Page 736 - Mechanical Engineers' Handbook (Volume 2)

3 State-Variable Selection and Canonical Forms 727

and coupling matrices for the controllable canonical form are listed in Table 1. The special
form of the A matrix is referred to as the companion form. Here, I n 1 is the (n 1)
(n 1) identity matrix.
For a SISO, LTI system described by Eqs. (6) and (7), the state transformation matrix
T in Eq. (14) which transforms the state-space equations into the controllable canonical form
1
exists if the controllability matrix P in Eq. (30) is nonsingular :
c
P [BAB ... A n 1 B] (30)
c
The transformation matrix T is defined in Table 2.
The observable canonical form is useful in state estimator or observer design applica-
tions. The observable canonical form for an nth-order, SISO, LTI system is described in
Table 1 in a manner similar to the controllable canonical form. The corresponding A matrix
is the transpose of the A matrix for the controllable canonical form and is also referred to
as a companion matrix. The state transformation matrix T in Eq. (14), which transforms
given state-space Eqs. (6) and (7) into the observable canonical form, exists if the observa-
bility matrix P is nonsingular :
1
0

C
CA
P (31)
0
CA n 1
State variables can also be chosen to diagonalize or nearly diagonalize the state matrix
A. The resulting state-space equations are completely or almost completely decoupled from
one another and hence show very clearly the effect of initial conditions or forcing inputs on
the different characteristic modes of the system response. The resulting physical insight into
the system behavior makes the corresponding form of the system equations, called the normal
form or diagonal Jordan form, valuable in vibration analysis applications and in control
applications involving modal control. The normal form of the state-space equations for a
SISO, LTI system with real, distinct characteristic roots is given in Table 1. The diagonal
elements of the A matrix in the table are the system characteristic roots or eigenvalues. The
state variables x , x , ..., x lie along the eigenvectors of the A matrix, in state space. As
n
1
2
the corresponding simulation diagram indicates, the behavior of each state variable is gov-
erned solely by one eigenvalue, the initial condition on that state variable, and the forcing
input.
If some of the distinct characteristic roots of a SISO, LTI system are complex, the
matrices A and C in Eqs. (6) and (7) have complex elements when represented in the normal
form just described. Since this could be inconvenient in subsequent matrix manipulations, a
nearly diagonal A matrix can be obtained for cases where the complex characteristic roots
occur in complex-conjugate pairs. This would be the case for system differential equations
with only real coefficients. The near-normal form of the system equations for a system with
one pair of complex-conjugate characteristic roots is given in Table 1. Extension to the case
of multiple complex root pairs is straightforward. The complex characteristic roots result in
a few nonzero off-diagonal elements in the A matrix; otherwise, the decoupled nature of the
system equations is retained.
If one characteristic root of a SISO, LTI system is real and repeated m times, the state
equations can only be partially decoupled by appropriate state-variable selection, as shown
in Table 1. The resulting state-space equations are said to be in the Jordan canonical form.
The corresponding A matrix has one submatrix with the repeated eigenvalue at the diagonal
positions, ones immediately to the right of the repeated diagonal elements within the sub-
matrix, and zero elements at all other nondiagonal positions. The A matrix is then said to
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