Question 3.15: Zero-Input Response of a Second-Order System with Repeated R......
Zero-Input Response of a Second-Order System with Repeated Roots
Consider a second-order difference equation with repeated roots:
\left(E^2+6 E+9\right) y[n]=\left(2 E^2+6 E\right) x[n]
Determine the zero-input response y_0[n] if the initial conditions are y_0[-1]=-1 / 3 and y_0[-2]=-2 / 9.
The characteristic polynomial is \gamma^2+6 \gamma+9=(\gamma+3)^2, and we have a repeated characteristic root at \gamma=-3. The characteristic modes are (-3)^n \text { and } n(-3)^n. Hence, the zero-input response is
y_0[n]=\left(c_1+c_2 n\right)(-3)^n
Although we can determine the constants c_1 \text { and } c_2 from the initial conditions following a procedure similar to Ex. 3.13, we instead use MATLAB to perform the needed calculations.
Script File
>> c = inv([(-3)^(-1) -1*(-3)^(-1);(-3)^(-2) -2*(-3)^(-2)])*[-1/3;-2/9]
c = 4
3
Thus, the zero-input response is
y_0[n]=(4+3 n)(-3)^n \quad n \geq 0