Finding Pole-zero pattern of transfer fcn and Stability of LTI system

[aguntuk]

Homework Statement

The transfer function of an LTI system H(s) = (s^2 + 2)/(s^3+2s^2+2s+1)
Find the followings

i) pole-zero pattern of H(s)
ii) Stability of the system
iii) Impulse response h(t)

Homework Equations

Zero for which H(s) = 0 & Pole is for which H(s) = ∞

The Attempt at a Solution

Finding Zeros:
Here H(S) =0, if s^2 + 2 =0, so how to find out the solution for s from the equation, I tried for different combination for imaginary values of i.

Finding Poles:
Here H(S) =∞, if s^3+2s^2+2s+1=0

so, s^3+2s^2+2s+1 =(s+1) (s^2+s+1) . Can anyone find me the solution here to find out the poles?

Please help me to find out the zeros & poles so that I can find out the stability of the system.

 

[donpacino]

there are 2 zeros,
hint (-2)*(-2)=(2)*(2)

also the zeros will be purely imaginary.


there are 3 poles.
one of them is -1

you should be able to find the other 2

 

[aguntuk]

Well, I found the zeros = √2*i & -√2*i
poles = -1, (-1+√3*i)/2 & -(1+√3*i)/2

Now, I am confused with the stability here? What type of stability is this LTI system? Anyone?

 

[donpacino]

take the inverse laplace transform of the transfer function, and evaluate it as t approaches inf. If the expression approaches infinity, then the system is unstable. If the system is purely sinusoidal then it is marginally stable. If not the system is stable.

Now the being said there is a shortcut. The poles of a system determine stability. If any pole is in the right half plane, the system is unstable. If any pole is on the y-axis the system is marginally stable. Else, the system is stable

 

[rezeew]

To find the pole-zero pattern of the transfer function H(s), we can rewrite it in its factored form as H(s) = (s+√2i)(s-√2i)/(s+1)(s^2+s+1). This shows that there are two zeros at s = ±√2i and three poles at s = -1, s = (-1±√3i)/2. These zeros and poles can be plotted on the s-plane to visualize the pole-zero pattern of the system.

To determine the stability of the system, we need to analyze the location of the poles. If all the poles lie in the left half of the s-plane, then the system is stable. In this case, we have one pole at s = -1 which lies in the left half of the s-plane, and two complex conjugate poles at s = (-1±√3i)/2 which also lie in the left half of the s-plane. Therefore, the system is stable.

To find the impulse response h(t), we can use the inverse Laplace transform of the transfer function H(s). This can be done by using partial fraction decomposition to rewrite H(s) in a form that can be easily inverted. Once we have the inverse Laplace transform, we can use the properties of the impulse function to find the impulse response h(t).