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
  

These Are Fundamental Properties of Dynamic Systems

- Given the System:

Controllability: The pair (A,B) is said to be controllable iff at the initial time there exist a control function which will transfer the system from its initial state to the origin in some finite time. If this statement is true for all time, then the system is "Completely Controllable".

Observability: The pair (A,C) is said to be observable at the time iff can be determined from the given outputs in some finite time . If this statement is true for all time, then the system is "Completely Observable".

The Nature of Subsystems in a System

Controllability Tests:

Systems with Distinct Eigenvalues

a) Find the System Eigenvectors:

b) Form the Modal Matrix :

c) Carry out the Transformation:

The Pair (A,B) is Controllable iff None of the Rows of are Zero.

More General Systems

Form the nxmn Controllability Matrix:

The Pair (A, B) is Completely Controllable iff the Controllability Matrix P has Rank n

Observability Tests:

Systems with Distinct Eigenvalues

Carry out the Transformation:

The Pair (A,C) is Observable iff None of the Columns of are Zero.

More General Systems

Form the nxrn Observability Matrix:

The pair (A, C) is Completely Observable iff the Observability Matrix Q has Rank n.