Process Dynamics and Control
2. Analysis of Single-Loop Control Systems
Actuator: the control valve, \(G_a\)
\[ q=C_vf(\ell)\sqrt{\frac{\Delta P_v}{g_s}} \]
\(C_v\): valve coefficient, \(\ell\): lift (0–1), \(f(\ell)\): valve characteristic.
The actuator must cover the range of the manipulated variable and respond faster than the process.
Fail-safe action NEW: air-to-open (fail closed) or air-to-close (fail open), chosen for safety when the air supply fails. It fixes the sign of the valve gain.
Lumping I/P, valve and steam coil: \[ G_a=G_v=\frac{K_v}{\tau_vs+1},\quad K_v=0.6\ ^\circ\text{C/mA},\ \tau_v=0.2\ \text{min} \]
The heater loop assembled NEW
\[ G_p=\frac{0.85}{20s+1},\quad G_v=\frac{0.6}{0.2s+1} \]
\(G_m=K_m=0.16\) mA/°C (the transmitter above).
With P control, \(G_c=K_c\) (mA/mA): \[ K_{OL}=K_cK_vK_pK_m=0.0816\,K_c \] \[ \text{offset}=\frac{1}{1+K_{OL}} \]
The loop gain is the product of all the gains, so a sluggish transmitter or valve gain must be made up by \(K_c\). P control alone always leaves an offset, which motivates integral action.
Proportional control
\[ p(t)=p^s+K_ce(t)\ \Rightarrow\ G_c(s)=\frac{P(s)}{E(s)}=K_c \] (in deviation variables, \(e^s=0\))
- \(K_c\): proportional gain (controller sensitivity)
- proportional band: \(\text{PB}=\dfrac{100}{K_c}\,\%\)
Action: NEW reverse acting (\(K_c>0\)): output falls as the measurement rises. Direct acting (\(K_c<0\)) is needed when the process gain is negative, so that \(K_cK_p>0\).
- larger \(K_c\): faster response, smaller offset
- too large \(K_c\): oscillation, eventually instability
- the output saturates at \(p_{\min}\), \(p_{\max}\)
- offset always remains: to hold \(e=0\) at a new steady state, \(p\) must move away from \(p^s\), but \(p-p^s=K_ce\) needs \(e\ne0\)
Proportional-integral control
\[ p(t)=p^s+K_c\left[e(t)+\frac{1}{\tau_I}\int_0^te\,dt\right]\ \Rightarrow\ G_c(s)=K_c\left[1+\frac{1}{\tau_Is}\right] \]
\(\tau_I\): integral (reset) time; \(1/\tau_I\) is the reset rate. Integral action is never used alone. The integral keeps changing as long as \(e\ne0\), so it can supply the new output that a steady state needs while \(e=0\): no offset.
Reset windup: when \(e\) persists (start-up, large set-point changes) the integral grows until the valve saturates, then must “unwind”, which causes a long overshoot. Anti-windup stops integrating while the output is saturated.
The price of integral action
- removing the offset usually costs a more oscillatory response: short rise time, but more overshoot or a longer settling time
- so \(K_c\) is usually lowered when integral action is added
- more integral action (small \(\tau_I\)) is faster but less damped
Derivative kick
A set-point step makes \(de/dt\) infinite, and the filtered D produces a spike of about \(K_c/\alpha\) in the controller output.
Acting on the measurement instead of the error removes it: \(R\) reaches the output only through the PI part. For a load change (\(R\) constant) nothing changes, since then \(de/dt=-dC/dt\).
The three modes side by side
| P | I | D | |
|---|---|---|---|
| speed | ↑ | ↑ | – |
| offset | ↓ but stays | removed | – |
| overshoot | ↑ | ↑ | ↓ |
| stability margin | ↓ | ↓ | ↑ (usually) |
| noise sensitivity | – | – | ↑ |
Arrows: effect of increasing \(K_c\), \(1/\tau_I\) and \(\tau_D\). NEW
Start simple: P, then PI; add D only if it helps.
Examples 2.1–2.5 at a glance
| Ex | \(G_c\) | \(G_p\) | CL order | offset |
|---|---|---|---|---|
| 2.1 | P | 1st | 1 | \(\dfrac{1}{1+K_cK_p}\) |
| 2.2 | P | 2nd | 2 | \(\dfrac{1}{1+K_cK_p}\) |
| 2.3 | PI | 1st | 2 | 0 |
| 2.4 | PD | 1st | 1 | \(\dfrac{1}{1+K_cK_p}\) |
| 2.5 | P | \(K_p/s\) | 1 | 0 (servo) |
Example 2.6 Closed-loop simulation
import numpy as np, control as ct
import matplotlib.pyplot as plt
Gp = ct.tf(0.8, [5, 1])
t = np.linspace(0, 6, 601)
Gc = 10 # P
Gcl = ct.feedback(Gp*Gc, 1) # 8/9
plt.plot(t, ct.step_response(Gcl, t).outputs)
Gc = ct.tf([10, 10], [1, 0]) # PI
Gcl = ct.feedback(Gp*Gc, 1)
plt.plot(t, ct.step_response(Gcl, t).outputs)
P: \(K_{cl}=8/9\), offset 0.11. PI: no offset, \(\zeta=0.71\), 16% overshoot. In Simulink the same loop is a Step, a Sum, two gains with an integrator (PI) and a Transfer Fcn block.
Interactive: try the modes yourself NEW
Interactive version: seoultechpse.github.io/ProCon/ch2/ch2_analysis.html — process, mode, \(K_c\), \(\tau_I\), \(\tau_D\), set-point or load step.
