Decentralized control and filtering in interconnected dynamical systems

Qd6. T

Decentralized Control

and Filtering in

Interconnected

Dynamical Systems

Magdi s. Mahmoud

Di 0v)G

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_________aV.J"______________________________________

I Librai|^^>ấ)(^^ạress Cataloging-in-Publication Data

Mahmoud, Magdi s.« a ,

Decentralized coftrõíVn.dtftteiìn^ýf interconnected dynamical systems / Magdi s.

Mahmoud.

p. cm.

Includes bibliographical references and index.

ISBN 978-1-4398-3814-3 (hardback)

1. Decomposition method. 2. Large scale systems—Mathematical models. 3.

Mechanics, Analytic—Mathematical models. 4, Intelligent control systems. I. Title.

QA402.2.M34 2010

003’.71015118-dc22 2010035473

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In the Name of

the All-Compassionate, the All-Merciful.

'‘And of knowledge, you (mankind) have been given only a

little.”

Dedicated

To The Memory of My Parents

(Source of My Big Dreams)

To My Family: Salwa, Medhat, Monda, Mohamed

(For Their Love and Gratitude)

To My Grandchildern: Malak, Mostafa

(Representative of New Generations)

MsM

Dhahran, Saudi Arabia, 2010

Contents

*reface

Vcknowledgments

dotations and Symbols

List of Acronyms

Vuthor

xxi

xvii

XXV

xxix

xxxi

I Introduction

1.1 O v erv iew ....................................................................................

1.1 Information structure.....................................................

1.2 System representation ..................................................

Hierarchical sy stem s.....................................................

Structure of interconnections........................................

Revived challenges........................................................ 10

Outline of the B o o k .................................................................... 12

1.2.1 M ethodology.................................................................. 12

1.2.2 Chapter organization..................................................... 13

Some N o te s ................................................................................. 15

.1.3

.1.4

.15

1.2

1.3

Mathematical Foundations 17

2.1 Basic Mathematical Concepts..................................................... 17

2. 1.1

2. 1.2

2.1.3

2.1.4

2.1.5

Euclidean space.............................................................. 17

Norms of vecto rs........................................................... 18

Convex s e ts ..................................................................... 19

Continuous functions..................................................... 20

Function norm s.............................................................. 20

2.2 Calculus and Algebra of Matrices ............................................ 21

Contents￾2.2.1 Fundamental subspaces................................................. 22

2.2.2 Calculus of vector-matrix functions of a scalar .... 23

2.2.3 Derivatives of vector-matrix p ro d u cts......................... 24

2.2.4 Positive definite and positive semidefinite matrices . . 26

2.2.5 Trace properties.............................................................. 26

2.2.6 Partitioned matrices .................................................... 28

2.2.7 Matrix inversion le m m a .............................................. 29

2.2.8 Singular value decom position..................................... 30

2.3 Directed Graphs.......................................................................... 31

2.4 Notes and References................................................................. 33

Historical Perspective and Scope 35

3.1 O verview ................................................................................... 36

3.2 Decomposition-Coordination M ethods..................................... 36

3.2.1 Hierarchical structures.................................................. 38

3.2.2 Decentralized control.................................................... 39

3.3 Multilevel Optimization of Nonlinear Systems......................... 40

3.3.1 Local-level optim ization.............................................. 40

3.4 Decentralized Nonlinear Systems.............................................. 44

3.4.1 Diagonal dominance condition...................................... 46

3.4.2 Static controller design................................................. 47

3.4.3 Illustrative example 3 .1 ................................................. 50

3.5 Nominally Linear Systems—Continuous C ase ......................... 51

3.5.1 Decentralized control..................................................... 54

3.5.2 Two-level control.......................................................... 55

3.5.3 Illustrative example 3 .2 ................................................. 57

3.6 Linear S ystem s.......................................................................... 61

3.6.1 Illustrative example 3 .3 ................................................. 66

3.7 A Perturbational A pproach........................................................ 66

3.7.1 Multilevel state regulation........................................... 70

3.7.2 Generalized scheme ....................................................... 73

3.7.3 Illustrative example 3 .4 ................................................ 77

3.8 Observation and Estimation........................................................ 79

3.8.1 Model se t-u p ................................................................. 80

3.8.2 Observation scheme I .................................................... 80

3.8.3 Observation scheme I I ................................................. 82

3.8.4 Estimation sch em e........................................................ 83

3.9 Interconnected Discrete-Time Systems..................................... 87

3.9.1 Mode separation in discrete system s............................. 89

3.10 Feedback Control D esign.......................................................... 92

Con ten is XI

3.10.1 State feedback design..................................................... 92

3.10.2 Design algorithm 1 ........................................................ 93

3.10.3 Illustrative example 3 .5 .................................................. 93

3.10.4 Observer design ........................................................... 95

3.10.5 Design algorithm I I ........................................................ 95

3.10.6 Observer-based control d e sig n ...................................... 96

3.10.7 Design algorithm I I I ..................................................... 97

3.10.8 Illustrative example 3 .6 .................................................. 97

3.10.9 Decentralized state reconstruction............................... 102

3.10.10 Illustrative example 3 .7 ................................................. 104

3.10.11 Linear quadratic control design ................................. 105

3.10.12 Illustrative example 3 .8 .................................................. 110

3.11 Applications................................................................................ Ill

3.11.1 Serially connected subsystem s..................................... 112

3.11.2 Liquid-metal cooled re a c to r......................................... 112

3.11.3 Boiler system .................................................................. 115

3.11.4 Hydraulic system with electronic test g e a r................... 123

3.12 Problem Set I .............................................................................. 124

3.13 Notes and References.................................................................. 129

Decentralized Servomechanism System 131

4.1 Introduction................................................................................. 132

4.2 A Class of Large-Scale S y ste m s............................................... 132

4.3 Decentralized Stabilization........................................................ 134

4.3.1 Fixed modes .................................................................. 134

4.3.2 Decentralized fixed m o d e s............................................ 140

4.4 Illustrative E xam ples.................................................................. 141

4.4.1 Illustrative example 4 .1 ................................................... 141

4.4.2 Illustrative example 4 .2 ................................................... 143

4.4.3 Illustrative example 4 .3 ................................................... 144

4.5 Decentralized R egulators........................................................... 145

4.5.1 The robust decentralized servomechanism problem . . 14Ế

4.5.2 The robust decentralized controller ............................ 14Í

4.5.3 Decentralized controller for interconnected systems . . 15C

4.5.4 Illustrative example 4 .4 ................................................... 152

4.6 Hierarchically Structured Servomechanism ............................ 154

4.6.1 Decentralized controller stru c tu re ............................... 158

4.6.2 Illustrative example 4 .5 ................................................... 16C

4.6.3 Illustrative example 4 .6 ................................................... 161

4.7 Problem Set I I .............................................................................. 162

xii Contents.

4.8 Notes and References................................................................. 164

5 Overlapping Control 167'

5.1 Decomposition .......................................................................... 168

5.1.1 Decoupled subsystem s.................................................... 168

5.1.2 Overlapping subsystem s............................................... 170'

5.1.3 Transformations ........................................................... 170'

5.1.4 Illustrative example 5 .1 .................................................... 172

5.2 Feedback Control D esign........................................................... 173

5.2.1 Linear-quadratic c o n tro l................................................. 174

5.2.2 Change of basis................................................................ 176

5.2.3 Improved expansion-contraction..................................... 177

5.2.4 Particular selection.......................................................... 181

5.2.5 Illustrative example 5 .2 .................................................... 183

5.3 LMI-Based Overlapping C ontrol............................................... 187

5.3.1 Design procedure............................................................. 189

5.3.2 Illustrative example 5 .3 .................................................... 190

5.4 Application to Unmanned Aerial Vehicles................................ 191

5.5 Problem Set III .................................. ................... . . 198

5.6 Notes and References................................................................. 201

6 Interconnected Time-Delay Systems 203

6.1 Introduction................................................................................ 203

6.2 Linear Interconnected Systems: Continuous-Time....................... 204

6.2.1 Local delay-dependent analysis ..................................... 206

6.2.2 Linear polytopic system s................................................. 212

6.2.3 Illustrative example 6 .1 .................................................... 214

6.2.4 State feedback stabilization.............................................. 215

6.2.5 Illustrative example 6 .2 ......................................................218

6.3 Nonlinear Interconnected System s............................................ 219

6.3.1 Problem statem ent.......................................................... 219

6.3.2 Local delay-dependent analysis ..................................... 220

6.3.3 State feedback stabilization.............................................. 226

6.3.4 Illustrative example 6 .3 .................................................... 229

6.4 Interconnected Discrete-Time-Delay S ystem s..............................231

6.4.1 Model description and preliminaries..................................232

6.4.2 Subsystem stability.......................................................... 234

6.4.3 State feedback design....................................................... 238

6.4.4 Bounded state feedback design........................................ 239

6.4.5 Resilient state feedback design........................................ 240

Con len is xiii

6.4.6 Output feedback stabilization ............................ . . . 244

6.4.7 Resilient output feedback stabilization............... . . 247

6.4.8 Illustrative example 6 .4 ........................................ . . . 250

6.5 Overlapping Control ........................................................ . . . 254

6.5.1 Systems and inclusions........................................ . . . 257

6.5.2 Inclusion of linear quadratic c o s ts ...................... . . . 259

6.5.3 Guaranteed cost control ..................................... . . . 264

6.5.4 Overlapping control design.................................. . 266

6.5.5 Illustrative example 6 .5 ........................................ . . . 269

6.6 Problem Set I V .................................................................. . . . 271

6.7 Notes and References........................................................ . . . 275

7 Decentralized Reliable Control 277

7.1 Interconnected Continuous S y ste m s............................... . . . 277

7.1.1 Problem description ........................................... . . . 279

7.1.2 Actuator failure m o d e l........................................ . . . 280

7.1.3 State-feedback reliable d e s ig n ............................ . . . 281

7.1.4 Feedback g a in s ..................................................... . . . 288

7.1.5 Interconnected uncertain systems ...................... . . . 289

7.1.6 Static output-feedback reliable d e sig n ............... . . . 291

7.1.7 Illustrative example 7 .1 ........................................ . . . 293

7.2 Application to Multi-Area Power S ystem s...................... . . . 296

7.2.1 Single-area AGC model ..................................... . . . 300

7.2.2 Simulation results ............................................... . . . 301

7.2.3 Reliable control against sensor failures ............ . . . 305

7.2.4 State-feedback reliable d e s ig n ............................ . . . 307

7.2.5 Dynamic output-feedback reliable design .... . . . 314

7.2.6 Illustrative example 7 .2 ........................................ . . . 318

7.3 Interconnected Discrete Delay Systems ......................... . . . 321

7.3.1 Problem form ulation............................................ . . . 321

7.3.2 Failure m o d el........................................................ . . . 322

7.3.3 State-feedback reliable d e s ig n ............................ . . . 323

7.3.4 Illustrative example 7 .3 ........................................ . . . 331

7.4 Reliable Control of Symmetric Composite Systems . . . . . . 335

7.4.1 Problem form ulation............................................ . . . 337

7.4.2 Decentralized Koc c o n tro l.................................. . . . 339

7.4.3 Controller d e sig n .................................................. . . . 343

7.4.4 Illustrative example 7 .4 ......................................... . . . 346

7.4.5 Illustrative example 7 .5 ........................................ . 346

7.4.6 Reliability analysis............................................... . . . 347

xiv Contents

7.4.7 Control input failures..................................................... 347

7.4.8 General failures.............................................................. .349

7.4.9 Illustrative example 7 .6 .................................................... 351

7.4.10 Illustrative example 7 .7 ........................................................351

7.4.11 Illustrative example 7 .8 ....................................................... 351

7.5 Problem Set V ..................................................................................353

7.6 Notes and References......................................................................357

Decentralized Resilient Control 359

8.1 Introduction.....................................................................................359

8.2 Problem Statem ent.........................................................................360

8.3 Resilient Decentralized Stabilization............................................. 362

8.3.1 Resilient state feedback...................................................... 363

8.3.2 Resilient static output feedback ....................................... 367

8.3.3 Resilient dynamic output feedback.................................... 374

8.3.4 Illustrative example 8 .1 ....................................................... 383

8.3.5 Illustrative example 8 .2 ....................................................... 384

8.3.6 Illustrative example 8 .3 ....................................................... 385

8.3.7 Illustrative example 8 .4 ....................................................... 385

8.4 Resilient Overlapping C o n tro l...................................................... 387

8.4.1 Feedback problem formulation.......................................... 390

8.4.2 LMI control design............................................................ 393

8.4.3 Application of the inclusion prin cip le ..............................394

8.4.4 Overlapping state feedback................................................ 400

8.4.5 Illustrative example 8 .5 ....................................................... 402

8.5 Resilient Stabilization for Discrete S y stem s.................................. 404

8.5.1 Problem statem ent............................................................ 405

8.5.2 Subsystem stability............................................................ 407

8.5.3 State feedback design......................................................... 409

8.5.4 Bounded state feedback design..........................................410

8.5.5 Resilient state feedback design.......................................... 411

8.5.6 Output feedback stabilization .......................................... 415

8.5.7 Resilient output feedback stabilization..............................417

8.5.8 Illustrative example 8 .6 ....................................................... 421

8.6 Problem Set V I ...............................................................................423

8.7 Notes and References..................................................................... 426

Contents XV

9 Decentralized Sliding-Mode Control 427

9.1 Reibust Sliding-Mode Control..................................... ............ 427

9.1.1 Introduction and preliminaries...................... ............ 428

9.1.2 System description........................................ ............ 429

9.1.3 Stability an aly sis........................................... ............ 431

9.1.4 Decentralized sliding-mode control ............ ............ 434

9.1.5 Illustrative example 9 .1 .................................. . . . . 437

9.2 Delay-Dependent Sliding-Mode C ontrol.................. ............ 440

9.2.1 System description........................................ ............ 441

9.2.2 Stability an aly sis........................................... ............ 444

9.2.3 Reachability analysis..................................... ............ 449

9.2.4 Illustrative example 9 .2 .................................. ............ 452

9.3 Problem Set V I I............ ... ............................... ............ 454

9.4 Notes and References.................................................. ............ 457

10 Decentralized Filtering-I 459

10.1 Introduction................................................................. ............ 459

10.2 Problem S tatem en t..................................................... ............ 460

10.3 Robust Subsystem F ilters........................................... ............ 461

10.3.1 Robust filter............................................... ............ 462

10.3.2 Illustrative example 1 0 .1 ............................... . 467

10.4 Resilient Subsystem F ilters........................................ ............ 468

10.4.1 Resilient fac filter........................................... ............ 470

10.4.2 Illustrative example 1 0 .2 ............................... ............ 473

10.5 Problem Set V I I I ........................................................ ............ 475

10.6 Notes and References.................................................. ............ 476

11 Decentralized Filtering-II 479

11.1 Decentralized Kalman Filtering.................................. ............ 479

11.1.1 Basis of filtering m eth o d ............................... ............ 480

11.1.2 The algebraic structure.................................. ............ 482

11.1.3 Application to multimachine system ............ ............ 48Í

11.2 Decentralized Fault Detection .................................. ............ 489

11.2.1 System m odel.................................................. ............ 490

11.2.2 Overlapping decomposition ......................... ............ 491

11.2.3 Fault detection technique............................... ............ 493

11.2.4 Simulation results ........................................ ............ 49Í

11.3 Problem Set I X ........................................................... ............ 507

11.4 Notes and References.................................................. ............ 509