3.5.31 · D1Lyapunov stability — Lyapunov function, positive definitenes
Intuition Yahan sab kuch ek hi idea pe tika haiAgar tum ek "height number" dhundh sako jo target pe sabse neeche ho aur system ke chalte chalte sirf neeche hi jaata ho , toh system zaroor usi target ki taraf roll karega — aur tumhe uski equations bilkul solve nahi karni padi. Yeh poora topic bas yahi hai ki height , sabse neeche , aur neeche jaana ko itna precise banao ki mathematical proof ban sake.
Parent note Lyapunov stability padhne se pehle tumhe un symbols ka fluent hona zaroori hai jo woh tumhare saamne phenkta hai. Hum har ek ko zero se banate hain, ek aisi order mein jahan har symbol earn karke hi use hota hai.
Koi bhi machine (spacecraft, pendulum, car) kisi bhi instant par numbers ki ek choti si list se poori tarah describe ho jaati hai: woh kahan hai aur har part kitni tezi se chal raha hai . Woh list hi state hai.
x x x stack hai — ek point jo poori machine ko abhi pin down karta hai. Agar n x = ( x 1 , x 2 , … , x n )
Picture: origin se ek dot tak ka arrow. n = 2 Topic ko yeh kyun chahiye: stability ek statement hai dot time ke saath kahan jaata hai ke baare mein. Dot nahi, track karne ke liye kuch nahi.
n R n n R n aisi saari possible stacks ka set — har woh dot jahan tum kabhi bhi ho sakte ho.
Picture: R 1 R 2 R 3 Kyun: "x ∈ R n
Upar wala red dot dekho: woh akela point hi is instant par poori machine hai. Time ke saath, red dot slideta hai aur ek curve trace karta hai.
State rukti nahi — woh chalti hai. Yeh baat karne ke liye ki kitni tezi se chalti hai, humein velocity chahiye.
x ˙ rate of change per second (velocity). x ˙ 1 x 1
Picture: red dot se attached ek chota sa arrow jo dikhata hai kis direction mein aur kitni tezi se woh move karne wala hai. Is arrow ko velocity vector kehte hain.Kyun: stability poochhti hai "dot kis taraf drift karta hai?" — yahi toh velocity arrow batata hai.
Ab, woh arrow decide kaun karta hai? Machine ki physics karta hai. Ek rule hota hai: "agar tum state x f
f ( x ) f machine hai jo state khaati hai aur uska velocity arrow ugalti hai . Equation of motion bas yeh hai:
x ˙ = f ( x ) f
Picture: sheet ke har dot par, woh arrow banao jo f vector field (figure dekho). Red dot ko chhodo aur woh arrows ke saath stream mein patte ki tarah beh jaata hai.Topic ko kyun chahiye: Lyapunov ki poori trick yeh hai ki in arrows ki direction padhi jaaye bina leaf ka poora safar compute kiye.
Parent kehta hai system autonomous hai: f kahan ho us par depend karta hai, kya time hai us par nahi. Arrow field frozen hai — yeh 9 AM aur midnight ko same dikhta hai. Isi liye V no explicit time dependence hoti hai.
Definition Equilibrium aur zero vector0 0 equilibrium woh state hai jahan velocity arrow ki length zero hai: f ( 0 ) = 0
Picture: ek dot jahan stream bilkul still hai. Patte ko bilkul wahan drop karo aur woh kabhi nahi hilta.Kyun: stability hamesha ek equilibrium ke baare mein hoti hai — woh "target" jahan machine ko wapas aana chahiye. Hum coordinates shift karte hain taaki yeh target origin par baithe, jo hamesha allowed hai (bas subtract karo jahan woh actually hai).
Yeh kehne ke liye ki "dot target ke paas rehta hai" humein measure karna hoga ki dot origin se kitna door hai.
∥ x ∥ ∥ x ∥ length hai — uski seedhi distance. x = ( x 1 , x 2 ) x 1 2 + x 2 2
Picture: origin se dot tak ruler rakhna. "∥ x ∥ < ε ε Kyun: parent ki stability ki ε δ δ ε
ε δ do choti radii ke liye khadi hain: ε δ tum jo bhi wander-limit chuno, ek aisi aim-tightness hoti hai jo use hamesha guarantee karti hai .
Definition Scalar functionV ( x ) V ek akela number — ek scalar — return karta hai. Isko har dot ko assigned ek height / altitude samjho.
Picture: states ki sheet ke upar bichha ek landscape — pahad aur ghatiyan. V ( x ) x Kyun: Lyapunov ka poora method yeh hai ki "ek bowl jaisa landscape dhundho jiska bottom target hai, aur check karo leaf hamesha neeche hi beh rahi hai." V
Bowl shape ka ek naam hai — positive definite — jise hum ab symbol by symbol kholte hain.
Definition Positive definite (PD), words meinV V ( 0 ) = 0 aur V ( x ) > 0
Picture: ek asli bowl — ek sabse neecha point, ground har direction mein usse door hote hote upar uthta hai.Contrast (semi-definite): agar ground sirf ≥ 0 flat rehta hai, toh woh sirf positive semi-definite hai — ek level trough wala bowl, jo channel ke saath roll karne wale marble ko trap nahi kar sakta.
"Neeche jaana" precisely kehne ke liye humein steepest uphill ki direction chahiye — yahi gradient hai.
∇ V ∇ V partial slopes ka vector hai: ∇ V = ( ∂ x 1 ∂ V , … , ∂ x n ∂ V ) ∂ x i ∂ V x i
Picture: landscape par ek arrow jo seedha upar ki taraf point karta hai, jahan slope sabse steep ho wahan sabse lamba.Kyun: yeh jaanne ke liye ki leaf neeche jaati hai ya nahi, uski velocity arrow f ∇ V dot product hai.
⊤ a ⊤ b ⊤ a ⊤ b dot product a 1 b 1 + a 2 b 2 + … do arrows kitna same direction mein point karte hain .
Picture: positive agar arrows roughly agree karein, zero agar perpendicular, negative agar oppose karein.Kyun yeh tool aur koi nahi: hum ek number chahte hain jo kahe "leaf ki motion uphill hai ya downhill?" Velocity f uphill ∇ V downhill .
V ˙ V ˙ Chain rule se (ek compound quantity ki rate = slope × speed, har direction mein sum karke):
V ˙ = ∑ i ∂ x i ∂ V x ˙ i = ∇ V ⊤ f ( x )
Picture: chalti hui leaf par khade ho aur apna altimeter dekho. V ˙ < 0 girta padh raha hai.Kyun chain rule aur ∂ V / ∂ t V moving dot ke through aata hai, jiska speed dynamics f x ˙ = f V ˙
Mnemonic Ek saans mein teen key symbolsf kahan jaati hai, V ucha hai, V ˙ = ∇ V ⊤ f neeche ja rahi hai ya nahi. Stability = bowl V V ˙
Parent ke quadratic bowls V = x ⊤ P x
Definition Symmetric matrixP P P = P ⊤ x ⊤ P x weighted sum of squares hai — sabse simple possible bowl.
Picture: P = I x 1 2 + x 2 2 P Kyun: aisi bowl check karna ki woh PD hai, yeh check karne mein reduce ho jaata hai ki P Positive Definite Matrices & Sylvester's Criterion aur Lyapunov equation .
dynamics f as arrow field
Har arrow ka matlab hai "right box samajhne ke liye left box zaroori hai." Do boxes positive-definite bowl aur downhill V ˙ seedhe Lyapunov ke theorem mein feed hote hain — wahi poora topic hai.
Right side dhako; kya tum padhne se pehle har ek ka jawab de sakte ho?
Bold x n R n
x ˙ Dot par velocity arrow; dynamics f x ˙ = f ( x )
Equilibrium ek picture mein kya hai? Ek dot jahan arrow ki length zero hai (f ( 0 ) = 0
∥ x ∥ Origin se dot tak seedhi-line distance (arrow length).
Stability ko ε δ Kisi bhi target ball ε δ δ ε
V ( x ) Ek scalar — har state ke liye ek number (ek height); state space par ek landscape.
V V ( 0 ) = 0 V ( x ) > 0
∇ V Landscape par seedha uphill, jahan steepest ho wahan sabse lamba.
V ˙ = ∇ V ⊤ f ∂ V / ∂ t V f
V ˙ Negative — velocity uphill gradient ka oppose karti hai.
