aaSection 5–5 / Transient-Response Analysis with MATLAB 205Response of Spring-Mass-Damper System to Initial ConditionAmplitude0.120.0200.080.040.060.1Time (sec)0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5Figure 5–31
Response of the
mechanical system
considered in
Example 5–8.
Response to Initial Condition (State-Space Approach, Case 1). Consider the
system defined by
(5–49)
Let us obtain the response x(t)when the initial condition x(0)is specified. Assume that there
is no external input function acting on this system. Assume also that xis an n-vector.
First, take Laplace transforms of both sides of Equation (5–49).
This equation can be rewritten as
(5–50)
Taking the inverse Laplace transform of Equation (5–50), we obtain
(5–51)
(Notice that by taking the Laplace transform of a differential equation and then by
taking the inverse Laplace transform of the Laplace-transformed equation we generate
a differential equation that involves the initial condition.)
Now define
(5–52)
Then Equation (5–51) can be written as
(5–53)
By integrating Equation (5–53) with respect to t, we obtain
(5–54)
where
B=x(0), u=1(t)
z
=Az+x( 0 ) 1 (t)=Az+Bu
z
$
=Az
+x(0)d(t)
z
=x
x
=Ax+x(0)d(t)
s X(s)=AX(s)+x(0)
s X(s)-x(0)=AX(s)
x
=Ax, x(0)=x 0