# lec6 - 1 Outline Â Motivation s tems stems Â Recap of Local...

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Unformatted text preview: 1 Outline Â¡ Motivation s tems stems Â¡ Recap of Local Stability Results Â¡ Global Stability Â¡ Invariant Set Theorems Â¡ Examples ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys Motivation Â¡ Local stability results only guarantee a finite (possibly small) domain of attraction stemsstems (possibly small) domain of attraction Â¡ Lyapunovâ€™s global stability theorems guarantee an infinite domain of attraction Â¡ The rate of change of Lyapunov function, dV/dt, for many asymptotically stable systems may not be negative definite ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys may not be negative definite Â¡ The invariant set results relaxes the negative definiteness requirement 2 Steps for Applying Lyapunovâ€™s Theorem Â¡ Choose a suitable l.p.d. (energy-like) function V(x) for the system s tems stems function V(x) for the system Â¡ Evaluate dV(x)/dt along the systemâ€™s trajectories: ( ) x f x x x âˆ‚ âˆ‚ = âˆ‚ âˆ‚ = = V V dt dV V & & ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys Â¡ dv/dt â‰¤ 0 â‡’ 0 is stable Â¡-dv/dt l.p.d. â‡’ 0 is asymptotically stable Damped Pendulum Eq. Of Motion: sin = Î¸ + Î¸ + Î¸ & & & stemsstems ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys 3 Alternative Lyapunov Function for the Pendulum System Â¡ Lyapunov Candidtate: s tems stems Â¡ The rate dV/dt along system trajectories: ( ) ( ) 2 2 1 2 2 1 2 1 2 1 cos 1 2 ) ( x x x x V + + + âˆ’ = x ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys Â¡ dv/dt is locally negative definite! 2 2 1 1 sin ) ( x x x dt dV âˆ’ âˆ’ = x How to Determine the DOA Â¡ The domain of attraction (DOM) of an stemsstems asymptotically stable system is DOA={x âˆˆ R n : x(0)=x â‡’ limx(t) â†’ 0} Â¡ The DOM can be estimated by finding the largest l for some l>0 such that the set Î© ={x âˆˆ R n : V(x)<l} is bounded.is bounded....
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lec6 - 1 Outline Â Motivation s tems stems Â Recap of Local...

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