RNN for Parameter Estimation

The idea is to estimate the parameters of a dynamic system

Подпись: (9.72)5c = Ax + Bu; x(0) = x0

Подпись: Measured response

image188,image189,image190,image191

Error

elements of A and B, and we obtain expressions dE/dA and dE/dB for A and B vectors, respectively, with ^ (•) = ^^=1 (•), after simplification:

 

(9.73)

 

(9.74)

 

dE

dA

dE

dB

 

— = ^xxT + ^ uxT — ^2 xxT
– A У^ xu + B Eu2 — E x u
In terms of individual elements of the matrices (for A(2,2) and B(2,1)), we obtain

 

(9.75)

 

dE d я11 dE d Я12 dE d Я21 dE d Я22

 

Я11 y^x^ + —-b1 E x1u x 1×1

ЯЦ УУ x1x2 + ••• + b^ УУ ux2 У^ x 1×2 Я21 Ex?+ ^+b2E ux1 — E x 2×1 Я21 E X1X2 ————————– b2 УУ ux2 — УУ. r2x2

 

(9.76)

 

dE

db1

dE

db2

 

Я11 УУ x1u ——————————— b1 Eu2 — E x 1 u

Я21 УУ x1u ——————————- b2 Eu2 — E x2u

 

(9.77)

 

j=1

Подпись:+ hi, j=i

and since bi = f (Xi), and Xi = (f-1)’b,• dPi

Подпись: 1 (f-1)'(Pi) Подпись: (9.82)

we have, (f 1)’/3,• = —; hence

Comparing expressions from Equations 9.76 and 9.77 to Equation 9.80, the expres­sions for the weight matrix W and the bias vector b are obtained as

Подпись: X21 X2X1 0 0 EUX1 0 EX1X2 X22 0 0 uX2 0 0 0 X21 X2X1 0 uX1 0 0 X1 X2 P X2 0 EUX2 X1 u EX2U 0 0 Eu2 0 0 0 X1 u EX2U 0 u2 EX 1x1 X]X 1X2 EX2X1 X]X2X2 EX 1U X]X 2И (9.83)

(9.84)

The algorithm for parameter estimation of a dynamic system is given as follows: (1) as the measurements of X, X, and u are available for a certain time interval T, compute W matrix and bias vector b; (2) choose initial values of b randomly; and (3) solve the following differential equation.

Подпись: db; dt Подпись: (9.85) (9.86)

Since bi = f (X;) and the sigmoid nonlinearities are known, by differentiating and simplifying, we get

Integration of Equation 9.85 would yield the solution to the parameter-estimation problem posed in equation error/RNN structure. Proper tuning of A and p is

essential. Often l is chosen as a small number, i. e., less than 1.0. The value of p is chosen such that when xt (of RNN) approaches ±1, f approaches ± p.