control//closed-loop stability
Closed-loop stability is the property of a feedback loop that returns to its equilibrium after a small push, judged on the interconnection of controller and plant rather than on either of them alone, and it is the first test any control design has to pass: a loop that is fast and accurate on paper and unstable in the field is a machine that shakes itself apart. The reason it needs its own name is that **feedback moves the eigenvalues** of the system it closes. That is its power and its danger. It can stabilize what is unstable on its own (a hovering drone does not come back to its spot by itself, its dynamics are a chain of integrators) and destabilize what was stable: a shower does not oscillate until someone tries to regulate it through three seconds of pipe.
Closed-loop stability is the property of a feedback loop that returns to its equilibrium after a small push, judged on the interconnection of controller and plant rather than on either of them alone, and it is the first test any control design has to pass: a loop that is fast and accurate on paper and unstable in the field is a machine that shakes itself apart. The reason it needs its own name is that feedback moves the eigenvalues of the system it closes. That is its power and its danger. It can stabilize what is unstable on its own (a hovering drone does not come back to its spot by itself, its dynamics are a chain of integrators) and destabilize what was stable: a shower does not oscillate until someone tries to regulate it through three seconds of pipe.
The test is the one for any linear system (equilibrium and stability): every eigenvalue with negative real part. What changes is the system it is applied to, which now contains the controller's gains and its own memory. For the altitude of a drone of mass mmm under a PID, looking for solutions eλte^{\lambda t}eλt of the closed loop gives
mλ3+Kdλ2+Kpλ+Ki=0,m\lambda^3+K_d\lambda^2+K_p\lambda+K_i=0,mλ3+Kdλ2+Kpλ+Ki=0,
and its three roots decide whether the drone settles, rings or climbs away. Without the integral term the polynomial is quadratic, mλ2+Kdλ+Kpm\lambda^2+K_d\lambda+K_pmλ2+Kdλ+Kp, and stable for any positive pair of gains; the integral raises the degree, and with it the possibility of instability.
Stability belongs to the loop, never to the plant or the controller alone.
The same PID that holds a drone steady destabilizes it once the integral gain passes a bound set by the other two gains and the mass, or once a heavy filter adds tens of milliseconds; the same plant is unstable open and stable closed. Every claim of stability names the configuration it is about.
The characteristic polynomial is where the loop's stability is written down. Its coefficients are made of gains and plant parameters, so it shows on one line which knob moves which root; it is the object to write first, before any criterion.
The Routh-Hurwitz criterion answers stable or unstable from those coefficients alone, without solving. It is the tool when a bound on a gain is wanted in closed form: for the cubic above it gives Ki<KdKp/mK_i<K_dK_p/mKi<KdKp/m, which falls when the drone picks up a payload.
The roots themselves are called poles in the language of the Laplace transform, and their position says how fast and how oscillatory the loop is, beyond the yes or no. Choosing where to put them is pole placement.
A stable loop can still be one push from the edge. The eigenvalues prove stability; the stability margins measure how much extra gain or delay the loop tolerates before losing it, which is what a plant that changes with load, battery or wear needs to know.
Eigenvalues of a linearization speak only near the equilibrium. For large excursions or a loop with saturation, the guarantee comes from a Lyapunov function, an energy-like quantity that always decreases along the motion.