Controllability Matrix and System Controllability

From Concept to Calculation: The Controllability Matrix

For a linear time-invariant (LTI) system described by $\dot{\mathbf{x}}=A\mathbf{x}+B\mathbf{u}$, where $\mathbf{x}$ is the n-dimensional state vector, there is a powerful algebraic test for controllability. This test revolves around a special matrix called the controllability matrix.

This matrix, often denoted as $\mathcal{C}$ or $P_C$, is constructed from the system's $A$ and $B$ matrices as follows:

Let's break down the components of this matrix:

• $B$: The columns of the $B$ matrix represent the directions in the state space that our control input $\mathbf{u}$ can directly influence or "push."
• $AB$: This term represents the directions the system can be pushed in after one infinitesimal time step. The system's internal dynamics, represented by $A$, act on the initial push from $B$.
• $A^{k}B$: This represents the directions reachable after $k$ such steps.

The columns of the entire controllability matrix $\mathcal{C}$ collectively span all the directions in the state space that can be reached by applying some control input.

To get a formal introduction to this matrix and the test that uses it, please watch the first part of the following video.

The Controllability Rank Condition

As the video explained, the definitive test for controllability is the rank condition.

A system is controllable if and only if the rank of its controllability matrix $\mathcal{C}$ is equal to $n$, the number of state variables.

$\text{rank}(\mathcal{C})=n⟺\text{The system is controllable}$

From your background in signal processing and linear algebra, you know that the rank of a matrix is the number of linearly independent columns (or rows) it contains.

• If $\text{rank}(\mathcal{C})=n$, the columns of $\mathcal{C}$ span the entire n-dimensional state space. This means there are no "hidden" or "unreachable" directions. We can steer the state anywhere.
• If $\text{rank}(\mathcal{C})<n$, the matrix is "rank-deficient." This means the columns only span a subspace of the full state space. There are directions the control input can never influence, making the system uncontrollable.

For a formal statement of this crucial theorem, please review the following document.

A Visual Example

Let's solidify this with a clear, static example. The image below shows the calculation for two different third-order systems.

Let's trace Example 1:

1. Identify A and B: The system matrices are given. $n=3$.
2. Form the Matrix: We need to compute $\mathcal{C}=\left[\begin{array}{ccc}B& AB& A^{2}B\end{array}\right]$.
   • $B$ is given: $\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]$
   • $AB=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -6& -11& -6\end{array}\right]\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]=\left[\begin{array}{c}0\\ 1\\ -6\end{array}\right]$
   • $A^{2}B=A(AB)=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -6& -11& -6\end{array}\right]\left[\begin{array}{c}0\\ 1\\ -6\end{array}\right]=\left[\begin{array}{c}1\\ -6\\ 25\end{array}\right]$
3. Construct $\mathcal{C}$: $$\mathcal{C}=\left[\begin{array}{ccc}0& 0& 1\\ 0& 1& -6\\ 1& -6& 25\end{array}\right]$$
4. Check the Rank: We can find the rank by checking for linear independence. The determinant of this matrix is non-zero ($\det (\mathcal{C})=-1$), which for a square matrix means it has full rank.
5. Conclusion: Since $\text{rank}(\mathcal{C})=3$ and $n=3$, the system is controllable.

Physical Intuition Behind the Rank Test

Now, let's connect this mathematical test back to the physical world. The following examples show how the structure of the system and its actuators directly impacts the rank of the controllability matrix.

These examples beautifully illustrate the link between the physical system and the abstract mathematics:

• An inability to actuate a certain part of the system (like the vertical motion in the first example) manifests as zero-rows in the controllability matrix, leading to a rank less than $n$.
• Adding more actuators (or placing them more effectively) adds more non-zero entries and linearly independent columns to $\mathcal{C}$, increasing its rank and potentially making the system controllable.

Conclusion

In this lesson, we have established the definitive mathematical test for controllability, a critical step in control system design.

Key Takeaways:

• The controllability matrix $\mathcal{C}=\left[\begin{array}{cccc}B& AB& \cdots & A^{n-1}B\end{array}\right]$ captures all the state-space directions reachable by the control input.
• The rank condition provides a clear, binary answer: the system is controllable if and only if $\text{rank}(\mathcal{C})=n$.
• This test can be performed by hand for small systems or with software like MATLAB (using rank(ctrb(A,B))) for larger, more complex systems like your inverted pendulum.

Preview of the Next Lesson:

We now have a tool to answer the question, "Can we control the state?" But this assumes we know what the state is. In many real systems, we can't measure every state variable directly (e.g., measuring velocity might be difficult or expensive, while position is easy). This raises a new question: "Can we determine the full state of the system by only looking at its outputs?" This is the question of observability, the dual concept to controllability, and it will be the subject of our next lesson.