Open loop transfer function - explaination

Question

Assume there is a simple closed-loop control system with G(s) representing the plant and H(s) representing the feedback element:

Could any one explain what exactly is open-loop transfer function ?

From what I have understood from textbook is that,

$$OLTF = G(s)*H(s)$$

From the internet,I found that, we cut the loop at the summing point and perform the reduction techinques to obtain OTLF.

The name is a bit confusing to understand and why do we even have to cut the loop in the first place? What is the real significance of obtaining OLTF? Please explain with any example/scenario.

Answer 1

Assume there is a simple closed-loop control system with G(s) representing the plant and H(s) representing the feedback element

schematic

^{simulate this circuit – Schematic created using CircuitLab}

The closed-loop transfer function (CLTF) is

$$ \frac{Y}{X}=\frac{G}{1+G\cdot H} $$

Could any one explain what exactly is open-loop transfer function ?

By definition, if you cut the loop right before the summing junction you'll get the open-loop transfer function:

schematic

^{simulate this circuit}

So the open-loop transfer function (OLTF) is

$$ OLTF = G\cdot H $$

... why do we even have to cut the loop in the first place? What is the real significance of obtaining OLTF?

If you look at the closed-loop system and its transfer function, the OLTP which is \$G\cdot H\$, appears there. So the OLTF plays a role on system response because, effectively, it gives you a lot of information about the system response. Therefore, by analysing the open-loop transfer function (i.e. finding the zeroes and poles) you can gather a lot of information about the system response such as stability and step response.

Please explain with any example/scenario.

Imagine you are riding a bike on a smooth tarmac road and trying to keep it on a straight line. Now the \$G\$ above is the combination of the road properties (e.g. whether it has any potholes along the straight line), bike properties (e.g. overall performance, max speed, etc) and your properties (e.g. how fast or how slow you can pedal, how fast or slow you can steer, etc); and \$H\$ is the combination of the feedback system which partially come from the bike itself (e.g. metal structure, tyres, etc to transfer the vibration to your body) and partially from yourself (e.g. hands, bottom, and eyes to feel the vibration, direction, and overall balance of the bike).

Here, \$G\cdot H\$ which is the open-loop transfer function gives you everything to analyse and ultimately determine if the closed-loop response is acceptable to keep the bike on the straight line. This analysis will also tell you what adjustments need to be made (i.e. whether zeroes or poles need to be added, or whether any of the proportional, derivative, or integral gains need to be changed) to keep the bike on the straight line.

If any of the variables in either \$G\$ or \$H\$ changes then the OLTF changes therefore the response changes. You can analyse the OLTF to see if the bike will be kept on the straight line. Examples:

• If you use a different bike (e.g. tricycle or a motorbike) then the OLTF changes because of the new \$G\$ (bike properties) and \$H\$ (structure, speedometer (if any), etc). OLTF analysis will tell you that you'll have to, for example, wear a helmet and adjust your steering and speeding/pedalling strategy accordingly to keep the bike on the straight line.
• If the road and the bike remain unchanged but you remove your helmet then again, OLTF changes because of the new \$H\$ (e.g. no visor). OLTF analysis will tell you what to do to keep the bike on the straight line under the new situations.

• Theja: To be clear a bit more, the OLTF (GH) is different from closed loop transfer function (G/(1+GH)) [At least visually]. So how do we get the properties of CLTF by *analysing OLTF?
• Rohat Kılıç: @Theja OLTF is the transfer function of the loop when it's broken just before the summing junction. So yes, OLTF and CLTF are totally different. So how do we get the properties of CLTF by *analysing OLTF? The answer is Root Locus Analysis. If you study RLA, you'll understand that the behaviour of a closed-loop system which is determined by its poles can be determined by the zeroes and poles of the open-loop system. This means that if you find the zeroes and poles of OLTF you can explain the behaviour of CLTF.