The stable manifold approach for optimal swing up and stabilization of an inverted pendulum with input saturation

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1 Conrol Sysems Enineerin Aerosace Enineerin UNIERSIY he sable manifold aroach for oimal swin u and sabilizaion of an invered endulum wih inu sauraion Ryu Fujimoo and Noboru Saamoo Naoya Universiy IFAC World Conress Auus 3

2 Ouline Bacround and roblem formulaion Nonlinear oimal conrol Aroimaion mehods for HJ equaions Sable manifold aroach Simulaion / Eerimen resuls Non-uniqueness of soluion in HJ equaions Concludin remars

3 Bacround/roblem saemen Invered endulum Sron nonlineariy Under acuaed sysem Benchmar roblem of nonlinear conroller desin Swin u and sabilizaion of invered endulum Swichin of swin u and sabilizaion conrollers Asrom & Furua ooloical issues / Aneli Asrom e.al 8 his al: Sinle oimal sae feedbac conrol Enhancemen of valid aroimaion rane for HJ eq.

4 Invered endulum sysem sin cos sin sin sin sin cos sin cos m l Mml m M J m l ml ml J C ml J m l Mml m M J ml M m m l C ml f sin sin cos m l Mml m M J C ml J m l Mml m M J C ml ] [ ] [ 4 3 u sa f ] 8[ ] 8[ min ma u u Inu limiaion: sa sa R R R Q R f sa Hamilon-Jacobi:

5 Aroimae soluions for HJ eq aylor aroimaion mehod: Canno handle non-analyic nonlineariies Comuaionally inefficien Small domain of converence Sable manifold aroach:n.saamoo and A.J.van der Schaf IEEE AC 8 is direcly comued ieraive mehod suiable for comuer imlemenaion Larer domain of converence ec

6 Sable manifold mehod for HJ eq.

7 Sable manifold mehod for HJ eq.

8 H anular velociyrad/s Comuaional resul Closed loo rajecories for differen ieraion number H f sa R Q sa R R sa R HJ eq..5.5 =linear =6 = =5 Iniial value 5 = =6 = = Iniial value endulum anlerad Hamilonian endulum anlerad rajecories in he θdθ/d-sace

9 Simulaion/Eerimen resuls 7.7[] Resonses simulaion Inu volae is under he limiaion 8[] Swin u wih swins Movie Robusness for arameer variaions % Movie

10 Wha do sauraion funcions do? Effec of sauraion funcion Resonses simulaion Sauraion is an ideniy funcion inside of limiaion he soluion solves he oriinal HJ eq as well?? Q R f sa sa sa R R R Hamilon-Jacobi eq wih sauraions Simulaion by HJ eq wihou sauraion funcion - Swin u conrol wih swin - Inu volae is far beyond 8[]

11 Uniqueness of soluion Analysis for a -dimensional model Comued sable manifold wihou sauraions wo layers of sable manifolds Closed loo rajecories

12 Uniqueness of soluion Analysis for a -dimensional model Comued sable manifold wih sauraions One sable manifold is filered ou Closed loo rajecories

13 Wih smaller inu consrain u u ma min 8[ ] 8[ ] u u ma min [ ] [ ].8[] Aly 4 ieraions of he sable manifold alorihm Feedbac funcion is eressed wih olynomials Swin u wih 3 swins Efficien sraey wih low volae hird sable manifold infinie layers Resonses simulaion Movie3

14 Concludin remars Oimal swin u and sabilizaion of invered endulum Sinle feedbac by solvin a Hamilon-Jacobi equaion Lare domain of validiy o include he endin osiion Elicily include sauraion funcions in he HJ equaion swin swin and 3 swin conrollers by chanin he value of inu limiaion An eamle of non-unique soluions o HJ equaion

15 han you for your aenion

16 Sable manifold alorihm N.Saamoo and A.J.van der Schaf 8 convere o a soluion on [ F F ds s s e ds s s e e e s F s F F A A Hamiloinan sysem Sable Fsmooh nonlineariies

17 Limiaion of he aylor mehod 3 u -dimensional oimal conrol roblem 3 3 H d u J Series soluions canno aroimae for.9 has comle zeros a 9. z z

18 Invered endulum seu