aa
Section 5–5 / Transient-Response Analysis with MATLAB 199Unit-Ramp Response of a System Defined in State Space. Next, we shall treat
the unit-ramp response of the system in state-space form. Consider the system described by
whereuis the unit-ramp function. In what follows, we shall consider a simple example
to explain the method. Consider the case where
When the initial conditions are zeros, the unit-ramp response is the integral of the unit-
step response. Hence the unit-ramp response can be given by
(5–44)
From Equation (5–44), we obtain
(5–45)
Let us define
Then Equation (5–45) becomes
(5–46)
Combining Equation (5–46) with the original state-space equation, we obtain
(5–47)
(5–48)
whereuappearing in Equation (5–47) is the unit-step function. These equations can be
written as
where
Note that x 3 is the third element of x. A plot of the unit-ramp response curve z(t)can
be obtained by entering MATLAB Program 5–11 into the computer. A plot of the unit-
ramp response curve obtained from this MATLAB program is shown in Figure 5–27.
BB= C
0
1
0
S = B
B
0
R, CC=[ 0 0 1 ], DD=[ 0 ]
AA= C
0
- 1
1
1
- 1
0
0
0
0
S = C
A
C
0
0
0
S
z=CCx+DDu
x# =AAx+BBu
z =[0 0 1]C
x 1
x 2
x 3
S
C
x# 1
x# 2
x# 3
S = C
0
- 1
1
1
- 1
0
0
0
0
SC
x 1
x 2
x 3
S + C
0
1
0
Su
x# 3 =x 1
z=x 3
z# =y=x 1
z=
3
t0ydt
C=[ 1 0 ], D=[ 0 ]
B= B x(0)= 0
0
1
A= B R,
0
- 1
1
- 1
R,