Page 360 - Mechanical Engineers' Handbook (Volume 2)

6 State-Variable Methods 351

Av v i
i
i
Note that the eigenvectors represent a set of special directions in the state space. If the state
vector is aligned in one of these directions, then the homogeneous state equation becomes
v i , implying that each of the state variables changes at the same rate determined
˙ v A˙v i
i
by the eigenvalue . This further implies that, in the absence of inputs to the system, a state
i
vector that becomes aligned with an eigenvector will remain aligned with that eigenvector.
The system eigenvalues are calculated by solving the nth-order polynomial equation
n
I A a n 1 n 1 a a 0
0
1
This equation is called the characteristic equation. Thus the system eigenvalues are the roots
of the characteristic equation, that is, the system eigenvalues are identically the system poles
defined in transform analysis.
Each system eigenvector is determined by substituting the corresponding eigenvalue into
the defining equation and then solving the resulting set of simultaneous linear equations.
Only n 1 of the n components of any eigenvector are independently defined, however. In
other words, the magnitude of an eigenvector is arbitrary, and the eigenvector describes a
direction in the state space.

Diagonalized Canonical Form
There will be one linearly independent eigenvector for each distinct (nonrepeated) eigenvalue.
If all of the eigenvalues of an nth-order system are distinct, then the n independent eigen-
vectors form a new basis for the state space. This basis represents new coordinate axes
defining a set of state variables z (t), i 1, 2, . . . , n, called the diagonalized canonical
i
variables. In terms of the diagonalized variables, the homogeneous state equation is
˙ z(t) z

where is a diagonal system matrix of the eigenvectors, that is,
1 0
0
0
0

2
0 0 n
The solution to the diagonalized homogeneous system is

t
z(t) ez(0)
where e t is the diagonal state transition matrix
e 0 t 1 e 0 t 2 0

0
e t
0 0 e t n
Modal Matrix
Consider the state equation of the nth-order system

˙ x(t) Ax(t) Bu(t)
which has real, distinct eigenvalues. Since the system has a full set of eigenvectors, the state
vector x(t) can be expressed in terms of the canonical state variables as