Linear Multivariable Systems 1 — Introduction 2 — Mathematical Preliminaries 2.1 Introduction 2.2 Linear Vector Spaces 2.3 Linear Operators 2.4 Scalar Matrices 2.5 Polynomial Matrices 2.6 Concluding Remarks and References 3 — The State Space 3.1 Introduction 3.2 State Representations 3.3 The Determination of eAt 3.4 Equivalent Systems 3.5 Controllability and Observability 3.6 Controllable and Observable Companion Forms 3.7 Concluding Remarks and References 4 — Frequency Domain Representations 4.1 Introduction 4.2 The Transfer Matrix 4.3 The Structure Theorem 4.4 Realization Theory (Time Domain Reduction) 4.5 Concluding Remarks and References 5 — Differential Operator Representations 5.1 Introduction 5.2 Transfer and Equivalence Relations 5.3 Differential Operator Controllability and Observability 5.4 Realization Theory (Frequency Domain Reduction) 5.5 System Invertibility and Functional Reproducibility 5.6 Concluding Remarks and References 6 — Linear State Variable Feedback 6.1 Introduction 6.2 Quadratic Optimization 6.3 Pole Assignment via the Controllable Companion Form 6.4 Asymptotic State Estimation 6.5 Concluding Remarks and References 7 — Frequency Domain Compensation 7.1 Introduction 7.2 Frequency Domain Implications of State Feedback 7.3 Frequency Domain State Estimation and Feedback 7.4 A General Compensation Technique 7.5 Concluding Remarks and References 8 — Design Objectives 8.1 Introduction 8.2 Arbitrary Pole Placement 8.3 Decoupling 8.4 Static Decoupling 8.5 Exact Model Matching 8.6 Concluding Remarks and References References. This text was developed over a three year period of time (1971 1973) from a variety of notes and references used in the presentation of a senior first year graduate level course in the Division of En­ gineering at Brown University titled Linear System Theory. The in­ tent of the course was not only to introduce students to the more modern state space approach to multivariable control system analysis and design as opposed to the classical frequency domain approach but also to draw analogies between the two approaches whenever and wherever possible. It is therefore felt that the material presented will have broader appeal to practicing engineers than a text devoted exclusively to the state space approach. It was assumed that students taking the course had also taken as a prerequisite an undergraduate course in classical control theory and also were familiar with certain standard linear algebraic notions as well as the theory of ordinary differential equations although a substantial effort was expended to make the material as self contained as possible. In particular Chapter 2 is employed to familiarize the reader with a good deal of the mathematical material employed through­ out the remainder of the text. Chapters 3 through 5 were drawn in part from a number of contemporary state space and matrix algebraic references as well as some recent research of the author especially those portions which deal with polynomial matrices and the differential operator approach.