26 CHAPTER 2. MATHEMATICAL MODELS IN CONTROL
C=[1000;
0010]
D = [0;
0]
The output gives the following state-space model
x ̇ 1 (t)
x ̇ 2 (t)
φ ̇ 1 (t)
φ ̇ 2 (t)
=
01 00
0 − 0 .1818 2.6727 0
00 01
0 − 0 .4545 31.1818 0
x 1 (t)
x 2 (t)
φ 1 (t)
φ 2 (t)
+
0
1. 8182
0
4. 5455
u(t)∑
y 1 (t)
y 2 (t)∏
=
∑
1000
0010
∏
x 1 (t)
x 2 (t)
φ 1 (t)
φ 2 (t)
+
∑
0
0
∏
u(t)Figure 2.6. Time simulation for a unit stepFigure 2.7. Original control structureFigure 2.6 shows the response of the open-loop system where the system
is unstable and Figure 2.7 illustrates the closed-loop control structure for this
problem. Note that the control objective is to bring the pendulum to the upright
position. As such, the output of the plant is tracking a zero reference with the