control//closed-loop stability//characteristic polynomial
The characteristic polynomial of a linear system is the polynomial whose roots are its eigenvalues, \(\det(\lambda I-A)\), and in control the one that matters is the closed loop's: its coefficients are made of plant parameters and controller gains, so it shows on a single line how each gain moves the roots that decide whether the loop settles, rings or diverges. It is the first thing written down in a stability study, before any criterion is applied to it.
The characteristic polynomial of a linear system is the polynomial whose roots are its eigenvalues, det(λI−A)\det(\lambda I-A)det(λI−A), and in control the one that matters is the closed loop's: its coefficients are made of plant parameters and controller gains, so it shows on a single line how each gain moves the roots that decide whether the loop settles, rings or diverges. It is the first thing written down in a stability study, before any criterion is applied to it.
It comes out of the differential equation by trying a solution eλte^{\lambda t}eλt, which turns every derivative into a factor λ\lambdaλ. A PD controller on the altitude of a drone of mass mmm gives my¨+Kdy˙+Kpy=Kprm\ddot y+K_d\dot y+K_py=K_prmy¨+Kdy˙+Kpy=Kpr, the equation of a mass on a spring with a damper, and its polynomial is
mλ2+Kdλ+Kp=0,ωn=Kp/m,ζ=Kd2mKp.m\lambda^2+K_d\lambda+K_p=0,\qquad \omega_n=\sqrt{K_p/m},\qquad \zeta=\frac{K_d}{2\sqrt{mK_p}}.mλ2+Kdλ+Kp=0,ωn=Kp/m,ζ=2mKpKd.
The proportional gain sets the natural frequency and the derivative gain the damping ratio of a second-order system. With m=1.2m=1.2m=1.2 kg and 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, the classic compromise with about 5 % overshoot. Without derivative action ζ=0\zeta=0ζ=0 and the ideal drone bounces for ever.
The controller's memory is part of the polynomial. Adding integral action adds a state, so the polynomial becomes the cubic mλ3+Kdλ2+Kpλ+Kim\lambda^3+K_d\lambda^2+K_p\lambda+K_imλ3+Kdλ2+Kpλ+Ki, and a cubic can have roots in the right half-plane even with every gain positive; the Routh-Hurwitz criterion says when.
Open-loop and closed-loop polynomials describe different systems. The plant's own polynomial (mλ2m\lambda^2mλ2 for the hovering drone, two roots at zero) says what happens with no control; feedback replaces it, and attributing either set of roots to the system without naming the configuration is the usual confusion (closed-loop stability).
Design runs it backwards. Choose where the roots should be, multiply out (λ−p1)⋯(λ−pn)(\lambda-p_1)\cdots(\lambda-p_n)(λ−p1)⋯(λ−pn), and match coefficients with det(λI−(A−BK))\det\bigl(\lambda I-(A-BK)\bigr)det(λI−(A−BK)) to get the gains: that is pole placement. In the Laplace transform the same polynomial appears as the numerator of 1+L(s)1+L(s)1+L(s), and its roots are called poles.
A dead time breaks the polynomial form. With a delay τ\tauτ the equation gains a factor e−λτe^{-\lambda\tau}e−λτ and has infinitely many roots, which is why delayed loops are judged on their stability margins instead.