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Polynomial Methods for Control Systems Design

The subject of this volume is the design of optimal and robust control systems using system models which are represented by polynomial matrices, the polynomial approach being close to the frequency domain design method that engineers use in practice. The subject matter ranges from fundamental systems theory to application problems. The text introduces the polynomial systems approach to solving both control and filtering problems. The benefits of frequency domain analysis and design procedures are demonstrated and the power of polynomial modelling procedures in applications is revealed. Both H2 and H (infinity) design methods are considered, involving both scalar and multivariable problem solutions.
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Table of Contents

Preface ix.- 1 A Tutorial on H2 Control Theory: The Continuous Time Case.- 1.1 Introduction.- 1.2 LQG control theory.- 1.2.1 Problem formulation.- 1.2.2 Finite horizon solution.- 1.2.3 Infinite horizon solution.- 1.3 H2 control theory.- 1.3.1 Preliminaries.- 1.3.2 State space solution.- 1.3.3 Wiener-Hopf solution.- 1.3.4 Diophantine equations solution.- 1.4 Comparison and examples.- 1.4.1 The LQG as an H2 problem.- 1.4.2 Internal stability.- 1.4.3 Solvability assumptions.- 1.4.4 Non-proper plants.- 1.4.5 Design examples.- 1.5 References.- 2 Frequency Domain Solution of the Standard H? Problem.- 2.1 Introduction.- 2.1.1 Introduction.- 2.1.2 Problem formulation.- 2.1.3 Polynomial matrix fraction representations.- 2.1.4 Outline.- 2.2 Well-posedness and closed-loop stability.- 2.2.1 Introduction.- 2.2.2 Well-posedness.- 2.2.3 Closed-loop stability.- 2.2.4 Redefinition of the standard problem.- 2.3 Lower bound.- 2.3.1 Introduction.- 2.3.2 Lower bound.- 2.3.3 Examples.- 2.3.4 Polynomial formulas.- 2.4 Sublevel solutions.- 2.4.1 Introduction.- 2.4.2 The basic inequality.- 2.4.3 Spectral factorization.- 2.4.4 All sublevel solutions.- 2.4.5 Polynomial formulas.- 2.5 Canonical spectral factorizations.- 2.5.1 Definition.- 2.5.2 Polynomial formulation of the rational factorization.- 2.5.3 Zeros on the imaginary axis.- 2.6 Stability.- 2.6.1 Introduction.- 2.6.2 All stabilizing sublevel compensators.- 2.6.3 Search procedure - Type A and Type B optimal solutions.- 2.7 Factorization algorithm.- 2.7.1 Introduction.- 2.7.2 State space algorithm.- 2.7.3 Noncanonical factorizations.- 2.8 Optimal solutions.- 2.8.1 Introduction.- 2.8.2 All optimal compensators.- 2.8.3 Examples.- 2.9 Conclusions.- 2.10 Appendix: Proofs for section 2.3.- 2.11 Appendix: Proofs for section 2.4.- 2.12 Appendix: Proof of theorem 2.7.- 2.13 Appendix: Proof of the equalizing property.- 2.14 References.- 3 LQG Multivariable Regulation and Tracking Problems for General System Configurations.- 3.1 Introduction.- 3.2 Regulation problem.- 3.2.1 Problem solution.- 3.2.2 Connection with the Wiener-Hopf solution.- 3.2.3 Innovations representations.- 3.2.4 Relationships with other polynomial solutions.- 3.3 Tracking, servo and accessible disturbance problems.- 3.3.1 Problem formulation.- 3.4 Conclusions.- 3.5 Appendix.- 3.6 References.- 4 A Game Theory Polynomial Solution to the H? Control Problem.- 4.1 Abstract.- 4.2 Introduction.- 4.3 Problem definition.- 4.4 The game problem.- 4.4.1 Main result.- 4.4.2 Summary of the simplified solution procedure.- 4.4.3 Comments.- 4.5 Relations to the J-factorization H? problem.- 4.5.1 Introduction.- 4.5.2 The J-factorization solution.- 4.5.3 Connection with the game solution.- 4.6 Relations to the minimum entropy control problem.- 4.7 A design example: mixed sensitivity.- 4.7.1 Mixed sensitivity problem formulation.- 4.7.2 Numerical example.- 4.8 Conclusions.- 4.9 Appendix.- 4.10 References.- 4.11 Acknowledgements.- 5 H2 Design of Nominal and Robust Discrete Time Filters.- 5.1 Abstract.- 5.2 Introduction.- 5.2.1 Digital communications: a challenging application area...- 5.2.2 Remarks on the notation.- 5.3 Wiener filter design based on polynomial equations.- 5.3.1 A general H2 filtering problem.- 5.3.2 A structured problem formulation.- 5.3.3 Multisignal deconvolution.- 5.3.4 Decision feedback equalizers.- 5.4 Design of robust filters in input-output form.- 5.4.1 Approaches to robust H2 estimation.- 5.4.2 The averaged H2 estimation problem.- 5.4.3 Parameterization of the extended design model.- 5.4.4 Obtaining error models.- 5.4.5 Covariance matrices for the stochastic coefficients.- 5.4.6 Design of the cautious Wiener filter.- 5.5 Robust H2 filter design.- 5.5.1 Series expansion.- 5.5.2 The robust linear state estimator.- 5.6 Parameter tracking.- 5.7 Acknowledgement.- 5.8 References.- 6 Polynomial Solution of H2 and H? Optimal Control Problems with Application to Coordinate Measuring Machines.- 6.1 Abstract.- 6.2 Introduction.- 6.3 H.2 control design.- 6.3.1 System model.- 6.3.2 Assumptions.- 6.3.3 The H2 cost function.- 6.3.4 Dynamic weightings.- 6.3.5 The H2 controller.- 6.3.6 Properties of the controller.- 6.3.7 Design procedure.- 6.4 H? Robust control problem.- 6.4.1 Generalised H2 and H? controllers.- 6.5 System and disturbance modelling.- 6.5.1 System modelling.- 6.5.2 Disturbance modelling.- 6.5.3 Overall system model.- 6.6 Simulation and experimental studies.- 6.6.1 System definition.- 6.6.2 Simulation studies.- 6.6.3 Experimental studies.- 6.6.4 H? control.- 6.7 Conclusions.- 6.8 Acknowledgements.- 6.9 References.- 6.10 Appendix: two-DOF H2 optimal control problem.

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