### Multivariable Feedback Systems

1. On the Advantages of Feedback.- 1.1. Introduction.- 1.2. Singular Value Decomposition of a Matrix.- 1.3. Large Loop Gain.- 2. Matrix Fraction Description of Transfer Functions.- 2.1. Introduction.- 2.2. Polynomials, Euclidean Rings, and Modules.- 2.3. Polynomial Matrices.- 2.3.1. Divisors, Coprimeness, Rank.- 2.3.2. Elementary Operations on Polynomial Matrices.- 2.3.3. Elementary Operations and Differential Equations.- 2.3.4. Standard Forms: Hermite and Smith Forms.- 2.3.5. The Solution Space of D(p) ?(t) = 9 t ? 0.- 2.3.6. Greatest Common Divisor Extraction.- 2.4. Matrix Fraction Descriptions of Rational Transfer Function Matrices.- 2.4.1. Coprime Fractions.- 2.4.2. Smith-Mcmillan Form Relation to Coprime Fractions.- 2.4.3. Proper Transfer Function Matrices.- 2.4.4. Poles and Zeros.- 2.4.5. Dynamical Interpretation of Poles and Zeros.- 2.5. Realization and Polynomial Matrix Fractions.- 3. Polynomial Matrix System Descriptions and Related Transfer Functions.- 3.1. Introduction.- 3.2. Dynamics of a PMD Redundancy.- 3.2.1. Dynamics of a PMD.- 3.2.2. Reachability of PDMs.- 3.2.3. Observability of PMDs.- 3.2.4. Minimality, Hidden Modes, Poles, and Zeros.- 3.3. Well-Formed and Exponentially Stable PMDs.- 3.3.1. Well-Formed PMDs.- 3.3.2. Exponentially Stable PMDs 121 3.4. Transfer Functions: Right-Left Fractions Internally Proper Fractions.- 4. Interconnected Systems.- 4.1. Introduction.- 4.2. Exponential Stability of an Interconnection of Subsystems.- 4.3. Feedback System Exponential Stability.- 4.4. Special Properties of Feedback Systems.- 5. Single-Input Single-Output Systems.- 5.1. Introduction.- 5.2. Problem Statement and Analysis.- 5.3. Design.- 6. The Closed-Loop Eigenvalue Placement Problem.- 6.1. Introduction.- 6.2. The Compensator Problem.- 7. Asymptotic Tracking.- 7.1. Introduction.- 7.2. Theory of Asymptotic Tracking.- 7.3. The Tracking Compensator Problem.- 8. Design with Stable Plants.- 8.1. Introduction.- 8.2. Q-Parametrization Design Properties.- 8.3. Q-Design Algorithm for Decoupling by Feedback.- 8.4. Two-Step Compensation Theorem for Unstable Plants.- Epilogue.- Appendices.- A. Rings and Fields.- B. Matrices with Elements in a Commutative Ring IK.- C. Division of a Polynomial Vector on the Left by a Polynomial Matrix.- References.- Symbols.