Inverse Optimal Control of Evolution Systems and Its Application to Extensible and Shearable Slender Beams
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摘要
An optimal(practical) stabilization problem is formulated in an inverse approach and solved for nonlinear evolution systems in Hilbert spaces. The optimal control design ensures global well-posedness and global practical K∞-exponential stability of the closed-loop system, minimizes a cost functional,which appropriately penalizes both state and control in the sense that it is positive definite(and radially unbounded) in the state and control, without having to solve a Hamilton-Jacobi-Belman equation(HJBE). The Lyapunov functional used in the control design explicitly solves a family of HJBEs. The results are applied to design inverse optimal boundary stabilization control laws for extensible and shearable slender beams governed by fully nonlinear partial differential equations.
An optimal(practical) stabilization problem is formulated in an inverse approach and solved for nonlinear evolution systems in Hilbert spaces. The optimal control design ensures global well-posedness and global practical K∞-exponential stability of the closed-loop system, minimizes a cost functional,which appropriately penalizes both state and control in the sense that it is positive definite(and radially unbounded) in the state and control, without having to solve a Hamilton-Jacobi-Belman equation(HJBE). The Lyapunov functional used in the control design explicitly solves a family of HJBEs. The results are applied to design inverse optimal boundary stabilization control laws for extensible and shearable slender beams governed by fully nonlinear partial differential equations.
An optimal(practical) stabilization problem is formulated in an inverse approach and solved for nonlinear evolution systems in Hilbert spaces. The optimal control design ensures global well-posedness and global practical K∞-exponential stability of the closed-loop system, minimizes a cost functional,which appropriately penalizes both state and control in the sense that it is positive definite(and radially unbounded) in the state and control, without having to solve a Hamilton-Jacobi-Belman equation(HJBE). The Lyapunov functional used in the control design explicitly solves a family of HJBEs. The results are applied to design inverse optimal boundary stabilization control laws for extensible and shearable slender beams governed by fully nonlinear partial differential equations.
引文
[1]B.D.O.Anderson and J.B.Moore,Linear Optimal Control.New Jersey:Prentice-Hall,1971.
[2]M.Hinze,R.Pinnau,M.Ulbrich,and S.Ulbrich,Optimization with PDEConstraints,Mathematical Modeling:Theory and Applications,vol.23,New York:Springer,2009.
[3]J.L.Lions,Optimal Control of Systems Governed by Partial Differential Equations.New York-Berlin:Springer-Verlag,1971.
[4]V.Barbu,E.N.Barron,and R.Jensen,“The necessary conditions for optimal control in Hilbert spaces,”Journal of Mathematical Analysis and Applications,vol.133,pp.151-162,1988.
[5]F.Troltzsch,Optimal Control of Partial Differential Equations:Theory,Methods and Applications,Graduate Studies in Mathematics,vol.112,American Mathematical Society,2010.
[6]I.Lasiecka,Mathematical Control Theory of Coupled PDEs.Philadelphia:SIAM,2002.
[7]M.D.Chekroun,A.Kroner,and H.Liu,“Galerkin approximations of nonlinear optimal control problems in Hilbert spaces,”Electronic Journal of Differential Equations,vol.2017,no.189,pp.1-40,2017.
[8]R.Sepulchre,M.Jankovic,and P.Kokotov′?c,Constructive Nonlinear Control.New York:Springer,1997.
[9]M.Krst?c and H.Deng,Stabilization of Nonlinear Uncertain Systems.London:Springer,1998.
[10]M.Krst?c and Z.H.Li,“Inverse optimal design of input-to-state stabilizing nonlinear controllers,”IEEE Transactions on Automatic Control,vol.43,no.3,pp.336-350,1998.
[11]M.Krst?c,“On global stabilization of Burgers equation by boundary control,”Systems&Control Letters,vol.37,pp.123-141,1999.
[12]K.D.Do,“Inverse optimal gain assignment control of evolution systems and its application to boundary control of marine risers,”Automatica,Submitted,2018.
[13]K.D.Do,“Stability of nonlinear stochastic distributed parameter systems and its applications,”Journal of Dynamic Systems,Measurement,and Control,vol.138,pp.1-12,2016.
[14]K.D.Do,“Modeling and boundary control of translational and rotational motions of nonlinear slender beams in three-dimensional space,”Journal of Sound and Vibration,vol.389,pp.1-23,2017.
[15]A.L.Zuyev,Partial Stabilization and Control of Distributed Parameter Systems with Elastic Elements.Switzerland:Springer International Publishing,2015.
[16]A.Cavallo,G.de Maria,and C.N.S.Pirozzi,Active Control of Flexible Structures.London:Springer,2010.
[17]T.Meurer,D.Thull,and A.Kugi,“Flatness-based tracking control of a piezoactuated Euler-Bernoulli beam with non-collocated output feedback:theory and experiments,”International Journal of Control,vol.81,pp.475-493,2008.
[18]F.F.Jin and B.Z.Guo,“Lyapunov approach to output feedback stabilization for the Euler-Bernoulli beam equation with boundary input disturbance,”Automatica,vol.52,pp.95-102,2015.
[19]K.D.Do,“Stochastic boundary control design for extensible marine risers in three dimensional space,”Automatica,vol.77,pp.184-197,2017.
[20]W.He,S.Nie,and T.Meng,“Modeling and vibration control for a moving beam with application in a drilling riser,”IEEE Transactions on Control Systems Technology,vol.25,pp.1036-1043,2017.
[21]W.He,T.M.D.Huang,and X.Li,“Adaptive boundary iterative learning control for an Euler-Bernoulli beam system with input constraint,”IEEETransactions on Neural Networks and Learning Systems,In press,2017.
[22]K.D.Do and A.D.Lucey,“Boundary stabilization of extensible and unshearable marine risers with large in-plane deflection,”Automatica,vol.77,pp.279-292,2017.
[23]K.D.Do and A.D.Lucey,“Stochastic stabilization of slender beams in space:Modeling and boundary control,”Automatica,vol.91,pp.279-293,2018.
[24]M.S.D.Queiroz,M.Dawson,S.Nagarkatti,and F.Zhang,LyapunovBased Control of Mechanical Systems.Boston:Birkhauser,2000.
[25]J.U.Kim and Y.Renardy,“Boundary control of the Timoshenko beam,”SIAM Journal of Control and Optimization,vol.25,pp.1417-1429,1987.
[26]O.Morgul,“Dynamic boundary control of the Timoshenko beam,”Automatica,vol.28,pp.1255-1260,1992.
[27]C.Mei,“Hybrid wave/mode active control of bending vibrations in beams based on the advanced Timoshenko theory,”Journal of Sound and Vibration,vol.322,pp.29-38,2009.
[28]G.Xu and H.Wang,“Stabilisation of Timoshenko beam system with delay in the boundary control,”International Journal of Control,vol.86,pp.1165-1178,2013.
[29]W.He,T.Meng,J.K.Liu,and H.Qin,“Boundary control of a Timoshenko beam system with input dead-zone,”International Journal of Control,vol.88,pp.1257-1270,2015.
[30]W.He,S.S.Ge,and C.Liu,“Adaptive boundary control for a class of inhomogeneous Timoshenko beam equations with constraints,”IETControl Theory&Applications,vol.8,pp.1285-1292,2014.
[31]T.Endo,F.Matsuno,and Y.Jia,“Boundary cooperative control by flexible Timoshenko arms,”Automatica,vol.81,pp.377-389,2017.
[32]M.Krst′?c,A.A.Siranosian,and A.A.Smyshlyaev,“Backstepping boundary controllers and observers for the slender Timoshenko beam:Part I control design,”in Proc.45th IEEE Conference on Decision and Control,San Diego,CA,pp.3938-3943,2006.
[33]M.Krst′?c,A.A.Siranosian,A.A.Smyshlyaev,and M.Bement,“Backstepping boundary controllers and observers for the slender Timoshenko beam:Part II stability and simulations,”in Proc.American Control Conference,Minneapolis,MN,pp.2412-2417,2006.
[34]M.Krst′?c and A.Smyshlyaev,Boundary Control of PDEs:A Course on Backstepping Designs.Philadelphia,PA:Society for Industrial and Applied Mathematics,2008.
[35]K.D.Do,“Stabilization of exact nonlinear Timoshenko beams in space by boundary feedback,”Journal of Sound and Vibration,vol.422,pp.278-299,2018.
[36]L.Gawarecki and V.Mandrekar,Stochastic Differential Equations in Infinite Dimensions with Applications to Stochastic Partial Differential Equations.Berlin:Springer,2011.
[37]P.L.Chow,Stochastic Partial Differential Equations.Boca Raton:Chapman&Hall/CRC,2007.
[38]G.Hardy,J.E.Littlewood,and G.Polya,Inequalities.Cambridge:Cambridge University Press,2nd ed.,1989.
[39]H.Khalil,Nonlinear Systems.Prentice Hall,2002.
[40]K.D.Do,“Global inverse optimal stabilization of stochastic nonholonomic systems,”Systems&Control Letters,vol.75,pp.41-55,2015.
[41]M.Krst′?c,I.Kanellakopoulos,and P.Kokotov′?c,Nonlinear and Adaptive Control Design.New York:Wiley,1995.
[42]A.Love,A Treatise on the Mathematical Theory of Elasticity.Cambridge University Press,3rd ed.,1920.
[43]R.A.Adams and J.J.F.Fournier,Sobolev Spaces.Oxford,UK:Academic Press,2nd ed.,2003.
[44]M.M.Bernitsas,J.E.Kokarakis,and A.Imron,“Large deformation three-dimensional static analysis of deep water marine risers,”Applied Ocean Research,vol.7,pp.178-187,1985.
[45]J.M.Niedzwecki and P.Y.F.Liagre,“System identification of distributed-parameter marine riser models,”Ocean Engineering,vol.30,no.11,pp.1387-1415,2003.
[2]M.Hinze,R.Pinnau,M.Ulbrich,and S.Ulbrich,Optimization with PDEConstraints,Mathematical Modeling:Theory and Applications,vol.23,New York:Springer,2009.
[3]J.L.Lions,Optimal Control of Systems Governed by Partial Differential Equations.New York-Berlin:Springer-Verlag,1971.
[4]V.Barbu,E.N.Barron,and R.Jensen,“The necessary conditions for optimal control in Hilbert spaces,”Journal of Mathematical Analysis and Applications,vol.133,pp.151-162,1988.
[5]F.Troltzsch,Optimal Control of Partial Differential Equations:Theory,Methods and Applications,Graduate Studies in Mathematics,vol.112,American Mathematical Society,2010.
[6]I.Lasiecka,Mathematical Control Theory of Coupled PDEs.Philadelphia:SIAM,2002.
[7]M.D.Chekroun,A.Kroner,and H.Liu,“Galerkin approximations of nonlinear optimal control problems in Hilbert spaces,”Electronic Journal of Differential Equations,vol.2017,no.189,pp.1-40,2017.
[8]R.Sepulchre,M.Jankovic,and P.Kokotov′?c,Constructive Nonlinear Control.New York:Springer,1997.
[9]M.Krst?c and H.Deng,Stabilization of Nonlinear Uncertain Systems.London:Springer,1998.
[10]M.Krst?c and Z.H.Li,“Inverse optimal design of input-to-state stabilizing nonlinear controllers,”IEEE Transactions on Automatic Control,vol.43,no.3,pp.336-350,1998.
[11]M.Krst?c,“On global stabilization of Burgers equation by boundary control,”Systems&Control Letters,vol.37,pp.123-141,1999.
[12]K.D.Do,“Inverse optimal gain assignment control of evolution systems and its application to boundary control of marine risers,”Automatica,Submitted,2018.
[13]K.D.Do,“Stability of nonlinear stochastic distributed parameter systems and its applications,”Journal of Dynamic Systems,Measurement,and Control,vol.138,pp.1-12,2016.
[14]K.D.Do,“Modeling and boundary control of translational and rotational motions of nonlinear slender beams in three-dimensional space,”Journal of Sound and Vibration,vol.389,pp.1-23,2017.
[15]A.L.Zuyev,Partial Stabilization and Control of Distributed Parameter Systems with Elastic Elements.Switzerland:Springer International Publishing,2015.
[16]A.Cavallo,G.de Maria,and C.N.S.Pirozzi,Active Control of Flexible Structures.London:Springer,2010.
[17]T.Meurer,D.Thull,and A.Kugi,“Flatness-based tracking control of a piezoactuated Euler-Bernoulli beam with non-collocated output feedback:theory and experiments,”International Journal of Control,vol.81,pp.475-493,2008.
[18]F.F.Jin and B.Z.Guo,“Lyapunov approach to output feedback stabilization for the Euler-Bernoulli beam equation with boundary input disturbance,”Automatica,vol.52,pp.95-102,2015.
[19]K.D.Do,“Stochastic boundary control design for extensible marine risers in three dimensional space,”Automatica,vol.77,pp.184-197,2017.
[20]W.He,S.Nie,and T.Meng,“Modeling and vibration control for a moving beam with application in a drilling riser,”IEEE Transactions on Control Systems Technology,vol.25,pp.1036-1043,2017.
[21]W.He,T.M.D.Huang,and X.Li,“Adaptive boundary iterative learning control for an Euler-Bernoulli beam system with input constraint,”IEEETransactions on Neural Networks and Learning Systems,In press,2017.
[22]K.D.Do and A.D.Lucey,“Boundary stabilization of extensible and unshearable marine risers with large in-plane deflection,”Automatica,vol.77,pp.279-292,2017.
[23]K.D.Do and A.D.Lucey,“Stochastic stabilization of slender beams in space:Modeling and boundary control,”Automatica,vol.91,pp.279-293,2018.
[24]M.S.D.Queiroz,M.Dawson,S.Nagarkatti,and F.Zhang,LyapunovBased Control of Mechanical Systems.Boston:Birkhauser,2000.
[25]J.U.Kim and Y.Renardy,“Boundary control of the Timoshenko beam,”SIAM Journal of Control and Optimization,vol.25,pp.1417-1429,1987.
[26]O.Morgul,“Dynamic boundary control of the Timoshenko beam,”Automatica,vol.28,pp.1255-1260,1992.
[27]C.Mei,“Hybrid wave/mode active control of bending vibrations in beams based on the advanced Timoshenko theory,”Journal of Sound and Vibration,vol.322,pp.29-38,2009.
[28]G.Xu and H.Wang,“Stabilisation of Timoshenko beam system with delay in the boundary control,”International Journal of Control,vol.86,pp.1165-1178,2013.
[29]W.He,T.Meng,J.K.Liu,and H.Qin,“Boundary control of a Timoshenko beam system with input dead-zone,”International Journal of Control,vol.88,pp.1257-1270,2015.
[30]W.He,S.S.Ge,and C.Liu,“Adaptive boundary control for a class of inhomogeneous Timoshenko beam equations with constraints,”IETControl Theory&Applications,vol.8,pp.1285-1292,2014.
[31]T.Endo,F.Matsuno,and Y.Jia,“Boundary cooperative control by flexible Timoshenko arms,”Automatica,vol.81,pp.377-389,2017.
[32]M.Krst′?c,A.A.Siranosian,and A.A.Smyshlyaev,“Backstepping boundary controllers and observers for the slender Timoshenko beam:Part I control design,”in Proc.45th IEEE Conference on Decision and Control,San Diego,CA,pp.3938-3943,2006.
[33]M.Krst′?c,A.A.Siranosian,A.A.Smyshlyaev,and M.Bement,“Backstepping boundary controllers and observers for the slender Timoshenko beam:Part II stability and simulations,”in Proc.American Control Conference,Minneapolis,MN,pp.2412-2417,2006.
[34]M.Krst′?c and A.Smyshlyaev,Boundary Control of PDEs:A Course on Backstepping Designs.Philadelphia,PA:Society for Industrial and Applied Mathematics,2008.
[35]K.D.Do,“Stabilization of exact nonlinear Timoshenko beams in space by boundary feedback,”Journal of Sound and Vibration,vol.422,pp.278-299,2018.
[36]L.Gawarecki and V.Mandrekar,Stochastic Differential Equations in Infinite Dimensions with Applications to Stochastic Partial Differential Equations.Berlin:Springer,2011.
[37]P.L.Chow,Stochastic Partial Differential Equations.Boca Raton:Chapman&Hall/CRC,2007.
[38]G.Hardy,J.E.Littlewood,and G.Polya,Inequalities.Cambridge:Cambridge University Press,2nd ed.,1989.
[39]H.Khalil,Nonlinear Systems.Prentice Hall,2002.
[40]K.D.Do,“Global inverse optimal stabilization of stochastic nonholonomic systems,”Systems&Control Letters,vol.75,pp.41-55,2015.
[41]M.Krst′?c,I.Kanellakopoulos,and P.Kokotov′?c,Nonlinear and Adaptive Control Design.New York:Wiley,1995.
[42]A.Love,A Treatise on the Mathematical Theory of Elasticity.Cambridge University Press,3rd ed.,1920.
[43]R.A.Adams and J.J.F.Fournier,Sobolev Spaces.Oxford,UK:Academic Press,2nd ed.,2003.
[44]M.M.Bernitsas,J.E.Kokarakis,and A.Imron,“Large deformation three-dimensional static analysis of deep water marine risers,”Applied Ocean Research,vol.7,pp.178-187,1985.
[45]J.M.Niedzwecki and P.Y.F.Liagre,“System identification of distributed-parameter marine riser models,”Ocean Engineering,vol.30,no.11,pp.1387-1415,2003.