Page 756 - The Mechatronics Handbook

We also observe that the energy, w(t), stored in the system is given by

1 1
T
2
2
(
(
wt() = -- Kdt()) + -- Mvt()) = x t() Lx t() (24.10)
2 2
Ï K M ¸
where ΛΛ ΛΛ is a diagonal matrix: ΛΛ ΛΛ = diag --- , ----- . ˝
Ì
2 2
Ó ˛
Finally, the nonuniqueness of the state vector can be appreciated if, instead of the choices made in
2¥2
x
(24.8), we choose a new state (t) related to x(t) by a nonsingular matrix T Π, i.e.,
x t() = Tx t() (24.11)
More on this will be said in subsection “State Similarity Transformation.”
Signals and State Space Description
The state space framework can also be used to describe a wide variety of signals using a model of the form

d x t()
-------------- = Ax t(), yt() = Cx t() for continuous-time signals (24.12)
dt
x t + 1] = A q x t[], yt[] = C q x t[] for discrete-time signals (24.13)
[
To illustrate the idea we consider a continuous-time signal given by

5t
5t –
ft() = 2 + 4cos () sin () (24.14)
This signal can be interpreted as the solution for the homogeneous differential equation

3
d ft() df t()
˙˙
------------- 25 ------------ = 0, subject to f 0() = 6, f 0() = – 5 and f 0() = – 100 (24.15)
˙
3 +
dt dt
˙˙
˙
f
f
If we now choose, as state variables, x 1 (t) = f(t), x 2 (t) = (t), and x 3 (t) = (t), then the state space
model for this signal is
0 1 0
dx t()
-------------- = 0 0 1 x t(), yt() = 1 [ 0 0] x t() (24.16)
dt
0 – 25 0
In this usage of state space models, the state variables have no particular physical meaning. However,
this description is particularly useful in signal reconstruction theory and when dealing with disturbances
in control system synthesis.

24.3 State Space Description for Continuous-Time Systems

In this section the state space description for continuous-time systems is presented. The analysis is focused
on the class of linear and time invariant systems; to do that, we first show how to build a linear model
from the nonlinear equations (24.1) and (24.2).
An additional restriction is that, at this stage, the systems under study have no pure time delays. This
feature generates an infinite dimensional state vector. However, we will see in section 24.4 that this class
of systems can be successfully dealt with using sampled data models.

©2002 CRC Press LLC