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PVDnotes - Numerical Linear Algebra for Signals Systems and...

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Numerical Linear Algebra for Signals Systems and Control Paul M. Van Dooren University of Louvain, B-1348 Louvain-la-Neuve, Belgium Draft notes prepared for the Graduate School in Systems and Control Spring 2003
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Contents 1 SCOPE AND INTRODUCTION 1 1.1 Scope of the Course . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 About Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Numerical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Basic Problems in Numerical Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2 IDENTIFICATION 23 2.1 SISO identification from the impulse response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2 State-space realizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.3 Balanced realizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.4 Pad´ e algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.5 Multi-input multi-output impulse response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.6 Input-Output Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.7 Recursive least squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.8 MIMO identification via I/O pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.9 Linear prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3 STATE SPACE ANALYSIS 67 3.1 Orthogonal State-Space Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.2 Condensed Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.3 Controllability, Observability and Minimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.4 Staircase form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.5 Subspaces and Minimal Realizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.6 Poles and Zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4 STATE SPACE DESIGN 91 4.1 State feedback and pole placement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.2 Multi-input feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.3 Observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.4 Lyapunov and Sylvester Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.5 Algebraic Riccati Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5 KALMAN FILTERING 123 5.1 Kalman filter implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.2 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 5.3 Experimental evaluation of the different KF’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 5.4 Comparison of the different filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 6 POLYNOMIAL VERSUS STATE-SPACE MODELS 139 6.1 System Models And Transformation Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 6.2 Stabilized Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 6.3 State-Space Realizations Of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6.4 Fast Versus Slow Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 iii
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Chapter 1 SCOPE AND INTRODUCTION 1.1 Scope of the Course This course looks at numerical issues of algorithms for signals, systems and control. In doing that a clear choice is made to focus on numerical linear algebra techniques for linear time-invariant, finite dimensional systems . At first hand, this may look as narrowing down the subject quite a bit, but there are simple reasons for this. The fact that we deal only with numerical methods for linear time-invariant, finite dimen- sional systems is not really as restrictive as it seems. One encounters in fact very much the same numerical problems when relaxing these constraints: problems in time varying systems are often based on the time-invariant counterpart because of the recursive use of time-invariant techniques or because one uses a reformulation into a time-invariant problem. There is e.g., hardly any difference between Kalman filtering for time- varying and time-invariant systems since they both lead to recursive least squares problems. Another typical example is that of periodic systems which can be rewritten as a time-invariant problem by considering the “lifted” system nonlinear systems are typically approximated by a sequence of linearized models for which standard techniques are then applied. Also the approximation is sometimes using techniques borrowed from linear systems theory infinite dimensional systems typically arise from partial differential equations. One then uses discretizations such as finite elements methods to represent the underlying operator by a (typically large and sparse) matrix.
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