gBSB Snapshots

Stanislaw H. Żak ^{∗}

April 12, 2001

Abstract Adaptive fuzzy tracking controller for a class of dynamical systems is analyzed. It is shown that the closed-loop system driven by this fuzzy adaptive controller stable and the adaptation parameters are bounded.

1 Introduction

Example 1 In this example, we use a gBSB neural net with the state in the hypercube {−1, 1}^{25}. We construct the weight matrix to store the four letters of alphabet whose bit maps are shown in Figure 1. The bit maps are 5 × 5 with elements from the set {−1, 1} with −1 corresponding to black and 1 to white. The pattern vectors are obtained from the bit maps by stacking their columns. For example, if

∗School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907

Figure 1: Bit maps of the characters A, B, C, and D stored in the gBSB neural net of Example 1.

Figure 2: Snapshots of the iterations of the gBSB net of Example 1 converging towards the letter A.

A = [ a_{1}, a_{2}, a_{3}, a_{4}, a_{5} ], where a_{i}, i = 1, 2, …, 5 is the i-th column of A1, then the corresponding pattern vector is obtained from A by applying the stacking operator s : ℝ^{5×5} → ℝ^{25} to get

s(A) = [ a_{1} ; ⋮ ; a_{5} ] ∈ {−1, 1}^{25}.

The weight matrix, W, of the gBSB net was constructed using the weight matrix construction algorithm, where

V = [ A(:) B(:) C(:) D(:) ] .

The values of other parameters chosen in this example were: D = 1.1I_{25}, Λ = −2I_{25}, b = 0.3A(:) + 0.1B(:), and α = 0.5. Snapshots of the network iterates for different initial patterns are shown in Figures 2–5.

Figure 3: Snapshots of the iterations of the gBSB net of Example 1 converging towards the letter B.

Figure 4: Snapshots of the iterations of the gBSB net of Example 1 converging towards the letter C.

Figure 5: Snapshots of the iterations of the gBSB net of Example 1 converging towards the letter D.