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.
  • Sweep Technique
• References
  

Sweep Technique for Soving Linear Two-Point Boundary-Value Problems

Given the Linear Two-Point Boundary-Value Problem

Since: , Consider a Solution of the Form:

S is an nxn Matrix. So that:

Or

Since this Expression Must hold for all X,

This is called the Matrix-Riccati Equation. Note that at ,. Thus,

Procedure for Obtaining Optimal Control:

1. Select Positive Semi-Definite A, Positive Definite B.

2. Formulate the Matrix Riccati Equation

3. Integrate the Matrix-Riccati Equation Backwards from to With the Boundary Condition .

4. Store S(t).

5. Compute Optimal Control as: . Defining a time Varying Gain Matrix: , The Optimal Control Law Becomes:

Note: The Optimal Control Law has a Time-Varying Feedback Gain.

A Time-Invariant Control Law can be Derived by Assuming that , so that . In this case, the Matrix Riccati Equation Degenerates to the Algebraic Riccati Equation:

The Solution to this Equation is a Symmetric, Positive Definite Matrix S.

Near Optimal, Time-Invariant Control Law is then Given by:

Where, the Control Law Gain is:

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