Sampled-Data Control Systems

1. Introduction.- 1.1 Sampling, Sampled-Data Controllers.- 1.2 Sampled-Data Systems.- 1.3 Design Problems for Sampled-Data Loops.- 1.4 Exercises.- 2. Continuous Systems.- 2.1 Modelling, Linearization.- 2.2 Basis of the State Space.- 2.3 System Properties.- 2.3.1 Eigenvalues and Stability.- 2.3.2 Transfer Function.- 2.3.3 Controllability.- 2.3.4 Controllable Eigenvalues.- 2.3.5 Linear Dependencies in the Controllability Matrix.- 2.3.6 Observability.- 2.3.7 Canonical Decomposition, Pole Zero Cancellations.- 2.4 Solutions of the Differential Equation.- 2.4.1 Calculation of the Transition Matrix.- 2.4.2 Impulse and Step Responses.- 2.5 Specifications.- 2.5.1 Steady-State Response and Integral Controller.- 2.5.2 Transients and Location of the Eigenvalues.- 2.6 Pole Shifting.- 2.6.1 Closed-loop Eigenvalues.- 2.6.2 Feedback Gains for Pole Placement.- 2.6.3 Fixed Eigenvalues.- 2.6.4 Fixed Gains.- 2.6.5 Pole Shifting in Multi-input Systems.- 2.7 Exercises.- 3. Modelling and Analysis of Sampled-Data Systems.- 3.1 Discretization of the Plant.- 3.2 Homogeneous Solutions: Eigenvalues, Solution Sequences.- 3.3 Inhomogeneous Solutions: z-Transfer Function, Impulse and Step Responses.- 3.4 Discrete Controller and Control Loop.- 3.4.1 Representation by the z-Transfer Function.- 3.4.2 Pole Assignment by Equating Coefficients.- 3.4.3 Integral Controller.- 3.4.4 State Representation of Controller and Control Loop.- 3.5 Root Locus Plots and Pole Specifications in the z-Plane.- 3.6. Time Domain Solutions and Specifications.- 3.6.1 Recursive Solution of the Vector Difference Equation.- 3.6.2 Numerical Inverse z-Transform.- 3.7 Behavior Between the Sampling Instants.- 3.8 Time-Delay Systems.- 3.9 Frequency Response Methods.- 3.9.1 Frequency Response Determination.- 3.9.2 Nyquist Criterion.- 3.9.3 Absolute Stability, Tsypkin and Circle Criteria.- 3.9.4 Other Graphical Frequency Response Methods.- 3.10 Special Sampling Problems.- 3.11 Exercises.- 4. Controllability, Choice of Sampling Period and Pole Assignment.- 4.1 Controllability and Reachability.- 4.2 Controllability Regions for Constrained Inputs.- 4.3 Choice of the Sampling Interval.- 4.3.1 Controllability Aspects.- 4.3.2 Bandwidth Aspects, Anti-Aliasing Filters.- 4.4 Pole Assignment.- 4.5 Exercises.- 5. Observability and Observers.- 5.1 Observability and Constructability.- 5.2 The Observer of Order n.- 5.3 The Reduced Order Observer.- 5.4 Choice of the Observer Poles.- 5.5 Disturbance Observer.- 5.6 Exercises.- 6. Control Loop Synthesis.- 6.1 Design Methodology.- 6.2 Controller Structures.- 6.2.1 Feedback, Prefilter and Disturbance Compensation.- 6.2.2 Discrete and Continuous Compensation.- 6.3 Separation.- 6.4 Construction of a Linear Function of the States.- 6.5 Synthesis by Polynomial Equations.- 6.6 Pole-Zero-Cancellations.- 6.7 Closed-loop Transfer Function and Prefilter.- 6.8 Disturbance Compensation.- 6.9 Exercises.- 7. Geometric Stability Investigation and Pole Region Assignment.- 7.1 Stability.- 7.2 Stability Region in P Space.- 7.3 Barycentric Coordinates, Bilinear Transformation.- 7.4 ?-Stability.- 7.4.1 Circular Eigenvalue Regions.- 7.4.2 Degree of Stability.- 7.4.3 Nice Stability Regions for Continuous Systems.- 7.4.4 General Eigenvalue Regions.- 7.5 Pole-Region Assignment.- 7.5.1 Mapping from P Space to K Space.- 7.5.2 Affine Mapping by Pole Assignment Matrix.- 7.5.3 K Space Boundaries for State Feedback.- 7.5.4 K Space Boundaries for Static Output Feedback.- 7.5.5 Pole-Region Assignment for Observer and Compensator Feedback.- 7.6 Graphic Representation in Two-dimensional Cross Sections.- 7.6.1 Linear and Affine Subspaces.- 7.6.2 Invariance Planes.- 7.7 Exercises.- 8. Design of Robust Control Systems.- 8.1 Robustness Problems.- 8.1.1 Sensitivity.- 8.1.2 Robustness.- 8.1.3 Multi-Model Problem Formulation.- 8.2 Structural Assumptions and Existence of Robust Controllers.- 8.2.1 Examples for Simultaneous Stabilization.- 8.2.2 Conflicting Real Root Conditions.- 8.2.3 Remarks and Recommendations.- 8.3 Simultaneous Pole Region Assignment.- 8.3.1 Graphic Solution.- 8.3.2 ?-Contraction.- 8.3.3 Summary of Graphical Approaches.- 8.3.4 Computational Solution.- 8.4 Selection of a Controller from the Admissible Solution Set.- 8.4.1 Simulation with Nonlinear Plant.- 8.4.2 Solutions with Small Loop Gains.- 8.4.3 Safety Margin from the Boundary Surfaces.- 8.4.4 Gain Reduction Margins.- 8.4.5 Robustness against Sensor Failures.- 8.5 Stabilization of the Short-period Longitudinal Mode of an F4-E with Canards.- 8.5.1 Design Specifications.- 8.5.2 Robustness with Respect to Flight Condition.- 8.5.3 Robustness against Sensor Failures.- 8.6 Design by Optimization of a Vector Performance Criterion.- 8.7 Exercises.- 9. Multivariable Systems.- 9.1 Controllability and Observability Structure.- 9.1.1 Feedforward Control.- 9.1.2 Controllability Indices.- 9.1.3 ? and ? Parameters, Input Normalization.- 9.1.4 Observability Structure.- 9.2 Finite Effect Sequences (FESs).- 9.2.1 Introduction to FESs.- 9.2.2 Use of FESs.- 9.2.3 Basic FESs.- 9.2.4 Matrix and Polynomial Notation of FESs.- 9.3 FES Assignment.- 9.3.1 Assignment Formula.- 9.3.2 Design Choices.- 9.3.3 Pole Region Assignment.- 9.4 Quadratic Optimal Control.- 9.4.1 Discrete-Time Systems.- 9.4.2 Sampled-Data Systems.- 9.5 Exercises.- Appendix A Canonical Forms and Further Results from Matrix Theory.- A.1 Linear Transformations.- A. 2 Diagonal and Jordan Forms.- A. 3 Frobenius Forms.- A.3.1 Controllability-Canonical Form.- A.3.2 Feedback-Canonical Form.- A.3.3 Observability-Canonical Form.- A.3.4 Observer-Canonical Form.- A.4 Multivariable Canonical Forms.- A. 4.1 General Remarks.- A.4.2 Luenberger Feedback-Canonical Form.- A. 4.3 Brunovsky Canonical Form.- A.5 Computational Aspects.- A.5.1 Elementary Transformations to Hessenberg Form.- A. 5. 2 HN Form.- A.6 Sensor Coordinates.- A.7 Further results from Matrix Theory.- A. 7.1 Notations.- A.7.2 Vector Operations.- A.7.3 Determinant of a Matrix.- A. 7.4 Trace of a Matrix.- A. 7.5 Rank of a Matrix.- A. 7.6 Inverse Matrix.- A. 7.7 Eigenvalues of a Matrix.- A.7.8 Resolvent of a Matrix.- A.7.9 Orbit and Controllability of (A, b).- A.7.10 Eigenvalue Assignment.- A. 7.11 Functions of a Matrix.- Appendix B The z-Transform.- B.1 Notation and Assumptions.- B.2 Linearity.- B.3 Right Shifting Theorem.- B.4 Left Shifting Theorem.- B. 5 Damping Theorem.- B.6 Differentation Theorem.- B.7 Initial Value Theorem.- B.8 Final Value Theorem.- B.9 The Inverse z-Transform.- B.10 Real Convolution Theorem.- B.11 Complex Convolution Theorem, Parseval Equation.- B.12 Other Representations of Sampled Signals in Time and Frequency Domain.- B.13 Table of Laplace and z-Transforms.- Appendix C Stability Criteria.- C.1 Bilinear Transformation to a Hurwitz Problem.- C.2 Schur-Cohn Criterium and its Reduced Forms.- C.3 Necessary Stability Conditions.- C.4 Sufficient Stability Conditions.- Appendix D Application Examples.- D.1 Aircraft Stabilization.- D.2 Track-Guided Bus.- Literature.