Process Dynamics and Control
3. Tuning of Single-Loop Control Systems
Fitting the FOPDT model NEW
Two-point method (Smith): read \(t_{28}\) and \(t_{63}\) at 28.3 % and 63.2 % of the final change \[ \tau=1.5\,(t_{63}-t_{28}),\qquad \theta=t_{63}-\tau,\qquad K=\frac{\Delta C_m}{\Delta P} \]
Tangent method: draw the tangent at the inflection point. Its intercepts give \(\theta\) and \(\tau\).
Simple, but it overestimates \(\tau\) and underestimates \(\theta\), so it gives aggressive settings.
Minimum error-integral criteria
\[ \mathrm{ISE}=\int_0^\infty e^2(t)\,dt \] \[ \mathrm{IAE}=\int_0^\infty |e(t)|\,dt \] \[ \mathrm{ITAE}=\int_0^\infty t\,|e(t)|\,dt \]
- ISE weights large errors heavily: fast initial response, but oscillatory.
- IAE weights all errors equally.
- ITAE weights errors that persist: it penalizes slow oscillation, so the settings are well damped and conservative. Recommended. FIX
Ziegler–Nichols ultimate-cycle method (closed-loop test)
For processes that are open-loop stable but can be destabilized by feedback.
- P-only control: set \(\tau_I\) to its maximum and \(\tau_D\) to its minimum if they cannot be switched off.
- Raise \(K_c\) slowly, with small set-point or load bumps, until the oscillation is sustained.
- Record the ultimate gain \(K_{cu}\) and the ultimate period \(T_u\).
These are exactly the \(K_{cu}\) and \(\omega_u=2\pi/T_u\) of Ch 4 (Routh, direct substitution) and Ch 5 (gain margin).
With valve saturation (d) the oscillation stays bounded and looks sustained even above \(K_{cu}\), so check that the valve is not hitting its limits.
Example: the Van de Vusse reactor
\[ G_p(s)=\frac{0.5858\,(-0.3549s+1)}{0.1858s^2+0.8627s+1}\qquad\text{(RHP zero: inverse response)} \]
Raising \(K_c\) under P control (ultimate-cycle figure, two slides back):
\[ K_{cu}=4.150,\qquad T_u=1.46 \] FIX (the original read \(T_u=1.45\) off the plot; exact: \(\omega_u=4.297\), \(T_u=2\pi/\omega_u=1.462\))
| \(K_c\) | \(\tau_I\) | \(\tau_D\) | |
|---|---|---|---|
| P | 2.07 | — | — |
| PI | 1.89 | 1.22 | — |
| PID | 2.49 | 0.73 | 0.18 |
- P: offset, steady state 0.55
- PI: no offset, slow and oscillatory
- PID: fastest, but the deep inverse response (\(-0.88\)) is the price of high gain on an RHP-zero process
Example 3.6: responses
Interactive: compare the rules yourself NEW
Interactive version: seoultechpse.github.io/ProCon/ch3/ch3_tuning.html — vary \(\theta/\tau\), PI/PID and the IMC \(\tau_c\).
Choosing \(\tau_c\) NEW
\(\tau_c\) is the single knob that trades speed for robustness:
- smaller \(\tau_c\): faster, more overshoot, less tolerant of model error
- larger \(\tau_c\): slower, more robust
Rule of thumb: \(\tau_c\approx\theta\). For the IMC-PI on an exact FOPDT the loop becomes \(e^{-\theta s}/(2\theta s)\): \[ \mathrm{GM}=\pi=3.1,\qquad\mathrm{PM}=61^\circ \]
Example 3.7 IMC for the Van de Vusse reactor NEW
\(\tilde G_{p+}=-0.3549s+1\) (RHP zero), \(\;\tilde G_{p-}=\dfrac{0.5858}{0.1858s^2+0.8627s+1}\)
\(\tilde G_{p-}\) has relative degree 2, so take \(r=2\): \[ \frac{C}{R}=\frac{-0.3549s+1}{(\tau_cs+1)^2} \] (3.2-8) gives a PID with a derivative filter: \[ G_c=\frac{0.1858s^2+0.8627s+1}{0.5858\,s\,(\tau_c^2s+2\tau_c+0.3549)} \] \(K_c=\dfrac{0.8627}{0.5858(2\tau_c+0.3549)},\ \tau_I=0.863,\ \tau_D=0.215\)
No overshoot, and \(\tau_c\) sets the depth of the inverse response: \(-0.67\) for \(\tau_c=0.15\) (\(K_c=2.25\)), \(-0.27\) for \(\tau_c=0.3\) (\(K_c=1.54\)), against \(-0.88\) for ZN PID.
