Consider a system described by

ẋ = Ax + Bu

y = Cx + Du

The system is completely output controllable if and only if

Where:

x = State vector (n-vector)

u = Control vector (r-vector)

y = Output vector (m-vector)

A = n × n matrix

B = n × r matrix

C = m × n matrix

D = m × r matrix

Concept:

Consider the system described by

x = Ax + Bu        --- (i)

y = Cx + Du        --- (ii)

Where x = state vector (n-vector)

u = control vector (r-vector)

y = output vector (m-vector)

A = n × n matrix

B = n × r matrix

C = m × n matrix

D = m × r matrix

The system described by equation (i) and (ii) is said to be completely output controllable if it is possible to construct an unconstrained control vector u(t) that will transfer any given initial output y(t₀) to any final output y(t₁) in a finite time interval t₀ ≤ t ≤ t₁.

It can be proved that the condition for complete output controllability is as follows: The system described by equations (i) and (ii) is completely output controllable if and only if the m × (n + 1)r matrix

[CB | CAB | CA²B | … | CA^n-1 B|D] is of rank m