Kaplan, Sargent and Goodman 
tion 
Vo avet py = of (35) 
and it is required to estimate the coefficients a and b, assuming that 
measurements are made of the time histories of y and y. Solutions 
were obtained on the digital computer for Equation (35) with known 
coefficients having the values a=0.1 and b= 0.4, and these were 
the input information to the estimator and gain equations. The output 
time histories of the estimated and observed trajectories of y and 
y are shown in Figure 1, showing the rapid convergence of the es- 
timated values to the observed data and the ''tracking"' of these es- 
timates to the true values. The coefficient estimate were started 
with initial conditions (or ''guess' values) at a= 0.6 and b= 10, 
and the rapid convergence to the true values is exhibited in Figure 2. 
The second problem is represented by the equation 
y + ay + ay + 2 Yad o 0 (36) 
and it is required to find the value of the coefficient ag when 
the other coefficients are known (the values chosen are a, = 0. 05, 
a, = 0.3, "0a, = OF 01). Defining the four state variables 
st Me ily eter aw ge ee ieee: a, (37) 
Pitas =? 
Sau) eo sae 
2 
: (38) 
x, = -0. 05x, - xX, - 0. Olx, 
x, =i) 
and the observed variable is y or x, , with the observations given 
by the solution of Equation (36) with all the known coefficient values. 
The results obtained from the digital computer solution of this system 
(4 state variables, 16 Pi; equations) are shown in Figure 3, where 
the error signal €= x, - &, and the coefficient estimate x, are 
4 
given. The initial condition for x. (the actual system response) is 
2 
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