 using the unit step and sinusoidal forces at node 20. Some simulation results displaying time in seconds as abscissa and controller force in lbm in./s2 as ordinate are shown in Figs. 4±7. These ®gures indicate almost a perfect match between the substructure-trained RBFNN controller and the LQR complete structural controller. Thus, the RBFNN controller can e€ectively replace the full-state LQR feedback controller of this large ¯exible structure.

Fig. 4. Controller outputs at node 10 due to a unit step force applied at node 20 in the x direction. The LQR complete structural controller (Ð) and the RBFNN controller (- - -).

Fig. 5. Controller outputs at node 13 due to a unit step force applied at node 20 in the y direction. The LQR complete structural controller (Ð) and the RBFNN controller (- - -).

580

M. Sunar et al. / Computers and Structures 78 (2000) 575±581

Fig. 6. Controller outputs at node 19 due to a unit step force applied at node 20 in the y direction. The LQR complete structural controller (Ð) and the RBFNN controller (- - -).

6. Conclusions A substructure-based RBFNN is proposed for the active control of large ¯exible structures. After decomposing a large ¯exible structure into substructures, subcontrollers are designed using the LQR control scheme and they are assembled to obtain the central global controller for the entire structure. This controller is then used to train the RBFNN via sensors collocated with actuators. The RBFNN controller is compared with the actual complete structural controller using various input forces to the structure and the results are found to be in good agreement. There are two distinct advantages of the proposed RBFNN controller. One is its substructure-based training, which yields much saving in computations. The other is the collocation of sensors and actuators, which is a desired property in control applications. Based on

Fig. 7. Controller outputs at node 10 due to a unit sine force applied at node 20 in the x direction. The LQR complete structural controller (Ð) and (- - -) the RBFNN controller.

numerical results, it is concluded that the RBFNN controller can replace the LQR complete structural controller on this large ¯exible structure. The use of LQR as the control law is without loss of generality. Substructural control schemes based on control laws other than LQR (e.g. H1 ) can be used to obtain the data for training the RBFNN controller. Alternatively, a substructural neural network can be designed with its own control architecture. Consequently, similar RBFNN approaches can be followed for other large ¯exible structures using other control strategies.

Acknowledgements The authors gratefully acknowledge the support provided by the King Fahd University of Petroleum and

M. Sunar et al. / Computers and Structures 78 (2000) 575±581

Minerals for carrying out this research. The authors also acknowledge the constructive comments given by the reviewers.

References [1] Lazarus KB, Crawley EF. Multivariable high-authority control of plate-like active structures. In: Proc AIAA/ ASME/ASCE/AHS/ASE 33rd Structures, Structural Dynamics, and Materials Conference, Part 2, vol. 2. AIAA, Washington, DC, 1992. p. 931±46. [2] Khot NS, Oz H. Structural-control optimization with H2 and H1 norm bounds. Optim Control Appl Meth 1997;18(4):297±311. [3] Campbell ME, Crawley EF. Classically rationalized loworder, robust structural controllers. J Guidance Control Dynam 1998;21(2):296±306. [4] Campbell ME, Crawley EF. Development of structural uncertainty models. J Guidance Control Dynam 1997; 20(5):841±9. [5] Sain PM, Sain MK, Spencer BF. Models for hysteresis and application to structural control. Proc Am Control Conf 1997;1:16±20. [6] Skelton RE. Some open problems in structural control. In: Proc Am Control Conf 1995;3:2339±43. [7] Crawley EF. Intelligent structures for aerospace: a technology overview and assessment. AIAA J 1994;32(8):1689± 99. [8] Sunar M, Rao SS. A substructures method for the active control of large periodic structures. Comput Struct 1997;65(5):695±701. [9] Sunar M, Rao SS. Substructure decomposition method for the control design of large ¯exible structures. AIAA J 1992;30(10):2573±5. [10] Su T-J, Babuska V, Craig Jr. RR. Substructure-based controller design method for ¯exible structures. J Guidance Control Dynam 1995;18(5):1053±1061.

581

[11] Efrati T, Flashner H. Neural network based tracking control for mechanical systems. In: Proc IEEE Conf Decision Control, vol. 3. IEEE, Piscataway, NJ, 1997. p. 2501±6. [12] Szepesvari C, Cimmer S, Lorincz A. Neurocontroller using dynamic state feedback for compensatory control. Neural Networks 1997;10(9):1691±708. [13] Yesildirek A, Vandergrift MW, Lewis FL. Neural network controller for ¯exible link robots. J Intelligent Robotic Sys: Theory and Applications 1996;17(4):327±49. [14] Kim YH, Lewis FL. Output feedback control of rigid robots using dynamic neural networks. In: Proc IEEE Int Conf Robotics Automation, vol. 2. IEEE, Piscataway, NJ, 1996. p. 1923±28. [15] Jung S, Hsia TC. New neural network control techniques for robot manipulators. Proc Am Control Conf 1995;1:878±82. [16] Jain S, Peng P-Y, Tzes A, Khorrami F. Neural network design with genetic learning for control of a single link ¯exible manipulator. J Intelligent Robotic Syst: Theory and Applications 1996;15(2):135±51. [17] Yuan F, Puskorius GV, Feldkamp LA, Davis Jr. LI. Neural network control of a four-wheel ABS model. In: Midwest Symp Circuits and Systems, vol. 2. IEEE, Piscataway, NJ, 1994, p. 1503±6. [18] Nordgren RE, Meckl PH. Analytical comparison of a neural network and a model-based adaptive controller. IEEE Trans Neural Networks 1993;4(4):685±94. [19] Guyan RJ. Reduction of sti€ness and mass matrices. AIAA J 1965;3:380. [20] Broomhead DS, Lowe D. Multivariable functional interpolation and adaptive networks. Complex Sys 1998;2: 321±55. [21] Moody J, Darken CJ. Fast learning in networks of locally tuned processing units. Neural Comput 1989;I: 2281±94. [22] Anderson EH, Hagood NW. Simultaneous piezoelectric sensing/actuation: analysis and application to controlled structures. J Sound Vib 1994;174(5):617±39.