Exercises — Proportional navigation guidance — N·V_c·λ̇, derivation

PN law teen cheezein multiply karta hai — gain, closing speed, sightline turn-rate: $a_c=N{V}_c\dot{\lambda }=4\times 800\times 0.015=48{\text{ m/s}}^2.$ "g's" mein padhne ke liye $g$ se divide karo: $\frac{48}{9.81}\approx 4.89g.$ Kaisa lagta hai: ek modest, comfortably-flyable command — sightline barely drift kar rahi hai.

Definition se $V_c=-\dot{R}=-(-1200)=1200\text{ m/s}$. Kyunki $V_c>0$ hai, range shrink ho rahi hai — missile closing kar raha hai. Minus sign kyun hai: $\dot{R}$ negative hota hai jab distance decrease ho; sign flip karne se "closing" ek positive, intuitive number ban jaata hai.

Law $a_c=N{V}_c\dot{\lambda }$ ko $\dot{\lambda }$ ke liye invert karo: $\dot{\lambda }=\frac{a_c}{N{V}_c}=\frac{90}{3\times 1000}=0.03\text{ rad/s}.$ Kaisa lagta hai: sightline $0.03$ rad/s par drift kar rahi hai — missile us drift ko erase karne ke liye turn kar raha hai.

Require karo $N{V}_c\dot{\lambda }\le a_{c,\max }$: $N\le \frac{a_{c,\max }}{V_c\dot{\lambda }}=\frac{100}{1000\times 0.04}=\frac{100}{40}=\mathrm{2.5.}$ Sabse bada integer $\le 2.5$ hai $N=2$ — lekin derivation se yaad karo ki convergence ko $N>2$ chahiye, toh $N=2$ stability boundary par baitha hai aur $\dot{\lambda }$ kabhi decay nahi karta. Yeh ek genuine design conflict hai: actuator limit aur stability requirement disagree karte hain. Navigation Constant Selection and Actuator Limits dekho. Kaisa lagta hai: tum dono yahan satisfy nahi kar sakte — honest answer hai "koi bhi integer $N$ aise nahi hai jo limit ke andar fly kare aur converge bhi kare; tumhe $N=3$ ke saath briefly saturated actuator accept karna hoga, ya better launch heading se initial $\dot{\lambda }$ reduce karna hoga."

Derivation se $\dot{\lambda }\propto R^{N-2}$, toh $\frac{\dot{\lambda }}{{\dot{\lambda }}_0}={\left(\frac{R}{R_0}\right)}^{N-2}.$ $R/R_0=1/20$ ke saath:

• A ($N=3$): $(1/20{)}^1=\mathrm{0.05.}$
• B ($N=4$): $(1/20{)}^2=\mathrm{0.0025.}$ Missile B ka residual LOS rate $20\times$ chhota hai — uski terminal geometry ek perfect collision course ke kaafi kareeb hai, toh yeh cleaner end karta hai (chhota miss). Neeche di gayi figure dono curves ko $R\to 0$ hote waqt collapse hote dikhati hai.

$a_c=N{V}_c\dot{\lambda }$ ka sign $\dot{\lambda }$ se inherit hota hai ($V_c,N>0$ ke saath). $N=4$ lo:

• $t_1$: $a_c=4\times 900\times (+0.02)=+72{\text{ m/s}}^2$ — increasing $\lambda$ ki taraf accelerate karo (CCW-drifting sightline ko chase karo).
• $t_2$: $a_c=4\times 900\times (-0.02)=-72{\text{ m/s}}^2$ — opposite direction mein accelerate karo, decreasing $\lambda$ ki taraf. Kaisa lagta hai: command exactly tab sign flip karta hai jab drift sign flip karta hai — PN ek negative-feedback loop hai $\dot{\lambda }$ par, hamesha use null karne ki koshish karta hai. Feedback Control — Nulling an Error Signal dekho.

Substitute karo ($V_c=-\dot{R}$, toh $a_c=N{V}_c\dot{\lambda }=-N\dot{R}\dot{\lambda }$): $R\ddot{\lambda }+2\dot{R}\dot{\lambda }=-(-N\dot{R}\dot{\lambda })=N\dot{R}\dot{\lambda }.$ Collect karo $\dot{R}\dot{\lambda }$ terms: $R\ddot{\lambda }=(N-2)\dot{R}\dot{\lambda }⟹\frac{\ddot{\lambda }}{\dot{\lambda }}=(N-2)\frac{\dot{R}}{R}.$ Integrate karo dono sides time mein (left se $\ln \dot{\lambda }$ milta hai, right se $(N-2)\ln R$): $\ln \dot{\lambda }=(N-2)\ln R+C⟹\dot{\lambda }\propto R^{N-2}.$ Condition: jab $R\to 0$ tab hume exponent positive chahiye, yaani $\boxed{N>2}$. $N=3$ check karo: exponent $=1>0$, toh $\dot{\lambda }\propto R\to 0$. Converge karta hai. ✓ $2$ ka factor (Coriolis term, Coriolis Term in Polar Coordinate Acceleration dekho) hi yahi hai jo boundary $N>2$ banata hai $N>1$ ki jagah.

$\ddot{\lambda }=0$ set karo toh $R\ddot{\lambda }$ drop out ho jaata hai: $2\dot{R}\dot{\lambda }=a_T-a_c=a_T+N\dot{R}\dot{\lambda }.$ $\dot{\lambda }$ collect karo: $2\dot{R}\dot{\lambda }-N\dot{R}\dot{\lambda }=a_T⟹(2-N)\dot{R}\dot{\lambda }=a_T.$ ${\dot{\lambda }}_{\text{ss}}=\frac{a_T}{(2-N)\dot{R}}.$ Interpretation: ek maneuvering target ek non-zero residual $\dot{\lambda }$ chhod jaata hai jise plain PN null nahi kar sakta — ek steady bias. Yahi woh gap hai jo Augmented PN $\frac{N}{2}{a}_T$ feed-forward term add karke close karta hai. (Note karo $\dot{R}<0$ closing ke dauran, toh $N>2$ ke saath ${\dot{\lambda }}_{\text{ss}}$ ka sign $a_T$ ko sensibly track karta hai.)

(a) $a_{c,0}=N{V}_c{\dot{\lambda }}_0=4\times 1000\times 0.05=200{\text{ m/s}}^2=200/9.81\approx 20.4g.$ (Ek steep launch command — sightline initially fast drift karti hai.) (b) $R/R_0=2000/8000=1/4$, exponent $N-2=2$: $\dot{\lambda }=0.05\times (1/4{)}^2=0.05\times 0.0625=0.003125\text{ rad/s}.$ (c) $a_c=4\times 1000\times 0.003125=12.5{\text{ m/s}}^2=12.5/9.81\approx 1.27g.$ (d) Jaise range collapse hoti hai, $\dot{\lambda }$ tezi se girta hai, toh command $\sim 20g$ se $\sim 1.3g$ tak drop hoti hai — PN effort front-load karta hai aur near-perfect terminal geometry mein coast karta hai. Figure $a_c$ ko $R$ ke versus trace karta hai.

Lateral demand $\propto \dot{\lambda }$ ($V_c,N$ equal ke saath). Ratio: $\frac{a_{\text{PN}}}{a_{\text{pursuit}}}=\frac{{\dot{\lambda }}_0(1/10{)}^2}{{\dot{\lambda }}_0}=(1/10{)}^2=\mathrm{0.01.}$ PN terminal lateral acceleration ka $\frac{1}{100}$ demand karta hai — 100× gentler. Yahi core reason hai ki PN pure pursuit ko beat karta hai: pursuit apna poora turn end ke liye save karta hai, jahan airframe ke paas sabse kam margin hota hai; PN use early mein spend karta hai aur relaxed arrive karta hai.