Filtering theory : with applications to fault detection, isolation, and estimation

As soon as we (AS and PS) completed writing the book on H Optimal Con- 2 trol, another task of equal magnitude was laid to our charge. This task was to work on ?ltering and related topics. This book releases us from this charge. In this endeavor, we are fortunate to have found a capable person in our friend and colleague (AAS) who helped us release our burden. The subject of ?ltering is indeed vast and immense, much more so than the subject of H Optimal Control. In this work, we have tried to present what we 2 believe to be the fundamental issues of ?ltering. The book is not intended to give a chronological development of ?ltering from a historical point of view. A vast number of books already do so. Our intent here is to develop from our perspective the complete theory of ?ltering and various design methodologies associated with it along with their practical implementations. In this respect, we present here a state-of-the-art view of exact and almost input-decoupled ?ltering, H ,and H 2 1 ?ltering and inverse ?ltering issues, and include an application of ?ltering and inverse ?ltering to fault detection, isolation, and estimation. Most of the work reported here arose out of the research conducted by one or more of us and so- times in collaboration with our students and colleagues. Supposedly, young F.

Preface	p. xiii
1 Introduction	p. 1
1.1 Introduction	p. 1
1.2 Filtering problems	p. 3
2 Preliminaries	p. 9
2.1 A list of symbols	p. 9
2.2 A list of acronyms	p. 10
2.3 Matrices, linear spaces, and linear operators	p. 11
2.4 Norms of deterministic signals	p. 16
2.5 Norms of stochastic signals	p. 18
2.6 Norms of linear time- or shift-invariant systems	p. 19
3 A special coordinate basis (SCB) of linear multivariable systems	p. 27
3.1 Introduction	p. 27
3.2 SCB	p. 27
3.2.1 Observability (detectability) and controllability (stabilizablity)	p. 33
3.2.2 Left- and right-invertibility	p. 34
3.2.3 Finite zero structure	p. 35
3.2.4 Infinite zero structure	p. 42
3.2.5 Geometric subspaces	p. 43
3.2.6 Miscellaneous properties of the SCB	p. 48
3.2.7 Additional compact forms of the SCB	p. 50
4 Algebraic Riccati equations and matrix inequalities	p. 53
4.1 Continuous-time algebraic Riccati equations	p. 54
4.1.1 Definition of a CARE and its subclasses	p. 55
4.1.2 The Hamiltonian matrix	p. 59
4.1.3 Stabilizing and semi-stabilizing solutions of a CARE	p. 61
4.1.4 Positive semi-definite and positive definite solutions	p. 80
4.1.5 Continuity properties	p. 89
4.1.6 Algorithms for the computation of stabilizing solutions	p. 91
4.1.7 Algorithms for the computation of semi-stabilizing solutions	p. 97
4.2 Standard and generalized discrete-time algebraic Riccati equations	p. 98
4.2.1 Definitions	p. 99
4.2.2 Basic structure of a GDARE	p. 103
4.2.3 Solutions of a DARE and deflating subspaces	p. 105
4.2.4 Connections between a DARE and its associated CARE	p. 111
4.2.5 Properties, existence, and computation of various types of solutions of a DARE	p. 117
4.2.6 Continuity properties of the H 2 DARE	p. 130
4.2.7 Connections between a GDARE and its associated DARE	p. 131
4.2.8 Properties, existence, and computation of various types of solutions of a GDARE	p. 137
4.2.9 Continuity properties of the H 2 GDARE	p. 139
4.3 Continuous-time linear matrix inequalities	p. 140
4.3.1 Connections between a CLMI and its associated CARE	p. 145
4.3.2 Properties, existence, and computation of various types of solutions of a CLMI	p. 152
4.3.3 Continuity properties of CLMIs	p. 154
4.4 Discrete-time linear matrix inequalities	p. 156
4.4.1 Connections between a DLMI and its associated DARE	p. 164
4.4.2 Properties, existence, and computation of various types of solutions of a DLMI	p. 170
4.4.3 Continuity properties of the DLMI	p. 172
4.5 Continuous-time quadratic matrix inequalities	p. 173
4.5.1 Connection between a CQMI and its associated CARE	p. 176
4.A Linear matrix equations	p. 180
4.B Reduction to the case that H has full normal rank	p. 185
4.C Matrix pencils and generalized eigenvalue problems	p. 188
5 Exact disturbance decoupling via state and full information feedback	p. 191
5.1 Introduction	p. 191
5.2 Problem formulation	p. 191
5.3 Solvability conditions for EDD	p. 197
5.4 Static state feedback laws and associated fixed modes and fixed decoupling zeros	p. 198
5.4.1 EDD algorithm-left-invertible case	p. 200
5.4.2 EDD algorithm-non-left-invertible case	p. 206
5.4.3 An algorithm for EDD with pole placement	p. 213
5.5 Dynamic state feedback laws and associated fixed modes and fixed decoupling zeros	p. 215
5.5.1 ¿sub is left-invertible	p. 215
5.5.2 ¿sub is not left-invertible	p. 219
5.6 Static and dynamic full information feedback laws and associated fixed modes and fixed decoupling zeros	p. 221
5.A Proofs of Theorems 5.11 and 5.25	p. 223
5.A.1 Proof of Theorem 5.11	p. 223
5.A.2 Proof of Theorem 5.25	p. 225
6 Almost disturbance decoupling via state and full information feedback	p. 229
6.1 Introduction	p. 229
6.2 Problem formulation	p. 230
6.3 Solvability conditions for ADD	p. 234
6.3.1 Solvability conditions for ADD-continuous time	p. 234
6.3.2 Solvability conditions for ADD-discrete time	p. 236
6.4 More on ADD finite asymptotic fixed modes	p. 237
6.5 H2 ADD-design	p. 239
6.5.1 Computation of ¿ s 2+ and designing sequences of static H2 ADD controllers-continuous time	p. 239
6.5.2 Computation of ¿ s 2+ and designing sequences of static H2 ADD controllers-discrete time	p. 252
6.6 H∞ ADD-design	p. 258
6.6.1 Computation of ¿ s 2+∞ and designing sequences of static H∞ ADD controllers-continuous time	p. 258
6.6.2 Computation of ¿ s 2+∞ and designing sequences of static H∞ ADD controllers-discrete time	p. 278
7 Exact input-decoupling filters	p. 293
7.1 Introduction	p. 293
7.2 Preliminaries	p. 294
7.3 Statement of EID filtering problem and its solvability conditions	p. 295
7.4 Uniqueness of EID filters in the sense of transfer function matrix	p. 300
7.5 Design of EID filters	p. 301
7.5.1 Strictly proper EID filters of CSS architecture	p. 302
7.5.2 Proper EID filters of CSS architecture	p. 308
7.5.3 Reduced-order EID filters of CSS architecture	p. 329
7.6 Fixed modes of EID filters with arbitrary architecture	p. 339
7.A Duality between filtering and control	p. 341
8 Almost input-decoupled filtering under white noise input	p. 347
8.1 Introduction	p. 347
8.2 Preliminaries	p. 348
8.3 Statement of AID filtering problem and its solvability conditions	p. 349
8.4 Existence conditions-continuous-time case	p. 351
8.5 Existence conditions-discrete-time case	p. 354
8.6 Design of a family of H 2 AID filters of CSS architecture	p. 356
8.6.1 A family of full-order strictly proper H 2 AID filters-CSS architecture	p. 357
8.6.2 A family of full-order proper H2 AID filters-CSS architecture	p. 361
8.6.3 A family of reduced-order proper H 2 AID filters-CSS architecture	p. 374
9 Almost input-decoupled filtering without statistical assumptions on input	p. 383
9.1 Introduction	p. 383
9.2 Preliminaries	p. 383
9.3 Statement of AID filtering problem and its solvability conditions	p. 384
9.4 Existence conditions for H∞ AID filters-continuous-time case	p. 386
9.5 Existence conditions for H∞ AID filters-discrete-time case	p. 391
9.6 Design of a family of H∞ AID filters of CSS architecture	p. 394
9.6.1 A family of full-order strictly proper H∞ AID filters-CSS architecture	p. 395
9.6.2 A family of full-order proper H∞ AID filters-CSS architecture	p. 398
9.6.3 A family of reduced-order proper H∞ AID filters-CSS architecture	p. 409
10 Optimally (suboptimally) input-decoupling filtering under white noise input-H2 filtering	p. 417
10.1 Introduction	p. 417
10.2 Preliminaries	p. 418
10.3 OID and SOID filtering problems with white noise input	p. 419
10.4 Connection between H 2 OID (H 2 SOID) and EID (H 2 AID) filtering problems-continuous-time case	p. 422
10.5 Computation of ¿* sp and ¿* p - continuous-time case	p. 430
10.5.1 Relationship between ¿* sp and ¿* p and the structural properties of ¿	p. 431
10.6 Existence of H 2 OID and SOID filters-continuous-time case	p. 437
10.7 Connection between H 2 OID (H 2 SOID) and EID (H 2 AID) filtering problems-discrete-time case	p. 439
10.8 Computation of ¿* sp and ¿* p -discrete-time case	p. 447
10.8.1 Relationship between ¿* sp and ¿* p and the structural properties of ¿	p. 448
10.9 Existence of H 2 OID and SOID filters-discrete-time case	p. 452
10.10 Uniqueness of H 2 OID filters	p. 455
10.11 Uniqueness of the transfer matrix of H 2 OID error dynamics	p. 456
10.12 Design of H 2 OID filters-continuous-time case	p. 457
10.12.1 Strictly proper H 2 OID filters of CSS architecture	p. 457
10.12.2 Proper H 2 OID filters of CSS architecture	p. 467
10.12.3 Reduced-order H 2 OID filters of CSS architecture	p. 478
10.13 Design of H 2 SOID filters-continuous-time case	p. 487
10.13.1 Strictly proper H 2 SOID filters of CSS architecture	p. 488
10.13.2 Proper H 2 SOID filters of CSS architecture	p. 490
10.13.3 Reduced-order H 2 SOID filters of CSS architecture	p. 494
10.14 Design of H 2 OID filters-discrete-time case	p. 499
10.14.1 Strictly proper H 2 OID filters of CSS architecture	p. 499
10.14.2 Proper H 2 OID filters of CSS architecture	p. 509
10.14.3 Reduced-order H 2 OID filters of CSS architecture	p. 521
10.15 Design of H 2 SOID filters-discrete-time case	p. 531
10.15.1 Strictly proper H 2 SOID filters of CSS architecture	p. 531
10.15.2 Proper H 2 SOID filters of CSS architecture	p. 533
10.15.3 Reduced-order H 2 SOID filters of CSS architecture	p. 537
10.16 Fixed modes of H 2 OID filters with arbitrary architecture	p. 542
10.17 Performance measure for unbiasedness of filters with CSS architecture	p. 543
10.17.1 Strictly proper filter of CSS architecture	p. 543
10.17.2 Proper filter of CSS architecture	p. 545
10.17.3 Reduced-order filter of CSS architecture	p. 547
11 Optimally (suboptimally) input-decoupled filtering without statistical information on the input-H∞ filtering	p. 551
11.1 Introduction	p. 551
11.2 Preliminaries	p. 552
11.3 OID and SOID filtering problems without statistical information on the input	p. 553
11.4 Computation of ¿* sp and ¿* p	p. 557
11.4.1 Explicit computation of ¿* sp and ¿* p -continuous-time systems	p. 557
11.4.2 Numerical computation of ¿* sp and ¿* p -continuous-time systems	p. 562
11.4.3 Explicit computation of ¿* sp and ¿* p -discrete-time systems	p. 567
11.4.4 Numerical computation of ¿* sp and ¿* p -discrete-time systems	p. 573
11.5 Design of ¿-level H∞ SOID filters-continuous-time systems	p. 577
11.5.1 Regular ¿-level H∞ SOID filters	p. 577
11.5.2 Singular ¿-level H∞ SOID filters-the system characterized by (A, B, C, D) has no invariant zeros on the imaginary axis	p. 599
11.5.3 Singular ¿-level H∞ SOID filters-the system characterized by (A, B, C, D) has invariant zeros on the imaginary axis	p. 608
11.6 Design of ¿-level H∞ SOID filters-discrete-time systems	p. 610
11.6.1 Regular ¿-level H∞ SOID filters	p. 610
11.6.2 Singular ¿-level H∞ SOID filters-the system characterized by (A, B, C, D) has no invariant zeros on the unit circle	p. 631
11.6.3 Singular ¿-level H∞ SOID filters-the system characterized by (A, B, C, D) has invariant zeros on the unit circle	p. 639
12 Generalized H 2 suboptimally input-decoupled filtering	p. 641
12.1 Introduction	p. 641
12.2 Preliminaries	p. 642
12.3 Problem statements	p. 644
12.4 Performance, existence, and uniqueness conditions, design, and fixed modes	p. 647
12.5 Dependence of performance, existence, and uniqueness conditions and fixed modes on the input u2	p. 653
12.5.1 Dependency of performance on the input u2	p. 654
12.5.2 Dependency of the solvability conditions on the input u2	p. 655
12.5.3 Dependency of the fixed modes on the input u2	p. 657
12.6 Performance limitations due to structural properties of a system	p. 660
12.6.1 Dependence of performance on structural properties of the given system	p. 660
12.6.2 Performance issues of generalized unbiased filtering	p. 661
12.6.3 Impact of the structural properties of ¿ on \tilde J^{{\ast g}}	p. 664
12.7 Generalized EID filtering problem	p. 667
12.8 Generalized H 2 AID filtering problem	p. 668
13 Generalized H∞ suboptimally input-decoupled filtering	p. 671
13.1 Introduction	p. 671
13.2 Preliminaries	p. 671
13.3 ¿-level generalized H∞ SOID filtering problem statement	p. 673
13.4 Computation of ¿* g,sp and ¿* g,p and the design of ¿-level generalized H∞ SOID filters	p. 675
13.5 Dependence of performance on the input u2	p. 678
13.6 Performance limitations due to structural properties of a system	p. 680
13.7 Generalized H∞ AID filtering problem	p. 686
14 Fault detection, isolation, and estimation-exact or almost fault estimation	p. 689
14.1 Introduction	p. 689
14.2 Problem formulation	p. 690
14.3 Solvability conditions and design of residual generator	p. 693
14.4 Discussion	p. 694
15 Fault detection, isolation, and estimation-optimal fault estimation	p. 697
15.1 Introduction	p. 697
15.2 Problem statements	p. 700
15.3 H2 and H∞ deconvolution	p. 704
15.4 Solvability conditions and design	p. 709
Index	p. 713
References	p. 717

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