& - Analysis and Reduction of Large- Scale Dynamic Systems in MATLAB
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1 & - Analysis and Reduction of Large- Scale Dynamic Systems in MATLAB Alessandro Castagnotto in collaboration with: Maria Cruz Varona, Boris Lohmann related publications: sss & sssmor: Analysis and reduction of large-scale dynamic systems in MATLAB, at Automatisierungstechnik (2017) EU-MORNET 2 nd Exploratory Workshop Luxembourg
2 Model Order MORLab LTI Systems Parametric and LTV Systems Nonlinear Systems Second Order Systems (linear and non-linear) Tools
3 The model order reduction setting 3
4 Model order reduction (MOR) MOR high-fidelity approximation preservation of properties numerically efficient Source(s): nasa.gov, wikimedia.org, dailymail.co.uk 4
5 Projective MOR Approximation in the subspace. Petrov-Galerkin Projection: (cf. projection by ) [DeVillemagne/Skelton 87] 5
6 Balanced Truncation Preserve state-space directions with highest energy transfer. Controllability and Observability from an energy perspective Balanced realization is system invariant [Moore 81, Penzl 00, Li/White 02, Saak 09] 6
7 Moment matching (rational interpolation) Moments of a transfer function original : Interpolation frequency (shift) : i-th moment about Magnitude / db Rational Krylov (RK) subspaces Choose V und W such that: Frequency / rad/ sec [Grimme 97, Gallivan et al. 04a] 7
8 -optimal model order reduction Iterative Rational Krylov Algorithm Gradient-based methods Expressions for gradient and Hessian can be derived and used for trust-region optimization [Meier/Luenberger 67, Gugercin/Antoulas/Beattie 08, Gugercin/Beattie 12, Panzer et al. 14, C. et al 16] 8
9 sss & sssmor MATLAB Toolboxes 9
10 Toolboxes for sparse, large-scale models in sys = sss(a,b,c,d,e); sysr = tbr(sys,n) sysr = rk(sys,s0) sysr = irka(sys,s0) sysr = cure(sys) sysr = cirka(sys,s0) bode(sys), sigma(sys) step(sys), impulse(sys) norm(sys,2), norm(sys,inf) c2d, lsim, eigs, connect, Powered by: M-M.E.S.S. toolbox [Saak, Köhler, Benner] for Lyapunov equations Available at [C./Cruz Varona/Jeschek/Lohmann: sss & sssmor: Analysis and Reduction of Large- Scale Dynamic Systems in MATLAB, 2017 at-automatisierungstechnik] 10
11 Main characteristics State-space models of very high order on a standard computer Many Control System Toolbox functions, revisited to exploit sparsity Allows system analysis in frequency (bode, sigma, ) and time domain (step,norm,lsim, ), as well as manipulations (connect,truncate, ) Is compatible with the built-in syntax New functionality: eigs, residue, pzmap, Classical (modalmor, tbr, rk, ) and state-of-the-art (isrk, irka, cirka, cure, ) reduction methods Both highly-automatized sysr = irka(sys,n) and highly-customizable Opts.maxiter = 100 Opts.tol = 1e-6 Opts.stopcrit = comball Opts.verbose = true sysr = irka(sys,n,opts) solvelse and lyapchol as core functions 11
12 Comprehensive documentation with examples and references sssmor App graphical user interface completely free and open source (contributions welcome)
13 Comprehensive documentation with examples and references sssmor App graphical user interface completely free and open source (contributions welcome)
14 Comprehensive documentation with examples and references sssmor App graphical user interface completely free and open source (contributions welcome)
15 sss & sssmor a test case Live demo available for download here 15
16 Summary sss & sssmor: Analysis and reduction of large-scale dynamic systems in MATLAB Model reduction as projection Balanced truncation, IRKA Live demo Control System Toolbox cannot handle sparsity sss allows analysis of large-scale, sparse state space models sssmor allows the automated reduction of sss models with a variety of methods ssred objects can be used in MATLAB tools to efficiently design reduced order controllers psssmor analysis and reduction of parametric models 16
17 & - Analysis and Reduction of Large- Scale Dynamic Systems in MATLAB Alessandro Castagnotto in collaboration with: Maria Cruz Varona, Boris Lohmann related publications: sss & sssmor: Analysis and reduction of large-scale dynamic systems in MATLAB, at Automatisierungstechnik (2017) EU-MORNET 2 nd Exploratory Workshop Luxembourg
18 References [C. et al. 17] [C./Panzer/Lohmann 16] [De Villemagne/Skelton 87] [Grimme 97] sss & sssmor: Analysis and reduction of large-scale dynamic systems in MATLAB Fast H2-optimal model order reduction exploiting the local nature of Krylov subspace methods Model reductions using a projection formulation Krylov projection methods for model reduction [Gugercin/Antoulas/Beattie 08] H2-optimal model reduction for large-scale linear dynamical systems [Gugercin/Beattie 09] [Meier/Luenberger 67] [Panzer et al. 14] A trust-region method for optimal H2 model reduction Approximation of linear constant systems Greedy rational Krylov method for H2-pseudooptimal model order reduction with preservation of stability 18
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