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Modeling and Complexity Reduction for Interconnected Systems

Modeling and Complexity Reduction for Interconnected Systems. Carolyn Beck University of Illinois at Urbana-Champaign. August 2, 2001. Local information Distributed computing and decision making Dynamical behavior. Communication constraints Robustness Uncertainty

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Modeling and Complexity Reduction for Interconnected Systems

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  1. Modeling and Complexity Reduction for Interconnected Systems Carolyn Beck University of Illinois at Urbana-Champaign August 2, 2001

  2. Local information Distributed computing and decision making Dynamical behavior Communication constraints Robustness Uncertainty Reconfiguration & recovery Overview HierarchicalandDistributedSystems DominantIssues: 1/5 August 2, 2001

  3. Recent and Ongoing Results • Heterogeneous distributed systems • C. Beck, Approximation Methods for Heterogeneous Distributed Systems, inpreparation • Spatially invariant distributed systems • C. Beck and R. D’Andrea, SimplificationofSpatiallyDistributedSystems, CDC 1999 (and in preparation) • Linear time-varying systems • C. Beck and S. Lall, Model Reduction Error Bounds for Linear Time-varying Systems, MTNS 1998 • S. Lall and C. Beck, Guaranteed Error Bounds for Model Reduction of Linear Time-varying Systems, Trans. on Automatic Control, in review • Systems with uncertainty • C. Beck, J. Doyle and K. Glover, Model Reduction of Multi-Dimensional and Uncertain Systems, Trans. on Automatic Control, 1996 • L. Andersson, A. Rantzer and C. Beck, Model Comparison and Simplification, Int. Journal of Robust and Nonlinear Control, 1999 2/5 August 2, 2001

  4. Objectives: Overview Reduce model complexity for analysis, design, simulation • Maintain system structure • Systematic approach to reduced model • Handle latency and uncertainty • Model varying levels of granularity 3/5 August 2, 2001

  5. Focus on: Controls and Dynamical Systems Optimization Communications Methods: Overview Utilizeideasfrom • Unifying mathematical framework • Computational tractability • Communications 4/5 August 2, 2001

  6. Overview Tools: • Multi-Dimensional Systems • Principal Component Analysis • Semi-Definite Programming • Communications • Protocols • Aggregation • Fluids models 5/5 August 2, 2001

  7. Spatially Distributed Systems Local dynamic interactions between neighboring subsystems lead to overall complex system behavior automobiles, formation flight, power networks, smart materials, temperature distribution 1/4 August 2, 2001

  8. Spatially Distributed Systems Formation Flight • Individual vehicles maintain local control • Aircraft interact physically via the fluid dynamics • Communication between individual controllers to maintain formation and performance 2/4 August 2, 2001

  9. Spatially Distributed Systems Power Networks • Large scale: approximately 15,000 generators in U.S. with 750,000 MW capacity • Generators, lines, loads are dynamic • Hierarchical control necessary • Control must be fault-tolerant • Control must be distributed • generators independently controlled • may be independently owned August 2, 2001 3/4

  10. Spatially Distributed Systems Control Strategies: Centralized Decentralized Distributed 4/4 August 2, 2001

  11. Modeling One-dimensional Systems State-space form: Operator: Shift operator: 1/4 August 2, 2001

  12. Multi-dimensional Systems Modeling State-space form: Shift operators: 2/4 August 2, 2001

  13. Modeling Example: 2D Heat Equations q( t, p1, p2 ) is temperature of plate; q = 0 at infinity Discretization: Rewrite: August 2, 2001 3/4

  14. Modeling Example: 2D Heat Equations Define: statevector shift operator Discretized Heat Equation is: are memoryless operators where August 2, 2001 4/4

  15. Model Complexity Reduction Spatially Distributed Systems: • Use multi-D realization matrices to form operator inequalities: • P and Q inherit structure from multi-D system: 1/3 August 2, 2001

  16. Model Complexity Reduction Spatially Distributed Systems: • Employ multi-D transform theory; operator inertia and congruence arguments; multi-D KYP lemma; LFT synthesis methods • Constraints on P and Q: • Apply multi-D principal component analysis 2/3 August 2, 2001

  17. distributed system structure is maintained error bound, e , determined before reducing Model Complexity Reduction A Priori Error Bounds: Given a distributed system G, find a lower dimension model Gr such that: 3/3 August 2, 2001

  18. Homogeneity/Symmetry individual subsystems identical infinite extent –or- periodic boundary conditions Apply standard Fourier methods linear algebra semi-definite programming (SDP) Spatially Distributed Systems Issues: 1/2 August 2, 2001

  19. Heterogeneity/Asymmetry individual subsystems may vary finite chains of subsystems where leading and trailing subsystems behave differently Apply system functions operator theory and analysis convex programming Spatially Distributed Systems Issues: 2/2 August 2, 2001

  20. Ongoing Research • Modeling multiple levels of granularity in interconnected systems • partitioned application of multi-D reduction methods • Robustness analysis • stability analysis of model-reduced subsystem interconnections • Networks • stability robustness analysis and scalability issues 1/1 August 2, 2001

  21. Multi-Level Granularity Subsystem S1 Subsystem S2 • Analysis, design, simulation focus on S1 • Reduce S2 1/1 August 2, 2001

  22. Robustness Analysis Model Reduction in Interconnected Systems Reduce: If then interconnection stable interconnection stable 1/1 August 2, 2001

  23. Next • Delays wide ranging and nonstationary • Networked Systems • limited bandwidth, topological issues 1/1 August 2, 2001

  24. Future Considerations • System Identification/Data-Based Models for Large Scale Systems • Subspace Identification (Principal Component Analysis) • Subsystem Identification (Multi-Level Granularity) • Real-time System Identification/Reduction: Reconfiguration and Recovery 1/1 August 2, 2001

  25. Additional Research Projects • HybridSystemsControl • J. Chudoung and C. Beck, An Optimal Control Theory for Nonlinear Impulsive Systems, in preparation • J. Chudoung and C. Beck, The Minimum Principle for Deterministic Impulsive Control Systems, to appear CDC 2001 • Multi-Dimensional Realization Theory • C. Beck and J. Doyle, A Necessary and Sufficient Minimality Condition for Uncertain Systems, Trans. on Automatic Control, 1999 • C. Beck, On Formal Power Series Representations for Uncertain Systems, Trans. on Automatic Control, 2001 • C. Beck and R. D’Andrea, Minimality, Reachability and Observability for a Class of Multi-Dimensional Systems, Int. Journal of Robust and Nonlinear Control, in review • Power Systems • P. Bendotti and C. Beck, On the Role of LFT Model Reduction Methods in Robust Controller Synthesis for a Pressurized Water Reactor, Trans. on Control Systems Technology, 1999 • Human Dynamics Modeling • C. Beck, R. Smith, H. Lin and M. Bloom, On the Application of System Identification and Model Validation Methods for Constructing Multivariable Anesthesia Response Models, CCA, 2000 • A. Mahboobin and C. Beck, Human Postural Control Modeling and System Analysis, in preparation 1/1 August 2, 2001

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