Section 9–6 / Controllability 679
we can rewrite Equation (9–58) as
If the elements of any one row of the n*rmatrixFare all zero, then the corresponding
state variable cannot be controlled by any of the ui.Hence, the condition of complete
state controllability is that if the eigenvectors of Aare distinct, then the system is com-
pletely state controllable if and only if no row of has all zero elements. It is im-
portant to note that, to apply this condition for complete state controllability, we must
put the matrix in Equation (9–58) in diagonal form.
If the Amatrix in Equation (9–56) does not possess distinct eigenvectors, then
diagonalization is impossible. In such a case, we may transform Ainto a Jordan canonical
form. If, for example,Ahas eigenvalues and has n-3distinct
eigenvectors, then the Jordan canonical form of Ais
The square submatrices on the main diagonal are called Jordan blocks.
Suppose that we can find a transformation matrix Ssuch that
If we define a new state vector zby
(9–59)
then substitution of Equation (9–59) into Equation (9–56) yields
(9–60)
The condition for complete state controllability of the system of Equation (9–56) may
then be stated as follows: The system is completely state controllable if and only if (1)
=Jz+S-^1 Bu
z# =S-^1 ASz+S-^1 Bu
x=Sz
S-^1 AS=J
J=I
l 1
0
0
0
1
l 1
0
0
1
l 1
l 4
0
1
l 4
l 6
0
ln
Y