Sampled-Data Control for Periodic Objects

This book is devoted to the problem of sampled-data control of finite-dimensional linear continuous periodic (FDLCP) objects. It fills a deficit in coverage of this important subject. The methods presented here are based on the parametric transfer matrix, which has proven successful in the study of sampled-data systems with linear time-invariant objects. The book shows that this concept can be successfully transferred to sampled-data systems with FDLCP objects. It is set out in five parts: · · an introduction to the frequency approach for the mathematical description of FDLCP objects including the determination of their structure and their representation as a serial connection of periodic modulators and a linear time-invariant object; · construction of parametric transfer matrix for different types of open and closed sampled-data systems with FDLCP objects; · the solution of problems of causal modal control of FDLCP objects based on the mathematical apparatus of determinant polynomial equations; · consideration of the problem of constructing a quadratic quality functional for the H 2 -optimization problem of a single-loop synchronous sampled-data system with control delay; · description of the general H 2 -optimization procedure. Necessary mathematical reference material is included at relevant points in the book. Sampled-Data Control for Periodic Objects is of use to: scientists and engineers involved in research and design of systems of systems with FDLCP objects; graduate students wishing to broaden their scope of competence; their instructors; and mathematicians working in the field of control theory. Preface Contents Part I The Frequency Approach to the Mathematical Description of Linear Periodic Objects 1 Discrete Operational Transformations of Continuous Argument Functions and Operator Description of Linear Time-Invariant Systems 1.1 Discrete Laplace Transformation of Continuous Argument Functions 1.2 Discrete Laplace Transform of the Image 1.3 Operator Description of a Finite-Dimensional Linear Time-Invariant System 1.4 Transfer of an Exponentially Periodic Signal Through a Linear Time-Invariant System 2 State-Space Analysis of Finite-Dimensional Linear Continuous Periodic Objects 2.1 State-Space Description of Periodic Objects 2.2 Transfer of Periodic Signals Through Periodic Objects 2.3 Transfer of Exponentially Periodic Signals Through Periodic Objects 2.4 Higher-Order Periodic Objects 3 The Frequency Method in the Theory of Periodic Objects 3.1 Frequency Description of Linear Periodic Operators 3.2 Linear Periodic Integral Operators 3.3 Operator Description of a Basic Periodic Object 3.4 Parametric Transfer Matrix of the Basic Periodic Object 3.5 Parametric Transfer Matrix of a Complemented Periodic Object 4 The Floquet–Lyapunov Decomposition and Its Application 4.1 Floquet–Lyapunov Transformation 4.2 Floquet–Lyapunov Decomposition and Its Parametric Transfer Matrix 4.3 Periodic Object with Delay 4.4 Low-Frequency Exponentially Periodic Excitation of the Floquet–Lyapunov Decomposition Part II The Parametric Transfer Matrix Approach to Sampled-Data Systems with Periodic Objects 5 Open-Loop Sampled-Data System with Periodic Object 5.1 Multivariable Zero-Order Hold 5.2 Linearized Model of the Digital Controller 5.3 Open-Loop System with Time-Invariant Object 5.4 Synchronous Open-Loop System with Periodic Object 5.5 Asynchronous Rising Open-Loop System with Periodic Object 5.6 Open-Loop System with Periodic Object and High-Frequency Hold 6 Open-Loop Sampled-Data System with Periodic Object and Delay 6.1 Open-Loop System with Linear Time-Invariant Object and Delay 6.2 Synchronous Open-Loop System with Periodic Object and Delay 6.3 Asynchronous Rising Open-Loop SD System with Periodic Object and Delay 6.4 Open-Loop System with Periodic Object, High-Frequency Hold and Delay 7 Closed-Loop Sampled-Data System with Periodic Object and Delay 7.1 Synchronous Closed-Loop System 7.2 Asynchronous Rising Closed-Loop System 7.3 Closed-Loop System with Periodic Object, High-Frequency Hold and Delay Part III Determinant Polynomial Equations, Sampled-Data Modal Control and Stabilization of Periodic Objects 8 Polynomial Matrices 8.1 General Definitions and Properties 8.2 Polynomial Matrices Equivalence 8.3 Latent Equation and Latent Numbers 8.4 Pairs of Polynomial Matrices 9 Rational Matrices 9.1 General Definitions and Properties 9.2 McMillan Canonical Form 9.3 Matrix Fraction Description 9.4 Strictly Proper Rational Matrices 9.5 Polynomial Pairs and Transfer Matrices 9.6 Polynomial Matrix Division and Reduction of the Degree of Polynomial Pairs 10 Determinant Polynomial Equations, Causal Modal Control and Stabilization of Discrete Systems 10.1 General Definitions and Problem Setting for Causal Modal Control 10.2 Solving Backward Modal Control and Stabilization Problems for Polynomial Pairs 10.3 Solving Modal Control and Stabilization Problems for Backward Processes in Polynomial Matrix Description 11 Synchronous Sampled-Data Stabilization of Periodic Objects 11.1 Synchronous Stabilization of a Single Periodic Object 11.2 Synchronous Stabilization of a Periodic Object with Prefilter 11.3 Synchronous SD Stabilization of an FDLCP Object with Control Delay 12 Asynchronous Sampled-Data Stabilization of Periodic Objects 12.1 Low-Frequency Stabilization of a Periodic Object 12.2 Low-Frequency Stabilization of a Periodic Object with Prefilter 12.3 Low-Frequency Stabilization of a Periodic Object with Control Delay 12.4 Stabilization of a Periodic Object Using a High-Frequency Hold 12.5 Stabilization of a Periodic Object with Control Delay Using a High-Frequency Hold Part IV Building the Quality Functional for the H2-Optimization Task of the Sampled-Data System mathscrSτ 13 General Parametric Transfer Matrix Properties of the Synchronous Open-Loop Sampled-Data System with Delay 14 Parametric Transfer Matrix of the Closed-Loop Sampled-Data System with Delay as Function of Argument s 15 Calculation of Matrices v0(s), ξ0(s), ψ0(s) 16 System Function 17 Representing the Parametric Transfer Matrix of System mathcalSτ by the System Function 18 H2-Norm of the Closed-Loop Sampled-Data System 19 Construction of the Quality Functional Part V H2-Optimization of the Closed-Loop Sampled-Data System 20 Scalar and Matrix Quasi-Polynomials 21 Minimization of a Quadratic Functional on the Unit Circle 22 Construction of Matrix η(s,t) 23 Construction of Matrix tildeCT (s, t) 24 Transformation of the Quality Functional 25 H2-Optimization of the System mathcalSτ 25.1 General Solution of the H2-Optimization Task 25.2 General Solution of the H2-Optimization Task for a First-Order Periodic Object Appendix References Index