Performance Characterisitics Given a Zero in the System

[Kuriger9]

I have the equations to determine the rise time, peak time, percent overshoot, and settling time for a generic second-order system with no zeros in the system. Given a unit step input for the open-loop transfer function G(s)=(s+1)/(s²+2s+1) how do I analytically determine the performance characteristics (aside from using MATLAB)?


Thanks in advance!

 

[milesyoung]

In general, you're going to have to find the inverse Laplace transform of the response you're interested in and do a bit of calculus to derive relationships for rise time etc. These can quickly become unwieldy, so you often find that people just simulate the response and tune the parameters to fit their specifications.

That's not to say they're doing it blind though. If you read up a bit on how zeros affect the transient behavior of systems, you can get the hang of predicting their influence.

 

[Kuriger9]

In general, you're going to have to find the inverse Laplace transform of the response you're interested in and do a bit of calculus to derive relationships for rise time etc. These can quickly become unwieldy, so you often find that people just simulate the response and tune the parameters to fit their specifications.

That's not to say they're doing it blind though. If you read up a bit on how zeros affect the transient behavior of systems, you can get the hang of predicting their influence.

Thank you this certainly helps!

 

[milesyoung]

If you really want to find those relationships, I recommend you follow a text where they derive them for a second order system using calculus.

Then you can try to do it for the system you're interested in and get help with the specifics here if you get stuck.

 

[alphy]

I am happy to provide a response to your question.

To determine the performance characteristics of a system with no zeros, we can use the equations for rise time, peak time, percent overshoot, and settling time for a second-order system. These equations are:

1. Rise time (tr) = 0.35/ωn

2. Peak time (tp) = π/ωd

3. Percent overshoot (PO) = e^(-ζπ/√(1-ζ^2)) * 100%

4. Settling time (ts) = 4/ζωn

Where ωn is the natural frequency and ζ is the damping ratio, both of which can be determined from the transfer function.

For the given transfer function, we can first find the natural frequency by solving the equation ωn^2 = 1, which gives ωn = 1 rad/s. Next, we can find the damping ratio by solving the equation ζ = 1/2√(1-ωn^2) = 1/√2. This gives us a damping ratio of ζ = 0.707.

Using these values, we can now plug them into the equations above to determine the performance characteristics:

1. Rise time (tr) = 0.35/1 = 0.35 seconds

2. Peak time (tp) = π/√(1-0.707^2) = 3.14 seconds

3. Percent overshoot (PO) = e^(-0.707π/√(1-0.707^2)) * 100% = 0%

4. Settling time (ts) = 4/(0.707*1) = 5.66 seconds

Therefore, the performance characteristics for this system with no zeros are: rise time of 0.35 seconds, peak time of 3.14 seconds, percent overshoot of 0%, and settling time of 5.66 seconds.

It's important to note that these equations assume a second-order system with no external disturbances or noise. In real-world scenarios, these performance characteristics may vary due to external factors. Additionally, these equations are just one way to analytically determine the performance characteristics and there may be other methods or tools available, such as MATLAB, that can provide more accurate or detailed results