During the first quarter of 1983, following numerous comments from industry and universities concerning the limited distribution of the results of research contracts financed by the DRET, the idea of publishing these results first took root. Using the fourteen chosen authors mentioned on the cover page, all of whom work under contract for the DRET, the research results have been brought together and elaborated on in this book. A second idea, reflected in the structure of this work, was the desire to make certain control techniques, which had hitherto been restricted to specialist researchers, available to a larger number of engineers and students. Thus the second part of this book, concerned with practical applications, illustrates the use of algorithms whose theory is described in the first part. The first part is the result of a large amount of work which brings together the many different strands of current work. The applications to machines and robots enhance the book by providing non-academic illustrations. Each application may form the subject of a case study in an engineering course. Readers engaged in teaching may contact the authors if they require more detailed information.

Table of Contents:
One.- to Automatic Control.- 1. The Representation of Signals and Systems.- 1.1 Introduction.- 1.2 The Representation of Deterministic Signals.- 1.2.1 Notes on Continuous Signals.- 1.2.2 Notes on Discrete Signals.- 1.3 Representation of Deterministic Linear Systems.- 1.3.1 Representation of Systems in Continuous Form.- 1.3.2 Representation of Discrete Systems.- 1.4 Introduction of Aleatory Signals into Linear Systems.- 1.4.1 Continuous Random Signals.- 1.4.2 Discrete Random Signals.- 1.4.3 Approximation of a Continuous Random Signal by a Discrete Signal.- 1.5 Conclusion.- References.- 2. From Continuous to Discrete Control.- 2.1 Synthesis of Continuous Laws.- 2.1.1 Modal Methods.- 2.1.2 Linear Quadratic Optimal Methods.- 2.1.3 Control of Systems with Time Delays.- 2.2 Digitisation of Continuous Laws.- 2.2.1 Discrete Equivalent of a Continuous Controller.- 2.2.2 The Techniques of Digitisation.- 2.2.3 Analysis of the Closed Loop System.- 2.3 Conclusion.- References.- 3. Digital Controllers of Deterministic Systems The Polynomial Approach.- 3.1 Introduction.- 3.2 Controllers for a Minimum Time Criterion.- 3.2.1 Stabilising Controllers.- 3.2.2 Synthesis of Discrete Controllers Minimising the Response Time.- 3.2.3 Synthesis of Discrete Controllers Minimising a Quadratic Criterion.- 3.2.4 Synthesis of Controllers for Multivariable Systems.- 3.3 Pole Placement Controllers.- 3.3.1 Disturbance-free Servo Problem.- 3.3.2 Special Case in which the Controller Cancels all the Process Zeros (Minimum Phase System).- 3.4 Control Laws Based on the Minimisation of a One Step Quadratic Criterion.- 3.4.1 Optimal k-step-ahead Predictor.- 3.4.2 Minimum Variance Regulator.- 3.4.3 Generalised Minimum Variance Controller.- 3.4.4 Numerical Example.- 3.5 Conclusion.- References.- 4. Principles of Internal Model Control.- 4.1 Introduction and Definitions.- 4.2 Transferring from Conventional Regulation to Internal Model Control (C.M.I.).- 4.2.1 Principle.- 4.2.2 Properties.- 4.2.3 Choice of the Control Law: Calculation of D(z).- 4.2.4 Consequences.- 4.3 Principles of Internal Model Control.- 4.3.1 Internal Model.- 4.3.2 Reference Trajectory.- 4.3.3 Control Algorithm.- 4.3.4 Stability and Robustness.- 4.4 Example of Application in the Case of a System Described by its State Equations.- 4.4.1 Definition of the Control Problem.- 4.4.2 Control Strategy.- 4.4.3 Transfer Matrix of the Control System.- 4.4.4 Study of Convergence.- 4.4.5 Study of the Dynamic Performances.- 4.5 Conclusion.- References.- 5. Discrete Optimal Control of Linear Stochastic Systems.- 5.1 Introduction.- 5.2 Optimal Control by the Techniques of Polynomial Algebra.- 5.2.1 Single-input, Single-output Case.- 5.2.2 The Multivariable Case.- 5.2.3 Conclusion.- 5.3 The State Vector Modelling Approach.- 5.3.1 Notes on the Principal Results Obtained in Discrete Optimal Stochastic Control.- 5.3.2 Special Formulations.- 5.3.3 Example.- 5.4 Comparison of the Two Approaches.- 5.4.1 Rational Matrices.- 5.4.2 Polynomial Matrices.- 5.5 Conclusion.- References.- 6. Adaptive Control of Stochastic Systems.- 6.1 Introduction.- 6.2 Classification of Self-adaptive Control Systems.- 6.2.1 Self-adaptive Stochastic Optimal Control.- 6.2.2 Definition of the Concepts.- 6.2.3 Example.- 6.2.4 Classification of Controllers.- 6.3 Introduction to Self-tuning Controllers.- 6.4 Weighted Least Squares Estimation Algorithm.- 6.4.1 Ordinary Least Squares Estimation Algorithm.- 6.4.2 Recursive Weighted Least Squares Estimation Algorithm.- 6.5 Description of Self-tuning Controllers.- 6.5.1 Minimum Output Variance Control Strategy.- 6.5.2 Minimum Generalised Output Variance Control Strategy.- 6.5.3 Numerical Example.- 6.6 Conclusion.- References.- 7. Brief Description of Algebraic and Geometrical Methods for Non-linear Control.- 7.1 Introduction.- 7.2 State Space in Continuous Time.- 7.2.1 On the Concept of Vector Fields.- 7.2.2 Lie Brackets.- 7.2.3 Controllability.- 7.3 Generating Series.- 7.3.1 Bilinear Systems in Discrete Time.- 7.3.2 Bilinear Systems in Continuous Time.- 7.3.3 Implementation.- 7.4 Further Information.- 7.5 Brief References.- Two: Applications of Digital Control to Vehicles and Robots.- 8. Digital Control of Systems at the Limit of Stability: Application to the Stabilisation of Satellite.- 8.1 Introduction.- 8.2 Stabilisation of Satellites in Rotation.- 8.2.1 Outline of the Problem.- 8.2.2 Short-term Control.- 8.2.3 Long-term Control.- 8.3 Three-Axis Stabilisation of a Satellite with Flexible Appendages.- 8.3.1 Outline of the Problem.- 8.3.2 Dynamic Model of the Satellite.- 8.3.3 Roll and Yaw Axes.- 8.3.4 Pitch Axis.- References.- 9. Digital Control System of a Launch Vehicle.- 9.1 Introduction.- 9.2 Modelling the System.- 9.2.1 Notation.- 9.2.2 Hypotheses.- 9.2.3 Equations.- 9.2.4 State Representation.- 9.3 Choice of the Criterion and of the Covariance Matrices.- 9.3.1 Optimisation Criterion.- 9.3.2 Covariance Matrix.- 9.3.3 Choice of Parameters.- 9.4 Determination of the Controller — Application.- 9.4.1 Control Gains.- 9.4.2 Kaiman Gains.- 9.4.3 Application.- 9.5 Comments.- 9.6 Conclusion.- References.- 10. Reduction of Disturbances: Application to Passenger Comfort on Aircraft.- 10.1 Introduction.- 10.2 Principle of Turbulence Absorption for the Improvement of Passenger Comfort.- 10.3 Model of a Rigid Aircraft in a Turbulent Atmosphere.- 10.4. Determination of the Control Laws.- 10.5 Control Structures.- 10.6 Digital Implementation of the Controller.- 10.7 Example of Application — Conclusion.- References.- 11. Application of Internal Model Control to the Automatic Steering of a Ship.- 11.1 Introduction.- 11.2 General Description.- 11.3 Description of the Pact System.- 11.3.1 Functional Characteristics of Pact.- 11.3.2 Performance.- 11.3.3 Developments and Prospects.- References.- 12. Control by Reference Model: Application to Decoupling and Manoeuvrability in a Helicopter.- 12.1 Introduction.- 12.2 General Description of the Helicopter.- 12.3 Principle of Determination of the Control Law.- 12.3.1 General Theory.- 12.3.2 Parameter Optimisation.- 12.4 Choice of the Reference Model.- 12.4.1 Manoeuvrability Standard.- 12.4.2 Design of the Reference Model.- 12.5 Adjustment of the Control Law.- 12.5.1 Blocks Ku, T, F, S.- 12.5.2 Block Km.- 12.5.3 Block K.- 12.5.4 Block C.- 12.5.5 Block Ku.- 12.6 Results.- 12.6.1 Frequency Results.- 12.6.2 Time Results.- 12.7 Conclusion.- References.- 13. Adaptive Control Applied to the Reduction of Vibrations in Helicopters.- 13.1 Introduction.- 13.2 Helicopter Model.- 13.3 Optimal Controller Synthesis.- 13.4 Estimation of the Parameters of the Matrix S.- 13.4.1 First Case: With Constant Parameters.- 13.4.2 Second Case: With Variable Parameters.- 13.5 Experimental Results.- 13.5.1 Constant Speed Configuration.- 13.5.2 Variable Speed Configuration.- 13.6 Conclusion.- References.- 14. Minimisation of the Operating Cost of an Aircraft Flight by Optimisation of the Trajectory.- 14.1 Introduction.- 14.2 Criterion and Mathematical Model.- 14.2.1 Criterion.- 14.2.2 Dynamic Model.- 14.2.3 Constraints.- 14.3 Choice of the Optimisation Method.- 14.3.1 Sequential Procedure.- 14.3.2 Parallel Procedure.- 14.4 Method of Forced Singular Perturbations.- 14.5 Application to the Present Problem.- 14.5.1 Reduced Solution (Cruising Phase).- 14.5.2 Solution for the First Initial Layer.- 14.5.3 Solution in the Second Layer.- 14.5.4 Synthesis of the Overall Trajectory.- 14.6 Conclusion.- References.- 15. Interception in Minimum Time with Specified Final Conditions.- 15.1 Description of the Problem.- 15.2 Optimality Equations.- 15.3 Determination of the Optimal Control.- 15.4 Determination of the End Point.- 15.5 Conclusion.- 16. Robot Control Techniques.- 16.1 General.- 16.1.1 General Notes.- 16.1.2 Principal Functions to be Performed.- 16.2 Use of Kinematic and Geometrical Models.- 16.2.1 Generation of Movements in the Configuration Space.- 16.2.2 Generation of Movements in the Operation Space.- 16.3 Use of Models of the Dynamics.- 16.3.1 Theoretical Controllers.- 16.3.2 Dynamic Control Approaches in the Operation Space.- 16.3.3 Conclusion on Dynamic Control.- 16.4 Controllers Driven by Sensors.- 16.4.1 Proximity Measurement Feedback.- 16.4.2 Force Feedback.- 16.4.3 Hybrid Control.- 16.5 Conclusion.- References.- A.1 Polynomial Matrices.- A.1.1 Basic Definitions Concerning the Matrices.- A.1.2 Principal Transformations.- A.1.3 Divisibility.- A.1.4 Applications.- A.2 Problems of Inverses.- A.2.1 Generalised Inverse.- A.2.2 Inverses of Polynomial Matrices.- A.2.3 The Equation M.D = Y.- A.3 Spectral Factorisation.