An Overview of Emerging Trends in Nonlinear Control

Outline

• Linearization
• State Transition Matrix, Eigenvalues and Eigenvectors
• Controllability and Observability
• State Variable and Output Feedback, Pole-Placement Design, General Eigenstructure Assignment problem
• Full-Order Observers and Reduced-Order Observers
• An Introduction to the Calculus of Variations
• The Minimum Principle
• Linear-Quadratic Optimal Control. Optimal Regulators and Observers.
• References
  

, Define a Nominal Condition: Such that

and:

Expand the Right Hand Sides of the F(.,.), G(.,.) in a Multi-Dimensional Taylor Series About the Nominal Condition To Yield:

Using the Fact: , at any Given Nominal Value , the Linearized System Dynamics is Given By:

Where A, B, C, D are Matrices of Partial Derivatives (Jacobian Matrix) Evaluated at the Nominal Values:

Jacobi was a Professor at the University of Konigsberg from 1826 to 1843. He was the son of a Berlin Banker. He spent some time in Italy trying to recover his health, and ended his carrier as a professor at the University of Berlin, dying at the age of 46. He was witty and liberal thinker, an inspiring teacher, and a scientist whose enormous energy and clarity of thought left few branches of mathematics untouched. Sylvester (English Mathematician, Poet, contemporary and collaborator of Arthur Cayley, spent most of his life in the Acturial Trade, also taught at the University of Virginia, and Johns Hopkins University) coined the name "Jacobian" to the functional determinant in order to pay respect to Jacobi's work on Algebra and Elimination Theory. The most popular work of Jacobi was his lecture series on dynamics. He also obtained important results on the "Sufficiant Conditions" for a minimum, which are still in use by the practitioners of Optimal Control Theory.