aa
Section 5–5 / Transient-Response Analysis with MATLAB 207Response to Initial Condition (State-Space Approach, Case 2). Consider the
system defined by
(5–56)
(5–57)
(Assume that xis an n-vector and yis an m-vector.)
Similar to case 1, by defining
we can obtain the following equation:
(5–58)
where
Noting that Equation (5–57) can be written as
(5–59)
By substituting Equation (5–58) into Equation (5–59), we obtain
(5–60)
The solution of Equations (5–58) and (5–60), rewritten here
where gives the response of the system to a given initial condi-
tion. MATLAB commands to obtain the response curves (output curves y1versust, y2
versust, ... , ymversust) are shown next for two cases:
CaseA. When the time vector t is not specified (that is, the time vector tis to be de-
termined automatically by MATLAB):
% Specify matrices A, B, and C
[y,z,t] = step(A,B,CA,CB);
y1 = [1 0 0 ... 0]*y';
y2 = [0 1 0 ... 0]*y';
ym = [0 0 0 ... 1]*y';
plot(t,y1,t,y2, ... ,t,ym)
B=x( 0 ) and u= 1 (t),
y=CAz+CBu
z
=Az+Bu
y=C(Az+Bu)=CAz+CBu
y=Cz
x=z
,
B=x(0), u=1(t)
z
=Az+x(0)1(t)=Az+Bu
z
=x
y=Cx
x
=Ax, x(0)=x 0