Question 3 of 8: Steady-State Error Design, Gain Margin, and Superposed Ramp Disturbance
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Paper format. 17-Phys-B5 Systems and Control, National Exams May 2018
— a three-hour closed-book examination with one double-sided handwritten formula/notes
sheet permitted and an approved calculator. Questions 1 and 2 are compulsory; candidates choose
three of the remaining six (3–8). Every question is nonetheless answered in full below so
the paper remains a complete study resource. All eight questions carry equal value (20 marks
each).
Reference texts. N. S. Nise, Control Systems Engineering, 7th ed.
(transient-response specifications, root locus, Routh–Hurwitz, frequency-response and lead
compensator design, state-space representation, steady-state errors); K. Ogata, Modern
Control Engineering, 5th ed. (root-locus and PID/rate-feedback compensator design,
controllability/observability, Mason's Gain Formula on signal-flow graphs).
Question 3: Steady-State Error Design, Gain Margin, and Superposed Ramp
Disturbance (20 marks)
Given. Same signal-flow graph and $G_{cl}(s),G_d(s)$ as Question 1
(general $K_p$); design target $e_{ss(ramp)}=0.5$ V/V (step error is already zero for any
stabilizing $K_p$, since the loop is Type 1).
Find. 1) $K_{op}$. 2) Stability and gain margin at $K_p=K_{op}$. 3)&4)
Total steady-state error under superposed reference and disturbance ramps, at $K_p=1$ and
$K_p=K_{op}$.
Approach. Use the general (symbolic-$K_p$) open-loop TF from Question 1
to get $K_v(K_p)$ and solve $1/K_v=0.5$ for $K_{op}$. The phase-crossover frequency of
$G_{open}$ does not depend on $K_p$ (a real scalar gain only shifts magnitude, never phase), so
it equals the $\omega_{osc}$ already found in Question 1 — giving the gain margin in
one line. For parts 3–4, superpose the reference-tracking error and the disturbance-induced
error via the final value theorem.
Part 1) — required $K_{op}$. From Question 1's signal-flow-graph
solution, the general (symbolic-$K_p$) open-loop transfer function (loop broken at the outer
summing node) is
$$G_{open}(s)=\frac{K_p(s+10)}{5s(s^2+2s+3)}\ \Rightarrow\
K_v=\lim_{s\to0}sG_{open}(s)=\frac{2K_p}{3}.$$
Setting $e_{ss(ramp)}=1/K_v=0.5$ gives $K_v=2\Rightarrow \boxed{K_{op}=3}$ — matching the
$K_{op}=3.0$ already used in Question 2, confirming that paper's data is exactly the
solution to THIS design requirement.
Part 2) — stability and gain margin at $K_p=3$. Since $0<3<3.75$
(Question 1's stable range), the system IS stable. The phase of $G_{open}(j\omega)$ does not
depend on $K_p$ (a positive real scalar), so the phase-crossover frequency is the same
$\omega_{pc}=\sqrt{15}/2=1.9365$ rad/s found as $\omega_{osc}$ in Question 1. There,
$|G_{open}(j\omega_{pc})|=K_{crit}\times(\text{unit gain factor})=1$ by definition of marginal
stability, i.e. the "unit-gain factor" equals $1/K_{crit}=1/3.75$. At the OPERATING gain
$K_p=3$: $$|G_{open}(j\omega_{pc})|=\frac{3}{3.75}=0.8\ \Rightarrow\
G_m=\frac{1}{0.8}=\boxed{1.25\ \text{V/V}}\ (\approx1.94\ \text{dB}).$$
Part 3) — total SSE at $K_p=1$. By superposition, the error is
$e(t)=\big[r(t)-y_r(t)\big]-y_d(t)$, where $y_r$ is the reference-driven output and $y_d$ the
disturbance-driven output. For a unit-feedback loop,
$$e_{ss}=\lim_{s\to0}s\left[\frac{R(s)}{1+G_{open}(s)}-G_d(s)T_d(s)\right].$$
With $R(s)=2/s^2$ (slope-2 ramp) this reference term is $2/K_v(K_p)$; with $T_d(s)=10/s^2$ the
disturbance term uses $G_d(s)=-(s^2+s)/[10s^3+20s^2+(30+2K_p)s+20K_p]$ from Question 1,
whose $s\to0$ limit gives $-\lim s\,G_d(s)T_d(s)=1/(2K_p)$. At $K_p=1$: $K_v=2/3$, so the
reference term is $2/(2/3)=3$, and the disturbance term is $1/2=0.5$; both errors add (the
disturbance opposes tracking in the same sense), giving
$$e_{ss,total}=3+0.5=\boxed{3.5}.$$
Part 4) — total SSE at $K_p=K_{op}=3$. Repeating with $K_p=3$:
$K_v=2$, reference term $=2/2=1$; disturbance term $=1/(2\times3)=1/6$; total
$$e_{ss,total}=1+\frac16=\boxed{\frac76\approx1.167}.$$
