Please help me in understanding the deep concept of the relation of BIBO stability and absolute summability.
1. The mathematical statement says,that, if the input is "bounded" that is within finite values ,then the output of the system, is also bounded. for a stable system.
2.In real systems , it is conceivable that the system is stable but the equipment fails if any component exceeds its specification. For example, in even a simple operational amplifier, if resistors are not chosen for proper heat dissipation , it might fail ,even though the circuit values assure a stable. response from a control point of view.. One other example is power system stability , where we encounter a breakdown if the torque angle exceeds a critical value during disturbances..
3 Another possibility is that the given definition ,could include limit cycles and attractors in chaos situations.
4 A possible application of summing n terms of input and output would be in Power system stability . Consider a sudden increase of load . in a power network. If we are able to sum power input and output at each node and impose some thresholds we can come up with a criteria for ensuring stable operation in real time.
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For a linear system, stability is usually defined in terms of the "bounded input/ bounded output" (BIBO) criteria.If a system is BIBO stable, then the output will be bounded for every input to the system that is bounded.
Based on our control systems , we know that a transfer function is a ratio of OP/IP . Be it OP or IP they are characterized by a differential equation . The solution to such a differential equation may end up have be infinity, the response needs to be finite and measurable , hence BIBO .
Denominator largely effect stability (1/0 ) , in the freq-domain (or s) it is the pole that give stability points (graphically)
In the time domain we convolve input and system definition or functioning ( simple is impulse ) to get the overall plant performance . This again needs to be within the limit to characterize it as time-invariant.
Graphically , summing would give you the area under the curve .
Electrical systems : we have many many limitations such as bandwidth / overshooting /undershooting limits ...
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