# Linear System Theory

### General Information

Instructor: Dr. Philippe MULLHAUPT, Chargé de cours, ME C2 391

Objective: Provide a solid treatment of linear system theory and design using polynomial matrices and state-space theory. Special emphasis is put on the multi-input multi-output case.

Key topics: Controllability and Observability, Coprime Fractions, State Feedback and Estimators, Realizations, Pole Placement and Model Matching.

### Mathematical Descriptions of Systems

Linear Systems
Linear Time-Invariant (LTI) Systems
Linearization
Discrete-Time Systems

### State-Space Solutions and Realizations

Solution of LTI State Equations
Equivalent State Equations
Realizations
Solution of Linear Time-Varying (LTV) Equations
Equivalent Time-Varying Equations
Time-Varying Realizations

### Stability

Input-Output Stability of LTI Systems
Internal Stability
Lyapunov Theorem
Stability of LTV Systems

### Controllability and Observability

Controllability
Observability
Canonical Decomposition
Conditions in Jordan-Form Equations
Discrete-Time State Equations
Controllability After Sampling
LTV State Equations

### Minimal Realizations and Coprime Fractions

Implications of Coprimeness
Computing Coprime Fractions
Balanced Realization
Realizations and Markov Parameters
Degree of Transfer Matrices
Minimal Realization – Matrix Case
Matrix Polynomial Fractions
Realizations from Matrix Coprime Fractions
Realizations from Matrix Markov Parameters

### Place Placement and Model Matching

Introduction
Unity-Feedback Configuration-Pole Placement
Regulation and Tracking
Robust Tracking and Disturbance Rejection
Embedding Internal Models
Implementable Transfer Functions
Model Matching-Two-Parameter Configuration
Implementation of Two-Parameter Compensators
Multivariable Unity-Feedback Systems
Regulation and Tracking
Robust Tracking and Disturbance Rejection
Multivariable Model Matching-Two-Parameter Configuration
Decoupling

### Books

Main Reference:

Chen, C.T. “Linear System Theory and Design” Oxford University Press, 1999

Linear Algebra:

Baer “Linear Algebra and Projective Geometry”, Academic Press, 1952.
Gantmatcher “Matrix Theory” Vol 1 & 2, AMS, Chelsea, (1959) 2000.
Godunov “Modern Aspects of Linear Algebra”, AMS, 1998.
Gohberg, Lancaster, Rodman “Matrix Polynomials”, Academic Press, 1982.
Golub, Van Loan, “Matrix Compuations”, Johns Hopkins University Press, 1996.
Householder, “The Theory of Matrices in Numerical Analysis”, Dover, 1964.
Katznelson & Katznelson “A (Terse) Introduction to Linear Algebra”, AMS, 2008.
Lancaster “Lambda-Matrices and Vibrating Systems”, Oxford, Pregamon, 1966.
Laub “Matrix Analysis for Scientists and Engineers”, SIAM, 2005.
Mirsky “An Introduction to Linear Algebra”, Oxford, Clarendon Press, 1955.
Watkins “The Matrix Eigenvalue Problem: GR and Krylov Subspace Methods”, SIAM, 2007.
Willkinson “The Algebraic Eigenvalue Problem”, Oxford University Press, 1965.

Multivariable Control:

Rosenbrock “State Space and Multivariable Theory”, Nelson, London, 1970.
Wonham “Linear Multivariable Theory. A Geometric Approach, 2nd ed. Springer, 1979.
Wolovich “Linear Multivariable Systems”, Springer, 1974.
Callier, Desoer “Multivariable Feedback Systems”, Springer, 1982.
Vardulakis “Linear Multivariable Control” John Wiley, 1991.
Zadeh, Desoer “Linear System Theory: The State Space Approach”, McGraw-Hill, 1963.
Kailath, “Linear Systems”, Prentice-Hall, 1980.