Controllability of a Linear System: The Controllability Matrix and the PBH Test

In this video we explore controllability of a linear system. We discuss two methods to test for controllability, the controllability matrix as well as the PBH test.

Topics and time stamps:
0:00 – Introduction and definition.
1:04 – Controllability of a dog.
3:48 – Controllability matrix.
7:18 – Example 1: Controllable system.
11:10 – Example 2: Uncontrollable system.
17:36 – Example 3: Make an uncontrollable system controllable.
22:11 – Example 4: System is controllable using single input.
28:54 – Example 5: Symmetry makes system uncontrollable with single input.
31:38 – PBH test history and background.
35:09 – PBH test statement and analysis.
45:31 – Example 6: PBH test.
1:29:22 – Example 7: System that needs multiple control inputs to be controllable.
1:35:46 – Summary and conclusions.

Additional videos in this series:
-Controllability of a Linear System: The Controllability Matrix and the PBH Test (   • Controllability of a Linear System: T...  )
-Introduction to Full State Feedback (   • Introduction to Full State Feedback C...  )
-Practical Implementation Issues with a Full State Feedback Controller (   • Practical Implementation Issues with ...  )
-Introduction to Linear Quadratic Regulator (LQR) Control (   • Introduction to Linear Quadratic Regu...  )

Overview of engineering videos and curriculum located at http://faculty.washington.edu/lum/Edu...

Additional resources:
-Steve Brunton’s discussion on controllability (   • Controllability [Control Bootcamp]  )
-Steve Brunton’s discussion on the PBH test (   • Controllability and the PBH Test [Con...   )
-Hautus Lemma - https://en.wikipedia.org/wiki/Hautus_...
-Rank Nullity Theorem (https://en.wikipedia.org/wiki/Rank%E2...)