Why do we need feedback in a control system?? (And basic functioning of an integrator)

 I still remember my introductory course on control systems. It was "not very well explained" to say the least. It was something which I learned as I needed to use these concepts in various advanced courses and projects. Just trying to share some of that.

So why do we need a feedback in a control system ?? A plain simple answer might be to sense how much deviation the system has from the desired response. Which is 100% correct. But how does this error help the "controller" actually control the output ?? I will take the example of a simple integrating controller (modeled by 1/s) to try and explain the function of feedback. 

 Let us say I give my system some input reference (ref) and the system now gives me some output (out).  Now ideally the output should be equal to the reference however due to  a multitude of reasons(system non-ideality, noise, or load applied to the system) in most real world scenarios the input will not match the output. That is why need a feedback and a controller. 

So how does this "controller" work?? For integrator it simply integrates the error and feeds that to the system. If the error is negative (ref < out) then that means we need to decrease the system output. The integrator does just that. till the time the error is negative it keeps on decreasing the output. (integration of negative numbers with time gives us a constantly decreasing output). The reverse is true in case the error is positive (ref > out). 

Let me try and and explain it via some equations : 

Now if my error is negative fcontrol will be a decreasing function with time (you can differentiate on both sides and verify that). Since error = (ref - out) the control is attempting to decrease the system output. If error is positive then the controller will be trying to increase the system output.  

I will now take a very simple example  to demonstrate this. For now my system is a system is a simple RC circuit. 

 

Fig 1 : The system (modeled as a RC circuit)

 Vref is the reference voltage that is provided and Vout is the output of the system. If a step voltage is applied to this system it will have a typical first order response which will saturate at the Vref value. This is called the open loop response of the system. Which is shown below:



The output starts from zero and approaches the reference with time. In this case the system has no load (or any noise) so the output automatically "follows" the input.   Now what happens if we include a load on the system. As a load I simply add a resistor to the system. 


If the load resistance is 10 (ohm) (the load resistance being higher means lower load current. then the system settles at a different level. In the figure below it can be seen that the system now settles at around 4.5V even though the reference is 5V.



 However we would want our system to still settle at 5V (as that is our reference). For this purpose we use a feedback and controller. The basic logic is we "feed" the error (ref - out) into the controller and then the controller is the one driving the system. A basic block diagram is shown below:


The Ki/s block is the integrating controller. the (1 / 1+ RCs) block is the main plant. Now I try to simulate this in a circuit. (how the integrator block and adder block are implemented are described in another post). but on simulation these give the following output : 

    legend :

                    v(ref) reference voltage

                    V(out) : output voltage with feedback

                    V(OL) : output of the open loop system

  In the above figure, it can be seen that even with the load the system with feedback gives the correct output (matching with reference) while the open loop system settles at a different level. 


In this post I have tried to explain  the need and reasoning behind feedback control. I hope it was helpful for you. If you have any comments and feedback please do leave a comment.


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