natural frequencyof the system. (a) Determine ananalytical expression for the tw

Question

natural frequencyof the system.

(a) Determine ananalytical expression for the two poles of this system in termsof zeta and Wn.Hint: use

thequadratic formula!

For the remainderof this project, assume that Wn = 1. You willinvestigate how the parameter zeta affects thesystem response.

(b) Answer thefollowing questions about the pole locations.

(i) What valuesof the parameter zeta result inthe two poles being real (the imaginary part is zero)?

(ii) What valuesof the parameter zeta result in the two poles being imaginary (the real part iszero)?

(iii) What valuesof zeta yield two equal poles?

(iv) What valuesof zeta yield a stable system? Recall that a causalLTI system is stable if its poles are

in the left halfplane.

(c) Now considerthe form of the step response of the system, i.e., the output of the system when the

input is equalto u(t). Whatis Y (s) (Laplace transform of the output) when u(t) is the input?Using

partial fractionexpansion of Y(s) followed by an inverse transform, you cancalculate the output of

the system.Determine the form of the step response for the followingcases:

(i) the poles arereal and unequal

(ii) the twopoles are equal

(iii) the twopoles are complex conjugates

Note: youdo not need tocompute the coefcients of the partial fraction expansion! All thatis required

is to determine the form ofthe time-domain solution.

Explanation / Answer

natural frequencyof the system. (a) Determine ananalytical expression for the tw

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