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
  

An Introduction to the Calculus of Variations

Simplest Problem in the Calculus of Variations:

This is called the Euler's Equation or the Euler-Lagrange Equation in the Calculus of Variations. The Solution to this Differential Equation must satisfy the given Boundary Conditions on x(.).

If one or both Boundary Conditions are Unspecified, the E-L Equation must also satisfy the Natural Boundary Conditions :

Any Trajectory Satisfying the Euler-Lagrange Equation and the Associated Boundary Conditions is a Candidate for a Weak Local Minimum. If this Trajectory also Satisfies the Legendre's Necessary Condition and Jacobi's Necessary Condition, then it Provides a Weak Local Minimum.

Additionally, if it satisfies the Weierstrass's Necessary Condition, the trajectory Provides a Strong Local Minimum. Finally, if is the only solution, it is safe to assume that it provides a Global Minimum.

Click Here for Brief Biographies of Euler, Lagrange, Legendre, and Weierstrss