LRCF Newton Method for the ARE Reordering Strategies ADI Shift Parameters Column Compression for the LRCF Generalized Systems Conclusions and Outlook Efficient Implementation of Large Scale Lyapunov and Riccati Equation Solvers Jens Saak joint work with Peter Benner (MiIT) Professur Mathematik in Industrie und Technik (MiIT) Fakult¨ at f¨ ur Mathematik Technische Universit¨ at Chemnitz Computational Methods with Applications Harrachov August 24, 2007 1/26 jens.saak@mathematik.tu-chemnitz.de Jens Saak Efficient Large Scale Lyapunov and ARE Solvers
LRCF Newton Method for the ARE Reordering Strategies ADI Shift Parameters Column Compression for the LRCF Generalized Systems Conclusions and Outlook Aim of this talk What is the aim of this talk? Promote the upcoming release 1.1 of the LyaPack software package. 2/26 jens.saak@mathematik.tu-chemnitz.de Jens Saak Efficient Large Scale Lyapunov and ARE Solvers
LRCF Newton Method for the ARE Reordering Strategies ADI Shift Parameters Column Compression for the LRCF Generalized Systems Conclusions and Outlook Aim of this talk What is the aim of this talk? Promote the upcoming release 1.1 of the LyaPack software package. What is LyaPack ? 2/26 jens.saak@mathematik.tu-chemnitz.de Jens Saak Efficient Large Scale Lyapunov and ARE Solvers
LRCF Newton Method for the ARE Reordering Strategies ADI Shift Parameters Column Compression for the LRCF Generalized Systems Conclusions and Outlook Aim of this talk What is the aim of this talk? Promote the upcoming release 1.1 of the LyaPack software package. What is LyaPack ? Matlab toolbox for solving large scale Lyapunov equations (applications like in M. Embrees plenary talk on Tuesday) Riccati equations linear quadratic optimal control problems 2/26 jens.saak@mathematik.tu-chemnitz.de Jens Saak Efficient Large Scale Lyapunov and ARE Solvers
LRCF Newton Method for the ARE Reordering Strategies ADI Shift Parameters Column Compression for the LRCF Generalized Systems Conclusions and Outlook Origin of the Riccati equations output equation semi discrete parabolic PDE x ( t ) = Ax ( t ) + Bu ( t ) ˙ x (0) = x 0 ∈ X . y ( t ) = Cx ( t ) (Cauchy) (output) cost function ∞ J ( u ) = 1 � < y , y > + < u , u > dt (cost) 2 0 and the linear quadratic regulator problem is LQR problem Minimize the quadratic (cost) with respect to the linear constraints (Cauchy),(output). 3/26 jens.saak@mathematik.tu-chemnitz.de Jens Saak Efficient Large Scale Lyapunov and ARE Solvers
LRCF Newton Method for the ARE Reordering Strategies ADI Shift Parameters Column Compression for the LRCF Generalized Systems Conclusions and Outlook Origin of the Riccati equations output equation semi discrete parabolic PDE x ( t ) = Ax ( t ) + Bu ( t ) ˙ x (0) = x 0 ∈ X . y ( t ) = Cx ( t ) (Cauchy) (output) cost function ∞ J ( u ) = 1 � < Cx , Cx > + < u , u > dt (cost) 2 0 and the linear quadratic regulator problem is LQR problem Minimize the quadratic (cost) with respect to the linear constraints (Cauchy),(output). 3/26 jens.saak@mathematik.tu-chemnitz.de Jens Saak Efficient Large Scale Lyapunov and ARE Solvers
LRCF Newton Method for the ARE Reordering Strategies ADI Shift Parameters Column Compression for the LRCF Generalized Systems Conclusions and Outlook Origin of the Riccati equations In the open literature it is well understood that the optimal feedback control is given as u = − B T X ∞ x , where X ∞ is the minimal, positive semidefinite, selfadjoint solution of the algebraic Riccati equation 0 = R ( X ) := C T C + A T X + XA − XBB T X . 4/26 jens.saak@mathematik.tu-chemnitz.de Jens Saak Efficient Large Scale Lyapunov and ARE Solvers
LRCF Newton Method for the ARE Reordering Strategies ADI Shift Parameters Column Compression for the LRCF Generalized Systems Conclusions and Outlook Outline LRCF Newton Method for the ARE 1 Reordering Strategies 2 ADI Shift Parameters 3 Column Compression for the low rank factors 4 Generalized Systems 5 Conclusions and Outlook 6 5/26 jens.saak@mathematik.tu-chemnitz.de Jens Saak Efficient Large Scale Lyapunov and ARE Solvers
LRCF Newton Method for the ARE Reordering Strategies Large Scale Riccati and Lyapunov Equations ADI Shift Parameters Newton’s method for solving the ARE Column Compression for the LRCF Cholesky factor ADI for Lyapunov equations Generalized Systems Conclusions and Outlook LRCF Newton Method for the ARE LRCF Newton Method for the ARE 1 Large Scale Riccati and Lyapunov Equations Newton’s method for solving the ARE Cholesky factor ADI for Lyapunov equations Reordering Strategies 2 ADI Shift Parameters 3 Column Compression for the low rank factors 4 Generalized Systems 5 Conclusions and Outlook 6 6/26 jens.saak@mathematik.tu-chemnitz.de Jens Saak Efficient Large Scale Lyapunov and ARE Solvers
LRCF Newton Method for the ARE Reordering Strategies Large Scale Riccati and Lyapunov Equations ADI Shift Parameters Newton’s method for solving the ARE Column Compression for the LRCF Cholesky factor ADI for Lyapunov equations Generalized Systems Conclusions and Outlook LRCF Newton Method for the ARE Large Scale Riccati and Lyapunov Equations We are interested in solving algebraic Riccati equations 0 = A T P + PA − PBB T P + C T C =: R ( P ) . (ARE) where A ∈ R n × n sparse, n ∈ N “large” C ∈ R p × n and p ∈ N with p ≪ n B ∈ R n × m and m ∈ N with m ≪ n 7/26 jens.saak@mathematik.tu-chemnitz.de Jens Saak Efficient Large Scale Lyapunov and ARE Solvers
LRCF Newton Method for the ARE Reordering Strategies Large Scale Riccati and Lyapunov Equations ADI Shift Parameters Newton’s method for solving the ARE Column Compression for the LRCF Cholesky factor ADI for Lyapunov equations Generalized Systems Conclusions and Outlook LRCF Newton Method for the ARE Large Scale Riccati and Lyapunov Equations We are interested in solving algebraic Riccati equations 0 = A T P + PA − PBB T P + C T C =: R ( P ) . (ARE) where A ∈ R n × n sparse, n ∈ N “large” C ∈ R p × n and p ∈ N with p ≪ n B ∈ R n × m and m ∈ N with m ≪ n and Lyapunov equations F T P + PF = − GG T . (LE) with F ∈ R n × n sparse or sparse + low G ∈ R n × m and m ∈ N with m ≪ n rank update, n ∈ N “large” 7/26 jens.saak@mathematik.tu-chemnitz.de Jens Saak Efficient Large Scale Lyapunov and ARE Solvers
LRCF Newton Method for the ARE Reordering Strategies Large Scale Riccati and Lyapunov Equations ADI Shift Parameters Newton’s method for solving the ARE Column Compression for the LRCF Cholesky factor ADI for Lyapunov equations Generalized Systems Conclusions and Outlook LRCF Newton Method for the ARE Newton’s method for solving the ARE Newton’s iteration for the ARE R ′ | P ( N l ) = − R ( P l ) , P l +1 = P l + N l , where the Frech´ et derivative of R at P is the Lyapunov operator R ′ | P : Q �→ ( A − BB T P ) T Q + Q ( A − BB T P ) , can be rewritten as one iteration step ( A − BB T P l ) T P l +1 + P l +1 ( A − BB T P l ) = − C T C − P l BB T P l i.e. in every Newton step we have to solve a Lyapunov equation of the form (LE) 8/26 jens.saak@mathematik.tu-chemnitz.de Jens Saak Efficient Large Scale Lyapunov and ARE Solvers
LRCF Newton Method for the ARE Reordering Strategies Large Scale Riccati and Lyapunov Equations ADI Shift Parameters Newton’s method for solving the ARE Column Compression for the LRCF Cholesky factor ADI for Lyapunov equations Generalized Systems Conclusions and Outlook LRCF Newton Method for the ARE Cholesky factor ADI for Lyapunov equations Recall Peaceman Rachford ADI: Consider Au = s where A ∈ R n × n spd, s ∈ R n . ADI Iteration Idea: Decompose A = H + V with H , V ∈ R n × n such that ( H + pI ) v = r ( V + pI ) w = t can be solved easily/efficiently. 9/26 jens.saak@mathematik.tu-chemnitz.de Jens Saak Efficient Large Scale Lyapunov and ARE Solvers
LRCF Newton Method for the ARE Reordering Strategies Large Scale Riccati and Lyapunov Equations ADI Shift Parameters Newton’s method for solving the ARE Column Compression for the LRCF Cholesky factor ADI for Lyapunov equations Generalized Systems Conclusions and Outlook LRCF Newton Method for the ARE Cholesky factor ADI for Lyapunov equations Recall Peaceman Rachford ADI: Consider Au = s where A ∈ R n × n spd, s ∈ R n . ADI Iteration Idea: Decompose A = H + V with H , V ∈ R n × n such that ( H + pI ) v = r ( V + pI ) w = t can be solved easily/efficiently. ADI Iteration If H , V spd ⇒ ∃ p j , j = 1 , 2 , ... J such that = 0 u 0 ( H + p j I ) u j − 1 = ( p j I − V ) u j − 1 + s (PR-ADI) 2 ( V + p j I ) u j = ( p j I − H ) u j − 1 2 + s converges to u ∈ R n solving Au = s . 9/26 jens.saak@mathematik.tu-chemnitz.de Jens Saak Efficient Large Scale Lyapunov and ARE Solvers
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