Kontrollbarhetsmatrisen

Kontrollbarhetsmatrisen, often referred to as the controllability matrix, is a fundamental concept in control theory. It is a mathematical tool used to determine whether a linear time-invariant (LTI) system is controllable. Controllability means that it is possible to steer the system from any initial state to any desired final state in a finite amount of time using the system's input signals.

For a single-input single-output (SISO) LTI system described by state-space equations $\dot{x}(t) = Ax(t) + Bu(t)$ and $y(t) = Cx(t) + Du(t)$, where $x(t)$ is the state vector, $u(t)$ is the input, $y(t)$ is the output, and $A$, $B$, $C$, and $D$ are matrices, the controllability matrix is constructed as follows:

The controllability matrix, denoted by $Q_c$ or $\mathcal{C}$, is formed by stacking the matrix $B$ and its successive products with the $A$ matrix:

$Q_c = [B \quad AB \quad A^2B \quad \dots \quad A^{n-1}B]$

where $n$ is the order of the system (the dimension of the state vector $x(t)$).

A key theorem states that the system is controllable if and only if the controllability matrix $Q_c$ has full rank, meaning its rank is equal to $n$. If the rank of $Q_c$ is less than $n$, the system is uncontrollable, implying that certain states cannot be influenced by the input. The controllability matrix is crucial for controller design, as it helps establish the conditions under which feedback control strategies can effectively manage the system's behavior.