Page 770 - The Mechatronics Handbook

0066_Frame_C24 Page 18 Thursday, January 10, 2002 3:44 PM

Further assume that A d ∈ n×n and that, for simplicity, it has n distinct eigenvalues h , with n linearly
independent eigenvectors v . Then there always exists a set of n constants a such that

n
x o ∑ a v , a ∈ C (24.107)
=
=1
k k , for k ∈
A well known result from linear algebra tells us that the eigenvalues of A d are h , with
corresponding eigenvectors v . The application of this result yields

n
n
x t[] = A d x o = A d∑ a v ∑ a A d V (24.108)
=
t
t
t
  
=1 =1 t
h v

n
x t[] = ∑ a h v (24.109)
t
=1
This equation shows that the unforced component of the state is a linear combination of natural
t
modes, { }, and each one is associated with an eigenvalue of A d , which are also known as naturalh
frequencies of the model. Thus, we again have that the matrix A d determines:
• the structure of the unforced response
• the stability (or otherwise) of the system
• the speed of response
Structure of the Unforced Response
In the absence of input, the state evolves as a combination of natural modes which belong to a defined
class of functions: the powers of the model eigenvalues, either real or complex. These modes are discrete
functions related to constants, real exponentials, pure sine waves, exponentially modulated sine waves,
and some other specials functions arising from repeated eigenvalues.
To illustrate these ideas and their physical interpretation consider the sampled system in Example 24.6.
If ∆ = 1, the state space matrices are

A d = 0.8913 0.5525 , B d = 0.3397 (24.110)
– 0.1768 0.2283 0.5525

Hence, the system eigenvalues are solutions to the equation

 
–
det hIA d ) = det h 0.8913 – 0.5525  (24.111)
(
–
 0.1768 h 0.2283 
–
(
= ( h 0.6703) h 0.4493) = 0 (24.112)
–
–
i.e., h 1 = 0.6703, h 2 = 0.4493, and the unforced response is
x u t[] = C 1 0.6702) + C 2 0.4493) t (24.113)
(
t
(
where C 1 and C 2 depend on the initial conditions only. We can observe that, when t tends to infinity, x u [t]
decays to zero, because |h 1,2 | < 1. Also these eigenvalues are positive real numbers, so there is no oscillation

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