control//step response//second-order system
A second-order system is a dynamical system with two states (a position and a velocity, or two coupled stores) whose response to a step is fixed by two numbers, a natural frequency and a damping ratio, and it is the shape control engineers design towards: specifications of overshoot and settling time are written for it, and pole placement aims at it. Its standard form is
A second-order system is a dynamical system with two states (a position and a velocity, or two coupled stores) whose response to a step is fixed by two numbers, a natural frequency and a damping ratio, and it is the shape control engineers design towards: specifications of overshoot and settling time are written for it, and pole placement aims at it. Its standard form is
x¨+2ζωn x˙+ωn2 x=ωn2 u.\ddot x+2\zeta\omega_n\,\dot x+\omega_n^2\,x=\omega_n^2\,u .x¨+2ζωnx˙+ωn2x=ωn2u.
ωn\omega_nωn is the natural frequency, the rate at which the system would oscillate with no damping, and ζ\zetaζ the damping ratio, which says how much of that oscillation survives. With ζ>1\zeta>1ζ>1 the system is overdamped and creeps to its target like two first-order lags in series; ζ=1\zeta=1ζ=1 is critically damped, the fastest response without overshoot; below 1 it is underdamped and rings at ωn1−ζ2\omega_n\sqrt{1-\zeta^2}ωn1−ζ2; at zero it oscillates for ever.
It appears the moment a controller acts on a mass. A PD controller on a drone's altitude pushes back in proportion to the error, like a spring of stiffness KpK_pKp, and in proportion to the vertical speed, like a damper KdK_dKd, so the closed loop obeys my¨+Kdy˙+Kpy=Kprm\ddot y+K_d\dot y+K_p y=K_p rmy¨+Kdy˙+Kpy=Kpr, the mass on a spring and damper:
ωn=Kp/m,ζ=Kd2m Kp.\omega_n=\sqrt{K_p/m},\qquad \zeta=\frac{K_d}{2\sqrt{m\,K_p}} .ωn=Kp/m,ζ=2mKpKd.
For a 1.2 kg drone with Kp=12K_p=12Kp=12 N/m, ωn≈3.2\omega_n\approx3.2ωn≈3.2 rad/s; Kd=5.3K_d=5.3Kd=5.3 N·s/m gives ζ≈0.7\zeta\approx0.7ζ≈0.7, and without the D term the ideal model bounces for ever.
A damping ratio near 0.7 is the classic compromise.
The overshoot stays around 5 % and the settling time is close to the shortest possible; lower damping buys speed with ringing, higher damping buys calm with sluggishness, and the overshoot formula in step response turns any specification into a value of ζ\zetaζ.
The two numbers are the position of a pair of eigenvalues, −ζωn±jωn1−ζ2-\zeta\omega_n\pm j\omega_n\sqrt{1-\zeta^2}−ζωn±jωn1−ζ2: the real part sets the decay (settling to 2 % takes about 4/(ζωn)4/(\zeta\omega_n)4/(ζωn)), the imaginary part the ringing. Choosing ωn\omega_nωn and ζ\zetaζ is choosing where the poles go, which on a double integrator gives the gains directly (pole placement).
Higher-order loops are judged as if they were second order when one complex pair sits much closer to the imaginary axis than everything else (modes). An extra slow pole, a zero or a delay breaks the formulas, and saturation of the actuator makes a large step overshoot more than a small one.
Faster is not free. Raising ωn\omega_nωn means larger gains, more amplified sensor noise through the D term and less tolerance to delay (delay margin), so the natural frequency is bounded by the slowest part of the chain, often the motor itself.