Sir, I am trying to understand intuitively frequency analysis of stability. My conjecture is that frequency methods of stability makes sense only in Delay Differential Equation(DDE), not ODE. Am I right? I believe I am right because at 8:22 even you said Linear system is either stable or unstable. Feedback is essentially Delay Differential Equation. Is Limit cycle same as oscillatory instability in DDE? My understanding is: Stability of ODE does not depend on external forcing term, but DDE's stability depends on the forcing term. So, we study the stability of DDE by subjecting them to various frequencies. Am I correct?
If by ODE you mean ordinary differential equations, frequency-domain methods of stability analysis were born from the study of linear, continuous-time, time-invariant systems -- which are the kinds of systems you'd describe with linear ODEs. There's a lot of congruence between discrete-time linear systems analysis and continuous-time linear systems analysis, with just enough differences in the details to trip you up and make your life interesting. I'm pretty sure that what you're calling a DDE is what I'd call a difference equation, for example something like x(n) = a x(n-1) + b x(n-2) + c y(n) + d y(n-1), etc. A limit cycle isn't exactly the same as oscillatory instability in a linear system -- linear systems know no bounds, so an oscillation grows forever. In the real world, a physical system that you might model with a system of linear difference equations (or ODEs) won't oscillate without bound -- it'll either hit some nonlinearity that creates a more-or-less steady oscillation (this is a 'traditional' limit cycle), or it'll damage itself and stop working (with side effects ranging from a very boring nothing to very exciting parts flying around the room, or smoke pouring out of the thing).
Does a hard limit cycle system only enter this limit cycle *under a large enough disturbance* or also under a large enough *driving input*? From the pendulum example it seems that the electrical input is the same throughout, including at rest, and it is the *disturbance* to the system that puts it into its limit cycle, but would something that entered a limit cycle under *certain inputs* be deemed a "hard" limit cycle system too?
Whether any particular input to the system is a disturbance or a driving input is really a matter of attitude, intent, and terminology -- intended inputs get called "driving inputs" or just "inputs", while undesired inputs get called "disturbances". There's really no fundamental difference in the mathematics that makes an input to the system one thing or another. I would consider a hard limit cycle to be any limit cycle that isn't reached from the system at rest, but which is out there someplace in the system dynamics, waiting to fire up if it is excited. I think that those times when I've encountered hard limit cycles in control systems, it's been about 50/50 whether the first occurrence was due to a large command input, or a large disturbance.
I should have noted that depending on what community you're working with the definitions may change: we like to think that scientific and technical terms are unchanging and consistent, but they can mean radically different things from one group to another. "Permeability", for instance, means two entirely different things to a circuit designer who's buying magnetic components and a civil engineer who's designing a septic system.
Sorta-kinda. Strictly speaking it's not a limit cycle if it dies down. If you supplement your LC circuit or mass-spring device with a feedback amplifier of the right sort to make sustained oscillations, then yes, it's a limit cycle.
Very well explained. Thank you!
Sir, I am trying to understand intuitively frequency analysis of stability.
My conjecture is that frequency methods of stability makes sense only in Delay Differential Equation(DDE), not ODE. Am I right?
I believe I am right because at 8:22 even you said Linear system is either stable or unstable. Feedback is essentially Delay Differential Equation.
Is Limit cycle same as oscillatory instability in DDE?
My understanding is: Stability of ODE does not depend on external forcing term, but DDE's stability depends on the forcing term. So, we study the stability of DDE by subjecting them to various frequencies. Am I correct?
If by ODE you mean ordinary differential equations, frequency-domain methods of stability analysis were born from the study of linear, continuous-time, time-invariant systems -- which are the kinds of systems you'd describe with linear ODEs. There's a lot of congruence between discrete-time linear systems analysis and continuous-time linear systems analysis, with just enough differences in the details to trip you up and make your life interesting.
I'm pretty sure that what you're calling a DDE is what I'd call a difference equation, for example something like x(n) = a x(n-1) + b x(n-2) + c y(n) + d y(n-1), etc.
A limit cycle isn't exactly the same as oscillatory instability in a linear system -- linear systems know no bounds, so an oscillation grows forever. In the real world, a physical system that you might model with a system of linear difference equations (or ODEs) won't oscillate without bound -- it'll either hit some nonlinearity that creates a more-or-less steady oscillation (this is a 'traditional' limit cycle), or it'll damage itself and stop working (with side effects ranging from a very boring nothing to very exciting parts flying around the room, or smoke pouring out of the thing).
thanks..very nice explanation Sir
Thank you Very Much , Proff
Does a hard limit cycle system only enter this limit cycle *under a large enough disturbance* or also under a large enough *driving input*? From the pendulum example it seems that the electrical input is the same throughout, including at rest, and it is the *disturbance* to the system that puts it into its limit cycle, but would something that entered a limit cycle under *certain inputs* be deemed a "hard" limit cycle system too?
Whether any particular input to the system is a disturbance or a driving input is really a matter of attitude, intent, and terminology -- intended inputs get called "driving inputs" or just "inputs", while undesired inputs get called "disturbances". There's really no fundamental difference in the mathematics that makes an input to the system one thing or another.
I would consider a hard limit cycle to be any limit cycle that isn't reached from the system at rest, but which is out there someplace in the system dynamics, waiting to fire up if it is excited. I think that those times when I've encountered hard limit cycles in control systems, it's been about 50/50 whether the first occurrence was due to a large command input, or a large disturbance.
Wescott Design Services Great answer, thank you!
I should have noted that depending on what community you're working with the definitions may change: we like to think that scientific and technical terms are unchanging and consistent, but they can mean radically different things from one group to another. "Permeability", for instance, means two entirely different things to a circuit designer who's buying magnetic components and a civil engineer who's designing a septic system.
Thanks. great video. Are the oscillations in LC circuits or mass-spring devices can be called limit Cycles if I am not wrong ?
Sorta-kinda. Strictly speaking it's not a limit cycle if it dies down. If you supplement your LC circuit or mass-spring device with a feedback amplifier of the right sort to make sustained oscillations, then yes, it's a limit cycle.
@@wescottdesignservices8928 If the oscillations dont die , will it not be called marginally stable systems
thanks alot ...