Digital Signal Processing Reference
In-Depth Information
second one, an algebraic equation, is called the
output equation
. The state equation
is a first-order matrix differential equation, and the state vector
x
(
t
) is its solution.
Given knowledge of
x
(
t
) and the input vector
u
(
t
), the output equation yields the
output
y
(
t
). The output equation is a linear algebraic matrix equation.
In the state equations (8.9), only the first derivatives of the state variables may
appear on the left side of the equation, and no derivatives of either the states or the
inputs may appear on the right side. No derivatives may appear in the output equa-
tion. Valid first-order coupled equations that model an LTI system may be written
without following these rules; however, those equations are not in the standard
form.
The standard form of the state equations, (8.9), allows more than one input
and more than one output. Systems with more than one input or more than one out-
put are called
multivariable systems
. For a single-input system, the matrix
B
is an
column vector and the input is the scalar
(n * 1)
u(t).
For a single-output system,
the matrix
C
is a
(1 * n)
row vector and the output is the scalar
y(t).
An example is
now given to illustrate a multivariable system.
State variables for a second-order system
EXAMPLE 8.1
Consider the system described by the coupled differential equations
y
#
1
(t) + 2y
1
(t) - 3y
2
(t) = 4u
1
(t) - u
2
t
and
y
#
2
(t) + 2y
2
(t) + y
1
(t) = u
1
(t) + 5u
2
(t),
where
u
1
(t)
and
u
2
(t)
are system inputs and
y
1
(t)
and
y
2
(t)
are system outputs. We define the
outputs as the states. Thus,
x
1
(t) = y
1
(t);
x
2
(t) = y
2
(t).
From the system differential equations, we write the state equations
x
#
1
(t) =-2x
1
(t) + 3x
2
(t) + 4u
1
(t) - u
2
(t)
and
x
#
2
(t) =-x
1
(t) - 2x
2
(t) + u
1
(t) + 5u
2
(t)
and the output equations
y
1
(t) = x
1
(t)
and
y
2
(t) = x
2
(t).
