Open vs Closed Loop

Part 9 · Control Loops

Key point: write \(T(s)\) low-entropy and you can sketch the open-loop Bode by inspection. Close the loop and the zeros stay put — the poles become the roots of \(1+T=0\), and that algebra is a different story.

Click a step, or Next, to set the lab. Slow / Mid / High walk the green zero; the closed-loop poles follow.

1
Open-loop is the stack you already know.
Origin pole, a low-frequency pole, a mid-band zero, a high-frequency pole. All four factors start on. Low-entropy: each factor is a breakpoint. ±20 dB/decade, ±90° per factor.
2
Open-loop Bode is \(T\). Closed-loop is \(G=T/(1+T)\).
At 10 Hz, \(T\) is large and ~−90° (the integrator). \(G\) sits at 0 dB and ~0°. That is tracking — do not read a 0 dB closed-loop Bode as “no gain.”
3
Slow zero — a CL pole follows.
\(G=N/(D+N)\). The numerator of \(T\) survives, so the zero is the same open and closed. A low \(f_z\) parks a closed-loop pole next to it. The step is slow.
4
Mid zero — the pair peels off.
\(f_z = 10\) kHz. Dominant closed-loop poles become a complex pair — two magenta × on the s-plane. Bode only sees \(f_n=\sqrt{\sigma^2+\omega^2}\), so |G| gets one × at that corner. The step rings.
5
High zero — barely moves the CL poles.
\(f_z = 40\) kHz, above \(f_c\). The zero stays; the closed-loop poles hardly care.

Open-loop you read off the page. Closed-loop needs a computer — expand \(D+N\) below. Next page is that cubic as \(K\) walks.

T(s) then T/(1+T)

Unity feedback · origin + LF pole + zero + HF pole

T(s) = 80.0 dB × 1 + s 2π · 10 kHz s · 1 + s 2π · 1.0 kHz · 1 + s 2π · 100 kHz

Gain K (dB) Pole at origin (1/s) LF pole kHz Mid-band zero kHz HF pole kHz

open-loop \(f_c\) → —

at 10 Hz, closed-loop —

step · —

Open-loop T

|T| phase × origin × LF ○ zero × HF

Closed-loop G = T/(1+T) · 0 dB / 0° at low f is tracking

|G| phase ○ zeros (stay) × CL poles

s-plane · log |σ| (kHz), LHP · jω linear

× origin × LF × HF ○ zeros × CL poles

Closed-loop step · dominant poles only

Zeros — —

Open-loop poles — —

Closed-loop poles (roots of \(1+T=0\)) — —

Grind G = T/(1+T) once

Open-loop you sketched by inspection. Closed-loop is a page of algebra that does not stay factored. The numbers below are live — drag \(f_z\) and watch the mess move. That is the point.

Write \(T = N/D\) low-entropy. \(K=10^{\mathrm{dB}/20}\). Here dB = 80.0, so \(K=\) 1.000e+4 (linear). Then \[ N(s)=K\Bigl(1+\frac{s}{\omega_z}\Bigr) \qquad D(s)=s\Bigl(1+\frac{s}{\omega_{p1}}\Bigr)\Bigl(1+\frac{s}{\omega_{p2}}\Bigr) \] with \(\omega_{p1}=2\pi\cdot\)1.00 kHz = —, \(\omega_z=2\pi\cdot\)10.00 kHz = —, \(\omega_{p2}=2\pi\cdot\)100.0 kHz = —.
Close the loop. Unity feedback: \(G = T/(1+T) = N/(D+N)\). The zeros of \(G\) are the zeros of \(N\) — same as open-loop. The poles of \(G\) are the roots of \(D+N=0\). That is no longer factored.
Expand \(D\). First the two parentheses: \[ \Bigl(1+\frac{s}{\omega_{p1}}\Bigr)\Bigl(1+\frac{s}{\omega_{p2}}\Bigr) = 1 + s\Bigl(\tfrac{1}{\omega_{p1}}+\tfrac{1}{\omega_{p2}}\Bigr) + \frac{s^2}{\omega_{p1}\omega_{p2}} \] then multiply by the origin pole \(s\): \[ D(s)= s + s^2\Bigl(\tfrac{1}{\omega_{p1}}+\tfrac{1}{\omega_{p2}}\Bigr) + \frac{s^3}{\omega_{p1}\omega_{p2}} \] Already uglier than \(s(1+s/\omega_{p1})(1+s/\omega_{p2})\).
Add \(N\). \(N(s)=K + (K/\omega_z)\,s\), so \(K/\omega_z=\) —. \[ D+N = \frac{s^3}{\omega_{p1}\omega_{p2}} + s^2\Bigl(\tfrac{1}{\omega_{p1}}+\tfrac{1}{\omega_{p2}}\Bigr) + s\Bigl(1+\tfrac{K}{\omega_z}\Bigr) + K \]
Clear the leftover \(\omega\)’s — multiply through by \(\omega_{p1}\omega_{p2}\). That product is —. You get a monic cubic. Coefficients are live, in rad/s: \[ s^3 + A s^2 + B s + C = 0 \] \(A=\omega_{p1}+\omega_{p2}=\) —, \(B=\omega_{p1}\omega_{p2}(1+K/\omega_z)=\) —, \(C=K\,\omega_{p1}\omega_{p2}=\) —. A cubic does not factor by inspection. Cardano, or a computer.
The computer factors it. Roots of \(D+N=0\), written as if they were pretty (they are not):
—
Drag Slow zero and watch a real root park next to \(\omega_z\). That dipole is the slow closed-loop mode. Open-loop never needed this page.

Moral: we shape \(T\), not \(G\). Next page (root locus) is this same cubic as \(K\) walks.