Multivariable Control

The foundation of linear systems theory goes back to Newton and has been followed over the years by many improvements such as linear operator theory, Laplace Transformation etc. After the World War II, feedback control theory has shown a rapid development, and standard elegant analysis and synthesis techniques have been discovered by control system workers, such as root-locus (Evans) and frequency response methods (Nyquist, Bode). These permitted a fast and efficient analysis of simple-loop control systems, but in their original "paper-and-pencil" form were not appropriate for multiple­ loop high-order systems. The advent of fast digital computers, together with the development of multivariable multi-loop system techniques, have eliminated these difficulties. Multivariable control theory has followed two main avenues; the optimal control approach, and the algebraic and frequency-domain control approach. An important key concept in the whole multivariable system theory is "ob­ servability and controllability" which revealed the exact relationships between transfer functions and the state variable representations. This has given new insight into the phenomenon of "hidden oscillations" and to the transfer function modelling of dynamic systems. The basic tool in optimal control theory is the celebrated matrix Riccati differential equation which provides the time-varying feedback gains in a linear-quadratic control system cell. Much theory presently exists for the characteristic properties and solution of this Riccati equation.

I General Topics.- 1 Applications of Algebraic Function Theory in Multivariable Control.- Preliminaries.- Classical Single-Loop Theory.- The Generalized Nyquist Stability Criterion.- Characteristic Gains and Fixed Modes.- Algebraic Functions and Riemann Surfaces.- The Riemann Surface Proof of the Generalised Nyquist Stability Criterion.- The Inverse Nyquist Stability Criterion and One-Parameter Families of Feedback Gains.- The Multivariab1e Root-Locus.- Poles and Zeros.- Approach and Departure Angles and Multivariab1e Pivots.- References.- 2 The Theory of Polynomial Combinants in the Context of Linear Systems.- Sets of Polynomials: Definitions and Normal Equivalence.- Almost Zeros: Definition Location and Computation.- Polynomial Combinants and the Exact and Approximate Zero Assignment Problems.- Strongly Nonassignab1e Sets and Almost Fixed Zeros.- Stability Aspects of Polynomial Combinants.- Conclusions.- References.- 3 The Occurrence of Non-Properness in Closed-Loop Systems and Some Implications.- Preliminary Observations.- Conditions for Closed-Loop Nonproperness.- Implications for Composite Systems.- Conclusions.- References.- 4 Skew-Symmetric Matrix Equations in Multivariable Control Theory.- Derivation of the Orthogonal Representation.- Skew-Symmetric Formulations.- Analytical Solutions in IR2x2 and IR3x3.- Analytical Solutions in IRnxn.- The Algebraic Riccati Generator.- Summary and Conclusions.- References.- 5 Feedback Controller Parameterizations: Finite Hidden Modes and Causality.- Main Results.- -Internal stability.- -Causality.- -Hidden Modes.- Conclusions.- References.- 6 Decompositions for General Multilinear Systems.- Time-Systems.- Linear and Multilinear Systems.- Decompositions for Multilinear Systems.- Conclusions.- References.- 7 Simplification of Models for Stability Analysis of Large-Scale Systems.- System Description and State-Space Representation.- Stability Analysis.- Application to the Definition of the Reduced Order System.- Algebraic Condition for the Application of the Results.- Methodology and Implementation.- Conclusion.- References.- II Uncertain Systems and Robust Control.- 8 Representations of Uncertainty and Robustness Tests for Multivariable Feedback Systems.- Representations of Uncertainty.- Robustness Tests.- Conclusions.- References.- 9 Additive,Multiplicative Perturbations and the Application of the Characteristic Locus Method.- Eigenvalue Inclusion Regions.- Stability and Design.- The Multiplicative Case.- A Robust Stability Study.- References.- 10 A Design Technique for Multi-Represented Linear Multi-Variable Discrete-Time Systems Using Diagonal or Full Dynamic Compensators.- Frequency Domain Bounds.- Choice of the Nominal Reference.- Realisation of the Compensator.- An Extension to Full Feedback Structures.- A Design Example.- References.- 11 Minimizing Conservativeness of Robustness Singular Values.- First and Second Derivatives of Singular Values.- Main Result.- Discussion.- Conclusions.- References.- III Algebraic and Optimal Controller Design.- 12 Frequency Assignment Problems in Linear Multivariable Systems:Exterior Algebra and Algebraic Geometry Methods.- Notation.- The Determinantal Pole,Zero Assignment Problems.- The Grassmann Representative of a Vector Space.- Plücker Matrices and the Linear Subproblem of DAP.- Decomposabi1ity of Multivectors: The Reduced Quadratic Plücker Relations.- Linearisation of the RQPRs and Feedback.- Conclusions.- References.- 13 On the Stable Exact Model Matching and Stable Minimal Design Problems.- Background.- Coprimeness in P of Proper and ?-Stable Rational Matrices.- Proper and ?-Stab1e,Minimal MacMillan Degree Bases of Rational Vector Spaces.- Stable Exact Model Matching and Stable Minimal Design Problems.- Conclusions.- References.- 14 Pole Placement in Discrete Multivariable Systems by Two and Three-Term Controllers.- Design of PID Controllers.- Design of PI Controllers.- Design of PD Controllers.- Conclusions.- References.- 15 Linear Quadratic Regulators with Prescribed Eigenvalues for a Family of Linear Systems.- Problem Formulation.- Sequential Pole Placement.- Example.- Application to Systems with Largely Varying Parameters.- Design Example.- Conclusions.- References.- 16 Sensitivity Reduction of the Linear Quadratic Optimal Regulator.- Sensitivity Reduction by Q Modification.- Extension for a Family of Plant Equations.- Conclusions.- References.- 17 Design of Low-Order Delayed Measurement Observers for Discrete Time Linear Systems.- Statement of the Problem.- Low-Order-Delayed Measurement Observer.- Example.- Conclusion.- References.- 18 Singular Perturbation Method and Reciprocal Transformation on Two-Time Scale Systems.- Presentation of the Singular Perturbations Method.- Definitions and Properties of the Reciprocal Transformation.- Resolution of a Singular Problem in Optimal Control by Reciprocal System.- Comparison on a Mechanical Example of Reduction Results Obtained by SP and (SP+R) Methods.- Conclusion.- References.- 19 Coordinated Decentralized Control with Multi-Model Representation (CODECO).- Tracking Approach to Coordinated Decentralized Control (CODECO).- Multimodel with Tracking Approach.- Algorithms Development.- Conclusion.- References.- 20 Design of Two-Level Optimal Regulators with Constrained Structures.- A Classification of Decomposition-Coordination Methods.- Principle of a New Method.- Application to Dynamic Processes.- Design of the First Leve1-Constrained Structures.- A Special Case for the Second Level:Disturbance Rejection.- Experiment Result.- Conclusion.- References.- IV Multidimensional Systems.- 21 A Canonical State-Space Model for m-Dimensional Discrete Systems.- m-D State-Space Model.- (m+l)-D State-Space Model.- Special Cases.- Example.- Conclusion.- References.- 22 Eigenvalue Assignment of 3-D Systems.- Problem Statement.- Problem Solution.- - Method 1.- - Method 2.- - Method 3.- References.- 23 Feedback Deadbeat Control of 2-Dimensional Systems.- Deadbeat Controller Design Using the Transfer Function Model.- Deadbeat Controller Design Using the State Space Model.- Example.- Appendices.- Conclusion.- References.- 24 State Observers for 2-D and 3-D Systems.- Observability and Controllability of 3-D Systems.- Transformation of Triangular 3-D Systems to Canonical Form.- The 2-D Parameter Identification Problem.- Identification via Model Reference Approachi:All States Accessible.- Adaptive Controller.- Adaptive Observer.- Description of 3-D State Observer.- State Observer Design for Triangular 3-D Systems.- Concluding Remarks.- References.- 25 Eigenvalue-Generalized Eigenvector Assignment Using PID Controller.- 1 — Continuous-time system.- Problem Statement.- Eigenvalue-Generalized Eigenvector Structure.- Numerical Examples.- 2 — Discrete-time system.- Problem Statement.- Eigenvalue-Generalized Eigenvector Assignment.- Illustrative Examples.- Conclusions.- References.