Editing Controllability (section)

== Output controllability ==

''Output controllability'' is the related notion for the output of the system (denoted ''y'' in the previous equations); the output controllability describes the ability of an external input to move the output from any initial condition to any final condition in a finite time interval. It is not necessary that there is any relationship between state controllability and output controllability. In particular:
* A controllable system is not necessarily output controllable. For example, if matrix ''D''&nbsp;=&nbsp;0 and matrix ''C'' does not have full row rank, then some positions of the output are masked by the limiting structure of the output matrix, and therefore unachievable. Moreover, even though the system can be moved to any state in finite time, there may be some outputs that are inaccessible by all states. A trivial numerical example uses ''D''=0 and a ''C'' matrix with at least one row of zeros; thus, the system is not able to produce a non-zero output along that dimension.
* An output controllable system is not necessarily state controllable. For example, if the dimension of the state space is greater than the dimension of the output, then there will be a set of possible state configurations for each individual output. That is, the system can have significant [[zero dynamics]], which are trajectories of the system that are not observable from the output. Consequently, being able to drive an output to a particular position in finite time says nothing about the state configuration of the system.

For a linear continuous-time system, like the example above, described by matrices <math>A</math>, <math>B</math>, <math>C</math>, and <math>D</math>, the <math>m \times (n+1)r</math> ''output controllability matrix''
:<math>\begin{bmatrix} CB & CAB & CA^2B & \cdots & CA^{n-1}B & D\end{bmatrix}</math>
has full row rank (i.e. rank <math>m</math>) if and only if the system is output controllable.<ref name="Ogata97" />{{rp|742}}