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Approximate abstractions of control systems with an application to aggregation. (English) Zbl 1451.93048

Summary: Previous approaches to constructing abstractions for control systems rely on geometric conditions or, in the case of an interconnected control system, a condition on the interconnection topology. Since these conditions are not always satisfiable, we relax the restrictions on the choice of abstractions, instead opting to select ones which nearly satisfy such conditions via optimization-based approaches. To quantify the resulting effect on the error between the abstraction and concrete control system, we introduce the notions of practical simulation functions and practical storage functions. We show that our approach facilitates the procedure of aggregation, where one creates an abstraction by partitioning agents into aggregate areas. We demonstrate the results on an application where we regulate the temperature in three separate zones of a building.

MSC:

93B11 System structure simplification
93D30 Lyapunov and storage functions
93C05 Linear systems in control theory

Software:

Mosek; YALMIP; Gurobi
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Full Text: DOI arXiv

References:

[1] Antoulas, A. C., Approximation of large-scale dynamical systems (2005), SIAM · Zbl 1112.93002
[2] Arcak, M.; Meissen, C.; Packard, A., Networks of dissipative systems: compositional certification of stability, performance, and safety (2016), Springer · Zbl 1343.93001
[3] Baier, C.; Katoen, J.-P., Principles of model checking (2008), MIT Press · Zbl 1179.68076
[4] Boyd, S.; El Ghaoui, L.; Feron, E.; Balakrishnan, V., Linear matrix inequalities in system and control theory (1994), SIAM · Zbl 0816.93004
[5] time-scale modeling of dynamic networks with applications to power systems (1982), Springer-Verlag: Springer-Verlag Berlin Heidelberg · Zbl 0498.93003
[6] Donzé, Alexandre, On signal temporal logic, (International conference on runtime verification (2013), Springer), 382-383
[7] Ferreira, A. S.R.; Arcak, M., A graph partitioning approach to predicting patterns in lateral inhibition systems, SIAM Journal on Applied Dynamical Systems, 12, 4, 2012-2031 (2013) · Zbl 1278.93029
[8] Girard, Antoine, A composition theorem for bisimulation functions (2013), arXiv preprint arXiv:1304.5153 · Zbl 1291.93156
[9] Girard, A.; Gössler, G.; Mouelhi, S., Safety controller synthesis for incrementally stable switched systems using multiscale symbolic models, IEEE Transactions on Automatic Control, 61, 6, 1537-1549 (2016) · Zbl 1359.93350
[10] Girard, A.; Pappas, G. J., Hierarchical control system design using approximate simulation, Automatica, 45, 2, 566-571 (2009) · Zbl 1158.93301
[11] Godsil, C.; Royle, G. F., Algebraic graph theory (vol. 207) (2013), Springer Science & Business Media
[12] Gurobi Optimization, L. L.C., Gurobi optimizer reference manual (2018)
[13] Lofberg, J., YALMIP: A toolbox for modeling and optimization in MATLAB, (Computer aided control systems design, 2004 IEEE international symposium on (2004), IEEE), 284-289
[14] Lofberg, J., Binmodel (2016), https://yalmip.github.io/command/binmodel/. (Accessed December 2017)
[15] Maler, Oded; Nickovic, Dejan, Monitoring temporal properties of continuous signals, (Formal techniques, modelling and analysis of timed and fault-tolerant systems (2004), Springer), 152-166 · Zbl 1109.68518
[16] MOSEK ApS, The MOSEK optimization toolbox for MATLAB manual. Version 8.1 (2017)
[17] Raman, V.; Donzé, A.; Maasoumy, M.; Murray, R. M.; Sangiovanni-Vincentelli, A. L.; Seshia, S. A., Model predictive control for signal temporal logic specification (2017), CoRR, abs/1703.09563. http://arxiv.org/abs/1703.09563
[18] Rungger, M.; Zamani, M., Compositional construction of approximate abstractions of interconnected control systems, IEEE Transactions on Control of Network Systems, 5, 1, 116-127 (2016) · Zbl 1507.93095
[19] Sandberg, H.; Murray, R. M., Model reduction of interconnected linear systems, Optimal Control Applications & Methods, 30, 3, 225-245 (2009)
[20] Smith, S. W.; Arcak, M.; Zamani, M., Hierarchical control via an approximate aggregate manifold, (American control conference, 2018 (2018), IEEE), 2378-2383
[21] Sontag, E. D., Mathematical control theory (vol. 6) (1998), Springer-Verlag: Springer-Verlag New York · Zbl 0945.93001
[22] van der Schaft, A., Equivalence of dynamical systems by bisimulation, IEEE Transactions on Automatic Control, 49, 12, 2160-2172 (2004) · Zbl 1365.93212
[23] Yin, H.; Bujarbaruah, M.; Arcak, M.; Packard, A., Optimization based planner tracker design for safety guarantees (2019), arXiv preprint arXiv:1910.00782
[24] Young, W. H., On classes of summable functions and their Fourier series, Proceedings of the Royal Society of London, Series A, 87, 594, 225-229 (1912) · JFM 43.0334.09
[25] Zamani, M.; Arcak, M., Compositional abstraction for networks of control systems: A dissipativity approach, IEEE Transactions on Control of Network Systems, 5, 3, 1003-1015 (2018) · Zbl 1515.93137
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