Ece 456 - embedded control and mechatronics final project

Question

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,      μ_n = 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, b_n = 0.05, and µ_n = 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, b_n, and µ_n.