Page 70 - Servo Motors and Industrial Control Theory -

62 3 State Variable Feedback Control Theory

If state variable feedback control strategy is used, only controllability must be
studied. First a single input and a single output state equation is considered as

d X := AXBu+
dt (3.27)
y := CX

X is the vector of n state variables A is the system matrix and B is the input vector
which will be a matrix for multi-input system. C is the output vector which will be
a matrix for multi-output system.
A transformation of state variables as

X := UZ (3.28)

is considered. U is the matrix of eigenvectors of A and Z is the transformed state
variables. Substituting Eq. (3.28) in Eq. (3.27) gives
d
U ⋅ Z =: AUZ Bu+
dt (3.29)
y =: CUZ
−1
Pre-multiplying the system equation by U gives
d Z := U AU + UBu
−1
−1
dt (3.30)
y := CUZ
It can be shown that the matrix U AU is a diagonal matrix with the diagonal terms
−1
being the eigenvalues ( λ ) of the matrix A. For proof, the readers are referred to
i
more advanced books. With this in mind, Eq. (3.30) becomes
d − 1

dt Z : λ= + U Bu (3.31)
=
y : CUZ
In Eq. (3.31), λ is a diagonal matrix with diagonal terms being the eigenvalues of
the system. Now the controllability and observability can be discussed. For a single
input the system equation is n uncoupled first order differential equations which
have a simple solution. The input u can be used to change the eigenvalues λ. The
system is controllable if all eigenvalues can be influenced by the input u. The matrix
−
U 1 B for single input is a vector and for controllability all values should be nonzero
meaning that that all eigenvalues can be influenced by the input u. If u is a vector
then U 1 B is a matrix of n × m. The system is then controllable if U 1 B has no
−
−
rows consisting entirely of zero elements. Most physical system is controllable by a
single input variable even for multi-input control systems.
The system is observable if the row vector CU has no zero elements. For multi-
output control system the matrix CU must have no columns consisting of entirely of
zero elements. In this way, the output will have contribution from all state variables Z.