1 Introduction, Motivation and Geometric Tools.- 1.1 Models of systems and system identification.- 1.2 A new generation of system identification algorithms.- 1.2.1 State space models are good engineering models.- 1.2.2 How do subspace identification algorithms work ?.- 1.2.3 What's new in subspace identification ?.- 1.2.4 Some historical elements.- 1.3 Overview.- 1.4 Geometric tools.- 1.4.1 Orthogonal projections.- 1.4.2 Oblique projections.- 1.4.3 Principal angles and directions.- 1.4.4 Statistical tools.- 1.4.5 Geometric tools in a statistical framework.- 1.5 Conclusions.- 2 Deterministic Identification.- 2.1 Deterministic systems.- 2.1.1 Problem description.- 2.1.2 Notation.- 2.2 Geometric properties of deterministic systems.- 2.2.1 Matrix input-output equations.- 2.2.2 Main Theorem.- 2.2.3 Geometric interpretation.- 2.3 Relation to other algorithms.- 2.3.1 Intersection algorithms.- 2.3.2 Projection algorithms.- 2.3.3 Notes on noisy measurements.- 2.4 Computing the system matrices.- 2.4.1 Algorithm 1 using the states.- 2.4.2 Algorithm 2 using the extended observability matrix.- 2.5 Conclusions.- 3 Stochastic Identification.- 3.1 Stochastic systems.- 3.1.1 Problem description.- 3.1.2 Properties of stochastic systems.- 3.1.3 Notation.- 3.1.4 Kalman filter states.- 3.1.5 About positive real sequences.- 3.2 Geometric properties of stochastic systems.- 3.2.1 Main Theorem.- 3.2.2 Geometrical interpretation.- 3.3 Relation to other algorithms.- 3.3.1 The principal component algorithm (PC).- 3.3.2 The unweighted principal component algorithm (UPC).- 3.3.3 The canonical variate algorithm (CVA).- 3.3.4 A simulation example.- 3.4 Computing the system matrices.- 3.4.1 Algorithm 1 using the states.- 3.4.2 Algorithm 2 using the extended matrices.- 3.4.3 Algorithm 3 leading to a positive real sequence.- 3.4.4 A simulation example.- 3.5 Conclusions.- 4 Combined Deterministic-Stochastic Identification.- 4.1 Combined systems.- 4.1.1 Problem description.- 4.1.2 Notation.- 4.1.3 Kalman filter states.- 4.2 Geometric properties of combined systems.- 4.2.1 Matrix input-output equations.- 4.2.2 A Projection Theorem.- 4.2.3 Main Theorem.- 4.2.4 Intuition behind the Theorems.- 4.3 Relation to other algorithms.- 4.3.1 N4SID.- 4.3.2 MOESP.- 4.3.3 CVA.- 4.3.4 A simulation example.- 4.4 Computing the system matrices.- 4.4.1 Algorithm 1: unbiased, using the states.- 4.4.2 Algorithm 2: biased, using the states.- 4.4.3 Variations and optimizations of Algorithm 1.- 4.4.4 Algorithm 3: a robust identification algorithm.- 4.4.5 A simulation example.- 4.5 Connections to the previous Chapters.- 4.6 Conclusions.- 5 State Space Bases and Model Reduction.- 5.1 Introduction.- 5.2 Notation.- 5.3 Frequency weighted balancing.- 5.4 Subspace identification and frequency weighted balancing.- 5.4.1 Main Theorem 1.- 5.4.2 Special cases of the first main Theorem.- 5.4.3 Main Theorem 2.- 5.4.4 Special cases of the second main Theorem.- 5.4.5 Connections between the main Theorems.- 5.5 Consequences for reduced order identification.- 5.5.1 Error bounds for truncated models.- 5.5.2 Reduced order identification.- 5.6 Example.- 5.7 Conclusions.- 6 Implementation and Applications.- 6.1 Numerical Implementation.- 6.1.1 An RQ decomposition.- 6.1.2 Expressions for the geometric operations.- 6.1.3 An implementation of the robust identification algorithm.- 6.2 Interactive System Identification.- 6.2.1 Why a graphical user interface ?.- 6.2.2 ISID: Where system identification and GUI meet.- 6.2.3 Using ISID.- 6.2.4 An overview of ISID algorithms.- 6.2.5 Concluding remarks.- 6.3 An Application of ISID.- 6.3.1 Problem description.- 6.3.2 Chain description and results.- 6.3.3 PIID control of the process.- 6.4 Practical examples in Matlab.- 6.5 Conclusions.- 7 Conclusions and Open Problems.- 7.1 Conclusions.- 7.2 Open problems.- A Proofs.- A.1 Proof of formula (2.16).- A.2 Proof of Theorem 6.- A.3 Note on the special form of the Kalman filter.- A.4 Proof of Theorem 8.- A.5 Proof of Theor
Subspace Identification for Linear Systems focuses on the theory, implementation and applications of subspace identification algorithms for linear time-invariant finite- dimensional dynamical systems. These algorithms allow for a fast, straightforward and accurate determination of linear multivariable models from measured input-output data.
The theory of subspace identification algorithms is presented in detail. Several chapters are devoted to deterministic, stochastic and combined deterministic-stochastic subspace identification algorithms. For each case, the geometric properties are stated in a main 'subspace' Theorem. Relations to existing algorithms and literature are explored, as are the interconnections between different subspace algorithms. The subspace identification theory is linked to the theory of frequency weighted model reduction, which leads to new interpretations and insights.
The implementation of subspace identification algorithms is discussed in terms of the robust and computationally efficient RQ and singular value decompositions, which are well-established algorithms from numerical linear algebra. The algorithms are implemented in combination with a whole set of classical identification algorithms, processing and validation tools in Xmath's ISID, a commercially available graphical user interface toolbox. The basic subspace algorithms in the book are also implemented in a set of Matlab files accompanying the book.
An application of ISID to an industrial glass tube manufacturing process is presented in detail, illustrating the power and user-friendliness of the subspace identification algorithms and of their implementation in ISID. The identified model allows for an optimal control of the process, leading to a significant enhancement of the production quality. The applicability of subspace identification algorithms in industry is further illustrated with the application of the Matlab files to ten practical problems. Since all necessary data and Matlab files are included, the reader can easily step through these applications, and thus get more insight in the algorithms.
Subspace Identification for Linear Systems is an important reference for all researchers in system theory, control theory, signal processing, automization, mechatronics, chemical, electrical, mechanical and aeronautical engineering.
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