" notice="" for="" common="" control="" problem,="" solution="" to="" get="" here,="" trajectory!="" feedback="" controller="" also="" affected="" by="" trajectory="" (u(alpha))="" however,="" form="" vc="" fixed="" then="" talk="" about="" path="" planning.,="" lagrangian="" ode="" solver,="" summary,="" guard="manifold" =="" switching="" variable,="" [answered]="" where="" do="" linearize="" system?="" i've="" been="" keep="" thinking="" conventional="" linearization="" (input-state)="" addressed="" adaptive="" control,="" subtract="" nonlinear="" term.="" that's="" why="" i="" couldn't="" find="" process="" being="" done="" here.="" io="" totally="" different="" as="" address="" dynamics.="" (a="" system.="" now="" don't="" care="" system="" nl="" or="" not,="" addresses="" be="" pd="" controller)="" we're="" confused="" because="" normally="" have="" 1="" over="" system,="" u.="" controls:="" desired="" part="" newly="" defined="" io-linearization="" define="" h="" based="" this.=">" other="" words,="" disregard="" we've="" known.="" new="" made="" outputs="" only.="" (via="" controller,="" state="" )="" put="" zero="" manifold.="" trajectory,="" this="" 'phi')="" so="" think="" these="" two="" concurrently.="" 'deffeomorphism'="" 'normal="" form'="" parts:="" one="" another="" former="" controlled="" (io="" latter="" polynomial,="" choosing="" initial="" inside="" domain="" attraction="" its="" convergence="" rate:="" optimization="" polynomials.="" middle="" elements="" includes:="" 1.="" energy="" efficiency="" cost="" 2.="" equality="" inequality="" constraints="" compute="" nl,="" ?="" than="" planning="" there="" any="" involving="" stability,="" limit="" cycle,="" convergence,="" lyapunov?="" should="" be.="" 3="" link="" walker="" has="" phase="" portaits="" cycles,="" w.="" holonomial="" conditions,="" virtual="" manifold,="" point,="" poincare="" map="" for:find="" stability="" point="" numerical="" analysis="" see="" eigen="" values="" are="" within="" unit="" circle,="" ct="" dynamics:="" ode,="" smilar="" above,="" did="" map?,="" via="" conservation="" moment="" +="" impact="" (grf),="" variable="" -="" dt="" dynamics,="" hybrid="" dyamics,="" transversality"="">