A First Course in FUZZY and NEURAL CONTROL

8.4. IDENTIFICATION OF TRASH IN COTTON 267

In many instances, the row vectors of matrixA(and the corresponding
elements ind) are obtained sequentially; hence, it is desirable to compute the
least squares estimate ofxin Equation 8.10 recursively. Let theithrow vector
of the matrixAdefinedinEquation8.10beaiand theithelement ofdbed(i);
thenx∗can be calculated recursively using the formulas

xi+1 = xi+Si+1aTi+1

≥

d(i+1)−ai+1xi

¥

(8.12)

Si+1 = Si−

SiaTi+1ai+1Si

1+ai+1SiaTi+1

,i=0, 1 ,∑∑∑,n− 1

x∗ = xn

with the initial conditions of

x 0 = 0

S 0 = γ∑I

whereγis a positive large number andIis the identity matrix.
The least squares estimate ofxin Equations 8.12 can also be interpreted as
aKalmanfilter for the process

x(k+1) = x(k) (8.13)

y(k)=A(k)x(k)+noise

wherex(k)≡xk, y(k)≡d(k),andA(k)=ak. Therefore, the formulas in
Equation 8.13 are usually referred to as a Kalmanfilter algorithm.
In the forward pass of the hybrid learning algorithm, node outputs go for-
ward until Layer 4 and the consequent parameters are identified by the least
squares method outlined above. In the backward pass, the signals that propa-
gate backwards are the error signals and the premise parameters are updated
by the gradient descent method. Table 8.4 summarizes the various activities
during each pass [36].

Table 8.4. Parameter update during the forward and backward

passes in the hybrid-learning procedure for ANFIS.

Signalflow direction Forward pass Backward pass

Consequent parameters Least-squares estimator Fixed

Premise parameters Fixed Gradient descent method

Signals Node outputs Error signals

ANFIS classification results The ANFIS algorithm was used to identify
the trash types, and its performance was evaluated as a classifier. The classi-
fication results were compared with the results obtained from fuzzy clustering
and backpropagation neural network algorithms. The inputs to the network
arearea,solidity,andEdifmeasures for the trash objects. These inputs form
the linguistic variables for the fuzzy ìIf...then...î rules and are assigned the
linguistic values as shown in Table 8.5.