Page 361 - Mechanical Engineers' Handbook (Volume 2)

352 Mathematical Models of Dynamic Physical Systems

x(t) v z (t) v z (t) v z (t) Mz(t)
11
nn
22
where M is the n n matrix whose columns are the eigenvectors of A, called the modal
matrix. Using the modal matrix, the state transition matrix for the original system can be
written as
t
(t) e t Me M 1
where e t is the diagonal state transition matrix. This frequently proves to be an attractive
method for determining the state transition matrix of a system with real, distinct eigenvalues.

Jordan Canonical Form
For a system with one or more repeated eigenvalues, there is not in general a full set of
eigenvectors. In this case, it is not possible to determine a diagonal representation for the
system. Instead, the simplest representation that can be achieved is block diagonal. Let L ( )
k
be the k k matrix
1 0 1
0

0
0

1
L ( ) 0
k

0 0 0 0
Then for any n n system matrix A there is certain to exist a nonsingular matrix T such
that
L ( ) L ( )
1
k 1
1
TAT k 2 2
L ( )
k r r
where k k k n and , i 1, 2,..., r, are the (not necessarily distinct)
r
1
2
i
eigenvalues of A. The matrix T AT is called the Jordan canonical form.
1
7 SIMULATION
7.1 Simulation—Experimental Analysis of Model Behavior
Closed-form solutions for nonlinear or time-varying systems are rarely available. In addition,
while explicit solutions for time-invariant linear systems can always be found, for high-order
systems this is often impractical. In such cases it may be convenient to study the dynamic
behavior of the system using simulation.
Simulation is the experimental analysis of model behavior. A simulation run is a con-
trolled experiment in which a specific realization of the model is manipulated in order to
determine the response associated with that realization. A simulation study comprises mul-
tiple runs, each run for a different combination of model parameter values and/or initial
conditions. The generalized solution of the model must then be inferred from a finite number
of simulated data points.
Simulation is almost always carried out with the assistance of computing equipment.
Digital simulation involves the numerical solution of model equations using a digital com-
puter. Analog simulation involves solving model equations by analogy with the behavior of
a physical system using an analog computer. Hybrid simulation employs digital and analog
simulation together using a hybrid (part digital and part analog) computer.