2.14 Analysis and Design of Feedback Control Systems
StateSpace Representation of LTI Systems
Derek Rowell
October 2002
1
Introduction
The classical control theory and methods (such as root locus) that we have been using in
class to date are based on a simple inputoutput description of the plant, usually expressed
as a transfer function. These methods do not use any knowledge of the interior structure of
the plant, and limit us to singleinput singleoutput (SISO) systems, and as we have seen
allows only limited control of the closedloop behavior when feedback control is used.
Modern control theory solves many of the limitations by using a much “richer” description
of the plant dynamics. The socalled statespace description provide the dynamics as a set
of coupled firstorder differential equations in a set of internal variables known as
state
variables
, together with a set of algebraic equations that combine the state variables into
physical output variables.
1.1
Definition of System State
The concept of the
state
of a dynamic system refers to a minimum set of variables, known
as
state variables
, that fully describe the system and its response to any given set of inputs
[13]. In particular a
statedetermined
system model has the characteristic that:
A mathematical description of the system in terms of a minimum set of variables
x
i
(
t
),
i
= 1
, . . . , n
, together with knowledge of those variables at an initial time
t
0
and the system inputs for time
t
≥
t
0
, are suﬃcient to predict the future system
state and outputs for all time
t > t
0
.
This definition asserts that the dynamic behavior of a statedetermined system is completely
characterized by the response of the set of
n
variables
x
i
(
t
), where the number
n
is defined
to be the
order
of the system.
The system shown in Fig. 1 has two inputs
u
1
(
t
) and
u
2
(
t
), and four output vari
ables
y
1
(
t
)
, . . . , y
4
(
t
).
If the system is statedetermined, knowledge of its state variables
(
x
1
(
t
0
)
, x
2
(
t
0
)
, . . . , x
n
(
t
0
)) at some initial time
t
0
, and the inputs
u
1
(
t
) and
u
2
(
t
) for
t
≥
t
0
is
suﬃcient to determine all future behavior of the system. The state variables are an
internal
description of the system which completely characterize the system state at any time
t
, and
from which any output variables
y
i
(
t
) may be computed.
Large classes of engineering, biological, social and economic systems may be represented
by statedetermined system models.
System models constructed with the pure and ideal
(linear) oneport elements (such as mass, spring and damper elements) are statedetermined
1
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Figure 1: System inputs and outputs.
system models. For such systems the number of state variables,
n
, is equal to the number of
independent
energy storage elements in the system. The values of the state variables at any
time
t
specify the energy of each energy storage element within the system and therefore
the total system energy, and the time derivatives of the state variables determine the rate
of change of the system energy.
Furthermore, the values of the system state variables at
any time
t
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 Spring '06
 TSAO
 Linear Algebra, Equations, LTI system theory

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