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

Overview Local information Distributed computing and decision making Dynamical behavior Communication constraints Robustness –Uncertainty –Reconfiguration & recovery Hierarchical and Distributed Systems Dominant Issues : August 2, 20011/5

Recent and Ongoing Results Heterogeneous distributed systems –C. Beck, Approximation Methods for Heterogeneous Distributed Systems, in preparation Spatially invariant distributed systems –C. Beck and R. D’Andrea, Simplification of Spatially Distributed Systems, 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 August 2, 20012/5

Overview Maintain system structure Systematic approach to reduced model Handle latency and uncertainty Model varying levels of granularity Objectives: Reduce model complexity for analysis, design, simulation August 2, 20013/5

Overview Utilize ideas from Methods: Focus on: Controls and Dynamical Systems Optimization Communications Unifying mathematical framework Computational tractability Communications August 2, 20014/5

Overview Multi-Dimensional Systems Principal Component Analysis Semi-Definite Programming Communications –Protocols –Aggregation –Fluids models Tools: August 2, 20015/5

Spatially Distributed Systems Local dynamic interactions between neighboring subsystems lead to overall complex system behavior automobiles, formation flight, power networks, smart materials, temperature distribution August 2, 20011/4

Spatially Distributed Systems Individual vehicles maintain local control Aircraft interact physically via the fluid dynamics Communication between individual controllers to maintain formation and performance Formation Flight August 2, 20012/4

Spatially Distributed Systems 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 Power Networks August 2, /4

Spatially Distributed Systems Control Strategies: CentralizedDecentralized Distributed August 2, 20014/4

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

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

Modeling August 2, 2001 Example: 2D Heat Equations 3/4 q( t, p 1, p 2 ) is temperature of plate; q = 0 at infinity Discretization: Rewrite:

Modeling Example: 2D Heat Equations August 2, /4 Define: state vector shift operator Discretized Heat Equation is: where are memoryless operators

Model Complexity Reduction Spatially Distributed Systems: August 2, 20011/3 Use multi-D realization matrices to form operator inequalities: P and Q inherit structure from multi-D system:

Model Complexity Reduction August 2, 2001 Spatially Distributed Systems: 2/3 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

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

Spatially Distributed Systems Homogeneity/Symmetry –individual subsystems identical –infinite extent –or- periodic boundary conditions Apply –standard Fourier methods –linear algebra –semi-definite programming (SDP) Issues: August 2, 20011/2

Spatially Distributed Systems 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 Issues: August 2, 20012/2

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 August 2, 20011/1

Multi-Level Granularity August 2, 20011/1 Subsystem S1 Subsystem S2 Analysis, design, simulation focus on S1 Reduce S2

Robustness Analysis August 2, 20011/1 Model Reduction in Interconnected Systems Reduce : thenIf interconnection stable

Next August 2, 20011/1 Delays wide ranging and nonstationary Networked Systems limited bandwidth, topological issues

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 August 2, 20011/1

Additional Research Projects Hybrid Systems Control –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 August 2, 20011/1