Page 749 - Mechanical Engineers' Handbook (Volume 2)

740 State-Space Methods for Dynamic Systems Analysis

(t t ) A (t t ) t t 0 (35)
0
0
with the initial condition
(t t ) (0) I (36)
0
0
It has the following properties:
(t ) (t) ( ) ( ) (t) (37)
1
(t) ( t) (38)
The following expressions for (t) can be verified and are useful in its evaluation:
33
22
A t A t
(t) e At I At (39)
2! 3!
and
1
1
(t) L [(sI A) ] (40)
1
where L denotes the inverse Laplace transform. Details related to Eqs. (35)–(40) have been
2
described by DeRusso et al. and Brogan. 4
Knowledge of the state transition matrix for a given system simplifies the task of de-
termining the response of the system to a variety of initial conditions x(t ) and forcing
0
functions u(t). A number of analytical and numerical techniques for its evaluation are avail-
able.
Equation (39) forms the basis for a numerical method of determining (t). Closed-form
evaluation of e At is possible only for special forms of the A matrix. For example, if A is a
diagonal matrix with diagonal elements equal to the eigenvalues s , it can be shown that (t)
i
4
is also diagonal and is given by
e st 1 e st 2 0
(t) e At 0 (41)
e st n
Closed-form evaluation of e is only slightly more complex if A is in the nondiagonal Jordan
At
4
canonical form. If the transformation matrix T [Eq. (14)] was used to obtain the diagonal
1
or nondiagonal Jordan matrix A, the transition matrix, for the original state vector T x,is
T e T.
1 At
Equation (40) provides the basis for an analytical evaluation of (t) that is suitable for
low-order dynamic systems. This method requires the inversion of the n n matrix sI
A, followed by the inverse Laplace transformation of the n elements. The matrix inversion
2
is especially cumbersome since the elements of the matrix are functions of s. The matrix
inversion can be avoided altogether by using simulation diagrams of the system, in con-
junction with block diagram reduction techniques, to determine elements of the matrix (sI
2
A) . Alternative analytical techniques for the evaluation of (t) based on Sylvester’s
1
2
theorem and the Cayley–Hamilton theorem have been described by DeRusso et al. and
Brogan. 4
Numerical evaluation of (t) for a specified value of t can be performed using Eq. (39)
and retaining a finite number of terms from the series expansion. The number of terms
retained increases with the desired degree of accuracy. An iterative procedure for determining
the number of terms to be retained for a specified degree of accuracy has been described by
Shinners. 5