Question 7 of 8: Lead controller design in the frequency domain
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 2016, 3 hours, closed book (approved Casio/Sharp calculator plus one signed, double-sided 8.5 × 11" formula sheet; a Laplace-transform table and the standard ζ–overshoot and ζ–resonant-peak design plots are supplied). Questions 1 and 2 are compulsory; a complete paper is five questions, so candidates choose three of 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) — time response and second-order specs (Ch. 4), block reduction (Ch. 5), Routh–Hurwitz (Ch. 6), steady-state error and static error constants (Ch. 7), root locus (Ch. 8), PID and lead–lag design (Ch. 9–11), frequency response, Bode, Nyquist, gain/phase margins (Ch. 10–11), state space, controllability/observability (Ch. 3, 12); K. Ogata, Modern Control Engineering (5th ed., Prentice Hall) — dominant-poles modelling. All step responses, block diagrams, pole–zero maps, root loci, the Bode sketches and the Nyquist plot below are redrawn as inline figures.
Reading the supplied design charts. The percent overshoot chart uses $PO=100\,e^{-\zeta\pi/\sqrt{1-\zeta^2}}$ and the second-order model is $G_m(s)=K_{dc}\,\dfrac{\omega_n^2}{s^2+2\zeta\omega_n s+\omega_n^2}$; the figures on the exam paper (step responses, root loci, Bode plots) are read graphically, so the values below carry normal chart-reading tolerance.
Question 7 — Lead controller design in the frequency domain [6 + 4 + 10]
Check / scope. The plant transfer function is not given — only the graphical Bode data on Fig Q7.1 — so this is a graphical lead design and the numbers below carry chart-reading tolerance. A different but internally consistent design results from other defensible reads of the phase curve.
Part (1) — position constant and DC gain parameter
Uncompensated position constant. Type-0 system, $K_{pos,u}=10^{2.5/20}=1.33$, so $e_{ss(step),u}=\dfrac{1}{1+1.33}=\boxed{42.9\%}$.
Required compensated constant. $e_{ss}\le25\%\Rightarrow 1+K_{pos,c}\ge4\Rightarrow K_{pos,c}\ge3$; take $K_{pos,c}=3$.
Controller DC gain. $K_{pos,c}=a_0K_{pos,u}\Rightarrow a_0=\dfrac{3}{1.33}=\boxed{2.25}$.
Part (2) — phase margin and crossover targets
Damping from overshoot. $PO\le20\%\Rightarrow\zeta\ge0.456$, so aim for a phase margin $\Phi_{m,c}\approx100\zeta\approx\boxed{45^\circ}$.
Crossover from settling. $T_{settle(\pm2\%)}\ge1$ s bounds $\zeta\omega_n\le4$; with $\zeta=0.456$ this keeps $\omega_n\lesssim8.8$ and the gain crossover modest. Choose $\boxed{\omega_{cp,c}\approx5\text{ rad/s}}$ ($\zeta\omega_n\approx2.8$, $T_{settle}\approx1.4$ s $\ge1$ s).
Q7.1: designed lead network — the zero at $-2.33$ is nearer the origin than the pole at $-10.7$, so $G_c$ adds phase.
Part (3) — remaining lead parameters
Phase to add. Boosting the gain by $a_0$ pushes the crossover higher (where the plant phase is more negative), so the lead must supply about $\phi_{max}\approx40^\circ$ to reach $\Phi_{m,c}=45^\circ$.
Lead ratio. $\alpha=\dfrac{1-\sin\phi_{max}}{1+\sin\phi_{max}}=\dfrac{1-\sin40^\circ}{1+\sin40^\circ}=\boxed{0.217}$.
Place the peak at the crossover. $\tau=\dfrac{1}{\omega_{cp,c}\sqrt{\alpha}}=\dfrac{1}{5\sqrt{0.217}}=0.429$, so zero at $-1/\tau=-2.33$, pole at $-1/(\alpha\tau)=-10.7$.
Controller parameters. $a_1=a_0\tau=2.25(0.429)=\boxed{0.97}$, $b_1=\alpha\tau=\boxed{0.093}$, giving $\boxed{G_c(s)=\dfrac{0.97\,s+2.25}{0.093\,s+1}}$. Zero nearer the origin than the pole $\Rightarrow$ a lead controller.
