Polynomial Methods for Control Systems Design

This monograph was motivated by a very successful workshop held before the 3rd IEEE Conference on Decision and Control held at the Buena Vista Hotel, lake Buena Vista, Florida, USA. The workshop was held to provide an overview of polynomial system methods in LQG (or H ) and Hoo optimal control and 2 estimation. The speakers at the workshop were chosen to reflect the important contributions polynomial techniques have made to systems theory and also to show the potential benefits which should arise in real applications. An introduction to H2 control theory for continuous-time systems is included in chapter 1. Three different approaches are considered covering state-space model descriptions, Wiener-Hopf transfer function methods and finally polyno­ mial equation based transfer function solutions. The differences and similarities between the techniques are explored and the different assumptions employed in the solutions are discussed. The standard control system description is intro­ duced in this chapter and the use of Hardy spaces for optimization. Both control and estimation problems are considered in the context of the standard system description. The tutorial chapter concludes with a number of fully worked ex­ amples.

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.