Imagine a generic feedback system, where A is forward gain and F is feedback gain. I will now go over two cases where in i will analyse this system.
- 1. Reference = 110, A=10, F=1
In this case, it can be easily thought that Error will be 10. Output will be 100. These values will not change as this condition is stable (maybe marginally stable, as we will see soon).Now, if i change Reference from 110 to 100, below things will happen as i move in time. Before that, to remind, this is negative feedback system as A and F are positive.
New error will be 100 - 100 = 0. As error is 0, output will be 0. As output is 0, output of feedback path will be 0. New error will be 100 - 0 = 100. For error of 100, output will be 1000. Feedback path output will thus be 1000. New error will be 100 - 1000 = -900. For -900 error, new output will be -9000. Hence, new feedback path output will be -9000. New error thus will be 100 - (-9000) = 9100. New output will be 91000.
Just to go over output sequence again, it is +0, 1000, -9000, +91000....
If i continue doing this, numbers are going to rise continuously. This system will not converge to any specific output value. So i believe output will diverge and if any saturation is there for let's say feed-forward gain A, then it will maximum reach that saturation value. Thus, what i have created is an oscillator.
To mention again, when reference was 110, output was stable. As soon as i changed step from 110 to 100, system starts to oscillate. This means system was marginally stable to begin with.
If i do that same analysis, but rather than A = 10, if i make A = 0.5 let's say (AF < 1), then systems becomes stable and output will always reach some steady state and it will not oscillate.
- 1. Reference = -90, A= -10, F=1
Notice that the only difference i have made is A is -10 rather than +10, and reference is -90 rather than 110 as in prev case.
In this case, i can calculate that error will be 10. Output will be -100. This condition is stable(again, marginally stable as we will see soon) and numbers will not change with time. Now if i change reference from -90 to -100, system will do through transitions as mentioned below.
New error will be -100 - (-100) = 0. New output will be 0. New feedback path output will be 0. New error will be -100 - 0 = -100. For this new error, output will be 1000. Feedback path output will be thus 1000. For this, new error will be -100 - 1000 = -1100. For this error, output will be 11000. For this new output, feedback path output is 11000. Thus new error is -100 - 11000 = -11100. For this, new output will be 111000.
Just to go over output sequence again, it is +0, +1000, +11000, +111000....
As noticed, in previous case output was diverging when |AF| > 1 and signs of each output sample were opposite. In this case, |AF| > 1 and signs of each output sample are same. This means in this case, output will go on rising without changing signs. If feedforward gain has saturation, it will saturate there and output will stay there. This is not an oscillator. Similarly, there can be another case where output goes to negative saturation limit and stays there.
In the above analysis, conclusions are:
1. If AF > 1
- If A*F is positive, one stability point can be easily found numerically. The system will be marginally stable. If i perturb the reference, output diverges and signs oscillates with consequent sample. So this is an oscillator.
- If A*F is negative, one stability point can be easily found numerically. The system will be marginally stable. If i perturb the reference, output diverges and signs don't change with consequent sample.
1. If AF < 1
- If A*F is positive, one stability point can be easily found numerically. If i perturb the reference, output converges to some output. So system is stable.
- If A*F is positive, one stability point can be easily found numerically. If i perturb the reference, output converges to some output. So system is stable.
Questions:
- Is this analysis valid for continuous time system as well ( I believe analysis i did is valid for discrete-time system).
- I think above mentioned conditions, if are somehow analysed in continuous time domain, will be stable(not marginally stable) and output will always converge to some output. It will never diverge like our analysis shows. Is this correct?
- If point 2 is correct, how can we think of continuous time system, by intuition?