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
  

and the Performance Index:

Find the Optimal Control u*(t) that Minimizes the Performance Index.

Derivation of the Necessary Conditions for Optimality:

1. Form the Variational Hamiltonian

2. Derive the Costate Equations

3. Derive the Optimality Conditions

4. Find the Costate Initial Conditions:

5. Find the Costate Final Conditions:

Use the Conditions in (2) through (5) Together with

and Specified Boundary Conditions on x for Computing the Optimal Control u*(t).

Note: This is a Nonlinear Two-Point Boundary-Value Problem, and Requires Sophisticated Numerical Methods for Solution. Some of the Well known Techniques are:

- Multiple Shooting

- Quasilinearization

An Excellent Reference: A. E. Bryson and Y. C. Ho, Applied Optimal Control, Hemisphere, New York, NY, 1975.

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