2.14 Analysis and Design of Feedback Control Systems
State-Space 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 input-output 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 single-input single-output (SISO) systems, and as we have seen
allows only limited control of the closed-loop 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 so-called state-space description provide the dynamics as a set
of coupled ±rst-order di²erential 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 DeFnition 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
[1-3]. In particular a
state-determined
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 de±nition asserts that the dynamic behavior of a state-determined system is completely
characterized by the response of the set of
n
variables
x
i
(
t
), where the number
n
is de±ned
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 state-determined, 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
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 state-determined system models. System models constructed with the pure and ideal
(linear) one-port elements (such as mass, spring and damper elements) are state-determined
1