Page 191 - Modern Control Systems

Section 3.2 The State Variables of a Dynamic System 165
We can rewrite Equations (3.6) and (3.7) as a set of two first-order differential
equations in terms of the state variables X\ and x 2 as follows:

f =-^4«W, (3.8)

and
dx 2 = + 1 R (3
^ „ x 9)
X
-£ z^ ~ * -
T
The output signal is then
*(0 = «o(0 = ^ 2 - (3.10)

Utilizing Equations (3.8) and (3.9) and the initial conditions of the network represented
c
W [*i(*o)> *2(/o)]> we a n determine the system's future behavior and its output.
The state variables that describe a system are not a unique set, and several alter-
native sets of state variables can be chosen. For example, for a second-order system,
such as the spring-mass-damper or RLC circuit, the state variables may be any two
independent linear combinations of Xi(t) and x 2(t). For the RLC circuit, we might
choose the set of state variables as the two voltages, v c{t) and v L(t), where v L is the
voltage drop across the inductor. Then the new state variables, x* and x^, are related
to the old state variables, X\ and x 2, as

x* = v c = x h (3.11)

and
= v c- = x\ - Rx 2. (3.12)
*2 = v L Ri L
Equation (3.12) represents the relation between the inductor voltage and the former
state variables v c and i L. In a typical system, there are several choices of a set of state
variables that specify the energy stored in a system and therefore adequately de-
scribe the dynamics of the system. It is usual to choose a set of state variables that can
be readily measured.
An alternative approach to developing a model of a device is the use of the bond
graph. Bond graphs can be used for electrical, mechanical, hydraulic, and thermal de-
vices or systems as well as for combinations of various types of elements. Bond
graphs produce a set of equations in the state variable form [7].
The state variables of a system characterize the dynamic behavior of a sys-
tem. The engineer's interest is primarily in physical systems, where the variables
are voltages, currents, velocities, positions, pressures, temperatures, and similar
physical variables. However, the concept of system state is not limited to the
analysis of physical systems and is particularly useful in analyzing biological, so-
cial, and economic systems. For these systems, the concept of state is extended be-
yond the concept of the current configuration of a physical system to the broader
viewpoint of variables that will be capable of describing the future behavior of
the system.