# Controllability

Controllability is an important property of a control system, and the controllability property plays a crucial role in many control problems, such as stabilization of unstable systems by feedback, or optimal control.

Controllability and observability are dual aspects of the same problem.

Roughly, the concept of controllability denotes the ability to move a system around in its entire configuration space using only certain admissible manipulations. The exact definition varies slightly within the framework or the type of models applied.

from Controllability - Wikipedia

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## State Space, Part 3: A Conceptual Approach to Controllability and Observabil...

13 min
Beginner
Video
Theory

This video helps you answer two really important questions that come up in control systems engineering: Is your system controllable? And is it observable? Assuming you have a good linear...

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## The Institute for Systems Theory and Automatic Control MATLAB Apps

Intermediate
App
Theory

The Institute for Systems Theory and Automatic Control offers 5 Matlab Apps on the topics of the Nyquist Criterion, Robustness and Stability, Loopshaping, Controllability and Observability...

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## Controllability and the Discrete-Time Impulse Response [Control Bootcamp]

5 min
Beginner
Video
Theory

This lecture derives the impulse response for a discrete-time system and relates this to the controllability matrix.

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## Controllability [Control Bootcamp]

32 min
Beginner
Video
Theory

This lecture explores when a linear system is controllable. We begin with the simple test in terms of the rank of the controllability matrix on a few intuitive examples.

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## Reachability and Controllability with Cayley-Hamilton [Control Bootcamp]

10 min
Beginner
Video
Theory

Here we use the Cayley-Hamilton Theorem to show that the full state space is reachable if and only if the system is controllable.

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## Degrees of Controllability and Gramians [Control Bootcamp]

15 min
Beginner
Video
Theory

This lecture discusses degrees of controllability using the controllability Gramian and the singular value decomposition of the controllability matrix.

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## Controllability, Reachability, and Eigenvalue Placement [Control Bootcamp]

10 min
Beginner
Video
Theory

This lecture explains the equivalence of controllability, reachability, and the ability to arbitrarily place eigenvalues of the closed loop system.

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## Controllability and the PBH Test [Control Bootcamp]

13 min
Beginner
Video
Theory

This video describes the PBH test for controllability and describes some of the implications for good choices of "B".

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## Controllability of a Linear System: The Controllability Matrix and the PBH T...

97 min
Beginner
Video
Theory

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 P...

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