How Do You Calculate K for a Closed Loop Damping Ratio in Higher Order Systems?

In summary, To find the value of K that gives a closed loop damping ratio of 1/sqrt2, we are given the complimentary sensitivity function T(S)=(2KS+4K)/(s^3+162S^2+(320+2K)S+4K). However, this equation does not fit into the general equation because of the s^3 term on the bottom and the s term on the top. Additionally, it is unclear if a damping ratio is defined for a third-order system. Wikipedia defines damping ratio as a parameter for a second-order ordinary differential equation. The complementary sensitivity function only represents the output/input of a noiseless system, so it does not provide any further help in finding the value of K.
  • #1
vvl92
13
0
From control systems:
I am asked to find the value of K that gives the closed loop damping ratio of 1/sqrt2.
The value for the complimentary sensitivity is
T(S)=(2KS +4K)/(s^3 +162S^2 +(320+2K)S +4K)
so how do I find the value for K?

I tried putting it in the general equation, but it won't fit since on the bottom, there is a s^3 term, and on the top, there is a s term.

If it is not allowed on physics forums to give a proper answer, then please can you just lead me in the right direction? My notes have nothing in and I can't find anything online!

Thanks!
 
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  • #2
I don't even know if a damping ratio is defined for a third-order system. I would ask what the definition of damping ratio is for such a system.

To quote Wikipedia: "The damping ratio is a parameter, usually denoted by ζ (zeta),[1] that characterizes the frequency response of a second order ordinary differential equation."

You could force the three poles to all be barely real but that would give you critical damping, not what is asked for. And factoring a 3rd order polynomial is a bear anyway ...

BTW I looked up "complementary sensitivity function" and it's just output/input of a noiseless system. So no help there.
 
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FAQ: How Do You Calculate K for a Closed Loop Damping Ratio in Higher Order Systems?

1. What is closed loop damping ratio?

Closed loop damping ratio is a measure of the stability of a control system. It represents the ratio of the actual damping to the critical damping of the system. It is used to determine the response of a system to an input or disturbance, and is an important factor in designing and analyzing control systems.

2. How is closed loop damping ratio calculated?

Closed loop damping ratio is calculated by dividing the actual damping of a system by the critical damping. The critical damping is the minimum amount of damping required for a system to return to its equilibrium state without oscillating. It can be calculated using the transfer function of the system or by measuring the response of the system to a step input.

3. What is the significance of closed loop damping ratio in control systems?

The closed loop damping ratio is an important factor in determining the stability and performance of a control system. A higher damping ratio indicates a more stable system with a faster response time to disturbances. It also helps in designing controllers that can improve the performance of the system by adjusting the damping ratio to meet specific requirements.

4. How does closed loop damping ratio affect the response of a system?

The closed loop damping ratio directly affects the response of a system to an input or disturbance. A higher damping ratio results in a faster response time and a more stable system. On the other hand, a lower damping ratio can result in oscillations and instability in the system.

5. Can the closed loop damping ratio be adjusted?

Yes, the closed loop damping ratio can be adjusted by changing the parameters of the control system, such as the gain and time constants. This can be done manually or through the use of controllers, such as a PID controller, which can automatically adjust the damping ratio to meet specific requirements and optimize the performance of the system.

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