Page 145 - MATLAB an introduction with applications

130 ——— MATLAB: An Introduction with Applications

3.10.2 Response to Initial Condition
Case 1: State Space Approach
Consider a system defined in state space given by
x = Ax ...(3.1)
x (0) = x o
Assuming that there is no external input acting on the system, the response x(t) knowing the initial condition
x(0) and that x is an n-vector, is obtained as follows:
Taking Laplace transform of both sides of Eq. (3.1), we obtain
s x(s) – x(0) = AX(s) ...(3.2)
Equation (3.2) can be rearranged as
s x(s) = AX(s) + x(0) ...(3.3)
Taking inverse Laplace transform of Eq. (3.3), we get
x = Ax + x(0) ä(t) ...(3.4)
Defining z = x, Eq. (3.4) can be written as
z = A z + x(0) ä(t) ...(3.5)
Integrating Eq. (3.5), we obtain
z = A z + x(0) 1(t) = Az + B u ...(3.6)
where B = x(0) and u = 1(t)
Noting that z = x and x(t) = z (t), we have
x = z = A z + B u ...(3.7)
The response to initial condition is obtained by solving Eqs. (3.6) and (3.7).
The corresponding MATLAB command used to obtain the response curves are given as follows:
[x, z, t] = step (A, B, A, B);
x = [1 0 0 …0] * x’;
1
x = [1 0 0 …0] * x’;
2
. .
. .
. .
x = [0 0 0 …1] * x’;
n
plot (t, x , x ,…, t, x
1 2 n
Case 2: State Space Approach
Consider the system defined in state space is by
x = Ax x(0) = x 0 ...(3.8)
y = Cx ...(3.9)
where x is an n vector and y is an m vector.
By defining z = x ...(3.10)

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