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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.
cart mass
M
rod mass
m
rod length
l
cart friction coefficient
μ
bearing friction coefficient
b
gravitational acceleration
g
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.
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