Characteristics of a Second Order Discrete-Data System

To determine the stability of a second order discrete-data system, we need to analyze its characteristic equation. The characteristic equation is obtained by setting the transfer function of the system equal to zero.

The given characteristic equation is:
F(z) = z^2 - z + 0.25 = 0

Stability Analysis

To analyze the stability of the system, we need to determine the roots of the characteristic equation. The roots of the equation will indicate the behavior of the system.

Routh-Hurwitz Criterion

The Routh-Hurwitz criterion is a method used to analyze the stability of a system based on the coefficients of its characteristic equation. According to this criterion, if all the coefficients of the characteristic equation have the same sign, then the system is stable.

Deriving the Routh Array

To apply the Routh-Hurwitz criterion, we need to construct the Routh array using the coefficients of the characteristic equation.

The Routh array is constructed as follows:

1. Write down the coefficients of the characteristic equation in descending order.

2. Create two rows of the Routh array. The first row contains the coefficients of the even powers of z, while the second row contains the coefficients of the odd powers of z.

3. Fill in the remaining elements of the Routh array using the following formulas:

a. For the first element of each row, use the coefficients of the characteristic equation.

b. For the remaining elements, use the formulas:

R(i,j) = -R(i-2,1)*R(i-1,j+1)/R(i-1,1), where i > 2 and j > 1

R(i,j) = -R(i-2,1)/R(i-1,1), where i > 2 and j = 1

Applying the Routh-Hurwitz Criterion

Once we have constructed the Routh array, we can determine the stability of the system by examining the sign patterns of the first column of the array.

If all the elements in the first column have the same sign, the system is stable. If there is a sign change, the system is unstable. If there are zero elements in the first column, the system is marginally stable.

Applying the Routh-Hurwitz Criterion to the Given System

Using the given characteristic equation, let's construct the Routh array:

R(1,1) = 1
R(1,2) = 0.25
R(2,1) = 1
R(2,2) = 0

Based on the Routh array, we can see that all the elements in the first column have the same sign (positive). Therefore, the system is stable.

Conclusion

In conclusion, the given second order discrete-data system is stable based on the Routh-Hurwitz criterion. The characteristic equation of the system does not have any sign changes in the first column of the Routh array, indicating that all the roots of the equation have negative real parts.