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Fundamentals of Matrix and LMIs

Fundamentals of Matrix and LMIs

  1. Basic Matrix Theory
  2. Notion of Matrix Positivity
  3. Matrix Inequalities and LMIs
  4. Convexity of LMIs
  5. Concatenation of LMIs

Properties of LMIs

  1. Change of Subject
  2. Congruence Transformation
  3. Young’s Relation (Completion of the Squares)
    1. Young’s Relation-Based Properties
    2. Special Cases of Young’s Relation
    3. Iterative Convex Overbounding
  4. Projection Lemma (Matrix Elimination Lemma)
    1. Strict Projection Lemma
    2. Non Strict Projection Lemma
    3. Reciprocal Projection Lemma
    4. Projection Lemma-Based Properties
  5. Ellipsoidal inequality
  6. Continuous Time Properties
    1. Schur Complement
      1. Strict Schur Complement
      2. Non-Strict Schur Complement
      3. Schur Complement Lemma-Based Properties
    2. Eigenvalue related Problems
      1. Matrix Eigen Value Minimization
      2. Continuous Time/Eigenvalue Problem
      3. LMI for Generalized eigenvalue problem
    3. Matrix Norm Minimization
    4. LMI for Generalized Eigenvalue Problem
    5. LMI for Linear Programming
    6. LMI for Feasibility Problem
    7. Continuous Time/Structured Singular Value
    8. LMI for Minimizing Condition Number of Positive Definite Matrix
    9. Continuous time quadratic stability
    10. Minimizing Norm by Scaling
  7. Discrete Time Properties
    1. Discrete Time Minimum Gain Lemma
    2. Discrete Time Modified Minimum Gain Lemma
  8. Finsler’s Lemma
  9. Dilation
  10. Tangential Nevanlinna Pick
  11. Nevanlinna Pick Interpolation with Scaling
  12. Deduced LMI Conditions for Hinf Index
  13. Deduced LMI Conditions for H2 Index
  14. Dissipativity of Systems
  15. Generalized H2 Norm
  16. Passivity and Positive Realness
  17. Non-Expansivity and boundedness Realness
  18. Variable Reduction Lemma
  19. D-stability Rise Poles
  20. D-stability Settling time poles
  21. D-stability Max Percent Overshoot Poles
  22. Iterative Convex Overbounding
  23. Frobenius Norm
  24. Dualization Lemma
  25. Submatrix Determinants and Imaginary and Real Parts
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