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Summary
Summary
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.
Table of Contents
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 |