Question 5 of 8: Signal-flow graph / Mason and proportional-control error
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Paper format. National Exams, 07-Elec-A2 Systems & Control, May 2014, 3 hours, closed book (approved calculator plus one signed 8.5 × 11" formula sheet). Questions 1 and 2 are compulsory; answer three of the remaining six (Q3–Q8). Each question is worth 20 marks. All eight are worked below so the set is a complete study resource.
Reference texts: N. S. Nise, Control Systems Engineering (7th ed., Wiley) — Routh–Hurwitz and stability (Ch. 6), steady-state error and error constants (Ch. 7), root locus (Ch. 8), PID and lag/lead design (Ch. 9–11), frequency response, Nyquist and gain/phase margins (Ch. 10–11), state space, controllability/observability and pole placement (Ch. 3, 12); K. Ogata, Modern Control Engineering (5th ed., Prentice Hall) — companion treatment of dominant-poles modelling and Mason’s rule; G. F. Franklin, J. D. Powell & A. Emami-Naeini, Feedback Control of Dynamic Systems. All block diagrams, pole–zero maps, root loci, Bode/Nyquist plots and signal-flow graphs below are redrawn as inline figures.
Reading the exam figures. Q1 supplies an open-loop Bode plot and Q6 supplies open-loop Bode plots of the process. Where a transfer function is given exactly in the text (Q1’s system is the same plant as Q2, stated exactly there), every graphical reading is also confirmed analytically and the exact model governs. Q6 is a pure graph-reading problem (no closed form given); its numeric reads are approximate and are flagged accordingly. Note the printed exam mislabels the header of the last two pages “May 2013” — this is the May 2014 sitting throughout.
Part A — Signal-flow graph & Mason verification [10]
Approach. Realise $G(s)=\dfrac{2s+50}{s^3+9s^2+26s+24}$ in controllable canonical form — a chain of three integrators with the denominator coefficients as feedback gains and the numerator coefficients as output feedforward taps — then confirm with Mason’s rule.
Completed signal-flow graph. Forward chain $U\!\to\!N2\!\to\!N3\!\to\!N4\!\to\!N5$ (gains $1,\tfrac1s,\tfrac1s,\tfrac1s$); feedback gains $-9,-26,-24$ from $N3,N4,N5$ to $N2$; output $Y=50\,N5+2\,N4$ (feedforward taps).
Denominator → feedback gains. $(s+2)(s+3)(s+4)=s^3+9s^2+26s+24$, so the three feedback branches into the summing node $N2$ are $-9$ (from $N3$), $-26$ (from $N4$) and $-24$ (from $N5$).
Numerator → feedforward gains. $2s+50$ has $b_1=2,b_0=50$, so the output taps are $2$ (from $N4$, the two-integrator state) and $50$ (from $N5$, the three-integrator state); the input branch $U\to N2$ has gain $1$.
Mason’s rule — forward paths & loops. Two forward paths $P_1=50/s^3$ (through all three integrators to the $50$ tap) and $P_2=2/s^2$ (to the $2$ tap). Three touching loops $L_1=-9/s,\ L_2=-26/s^2,\ L_3=-24/s^3$ (all share $N2$), so $\Delta=1+\tfrac{9}{s}+\tfrac{26}{s^2}+\tfrac{24}{s^3}$ and $\Delta_1=\Delta_2=1$.
Check: SFG drawn in controllable canonical form. The exam prints a partially-labelled graph; the completed graph above is the standard controllable-canonical realisation (input gain $1$; feedback $-9,-26,-24$ from the three integrator outputs; output feedforward $2$ and $50$). Mason’s formula reproduces $G(s)$ exactly, which is what Part A requires.
Part B — Proportional-control range for stability and $e_{ss}\le8\%$ [10]
Approach. Routh test bounds $K_p$ above; the position error constant bounds it below.
Practical range. Both conditions together: $$\boxed{5.52\le K_p\lt6.5625.}$$ A narrow but non-empty window — the error spec needs high gain while stability caps it.
