Bode Plot and Gain-Margin Design for a Dead-Time-Plus-Lag Process: 20-Bio-A2 Process Dynamics and Control, Undated paper, Q8

Problem 8: Bode Plot and Gain-Margin Design for a Dead-Time-Plus-Lag Process (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. $G_p(s)=\dfrac{e^{-0.2s}}{0.5s+1}$ ($\theta=0.2\ \text{s}$ dead time, $\tau=0.5\ \text{s}$ lag); proportional controller $K_c$.

Find. (a) the qualitative Bode magnitude and phase plots (slopes, corner frequency, extreme values); (b) $K_c$ such that the gain margin $=1.7$.

Bode magnitude: flat at $0\ \text{dB}$ for $\omega\ll2\ \text{rad/s}$ (dead time never attenuates magnitude), corner frequency $\omega_c=1/\tau=2\ \text{rad/s}$, rolling off at $-20\ \text{dB/decade}$ beyond the corner.

Bode phase: starts at $0^{\circ}$ ($\omega\to0$), passes $-90^{\circ}$ near the corner frequency, and (unlike a pure lag) keeps decreasing without bound as $\omega\to\infty$ because the dead time contributes an unbounded $-0.2\omega\ \text{rad}$ term — crossing $-180^{\circ}$ at $\omega_{pc}=8.953\ \text{rad/s}$.

Approach. Write exact magnitude/phase of $G_p(j\omega)$ (the dead time contributes phase only), read off the corner frequency and asymptotic slopes for the qualitative Bode plot, solve the transcendental phase-crossover condition $\angle G_p(j\omega_{pc})=-180^{\circ}$ numerically, then use $\text{GM}=1/[K_c|G_p(j\omega_{pc})|]$ to solve for $K_c$.

(a) Frequency response. $$|G_p(j\omega)|=\frac{1}{\sqrt{(0.5\omega)^2+1}},\qquad \angle G_p(j\omega)=-0.2\omega\ (\text{rad})-\arctan(0.5\omega).$$
(a) Extremes and corner. At $\omega=0$: $|G_p|=1$ ($0\ \text{dB}$), phase $=0^{\circ}$. Corner frequency (lag break point) $\omega_c=1/0.5=2\ \text{rad/s}$: magnitude flat below $\omega_c$, rolling off at $-20\ \text{dB/decade}$ above it. As $\omega\to\infty$: $|G_p|\to0$, and phase $\to-\infty$ (unbounded) — the dead-time term $-0.2\omega\ \text{rad}=-11.46\omega\ \text{deg}$ dominates and never saturates, unlike the lag's phase which alone would saturate at $-90^{\circ}$.
(b) Phase crossover $\omega_{pc}$. Solve $-0.2\omega-\arctan(0.5\omega)=-\pi$ numerically (bisection): $$\boxed{\omega_{pc}=8.953\ \text{rad/s}}\quad(\text{check: }-0.2(8.953)-\arctan[0.5(8.953)]=-180.0^{\circ}\ \checkmark).$$
(b) Magnitude at $\omega_{pc}$ (dead time contributes no attenuation). $$|G_p(j\omega_{pc})|=\frac{1}{\sqrt{(0.5\times8.953)^2+1}}=0.2180.$$
(b) Solve for $K_c$. Gain margin $\text{GM}=\dfrac{1}{K_c|G_p(j\omega_{pc})|}$; setting $\text{GM}=1.7$: $$\boxed{K_c=\frac{1}{1.7\times0.2180}=2.696.}$$ (Check: $K_c|G_p(j\omega_{pc})|=2.696\times0.2180=0.588=1/1.7$ $\checkmark$.) Note that a dead time of $\theta=0.1\ \text{s}$ instead of the $0.2\ \text{s}$ here would raise $\omega_{pc}$ ($16.887$ vs. $8.953\ \text{rad/s}$) and roughly double the achievable $K_c$ for the same gain margin ($5.00$ vs. $2.696$) — more dead time costs allowable gain.

Result	Value
Corner frequency	$\omega_c=2\ \text{rad/s}$; slope $0\to-20\ \text{dB/decade}$
DC / high-freq magnitude	$0\ \text{dB}$ at $\omega=0$; $\to-\infty$ as $\omega\to\infty$
Phase behaviour	$0^{\circ}\to-\infty$ (unbounded, dominated by dead time)
Phase-crossover frequency	$\omega_{pc}=8.953\ \text{rad/s}$
$K_c$ for GM $=1.7$	$K_c=2.696$

Concept & Theory

Topic. Bode-plot construction and gain-margin-based controller tuning for a dead-time-plus-first-order-lag process.

Key relations. Dead-time phase contribution $-\theta\omega$ (radians, unbounded); first-order lag magnitude/phase; $\text{GM}=1/|L(j\omega_{pc})|$ where $\angle L(j\omega_{pc})=-180^{\circ}$.

Why this works. A proportional controller scales $|G_p|$ uniformly at every frequency without changing phase, so the phase-crossover frequency $\omega_{pc}$ is $K_c$-independent and can be solved once from $G_p$ alone; the required $K_c$ then follows directly from the reciprocal-magnitude definition of gain margin at that frequency.

Common pitfall. Forgetting that dead time contributes only phase (never attenuates magnitude) when computing $|G_p(j\omega_{pc})|$, or mixing radians and degrees when evaluating the $-0.2\omega$ phase term.

Source. Seborg et al., Process Dynamics and Control, Ch. 14 (Bode stability criterion, gain margin, dead-time processes).

Question 8 of 8: Bode Plot and Gain-Margin Design for a Dead-Time-Plus-Lag Process

Problem 8: Bode Plot and Gain-Margin Design for a Dead-Time-Plus-Lag Process (20%)