Subsystem Planning Determine how to break the total system into subsystems and assign appropriate sampling rates Calculation using SF init(max SM=I dSF down sample factor SM start mode EM end mode TM total mode bz block size SF= DSF /2 or EM-SM+1≥BZ DSF= DSF EM-SM≥B EM=SM+ BZ max-I or TM SM=EM+I EM= EM or TM SM=EM+I DSF= DSF/2 or Fast discretization Discretization is to compute matrix exponential I+at+ 2 Block diagonal A A,T 0 0 matrix allows fast exp 0 A,T discretization Generic algorithm for fast discretization for i= 1 to total number of subsystems index= location of subsystem; DT_ system(index)=C2d(subsystem) end Computation time for matrix exponential of a 1000 by 1000 diagonal matrix tells the story dense matrix: 27 11 seconds(on the test machine) sparse matrix 0.031 seconds(on the test machineSubsystem Planning DSF down sample factor SM start mode EM end mode TM total mode BZ block size DSF_init (max) SM = 1 EM = ? EM ≥ TM? EM – SM + 1 ≥ BZ_min? DSF = DSF EM – SM ≥ BZ_max? EM = EM or TM SM = EM + 1 DSF = DSF / 2 or 1 EM = SM + BZ_max - 1 or TM SM = EM + 1 DSF = DSF / 2 or 1 Calculation using bisection method Y N N N Y Y Terminate • Determine how to break the total system into subsystems and assign appropriate sampling rates. Fast Discretization • Discretization is to compute matrix exponential. 22 33 2 3! AT d AT AT A e I AT = =+ + + +L • Block diagonal A matrix allows fast discretization. 1 2 3 1 2 0 0 exp 0 0 A T A T A T AT e AT e e ⎛ ⎞ ⎡ ⎤ ⎡ ⎤ ⎜ ⎟ ⎢ ⎥ ⎢ ⎥ = ⎜ ⎟ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎝ ⎠ ⎣ ⎦ ⎣ ⎦ L L O O M OO M O • Generic algorithm for fast discretization for i = 1 to total number of subsystems index = location of subsystem; DT_system(index) = c2d(subsystem); end • Computation time for matrix exponential of a 1000 by 1000 diagonal matrix tells the story: • dense matrix: 27.11 seconds (on the test machine) • sparse matrix 0.031 seconds (on the test machine)