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 Quadratic Performance Index:

, A(t): Positive Semi-Definite

B(t): Positive Definite

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

Derivation of the Necessary Conditions for Optimality

1. Form the Variational Hamiltonian

2. Derive the Costate Equations

3. Derive the Optimality Conditions

Or:

4. Find the Costate Initial Conditions:

5. Find the Costate Final Conditions:

Conditions in (1) through (5) can be used to obtain the Optimal Control.

Solve the Two-Point Boundary Value Problem

for . Find the Optimal Control using:

Note: is Unknown.

Several Methods can be setup to solve this Linear Two-Point Boundary-Value Problem. The Sweep Technique (Click here for a Description of the Sweep Technique) Produces a Feedback Solution.

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