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What is actually meant by stability of a control system in practice? I searched many forums but most of them describe stability in terms of equations and definitions, but they don't describe what happens in a real-world system.
Consider above LRC System. What happens when a physical implementation of this system becomes unstable?
Imagine you're driving down the road, and trying to stay in a lane. Neither your car nor the lane is perfect, so you occasionally have to nudge the steering wheel to keep between the lines.
As long as you're just nudging the steering wheel, you're fine. But imagine now that your steering wheel was so sensitive that the smallest push you give it would turn the wheels too far. You drift a little right, so you turn the wheel left, but instead of returning to the middle you go too far. You're now too far left, so you turn the wheel right, but you push it too hard so you go even further to the right. Now you're further to the right, so you push it even harder to the left. And so on, until your car either crashes into the guardrails or flips.
This analogy describes, more or less, how a proportional controller works. A well-tuned proportional controller drives the system towards a desired setpoint, like the car between the lane lines. A poorly-tuned proportional controller (with \$K_P\$ too large) drives the system too hard, creating growing oscillations around a setpoint until the system either oscillates between maximum values, or explodes. In other words: the control system is making the system unstable.
As for your schematic: the LRC system you've posted has no active components, and will never go unstable. No matter what components you choose for L, R, and C, for any bounded input you'll have a bounded output. It is, at worst, marginally-stable when R = 0 (which never happens in reality). You can prove this by deriving the differential equation for this system, and proving its bounded-input-bounded-output (BIBO) stability, by either confirming that the limit of the impulse response at \$t=\infty\$ is finite, or by taking the Laplace transform of the differential equation and evaluating it at \$s=0\$.
Physical real-world manifestations of system instability are that some parameter (velocity, position, temperature, whatever) of the system grows until it is limited in some undesirable fashion, such as motors overheating or parts breaking or things burning up.
In mechanical systems this exhibits and buzzing or singing or lurching back and forth (if it's an oscillating instability), or in the system "running away" until it hits a stop or something breaks or whatever.
Note that different systems may exhibit substantially equivalent behaviors, and yet one system will be considered just fine, while the other could be considered to be horribly unstable.
As an example, consider a mechanical pendulum clock. If that pendulum isn't swinging back and forth* steadily, then that clock is broken and needs to be fixed.
On the other hand, if someone made an automatic control system for a crane, and any time it hoisted a load that load were to swing back and forth exactly like a pendulum clock, then you'd have construction crews leaving the area as fast as they could, and lawyers for the crane owner serving papers to lawyers for the crane manufacturer.
So it's complicated, and a matter of interpretation.
* In nonlinear dynamic systems theory the pendulum swinging is a "limit cycle".
An official control systems definition is that a stable system produces a bounded output for a bounded input (or a certain range of inputs for a conditionally stable system).
Not a particularly useful definition for a real system since everything real is bounded (if only by the system destroying itself), but okay for mathematical analysis. Note that a system that hunts or oscillates around a control point has a bounded output so it's 'stable' but maybe not ideal. A nuclear power plant is inherently unstable since you'll get an unbounded output to the point of self-destruction if the control rods are withdrawn a bit too much. An ideal series-wound DC motor is similarly unstable. As are some aircraft. Using a controller the unstable open-loop system can often be stabilized.
Your RLC circuit has a bounded output for a bounded input because there are losses in the R. Any real system with just passive components is stable, and any theoretical one with even a tiny bit of loss.
You may want to think of non-official inputs such as noise too. An op-amp connected as so:
simulate this circuit – Schematic created using CircuitLab
.. where the input is the power supply voltage, may be theoretically stable since 0V input will result in 0V output for an ideal op-amp with finite gain.
However the slightest bit of noise or offset voltage will result in the output deviating from 0V and picking up speed until something stops it (in this case, when it hits the supply rails). It's like something carefully balanced on a knife-edge (in a real case there's a bit of friction or whatever holding it from falling from tiny air currents or thermal noise). It is stable (well, until it isn't) so qualifies as "conditionally stable".
Stability is defined as the ability for all state variables to reach a constant value (not necessarily the set-point value).
Systems with damped oscillations are not unstable.
Systems with constant amplitude oscillations are marginally stable also called sinusoidally stable
The op's RLC circuit is unconditionally stable, but will have damped oscillations for a control ratio less than unity.
Specifically for your case the states are the capacitor voltage and the inductor current. (Other state variables can be chosen but I like these.
Contrary to some comments, this RLC circuit can be demonstrated as a closed loop feedback control system as follows:
simulate this circuit – Schematic created using CircuitLab
