PSI - Issue 14

J. Prawin et al. / Procedia Structural Integrity 14 (2019) 234–241 J. Prawin et.al.,/ Structural Integrity Procedia 00 (2018) 000–000

236

3

studies (Prawin J et. al., 2017 ). 2. Adaptive Volterra Filter Concept

Adaptive algorithms are used for kernel estimation in the Volterra approximation. Several adaptive algorithms (Paulo 2008; Budra et. al., 2005, Prawin et.al., 2017 & 2018) such as least mean square, recursive least mean square etc. are reported in the literature. In the present work, Least Mean Square algorithm (LMS) is preferred as it is straightforward to implement and involves least computations. For a general single/multi-degree of freedom system, the adaptive Volterra filter input and output functional representation can be written as (5) Where ( ) x n  is the estimated adaptive filter response, H(n) indicates the Volterra kernel coefficient vector and ( ) e f n indicates the force vector. The error signal e(n) is formed by subtracting adaptive filter output ( ) x n  from the actual response x(n), ( ) ( ) ( ) ( ) ( ) ( )  T e n =x n - x n = x n -H n f e n (6) ( )  T x n = H n f e n ( ) ( )

The objective of the adaptive Volterra filter model is to identify the Volterra kernel coefficients from the known input and output vector by minimizing the following equation

( ) 2 T E e n = E ( x n - x n )( x n - x n ) é ù é ù ê ú ê ú ë û ë û ( ) ( ) ( ) ( )  

(7)

Where the Volterra coefficients vector can be updated by the gradient of the mean square error as ( ) ( ) ( ) { } 2 1 μ H n H n E e n + = +  é ù ê ú ë û

(8)

é ê ê ê ê ê ê ê ê ê ê ê ê ê ë

ù ú ú ú ú ú ú ú ú ú ú ú ú ú û

( ) e n h n e n h n ( ) ( ) 0 ( )

¶

¶

¶

{

}

( ) { } 2 e n ( ) ( ) { } e n e n @  =  = 2 .

2

é ê ë

ù ú û

( )

( )

( ) e n f n ( )

2

2

E e n

e n

(9)



=

¶

e

1



( ) e n h L n

¶ ¶ -

( )

1

The recursive update equation from Eqn. (8) can be written as

1

1

( ) ( ) H n+1 = H n + μ e n f n , where 0 < μ < ( ) ( )

(10)

<

e

trace(R)

λ max

where R is the autocorrelation of the input force and µ is small positive constant (referred to as the step size) that determines the speed of convergence and also the stability of the LMS algorithm. For a single degree of freedom system with chosen order ‘p’ and same memory depth ‘M’, the number of