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ARTICLE IN PRESS

Engineering Applications of Artiﬁcial Intelligence 16 (2003) 395–405

Learning soft computing control strategies in a modular neural network architecture S.K. Sharmaa,*, G.W. Irwina, M.O. Tokhib, S.F. McLoonec b

a School of EE Engineering, Queen’s University, Belfast BT9 5AH, UK Department of Automatic Control and Systems Engineering, University of Sheffield, S1 3JD, UK c Department of Electronic Engineering, National University of Ireland, Maynooth, Ireland

Received 29 May 2002; received in revised form 11 June 2003; accepted 11 June 2003

Abstract Modelling and control of nonlinear dynamical systems is a challenging problem since the dynamics of such systems change over their parameter space. Conventional methodologies for designing nonlinear control laws, such as gain scheduling, are effective because the designer partitions the overall complex control into a number of simpler sub-tasks. This paper describes a new genetic algorithm based method for the design of a modular neural network (MNN) control architecture that learns such partitions of an overall complex control task. Here a chromosome represents both the structure and parameters of an individual neural network in the MNN controller and a hierarchical fuzzy approach is used to select the chromosomes required to accomplish a given control task. This new strategy is applied to the end-point tracking of a single-link ﬂexible manipulator modelled from experimental data. Results show that the MNN controller is simple to design and produces superior performance compared to a single neural network (SNN) controller which is theoretically capable of achieving the desired trajectory. r 2003 Elsevier Ltd. All rights reserved. Keywords: Fuzzy logic; Genetic algorithms; Modular neuro-control; Flexible manipulator control

1. Introduction There is a high degree of uncertainty in using reducedorder approximate, dynamic models for analysis and control synthesis, particularly with respect to parameter variations and un-modelled, high-frequency dynamics. For this reason, neural network based learning has generated considerable interest due to the excellent capability of multi-layered neural networks for representing nonlinear dynamical systems (Zamarreno et al., 2000; Seng et al., 1999; Lu and Spurgeon, 1998). Neural network applications (Bishop, 1995) often incorporate a large number of neurons, thus requiring a great deal of computation for training and causing problems for error reduction. A recent trend in neural network design for large-scale problems is to split the original task into simpler subtasks, and use a subnetwork module for each one (Happel and Murre, 1994; *Corresponding author. Tel.: +44-289-027-4480; fax: +44-289-0664265. E-mail addresses: [email protected] (S.K. Sharma), [email protected] (M.O. Tokhi). 0952-1976/$ - see front matter r 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0952-1976(03)00070-8

Jacobs and Jordan, 1993; Kecman, 1996; Hodge et al., 1999). This divide-and-conquer strategy then leads to super-linear speedup in training. Further, it has been suggested that by combining the output of several subnetworks in an ensemble, one can improve the generalisation ability over that of a single large network (Jacobs and Jordan, 1993; Hanson and Salamon, 1990). It is believed that the use of a modular neural network (MNN) enables a wider use of artiﬁcial neural networks for large-scale systems. Embedding modularity into a neural network to perform local and encapsulated computation produces many advantages over a single network. It is also easier to encode a priori knowledge in a modular framework. In general a MNN is constructed from two types of network, as shown in Fig. 1, namely expert networks and a gating network (Jacobs and Jordan, 1993; Hodge et al., 1999). Expert networks compete to learn the training patterns and the gating network mediates this competition. During training, the weights of the expert and gating networks are adjusted simultaneously using the backpropagation algorithm (Rumelhart et al., 1986). In this paper the MNN learns to partition an input task

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Expert Network 1 y1

1

Expert Network 2 y2

α=3

0.8

g1

Gating Network

g2

α=2

0.6

α=1

0.4 0.2 tanh (αx)

y=g 1y1+g 2y2

Fig. 1. General MNN architecture.

0 -0.2 -0.4

NN1

-0.6

tanh(α 1 x)

-0.8

Input

Σ NNm

Output

-1 -1

-0.8

-0.6

-0.4

-0.2

0 x

0.2

0.4

0.6

0.8

1

Fig. 3. Activation function, tanh(ax) at the output of NN.

tanh(α m x) MNN

Fig. 2. A modular neural network (MNN) architecture.

into sub-tasks, allocating a different neural network (NN) to learn each one, as shown in Fig. 2. However accuracy of the MNN depends greatly on accurate fusion of the individual networks as decided by a gating network (Hodge et al., 1999). This paper presents a new method, using GAs (Goldberg, 1989; Holland, 1992) which removes the need for a gating network. Fusion of individual networks is decided by optimum slope selection of the activation function. The GA also optimises the structure and weights of the individual networks in the MNN. The paper is organised as follows. The modular neural network and the genetic learning process are described in Section 2. Section 3 deals with the experimental ﬂexible manipulator system which constitutes the application. Section 4 presents a set of results for neural control of the ﬂexible manipulator. These results show that the MNN controller is simple to design and produces superior performance compared to a single neural network (SNN) controller which is theoretically capable of achieving the desired trajectory. The conclusions are given in Section 5.

2. Genetic neural network architecture In the MNN architecture shown in Fig. 2, tanhðxÞ is the activation function in all the hidden neurons, while different activation functions tanhðai xÞ; i ¼ 1; 2; y; m are used in the output neurons. The overall MNN output is the sum of the outputs from all these subnetworks. An alternative to the MNN architecture is to use a single neural network (SNN) with the same overall number of neurons and weights.

The hyperbolic activation function, appearing in Fig. 3 for a=3.0, 2:0 and 1:0; shows that by increasing a; a larger neural response is achieved for the same input value. Thus, changing the ai values in the MNN regulates the individual contributions from each NN to the output. The next section describes the chromosome encoding of the MNN and equivalent SNN architectures. 2.1. Genetic encoding of neural networks Structured genetic coding is used to encode both the structure and the parameters of the MNN in a single chromosome. The genetic representation of the MNN is shown in Fig. 4. Each individual in the population is composed of three segments of strings to represent the modular network architecture; the ﬁrst deﬁnes the connectivity, the second encodes the connection weights and biases while the last segment denotes the activation function slope parameters ai : In the ﬁrst connection segment, each bit represents an individual connection and takes the value of 1 or 0 to indicate whether or not a connection exists. The total number of bits in this ﬁrst segment equals the total number of possible connections between the input and the hidden layer and between the hidden and the output layer. Since the weights, biases and slope parameters are numeric values, the second and last segments are encoded as a ﬁxed-length real-number string rather than as a bit string. Initially the connections are encoded randomly using binary bits 1 or 0 while the weights, biases and slope parameters are chosen randomly within the range 71.0. The corresponding SNN is genetically coded in the same way. Using GAs, the optimal network architecture and corresponding weight, biases and slope parameters can now be determined simultaneously. The next section describes the genetic learning scheme applied to the encoded chromosomes.

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2

1 segment 01011 NN 1

1101

----------

01111

NN2

1.5 - 0.2

0.6 - 0.1

NN 1

NN 2

NNm

Connectivity Matrix

nd

segment

3

--------------

0.3 - 0.4

α1

rd

397 segment α 2 -- -----

αm

NN m Slope parameter

Weights and Biases

Fig. 4. Genetic representation of the MNN.

Random Initialisation of Encoded Chromosomes

Ranking of Chromosome • Hierarchical Fuzzy Logic selection

No

Is maximum number of generations completed or error converged?

•

GA Operators Steady state GA

•

Crossover

•

Mutation

→ Uniform → Arithmetic → Biased → Unbiased

Yes Best chromosome is selected as the optimum representation. Fig. 5. Genetic learning scheme for neural networks.

2.2. Genetic learning for neural network The genetic learning scheme is illustrated in Fig. 5. The encoded chromosomes representing the neural network parameters are randomly initialised for genetic evaluation. The choice of ﬁtness function for the GA is central to this evolutionary design of a neural network architecture. The ﬁtness function not only constraints the set of neural architectures that can be represented, but also determines the efﬁciency and time-space complexity of the evolutionary design procedure as a whole. The GA is being used to ﬁnd both an optimal architecture as well as the weights, biases and slopes of the activation functions. Hence, the ﬁtness function must include not only a measure of accuracy (i.e. meansquare error) but also a feasibility measure of the network structure and its complexity in terms of the number of nodes and their connectivity. The resulting ﬁtness function should therefore be able to select a feasible network which satisﬁes these criteria and usually incorporates several constraints to achieve that. This can be achieved as follows: Suppose four constraints, namely number of correctly classiﬁed data points in training (b), change in error (’e), error (e) and number of connections (f), are chosen to deﬁne the optimum neural network architecture. Here the number of correctly classiﬁed data points are deﬁned when the actual output is within a certain desired limit from the required output. The ﬁrst three of these

constraints are responsible for tuning the neural network weights, whereas the last one is for optimising the neural structure. Now in general, the analytical expression for the ﬁtness function of an individual i in the population is of the form fitðiÞ ¼

k : k1 b þ k2 e’ þ k3 e þ k4 f

ð1Þ

Choosing an inappropriate value for the constants (k; k1 ; k2 ; k3 and k4 ) in the ﬁtness function can lead to a nearly infeasible structure or non-optimal solution. As the number of problem constraints increases it becomes increasingly difﬁcult to select such constants. Moreover, no consideration is given to priority in selection nor direction provided as to how a particular chromosome is moving in the problem domain. Such guidance is necessary for dynamic adaptation of a chromosome in the genetic evaluation. Here the ﬁrst priority is to select a chromosome associated with a feasible network and then to optimise f; the number of connections. Sharma and Tokhi (2000) recently have shown that, in this situation, a hierarchical fuzzy method can work better than the other conventional methods and is simple to apply. A hierarchical fuzzy method dynamically allocates a priority as well as optimising the direction of movement of a chromosome by dividing the multiple constraints into different tiers on the basis of priority. Fig. 6 shows a hierarchical fuzzy structure containing two such tiers. In the ﬁrst fuzzy logic level the number of

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β

Fitness Value

T1M1

. e (PM, NM)

First level

(PM, ZR)

(PM, NS)

e φ

T2M1

T2M3

T2M2

Fitness Value Second level

Fig. 6. Hierarchical fuzzy logic approach.

correctly classiﬁed data points (b) and the change in error (’e) are used as antecedents, while the ﬁtness value forms the consequent. The error (e) and the number of MNN connections (f) constitute antecedents at the second level. Here the change in error is deﬁned as e’n ¼ en en1 ;

ð2Þ

where en and en1 are the mean square errors at the nth and ðn 1Þth generations respectively. In Fig. 6, the ﬁrst level has one module T1M1 and the second level has three modules T2M1, T2M2 and T2M3. The fuzzy inference rules, corresponding to ﬁrst and second level modules, are shown in Tables 1 and 2 respectively. The ﬁrst level of fuzzy logic selects all chromosomes with acceptable ﬁtness in terms of the number of correctly classiﬁed data points and decreasing error. From these chromosomes those with fewer connections and less error are selected at the next fuzzy logic level. The second level in the fuzzy hierarchy thus handles chromosomes which have already been reﬁned previously. Since the ﬁrst priority is to select the chromosome with the maximum number of correctly classiﬁed data points and with decreasing error, the best chromosomes in every generation always come with a feasible solution. The number of data to be classiﬁed and the maximum number of connections are known at the beginning and the range of their membership functions (rmin, rmax ) can be pre-selected (Fig. 7). The ranges for the membership function e and e’ are set dynamically in such a way that a considerable number of chromosomes is always near to the solution. Figs. 8 and 9 show the respective membership functions of e’ and ﬁtness values. In this case, the dynamic range of the e is three times the minimum error in the population at every generation, whereas it is four times the minimum value for the e’: The minimum and maximum values for the membership function of f and b are 30% and 70% of the total. Whenever any chromosome goes out of range a penalty is automatically provided by the shape of membership function. A steady-state GA is applied to selected chromosomes (Michalewicz, 1994). The whole population is subjected to crossover and mutation. Uniform crossover (Wasser-

Table 1 Fuzzy inference rules for ﬁrst level module e’

NM NS ZR PS PM

b ZR

PS

PM

PB PM PS ZR ZR

PB PB PM PS ZR

T2M1 T2M2 T2M3 PM PS

Table 2 Fuzzy inference rules for the three second level modules f

ZR PS PM

e’ ZR

PS

PM

PB PB PM

PB PM PS

PM PS ZR

ZR

PS

rmin

PM

rmax

Fig. 7. Membership function of e; b and f:

man, 1993) is applied to the binary coded part of a chromosome and arithmetic crossover is applied to the decimal coded part (Michalewicz, 1994). Half of the remaining population is subjected to biased mutation, with 1 replaced by 0 and vice versa for binary coding and a small random number being added in the case of decimal encoding. The other half of the population is subjected to unbiased mutation with selected bits and weights being randomly replaced. Biased mutation provides uniformity and unbiased mutation ensures diversity in the population, leading to a better overall representation. The next section describes the

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accelerometer at the end-point of the ﬂexible arm measures the end-point acceleration. The ﬂexible arm is constructed using a piece of thin aluminium alloy and its physical parameters and characteristics are given in Table 3. It has been found that the vibration of the ﬂexible arm is dominated by the ﬁrst two or three resonance modes (Sharma, 2000). The hub-angle deﬁned the tip position, hub-velocity determined its angular velocity while end-point-acceleration provided the vibration associated with the ﬂexible manipulator movement. For the subsequent work discussed here, the ﬂexible manipulator was modelled using three separate SISO models with torque as a common input and hub-angle, hub-velocity and end-point acceleration as outputs. In this paper dynamic feedback from the manipulators is used as the basis for MNN and SNN based nonlinear controller designs. Some manipulator control schemes incorporating dynamic feedback for stabilisation and trajectory control is reported in DeLuca et al. (1985a, b), Spong et al. (1987) and Sira-Ramirez et al. (1990). In Stadenny and Belanger (1986) it has been shown that, if a state-space description of the closedloop dynamics is available, it is possible to use acceleration feedback to stabilise a manipulator. Kotnik and co-workers have carried out a comparative study on the control of a ﬂexible manipulator system using

application of this new soft computing technique for MNN control.

3. Flexible manipulator application A ﬂexible manipulator is a highly nonlinear system where position, velocity and acceleration feedback can be grouped in modular form, thus offering a suitable case study for MNNs. Fig. 10 shows the experimental rig consisting of a ﬂexible arm and associated measuring devices. The arm contains a ﬂexible link driven by a printed armature motor at the hub. The measuring devices are a shaft encoder, a tachometer, an accelerometer and strain-gauges along the length of the arm. Measurements from the strain-gauges were not employed in this study. The shaft encoder measures the manipulator hub angle and the tachometer measures its velocity. The

µ PM

PS

ZR

NS

NM

399

r max

r min range

Table 3 Parameters of the ﬂexible manipulator system

Fig. 8. Membership function of e’:

ZR

PS

PB

PM

Fmax

Fmin

Fig. 9. Membership function of ﬁtness value.

Parameter

Value

Length Width Thickness Mass density per unit volume Second moment of inertia, I Young’s modulus, E Moment of inertia, Ib Hub inertia, Ih

960.0 mm 19.008 mm 3.2004 mm 2710 kg m3 5.1924 1011 m4 71 109 N m2 0.04862 kg m2 5.86 104 kg m2

Shaft encoder HUB Accelerometer MOTOR

Torque input

Motor current Amplifier

Tachometer

Output

LPF 1

Input 1

LPF 2

Input 2

A/D & D/A

Digital to Hub-angle LPF Voltage 3 Processor

Input 3

RTI 815

End-point BPF Residuals

Input 4

Linear amplifier

Linear amplifier

Hub velocity

ISA-bus data communication

Fig. 10. Experimental setup of a ﬂexible manipulator.

COMPUTER 486DX2-50

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can thus be seen that, right from the beginning, the hierarchical fuzzy approach of selects the chromosomes with the desired criteria.

acceleration feedback (Kotnik et al., 1988). In all the above approaches, decoupling of dynamic feedback in modular form was not considered which is important to get the optimum effects of individual feedback on the overall control trajectories. Here a MNN using decoupled feedback on the basis of position, velocity and end-point acceleration of the ﬂexible manipulator system is considered and its performance compared to that of a SNN controller and shown to have better trajectory regulation. Two scenarios are analysed. In the ﬁrst the controller is designed to follow desired tip position with low vibration while in the second the controller tracks desired tip position and velocity with low vibration. The ﬂexible manipulator is considered with no payload for both cases, as described next.

3.2. Combined tip position and velocity tracking In controlling a ﬂexible manipulator it is also desirable to have a controller for tracking both the desired tip position and velocity with low vibration. To account for this additional desired velocity term within the genetic learning process, the GA selection method needs to be modiﬁed. This is achieved by incorporating the desired input/output velocity pairs from the hubvelocity model output into the correctly classiﬁed data along with the desired input/output tip position pairs and the absolute value of end-point acceleration for vibration suppression. In this manner, the hierarchical fuzzy approach of Section 2.2 searches for the chromosomes which have desirable features relating to tracking the tip position and velocity with reduced vibration. The implementation of both these controllers is considered next.

3.1. Tip position tracking The end-point acceleration model provides information about the vibration associated with the ﬂexible manipulator application. To reduce system vibration the absolute value of the end-point acceleration model output is required to be within a prescribed minimum limit. Accordingly the correctly classiﬁed data (b) includes information from the output of the end-point acceleration model for the reduction of vibration, along with the hub-angle model output required for the tracking of a desired tip position. A NN controller will be deemed to have a larger b value if it classiﬁes the desired tip position correctly and the absolute value of end-point acceleration output to less than say 1%. Accordingly, a NN controller which meets these criteria more often will have larger value of b and thus will be more acceptable at the ﬁrst level of the hierarchical fuzzy selection process (Fig. 6). Thus, the ﬁrst level ﬁlters the chromosomes which follow the desired tip position closely and with less vibration. The second level then selects the chromosomes from this subset which have fewer connections and less error. It

z-1

d

(k )

+

Σ -

Genetic learning

z-1

z-1

4. Implementation and simulation results Fig. 11 shows the MNN as a nonlinear controller for the ﬂexible manipulator system. Here three small subnetworks were employed. The MNN controller was used in series with the ﬂexible manipulator SISO model and genetic learning was applied to optimisation of the MNN parameters. The grouping of dynamic feedback in modular form was achieved as follows. The desired position yd ðkÞ and hub-angle feedback signal yðkÞ constituted the inputs to the ﬁrst NN controller ’ (NN1 ). The hub-velocity yðkÞ formed the input to the second NN controller (NN2 ), and the end-point accel. eration yðkÞ constituted the input to the third NN controller (NN3 ). Each NN was a multilayer perceptron

z-1 z-1

z-1 z-1

NN2

NN1

tanh(α 1 x)

MNN Controller

Σ

(k)

Hub angle Model

NN3

tanh(α 2 x)

+

z-1

tanh(α3 x)

Hub-velocity Model

Vin

End point acc. Model Flexible Manipulator

Fig. 11. Modular neural network controller for the ﬂexible manipulator.

. (k) .. (k)

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with one hidden layer. Fig. 12 illustrates the SNN controller with the same inputs as the MNN one and with tanhða xÞ as the activation function on the output neuron. A suitable combination of a1 ; a2 and a3 was required to achieve the desired closed-loop control performance. In the case of the SNN controller only, a was adjusted to achieve the required response. Both the SNN and MNN had eight inputs. In the MNN controller, the ﬁrst neural controller (NN1 ) had four inputs x% 1 where x% 1 ¼ ½ydk ; ydk1 ; yk1 ; yk2 : The second neural controller (NN2 ) had x% 2 ¼ ½y’ k1 ; y’ k2 and the third controller (NN3 ) had x% 3 ¼ ½y. k1 ; y. k2 as inputs. All the eight inputs to NN1 ; NN2 and NN3 were fed to a single large neural network in the SNN case. The number of hidden neurons was set to three for each NN in the MNN controller, while nine neurons were used in the SNN. Next, 20 chromosomes as represented in Fig. 4, were evaluated for 5 runs of 200 generations each. The resulting best chromosome was then selected to decode the controller and the a; a1 ; a2 and a3 parameters. In the MNN the maximum number of genes in a chromosome was chosen as 33 for the connectivity matrix and 36 for the weights/biases and slope parameters. In both cases a steady-state GA with crossover probability pc ¼ 0:65; biased mutation with pm ¼ 0:05 and unbiased mutation with pm ¼ 0:3 was used to optimise the controller parameters. Throughout this application study the experimental ﬂexible manipulator model was used for ofﬂine testing of the algorithm with 70% of the data used for training, the remaining 30% for testing. The nonlinear controller was then validated on-line for different trajectories.

401

0.8 rad. The performance of both SNN and MNN controller is now analysed and discussed. Fig. 13 shows the variation in mean-square-error over the generations and Fig. 14 illustrates the tip position (shaft angle) response of the manipulator for the SNN controller. Figs. 15 and 16 demonstrate the corresponding responses using the MNN controller. Note that, with the SNN controller, the desired angular position of 0.8 rad is not achieved after training the network for 200 generations and the response is also not sufﬁciently damped. It follows from these results that the proposed MNN control has good trajectory tracking and active damping performance. This may be due to the fact that there is less feedback of information in the SNN or it may require more generations for the GA to converge. The MNN is thus able to achieve a better trajectory regulation compared to the SNN with fewer parameters and with faster convergence of the GA. In this case the correctly classiﬁed pattern (b) in the hierarchical fuzzy selection approach only contains the information about desired tip position. In the next section, the effectiveness

0.02 0.018 0.016

Mean square error

0.014 0.012 0.01 0.008 0.006

4.1. Performance of MNN and SNN controllers 0.004

Three modal frequencies 11.11, 36.1 and 63.8 Hz were retained in the system model. These incorporate the vibrational energy content and their effect on the trajectory control is more dominant. The reference tip trajectory (yd ) to be tracked was a unit step command of

z-1

d

(k )

+ -

Genetic learning

z-1

z-1

z-1

0.002 0

20

40

60

80

120

140

160

Fig. 13. Mean square error of SNN.

z-1 z-1

z-1

z-1

Hub angle Model NN

V in SNN Controller

100

Number of generations

tanh (α x)

Hub-velocity Model End point acc. Model Flexible Manipulator

Fig. 12. SNN controller for ﬂexible manipulator.

(k) . (k) .. (k)

180

200

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4.2. Trajectory control by MNN

0.9 0.8

The modular neural network controller was now studied under two different operating conditions, each of which is now described in turn.

0.7

Shaft angle (rad)

0.6 0.5 0.4 0.3 0.2 0.1 0 - 0.1

0

0.5

1.0

1.5

2.0

Time (seconds)

Fig. 14. Tip position and shaft angle response of SNN. 0.005

Mean square error

0.004

0.003

0.002

0.001

0 20

40

60

80

100

120

140

160

180

200

Number of generations

Fig. 15. Mean square error of MNN. 1.2

1

4.2.1. Condition 1 Here the MNN was trained to track the desired position with low vibration and b thus had the desired input/output pairs for tip position and the absolute value of the end-point acceleration model for GA selection. A total elapsed time of 6 s with a sample time of 5 ms was used for testing. The trajectory followed and the vibration associated with the movement were recorded in each case. The required trajectory was set at 0 radian for the ﬁrst 1.5 s, then to 0.8 rad for the next 1.5 s, to 0.8 rad from 3 to 4.5 s and back to 0 rad beyond 4.5 s. The reference trajectory and the actual trajectory followed by the ﬂexible manipulator are shown in Fig. 17. It is clear that the ﬂexible manipulator was able to follow the desired trajectory very closely under MNN control. The end-point acceleration of the ﬂexible manipulator with, and without, the MNN controller is shown in Fig. 18. This shows that a signiﬁcant reduction in the vibration was achieved with the MNN controller. The results in Figs. 17 and 18 conﬁrm that optimising a1 ; a2 and a3 by genetic evaluation, with a priority-based approach through fuzzy logic based selection, will result in a MNN controller which produces better trajectory regulation with reduced system vibration. In this case b also included the low vibration information. To further analyse the MNN performance and to achieve a desired tip position at a required velocity and low vibration, an additional term relating to the required velocity was now included in b: The effectiveness of the resulting MNN controller after training will be shown next.

Shaft angle (rad)

0.8

0.6

1 desired actual

0.4

0.5 Hub-angle (rad)

0.2

0

-0.2 0

0.5

1.0

1.5

2.0

Time (seconds)

0

-0.5

Fig. 16. Tip position and shaft angle response of MNN.

of the proposed MNN is further investigated where b will also include knowledge of desired position, velocity and vibration as explained earlier.

-1

0

1

2

3 Time (sec)

Fig. 17. Hub-angle.

4

5

6

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With control

0.1 0 -0.1 -0.2 -0.3 -0.4

(a)

Without control

With control

0.2

-0.5 0

-10

Without control

Power SpectrumMagnitude (dB)

End-point acceleration (rad/sec2)

0.3

-20 -30 -40 -50 -60 -70

1

2

3

4

5

Time (sec)

403

6

1

20

40

60

80

100

Frequency (Hz)

(b)

Fig. 18. End point acceleration. (a) Time-domain. (b) Spectral density.

5. Concluding discussion A novel approach to the design of a modular neural controller with genetic learning has been described and investigated. Although the GA takes more time in training compared to backpropagation learning, it is a more general and ﬂexible approach to the construction of a modular neural network controller. It has been shown that it is relatively straightforward to design a NN in modular form and that the scheme requires an insigniﬁcant level of human intervention. The off-line construction of a modular NN controller for trajectory control of a ﬂexible manipulator has been demonstrated, compared to a single NN controller and shown to produce faster GA convergence. In the regular SNN more parameters are required to design a nonlinear controller. The modular aspect of the NN controller has

desired actual

35 Hub-angle (degree)

was used. Here, T is the total elapsed time of the desired trajectory. The control strategy was tested with T=1 s, y0=0 and yf=45 and a sample time of 5 ms. The required and actual positions achieved by the system under MNN control are shown in Fig. 19. The velocity of the ﬂexible manipulator is shown in Fig. 20 and the corresponding system vibration at the end-point of the ﬂexible manipulator in Fig. 21. Good position and velocity tracking were achieved and the system vibrations were reduced to a minimum within 0.03 s. The results in Figs. 19–21 demonstrate that very close trajectory tracking with reduced vibration is achieved, with different types of desired trajectories, using the MNN control strategy.

45 40

30 25 20 15 10 5 0 -5

0

0.25

0.5 Time (sec)

1.0

0.75

Fig. 19. Hub-angle of the ﬂexible manipulator.

90

desired actual

80 70 Hub-velocity (deg/sec)

4.2.2. Condition 2 To investigate combined position and velocity tracking for the ﬂexible manipulator application a sinusoidal acceleration of the form 2pt y. d ¼ a sin ð3Þ T

60 50 40 30 20 10 0 -10 0

0.25

0.5 Time (sec)

0.75

1.0

Fig. 20. Hub-velocity of the ﬂexible manipulator.

been analysed in detail. It has been shown that dominant factors can be incorporated easily into a modular neural network, and that the corresponding efforts can be optimised through genetic evolution so as to achieve smooth control performance at the desired

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404 0.025 0.02

End-point acceleration (deg/sec2)

0.015 0.01 0.005 0 -0.005 -0.01 -0.015 -0.02

0

0.25

0.5 Time (sec)

0.25

1.0

Fig. 21. End-point acceleration.

level. The dominant effect of any control term can easily be added or removed by just adding/removing a neural network into/from the control structure. The proposed control strategy can be applied to a wide range of applications. The scheme only requires prior knowledge of the grouping of the dominant control terms which is often known for a large number of applications. For example, the architecture is equally applicable to the problem of learning a piecewise feedforward control law (Jacobs and Jordan, 1993). Furthermore, the architecture may be useful for piecewise state reconstruction, system identiﬁcation, and for indirect approaches to learning control (Jordan and Rumelhart, 1992; Narendra and Parthasarathy, 1990). Also, some control problems may require learning models of different complexity in different regions of the plant’s parameter space. In this case, a MNN may contain expert networks with different characteristics (e.g., networks with different topologies or different regularisers) so that the architecture can allocate a network with an appropriate complexity to each region. Modular concepts can also be used in more classical approaches like open-loop control utilising shape command inputs, where the modular structure can be designed to removes the undesirable frequencies in the controller (Poerwanto, 1998). Future work will include the application of MNN in modelling and control of a pH neutralisation plant which is a highly nonlinear process with several distinct operating regions (Sharma et al., 2001) that are suitable for grouping in modular form.

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