Bode Plot and Gain Margin of an FOPDT Loop: 23-Chem-A6 Process Dynamics and Control, December 2014, Q8

Problem 8: Bode Plot and Gain Margin of an FOPDT Loop (20%)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. First-order-plus-dead-time loop $L=k_c\dfrac{e^{-0.1s}}{0.5s+1}$ (pole $\tau_p=0.5$, dead time $\theta=0.1$).

Find. (a) qualitative Bode asymptotes; (b) $k_c$ giving gain margin $1.7$.

Approach. Read the magnitude and phase asymptotes from the pole and all-pass delay, locate the phase-crossover frequency $\omega_{co}$ where $\angle L=-\pi$, then set the gain margin $=|L|^{-1}$ at $\omega_{co}$ to solve for $k_c$.

(a) Bode characteristics. $|L|=\dfrac{k_c}{\sqrt{1+0.25\omega^2}}$ (the dead time is all-pass) and $\varphi=-\arctan(0.5\omega)-0.1\omega$ (rad). Corner frequency $\omega=1/0.5=2$ rad/s. Low-frequency: AR$\to k_c$ (0 dB at $k_c=1$), slope 0, phase$\to0^\circ$. High-frequency: slope $-1$ ($-20$ dB/dec) from the single pole; phase falls without bound because the dead-time term $-0.1\omega$ keeps subtracting, crossing $-180^\circ$ at a finite frequency.
(b) Phase-crossover frequency. Set $\varphi=-\pi$: $\arctan(0.5\omega)+0.1\omega=\pi$. Solving numerically (Newton/bisection) gives $\omega_{co}\approx16.9$ rad/s. The proportional gain does not affect phase, so $\omega_{co}$ is fixed.
(b) Gain for GM = 1.7. $$\text{GM}=\frac{1}{|L(j\omega_{co})|}=\frac{\sqrt{1+0.25\omega_{co}^2}}{k_c}=1.7\ \Longrightarrow\ \boxed{k_c=\frac{\sqrt{1+0.25\omega_{co}^2}}{1.7}\approx5.0.}$$ A proportional gain of about 5 leaves a factor-of-1.7 cushion below the stability limit (the ultimate gain, GM$=1$, would be $k_{cu}\approx8.5$).

Problem 8(a): open-loop Bode plot (drawn at $k_c=1$). Magnitude is flat then breaks to $-20$ dB/dec at the corner $\omega=2$; phase starts at $0^\circ$ and, driven past $-90^\circ$ by the dead time, crosses $-180^\circ$ at $\omega_{co}\approx16.9$ — the frequency that sets the gain margin.

Quantity	Value
Corner frequency	$\omega=2$ rad/s
HF slope	$-20$ dB/decade
Phase-crossover $\omega_{co}$	$\approx16.9$ rad/s
(b) $k_c$ for GM$=1.7$	$\approx5.0$ ($k_{cu}\approx8.5$)

Concept & Theory

Topic. FOPDT Bode plot; designing a proportional controller to a target gain margin.

Key relations. AR$=k_c/\sqrt{1+(\tau_p\omega)^2}$; $\varphi=-\arctan(\tau_p\omega)-\theta\omega$; gain margin GM$=1/|L(j\omega_{co})|$ at $\angle L=-180^\circ$.

Why this works. Gain margin measures how much loop gain can be multiplied before marginal stability, read entirely at the phase-crossover frequency. Because a proportional controller scales the magnitude without shifting phase, $\omega_{co}$ is a property of the process alone, and choosing $k_c$ to hit a target GM is a one-line division.

Common pitfall. Working the dead-time phase in degrees while the term $\theta\omega$ is in radians; assuming the corner frequency (not the phase crossover) sets the gain margin.

Source. Seborg et al., Ch. 14 (Bode stability, gain/phase margins); Stephanopoulos, frequency-response design.

Question 8 of 8: Bode Plot and Gain Margin of an FOPDT Loop

Problem 8: Bode Plot and Gain Margin of an FOPDT Loop (20%)