Question 4.11: Modeling a simple feedback system with excitation Find the r......
Modeling a simple feedback system with excitation
Find the response of the system in Figure 4.42 if a = 1, b = −1.5, x[n] = δ[n] and the system is initially at rest.
The difference equation for this system is
y[n] = a(x[n]− b y[n −1]) = x[n]+1.5 y[n −1].
The solution for times n ≥ 0 is the homogeneous solution of the form K_hz^n . Substituting and solving for z, we get z = 1.5. Therefore y[n]= K_h ( 1.5 )^n , n ≥ 0. The constant can be found by knowing the initial value of the response, which, from the system diagram, must be 1. Therefore
y[0] = 1 = K_h (1.5)^0⇒ K_h = 1
and
y[n] = (1.5)^n , n ≥ 0.
This solution obviously grows without bound so the system is unstable. If we chose b with a magnitude less than one, the system would be stable because the solution is of the form y[n] = b^n , n ≥ 0.