Linear System Theory and Design

1. Introduction; 1.1 Introduction; 1.2 Overview; 2. Mathematical Descriptions of Systems; 2.1 Introduction; 2.1.1 Causality and Lumpedness; 2.2 Linear Systems; 2.3 Linear Time-Invariant (LTI) Systems; 2.4 Linearization; 2.5 Examples; 2.5.1 RLC Networks; 2.6 Discrete-Time Systems; 3. Linear Algebra; 3.1 Introduction; 3.2 Basis, Representation, And Orthonormalization; 3.3 Linear Algebraic Equations; 3.4 Similarity Transformation; 3.5 Diagonal Form And Jordan Form; 3.6 Functions Of A Square Matrix; 3.7 Lyapunov Equation; 3.8 Some Useful Formula; 3.9 Quadratic Form and Positive Definiteness; 3.10 Singular Valve Decomposition; 3.11 Norms of Matrices; 4. State-Space Solutions And Realizations; 4.1 Introduction; 4.2 Solution Of LTI State Equations; 4.2.1 Discretization; 4.2.2 Solution of Discrete-Time Equations; 4.3 Equivalent State Equations; 4.3.1 Canonical Forms; 4.3.2 Magnitude Scaling In Op-Amp Circuits; 4.4 Realizations; 4.5 Solution Of Linear Time-Varying (LTV) Equations; 4.5.1 Discrete-Time Case; 4.6 Equivalent Time-Varying Equations; 5. Stability; 5.1 Introduction; 5.2 Input-Output Stability Of LTI Systems; 5.2.1 Discrete-Time Case; 5.3 Internal Stability; 5.3.1 Discrete-Time Case; 5.4 Lyapunov Theorem; 5.5 Stability Of LTV Systems; 6. Controllability And Observability; 6.1 Introduction; 6.2 Controllability; 6.2.1 Controllability Indices; 6.3 Observability; 6.3.1 Observability Indices; 6.4 Canonical Decomposition; 6.5 Conditions In Jordan-Form Equations; 6.6 Discrete-Time State Equations; 6.6.1 Controllability To The Origin And Reachability; 6.7 Controllability After Sampling; 6.8 LTV State Equations; 7. Minimal Realizations And Coprime Fractions; 7.1 Introduction; 7.2 Implications Of Coprimeness; 7.2.1 Minimal Realizations; 7.3 Computing Coprime Ffractions; 7.3.1 QR Decomposition; 7.4 Balanced Realization; 7.5 Realizations From Markov Parameters; 7.6 Degree Of Transfer Matrices; 7.7 Minimal Realizations - Matrix Case; 7.8 Matrix Polynomial Fractions; 7.8.1 Column And Row Reduceness; 7.8.2 Computing Matrix Coprime Fractions; 7.9 Realization From Matrix Coprime Fractions; 7.10 Realizations From Matrix Markov Parameters; 7.11 Concluding Remarks; 8. State Feedback And State Estimators; 8.1 Introduction; 8.2 State Feedback; 8.2.1 Solving Lyapunov Equation; 8.3 Regulation And Tracking; 8.3.1 Robust Tracking And Disturbance Rejection; 8.3.2 Stabilization; 8.4 State Estimator; 8.4.1 Reduced-Dimensional State Estimator; 8.5 Feedback From Estimated States; 8.6 State Feedback-Multivariable Case; 8.6.1 Cyclic Design; 8.6.2 Lyapunov-Equation Method; 8.6.3 Canonical-Form Method; 8.6.4 Effect Of Transfer Matrices; 8.7 State Estimators-Multivariable Case; 8.8 Feedback From Estimated States-Multivariable Case; 9. Pole Placement And Model Matching; 9.1 Introduction; 9.1.1 Compensator Equation-Classical Method; 9.2 Unity-Feedback Configuration-Pole Placement; 9.2.1 Regulation And Tracking; 9.2.2 Robust Tracking And Disturbance Rejection; 9.2.3 Embedding Internal Models; 9.3 Implementable Transfer Functions; 9.3.1 Model-Matching Two-Parameter Configuration; 9.3.2 Implementation Of Two-Parameter Compensators; 9.4 Multivariable Unity Feedback Systems; 9.4.1 Regulation And Tracking; 9.4.2 Robust Tracking And Disturbance Rejection; 9.5 Multivariable Model Marching-Two-Parameter Configuration; 9.5.1 Decoupling; 9.6 Concluding Remarks; References