Foundations — Root locus — Evans' method, rules for sketching

3.5.40 · D1 · Physics › Guidance, Navigation & Control (GNC) › Root locus — Evans' method, rules for sketching

Ye page parent note mein use kiye gaye har symbol ko bilkul zero se, usi order mein build karta hai jis order mein tumhe unki zaroorat hai. Baad mein koi bhi cheez kisi aisi cheez par depend nahi karti jo abhi tak build nahi hui.

1. Complex plane — woh map jahan sab kuch rehta hai

Poles ke move hone ki baat karne se pehle, humein us jagah ki zaroorat hai jahan woh move hote hain.

Humein iska kya kaam hai? Ek real number ek line par ek point hota hai — ek 1D cheez. Lekin is topic ki equations ke answers (poles) line se bahar ho sakte hain: woh oscillate karte hain. "Kitni tezi se decay hota hai" ($\sigma$) aur "kitni tezi se wiggle karta hai" ($\omega$) — dono store karne ke liye tumhe do numbers chahiye, isliye tumhe ek 2D map chahiye. Woh map hi complex plane hai.

Ek complex number ke do aur descriptions bhi hain jo humein baar baar chahiye honge:

Yeh dono kyun? Kyunki parent ki poori method — "angle condition" aur "magnitude condition" — ek khaas quantity ke liye in dono numbers ko read karna hi hai. Hum inhe §6 mein dobara dekhenge.

2. $s$ ke functions: $N(s)$, $D(s)$, aur ratio $G(s)H(s)$

Sirf ek polynomial kyun nahi, ratio kyun? Kyunki do alag cheezein ek system ko break karti hain: recipe ka blow up hona ($D=0$) aur recipe ka vanish hona ($N=0$). Numerator aur denominator ko alag rakhne se hum dono ke baare mein alag baat kar sakte hain. Un dono events ke naam hain — poles aur zeros — jo aage aate hain.

3. Poles aur zeros — woh pins jिनके beech locus stretch hoti hai

Poles ki itni parwah kyun? Ek pole ki location hi system ka behaviour hai:

• Pole far left → response fast marta hai (stable, snappy).
• Pole middle line ke paas → response dheere marta hai.
• Pole ki nonzero height $\omega$ ho → response us frequency par oscillate karta hai.
• Pole right par → response grow karta hai → machine hilke toot jaati hai.

Toh "poles kahan hain?" ka jawab hai "kya mera drone stable hai aur yeh kaise fly karta hai?" — GNC control loops ka poora kaam.

4. Feedback aur gain knob $K$

Sirf ek knob kyun, zyada kyun nahi? Kyunki ek knob ke liye pole-paths curves (1D) hote hain, jinhe tum haath se draw kar sakte ho. Evans ka poora method is one-parameter sweep par built hai. Dekho PID / PD controller design ki extra knobs un curves ko reshape karne ke liye zeros kaise add karte hain.

5. Characteristic equation — master equation

Parent jo kuch bhi karta hai woh sab ek line se aata hai. Yahan dhheere dhheere build karte hain.

Closed-loop transfer function (output over input, feedback wired up ke saath) yeh hai:

Right-hand form $D(s)$ se multiply karke aata hai: kyunki $GH=N/D$,

Yeh woh polynomial hai jिसकी roots closed-loop poles hain — woh cheez jिसके baare mein Characteristic equation & closed-loop poles hai. Jab $K$ vary karta hai, iske roots locus trace karte hain.

6. $L(s)=-1$ ko do conditions ki tarah read karna

Characteristic equation rearrange hoti hai $KG(s)H(s)=-1$. Ab $-1$ khud bhi ek complex number hai — §1 ki picture par wapas jao aur use dhundho: yeh centre se ek unit left par baitha hai. Matlab:

• iska magnitude $1$ hai (centre se distance),
• iska angle $180°$ hai (seedha left ki taraf pointing) — ya $180°$ plus koi bhi full turn, yaani $\pm 180°(2k+1)$.

Kisi point $s$ ke pole hone ke liye, $L(s)$ ka $-1$ ke barabar hona zaroori hai, toh iska magnitude aur angle dono match karne chahiye:

Kisi point ka angle compute kaise karo? Har pole aur har zero se apne test point $s$ tak ek arrow draw karo. Har arrow ka angle measure karo. Zero-angles add karo, pole-angles subtract karo — woh sum hai $\angle G(s)H(s)$. Har sketching rule (real-axis test, departure angle, asymptotes) is arrow-arithmetic ka hi disguise hai.

7. Do tools jo parent use karta hai

8. Yeh sab topic ko kaise feed karta hai

Isko top-down padho: map (§1) poles/zeros (§3) ko host karta hai, jo transfer function (§2) build karte hain. Feedback (§4) aur woh function milke master equation (§5) dete hain. $-1$ ko size aur direction mein split karna (§6) do conditions deta hai, aur woh — derivative aur Routh (§7) ki madad se — parent note mein har sketching rule ko power dete hain.

Equipment checklist

Connections

• Root locus — Evans' method, rules for sketching — woh parent jिसके liye yeh page tumhe tayaar karta hai
• Characteristic equation & closed-loop poles
• Routh–Hurwitz stability criterion
• PID / PD controller design
• Bode plot & frequency response
• Nyquist criterion
• Damping ratio and settling time
• GNC control loops