Comparison of Different Growing Radial Basis Functions Algorithms

Comparison of Different Growing Radial Basis Functions Algorithms for Control Systems Applications Mario Luca Fravolini+ #, Giampiero Campa* #, Marcello Napolitano* @, Michele La Cava+ @ +

Department of Electronic and Information Engineering Perugia University, 06100 Perugia, Italy * Department of Mechanical and Aerospace Eng. West Virginia University, Morgantown, WV 26506/6106

Abstract Supervised Growing Neural Networks (SGNNs) are a class of self-organizing maps without a predefined structure. In fact the structure of the approximators is generated autonomously from a set of training data. Recently, new algorithms for SGNNs have been proposed with the objective to provide improved performance for on-line sequential learning. In this work two important class of algorithms for SGNNs are compared: Resource Allocating Networks (RAN) and Dynamic Cell Structures (DCS). The main objective is to provide a clear comparative study, which could help to assess the performance among the different algorithms for on-line real time application purposes. The main performance criteria are: the accuracy following the same amount of training - in terms of standard deviation and estimation error trends - and the computational complexity of the algorithm. The comparison has been performed through two different studies. The 1st study is relative to the learning of a nonlinear 3-D function. The 2nd study is relative to the learning of a 3-D look-up table of a specific aerodynamic parameter of an aircraft

List of Acronyms DCS Dynamic Cell Structure EMRAN Extended Minimal Resource Allocating Networks GRBF Growing Radial Basis Function LDCS Linear Dynamic Cell Structure MRAN Minimal Resource Allocating Networks NN Neural Networks RAN Resource Allocating Networks RBF Radial Basis Function SGNN Supervised Growing Neural Networks

1. Introduction The need to control non-linear systems of increasing complexity creates challenges for the control engineer. The traditional approach based on the linearization of a nominal non-linear model followed by the design of linear control laws integrated through gain scheduling often provides inadequate performance. Therefore the introduction of specific non-linear control approaches might be required. # @

Research Assistant Professor Professor

One of the problems for the control of complex non-linear systems is often the difficulty in the identification of an accurate mathematical model of the system. To address this issue, in recent years adaptive 1,2 and online approximation methodologies 3,4,5 are receiving considerable attention. In this context, Neural Networks (NNs) are excellent candidate to deal with online approximation problems thanks to their functional approximation capabilities and to the availability of effective learning algorithms. However, real-time, on-line learning of non-linear functions requires additional developments in neural learning algorithms. In particular, guidelines for the selection of the approximation algorithm are very important since they can dramatically influence the performance of the controlled system. Therefore, aspects such as computational complexity, mapping accuracy generalization capabilities, and stability properties of the learning algorithm should be clearly addressed. This paper focuses on the comparison of the performance provided by different classes of recently developed NN approximators belonging to the category of Growing Radial Basis Function (GRBF) NNs6. In particular, the study is relative to Resource Allocating Networks (RANs) 7 and Dynamic Cell Structures (DCSs)8. GRBF-NNs proved to be very effective for on-line sequential learning and system identification purposes 9,10,11. The detailed comparison has been performed through 2 studies: The 1st study is relative to the learning of a non-linear 3-D function; the 2nd study is relative to the learning of a 3-D look-up table of an aircraft stability derivative.

2. Growing Radial Basis Functions A class of function approximators is given by the Radial Basis Functions Neural Networks (RBF-NNs) 12. RBF-NNs have been extensively used as the basic structure of NNs for non-linear system identification and on-line sequential learning because of their local specialization and global generalization capabilities. In a typical RBF-NN the nonlinear function y(•) is approximated through a weighted superimposition of basis functions. In the case of Gaussian basis functions, a RBF-NN is expressed as: M

yˆ ( x,θ ) =

∑w

i

i =1

 x−µ 2  i    2σ i2    ⋅e

θ = [ w1 ,..., wN , µ1 ,..., µ N ,σ 1 ,...,σ N ]

(1)

where x is the input vector and θ comprises the set of parameters to be tuned by the learning algorithm. While constructive procedures for determining the center positions and the variances are well known, the main problem of RBF is that the total number of neurons tends to grow dramatically with the input dimension. This problem becomes particularly important when large dimension RBFNNs have to be used in real-time problems. Thus the attention focused on a recently introduced class known of approximators known as Growing RBF-NNs 6. The main advantage of these architectures is that their dimensionality is not pre-defined but grows incrementally along with the complexity of the model. In fact the neurons are added only in the regions of the input domain where the mapping accuracy is low, leading to a compact final structure that promises to be particularly suitable for online implementation. Therefore the resulting NNs finds automatically the appropriate dimension which is functionally related to the complexity of the problem. In this contest, two important subclasses of algorithms are the Resource Allocating Network (RAN) 7 and the Dynamic Cell Structure (DCS) 8. In the following a short review of the algorithms is given. 2.1. Resource Allocating Networks This class of NNs 13 has shown to be particularly suitable for on-line system identification problems and sequential learning 7. In RANs the neurons are added only in the regions of the input domain where the mapping accuracy is low, leading to a compact final structure. The basics principles of RANs 7 are briefly reviewed below. During the on-line learning, as the input and output pairs (x(k), y(k)) are sequentially sampled, the dimension of the RAN grows according to the fulfillment of the following criteria: • Current estimation error criteria: the estimation error must be bigger than a threshold E1: e(k ) = y (k ) − yˆ (k ) > E1 (2) • Novelty criteria: the nearest center distance must be bigger than a threshold E2: M

inf x (k ) − µ j (k ) ≥ E2 j =1

• 1 N

(3)

Windowed mean error criteria: the windowed error must be bigger than a threshold E3:

∑

N

 y ( k − N + i ) − yˆ ( k − N + i )  ≥ E3

i =0 

(4)

When all the three criteria are satisfied a new neuron (M+1) is added to the network. This new neuron is initialized with the following center, variance and weight respectively: (5) µ M +1 (k ) = x ( k ) M

σ M +1 = λ inf x (k ) − µ j (k )

(6)

wM +1 (k ) = e(k ) = y ( k ) − yˆ ( k )

(7)

j =1

When one of the criteria is not satisfied, the vector θ(k) containing the tuning parameters of the RBF-NN is updated using the following relationship: θ (k + 1) = θ (k ) − η

∂yˆ (k ) ⋅ e( k ) ∂θ ( k ) ( k )

(8)

where e(k) is the prediction error and η is the learning rate. As for a fixed architecture RBF, following a Lyapunov analysis is it possible to define a dead zone term to (8) that assures the ultimate boundness of both the estimation error e(k) and the weights θ(k) in presence of modeling error 14. Furthermore, a pruning strategy is implemented to avoid an excessive increase of the size of the NN; the resulting algorithm is called the Minimal RAN (MRAN). A further extension of the algorithm named Extended-MRAN (EMRAN) was proposed 7. In EMRAN only the parameters of the most activated neuron are updated, while all the other are unchanged. This strategy implies a significant reduction of the number of parameters to be updated online. 2.2. Dynamic Cell Structures DCSs are significantly more complex than RANs and standard fixed dimension RBFs 8. In fact, in addiction to the basic interpolating structure (1) for the output estimation of the nonlinear function y(•), DCSs are characterized by an additional lateral connection structure between the neural units of the hidden layer. This structure is used to reflect the topology of the input manifold. In order to perform perfect topology learning, a competitive Hebbian learning is commonly used. Differently form RANs, in standard DCSs the centers are updated through competitive learning in Kohonen-like fashion 15 in order to learn the spatial input samples distribution. During the supervised learning the estimation error e(k) is accumulated locally and used to determine where to insert the next unit. This insertion criterion is quite different from the RAN one that is based on the criteria in (2-4). Originally developed for off-line applications, the DCS algorithm was recently extended to on-line supervised learning by using an estimation error modulated Kohonen Rule for centers adaptation 10. The sequential steps of the extended version for the on-line learning 10 of the basic DCS supervised algorithm 8 are reported below. 1. Start with two gaussian units (centers) a and b in random positions µa and µb and random activations wa and wb. Define the widths: σa= σb= || µa- µb||/2. 2. Read an input-output pair [x(k), y(k)]. 3. Find the nearest unit s1 and the second-nearest unit s2 to the pair [x(k), y(k)]. 4. Increment the age of all the edges associated with s1. 5. Evaluate the current estimation yˆ ( k ) using (1). 6. Add the absolute estimation error to the error variable associated to s1 : ∆err ( s1 ) = y ( k ) − yˆ ( k ) .

7. Move s1 and its topological neighbors sn toward x(k) using the fallowing error modulated Kohonen rule: µ s1 ( k + 1) = µ s1 ( k ) + g ( err / err ( s1 ) ) ⋅ ε1 ⋅ ( x − µ s1 ) (9) µ sn ( k + 1) = µ sn ( k ) + g ( err / err ( s1 ) ) ⋅ ε n ⋅ ( x − µ sn ) ∀n ∈ Neigh.

where g(●) is a positive monotonically decreasing function with g(0)=1 and g(1)=0 respectively; ε1 and εn are adaptation coefficients (ε1>> εn). 8. If s1 and s2 are connected by an edge, set the strength of this edge to 1. If this edge does not exist, create it an set the strength of this edge to 1 9. Decrease the strength of the other connections by a decay constant α (