Terminal and Order Reduction of Multi-Input/Output LTI Systems Andr - PowerPoint PPT Presentation

Summer School on Numerical Linear Algebra for Dynamical and High-Dimensional Problems, Trogir, Croatia, October 12, 2011 Terminal and Order Reduction of Multi-Input/Output LTI Systems Andr´ e Schneider Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg, Germany System Reduction for Nanoscale MAX PLANCK INSTITUTE IC Design FOR DYNAMICS OF COMPLEX TECHNICAL SYSTEMS MAGDEBURG Max Planck Institute Magdeburg A. Schneider, Terminal and Order Reduction of MIMO LTI Systems 1/9

Overview The Setting Linear Time Invariant Descriptor System E ˙ x ( t ) = Ax ( t ) + Bu ( t ) , x (0) = x 0 , y ( t ) = Cx ( t ) + Du ( t ) , with A , E ∈ R n × n , B ∈ R n × m , C T ∈ R n × p , D ∈ R p × m (mostly in application D ≡ 0), x ∈ R n containing the generalized state space variables, u ∈ R m the vector of input variables, y ∈ R p the output vector, and x 0 ∈ R n the initial value. Max Planck Institute Magdeburg A. Schneider, Terminal and Order Reduction of MIMO LTI Systems 2/9

Overview The Setting Linear Time Invariant Descriptor System E ˙ x ( t ) = Ax ( t ) + Bu ( t ) , x (0) = x 0 , y ( t ) = Cx ( t ) + Du ( t ) , with A , E ∈ R n × n , B ∈ R n × m , C T ∈ R n × p , D ∈ R p × m (mostly in application D ≡ 0), x ∈ R n containing the generalized state space variables, u ∈ R m the vector of input variables, y ∈ R p the output vector, and x 0 ∈ R n the initial value. Either m , p ∈ O ( n ), Max Planck Institute Magdeburg A. Schneider, Terminal and Order Reduction of MIMO LTI Systems 2/9

Overview The Setting Linear Time Invariant Descriptor System E ˙ x ( t ) = Ax ( t ) + Bu ( t ) , x (0) = x 0 , y ( t ) = Cx ( t ) + Du ( t ) , with A , E ∈ R n × n , B ∈ R n × m , C T ∈ R n × p , D ∈ R p × m (mostly in application D ≡ 0), x ∈ R n containing the generalized state space variables, u ∈ R m the vector of input variables, y ∈ R p the output vector, and x 0 ∈ R n the initial value. Either m , p ∈ O ( n ), or m ∈ O ( n ) and p ≪ n , Max Planck Institute Magdeburg A. Schneider, Terminal and Order Reduction of MIMO LTI Systems 2/9

Overview The Setting Linear Time Invariant Descriptor System E ˙ x ( t ) = Ax ( t ) + Bu ( t ) , x (0) = x 0 , y ( t ) = Cx ( t ) + Du ( t ) , with A , E ∈ R n × n , B ∈ R n × m , C T ∈ R n × p , D ∈ R p × m (mostly in application D ≡ 0), x ∈ R n containing the generalized state space variables, u ∈ R m the vector of input variables, y ∈ R p the output vector, and x 0 ∈ R n the initial value. Either m , p ∈ O ( n ), or m ∈ O ( n ) and p ≪ n , or vise versa. Max Planck Institute Magdeburg A. Schneider, Terminal and Order Reduction of MIMO LTI Systems 2/9

Overview The Setting Linear Time Invariant Descriptor System E ˙ x ( t ) = Ax ( t ) + Bu ( t ) , x (0) = x 0 , y ( t ) = Cx ( t ) + Du ( t ) , with A , E ∈ R n × n , B ∈ R n × m , C T ∈ R n × p , D ∈ R p × m (mostly in application D ≡ 0), x ∈ R n containing the generalized state space variables, u ∈ R m the vector of input variables, y ∈ R p the output vector, and x 0 ∈ R n the initial value. Either m , p ∈ O ( n ), or m ∈ O ( n ) and p ≪ n , or vise versa. Applying the Laplace transform leads to Transfer Function H ( s ) = C ( sE − A ) − 1 B , s ∈ C . Max Planck Institute Magdeburg A. Schneider, Terminal and Order Reduction of MIMO LTI Systems 2/9

Overview Moore’s Law Transistor counts for integrated circuits plotted against their dates of introduction. The curve shows Moore’s law - the doubling of transistor counts every two years. Source: Wikipedia Max Planck Institute Magdeburg A. Schneider, Terminal and Order Reduction of MIMO LTI Systems 3/9

Overview An Old Slide of 2008 Wafer Development Integrated circuits (ICs) are miniaturized electronic devices. Recently, use of nanometer-scale chip manufacturing process, increasing number of (parasitic) elements, and production of multi-layered ICs. source: Qimonda AG Today, the Intel Core 2 Extreme (Conroe XE) CPU has about 291 million transistors, produced with a 65 nanometer chip manufacturing process. Max Planck Institute Magdeburg A. Schneider, Terminal and Order Reduction of MIMO LTI Systems 4/9

Overview Version 10/2011 Wafer Development Integrated circuits (ICs) are miniaturized electronic devices. Recently, use of nanometer-scale chip manufacturing process, increasing number of (parasitic) elements, and production of Altera Stratix IV EP4SGX230 FPGA on a PCB, multi-layered ICs. source: Altera Corporation Today: 2011 CPU - Intel Xeon Westmere-EX - 2.6 billion transistors (32nm), 2010 GPU - Nvidia GF100 - 3.0 bn (40nm), 2011 FPGA - Altera Stratix V - 3.8 bn (28nm). Max Planck Institute Magdeburg A. Schneider, Terminal and Order Reduction of MIMO LTI Systems 5/9

Overview Animation of a multi-layered IC wiring Multi-layered IC wiring leads to: very complex structure, a lot of interactions within the IC, large parasitic subcircuits with a large number of terminals, a system described at the beginning of the talk. Wikipedia image: Silicon chip 3d.png Examples are power grids and clock distribution networks. Max Planck Institute Magdeburg A. Schneider, Terminal and Order Reduction of MIMO LTI Systems 6/9

Overview Schematic Representation inputs u ( t ) outputs y ( t ) H ( s ) Max Planck Institute Magdeburg A. Schneider, Terminal and Order Reduction of MIMO LTI Systems 7/9

Overview Schematic Representation u ( t ) y ( t ) H r ( s ) Max Planck Institute Magdeburg A. Schneider, Terminal and Order Reduction of MIMO LTI Systems 7/9

Overview Schematic Representation ˜ u ( t ) y ( t ) H r ( s ) Max Planck Institute Magdeburg A. Schneider, Terminal and Order Reduction of MIMO LTI Systems 7/9

Overview Schematic Representation u ( t ) ? ˜ y ( t ) H r ( s ) ? Max Planck Institute Magdeburg A. Schneider, Terminal and Order Reduction of MIMO LTI Systems 7/9

Overview Existing Methods Methods which try to get a hand on the problem of many terminals: First ideas: RLP-Algorithm [Jain et al. 2004] , RecMOR, SVDMOR [Feldmann, Liu 2004] , Max Planck Institute Magdeburg A. Schneider, Terminal and Order Reduction of MIMO LTI Systems 8/9

Overview Existing Methods Methods which try to get a hand on the problem of many terminals: First ideas: RLP-Algorithm [Jain et al. 2004] , RecMOR, SVDMOR [Feldmann, Liu 2004] , ESVDMOR [Liu, Tan et al. 2007] = ⇒ SyreNe , Max Planck Institute Magdeburg A. Schneider, Terminal and Order Reduction of MIMO LTI Systems 8/9

Overview Existing Methods Methods which try to get a hand on the problem of many terminals: First ideas: RLP-Algorithm [Jain et al. 2004] , RecMOR, SVDMOR [Feldmann, Liu 2004] , ESVDMOR [Liu, Tan et al. 2007] = ⇒ SyreNe , TermMerg approach [Liu, Tan et al. 2007] , Max Planck Institute Magdeburg A. Schneider, Terminal and Order Reduction of MIMO LTI Systems 8/9

Overview Existing Methods Methods which try to get a hand on the problem of many terminals: First ideas: RLP-Algorithm [Jain et al. 2004] , RecMOR, SVDMOR [Feldmann, Liu 2004] , ESVDMOR [Liu, Tan et al. 2007] = ⇒ SyreNe , TermMerg approach [Liu, Tan et al. 2007] , Interpolation based methods [Antoulas, Lefteriu 2009] , Max Planck Institute Magdeburg A. Schneider, Terminal and Order Reduction of MIMO LTI Systems 8/9

Overview Existing Methods Methods which try to get a hand on the problem of many terminals: First ideas: RLP-Algorithm [Jain et al. 2004] , RecMOR, SVDMOR [Feldmann, Liu 2004] , ESVDMOR [Liu, Tan et al. 2007] = ⇒ SyreNe , TermMerg approach [Liu, Tan et al. 2007] , Interpolation based methods [Antoulas, Lefteriu 2009] , Approaches based on graph theory [Rommes, Schilders, Ionutiu 2010] , Max Planck Institute Magdeburg A. Schneider, Terminal and Order Reduction of MIMO LTI Systems 8/9

Overview Existing Methods Methods which try to get a hand on the problem of many terminals: First ideas: RLP-Algorithm [Jain et al. 2004] , RecMOR, SVDMOR [Feldmann, Liu 2004] , ESVDMOR [Liu, Tan et al. 2007] = ⇒ SyreNe , TermMerg approach [Liu, Tan et al. 2007] , Interpolation based methods [Antoulas, Lefteriu 2009] , Approaches based on graph theory [Rommes, Schilders, Ionutiu 2010] , Balanced truncation for many terminals (BTMT) [Benner, S. 2011] . Max Planck Institute Magdeburg A. Schneider, Terminal and Order Reduction of MIMO LTI Systems 8/9

Overview Schematic Representation of TermMerg inputs u ( t ) outputs y ( t ) H ( s ) Max Planck Institute Magdeburg A. Schneider, Terminal and Order Reduction of MIMO LTI Systems 9/9

Overview Schematic Representation of TermMerg H ( s ) Max Planck Institute Magdeburg A. Schneider, Terminal and Order Reduction of MIMO LTI Systems 9/9

Overview Schematic Representation of TermMerg H r ( s ) Max Planck Institute Magdeburg A. Schneider, Terminal and Order Reduction of MIMO LTI Systems 9/9