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:

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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

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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

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Overview

Utilize ideas from

Methods:

Focus on:

• Controls and Dynamical Systems• Optimization• Communications

• Unifying mathematical framework• Computational tractability• Communications

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Overview

• Multi-Dimensional Systems• Principal Component Analysis• Semi-Definite Programming• Communications

– Protocols

– Aggregation

– Fluids models

Tools:

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Spatially Distributed Systems

Local dynamic interactions between neighboring subsystems lead to overall complex system behavior

automobiles, formation flight, power networks, smart materials, temperature distribution

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0G1G2G 1G 2G

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

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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

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Spatially Distributed SystemsControl Strategies:

Centralized Decentralized

Distributed

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2G0G

2G

2G

0G

0G1G1G

1G

1G

1G

1G

0K

0K

1K

1K

1K

1K

2K

2K

K

Modeling One-dimensional Systems

1/4

State-space form: nkx R

kkkkk

kkkkk

uDxCy

uBxAx

1

Shift operator:22: Z

Operator:

),,,,0(),,,( 210210 xxxxxxZ

22: G

G

ku

ky

DuCxy

BuZAxZx

August 2, 2001

ModelingMulti-dimensional SystemsState-space form:

Shift operators:

),,1,,,(),,,( 11 LiLi pppkxppkxS

DuCxy

BuSAxSx

22: iS

pG1pG 1pG

),( pku

),( pky )1,( pky

)1,( pku)1,( pku

)1,( pky

),(),(),(),(),(

),(),(),(),(

)1,,,(

),,1,(

),,1,(

),,,1(

1

12

11

10

pkupkDpkxpkCpky

pkupkBpkxpkA

ppkx

ppkx

ppkx

ppkx

Lm

L

L

L

2/4 August 2, 2001

Modeling

August 2, 2001

Example: 2D Heat Equations

3/4

wp

q

p

q

t

q

22

2

21

2

q( t, p1, p2 ) is temperature of plate; q = 0 at infinity

Discretization:),1,(),1,(),,(),,1( 21212121 pptqpptqpptqpptq

ZwqSSSSZZqq )4( 122

111

Rewrite:

),,(),,(4)1,,()1,,( 21212121 pptwpptqpptqpptq

ModelingExample: 2D Heat Equations

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Define: state vector

shift operator ),,,,(diag 122

111

SSSSZS

Discretized Heat Equation is:

)( BwAxSx

where are memoryless

operators

0

0

0

0

1

),(,

00001

00001

00001

00001

11113

),( pkBpkA

TqSqSqSqSqx 122

111

Model Complexity ReductionSpatially Distributed Systems:

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• Use multi-D realization matrices to form operator inequalities:

• P and Q inherit structure from multi-D system:

0

0***

****

CCQSAQSA

SSBBPSAPSA

1P

2P

mP

P )),((dim)(dim where pkxP ii

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

rnonsingula Spatial

definite positive Temporal

i

i

P

P

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 Gr such that:

3/3

1G 0G 1G 2G2G

rGG

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:

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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:

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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

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Multi-Level Granularity

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Subsystem S1 Subsystem S2

• Analysis, design, simulation focus on S1• Reduce S2

dimensions reduce;then, stateSet 22

1

2

1S

S

S

S

S xP

PP

x

xx

Robustness Analysis

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S1 S2

Model Reduction in Interconnected Systems

QRHQ 21inf SS

Reduce: rSS 2

then

rSS 1

If rSS1

interconnection stable

21 SS interconnection

stable

Next

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• Delays wide ranging and

nonstationary

• Networked Systems limited bandwidth,

topological issues

0G1G2G 1G 2G

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

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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

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