લ્યાપુનોવ (Liapunov) વિધેયનો ઉપયોગ કરીને સ્થિરતા તપાસવાની પદ્ધતિ _____________ છે.
જવાબ અને પગલાવાર ઉકેલ
લ્યાપુનોવ વિધેય V(x) બનાવી સિસ્ટમના માર્ગો પર તેના વિકલનની તપાસ કરીને સ્થિરતા જાણવી તેને લ્યાપુનોવની પ્રત્યક્ષ (દ્વિતીય) પદ્ધતિ કહે છે. તેમાં અરેખીય વિકલ સમીકરણો સ્પષ્ટ રીતે ઉકેલવાની જરૂર પડતી નથી.
વિગતવાર ઉકેલ (મૂળ ભાષામાં)
Concept & Reasoning
A suitable Lyapunov function is positive definite near the equilibrium and its time derivative is negative definite for asymptotic stability (with standard qualifications). The indirect method instead linearizes the system near an equilibrium and inspects the linearized eigenvalues.
How to Solve It in the Exam
Lyapunov function itself → direct method.
Important Exam Point
Indirect method = linearization/eigenvalues.
Quick Trick
Lyapunov function itself → direct method.
Common Mistake
Calling it indirect merely because V̇ is computed indirectly from dynamics.
The method of investigating the stability using Liapunov function is _____________
Answer & step-by-step solution
Investigating stability by constructing a Lyapunov function V(x) and examining its derivative along system trajectories is called Lyapunov's direct (second) method. It avoids explicitly solving the nonlinear differential equations.
Concept & Reasoning
A suitable Lyapunov function is positive definite near the equilibrium and its time derivative is negative definite for asymptotic stability (with standard qualifications). The indirect method instead linearizes the system near an equilibrium and inspects the linearized eigenvalues.
How to Solve It in the Exam
Lyapunov function itself → direct method.
Important Exam Point
Indirect method = linearization/eigenvalues.
Quick Trick
Lyapunov function itself → direct method.
Common Mistake
Calling it indirect merely because V̇ is computed indirectly from dynamics.
लियापुनोव (Liapunov) फलन का उपयोग करके स्थिरता की जाँच करने की विधि _____________ है।
उत्तर और चरणबद्ध हल
लियापुनोव फलन V(x) बनाकर सिस्टम पथों पर उसके अवकलज की जाँच से स्थिरता ज्ञात करना लियापुनोव की प्रत्यक्ष (द्वितीय) विधि कहलाती है। इसमें अरैखिक अवकल समीकरणों को स्पष्ट रूप से हल करने की आवश्यकता नहीं होती।
विस्तृत हल (मूल भाषा में)
Concept & Reasoning
A suitable Lyapunov function is positive definite near the equilibrium and its time derivative is negative definite for asymptotic stability (with standard qualifications). The indirect method instead linearizes the system near an equilibrium and inspects the linearized eigenvalues.
How to Solve It in the Exam
Lyapunov function itself → direct method.
Important Exam Point
Indirect method = linearization/eigenvalues.
Quick Trick
Lyapunov function itself → direct method.
Common Mistake
Calling it indirect merely because V̇ is computed indirectly from dynamics.
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