control//closed-loop stability//pole

A pole of a linear system is a root of the denominator of its transfer function, the same number as an eigenvalue of its state matrix, and engineers use poles as the compact way to describe and to specify how a loop responds: where the closed-loop poles sit says how fast it settles, whether it oscillates and whether it is stable at all. A specification such as *settle in half a second without ringing* is, in this language, a region of the plane where the poles must lie.

A pole of a linear system is a root of the denominator of its transfer function, the same number as an eigenvalue of its state matrix, and engineers use poles as the compact way to describe and to specify how a loop responds: where the closed-loop poles sit says how fast it settles, whether it oscillates and whether it is stable at all. A specification such as settle in half a second without ringing is, in this language, a region of the plane where the poles must lie.

Each pole is a complex number p=−σ+jωp=-\sigma+j\omegap=−σ+jω in the s-plane, and it contributes a mode epte^{pt}ept to the response. The real part sets the decay: a pole at −8-8−8 rad/s dies with a time constant of 1/81/81/8 s and is practically gone (within 2 %) after about 4/σ=0.54/\sigma=0.54/σ=0.5 s. The imaginary part sets an oscillation: a pair at −2±3j-2\pm3j−2±3j rings at 3 rad/s, a period of about 2.1 s, inside an envelope that decays with a time constant of 0.5 s. A pole with positive real part grows instead of decaying, so the left half-plane is the stable region. The damping ratio of a pair is ζ=σ/∣p∣\zeta=\sigma/|p|ζ=σ/∣p∣, the cosine of its angle from the negative real axis.

Feedback moves the poles, and that is what control design does. The plant's open-loop poles describe it alone; closing the loop puts the poles at the roots of the characteristic polynomial, and raising a gain slides them along paths (the root locus) that can cross into the right half-plane (closed-loop stability).

The slowest poles dominate. Modes from fast poles die early, so a loop is usually described by its pair closest to the imaginary axis, the dominant poles, and read as a second-order system.

Fast poles are not free. Pushing them left needs larger gains, which means more actuator effort, more amplified noise and less tolerance to delay; with a motor whose time constant is 30 ms, asking for poles at −200-200−200 rad/s is fiction. Choosing them sensibly is the problem pole placement leaves open and LQR answers.

A zero, a root of the numerator, does not decide stability but shapes the response. A zero in the right half-plane makes the output start the wrong way, as the level of a boiler drum swells before it falls when steam demand rises (plant).

In a sampled loop the stable region is the inside of the unit circle, with z=esTz=e^{sT}z=esT mapping one picture to the other (digital control).