Lecture Notes: Control Theory and Controllability

Overview of Linear Systems

• We have been examining the equation: ẋ = Ax
• Future state of the system is determined primarily by the eigenvalues of matrix A.
  • If eigenvalues are unstable, the system is unstable.
  • If eigenvalues are stable, the system is stable.

Introducing Control

• We introduce control to manipulate the system: ẋ = Ax + Bu
• u can be a vector in Rⁿ (with Q inputs).
• A is Rⁿ x n, B is Rⁿ x q (n tall).
• The goal is to shape the eigenvalues of A using control.

Control Law Design

• Control law: u = -Kx (where K is a matrix).
  • This method will optimally stabilize linear systems.
  • Full state feedback is assumed, meaning all states can be measured.

Controllability

• A system is controllable if:
  • You can manipulate the input u to place eigenvalues anywhere you want.
  • You can steer the state x to any desired point in the state space.
• Controllable systems can be represented mathematically by the pair (A,B).

Example of Controllability

• Consider a system: ẋ = Ax + Bu
  • A controllable system allows manipulation through inputs.
  • An uncontrollable system has states that cannot be influenced by inputs.

Testing for Controllability

• Use the controllability matrix:
  • C = [ B, AB, A²B, ..., Aⁿ⁻¹B ]
  • If C has full column rank (equal to n), the system is controllable.

MATLAB Implementation

• Use CTRAB(A,B) in MATLAB to create the controllability matrix.
  • Check the rank of this matrix to determine controllability.

Examples

Uncontrollable Example

• A = [1, 0; 0, 2], B = [0; 1]
  • x1 is uncoupled and cannot be controlled; hence, the system is uncontrollable.

Controllable Example

• Modify B: B = [1; 1]
  • Now both states can be influenced, making the system controllable.

Coupling Example

• A = [1, 1; 0, 2], B = [0; 1]
  • This system is controlled through coupling, showing it is controllable.

Mathematical Justification

• The controllability matrix helps probe the system.
  • If all directions in Rⁿ are reachable, the system is controllable.
• Rank deficiency indicates uncontrollable directions.

Singular Value Decomposition (SVD)

• Analyzing the SVD of the controllability matrix can provide insight into which states are more controllable than others.

Summary

• You can determine controllability using a simple MATLAB command and checking the rank of the controllability matrix.
• If controllable, we can design controllers to achieve desired system dynamics.

Next Steps

• Explore implications of controllability and how to design control laws in future lectures.