⟿ G(s) = K ωp² / (s² + 2ζpωp s + ωp²) // CLOSED LOOP PID TRACKING

PID Control Design Lab

AUTOMATIC PI TUNING · POLE PLACEMENT · BODE & TIME RESPONSE
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PLANT MODEL
PLANT PARAMETERS (2nd-ORDER)
DESIRED CLOSED-LOOP
REFERENCE INPUT
TIME RESPONSE y(t)
CONTROL SIGNAL u(t)
GAIN & PHASE MARGINS (OPEN-LOOP BODE L(jω))
GAIN MARGIN GRAPH (Bode Magnitude |L(jω)|)
PHASE MARGIN GRAPH (Bode Phase ∠L(jω))
CLOSED-LOOP BODE RESPONSE (T(jω))
CLOSED-LOOP BODE MAGNITUDE |T(jω)|
CLOSED-LOOP BODE PHASE ∠T(jω)
STABILITY MARGINS (OPEN-LOOP L(s))
GAIN MARGIN (GM)
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phase crossover gain limit
PHASE MARGIN (PM)
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at gain crossover frequency
GAIN CROSSOVER (ωgc)
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frequency where |L| = 1 (0 dB)
PHASE CROSSOVER (ωpc)
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frequency where ∠L = -180°
ACTIVE DESIGN PID COEFFICIENTS
PROP. GAIN Kp
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proportional term
INT. GAIN Ki
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integral term
DERIV. GAIN Kd
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derivative term
THIRD REAL POLE -p3
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CLOSED-LOOP ZEROS
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GAINS & MARGINS ESTIMATION FOR FIXED DESIRED ωn AND DIFFERENT DAMPING RATIOS
Damping Ratio (ζ) Regime Kp Ki Kd Third Pole -p3 CL Zeros Gain Margin (GM) Phase Margin (PM) Stability Status
ⓘ Design Methodology: Pole Placement & PI Zero Effect (2nd-Order Plant)
The open-loop plant is a 2nd-order system: G(s) = K·ωp² / (s² + 2ζp·ωp·s + ωp²). Applying a PI controller C(s) = Kp + Ki/s yields a 3rd-order closed-loop system: T(s) = K·ωp²·(Kp·s + Ki) / (s³ + 2ζp·ωp·s² + ωp²(1 + K·Kp)·s + K·ωp²·Ki).
By placing the closed-loop poles at the desired complex roots s² + 2ζ·ωn·s + ωn² = 0 and a third real pole at -p3, the characteristic polynomial matches (s + p3)(s² + 2ζ·ωn·s + ωn²) = s³ + (2ζ·ωn + p3)s² + (ωn² + 2ζ·ωn·p3)s + p3·ωn² = 0.
Since the s² coefficient of the closed-loop is fixed by the plant parameters 2ζp·ωp, the third pole is determined by: p3 = 2ζp·ωp - 2ζ·ωn.
For stability, the third pole must be in the LHP (p3 > 0), which imposes a strict boundary on the design: ζ·ωn < ζp·ωp. If this boundary is violated, the system has an unstable pole, resulting in an unstable closed-loop response. The PI controller introduces a zero at z = -Ki/Kp, which typically increases the transient overshoot.