# What is wrong in this 2nd order transfer function?

• Engineering
• cnh1995
In summary, the person has attached their attempt at a solution for a 2nd order system with a steady state dc gain of 0.9. They are unsure if they have correctly modified the transfer function to account for this dc gain and have also calculated the damping ratio and natural frequency. They are seeking help in correctly accounting for the dc gain and have since found out that their transfer function is correct, but the formula they used for the resonant peak was incorrect.

#### cnh1995

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Homework Statement
Find the parameters mentioned in the problem statement image i.e. peak time, peak overshoot, steady state error for unit step input.
Relevant Equations
Standard second order system transfer function.
I have attached my attempt at a solution.
In the solution image, I have computed 3 things:
1. System transfer function based on my understanding of the problem statement.
This is a 2nd order system with steady state dc gain=0.9. So I wrote the transfer function accordingly.
However, I strongly feel it is incorrect since this is not a "standard" 2nd order system anymore as the dc gain isn't 1. So I am not sure if I have modified the transfer function correctly to account for this dc gain of 0.9.

2. Damping ratio z: This I have computed again using the formula for "standard" 2nd order system's frequency response. But if the transfer function itself is incorrect, I feel this calculation is too.

3. Natural frequency: Same as 2.

How do I correctly account for this 0.9 dc gain and write the correct transfer function? What am I missing here? (Whatever it is, I am sure it's pretty fundamental).
Once I write the correct transfer function, other parameters can be computed easily.

Any help is appreciated!

#### Attachments

• 20220912_143332.jpg
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• 20220912_174427.jpg
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I have found your question only today (3 weeks after it was posred).
Question: Are you still interested in an answer?

I have solved this question, and got all answers correctly finally.
Turns out there is nothing wrong with my transfer function. Actually, the resonant peak formula that I used is wrong for this particular problem. It should have 0.9 in the numerator instead of 1.

## 1. What is a 2nd order transfer function?

A 2nd order transfer function is a mathematical representation of a system's input-output relationship, where the output depends on both the input and the system's dynamics. It is commonly used in control engineering and signal processing to model systems with two energy storage elements.

## 2. How is a 2nd order transfer function different from a 1st order transfer function?

A 1st order transfer function only has one energy storage element, while a 2nd order transfer function has two. This means that a 2nd order transfer function can capture more complex dynamics and has a higher degree of freedom in modeling a system's behavior.

## 3. What is wrong in a 2nd order transfer function if it has a pole at the origin?

A pole at the origin in a 2nd order transfer function means that the system has a time delay of zero. This is physically unrealistic and can lead to unstable behavior, as the system's output will become infinite for any non-zero input.

## 4. Can a 2nd order transfer function have complex poles?

Yes, a 2nd order transfer function can have complex poles, which represent oscillatory behavior in the system. However, the poles must be complex conjugates in order for the transfer function to be physically realizable.

## 5. How can I determine the stability of a 2nd order transfer function?

The stability of a 2nd order transfer function can be determined by looking at the location of its poles in the complex plane. If all the poles are in the left half-plane, the system is stable. If any poles are in the right half-plane, the system is unstable. Additionally, the damping ratio and natural frequency of the system can also affect its stability.