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
  

· All the System States are not Measured.

· Using a Knowledge of the System Inputs and Available Outputs, an Observer Generates the Estimates of the System States.

Click on the Observer Structure to See How this Result Was Obtained

Reduced-Order Observers

· Reduced-Order Observers can be Constructed If Some of The States Are Accurately Measured.

· The Design of these Observers are Based on Partitioned Form of the System Dynamics

Full-Order Observer: (p+q)th Order

Reduced-Order Observer: qth Order.

Basis for Reduced-Order Observer Design: The State Vector is Assumed to be Measured, The State Vector is to be Estimated.

General Idea: Convert the State Equation of to Serve as a Measurement Equation for .

Since and u are Known, Rewrite This Equation As:

i.e., A new Measurement Equation.

· Since the State Equation for is:

The qth-order Reduced-Order Estimator with z as the Measurement Vector is:

Estimator Gain L can be Selected Using Pole Placement Method.

· Note that the z Vector Contains Which is Not Available. This Difficulty can be Resolved by Redefining The Reduced-Order Estimator States in terms of a New State Vector:

So That

· Steps in Implementing the Reduced-Order Estimator:

1. Estimate Using and u.

2. Compute From:

· Note: Reduced-Order Estimator Accuracy Depends Not Only on the Accuracy of Measuring , But also on the Accuracy of the Matrices .

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