Subspace Identification for Linear Systems

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 Theorem 9.- A.6 Proof of Theorem 11.- A.7 Proof of Theorem 12.- A.8 Proof of Lemma 2.- A.9 Proof of Theorem 13.- A.10 Proof of Corollary 2 and 3.- A.11 Proof of Theorem 14.- B Matlab Functions.- B.1 Getting started.- B.2 Matlab Reference.- B.2.1 Directory: 'subfun'.- B.2.2 Directory: 'applic'.- B.2.3 Directory: 'examples'.- B.2.4 Directory: 'figures'.- C Notation.- References.