Page 750 - Mechanical Engineers' Handbook (Volume 2)

4 Solution of System Equations 741

For linear, time-varying systems described by state-space equations (4) and (5), the
solution is given by 2,4
t
x(t) (t, t )x(t ) (t, )B( )u( ) d t t 0 (42)
0
0
t 0
t
y(t) C(t) (t, t )x(t ) C(t) (t, )B( )u( ) d D(t)u(t) (43)
0
0
t 0
where the n n state transition matrix (t, t 0 ) for the time-varying system depends on both
arguments t and t and not merely on the difference between these two time instants as in
0
the time-invariant system.
The state transition matrix (t, t ) is the solution of the partial differential equation 2,4
0
(t, t ) A(t) (t, t ) (44)
0
t 0
with the initial condition
(t , t ) I (45)
0
0
It has the following properties:
(t , t ) (t , t ) (t , t ) (46)
2 0 2 1 1 0
1
(t , t ) (t , t ) (47)
1 0 0 1
Techniques for evaluating the state transition matrix for time-varying systems are con-
siderably more involved than for time-invariant systems and are less widely applicable. A
number of analytical methods for determining (t, t ) for special cases of linear time-varying
0
systems have been described by DeRusso et al. 2
6
A simple numerical procedure has been suggested by Palm for computing the transition
matrix when analytical determination is not possible. Let the ith column of (t, t ) be denoted
0
by (t) for a specified value of t . The matrix partial differential equation (44) becomes n
i
0
vector ordinary differential equations:
(t) A(t) (t) i 1,..., n t t 0 (48)
i
i
with the initial conditions
[ (t ) (t ) (t )] I (49)
0
20
n
10
Numerical solution of the ordinary differential equations gives (t) and hence (t, t ). Note
i 0
that the computed (t, t ) would be different for different values of t for time-varying
0 0
systems.
4.2 Discrete-Time Systems
The solutions of the system equations for linear, discrete-time systems described either by
Eqs. (10) and (11) or by Eqs. (12) and (13) are given in Table 5. Expressions for the state
transition matrix (k k ) for time-invariant systems and (k, k ) for the time-varying
0
0
systems and the properties of the state transition matrix are also included in the table. These
results can be found in standard textbooks on state-space methods. 2,4
The state transition matrix (k k ) for a time-invariant system depends only on the
0
difference in the sequence number (k k ). The transition matrix is given by
0