Embedded Control and Mechatronics Final Project-
Figure 1 shows an inverted pendulum on a cart with parameters specified in Table 1. This system is represented by a set of state-space equations with the state and control variables given in Table 2.
| Table 1 |
|
cart mass
|
M
|
|
rod mass
|
m
|
|
rod length
|
l
|
|
cart friction coefficient
|
μ
|
|
bearing friction coefficient
|
b
|
|
gravitational acceleration
|
g
|
| Table 2 |
|
state I
|
rod angular position
|
θ (t)
|
|
state II
|
rod angular velocity
|
ω (t)
|
|
state III
|
cart velocity
|
v (t)
|
|
input
|
external force to cart
|
F (t)
|
The normalized form of these state-space equations is given by
with the normalized parameters
k = M/m, bn = b/ml2ω0, μn = μ/mω0
and assuming that
ω0 = √(g/l) = 1.
This system is known to be unstable. Assuming that the state vector can be fully measured, design a controller to stabilize the system around the state (0, 0, 0). Take the parameter values k = 4, bn = 0.05, and µn = 0.2 for your design. Demonstrate the performance of your designed controller by a series of simulations. Through these simulations specify the range of the initial states for which the system is stable. Examine the performance of the closed-loop system against variations in the system parameters k, bn, and µn.